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On Liapunov's direct method

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Title:
On Liapunov's direct method a unified approach to hydrodynamic stability theory
Creator:
San Giovanni, John Paul, 1941- ( Dissertant )
Reed, X. B. ( Thesis advisor )
Fahien, R. W. ( Thesis advisor )
Siekmann, J. ( Reviewer )
Walker, R. D. ( Reviewer )
Place of Publication:
Gainesville, Fla.
Publisher:
University of Florida
Publication Date:
Copyright Date:
1969
Language:
English
Physical Description:
1 v. (various pagings) : ill. ; 29 cm.

Subjects

Subjects / Keywords:
Abstract spaces ( jstor )
Differentials ( jstor )
Equations ( jstor )
Mathematical integrals ( jstor )
Mathematical variables ( jstor )
Mathematics ( jstor )
Topological theorems ( jstor )
Trajectories ( jstor )
Velocity ( jstor )
Velocity distribution ( jstor )
Chemical Engineering thesis, Ph.D ( lcsh )
Dissertations, Academic -- UF -- Chemical Engineering ( lcsh )
Lyapunov stability ( lcsh )
Stability ( lcsh )
Genre:
bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Abstract:
The objectives of this dissertation are two-fold: 1) to develop a unified approach to stability problems for systems described by operator equations of evolution, 2) to generalize the direct method of Liapunov. Objective (2) offers the possibility of a more discriminating treatment of objective (1), although here we only initiate a study of such implications. These objectives are accomplished by a utilization of functional analytical techniques of modern mathematics. Regarding (1) we concern ourselves primarily with physicochemical systems modeled as continuous media. From a somewhat novel formulation of continuum mechanics a versatile model is developed whose mathematical and physical complexity is regulated by: (i) specifying the state space, i.e., the number and nature of the state variables, and (ii) specifying the operator equation of evolution, i.e., the significant mechanisms for transport, the internal interactions, and the system's interaction with both distant and contiguous surroundings. The formalism is illustrated by considering a subclass of physical systems for which the describing equations are the balance of mass and linear momentum and for which the state is specified by the velocity field. Relevant stability analysis equations for the entire class of parallel flows are developed (i) for constitutive operators with particular mathematical characteristics, and then (ii) for several classes of ideal materials. Stability equations pertinent to any particular parallel flow are precipitated from these by the specification of a coordinate system and the components of the basic velocity field. Regarding (2) , the Liapunov operators in our generalization need not have the totally ordered positive portion of the real line as their range, rather their values may be in a positive cone in an abstract space - thereby offering possibilities for more subtle, delicate, and sophisticated distinctions in the state spaces of complex systems. As with the classical method the principal difficulty in applications is finding a suitable Liapunov operator; thus, we have also generalized a classical technique utilizing the theory of quadratic operators.
Thesis:
Thesis (Ph.D.)--University of Florida, 1969.
Bibliography:
Includes bibliographical references.
General Note:
Vita.
General Note:
Typescript.
Statement of Responsibility:
by John Paul San Giovanni.

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University of Florida
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University of Florida
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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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ALE1738 ( NOTIS )

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ON LIAPUNOV'S DIRECT METHOD: A
UNIFIED APPROACH TO HYDRODYNAMIC
STABILITY THEORY










By

JOHN PAUL SAN GIOVANNI


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY










UNIVERSITY OF FLORIDA
1969

































Copyright by

John Paul San Giovanni

1970


































my wife, Marie,

my sons, John Paul and Thomas

and my parents.








"The mere formulation of a problem is far more often

essential than its solution which may be merely a matter of mathemati-

cal or experimental skill. To raise new questions, new possibilities,

to regard old problems from a new angle requires creative imagina-

tion and marks the real advances in science."


A. Einstein








"My only purpose in this work is to explain what I succeeded

to do towards solving the problem I formulated and what may serve

as a starting point for further research of a similar character."

In the introduction to "General Theory of Stability" by


A. M. Liapunov










ACKNOWLEDGEMENTS


The author wishes to express his sincere appreciation to the

following persons and organizations:

Dr. X. B. Reed: for his guidance and assistance in this

investigation and the preparation of this dissertation,

Dr. R. W. Fahien: for his guidance throughout the author's

graduate career and assistance in the initial stages of this investi-

gation,

Dr. J. Siekmann: for his interest in the author's studies,

Professor R. D. Walker: for his interest,

Marie San Giovanni: for her typing of the rough drafts of

this manuscript; her many sacrifices and encouragement which led to

the realization of this dissertation,

Ford Foundation and the Chemical Engineering Department: for

financial assistance.









TABLE OF CONTENTS

Page

ACKNOWLEDGEMENTS ............................................... v

LIST OF TABLES ................................................. x

LIST OF FIGURES ................................................ xi

ABSTRACT. ....................................................... xii

CHAPTERS:

I. INTRODUCTION...................................... I.I1

1.1 The State Space Approach and Stability....... I.1

1.2 Hydrodynamic Stability Theory................. 1.7

1.3 Liapunov's Direct Method ...................... 1.15

1.4 Scope of the Present Investigation........... 1.24

Bibliography....................................... 1.28

A. On Liapunov's Direct Method................... 1.28

B. Hydrodynamic Stability........................ 1.31

II. THE GENERAL STABILITY PROBLEM...................... II.1

II.0 Prolegomena................................... II.1

II.0.a Introduction......................... II.1

II.O.b On the Scope of the Theory........... 11.4

II.1 The Space of States.......................... II.(

Addendum on the Generation of Perturbations
through Environmental Influences.................. 11.17

11.2 Constraints upon Possible States.............. 11.22

II.2.a Internal Constraints.................. 11.23

II.2.b Constraints Based Upon the Nature of
the Medium........................... 11.41

II.2.c External Constraints ................. 11.46






TABLE OF CONTENTS (Continued)


Page

11.3 Perturbation Problems Associated with
Dynamical Processes.......................... 11.48

II.3.a Perturbation of the Dynamical Pro--
cess Itself........................ 11.49

II.3.b The Stability Problem.............. 11.53

11.4 The General Stability Problem for a Dyna-
mical Process Associated with a Continuous
Medium....................................... 11.58

Bibliography........................................ 11.75

III. POSITIVE CONES AND LIAPUNOV OPERATORS.............. III.1

III.1 Introduction................................ III.1

111.2 Preliminaries: Relevant Definitions,
Properties, and Concepts .................... 111.4

111.3 Liapunov's Stability Theorems and Positive
Cones........................................ 111.12

111.4 Concluding Remarks .......................... 111.24

Bibliography........................................ 111.26

IV. QUADRATIC OPERATORS AND LIAPUNOV OPERATORS......... IV.1

IV.1 Introduction ................................ IV.1

IV.2 Definitions and Preliminaries ............... IV.2

IV.3 The Method of Squares ....................... IV.7

IV.4 The Method of Squares for Simple Bilirear
Operators................................... IV.11

IV.4.a Banach Space with a Positive
Multiplication ..................... IV.13

IV.4.b Hilbert Space ...................... IV.16

IV.4.c N-dimensional Hilbert Space........ IV.21

Bibliography. ....................................... IV.27

V. DIFFERENCE EQUATIONS FOR A CLASS OF BASIC FLOWS.... V.1

V.1 A Class of Basic Flows...................... V.2








TABLE OF CONTENTS (Continued)


V.2 Equations Describing Velocity and
Vorticity of Difference Motions...........

V.3 Form of Governing Equations for Particular
Classes of Fluids.........................

V.3.a Newtonian Fluids .................

V.3.b Stokesian Fluids with Constant
Coefficients.....................

V.3.c Finite Linear Viscoelastic Fluids

V.3.d Simple Fluids ....................

V.4 Equations Governing Difference Fields for
Parallel Flows ............................

V.4.a Newtonian Fluids .................

V.4.b Stokesian Fluids .................

V.4.c Finite Linear Viscoelastic Fluids

V.4.d Simple Fluids....................

Bibliography .....................................


APPENDICES:

A. ON THE PHYSICAL INTERPRETATIONS OF MATHEMATICAL
STABILITY ........................................

B. ON THE CALCULUS IN ABSTRACT SPACES ...............

B.1 Differentiation...........................

B.l.a Some Concepts of Abstract
Differentiation..................

B.l.b Historical Note..................

B.l.c 5Examples of Abstract Differen-
tiation ..........................

B.2 Abstract Integration.......................

B.2.a ftiout the Lebesgue Integral
-n, Its Generalizations..........


viii


Page


V.6


V.17

V.22


V.24

V.28

V.32


V.35

V.36

V.37

V.38

V.4C

V.42





A.I

B.1

B.3


B.7

B.13


B. 1,

B.22


B.27









TABLE OF CONTENTS (Continued)


B.2.b

B.3 Methods

B.3.a


Daniell's Theory of Integration...

of Solution of Operator Equations..

The Method of Contracting Opera-
tors ..............................


B.3.b Implicit Function Technique.......

B.3.c Newton-Raphson-Kantorovich Method.

B.3.d Method of Steepest Descent........

B.3.e Method of Weighted Residuals......

Bibliography.................................... ..

BIOGRAPHICAL SKETCH ............................................


Page

B.29

B.32


B.32

B.34

B.36

B.42

B.45

B.48

B.54








LIST OF TABLES


Table Page

1.1 General Balance Equation and the Fundamental
Principles of Physics ...- ........ .. .......... ....... II.42

V.I Calculated Quantities for Parallel Flows............ V.4

V.2 Key for Equations..... ....... .. .................. V.12

V.3 Useful Forms of Describing Equations ,................ V.15

V.4 Integral Formulation for Velocity Equations......... V.18

V.5 Integral Formulation for Vorticity Equation.......... V.19

V,6 Integral Formulation for Velocity Variance ............ V20

V.7 Integral Formulation for Vorticity Variance .,....... V.21

A.1 State Spaces, Topologies, and Concepts of Stability... A.2

B.1 Applications of Daniell's Formulation by Shilov and
Gurevich (1966) .... .......... ........ .. ............. B.31












LIST OF FIGURES


Figure Page

I.1 Geometrical Interpretation -. .............. .. ... 1.19

I.2 Methods of Hydrodynamic Stability Analysis .......... 1.27

II.1 Plane Couette Motion. ... .-----..-................ 11.19

11.2 Pulsed Alteration of Strain Field .................. 11,21

11.3 Nomenclature-...--.... ....... . ...- ............... 11.26

I 4 The Classical Tetrahedron. ..... ...... ............ II.32

B,l Newton's Method for Roots of an Algebraic Equation... B.37








Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy




ON LIAPUNOV'S DIRECT METHOD: A UNIFIED
APPROACH TO HYDRODYNAMIC STABILITY


By


John Paul San Giovanni


March, 1969


Chairman: R. W. Fahien
Co-Chairman: X. B. Reed, Jr.
Major Department: Chemical Engineering Department


The objectives of this dissertation are two-fold:

1) to develop a unified approach to stability problems for

systems described by operator equations of evolution,

2) to generalize the direct method of Liapunov.

Objective (2) offers the possibility of a more discriminating treatment

of objective (1), although here we only initiate a study of such implica-

tions. These objectives are accomplished by a utilization of functional

analytical techniques of modern mathematics.

Regarding (1) we concern ourselves primarily with physico-

chemical systems modelled as continuous media. From a somewhat novel

formulation of continuum mechanics a versatile model is developed whose

mathematical and physical complexity is regulated by: (i) specifying

the state space, i.e., the number and nature of the state variables, and








(ii) specifying the operator equation of evolution, i.e., the signifi-

cant mechanisms for transport, the internal interactions, and the system's

interaction with both distant and contiguous surroundings.

The formalism is illustrated by considering a subclass of physi-

cal systems for which the describing equations are the balance of mass

and linear momentum and for which the state is specified by the velocity

field. Relevant stability analysis equations for the entire class

of parallel flows are developed (i) for constitutive operators with

particular mathematical characteristics, and then (ii) for several classes

of ideal materials. Stability equations pertinent to any particular

parallel flow are precipitated from these by the specification of a

coordinate system and the components of the basic velocity field.

Regarding (2), the Liapunov operators in our generalization need

not have the totally ordered positive portion of the real line as their

range, rather their values may be in a positive cone in an abstract

space thereby offering possibilities for more subtle, delicate, and

sophisticated distinctions in the state spaces of complex systems.

As with the classical method the principal difficulty in applications is

finding a suitable Liapunov operator; thus, we have also generalized

a classical technique utilizing the theory of quadratic operators.


xiii










CHAPTER I

INTRODUCTION

I.i. The State Space Approach and Stability

The goal of this dissertation is a unified theory of stabi-

lity analysis utilizing Liapunov's direct method. We do not restrict

ourselves to specific physical systems, nor do we attempt to pre-

sent a theory which is all inclusive. Rather, we present a theory

which describes a significant class of physico-chemical systems. We

require only the weak restriction that the mathematical description

of the system be in the form of an operator equation of evolution,

that is, of the quite general form of a balance equation


Rate of f Net Rate + Rate of
Accumulation of Input3 Generation


Although this includes a panorama of mathematical models ranging

from kinetic theory to continuum mechanics, we will investigate in
*
detail only the models in continuum mechanics. However, if such a

specific formulation is to be at all ambitious in the sense of

describing several coupled phenomena occurring in a physical system,

then we would expect not simply one, but several coupled equations

of this general form. We have achieved just such quantitative



In Chapters III and IV, however, we do not limit ourselves
by this restriction. The results of these chapters are completely
general, subject only to the condition that the describing equation
is in the form of a matrix operator equation of evolution in a
Banach space.







1.2


descriptions within our formulation by merely treating the above

equation as an equation for a state vector; the elements of which,

relative to an appropriate basis, are column matrices, the components

of which are not simply numbers, however, but rather members of

suitable function spaces. The formulation for our unified theory

thereby provides the capability for treating a wide variety of

currently popular mathematical models.

The initial steps in the formulation of a mathematical model

for any specific physical system or class of physical systems are

the selection of an appropriate state space and the selection of an

appropriate operator equation of evolution. Although one often

discusses these elements of a mathematical model as if they were

independent of one another, when in actuality they are not, the

selection of the most suitable state space and of the operator

equation describing the evolution of states within that space may

not be divorced. Indeed, even the mere choice of whether to use

an integral or differential formulation of the general balance

equation makes a qualitative difference in the selection of the

appropriate state space.

The relationship between these two elements of a mathematical

model is considerably deeper. The selection of a state space

involves at its most elemental yet among many other things, the

selection of the minimum number of variables which characterize

the system. This may not be done, however, until the operator

equation of evolution, which specifies all significant mechanisms





1.3



for transport and transformation within the physical system, is

known. Conversely, one may take the point of view that the selection

of the state variables dictates the mechanisms for transport and

transformation which are significant by requiring, simply, that they

be consistent with the choice of the state variables and thus also

the state space. This latter tack, at least without sufficient

physical motivation, can tend to become a vacuous exercise.

The selection of a topology for the state space is also a

step of considerable content, for until a mathematical structure in

the form of a topology is imposed upon the state space, it is an

amorphous collection of elements, with only the possibility of some

algebraic operations defined over the collection. These algebraic

operations are necessarily defined so that the operator equations

of evolution are meaningful, and therefore they usually consist of

the operations of addition of elements of the linear space and of

multiplication of them by scalars conveniently defined so as to

give the set the algebraic structure of a linear space. The topology

is then imposed upon this algebraic structure in a manner such that

the algebraic operations are continuous. In this manner the topology

provides the algebraic structure (the set and the algebraic opera-

tions) with some sustenance as is evidenced by the vastness of the

theory of linear topological spaces. In dealing with mathematical

models of physical systems, this topology is often that induced by

an appropriate selection of a metric, that is, a measure of the












distance between any two states. There are an embarrassing variety

of metrics for any metrizable space, 2 where T is the cardinality

of the set, thereby permitting a considerable latitude in theory;

in actual practice, there are considerably fewer choices because

of various mathematical dictates and physical considerations.

The selection of a particular metric among all possible metrics is

based upon the interplay of physical intuition and mathematical

acceptability of the definition on one hand versus the analytical

utility and the physical accuracy of the model on the other hand.

The utility of a particular metric arises primarily because

of the topological properties it may give to the state space as well

as the effect it may have upon the properties of the operators in

the equation of evolution. Depending upon the metric selected, the

metrized state space may be complete, separable, or compact. Further-

more, depending upon the metric selected the operators in the equa-

tion of evolution may be continuous, compact, or completely continu-

ous. This latter fact is of significance not only in the proof of

various existence and uniqueness theorems for the operator equations

of evolution, but also in the application of various exact and

approximate methods of solution for these equations. Thus the act

of selecting a metric is much more important than it sometimes

appears.



The reader is referred to Appendix B or any standard
functional analysis text for these concepts.




1.5



Just as a metric for the measurement of distance between

elements of a state space is important, so also is it often important

to introduce notions of distance between sets of elements of a

state space. In fact, this notion is essential in stability theory

where we are interested in the closeness of trajectories rather

than the closeness of elements in the state space.

One point of view in defining such a metric is to look

upon the elements of the state space as functions of the variables

of physical space and time; the state space is to be taken as a

psuedo-metric space with the psudeo-metric being a continuous posi-

tive-definite function of time. This point of view would allow us

to view the trajectories as a whole that is, a single unit as

well as to consider with probably greater facility nonstationary

basic states as well as stationary basic states.

We have chosen, however, to take the more conventional point

of view, namely, to look upon time as a parameter; the state as a

function of the variables of physical space alone; and to investigate

the magnitude of the metric instantaneously. However, when we take

this second point of view we must explicitly state what is meant by

stability and asymptotic stability. In tnis context stability

means that if a trajectory has started within some bounded neighbor-

hood of the basic trajectory then at all subsequent instants of

times the trajectory is within this bounded region. Asymptotic

stability, on the other hand, requires in addition to stability, the

condition that all the trajectories beginning in some bounded






1.6


neighborhood will actually converge to the basic trajectory as time

grows without bound. It is referred to as uniform asymptotic sta-

bility if the convergence in time is uniform rather that pointwise.

In stating these notions of stability, we have spoken as

if we had existence and uniqueness in a bounded neighborhood of the

basic trajectory. For example, what does the above intuitive notion

of stability mean if it is possible to have multiple trajectories

emanating from the same point in event space? Some of these tra-

jectories may be such that they would imply stability by the above

intuitive notion while others would imply instability. Thus, in the

mathematical models for which multiple trajectories may be emanating

from some points of state space, a precise mathematical definition

based upon the above intuitive notion would be equivocal. One

possible way out of this dilemma, similar to the technique used

in control theory, is to further refine this intuitive notion for the

case when multiple trajectories exist. In particular, if, in the

conventional case (single trajectories) a set "S" is said to have the

quality "Q" when the trajectories originating at the points of "S"

possess the property "P," then in the more unconventional case

(multiple trajectories) the set "S" is said to be "strongly N' if

all the trajectories originating at the points of "S" possess the

property "P," and the set "S" is said to be "weakly N" if at each

point of "S" there exists some trajectory which possesses the property

"P." However, in this dissertation, we will consider only the

conventional case since we suppose as a matter of course that the





I.7


physical system is so formulated that the mathematical problem is

well-posed, that is, the operator equations of evolution and the

necessary auxiliary conditions (initial and boundary conditions) are

such that the solution (i) exists, (ii) is unique, and moreover

(iii) depends continuously upon the loundary data. We should empha-

size that this differs from the more standard usage. Well-posedness,

as the term is generally used, also includes a fourth condition that

the solution depends continuously upon the initial data as well as

the boundary data. But stability theory is the study of the behavior

of trajectories that initiate in a given neighborhood and hence is

naturally kept distinct from questions of well-posedness in the

sense of (i)-(iii). There is a larger sense in which stability

theory could become sterile and well-posedness could incorporate

continuous dependence upon both boundary and initial data, were it

possible to develop a complete physical theory that would explain

how "perturbations" arise. Well-posedness would then be truly the

measure of the realism of a mathematical model. This matter is

pursued no further in this dissertation, although brief sorties

against the origins of physical perturbations are cursorily made.


1.2. Hydrodynamic Stability Theory

The remarks made above about stability apply equally well

to all types of stability problems. In this dissertation we con-

centrate almost entirely on stability problems associated with a

special class of mathematical models, involving the flow of continuous







1.8


media. We also attempt a unification theme: the formulation is

intended to be sufficiently general as to unify a wide range of

stability phenomena, and the approach is a unified one based upon

Liapunov's direct method. In order to place this formulation in

sharper focus, at this juncture we provide a brief review of the

structure and composition of what might be termed classical hydro-

dynamic stability.

The immediate objectives of classical hydrodynamic stability

are to understand the mechanism of the instability of laminar flows

and to obtain criteria for their occurrence. The more fundamental -

and therefore more ambitious objectives of this theory are to

understand why, how, and under what circumstances turbulence arises

from laminar flow instability. In every system of which we have

knowledge, in fact, the transition to turbulence from a laminar

instability is by means of a sequence of stages which are in some

cases easily observed whereas in others they are almost unobservable.

Coles (1965) has demonstrated experimentally, for instance, that

the transition in a Couette cell may be from the basic laminar flow

to one of several types of laminar flow regimes and the ostensible

transition directly to turbulence occurs only under certain cir-

cumstances. Qualitatively similar results have been reported in

investigations of boundary layer instability phenomena and the

transition to turbulence (see, Benney, 1964; Emmons, 1951; Elder,

1960; Greenspan and Benney, 1963; Klebanoff, Tidstrom, and Sargent,

1962; Kovasznay, Kamoda, and Vasudeva, 1962; Miller and Fejer, 1964),









where the sequential transition to turbulence is even more striking,

proceeding from a laminar boundary layer, to Tollmein-Schlichting

waves, to layers of concentrated vorticity, to spots of "turbulent

bursts," and finally to a turbulent boundary layer. Similar results,

both experimental and theoretical, have also been reported for

transition from laminar flows in other systems (see, e.g., Gill, 1965;

Howard, 1963; Malkus and Veronis, 1958; Palm, 1960; Palm and Qiann,

1964; Sch'lter, Lortz, and Busse, 1965; Tippleskirch, 1956, 1957;

Veronis, 1965).

The mathematical formulation of the general problem of

classical hydrodynamic stability is obtained by taking a (generally

steady) solution of the relevant describing equations, by super-

imposing a perturbation upon the basic solution, and then by sub-

stituting the resultant disturbed motion into the relevant des-

cribing equations. A set of nonlinear equations of evolution for the

growth of the disturbance results. As expected, the difficulties

in the classical theory of hydrodynamic stability arise almost

exclusively because the basic equations are nonlinear.

The formulation of the describing equations has taken two

distinctive forms in the literature: (i) a partial differential

equation formulation (see, e.g. Lin, 1955) and (ii) an integral

equation formulation (see, e.g., Serrin, 1959). The former

intensifies the difficulties due to the nonlinear nature of the

equations whereas the latter tends to diminish possible mathematical

difficulties due to the nonlinearity. The partial differential







I.10


equation formulation is arrived at by assuming that, for example,

the difference velocity field


D = v B


where v : perturbed velocity field

v : the basic velocity field whose stability is under
investigation


may be expressed as an expansion in a complete system of normal modes,

or eigenfunctions. The substitution of this expansion into the

equations describing the rate of growth of the disturbance leads

to an infinite system of coupled nonlinear ordinary differential

equations for the amplitudes associated with the normal modes. The

well-recognized source of the nonlinearity coupling these is the

inertial term in Cauchy's First Law, voVv. Thus, if the expansion

in normal modes is substituted for VD, and if the inertial term is

isolated, two infinite series are then multiplied together;

therefore, the ordinary differential equation for the amplitude

associated with the ith mode is coupled with the amplitudes for

all the other modes.*

Difficulties of this sort occur in many familiar mathematical

models in varied disciplines. It is, therefore, reasonable to expect

that approximation techniques have been proposed to deal with them,



It is of some interest to note that Lamb (1945) ascribes
this difficulty to a "mathematical disability."





I.11


and indeed a majority of them have been developed and applied in

hydrodynamic stability theory.

The early theoretical attempts at hydrodynamic stability

theory (e.g., Rayleigh, 1880, 1887, 1895; Thompson, 1887a, 1887b;

Taylor, 1923) attempted to circumvent these difficulties by concern-

ing themselves with infinitesimaldisturbances so that the coupling

term

vD D

was assumed to be of negligible importance. Because of this assump-

tion, the time-dependent part of the disturbance may be taken as an

exponential form, exp(kt). If perturbations at the boundary are

excluded, the boundary conditions on the disturbance are homogeneous,

and one arrives at an eigenvalue problem for the parameter k.

In this linearized theory, therefore, the flow is said to be unstable

if it is possible for k to have a positive real part; otherwise, it

is said to be stable.

If k does have a positive real part, then the amplitude

of the disturbance grows with time until the coupling term may no

longer be neglected. The so-called "nonlinear" theories of hydro-

dynamic stability are constructed to account for this coupling term

in some approximate manner. The two most frequently used techniques



The interested reader may consult the books of Lin (1955)
and Chandrasekar (1961) for further examples, discussion, and
references.







1.12


are: (i) the truncated modal evolution, in which only a fixed

number, N, of modes are assumed to be of importance, and (ii) the

normal mode cascade, in which it is assumed that initially there are

only a finite number of "primary modes" and that all higher harmonics

are formed by interactions of these primary modes. The interested

reader will find examples of the truncated modal evolution in the

works of Dolph and Lewis (1958), Lorentz (1962), Meister (1963),

and Veronis (1963). Examples of the technique of normal mode

cascade will be found in the works of Eckhaus (1965), Palm (1960),

Segel (1962, 1965a, 1965b), Stuart (1960a), and Watson (1960).

Both these techniques accomplish the same thing in that they make

the mathematical problem of stability tractable the truncated modal

evolution by reducing the infinite system of equations to a finite

number of equations while the normal mode cascade allows us to solve

an infinite system successively.

Closely related to both the linear and nonlinear theory is

the "method of parametric expansion." This method is a linearized

stability analysis of a steady-state solution of the equations

describing the evolution of the difference state. Malkus and

Veronis (1958) applied this technique to a particular finite ampli-

tude solution for the Benard problem. SchUlter, Lortz, and Busse

(1965) extended the analysis of Malkus and Veronis to a larger

class of possible finite amplitude solutions. This latter research

is, of course, a theoretical attempt to explain why one or another

of the possible finite amplitude solutions (mode of convection in




1.13


the B&nard problem) is preferred.

All the methods of classical hydrodynamic stability discussed

to this point are based upon the partial differential equation for-

mulation, and they characteristically contain a number of assumptions

about either the magnitude of the disturbance or the coupling of

various normal modes of the disturbance; these assumptions alone

enable them to achieve the primary objectives of hydrodynamic

stability theory. In contrast, the integral equation formulation and

the associated "generalized energy" method require no such assump-

tions of this nature to reach these objectives. In particular, the

nonlinear terms may be fully accounted for. The integral equation

formulation is thus a far more natural framework within which to

view arbitrary but physically realistic perturbations. Unhappily, to

treat a wide variety of disturbances requires that one sacrifice the

more detailed results obtained from the partial differential equation

formulation. Because the purview of the integral equation formula-

tion is that of global properties, that is, of integrals over the

flow regime, the details of the flow pattern are necessarily lost.

Although this is a weakness of the "generalized energy" method, it

is precisely the global approach, which deals with the overall

effects rather than in details, that allows the method to be applied

to disturbances of any magnitude. For example, if by critical we

mean the onset of instabilities as predicted by linear theory, then

Joseph (1965) has used the integral equation formulation to show the

nonexistence of these subcritical instabilities for arbitrary







1.14


spatially periodic disturbances. To be redundant for emphasis,

the nonlinearity is retained and results far beyond those of linear

hydrodynamic stability theory may be obtained. Moreover, because

the method does not require a discussion of interactions of normal

modes (the physical significance of which may not be all obvious)

the physics described by the equations is readily apparent.

The integral equation formulation and the associated

"generalized energy" method seems to have originated in the work of

Reynolds (see, e.g., Reynolds, 1895) and Orr (see, e.g., Orr, 1907).

Although their basic approach to stability problems has been used

through the years (see, e.g., Hamel, 1911; Serrin, 1959; Synge,

1938a, 1938b; Thomas, 1943; von KArman, 1924), it received only a

small fraction of the use and the attention that the partial differen-

tial equation formulation received. Interest in the method, however,

has risen steeply since the publication of Serrin in 1959. The

work on the method since then may be placed into two classifications:

(i) an improvement of the bounds used by Serrin (see, Velte, 1962;

Sorger, 1966), and (ii) treatment of more complex problems (see,

Conrad and Criminale, 1965; Joseph, 1965, 1966). Thus, Conrad

and Criminale treated the case of time dependent basic velocity

fields whereas Joseph treated the Benard problem with the

Boussinesq equations.

In all these various extensions and generalizations of

Serrin's work, the key lies in the positive-definite property of





1.15


the "generalized energy" and the inequalities which are used to

obtain sufficient conditions for the time derivative of this "energy"

to be negative-definite. The essential ingredient for stability is,

to summarize, a generalized energy that is positive-definite yet

with a negative-definite total time derivative.

The properties of the "generalized energy" used by Serrin

(1959) are precisely those of a Liapunov functional, and, in fact,

Serrin's use of the "generalized energy" to ascertain sufficient

conditions for stability bears a striking resemblence to the use

of Liapunov functionals in Liapunov's direct method. Because

of this similarity and because one of the major contributions of

this dissertation is a generalization of Liapunov's direct method,

it is appropriate that we now discuss the method, its essential

ingredients, its interpretation, its advantages, its limitations,

and recent research work involving it.


1.3. Liapunov's Direct Method


Mathematics is at times the language of science and at other

times it is the queen of science (Bell, 1955). In its latter role

mathematics does well in its imposition of a logical structure upon

mathematical models of physico-chemical systems. On other occasions,

however, it fails in both its roles, notably in dealing adequately

with nonlinear problems. As an unhappy lesson of experience, more-

over, the more accurate the model desired or the wider the range of

theory sought, the more probable it is that the formulation will be







1.16


nonlinear. And it long ago became clear that solutions to nonlinear

operator equation are generally unobtainable, at least in the style

to which science became accustomed to when it dealt only with linear

problems. Analytical solutions seldom exist and quantitative

methods of analysis, while helpful, in no way suffice as they do

for linear problems. Qualitative methods, on the other hand, may

tell us much about a nonlinear system, they may even tell us all

that we need to know; but even when analytical methods provide us

with some useful information, qualitative methods will always serve

to complement them.

Within the larger framework of qualitative methods in

mathematics, the names of Henri Poincar6 and Alexander Mikhailovich

Liapunov occupy a revered position. They were among the first to

recognize the futility of attempting to obtain explicit solutions

to all differential equations leading the trend toward obtaining

qualitative characteristics of families of solutions of a differen-

tial equation. Whereas Poincare's interests in this area centered

primarily upon the existence of periodic solutions and the geo-

metrical properties of families of solutions generated by pertur-

bations of an operator, Liapunov's primary contribution was the

development of a method for studying the geometrical properties of

a family of solutions of a given equation, relative to a basic

solution that need not be known. This last is the source of the

terminology, Liapunov's direct method.

The direct method is a generic term for a number of theorems





1.17


centered about several precise definitions of stability and certain

real scalar-valued functionals that have come to be called Liapunov

functionals. These functionals have the nature of a generalized

"distance" or "energy:" (i) in much the same manner that a compari-

son of the norms of each of two elements of a linear space indicates

which is closer to the origin, even though the elements are in

different "directions'," so also may different points of the state

space be distinguished from one another, relative to the origin, by

different values of their Liapunov functional and (ii) in much the

same way that a redistribution of energy provides a means of con-

sidering transient phenomena, so also may local changes in the

values of Liapunov functional indicate that a trajectory in the

state space locally is tending toward or away from the basic tra-

jectory (which may be degenerate, in that it may be a basic state).

Loosely speaking, Liapunov functionals provide a partial ordering of

the admissible states of the system, as well as of trajectories in

the state space (Chapter III).

If we consider the Liapunov operators as real scalar-

valued functionals, this geometrical interpretation may be easily

illustrated (Elgerd, 1967; Hahn, 1963, 1967; Lasalle and Lefschetz,

1961). From the statements of the relevant theorems, a Liapunov

functional, V<.>, for an autonomous system described by


d_ = f(x), f(O) = 0
stability, for example, if it is a positive

insures asymptotic stability, for example, if it is a positive






1.18


definite functional, and if its temporal variation along admissible

trajectories of the system must be negative-definite. For ease of

visualization, we suppose that the state of the system may be

represented as a point x in I2R a particular state space generally

referred to as the phase plane. We then erect a perpendicular to

this plane at the origin to indicate the range space of the Liapunov

functional. In this 3 a Liapunov functional may be represented as

an open-ended, bowl-shaped surface as indicated in Figure I.la, and

in particular that is the graph of a single-valued function. Thus

the projections of the

Vax> = const. (I.1)


loci onto the phase plane generate a system of closed, non-

intersecting curves inR 2 (see Figure I.1), and it is an intuitively

obvious and weil-known result of elementary topology that a closed

curve in a plane divides the plane. Mathematically, the criterion

of negative-definite temporal variations along all admissible

trajectories may be expressed by


dV = -d VV < 0
dt dt -

Now the classical geometric interpretation of the gradient is that

of a vector, the direction field of which is everywhere perpendicular

to loci given by equation (I.1), and whose magnitude is a measure

of the spatial variation of V<.>. Geometrically, therefore, the

mathematical inequality noted above may be interpreted as a require-

ment that the tangent to the field of trajectories f(x) in the phase





1.19


V





I












_a) x-


x2

V(x)-3


V(x) 2


V x)1




dx
\ dt




.__A TYPICAL TRAJECTORY


FIGURE 1.1. GEOMETRICAL INTERPRETATION




1.20


plane must always and everywhere have a negative projection upon

the corresponding gradient vector (see Figure I.l.b). Therefore

the system is asymptotically stable if the admissible trajectories

are such that succeeding states along them correspond to lower

values of the Liapunov function. And in this sense, the succeeding

states may be said to be "closer" to the origin.

The framework of this geometrical interpretation of Liapunov's

direct method may also be used to emphasize an important point

about Liapunov's direct method: it provides only sufficient conditions

for stability which is, incongruously, both a strength and a weak-

ness, of which we shall have more to say later. If we had plotted

in this phase plane the vector field, f(x), in some neighborhood of

the origin (the basic solution), we could, by the method of iso-

clines and visual observation, determine the behavior of trajectories

in that neighborhood of the origin; it would, however, still remain

to find a suitable Liapunov functional if stability were the case

Conversely, if we had a Liapunov functional with the required

properties in some neighborhood of the origin, then we could avoid

che graphical construction of the method isoclines in that neigh-

borhood and yet be assured of stability. Now in state spaces of

higher dimension the first alternative is not a viable one, and

if we were also unable to find a suitable Liapunov functional, then

we would be unable to draw any conclusions about the stability of

the origin.




1.21


This sufficient nature of Liapunov's direct method permeates

every part of its structure and is at once both the greatest

strength and the most significant weakness of the method. We say

strength because, as a consequence of this sufficient nature, one

has an analytical tool powerful enough to assure us of stability

in some neighborhood of the origin not only for one equation but for

a class of equations a monumental accomplishment, Moreover, this

may be accomplished without the necessity of the many oversimplifica-

tions usually necessary to obtain an explicit solution to just one

number of this class of equations. On the other hand we say that

this sufficient nature is also a weakness of the method because if

we are unable to discover a suitable Liapunov functional, even after

an extensive search, we know no more abouc the stability of the

origin than before we began. Also, even if a suitable Liapunov

functional has n found and an associated neighborhood of stability

determined, this does not say that the portion of state space outside

this r. orhood is a r -n tof instability, Thus the major

dif-icul, in appi Liapunov's direct method to a particular

n Lem or class of --tems is in obtaining a Liapunov functional

with the required pr -ties. Consequently, it is reasonable to

expect that much of the research on the method is devoted to techniques

tor construct suiLable Liapunov functionals (see, e.g., Brayton


Another not in onsequential advantage of Liapunov's direct
M ehod, it seems to us, is its simple physical and geometrical
interpretation (cf. above)






1.22


and Miranker, 1964; Hahn, 1963, 1967; Ingwerson, 1961; Leighton,

1963; Letov, 1961;Luecke and McGuire, 1967; Krasovskii, 1963;

Peczkowski and Liu, 1967; Schultz and Gibson, 1962; Szego, 1962;

Walker and Clark, 1967; Zukov, 1961). Generally speaking, the proposed

methods may be classified into three categories: (1) those which

assume a certain form for the gradient of the Liapunov functional,

(2) those which assume a certain form for the Liapunov functional or

its time derivative, and (3) those which make use of the similarity

of two systems, for one of which a Liapunov functional is already

known, with the standard application being an extension from a linear

system to a "slightly" nonlinear system.

Zubov's now-famous method falls under the first classifica-

tion (see, e.g., Hahn, 1963; Zubov, 1961). Essentially, it makes

use of the geometric interpretation of a first order, partial differ-

ential equation. If the existence of a solution can be proved for

the partial differential equation, then one will have proved the

existence of a Liapunov functional for the system of ordinary

differential equations. Moreover, Zubov's method actually provides

a constructive method for Liapunov's method based upon the solution

of the related first order inhomogeneous partial differential equation.

The method of "variable gradients" (see, e.g., Hahn, 1967; Schultz

and Gibson, 1962) also belongs to this first classification. In


The reader is referred to any standard textbook in the theory
of partial differential equations.





1.23


this method one assumes a form for the gradient of the Liapunov

functional and requires that the functional be single-valued. This

latter requirement allows us to find the Liapunov functional by a

line integration along any convenient path. The methods of "separa-

tion of variables (Letov, 1961), "canonical variables" (see, e.g.,

Brayton and Miranker, 1964; Letov, 1961; Zubov, 1961), and "squares"

(see, e.g., Hahn, 1963, 1967; Letov, 1961; Krasovskii, 1963) all

belong to the second classification. In particular, the method of

squares focuses upon the fact that any positive-definite form is a

possible Liapunov functional, an essential feature to which we

return in Chapter IV. The most widely used method under the third

classification at least among engineers, is the one using a Liapunov

functional for a linear system to determine the region of asymptotic

stability for a nonlinear system which is somehow close to the linear

system (see, e.g., Krasovskii, 1963, Chapter IV).

These techniques of constructing Liapunov functionals -

and to a lesser extent the theorems themselves had, until 1960,

been applied mainly to stability problems associated with systems

of ordinary differential equations, or in engineering parlance, to

lumped parameter systems. However, since the appearance of Zubov's

monograph (see, Zubov, 1961), there has been an increased interest

in applying the method to mathematical models involving difference-

differential equations (see, e.g., EL's gol'c 1964, 1966; Krasovskii,

1963) and to partial differential equations (see, e.g., Brayton and

Miranker, 1964; Hsu, 1967; Mochvan, 1959, 1961; Parks, 1966;







1.24


Wang, 1964, 1965), that is to so-called distributed parameter systems.


1.4. Scope of the Present Investigation


Although our interests lie generally with systems that are

described by the general equation of balance, in this dissertation

we consider, almost exclusively, systems arising from continuum

mechanics. The primary objectives, then, of this dissertation are

to present a unifying formulation of stability problems associated

with continuous media and a unified approach to their analysis by

means of Liapunov's direct method. The methods and concepts which

have been employed in realizing these objectives are those of modern

mathematics, particularly, functional analysis.

The main significance of this unifying formulation and unified

approach is the fresh outlook from which to view not only the very

old problems of hydrodynamic stability but also those of arbitrary

physical and mathematical complexity. This dissertation presents a

systematic way of proceeding from the hypothetical mathematical

model of a physical system, through the corresponding operator

equations of evolution for the difference state, and, finally,

through operator equations whose solution will be a Liapunov opera-

tor for that particular physical system. In effect, therefore, we

have divorced the physical problems involved with modelling from the

mathematical problems involved with solving an operator equation for

the Liapunov operator. From another point of view, however, we

have directly related the physical problems to the mathematical





1.25


problems, in that once the mathematical model for a physical system

is selected, the corresponding operator equations for a Liapunov

operator may be immediately obtained by mere substitution in the

relevant equations. Again, from a still more distant vantage point,

we have actually provided but a simple illustration of the power

and versatility that the abstract spirit of modern mathematics may

bring to bear on the complicated problems facing scientists and

engineers of today.

Chapter II is devoted to the preliminaries necessary to

optimally utilize the power latent in Liapunov's direct method.

Thus, Chapter II relates and discusses a state approach to the

models of continuum mechanics and the place of stability problems

within the class of perturbation problems. In tone if not in

accomplishment, this chapter provides a somewhat novel approach to

continuum mechanics. We close the chapter with a natural generaliza-

tion of some of the work of Serrin (1959) and Joseph (1965, 1966)

in order to illustrate the advantages of an abstract approach to

stability problems by using the simplest type of Liapunov operator -

a quadratic form to arrive at sufficient conditions for stability,

one of our few concrete results.

In Chapter III, we turn to a consideration of Liapunov's

direct method and develop a generalization of the method. Instead

of restricting ourselves to the real line as the range space for

Liapunov functionals, we consider Liapunov operators with partially

ordered linear topological spaces and, in particular,positive







1.26


cones in these spaces as the range spaces. It is reasonable to expect

that the richer and more delicately structured the domain of the

Liapunov operators is because this is the state space the more

desirable it becomes for us to consider range spaces richer than

the real line if we are to provide a more highly discriminating

ordering of the elements and trajectories in the state space by

means of these operators. In other words, if one considered the

set of all possible Liapunov operators, it would be greatly enriched.

Moreover, it is likely that certain systems may be especially apt

for the application of these Liapunov operators, whereas they may

have hitherto been unamenable to analysis by means of Liapunov

functionals, and because of the sufficient nature of the method they

may have gone unanalyzed by thedirect method of Liapunov. Of the

many theorems that could have been developed from this conceptual

breakthrough, we emphasize only generalizations of the main theorems

on stability, asymptotic stability, instability, unbounded instability,

and also the conditional nature of the new method.

In Chapter IV we employ these theorems and a slight generaliza-

tion of the notion of a quadratic form to arrive at operator equations

for the construction of these Liapunov operators, and in Chapter V

we take a very special class of physical systems, namely, those whose

state is specified by the velocity field to illustrate the versa-

tility of our formulation and its possible range of application.

For the convenience of the reader in investigating the refer-

ences, we present a block diagram of Hydrodynamic Stability Theory in

Figure 1.2.




1.27








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1.28




1.29


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Publications of Amer. Math. Soc., 2 (Addresses), 227.

1938b, The stability of plane Poiseuille motion, Proc.
Fifth Inter. Cong. Appl. Math., 326, Cambridge, USA.







1.34


Taylor, G. I. 1923, Stability of a viscous liquid contained between
rotating cylinders, Phil. Trans. A, 223, 289.

Thomas, T. Y. 1943, On the uniform convergence of the solutions of
the Navier-Stokes equations, Proc. Nat. Acad. Sci. USA, 29, 243.

Thompson, J. J. 1887a, Rectilinear motion of a viscous fluid between
parallel planes, Mathematical and Physical Papers, 4, 321,
Cambridge Univ. Press, Cambridge.

1887b, Broad river flowing down an inclined plane bed,
Mathematical and Physical Papers, 4, 330, Cambridge Univ.
Press, Cambridge.

Tippleskirch, H. 1956, Uber Konvektionzellen, imbesondere im
flussigen Schwefel, Beitr. Phys. frei Atmos., 29, 37.

Tippleskirch, H. 1957, Uber die Benard-Stromung in Aerosolen,
Beitr. Phys. frei Atmos., 29, 219.

Velte, W. 1962, Uber ein Stabilitatskriterium der Hydrodynamik,
Arch. Rat. Mech. Anal., 9, 9.

Veronis, G. 1963, An analysis of wind-driven ocean circulation with
a limited number of Fourier components, J. Atmos. Sci., 20,
277.

Veronis, G. 1965, On finite amplitude instability in thermohaline
convection, J. Marine Res., 23, 1.

von Karman, Theodore 1924, Uber die Stabilitat der Laminarstromung
und die Theorie der Turbulenz, Proc. First Inter. Cong.
Appl. Mech., 97, Delft.

Watson, J. 1960, On the nonlinear mechanics of wave disturbances
in stable and unstable flows II, J. Fluid Mech., 9, 371.








CHAPTER II

THE GENERAL STABILITY PROBLEM



11.0. Prolegomena

II.O.a. Introduction

Given a physical system, the selection of an appropriate state

space and an operator equation are primarily questions of utility and

of accuracy. As important as such questions are in formulating a

mathematical model of a physical system, we shall defer them until

Sections 11.2 and 11.3. The operator equation of evolution may be

looked upon as an abstract operator that maps a given initial state

s into the sequence of states of the system; a sequence of states

{s/ fIso generated will be referred to as a state space trajectory.

In actuality, however, if we write the operator equation in the form





then the operator S may be viewed as a temporal connection* that

maps the state of the system at one instant of time, s_ into the

state of the system at the "next" instant of time, s t. p This sug-

gestive terminology for the right side of the equation of evolution

will often be used in the sequel, but it must be kept in mind that


.*The interpretation given to the words "temporal connection"
is analogous to the use of the words connection coefficient in
differential geometry. The operator S<.> may also be looked upon as
a map from the state space to the space of linear and continuous
operators from the real line to the space state.


11.1






11.2


the temporal connection will be single-valued only if the boundary

conditions are incorporated into S<.>-- which means that the equation

must be reinterpreted, say as an integral operator incorporating the

boundary conditions -- and if the problem were well-posed to begin with h

so that there are single trajectories emanating from each element of

the state space.

It is often the situation that arbitrary initial states or

arbitrary classes of initial states are to be considered, in which case

we speak of the flow of a set of states from the initial set.

The classical theory of stability is the study of the fate of

a perturbation superimposed upon a given state space trajectory, or it

may be the study of a state space trajectory acted upon by intermittently

or continuously acting perturbations. Or, more generally, one can study

a class of perturbations acting either once and for all, intermittently,

or continuously but superimposed on either a state space trajectory or

upon a flow of state space elements.

If the evolution of the state of a system can be described ac-

curately by a finite set of ordinary differential equations, then the

state space is finite-dimensional. There are standard definitions of

the-different types of stability in finite-dimensional state spaces

(see, e.g., Kalman and Bertram, 1960; Hahn, 1963). If the changes of

state of a system must be described by a system of partial differential

equations to achieve the desired accuracy, then the state space is

infinite-dimensional. There also are available standard definitions of

stability in infinite-dimensional state spaces (see, e.g., Zubov, 1961;




11.3


Hahn, 1963, 1967). In stability theory, as in any mathematical formu-

lation of a specific physical system, the proper selection of a specific

metric, or at least a specific topology is of utmost importance. This

is especially true in stability theory for we are interested in the

closeness of both trajectories and states. Conversely, the selection

of a specific metric may limit a theoretical analysis needlessly, No

specific metric or other type of topology is considered explicitly: we

consider abstract spaces as state spaces and an intuitive (essentially

topological) notion of stability and asymptotic stability,*

Not content with this level of generality, we shall consider

(see Sections II.3.a., II.3.b., II.3.c., and 11.2) the formulation in

the context of a general dynamical process -- by which we mean the set

of state space elements of a flow, the operator equation of evolution

generating that flow, and the boundary and initial conditions delineat-

ing the succession of state space elements. By considering perturba-

tions of dynamical processes as well as perturbations of basic states

(please see Section 11.3), classical perturbation theory and stability

theory under perturbations to both boundary conditions and initial

conditions are brought under the same province as classical stability

theory. There is, of course, a certain unity to be gained from this.

It is to this and to the abstract formulation with an "arbitrary" set


*In Appendix A, however, we do consider, for illustrative
purposes,.specific normed linear spaces appropriate to specific
equations drawn from continuum mechanics, and we consider the physi-
cal meaning of different norms and their relations to different
types of stability.






II.4


of general balance equations that we refer when we speak of a unified

stability theory.



II.O.b. On the Scope of-the Theory

Although the fate of a perturbation, loosely speaking, is our

ultimate aim, it is first necessary to attempt to say what is meant .by.

the terms "state space," "perturbation" and "relevant describing

equations."

Thus, with regard to the last, the derivation of a quite general

form for the operator equations of evolution in terms of essentially

three possible representations (integral equations, differential

equations, difference equations and combinations thereof) is discussed.

The two factors which are helpful in making possible the formulation

of a unified approach to stability theory based upon the general

balance equations are: (i) each of the principles of physics -- that

is, laws of our method of representation -- has the form of this

equation but with a different physical interpretation attached to the

mathematical symbols in each principle, and (ii) the facility of Li-

apunov's direct method for dealing with classes of equations as well

as with specific equations. In particular, we state at the onset

that the mathematical terms of the general balance equations are to

be interpreted within the context of continuum mechanics, thereby,

also including mass point mechanics.* The unification of the various


*Hamel (1908) has shown that when the motion of a body about
and relative to the center of mass may be neglected, then the equations
of continuum mechanics reduce to those of mass point mechanics. In his
formulation it is stress rather than force which is of central impor-
tance. See also Noll (1959).




11.5


principles of continuum physics in the form of a single general balance

equation emphasizes the now traditional, essentially axiomatic formu-

lation of physics. In this way we are able to present a unified treat-

ment of the class of stability problems associated with the objects

and phenomena within the range of the physical theories referred to as

continuum physics. Accordingly, subclasses of stability problems

associated with physical systems which involve continuous media,

classes of phenomena, classes of materials, and classes of state

space trajectories may be delineated by the specification of a set of

hypotheses.

It is clear, moreover, that the selection of the relevant

describing equations cannot be divorced from the selection of the

state space, nor can the selection of a state be separated from the

selection of a set of describing equations. The two are but different

aspects of a whole, the description of the processes occurring in a

given physical system. It is-nevertheless useful to make the dis-

tinction, however useless it may seem, in formulating the problem,

for it is the very essence of solving the problem.

In other words, the stability problem is so formulated that

its physical -- and mathematical-- complexity may be regulated by

specifying the number and nature of the state variables of the sys-

tem, by specifying the nature of all significant mechanisms for trans-

port -- the internal couplings in the system, by specifying the nature

of volume interactions with the surroundings, and by specifying the

nature of the interactions of the system with the continuous sur-






11.6


roundings. This formulation is accomplished by simultaneously con-

sidering a finite number of equations of the form of the general

balance equation, by considering the basic state whose stability is

being investigated as a solution to this set of equations, and finally

by employing slight revisions of some familiar mathematical relations

in order to obtain a general operator equation of evolution for the

difference between a perturbed state and a basic state.



II.l. The Space of States

Because we have chosen to interpret the terms in the general

operator equation of evolution within the context of continuum mechan-

ics, the state variables naturally appear as fields defined over a re-

gion of physical space. The values of these fields at a point in the

region of physical space -- that is, the physical properties such as

temperature, velocity, stress, density, concentration, polarization

density, etc. -- describe, indicate, and provide some measure of the

physical situation at that point. To be more precise, we have

Definition B.l.I: At an instant of time, the local state of a point

Ln the region of physical space,R is said to be known if the values

of an independent set of these physical properties are known.

The members of this independent set are referred to as the

local state variables. In a particular description of some physical

phenomena, the meaning of an "independent" set is intimately linked

to both the nature of the continuous medium occupying the region of





11.7


interest, R of physical space and the physical theory being used to

describe the phenomena. Alternatively, the choice of what constitutes

an independent set of these physical properties.may be used to charac-

terize the range of a proposed physical theory (cf. e.g., Toupin's

theory of an elastic dielectric [Toupin, 1956]; Coleman and Noll's

theory of thermostatics [Coleman and Noll, 1958]). We do not take up

the general and profound question of what constitutes an independent

set of state variables, although we do take it up peripherally as we

consider specific (but still somewhat general) situations of the basic

equations that describe them.

Continuum mechanics is constructed, however, so as to deal

directly with gross phenomena occurring in finite portions of physical

space. For this reason it deals with fields whose independent var-

iables are the points within the region of interest, R and whose

values are the various physical properties. Thus, the specifications

of a field is actually a specification of a continuum of local proper-

ties. In view of this, we define,

Definition 11.1.2: At any instant of time, the global state of a

region of physical space, k is said to be known if the field

yielding the local state at each point of 'R is known.

Now corresponding to local and global states, two types of

state space may be constructed. In particular, the local state at

some point, x in the region, 1 may be of particular interest.

The instantaneous local state at that point, x may be given as

an N-tuple whose elements are the values of an independent set of the






11.8


physical properties. Some of these elements will be real numbers,

some vectors, some dyadics, etc., but for convenience, this N-tuple

may be represented as a point in an appropriate state space formed by

taking (N) direct products of spaces to which each of the components

of the N-tuple belong. In general, the local state at the point,

x will be different at different times. It would, therefore, be

represented by different points in the local state space of the

point, xo of $ The totality of all the local states associated

with a point, xo of #1'as time elapses is a curve in this local

state space. This curve is called the trajectory of local states

associated with the point, x of 1 .

On the other hand, our interest may be in the instantaneous

global state of a region, tR In order to specify the global state,

a continuum of these local states must be specified. The concept of

a field, however, precisely specifies this global state by means of an

N-tuple of fields -- one element corresponding to each of the local

state variables. Again, for convenience, one may look upon this

N-tuple of fields as a point in an appropriate state space. This

state space will now be formed, however, by (N) direct products of

appropriate function spaces (or, in general, operator spaces). The

choice of the particular types of function spaces involved in the

(N) direct product operations depends upon the nature of the physical

properties associated with the fields describing the system and in-

volved in the N-tuple. Because the local states may vary with time,

the global states of necessity will also vary with time. Different




11.9


global states associated with different times are represented by

different points in this global state space. The totality of all

these points as time is varied continuously is called the trajectory

of global states. Any point on this curve, along with the associated

instant of time, is referred to as an event. An event is thus speci-

fied by an (N+l)-tuple.

Intimately connected with this concept of "state" is the con-

cept of "system." The conceptual division of the universe into sys-

tem and surroundings allows a discussion of process of primary

concern without discussing all processes in the entire universe, a

problem of trivially insurmountable proportions. Although this

division of the universe is arbitrary, a system may only be isolated

for study if the influence of the rest of the universe -- the sur-

roundings -- may be adequately controlled or described or both. The

intimate relation between the state of a system and the surroundings

of a system thus occupies a central position in the construction of

any meaningful theory.

Another distinction which will be found convenient because

we intend to deal with stability problems is that of environment of

a system, by which we mean all alterable (i.e., controllable) ex-

cernal influences on the system. Suppose, for example, that a

particular physical system has been designated for study. The sur-

roundings are immediately fixed by this and an appropriate selection

of the state variables. However, the environment is not automatically

fixed. For example, one might only be interested in and able to






II.10


control the effect of alterations of only a specific type of external

influence (e.g., an external electric field or a condition at the

physical boundary), in which case only that external influence would

constitute the environment. The environment is thus a controllable

subset of the surroundings of a given system. Needless to say, fields

that are parts of the environment in one class of systems may be parts

of the system or nonenvironmental parts of the surroundings in other

classes of systems.

The choice, then, of what constitutes the environment is

directly related to the question "stability with respect to what?"

because it may specify some of the ways in which perturbations may

be generated. Thus, as observers we are necessarily a part of the

surroundings and, presumably, are the modus operandi behind alter-

ations of the controllable external influences that can perturb the

state of a system. In classical stability theory, no allowance is

made for perturbing the system to its new state nor of permitting

the system any "inertia" in that direction. One simply assumes that

perturbations are achieved instantaneously and then the system ceases

co be acted upon by the external influences that provided the original

(step) change of state. The tacit assumption, of course, is that by

considering all possible perturbations (at least of a given class),

one accounts for many of these effects. More generally, intermittent

or continuous perturbations provide a more reasonable description

of these effects. (A further discussion of perturbations arising

from environmental influences is presented in an addendum at the end





II.11


of this section). There are, however, uncontrollable external in-

fluences that can perturb that state. Worse, there are molecular

sources providing for random fluctuations of the continuum (or

macroscopic) variables, and these sources of perturbations lie outside

the framework of the commonly-used theories. By the same token,

there has been little study of the influence of macroscopic fields

upon the microscopic processes, for example, upon the theory of

fluctuations. Some usual choices for the environment are: (1) cer-

tain classes of boundary conditions (the specification of fields or

of an operator equation for the fields on a mathematical surface or

surfaces bounding the region and representing the interaction of

the system with its contiguous surroundings); (2) certain classes

of body force fields (the specification of the interaction of system

and external influences acting throughout the regionP ),

A variety of terms such as physical property, global state,

event, trajectory, system, surroundings, and environment, have been

introduced. They have been discussed separately, and the relation-

ships between them have not been emphasized. To provide a degree of

unity another concept must be introduced, that of a dynamical process.

The dynamical process (DP) consists of all the information that is

needed to transform the present state into the next state in a manner

which conforms with physical reality.* Consequently, the particular


*This is the ultimate criterion of any physical theory, and
although it is universally recognized, it should nevertheless not
go unstated.






1112


elements in a dynamical process depends upon the previous selection

of (i) the state variables (ii) system and, (iii) the surroundings,

including the environment.

Although the choice of specific quantities for these three

elements of a dynamical process should be made simultaneously, let us

discuss separately how the "next" state is affected by them, A

choice of the state variables (i.e. an independent set of fields)

limits the class of possible "next" states in three ways. First, it

indicates the types of physical phenomena which are included in the

study (the range of a physical theory). Second, it suggests which

of the principles of physics will supply an independent set of

governing equations. Thus, if the state variables for a particular

problem have been correctly chosen, the relevant principles of physics

are precisely those sufficient to describe the evolution in time of

these state variables.* The third effect precipitated by a choice

of the state variables is a restriction it places upon the choice of

an environment.**


*As a simple illustration, the models of isothermal and
aonisothermal flow of a single-component newtonian fluid may be
considered. In the model of isothermal flow the Navier-Stokes
equations are taken to be the describing equations. On the other
hand, in the nonisothermal model these equations must be considered
simultaneously with an equation governing the temperature field.
**Again, the model of isothermal flow of an incompressible
pure newtonian fluid may be used as an illustration. If the den-
sity and velocity fields are selected as the state variables, then
it will not be consistent to choose for the environment -- i.e.,
the alterable external influences -- anything that may significantly
alter the temperature field.






11.13


Because the selection of a system and its environment are

intimately related, the question of specifying the state variables

as distinct from the environmental influences is quite delicate. It

traditionally hinges more upon theoretical and experimental (or both)

tractability than upon a bona fide effort at dealing with coupled

fields. For example, suppose we have a newtonian fluid flowing, g

subject to an electromagnetic, as well as a gravitational, field.

The describing equations are the Navier-Stokes equations (with Max-

well stresses) and Maxwell equations for a flowing (continuous)

medium. The relevant boundary conditions plus the (applied) body

force fields constitute the surroundings, and the problem is then

formulated. Unfortunately, we are not yet in a position to solve

nonlinear partial differential equations, much less highly coupled,

nonlinear sets of such equations. The uncoupling device is the

following: assume electromagnetic fields that result as solutions

of Maxwell's equations for certain similar geometries, etc., and use

them as the entries in the Navier-Stokes equations. In the latter

situation, then, electromagnetic fields are taken to be part of the

environment, whereas in the former they are part of the system and

are state variables. It is clear, then, that were we able to solve

the full, coupled, set of equations, the state space trajectories

would provide us a full description, but because we are mathemati-

cally inept, we must resort to a specification of certain of the

state variables once and for all in order to find the approximate

temporal and spatial variations of the remaining fields. Clearly,






11.14


the selection of a "smaller" state space will strongly prejudice the

accessible next states* of the system, simply by excluding the full

range of values for those "state variables" that we have been forced

to specify as part of the environment.

The way in which the environment influences the next state

may be loosely described as follows. In the absence of environmental

influences and presuming uniqueness (uniqueness is assumed throughout

the discussion; the argument may be generalized to nonunique situations),

there will be a single trajectory passing through a given state, with

the "next" state being thus defined. For a given non-zero value of

the environment, a different trajectory will in general pass through

that state, and as the environment is varied, still other trajectories

will result. With each change of trajectory, there will be, of course,

a change of "next" states.

The class of possible "next" states is, of course, also af-

fected by the past history of the system simply because we consider

a state space trajectory as beginning somewhere, at some initial

event. In general, if the system passed through different initial

events in the past, they will occupy different current events at

present and therefore have different future events. For certain

classes of equations of evolution, however, the past history of a

system assumes an even more significant role, in that a complete


*To reiterate, the colloquialism "next" state is used solely
for descriptive purposes. Time is taken throughout to be a continu-
ous, not a quantified, independent variable,





II.15


description requires not simply an initial event, but rather an initial

history.* The equations of evolution appropriate for the model of a

material with a memory, for example, requires an initial history.

In closing, we should note that if, rather than proceeding

from a set of hypotheses about the state variables of a system, the

particular form of the principles of physics, and the environment, we

had started from a particular physical situation and attempted to de-

termine the elements of that particular dynamical process, then the

following steps would have been taken:

(1) A consideration of the class of physical systems and the

range of the physical theory necessary to describe the events that

can take place in the system.

(2) A system which is consistent with (1).

(3) An environment which is consistent with both (1) and (2).

(4) and based upon (1) (3), decide upon the minimum a-

mount of information needed to predict subsequent events in a manner

such that at least for a certain range of the variables (i.e&, level

of operation of the system) this prediction (i) exists, (ii) is

unique, and (iii) depends continuously upon prescribed data.**

The decision mentioned in (1) amounts to a specification of

the type of phenomena under consideration and hence the relevant form


*An example of this situation is a situation in which the
relevant describing equations are differential equations with retarded
arguments. The interested reader is referred to El'sgol' (1964, 1966)
for a more detailed discussion.
**These three conditions are classically referred to as Hada-
mard's conditions for a well-posed problem. The interested reader may
wish to consult a text in partial differential equations (e.g. Garabed-
ian, 1964; Courant and Hilbert, 1952).







11.16


of the principles of physics involved as well as those principles to

be used. Thus, from (1) and (2) one-may find which and how many of

the principles of physics lead to an independent set of equations of

evolution (as well as the conditions imposed by the surroundings).

The part of the-surroundings which will be -- or may be -- considered

alterable.is-specified in (3).

If the above steps are used to determine a dynamical process

for phenomena associated with continuous-media, the following con-

siderations result:

-.() the principles of physics provide the governing equa-

tions and their specific form.

(2) the state variables-provide, together with (1), the

knowledge about an independent set of above equations.

(3)- the past history and the initial state provide initial

data-for equations-of evolution.

(4) the surroundings and environment provide the boundary

conditions and body force terms for the equations.

Alternatively, the above four points may be viewed as a

.larification .of points concerning the necessary elements of a set of

hypotheses-that delineate a class of stability problems associated

with a particular dynamical process or a class of dynamical processes

from a still larger class of problems.






11.17


Addendum on the Generation of Perturbations

through Environmental Influences



The fate of perturbations superimposed upon solutions of rele-

vant describing equations is indeed the main topic of stability analysis,

However, another important question which naturally arises is: "how

was the basic flow actually perturbed?" We, as observers of the sys-

tem, are part of the surroundings. Consequently, in order to exert

our influences upon the internal fields in the system -- in the form of

perturbations -- we must be able to affect the interaction of the sys-

tem with its surroundings. In this chapter interactions were divided

into two classes: (i) volume or body interactions and influences,

and (ii) surface or contact interactions and influences. They are,

therefore, the means by which we may perturb the existing internal

fields.

As may be inferred from the above, this question "how" pro-

vides one link between the physics and the mathematics of hydrodynamic

stability theory. The relationship and the inherent difficulties may

be therefore illustrated most easily by the examination of a concrete

example. Therefore, let us focus attention upon the idealization re-

ferred to as plane Couette flow. The corresponding physical situation

is a fluid contained between two infinite parallel plates separated

by a small gap(Figure II.1). The basic velocity field is generated

by translating the upper plate at a constant velocityU, parallel

to itself,






11.18


Suppose the region over which are stability question is posed

is that contained within the dotted lines on Figure II.1, The boun-

dary of this region may be decomposed into the six parts which are

also illustrated on the figure. Two of these surfaces, namely W

and 31?4, are at the interface between the fluid medium and the solid

boundary. Another difference between Y and v)4 and the other sur-

faces is that they are exposed to us the experimentalist, at least if

he. so designs the equipment. If we, as experimenters, decide to per-

turb the internal field or fields through contact influences it must

be done on the surfaces R3 or 41 In particular, suppose that we

decide upon this approach and instantaneously pulse the velocity of

the upper plate from U1 to 1U and then back to MU .I Because

real actions always take a finite time interval to perform, we have

already an "almost" type of experiment, which we will generally

idealize as a pair of step changes, thereby introducing a certain

error, or difference, into our mathematical model of the experiment.

There are several other important features of this "almost"

experiment. The act of pulsing the velocity of the upper plate alters

che strain field along This, in turn, alters the stress field

along which in turn affects the velocity field throughout

Consequently, the velocity field within R has been perturbed by

altering conditions -- the strain field or the stress field -- at one

of the exposed surfaces. In this case it was the strain field along

) which was directly altered, whereas a pressure pulse in the case

of plane Poiseuille flow is an example of an alteration of the stress

field.






11.19






11.20


Difficulties,- however, still remain in the details of our

"almost" experiment. Essentially they have arisen because our system

is: (i) an open system, and (ii) because we have direct control over

only two of the surfaces, namely, A 3 and I4" The difficulty is

"what are the conditions along )R and 1 ?" (Figure V.2). Al-

thought the velocity field is specified directly on 1?3and DIR ,

all that is known about the velocity field on R and 2-after the

pulse is that it must satisfy the describing equations at each of

their points. Because the basic velocity field does satisfy the

describing equations, the special case might be thought of where the

velocity field along 1?I and) is the basic velocity field. For

plane Couette motion, this can only be a "thought" experiment which

may be imagined while for some other flows, such as plane Poiseuille

flow, it might actually be obtained in practice if '1 and were

chosen carefully.

Only plane Couette flow has been discussed, and similar diffi-

culties arise in analyzing other members of the class of parallel flows,

For still more complicated flows the situation is still more compli-

cated, for as one attempts to develop a tractable theoretical model

that is useful in analyzing a given experiment, more and more dis-

crepancies can arise. Thus, as we investigate the stability of even

this simplest class of basic flows, it is well to keep in mind that

the stability investigation is above all a "Gedanken" experiment.





11.21


(a)+t

(b) t=t0


(c) t=to+e


- I I -


(d) tr-t1>to


FIGURE 11.2. PULSED ALTERATION OF STRAIN FIELD






II.22


11.2. Constraints upon Possible States

Of all the states, state space or trajectories, in a given

state space, only a limited number yield identities upon insertion

into the describing equations; that is, only a small subset of states

or trajectories are solutions of the operator equations. A state

that does satisfy the basic equations is termed an admissible state

or an admissible trajectory, as it is a solution of the steady or in-

steady equations. The equations of evolution may thus be said to

constrain the system, being in fact sometimes described as governing

equations. From a consideration of the general form of a dynamical

process associated with a continuous medium, two different classes

of constraints may be distinguished, namely, internal and external

constraints, of which the basic equations are of the former class

and the body forces couples,and, the boundary conditions imposed on

the system (reflecting the influence of the surroundings) are of the

latter class. We hasten to add three points: first, that the

division is artificial because the body forces and couples are present

in the basic equations; second, because the boundary conditions are

also in an integral formulation of the basic equations, and third,

that our use of the terminology of internal and external constraints

as distinct follows that of, for example, Noll (1959). Our usage

of the qualifiers internal and external is, however, clear, and the

notion of the term constraint is equally transparent, for constraints

follow the class of admissible states and they serve to select the

next state or states, depending upon whether or not uniqueness prevails.





11.23


II.2.a. Internal Constraints

Within the class of internal constraints, there are also two

major subdivisions: those arising from the fundamental principles of

physics, by which we shall always mean classical physics, and those

arising because of the nature of the material within the region P ,

of interest. In the first subdivision there are essentially six of

these fundamental principles of physics:

(1) Balance of Total Linear Momentum

(2) Balance of Total Angular Momentum

(3) Balance of Total Energy

(4) Balance of Total Mass

(5) Balance of Total Electric Charge

(6) Balance of Magnetic Flux

These principles -- or statements -- when expressed in a mathematical

form yield the operator equations of evolution (in more standard but

inverted terminology, the governing equations) which must be satis-

fied by the admissible states. The mathematical representations of

these principles may all be placed within the framework of the general

equation of balance, i.e.,

Rate of Accumulation Net influx of the specified
of a specified quantity quantity through the surface(
in a given control volume) bounding the control volume

+ Rate of Generation of the
specified quantity within (II.1)
the control volume

Conversely, these principles provide the specific items to appear in

the braces. In itself the equation of balance is nothing more than a







11.24


bit of common sense; it is thus only a skeleton -- the riesh is

supplied by the principles of physics.* To be embarrassingly speci-

fic, the principles are, in this context, what the flux of the

specified quantity is and how the specified quantity may be generated

within the volume. To reiterate, the mathematical representation of

the principles of physics, in the form of equations of balance, re-

stricts the class of all possible next states and delineates the

class of admissible states.

Among the various possible ways of interpreting this general

equation of balance we have selected the viewpoint provided by con-

tinnum mechanics.** However, rather than the usual approach taken

therein of dealing with body manifolds*** (see, e.g., Noll, 1959;

Truesdell, 1960), we choose to focus our attention upon a certain

region of physical space -- the control volume, b. Furthermore,

specific representations of the principles of physics are viewed as

plausible postulates, rather than hard and fast laws,


*In speaking of this equation, Truesdell (1960) says, "This
statement is sufficiently general in itself; it doesn't say anything
ac all."
**In that we will use its defining concept (to again borrow
the words of C. Truesdell) of a generalization of the "stress hy-
pothesis of Cauchy" -- or in our terminology, "the equipollence
hypothesis."
***In this way we have avoided any questions dealing with the
"actual" structure of the body -- that is, about the connections be-
tween "material points" within the medium -- which are necessary
hurdles to overcome in defining the mathematical operations inte-
gration over a set of material points. We feel that the approach
we have taken is a more practical and a more convenient -- and
further, one that is still consistent with the fundamental goal of
continuum mechanics -- that is; to explain, represent, and predict
gross phenomena in a macroscopic portion of physical space.






11.25


Thus, for purposes of description at the outset, we are inter-

ested in phenomena occurring in some compact region, b, of physical space.

We adopt the viewpoint that the phenomena occurring fall within the

range of physical theories referred to as continuum mechanics (so that

the local physical properties may be specified by their associated phy-

sical fields), and we assume that what we call an equipollence hypothesis

may be used to characterize internal interactions* within the region of

interest, b.

Let us denote by q) a "typical"** physical field (of the per unit
()
volume nature), which is used to describe the phenomena of interest. Let

a continuous medium,*** 33 occupy the spatial regionPBCE (physical

space) at time t, and letdu denote the surface in ET coinciding with

the elements of the material surface O3 bounding the body 13 at time t

(see Figure 11.3). Further, suppose that the influence of the universe

exterior to ID which we will assume is also occupied by a continuous

medium, upon the rate of change of the field 7T)may be characterized

specifying: (i) a surface influence, I of the exterior to 3 that

.ts upon 12 by acting only upon (and being defined upon)3 B and

ii) a body influence that acts through 13 (being defined per unit volume),




*This is not to say that one must consider the media which are the
subject of continuum mechanics to have no molecular structure. Rather one
avoids the explicit use of any molecular structure they may have, taking
iL into account, loosely speaking, only implicitly through its macroscopic
effects. To this end, molecular theories may and do play a qualitative
role in continuum physics.
**The quotes are to remind the reader that the principles of phy-
sics are unified by eq. (II.1).
***By a continuous medium we mean any matter which, as far as the
phenomena being considered areconcerned, falls under the range of the phy-
sical theory called continuum mechanics (cf., our previous remarks on the
range of continuum mechanics above).





11.26


, i i=h(, t; n2);h3b
\


I
/


FIGURE 11.3. NOMENCLATURE


n2






11.27


Because at time t, 3 occupies the region T) we may view J ) as

being defined on)CEBbyl ( ), and +t )as defined in

at the instant t. Further, suppose that at time t an observer

selects an arbitrary control volume, b as indicated in Figure

11.3. This control volume, bounded by the mathematical surface b ,

need not be contained entirely within, nor need it contain entirely

within itself, the spatial region > ; but, again, the matter within

is to be viewed as a continuous medium. Thus, the influence of the

surroundings upon bB may be characterized by: (i) a volume in-

fluence per unit volume,f[r and (ii) a surface influence,tLb

defined over the surface 46B.

This surface influence, L, unlike the surface influence,

I ', which is imposed by some external agent -- reflects the in-

ternal interactions between the various material points.* In the

special case of the balance of linear momentum this hypothesis is

usually referred to as the stress hypothesis. In its general con-

text, we refer to it as the equipollence hypothesis, for it asserts

hat -the interaction of the material points external to a4(D

a~ time t upon those within (c ) R )3 at time t is

equipollent -- as far as its contribution to the temporal rate of


*It is of interest to note that Poincare in his writings on
the philosophy of science argued against the use of the stress hy-
pothesis for exactly this reason. He felt that the concepts of physics
should all be real -- that is, observable -- and not owe their exis-
tence to a hypothesis which may not be directly tested. The reader
is referred to Synge (1960, pp.4-5) and Truesdell and Toupin (1960,
p.229) references contained therein for further discussion of this
"operational" philosophy of science and its ramifications.






1128


change of '() in 6(B is concerned -- to a field ib( )defined on the

surfaceoOE)B. Further, .it is assumed that this field is dependent*

only upon the position on the surface, time, and tangent plane to the

surface at that point, that is,







A
where FXIEand is the normal to this surface pointing away

from bjs.

By the use of the equipollence hypothesis again, this time

to characterize the internal interaction of material points in ced
at time t upon material points of tlIn ,** we obtain the field

_ (,)defined on the surface Db s Again,V 1 is assumed to

depend only upon position on the surface, time, and the tangent

plane to the surface at that point, that is,




A
where X _bl and is the normal to this surface


*We have followed the conventional presentation (e.g., Noll,
1959) by enumerating the dependence of hbupon a single geometric pro-
perty of the surface, the normal. This is often accurate assumption
at physical boundaries, but the recent work of Scriven, Eliassen, and
Murphy (Eliassen, 1963; Murphy, 1965; Murphy and Scriven, 1968) demon-
strates that other geometric properties enter if the boundaries of
the control volume are of high curvature. These points must therefore
be kept in mind when selecting a control volume.
**It should be mentioned that because of the conventions we
have chosen, any internal interaction of matter exterior to o --
other than material points in 1 at time t -- upon material in 6b(1
ar time t must be accounted for in the volume interaction term. This,
however, is consistent with the usual convention in electromagnetic
theory (see, e.g., Elsasser, 1956; Sommerfeld, 1952; Toupin, 1956).





11.29


pointing into P&6 For convenience, we define the following fields,


t) I~ T5(7)

(TT~ ) ) (~L&YT M6 V
0 ^.; ^s


((J)

so that equation (II.1) when applied to the region, b -- subject to the

above decompositions of influences and interactions -- becomes, at time t,




x +b
br}3 1 (11.2)

In words, the above equation and proceeding discussion may be collected

in the form


Rate of Accumulation)
of the quantity C
I within the control
volume b


Net influx of ()
= due to internal 4+
)interactions sub-
ject to equipollenc,
hypothesis


Net increase of )
& e) to the
surface sources
due to surround-
ing actions uponN
the body 3


Rate of Generationl
+ of within b
due to volume in-
Ifluences
Here, x and 3 have been used to indicate dummy integration variable

along the various surfaces (see, Figure 11.1) and

: element of volume in b,

S : element of surface area on b ,

Bj : element of surface area onD .






11.30


The term





deserves further comment, for, although an analogous term commonly

appears in the balances taken in electrostatics, it is scarcely seen

in the balances taken in other fields of continuum mechanics. This

term plays a versatile role in the balance equation, depending upon

the physical situation at hand; at times it may represent the contri-

bution from a flux which is presented at the boundary of B (possibly

by the dictation of some external agent), represent the contribution

from a surface source, or both of these. As an example of its rep-

resentation as a surface source term we may imagine a mass transfer

problem in which a chemical reaction is occurring only at the sur-

face of a catalyst particle and we have selected our control volume

to include some of the catalyst as well as some of the surrounding

fluid mixture. In this type of physical situation the term


J ( ) '


would represent the rate of generation of material within the con-

trol volume .due to chemical reaction at the surface of the catalyst

particles. On the other hand, physical situations where it repre-

sents the contribution from a prescribed flux field are most easily

found in heat transfer problem, e.g., a solid whose surface is com-

pletely covered by heating wires. It is of importance to note that

this term does not account 'for all the flux of heat across the





11,31


surface of the solid -- just that which is externally prescribed,

If the volume and surface integrals of eq. (11.2) are placed on

different sides of the equation and -- is brought within the in-
dt
tegral, there results



b 1u (11.3)

We now invoke a general form of the classical tetrahedron

argument* whereby it is demonstrated that the internal interaction

at time C between a material point at x>l and the material points

in U-9 -- as given by (x C' -- is the value of a linear oper-

ation on the vector VT The argument goes as follows. Suppose

that the control volume is selected such that lr}%)r. G." the null

set -- that is, b is either completely within or completely external

to B. Further, suppose that the control volume, b, is selected to

be a small tetrahedron of volume, Ub( and bounded by a surface of

area, A.0 which does not contain any points of )B (see, Figure

11.4). Then the mean value theorem for volume integrals is invoked,

~''-i:h leads to the equation





At this juncture, the classical derivation divides this re-

sult by'A;c6 and takes the limit of both sides as AL->O ; thereby,


*The reader unfamiliar with the following development is re-
ferred to the discussion of the stress tensor as given by Truesdell
and Toupin (1960); Eringen (1962),and Aris (1962).






11.32


THE CLASSICAL TETRAHEDRON


FIGURE II.4.






II.33


obtaining the classical result






where Lim = 0Q has been used






By following the classical development, this surface integral may be

decomposed into four contributions,








(11.5)


-th
3

where x = dummy integration variable indicating points on i
L
coordinate surface



= element of surface area .on side of tetrahedron which

is perpendicular to ith direction



= unit vector in i coordinate direction.



In the limiting situation as the tetrahedron shrinks to a point (see,

e.g., Cesari, 1956) it is assumed that

P- 5 10,1






11.34


or





where Kt denotes the normal to the slanted surface of the tetrahedron
v-4
and is an element of surface area on this face. If one substitutes

these relations into eq.(II.5), the following equation results


i Yh(T) T) tJ 0eVjo^ j-jt

At this point the classical derivation draws the implication

from this equation that





where x*denotes some point within the infinitesmal volume. Now the

quantity, ) is defined by





which allows us to rewrite eq. (11.6) as





This seemingly innocent relation is actually one -of the most pro-

found relations of continuum mechanics. In essence it implies that

it is not necessary-to specify the value \ 5 for each of all

the possible mathematical surfaces that may be constructed through

X rather all these possible values may be characterized by the

specification of a quantity, which is independent of the





11.35



mathematical surface -- thus, loosely speaking, this relation

separates the mathematical geometry (as embodied ingY\) from the

physics (as embodied in ).

Because the e 5 occurring in the definition of (T> de-

pend, at time T, only upon x the following identifications* may

be made:





and
L




By combining these identifications with eq. (11.7), one is led to





at each instant t. If eq. (11.9) is substituted into eq. (11.8),

one obtains the representation





Ihis representation may be viewed in either of two ways.** It may

be looked upon as defining a linear operator (p C .)- 7 at the place

x at time which operates on n to yield the local influence\ i C ,



*The reader looking for analogies is referred to similar
identifications made in developing the concept of a stress tensor
(see, e.g., Truesdell and Toupin, 1960; Eringen, 1962; Aris, 1962).
**The reader is reminded of the distinctions made in finite-
dimensional and abstract spaces between the differential, the
derivative, and the two possible interpretations of the derivative
(please see Appendix B, section II.l.a).






11.36


which is a 2.-form-in E3 in-the-mathematical-theory of differential

forms (see, e.g., Buck, 1956; Guggenheimer, 1963; Spivak, 1965;

Willmore, 1959). An alternative interpretation is obtained by viewing

it as a definition of a field C(.TC'V -- that is, a global quan-

tity -- at time t associated with the quantity y*) whose values at a

point Xt-B are the linear operators yielding local values of the

associated quantity, 1 ]. In the classical context of Cauchy's law,

where T .') assumes the role of andtC!*) assumes the role of

Vi ( )T' eq. (II.10) is referred to as the fundamental theorem

of Cauchy which asserts the existence of a stress tensor field,

T7 ) which characterizes the local stress vectors, ( xT--v).

In its more general context we refer to it as the generalized funda-

mental theorem of Cauchy,

Within this general context, the primary utility of the gen-

eralized fundamental theorem of Cauchy -- and therefore, also the

classical theorem -- becomes strikingly apparent. Because YC ) is a

2-form in E3 the analogs of the classical integral theorems of vector

analysis may be applied to it. The classical divergence theorem may

be applied to the surface integral of ) with an impact upon con-

tinuum mechanics at the deepest level. Before proceeding with the

development, however, it is well to list the five key points which

led to eq.(II.10); (i) the form of the general balance equation,

(ii) the equipollence hypothesis, (iii) the assumption that the value

of r 7at a point on )b depended only upon the point and the tangent

plane to b at that point, (iv) the fact that control volume b





11.37


b contained no points of the surface B where there was an imposed

surface traction, and (v) the assumption that the passage to the

limit in the generalized tetrahedron argument is a valid procedure.

These points must be kept in mind as we now make use of the repre-

sentation of n as a 2-form in E
(7)
These points must be kept in mind as we now apply the volume

to surface integral theorem* (see, e.g., Bergman and Schiffer, 1953;

Buck, 1956; Kellogg, 1929; Sommerfeld, 1952) by selecting a control

volume such that b == and by substituting eq. (II.10) into

eg. (II.3)to arrive at





Then applying the volume to surface integral theorem under suitable

conditions on 3 and 30 (see, e.g., Kellogg, 1929) -- for example,

providing ) is a continuously differentiable field and b is a

regular surface** -- eq. (II,11) may be expressed solely in terms of

volume integrals, that is



-m f) ^,t( (T ()

On the other hand, if we impose weaker conditions on the field ( ,

namely, that it may be discontinuous on a set of volume measure zero,


*This theorem -is alternatively referred to as Divergence
Theorem, Gauss Divergence Theorem, Ostrogadskii's Theorem, Green's
Theorem, and various permutations of these names. Not wishing to
enter the controversy we refer to it simply as the volume to surface
integral theorem.
**Kellogg (1929) defines a regular surface as one which may be
decompressed into a finite number of surfaces which have a continuous
normal.






11.38


then a suitably revised form-of the volume to surface integral theorem

(see, e.g., Bergman and Schiffer, 1953, p. 363; Truesdell and Toupin,

1960, p. 526) leads to the equation,





where ( is the k+ region of volume measure zero on which dp is

discontinuous; I is the jump in the field D across ( is
(t 1 (M (-kj. -) T is
the outward normal to positive when pointing from )0 to G in

relation to the convention chosen for I) It should be noted

that eq. (11.12) is a special case of eq. (11.13) when the field is

continuous throughout b. We have, in deriving eq. (11.13), allowed

the fields to be discontinuous on some surface or surfaces. Although

it may be argued that these discontinuous fields are not in the true

spirit of continuum mechanics, they are frequently convenient and

sometimes necessary idealizations; indeed, they are necessary if one

wishes to analyze phenomena such as shock waves within the realm of

continuum mechanics. Furthermore, just as surfaces of discontinuity

have sometimes proved convenient, so also have the idealizations of

lines and even points of discontinuity.* We do, however, limit our

development only to cases of surfaces of discontinuity.


*Although these idealizations are abundantly used in the
theory of electrostatics, they are also used in many other
disciplines. In mass:transfer, for example, the model of diffusion
from a point source to a moving stream is just such an idealization.
Often these idealizations have experimental significance; the analysis
of the example cited is used to determine "eddy diffusivities (see,
Sherwood and Pigford, 1952, pg. 42).






11.39


The .quantity' maybe-decomposed-into two-parts: a convective
kT)
contribution due to a transport of the physical properties associated

with material points as the material points themselves move across the

surface b into b and a diffusive contribution due to-the transport,

across o into 6 of the values of the physical properties associated

with material points which-need-not cross b that is,





The classical-and-currently modern theories all suppose that the dif-

fusive contributions are due only to physical properties associated

with material points in a small neighborhood of b .* The plausibility

of this classical supposition finds support from the molecular view-

point under a wide range of circumstances because of the effectively

short distances over which intermolecular forces act. The extent (in

terms of distances between interacting material points) and the

strength (in terms of how strong the interaction is between the

material points) of the diffusive contribution depends upon the na-

ture of the medium which, in turn, is mathematically described by

making a constitutive assumption. Thus, ) generally takes on

a different form for each material or class of materials. On the

other hand,dC depends only upon the density 0, kinematical quan-

city, the velocity of the material point instantaneously occupying


*This, in fact, was one of the reasons we choose to charac-
terize the internal interactions of Be o upon bG)B as being
volume forces, e.g. B is a dielectric and surroundings is some non-
polarizable medium.






11.40


the place x, and the particular physical property in the multilinear

combination,





If this relation -- namely that Ptxk)v(x,1) is the linear operator

on 4J whose value is the convective contribution mentioned above --
J)
is substituted into eqs. (11.12) and (11.13), one obtains



t[V ^,( ^0)*^- -, (11.14)
and



j kk. "oo ( 1 1 1 5 )
Because these equations have been developed for an arbitrary

control volume, b, the familiar assumptions and arguments (see, e.g.,

Truesdell and Toupin, 1960; and similar arguments in the calculus of

variations used to obtain the Euler-Lagrange equations, e.g., Courant

and Hilbert, 1952) used to derive local relations (differential

equations) from global relations (integral equations) may be applied

:o obtain the local relations





and



4Q (11.17)





11.41


Each of the fundamental principles-of physics may be placed

within the framework developed in eqs. (11.14) to (11.17) by different

interpretations of the quantities p 4 They are

cataloged in Table (11.1). For a given physical system, the number

of equations of this form which constitute an independent set of

equations of evolution depends upon the range of the physical theory,

and both the range and nature of activity within the associated dy-

namical process. Thus, an explicit expression p by an enumera-

tion of its independent variables, is directly related to what

constitutes this independent set -- that is, the nature of the

medium has an effect in determining the state variables.



II.2.b. Constraints Based Upon the Nature of the Madium

An explicit expression for the constraints based upon the

nature of the medium -- that is, a constitutive assumption -- serves

a two-fold purpose, namely, it describes the significant mechanisms

of diffusive transport, and it relates the constructs, to

he physical observables of a system. Moreover, the selection of

explicit expressions for has three major effects:

(i) It quantifies our physical intuition* about the inter-

actions (or couplings) between various fields;


*The interested reader is referred to Birkhoff (1960) for
discussion of the role played by intuition in science, in general,
and in hydrodynamics, in particular.















I 4 C 0
0 0 U-1
0 -4 N
0 10 0

M 0




b0 4





0 q *H U




+ -H *H 0





N- O4
"o


0 w0
0








mn


So
41



4J I
0 0
*H











,- ,-
4-1


w
0 u


S.- 3t
w 0









o W
I aj 0 1









, p
0) b


>W


I




r4
.-- 0


11.42


1 w
-i *'-i
0
0



r-1 *
4J

0 u



,-4
0
a4-











>


r4 -4



0
4-s




> 0





4 *H
4J -

AC


'-N

e-


a *He
: 4J r-I
0 *Hr 0
c t >


.r- U
W m











C 0


r-


CiL CiL4


0

0) 3











o W
(V W
w n
W
01













o 0
, o


0 0


I W
0 >{ 0 4Ji


6 0 (3
0 4J C


w4










II.43


0
o I
*OH
0 IW C

) ,*4 4n M4 w 4 14
H Wu aw) a) ( r-H t C 0r
4 H I -4 r. I= i r- C a) 4 C 4x 3 0

CO a O ** o 4 u o o
o) WU1 Y tfh ( 4-' (0. W n 0i- >i C s C^ 3>''4141o
SW ** o m i-' 0 0 %
>-l ~ ~ !q l (l | J 4-1 I--, t" 01% l) ^ (

4-)O 0 T fl. o W -T0 4- ) > o To 44o
4- C ZU rVI aW F= 4- 0 4-1 W4-i 4HC(
C m w r- 00 44. 00 C 4-
0 H u i -i 0 0 et coWT-4
0-



4 W 0 I ** i4
,AWC 0 Ha)04 I W0


L 4- U C 3 (U *.-4 H -2-
W *H 1 l r CW t -
'1 n > 4 tos o > 4
cu ii ni p w- co _z co 4- a)> U En pcs ^-W



(U W U0 U= *r c
U *H 0 w& X w w *0 *H a) H 44




W'-4 u 41
- ) 4 P 4 4 4

ow > w 0 >
M H -W r 3 uo 41 0 W
r) w ci C a) r-C ,.
oO Z) w Q 0 O- r* 0 Q) (0



-1 -4CZ VC 0 -A Q 0o *0





'-4 I 1- W
P ^ro 1 Hj3 ct4 nE l--




Q) W3441 X 00

-0 ~ 0 W 0 Q **OCH
u Hl) C C) 00 rA 41
0 *Uo 0 5 Q0



W0 :3 L r -4 0 W ca



Cr 00-0 0
0 Lo 43 >1 -p
u Q





) 0 i 0 CO( UC
I-1 : ) -1 1; El 44 C a




> H4) $ U 401
O w 4 C.) W (1) 4-W O (1)



(ao *C (1)
p( 0 0 )
M n V jo i lz. o1 4- >,e & tn*




31-4 W.Hz-4
4-1Q (U 4 (1)-|^ <





4-1 4 H -4 0
0 n

U *w ci ) ** 0 ** 0
,C >- c1 /^ -i- c 41 0,


4J <1 L 41CO
(4 1 H 1M
0 0 c 0

P14J
0) 4 4-

u .r 0 P C 0 .4
C 4 00 C w to4
M & 0 W> w U CU
CL 4 co <^ 00 0o 4
0 0 OO
0- r-l = 0 -4
441 A41) E(4w u1qw0 0U






11.44


(ii) it restricts-attention to-a specific ideal material or

class of ideal materials;

(iii) it introduces into the formulation various phenomeno-

logical coefficients as parameters describing the "level of operation"

of a dynamical process within a given class of ideal materials.

Effects (i) and (ii) are complementary in that an explicit

expression for ) will exclude possible couplings between a set

of fields q within the constitutive assumptions. To be sure,

the velocity field is linearly coupled to all other fields by its

appearance in the convective terms of the field equations of the

dynamical process, so that it occupies regal position within the

class of all physical fields, as does the density field. The three

effects may be illustrated by considering the constitutive assumption

for an incompressible newtonian fluid, namely,





where T is the total stress dyadic, r is the viscosity coefficient,

and b) is the deformation rate dyadic. This expression conveys the

idea that the material depends upon the deformation rate and does so

in a linear manner; therefore, we have focused attention upon a

certain class of ideal materials -- ideal because a specific real

material may behave as indicated under a certain set of circumstances,

and yet behave differently under;a different set of circumstances.

Furthermore, if one places the restriction upon that it is a

constant, then this class of ideal materials does not have the










possibility.of any explicit-coupling between the other physical

fields, e.g., temperature, composition -- except, of course,

through the coupling of..the velocity-field with all others in the re-

maining equations of balance. This parameter, t. may be used as an

example to demonstrate the third effect, in-that a specific value for

it characterizes the strength of response to a certain stimulus --

within the class of incompressible newtonian fluids. In some de-

tail, if we considered two different -members of this class, A and B,

with their corresponding viscosity coefficients such that





then they will have different values for -the stress field for the

same deformation rate field and vice versa; thus, the stress field

may be viewed as the stimulus or the response and the formation

rate as the converse. If the stress is viewed as the stimulus, al-

though the mechanism for the response; -as characterized by both the

form of ( and its arguments, is the same for fluids A and B, it may

be said the A reacts.more-strongly than B. In a sense, then, a value

of f.-- the-phenomenological coefficient of this ideal material --

determines a "level of operation" within the class of incompressible

newtonian fluids.

This "level of operation" is often expressed in a di-

mensionless form, as again may be illustrated by considering the

example of incompressible newtonian fluid. In particular, if this

constitutive assumption-is substituted into the balance of linear






II.46


momentum, which then becomes-the-Navier-Stokes equation,





which may be made dimensionless by taking




L L
where u. i ,7Tj p C are the dimensionless variables, thus

giving rise to the dimensionless parameter, the Reynolds number.

This parameter, a number, reflects "level of operation" of a material

within the class of ideal materials mentioned above. It may be used to

do much more than that. Indeed the Reynolds number may serve to in-

dicate the "level of operation" of a system, an ability obtained by

including characteristic geometrical kinematical, as well as the

physical properties, of the material. The reader wishing to pursue

the subject of similarity and modelling in greater detail is referred

to the delightful little book by Birkhoff (1960) and references

therein.



II.2.c. External Constraints

The interaction of the system with its surroundings comprises

the external constraints. External constraints may be of three basic

types: (i) those due to configuration or kinematical conditions (e.g.,

conditions on the location or movement of the bounding surfaces);

(ii) those due to the interaction with or influence from contiguous

surroundings, in the form of boundary conditions; (iii) the influence










of distant surroundings upon the system, in the form of body forces

acting throughout the system. The last have already been accounted

for, having appeared as source terms in the general balance equations,

and whereas boundary conditions may sometimes describe a genuine coup-

ling between the system and its (contiguous) surroundings and some-

times not, body forces always influence the system, yet are themselves

uninfluenced by the system and its behavior. The division into in-

ternal and external constraints is, to be sure, somewhat artificial,

and depending upon whether the equations are formulated and con-

sidered in integral or differential form, it may seem more or less

artificial, depending on one's taste. Thus, as a set of partial

differential equations, the field equations require appropriate

boundary conditions that appear separately, yet the body forces occur

in the equations themselves. Conversely, in integral form the basic

equations incorporate both external and internal constraints, both

body forces and boundary conditions on an equal footing. As a final

point with regard to all three kinds of external constraints that is

perhaps worth recalling, they may each be subdivided into those con-

ditions that are susceptible to control ("pure influence") and those

that are not (an interaction, or coupling).

The boundary conditions may be made dimensionless in the same

manner-as the partial differential field equations. But whereas the

dimensionless numbers that appeared there measured, or were at least

indicative of, the level of operation of the system, the dimensionless

parameters that arise from nondimensionalizing the boundary conditions

describe the level of communication between system and surroundings.











11.3. Perturbation Problems Associated with Dynamical Processes


Because stability theory is the topic of this dissertation

and because stability problems are concerned with the fate of per-

turbations, it is appropriate that the position occupied by stability*

problems within the class of perturbation problems be considered.

Moreover, the vantage point constructed in Section II.1 -- that is,

the concepts of a state space and a dynamical process -- will allow

us to distinguish the classes of perturbation problems in a straight-

forward manner. Once the class of stability problems is clearly

distinguished from other types of perturbation problems, we will

concentrate upon a specific class of dynamical processes (in Section

11.4), those for which the equations of evolution are of the form

of the general balance equation of Section 11.2.

In this classification of perturbation problems associated

with dynamical processes, it is advantageous to view the dynamical

process as an operation, (DP), which transforms the present state of

a system into a succeeding state, both states, of course, belonging

to the state space associated with the system. Among all the ele-

ments of this state space, a particular set of elements may be dis-

tinguished, those that are transformed into themselves by the

dynamical process -- that is, they are the set of fixed points of

the operation (DP). Any element which belongs to this set is referred


*By stability problems we mean not the classical problems of
stability with respect to boundary perturbations, or with respect to
perturbations in the body force field; rather we mean the stability
of a basic state of a particular dynamical process (see, e.g., Lin,
1955).






II.49



to as a basic state of that dynamical process. One type of pertur-

bation problem is concerned with perturbations of the basic states

of a specific dynamical process, while the other is concerned with

perturbations of the operation -- the dynamical process (DP) --

itself. In the first type of perturbation problem, which we have

taken as our prototype stability problem, is usually centered upon

a particular basic state without regard for the method in which

this basic state will be perturbed. On the other hand, in the

second type of perturbation problem -- that is, perturbation of the

dynamical process -- the method of perturbation is of primary concern.



II.3.a. Perturbation of the Dynamical Process Itself

Although -the main topic of this dissertation is with the

perturbation of a basic state of a dynamical process, we would like

to first briefly discuss the class of problems which involve the

perturbation of the dynamical process itself. This class of per-

turbation problems, which includes perturbation of boundary con-

ditions, perturbations of the shape of the physical boundaries,

perturbations of the external influence fields, and perturbations of

the operator in the constitutive assumption, is currently a topic of

interest in both mathematics and engineering (see, e.g., Van Dyke,

1964; Seeger and Temple, 1965).

In its broadest sense, the question raised in this type of

perturbation problem is: if one is given or can easily find the

trace of a particular dynamical process, (DP), as t is varied in
I







11.50


event space, then how may-the trace of some other dynamical process,

(DP ) -- which is close in some sense to (DP )4 -- be expressed in

terms of a modification of that of (DP ). Perhaps the most straight-

forward modification one could imagine is a series expansion in terms

of a scalar perturbation parameter. Indeed, this is the technique

proposed by Poincare in 1892 -- when such a modification was not so

obvious. Essentially Poincar4's method consists of expanding the

dependent variables (e.g., the state variables) of the relevant

governing equations in a power series in the perturbation parameter;

substituting the series into the relevant equations; equating terms

of similar powers in the perturbation parameter (powers of a scalar

real variable are linearly independent);-then solving the resulting

system of equations successively. The equation corresponding to

the zero power in the perturbation parameter is the governing equation

for (DP ). It is found that for a sizeable class of perturbation

problems -- referred to as regular perturbation problems -- this tech-

nique provides a valid representation of the trace of (DP ) based

upon that of (DP ) throughout physical space.

However, there exists an important class of dynamical pro-

cesses in mathematical physics for which this technique is not ade-

quate. For example, if the governing equations are differential

equations and the scalar parameter appears as a coefficient of the

highest order derivative, then Poincare's technique does not yield a

valid representation throughout all of physical space. The most fre-

quently used techniques of handling this type of perturbation problem






11.51


for differential equations are: (i) Prandtl's method of inner and

outer expansions, and (ii) Lighthill's method (see, e.g., Van Dyke,

1964). In Prandtl's method, series expansions are obtained that are

for valid (approximate) representations of different (hypothetical)

dynamical.process occurring in-different regions of physical space,

and it is assumed that there is a region of physical space in which

both expansions are valid and can be matched. The result, if suc-

cessful, is a uniformly valid approximation to the trace of the

actual dynamical process. Prandtl's original development was based

solely on physical intuition and certain orderly arguments in the

Navier-Stokes equation. It was half a century before any degree of

mathematical vigor could be given to Prandtl's arguments, despite

the pioneering efforts of Lagerstrom, Kaplun, and Cole (see, e.g.,

Lagerstrom and Cole, 1955; Kaplun and Lagerstrom, 1957; Kaplun,

1954), many open questions remain. In Lighthill's technique, on the

other hand, the dependent as well as the independent variables

(e.g., positions in physical space) are expanded in terms of a per-

turbation series in a third set of variables. The choice of ex-

plicit relations for this third set of variables is subject to a set

of guidelines that lead, under certain conditions, to a uniformly

valid representation of (DP) These guidelines provide a systematic

scheme for finding a solution by a change of variable -- as evidenced

by the expansion of the independent variables as well as the de-

pendent variables.







11.52


However, the validity of using the various techniques of

handling perturbation problems of this type rests upon the answer

to a more basic question: "Is the problem correctly set (or well-

posed)?" This question is a major consideration in determining the

elements of a dynamical process as well as the validity of any of the

above perturbation solution techniques. It is concerned with the

validity of a mathematical representation and thus is inherently

based upon the belief that if the mathematical representation of

some stable (in the physical sense) physical phenomena is suffici-

ently well formulated, then the presence of small errors in the

prescribed data should result in small changes in the solutions.

On the other hand, if the physical situation was physically stable,

yet the mathematical model did not behave in this manner, then one

would conclude that the model was badly formulated. Consequently,

if we wish to relate physical instability to the instability of a

dynamical process, it is first necessary that the dynamical process

provide a valid description of the physical situation. By valid, we

mean in the sense that the dynamical process provides a mathematical

formulation for which a solution (i) exists, (ii) is unique, and

(iii) depends continuously upon prescribed data. In order that we

may attach some physical significance to any stability analysis of

a dynamical process, we must suppose that it is a valid represen-

tation of the physical situation.




Full Text

PAGE 1

ON LIAPUNOV'S DIRECT METHOD: A UNIFIED APPROACH TO HYDRODYNAMIC STABILITY THEORY By JOHN PAUL SAN GIOVANNI A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUEREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1969

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UNIVERS 'TV OF FL ORIDA

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Copyright by John Paul San Giovanni 1970

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To my wife, Marie, my sons, John Paul and Thomas and my parents.

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"The mere formulation of a problem is far more often essential than its solution which may be merely a matter of mathematical or experimental skill. To raise new questions, new possibilities, to regard old problems from a new angle requires creative imagination and marks the real advances in science." A. Einstein "My only purpose in this work is to explain what I succeeded to do towards solving the problem I formulated and what may serve as a starting point for further research of a similar character." In the introduction to "General Theory of Stability" by A. M. Liapunov

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ACKNOWLEDGEMENTS The author wishes to express his sincere appreciation to the following persons and organizations: Dr. X. B. Reed: for his guidance and assistance in this investigation and the preparation of this dissertation, Dr. R. W. Fahien: for his guidance throughout the author's graduate career and assistance in the initial stages of this investigation, Dr. J. Siekmann: for his interest in the author's studies, Professor R. D. Walker: for his interest, Marie San Giovanni: for her typing of the rough drafts of this manuscript; her many sacrifices and encouragement which led to the realization of this dissertation, Ford Foundation and the Chemical Engineering Department: for financial assistance.

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TABLE OF CONTENTS Page ACKNOWLEDGEMENTS v LIST OF TABLES x LIST OF FIGURES xi ABSTRACT xii CHAPTERS : I. INTRODUCTION 1.1 1.1 The State Space Approach and Stability 1.1 1.2 Hydrodynamic Stability Theory 1.7 1 . 3 Liapunov ' s Direct Method 1.15 1.4 Scope of the Present Investigation 1.24 Bibliography 1 . 28 A . On Liapunov ' s Direct Method 1.28 B . Hydrodynamic Stability 1.31 II . THE GENERAL STABILITY PROBLEM II . 1 II . Prolegomena II . 1 II. 0. a Introduction II. 1 Il.O.b On the Scope of the Theory II. 4 II . 1 The Space of States II . (. Addendum on the Generation of Perturbations through Environmental Influences » . 11.17 II. 2 Constraints upon Possible States 11.22 II. 2. a Internal Constraints 11.23 II. 2. b Constraints Based Upon the Nature of the Medium II .41 II. 2. c External Constraints 11.46 VI

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TA3LE OF CONTENTS (Continued) Page 11. 3 Perturbation Problems Associated with Dynamical Processes II . 48 II. 3. a Perturbation of the Dynamical Process Itself 11.49 II. 3. b The Stability Problem 11.53 11. 4 The General Stability Problem for a Dynamical Process Associated with a Continuous Medium 11.58 Bibliography 11.75 III . POSITIVE CONES AND LIAPUNOV OPERATORS Ill . 1 111.1 Introduction III.l 111. 2 Preliminaries: Relevant Definitions, Properties, and Concepts III. 4 111. 3 Liapunov's Stability Theorems and Positive Cones Ill . 12 111. 4 Concluding Remarks III. 24 Bibliography Ill . 26 IV. QUADRATIC OPERATORS AND LIAPUNOV OPERATORS IV. 1 IV. 1 Introduction IV. 1 IV. 2 Definitions and Preliminaries IV. ? IV. 3 The Method of Squares IV. 7 IV. 4 The Method of Squares for Simple Bilirear Operators IV . 11 IV. 4. a Banach Space with a Positive Multiplication IV. 13 IV. 4. b Hilbert Space IV. 16 IV. 4. c N-dimensional Hilbert Space IV. 21 Bibliography IV . 27 V. DIFFERENCE EQUATIONS FOR A CLASS OF BASIC FLOWS.... V.l V.l A Class of Easic Flows V.2 vit

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APPENDICES: TABLE OF CONTENTS (Continued) Page V.2 Equations Describing Velocity and Vorticity of Difference Motions V.6 V.3 Form of Governing Equations for Particular Classes of Fluids V.17 V . 3 . a Newtonian Fluids V . 22 V.3.b Stokesian Fluids with Constant Coefficients V.24 V.3.c Finite Linear Viscoelastic Fluids V.28 V.3.d Simple Fluids V.32 V.4 Equations Governing Difference Fields for Parallel Flows V.35 V . 4 . a Newtonian Fluids V.36 V.4.b Stokesian Fluids V.37 V.4.c Finite Linear Viscoelastic Fluids V.38 V.4.d Simple Fluids V.4C Bibliography V.42 A. ON THE PHYSICAL INTERPRETATIONS OF MATHEMATICAL STABILITY A.l B . ON THE CALCULUS IN ABSTRACT SPACES B . 1 B.l Differentiation B.3 B.l.a Some Concepts of Abstract Differentiation B.7 B.l.b Historical Note. B.13 B.l.c L'xamples of Abstract Differentiation . . B.14 B.2 Abstract Integration,. B.22 B.2.a 'lout the Lebesgue Integral and Its Generalizations. B.27 vm

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TABLE OF CONTENTS (Continued) Page B.2.b Daniell's Theory of Integration... B.29 B.3 Methods of Solution of Operator Equations.. B.32 B.3.a The Method of Contracting Operators B.32 B.3.b Implicit Function Technique B.34 B.3.C Newton-Raphson-Kantorovich Method. B.36 B.3.d Method of Steepest Descent... B.42 B.3.e Method of Weighted Residuals B.45 Bibliography „ . B.48 BIOGRAPHICAL SKETCH B.54 IX

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LIST OF TABLES Table Page II. 1 General Balance Equation and the Fundamental Principles of Physics .................a.....,.....'..... 11,42 V.l Calculated Quantities for Parallel Flows....... , . V.4 V.2 Key for Equations. ................... a .........••'.. e • • V.12 V.3 Useful Forms of Describing Equations, ................ . V.15 V.4 Integral Formulation for Velocity Equations........... V.18 V.5 Integral Formulation for Vorticity Equation. ......... . V.19 v -6 Integral Formulation for Velocity Variance..,,,....... V-20 V.7 Integral Formulation for Vorticity Variance...-., V.21 A.l State Spaces, Topologies, and Concepts of Stability... A. 2 B.l Applications of Darnell's Formulation by Shilov and Gurevich (1966) ...... B.31

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LIST OF FIGURES Figure Page 1.1 Geometrical Interpretation* . . ,....'... .-. 1.19 1.2 Methods of Hydrodynamic Stability Analysis..-...,.,,. 1.27 11. 1 Plane Couette Motion. 11.19 11 . 2 Pulsed Alteration of Strain Field II . 21 11. 3 Nomenclature. ......-....'....-....'....................... II .26 11. 4 The Classical Tetrahedron............. 11.32 B,l Newton's Method for Roots of an Algebraic Equation... B.37 XI

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Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ON LIAPUNOV'S DIRECT METHOD: A UNIFIED APPROACH TO HYDRODYNAMIC STABILITY By John Paul San Giovanni March, 1969 Chairman: R. W. Fahien Co-Chairman: X. B. Reed, Jr. Major Department: Chemical Engineering Department The objectives of this dissertation are two-fold: 1) to develop a unified approach to stability problems for systems described by operator equations of evolution, 2) to generalize the direct method of Liapunov. Objective (2) offers the possibility of a more discriminating treatment of objective (1), although here we only initiate a study of such implications. These objectives are accomplished by a utilization of functional analytical techniques of modern mathematics. Regarding (1) we concern ourselves primarily with physicochemical systems modelled as continuous media. From a somewhat novel formulation of continuum mechanics a versatile model is developed whose mathematical and physical complexity is regulated by: (i) specifying the state space, i.e., the number and nature of the state variables, and XII

PAGE 14

(ii) specifying the operator equation of evolution, i.e., the significant mechanisms for transport, the internal interactions, and the system's interaction with both distant and contiguous surroundings. The formalism is illustrated by considering a subclass of physical systems for which the describing equations are the balance of mass and linear momentum and for which the state is specified by the velocity field. Relevant stability analysis equations for the entire class of parallel flows are developed (i) for constitutive operators with particular mathematical characteristics, and then (ii) for several classes of ideal materials. Stability equations pertinent to any particular parallel flow are precipitated from these by the specification of a coordinate system and the components of the basic velocity field. Regarding (2) , the Liapunov operators in our generalization need not have the totally ordered positive portion of the real line as their range, rather their values may be in a positive cone in an abstract space thereby offering possibilities for more subtle, delicate, and sophisticated distinctions in the state spaces of complex systems. As with the classical method the principal difficulty in applications is finding a suitable Liapunov operator; thus, we have also generalized a classical technique utilizing the theory of quadratic operators. xxii

PAGE 15

CHAPTER I INTRODUCTION 1.1. The State Space Approach and Stability The goal of this dissertation is a unified theory of stability analysis utilizing Liapunov's direct method. We do not restrict ourselves to specific physical systems, nor do we attempt to present a theory which is all inclusive. Rather, we present a theory which describes a significant class of physico-chemical systems. We require only the weak restriction that the mathematical description of the system be in the form of an operator equation of evolution, that is, of the quite general form of a balance equation i Rate of Accumulation Net Ratel. 1 Rate of | of Input \ ] Generation Although this includes a panorama of mathematical models ranging from kinetic theory to continuum mechanics, we will investigate in detail only the models in continuum mechanics. However, if such a specific formulation is to be at all ambitious in the sense of describing several coupled phenomena occurring in a physical system, then we would expect not simply one, but several coupled equations of this general form. We have achieved just such quantitative In Chapters III and IV, however, we do not limit ourselves by this restriction. The results of these chapters are completely general, subject only to the condition that the describing equation is in the form of a matrix operator equation of evolution in a Banach space. 1.1

PAGE 16

1.2 descriptions within our formulation by merely treating the above equation as an equation for a state vector; the elements of which, relative to an appropriate basis, are column matrices, the components of which are not simply numbers, however, but rather members of suitable function spaces. The formulation for our unified theory thereby provides the capability for treating a wide variety of currently popular mathematical models. The initial steps in the formulation of a mathematical model for any specific physical system or class of physical systems are the selection of an appropriate state space and the selection of an appropriate operator equation of evolution. Although one often discusses these elements of a mathematical model as if they were independent of one another, when in actuality they are not, the selection of the most suitable state space and of the operator equation describing the evolution of states within that space may not be divorced. Indeed, even the mere choice of whether to use an integral or differential formulation of the general balance equation makes a qualitative difference in the selection of the appropriate state space. The relationship between these two elements of a mathematical model is considerably deeper. The selection of a state space involves at its most elemental, yet among many other things, the selection of the minimum number of variables which characterize the system. This may not be done, however, until the operator equation of evolution, which specifies all significant mechanisms

PAGE 17

1.3 for transport and transformation within the physical system, is known. Conversely, one may take the point of view that the selection of the state variables dictates the mechanisms for transport and transformation which are significant by requiring, simply, that they be consistent with the choice of the state variables and thus also the state space. This latter tack, at least without sufficient physical motivation, can tend to become a vacuous exercise. The selection of a topology for the state space is also a step of considerable content, for until a mathematical structure in the form of a topology is imposed upon the state space, it is an amorphous collection of elements, with only the possibility of some algebraic operations defined over the collection. These algebraic operations are necessarily defined so that the operator equations of evolution are meaningful, and therefore they usually consist of the operations of addition of elements of the linear space and of multiplication of them by scalars conveniently defined so as to give the set the algebraic structure of a linear space. The topology is then imposed upon this algebraic structure in a manner such that the algebraic operations are continuous. In this manner the topology provides the algebraic structure (the set and the algebraic operations) with some sustenance as is evidenced by the vastness of the theory of linear topological spaces. In dealing with mathematical models of physical systems, this topology is often that induced by an appropriate selection of a metric, that is, a measure of the

PAGE 18

1.4 distance between any two states. There are an embarrassing variety of metrics for any metrizable space, 2 where y> is the cardinality of the set, thereby permitting a considerable latitude in theory; in actual practice, there are considerably fewer choices because of various mathematical dictates and physical considerations. The selection of a particular metric among all possible metrics is based upon the interplay of physical intuition and mathematical acceptability of the definition on one hand versus the analytical utility and the physical accuracy of the model on the other hand. The utility of a particular metric arises primarily because of the topological properties it may give to the state space as well as the effect it may have upon the properties of the operators in the equation of evolution. Depending upon the metric selected, the metrized state space may be complete, separable, or compact. Furthermore, depending upon the metric selected, the operators in the equation of evolution may be continuous, compact, or completely continuous. This latter fact is of significance not only in the proof of various existence and uniqueness theorems for the operator equations of evolution, but also in the application of various exact and approximate methods of solution for these equations. Thus the act of selecting a metric is much more important than it sometimes appears. * The reader is referred to Appendix B or any standard functional analysis text for these concepts.

PAGE 19

1.5 Just as a metric for the measurement of distance between elements of a state space is important, so also is it often important to introduce notions of distance between sets of elements of a state space. In fact, this notion is essential in stability theory where we are interested in the closeness of trajectories rather than the closeness of elements in the state space. One point of view in defining such a metric is to look upon the elements of the state space as functions of the variables of physical space and time; the state space is to be taken as a psuedo-metric space with the psudeo-metric being a continuous positive-definite function of time. This point of view would allow us to view the trajectories as a whole that is, a single unit as well as to consider with probably greater facility nonstationary basic states as well as stationary basic states. We have chosen, however, to take the more conventional point of view, namely, to look upon time as a parameter; the state as a function of the variables of physical space alone; and to investigate the magnitude of the metric instantaneously. However, when we take this second point of view we must explicitly state what is meant by stability and asymptotic stability. In tnis context stability means that if a trajectory has started within some bounded neighborhood of the basic trajectory then at all subsequent instants of times the trajectory is within this bounded region. Asymptotic stability, on the other hand, requires in addition to stability, the condition that all the trajectories beginning in some bounded

PAGE 20

1.6 neighborhood will actually converge to the basic trajectory as time grows without bound, It is referred to as uniform asymptotic stability if the convergence in time is uniform rather that pointwise. In stating these notions of stability, we have spoken as if we had existence and uniqueness in a bounded neighborhood of the basic trajectory. For example, what does the above intuitive notion of stability mean if it is possible to have multiple trajectories emanating from the same point in event space? Some of these trajectories may be such that they would imply stability by the above intuitive notion while others would imply Instability. Thus, in the mathematical models for which multiple trajectories may be emanating from some points of state space, a precise mathematical definition based upon the above intuitive notion would be equivocal. One possible way out of this dilemma, similar to the technique used in control theory, is to further refine this intuitive notion for the case when multiple trajectories exist. In particular, if, in the conventional case (single trajectories) a set "S" is said to have the quality "Q" when the trajectories originating at the points of "S" possess the property "P," then in the more unconventional case (multiple trajectories) the set "S" is said to be "strongly N' if all the trajectories originating at the points of "S" possess the property "P," and the set "S" is said to be "weakly N" if at each point of "S" there exists some trajectory which possesses the property "P." However, in this dissertation, we will consider only the conventional case since we suppose as a matter of course that the

PAGE 21

1.7 physical system is so formulated that the mathematical problem is well-posed, that is, the operator equations of evolution and the necessary auxiliary conditions (initial and boundary conditions) are such that the solution (i) exists, (ii) is unique, and moreover O'ii) depends continuously upon the 1 oundary data. We should emphasize that this differs from the more standard usage. Well-posedness, as the term is generally used, also includes a fourth condition that the solution depends continuously upon the initial data as well as the boundary data. But stability theory is the study of the behavior of trajectories that initiate in a given neighborhood and hence is naturally kept distinct from questions of i^ell-posedness in the sense of (i)-(iii). There is a larger sense in which stability theory could become sterile and well-posedness could incorporate continuous dependence upon both boundary and initial data, were it possible to develop a complete physical theory that would explain how "perturbations" arise. Well-posedness would then be truly the measure of the realism of a mathematical model. This matter is pursued no further in this dissertation, although brief sorties against the origins of physical perturbations are cursorily made. 1.2. Hydrodynamic Stability Theory The remarks made above about stability apply equally well to all types of stability problems. In this dissertation we concentrate almost entirely on stability problems associated with a special class of mathematical models, involving the flow of continuous

PAGE 22

1.8 media. We also attempt a unification theme: the formulation is intended to be sufficiently general as to unify a wide range of stability phenomena, and the approach is a unified one based upon Liapunov's direct method. In order to place this formulation in sharper focus, at this juncture we provide a brief review of the structure and composition of what might be termed classical hydrodynamic stability. The immediate objectives of classical hydrodynamic stability are to understand the mechanism of the instability of laminar flows and to obtain criteria for their occurrence. The more fundamental and therefore more ambitious objectives of this theory are to understand why, how, and under what circumstances turbulence arises from laminar flow instability. In every system of which we have knowledge, in fact, the transition to turbulence from a laminar instability is by means of a sequence of stages which are in some cases easily observed whereas in others they are almost unobservable. Coles (1965) has demonstrated experimentally, for instance, that the transition in a Couette cell may be from the basic laminar flow to one of several types of laminar flow regimes and the ostensible transition directly to turbulence occurs only under certain circumstances. Qualitatively similar results have been reported in investigations of boundary layer instability phenomena and the transition to turbulence (see, Benney, 1964; Emmons, 1951; Elder, 1960; Greenspan and Benney, 1963; Klebanoff, Tidstrom, and Sargent, 1962; Kovasznay, Kamoda, and Vasudeva, 1962; Miller and Fejer, 1964),

PAGE 23

1.9 where the sequential transition to turbulence is even more striking, proceeding from a laminar boundary layer, to Tollmein-Schlichting waves, to layers of concentrated vorticity, to spots of "turbulent bursts," and finally to a turbulent boundary layer. Similar results, both experimental and theoretical, have also been reported for transition from laminar flows in other systems (see, e.g., Gill, 1965; Howard, 1963; Malkus and Veronis, 1958; Palm, 1960; Palm and Qiann, 1964; Schulter, Lortz, and Busse, 1965; Tippleskirch, 1956, 1957; Veronis, 1965). The mathematical formulation of the general problem of classical hydrodynamic stability is obtained by taking a (generally steady) solution of the relevant describing equations, by superimposing a perturbation upon the basic solution, and then by substituting the resultant disturbed motion into the relevant describing equations. A set of nonlinear equations of evolution for the growth of the disturbance results. As expected, the difficulties in the classical theory of hydrodynamic stability arise almost exclusively because the basic equations are nonlinear. The formulation of the describing equations has taken two distinctive forms in the literature: (i) a partial differential equation formulation (see, e.g. Lin, 1955) and (li) an integral equation formulation (see, e.g., Serrin, 1959). The former intensifies the difficulties due to the nonlinear nature of the equations whereas the latter tends to diminish possible mathematical difficulties due to the nonlinearity . The partial differential

PAGE 24

1. 10 equation formulation is arrived at by assuming that, for example, the difference velocity field V = V . — V ~D ** ~B where ^ : perturbed velocity field v„ : the basic velocity field whose stability is under ""B ... investigation may be expressed as an expansion in a complete system of normal modes, or eigenf unctions . The substitution of this expansion into the equations describing the rate of growth of the disturbance leads to an infinite system of coupled nonlinear ordinary differential equations for the amplitudes associated with the normal modes. The well-recognized source of the nonlinearity coupling these is the inertial term in Cauchy's First Law, voVv. Thus, if the expansion in normal modes is substituted for v., and if the inertial term is "**D isolated, two infinite series are then multiplied together; therefore, the ordinary differential equation for the amplitude associated with the ith mode is coupled with the amplitudes for all the other modes • Difficulties of this sort occur in many familiar mathematical models in varied disciplines. It is, therefore, reasonable to expect that approximation techniques have been proposed to deal with them, It is of some interest to note that Lamb (1945) ascribes this difficulty to a "mathematical disability."

PAGE 25

1. 11 and indeed a majority of them have been developed and applied in hydrodynamic stability theory. The early theoretical attempts at hydrodynamic stability theory (e.g., Rayleigh, 1880, 1887, 1895; Thompson, 1887a, 1887b; Taylor, 1923) attempted to circumvent these difficulties by concerning themselves with infinitesimal disturbances so that the coupling term v °Vv was assumed to be of negligible importance. Because of this assumption, the time-dependent part of the disturbance may be taken as an exponential form, exp(kt). If perturbations at the boundary are excluded, the boundary conditions on the disturbance are homogeneous, and one arrives at an eigenvalue problem for the parameter k. In this linearized theory, therefore, the flow is said to be unstable if it is possible for k to have a positive real part; otherwise, it is said to be stable. If k does have a positive real part, then the amplitude of the disturbance grows with time until the coupling term may no longer be neglected. The so-called "nonlinear" theories of hydrodynamic stability are constructed to account for this coupling term in some approximate manner. The two most frequently used techniques The interested reader may consult the books of Lin (1955) and Chandrasekar (1961) for further examples, discussion, and references .

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1.12 are: (i) the truncated modal evolution, in which only a fixed number, N, of modes are assumed to be of importance, and (ii) the normal mode cascade, in which it is assumed that initially there are only a finite number of "primary modes" and that all higher harmonics are formed by interactions of these primary modes. The interested reader will find examples of the truncated modal evolution in the works of Dolph and Lewis (1958), Lorentz (1962), Meister (1963), and Veronis (1963) . Examples of the technique of normal mode cascade will be found in the works of Eckhaus (1965) , Palm (1960) , Segel (1962, 1965a, 1965b), Stuart (1960a), and Watson (1960). Both these techniques accomplish the same thing in that they make the mathematical problem of stability tractable the truncated modal evolution by reducing the infinite system of equations to a finite number of equations while the normal mode cascade allows us to solve an infinite system successively. Closely related to both the linear and nonlinear theory is the "method of parametric expansion." This method is a linearized stability analysis of a steady-state solution of the equations describing the evolution of the difference state. Malkus and Veronis (1958) applied this technique to a particular finite amplitude solution for the Benard problem. Schulter, Lortz, and Busse (1965) extended the analysis of Malkus and Veronis to a larger class of possible finite amplitude solutions. This latter research is, of course, a theoretical attempt to explain why one or another of the possible finite amplitude solutions (mode of convection in

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1.13 the Benard problem) is preferred. All the methods of classical hydrodynamic stability discussed to this point are based upon the partial differential equation formulation, and they characteristically contain a number of assumptions about either the magnitude of the disturbance or the coupling of various normal modes of the disturbance; these assumptions alone enable them to achieve the primary objectives of hydrodynamic stability theory. In contrast, the integral equation formulation and the associated "generalized energy" method require no such assumptions of this nature to reach these objectives. In particular, the nonlinear terms may be fully accounted for. The integral equation formulation is thus a far more natural framework within which to view arbitrary but physically realistic perturbations. Unhappily, to treat a wide variety of disturbances requires that one sacrifice the more detailed results obtained from the partial differential equation formulation. Because the purview of the integral equation formulation is that of global properties, that is, of integrals over the flow regime, the details of the flow pattern are necessarily lost. Although this is a weakness of the "generalized energy" method, it is precisely the global approach, which deals with the overall effects rather than in details, that allows the method to be applied to disturbances of any_ magnitude . For example, if by critical we mean the onset of instabilities as predicted by linear theory, then Joseph (1965) has used the integral equation formulation to show the nonexistence of these subcritical instabilities for arbitrary

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1.14 spatially periodic disturbances. To be redundant for emphasis, the nonlinearity is retained and results far beyond those of linear hydrodynamic stability theory may be obtained. Moreover, because the method does not require a discussion of interactions of normal modes (the physical significance of which may not be all obvious) the physics described by the equations is readily apparent. The integral equation formulation and the associated "generalized energy" method seems to have originated in the work of Reynolds (see, e.g., Reynolds, 1895) and Orr (see, e.g., Orr, 1907). Although their basic approach to stability problems has been used through the years (see, e.g., Hamel, 1911; Serrin, 1959; Synge, 1938a, 1938b; Thomas, 1943; von Karman, 1924), it received only a small fraction of the use and the attention that the partial differential equation formulation received. Interest in the method, however, has risen steeply since the publication of Serrin in 1959. The work on the method since then may be placed into two classifications: (i) an improvement of the bounds used by Serrin (see, Velte, 1962; Sorger, 1966), and (ii) treatment of more complex problems (see, Conrad and Criminale, 1965; Joseph, 1965, 1966). Thus, Conrad and Criminale treated the case of time dependent basic velocity fields whereas Joseph treated the Benard problem with the Boussinesq equations. In all these various extensions and generalizations of Serrin's work, the key lies in the positive-definite property of

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1.15 the "generalized energy" and the inequalities which are used to obtain sufficient conditions for the time derivative of this "energy" to be negative-definite. The essential ingredient for stability is, to summarize, a generalized energy that is positive-definite yet with a negative-definite total time derivative. The properties of the "generalized energy" used by Serrin (1959) are precisely those of a Liapunov functional, and, in fact, Serrin' s use of the "generalized energy" to ascertain sufficient conditions for stability bears a striking resemblence to the use of Liapunov functionals in Liapunov 's direct method. Because of this similarity and because one of the major contributions of this dissertation is a generalization of Liapunov 's direct method, it is appropriate that we now discuss the method, its essential ingredients, its interpretation, its advantages, its limitations, and recent research work involving it. 1.3. Liapunov 's Direct Method Mathematics is at times the language of science and at other times it is the queen of science (Bell, 1955) . In its latter role mathematics does well in its imposition of a logical structure upon mathematical models of physico-chemical systems. On other occasions, however, it fails in both its roles, notably in dealing adequately with nonlinear problems. As an unhappy lesson of experience, moreover, the more accurate the model desired or the wider the range of theory sought, the more probable it is that the formulation will be

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1.16 nonlinear. And it long ago became clear that solutions to nonlinear operator equations are generally unobtainable, at least in the style to which science became accustomed to when it dealt only with linear problems. Analytical solutions seldom exist and quantitative methods of analysis, while helpful, in no way suffice as they do for linear problems. Qualitative methods, on the other hand, may tell us much about a nonlinear system, they may even tell us all that we need to know; but even when analytical methods provide us with some useful information, qualitative methods will always serve to complement them. Within the larger framework of qualitative methods in mathematics, the names of Henri Poincare and Alexander Mikhailovich Liapunov occupy a revered position. They were among the first to recognize the futility of attempting to obtain explicit solutions to all differential equations leading the trend toward obtaining qualitative characteristics of families of solutions of a differential equation. Whereas Poincare's interests in this area centered primarily upon the existence of periodic solutions and the geometrical properties of families of solutions generated by perturbations of an operator, Liapunov 's primary contribution was the development of a method for studying the geometrical properties of a family of solutions of a given equation, relative to a basic solution that need not be known. This last is the source of the terminology, Liapunov' s direct method. The direct method is a generic term for a number of theorems

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1.17 centered about several precise definitions of stability and certain real scalar-valued functionals that have come to be called Liapunov functionals. These functionals have the nature of a generalized "distance" or "energy:" (i) in much the same manner that a comparison of the norms of each of two elements of a linear space indicates which is closer to the origin, even though the elements are in different "directions" so also may different points of the state space be distinguished from one another, relative to the origin, by different values of their Liapunov functionals and (ii) in much the same way that a redistribution of energy provides a means of considering transient phenomena, so also may local changes in the values of Liapunov functional indicate that a trajectory in the state space locally Is tending toward or away from the basic trajectory (which may be degenerate, in that it may be a basic state). Loosely speaking, Liapunov functionals provide a partial ordering of the admissible states of the system, as well as of trajectories in the state space (Chapter III) . If we consider the Liapunov operators as real scalarvalued functionals, this geometrical interpretation may be easily illustrated (Elgerd, 1967; Hahn, 1963, 1967; Lasalle and Lefschetz, 1961). From the statements of the relevant theorems, a Liapunov functional, V<.>, for an autonomous system described by ^ = f(x), f(0) = insures asymptotic stability, for example, if it is a positive

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1.18 definite functional, and if its temporal variation along admissible trajectories of the system must be negative-definite. For ease of visualization, we suppose that the state of the system may be represented as a point x in IfS 9 > a particular state space generally referred to as the phase plane. We then erect a perpendicular to this plane at the origin to indicate the range space of the Liapunov functional. In this IK,, a Liapunov functional may be represented as an open-ended, bowl-shaped surface as indicated in Figure I. la, and in particular that is the graph of a single-valued function. Thus the projections of the V^x> = const. (1.1) loci onto the phase plane generate a system of closed, nonintersecting curves infR„ (see Figure 1.1), and it is an intuitively obvious and weil-known result of elementary topology that a closed curve in a plane divides the plane. Mathematically, the criterion of negative-definite temporal variations along all admissible trajectories may be expressed by dV = d£ . w < dt dt Now the classical geometric interpretation of the gradient is that of a vector, the direction field of which is everywhere perpendicular to loci given by equation (1.1), and whose magnitude is a measure of the spatial variation of V<.>. Geometrically, therefore, the mathematical inequality noted above may be interpreted as a requirement that the tangent to the field of trajectories f(x) in the phase

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1.19 A TYPICAL TRAJECTORY (b) FIGURE I.I, GEOMETRICAL INTERPRETATION

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1.20 plane must always and everywhere have a negative projection upon the corresponding gradient vector (see Figure I.l.b). Therefore the system is asymptotically stable if the admissible trajectories are such that succeeding states along them correspond to lower values of the Liapunov function. And in this sense, the succeeding states may be said to be "closer" to the origin. The framework of this geometrical interpretation of Liapunov 's direct method may also be used to emphasize an important point about Liapunov' s direct method: it provides only sufficient conditions for stability which is, incongruously, both a strength and a weakness, of which we shall have more to say later. If we had plotted in this phase plane the vector field, _f(x), in some neighborhood of the origin (the basic solution), we could, by the method of isoclines and visual observation, determine the behavior of trajectories in that neighborhood of the origin; it would, however, still remain to find a suitable Liapunov functional if stability were the case Conversely, if we had a Liapunov functional with the required properties in some neighborhood of the origin, then we could avoid che graphical construction of the method isoclines in that neighborhood and yet be assured of stability. Now in state spaces of higher dimension the first alternative is not a viable one, and if we were also unable to find a suitable Liapunov functional, then we would be unable to draw any conclusions about the stability of the origin.

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1.21 This sufficient nature of Liapunov's direct method permeates every part of its structure and is at once both the greatest strength and the most significant weakness of the method. We say strength because, as a consequence of this sufficient nature, one has an analytical tool powerful enough to assure us of stability in some neighborhood of the origin not only for one equation but for a class of equations a monumental accomplishment. Moreover, this may be accomplished without the necessity of the many oversimplifications usually necessary to obtain an explicit solution to just one number of this class of equations. On the other hand we say that this sufficient nature is also a weakness of the method because if we are unable to discover a suitable Liapunov functional, even after an extensive search, we know no more about the stability of the origin than before we began. Also, even if a suitable Liapunov functional has been found and an associated neighborhood of stability determined, this does not say that the portion of state space outside this neighborhood is a region of instability, Thus the major iculty in applying Liapunov's direct method to a particular system or class of systems is in obtaining a Liapunov functional with the required properties, Consequently, it is reasonable to expect that much of the research on the method is devoted to techniques tor constructing suitable Liapunov functionals (see, e.g., Brayton * Another not inconsequential advantage of Liapunov's direct method, it seems to us, is its simple physical and geometrical interpretation (cf . above;

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1.22 and Miranker, 1964; Hahn, 1963, 1967; Ingwerson, 1961; Leighton, 1963; Letov, 1961; Luecke and McGuire, 1967; Krasovskii, 1963; Peczkowski and Liu, 1967; Schultz and Gibson, 1962; Szego, 1962; Walker and Clark, 1967; Zukov, 1961). Generally speaking, the proposed methods may be classified into three categories: (1) those which assume a certain form for the gradient of the Liapunov functional, (2) those which assume a certain form for the Liapunov functional or its time derivative, and (3) those which make use of the similarity of two systems, for one of which a Liapunov functional is already known, with the standard application being an extension from a linear system to a "slightly" nonlinear system. Zubov's now-famous method falls under the first classification (see, e.g., Hahn, 1963; Zubov, 1961). Essentially, it makes use of the geometric interpretation of a first order, partial differ* ential equation. If the existence of a solution can be proved for the partial differential equation, then one will have proved the existence of a Liapunov functional for the system of ordinary differential equations. Moreover, Zubov's method actually provides a constructive method for Liapunov' s method based upon the solution of the related first order inhomogeneous partial differential equation. The method of "variable gradients" (see, e.g., Hahn, 1967; Schultz and Gibson, 1962) also belongs to this first classification. In The reader is referred to any standard textbook in the theory of partial differential equations.

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1.23 this method one assumes a form for the gradient of the Liapunov functional and requires that the functional be single-valued. This latter requirement allows us to find the Liapunov functional by a line integration along any convenient path. The methods of "separation of variables (Letov, 1961), "canonical variables" (see, e.g., Brayton and Miranker, 1964; Letov, 1961; Zubov, 1961), and "squares" (see, e.g., Hahn, 1963, 1967; Letov, 1961; Krasovskii, 1963) all belong to the second classification. In particular, the method of squares focuses upon the fact that any positive-definite form is a possible Liapunov functional, an essential feature to which we return in Chapter IV. The most widely used method under the third classification at least among engineers, is the one using a Liapunov functional for a linear system to determine the region of asymptotic stability for a nonlinear system which is somehow close to the linear system (see, e.g., Krasovskii, 1963, Chapter IV). These techniques of constructing Liapunov functionals and to a lesser extent the theorems themselves had, until 1960, been applied mainly to stability problems associated with systems of ordinary differential equations, or in engineering parlance, to lumped parameter systems. However, since the appearance of Zubov' s monograph (see, Zubov, 1961), there has been an increased interest in applying the method to mathematical models involving differencedifferential equations (see, e.g. , EL' s gol'c 1964, 1966; Krasovskii, 1963) and to partial differential equations (see, e.g., Brayton and Miranker, 1964; Hsu, 1967; Mochvan, 1959, 1961; Parks, 1966;

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1.24 Wang, 1964, 1965), that is to so-called distributed parameter systems. 1.4. Scope of the Present Investigation Although our interests lie generally with systems that are described by the general equation of balance, in this dissertation we consider, almost exclusively, systems arising from continuum mechanics. The primary objectives, then, of this dissertation are to present a unifying formulation of stability problems associated with continuous media and a unified approach to their analysis by means of Liapunov's direct method. The methods and concepts which have been employed in realizing these objectives are those of modern mathematics, particularly, functional analysis. The main significance of this unifying formulation and unified approach is the fresh outlook from which to view not only the very old problems of hydrodynamic stability but also those of arbitrary physical and mathematical complexity. This dissertation presents a systematic way of proceeding from the hypothetical mathematical model of a physical system, through the corresponding operator equations of evolution for the difference state, and, finally, through operator equations whose solution will be a Liapunov operator for that particular physical system. In effect, therefore, we have divorced the physical problems involved with modelling from the mathematical problems involved with solving an operator equation for the Liapunov operator. From another point of view, however, we have directly related the physical problems to the mathematical

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1.25 problems, in that once the mathematical model for a physical system is selected, the corresponding operator equations for a Liapunov operator may be immediately obtained by mere substitution in the relevant equations. Again, from a still more distant vantage point, we have actually provided but a simple illustration of the power and versatility that the abstract spirit of modern mathematics may bring to bear on the complicated problems facing scientists and engineers of today. Chapter II is devoted to the preliminaries necessary to optimally utilize the power latent in Liapunov' s direct method. Thus, Chapter II relates and discusses a state approach to the models of continuum mechanics and the place of stability problems within the class of perturbation problems. In tone if not in accomplishment, this chapter provides a somewhat novel approach to continuum mechanics. We close the chapter with a natural generalization of some of the work of Serrin (1959) anH Joseph (1965, 1966) in order to illustrate the advantages of an abstract approach to stability problems by using the simplest type of Liapunov operator a quadratic form to arrive at sufficient conditions for stability, one of our few concrete results. In Chapter III, we turn to a consideration of Liapunov 's direct method and develop a generalization of the method. Instead of restricting ourselves to the real line as the range space for Liapunov functionals , we consider Liapunov operators with partially ordered linear topological spaces and, in particular, positive

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1.26 cones in these spaces as the range spaces. It is reasonable to expect that the richer and more delicately structured the domain of the Liapunov operators is because this is the state space the more desirable it becomes for us to consider range spaces richer than the real line if we are to provide a more highly discriminating ordering of the elements and trajectories in the state space by means of these operators. In other words, if one considered the set of all possible Liapunov operators, it would be greatly enriched. Moreover, it is likely that certain systems may be especially apt for the application of these Liapunov operators, whereas they may have hitherto been unamenable to analysis by means of Liapunov functionals, and because of the sufficient nature of the method they may have gone unanalyzed by the direct method of Liapunov. Of the many theorems that could have been developed from this conceptual breakthrough, we emphasize only generalizations of the main theorems on stability, asymptotic stability, instability, unbounded instability, and also the conditional nature of the new method. In Chapter IV we employ these theorems and a slight generalization of the notion of a quadratic form to arrive at operator equations for the construction of these Liapunov operators, and in Chapter V we take a very special class of physical systems, namely, those whose state is specified by the velocity field to illustrate the versatility of our formulation and its possible range of application. For the convenience of the reader in investigating the references, we present a block diagram of Hydrodynamic Stability Theory in Figure 1.2.

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1.27 « uj o — >C3 —I 13 C

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BIBLIOGRAPHY A. ON LIAPUNOV'S DIRECT METHOD Antosiewicz, H. A. 1959, A survey of Liapunov's second method, in Contr. to Nonlinear Oscillations , S. Lefschetz (Ed.), IV, 141. Antosiewicz, H. and Davis, P. 1954, Some implications of Liapunov's conditions of stability, J. Rat. Mech. Anal. , _3, 447. Aris, R. 1965, Dynamic Programming , Coll. Lectures in Pure and Applied Science, no. 8, Socony Mobil Oil Co., Dallas, Tex. Brayton, R. K. and Miranker, W. L. 1964, A stability theory for nonlinear mixed initial value problems, Arch. Rat. Mech. Anal . , 17, 358. Cartwright, M. L. 1956, The stability of solution of certain equations of fourth order, Quar. J. Mech. Appl. Math. , 9_, 185. Chetaev, N. G. 1961, The Stability of Motion , Pergamon Press, London. Cunningham, W. J. 1958, Introduction to Nonlinear Analysis , McGrawHill Book Co. , N.Y. Elgerd, 0. I. 1967, Control Systems Theory , McGraw-Hill Book Co., N.Y. El'sgol'c, L. E. 1964, Qualitative Methods in Mathematical Analysis , American Mathematical Society, Providence, R.I. Ergen, W. K. , Lipkin, H. J. and Nohel, J. A. 1957, Applications of Liapunov's second method in reactor dynamics, J. Math. Phys. , 36, 36. Hahn, W. 1955, Uber Stabilitat bei nichlinearen Systemen, ZAMM 35, 459. , 1957, Uber Dif ferential-Diff erenzengleichungen, Math. Ann. , 133 , 251. 1963, Theory and Applications of Liapunov's Direct Method , Prentice-Hall, Inc., Englewood Cliffs, N.J. 1967, Stability of Motion , Springer Verlag, Inc., N.Y. Hsu, C. 1967, Stability analysis of reactor systems via Liapunov's second method, J. Basic Eng. , Trans. ASME , ser. D , preprint no. 66-WA/Aut-F. 1.28

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1.29 Ingwerson, D. R. 1961, A modified Liapunov method for nonlinear stability problems, IRE Trans, on Automatic Control , May , 199, Kalman, R. E. and Bertram, J. E. 1960, Control analysis and design via the second method of Liapunov, J. Basic Eng . , Trans. ASME , ser. D , 82 , 371. Krasovskii, N. N. 1959, On the theory of optimal control, Appl. Math . Mech. , 2_3, 624. , 1963, Stability of Motion, Stanford Univ. Press, Stanford, Calif. Lasalle, J. P. and Lefschetz, S. 1961, Stability by Liapunov 's Direct Method with Applications , Academic Press, N.Y. Leighton, W. 1963, On the construction of Liapunov functions for certain autonomous nonlinear differential equations, Contrib. to Diff. Eq . , 2, 10. Letov, A. M. 1961, Stability in Nonlinear Control Systems , Princeton Univ. Press, Princeton, N.J. Liapunov, A. M. 1967, General Problem of Stability of Motion , Academic Press, N.Y. Luecke, R. H. and McGuire, M. L. 1967, Stability analysis by Liapunov's direct method, IEC Fundamentals , 6_, 432. Malkin, I. G. 1950, Certain questions in the theory of stability of motion in the sense of Liapunov, Amer . Math. Soc . Translations, 2, 20. , 1959, Theorie der Stabilitat einer Bewegung , Verlag R. Oldenbourg, Muchen. Movchan, A. A. 1959, The direct method of Liapunov in stability problems of elastic systems, Appl. Math. Mech. , 23 , 483. , 1961, Stability of processes with respect to two matrics, Appl. Math. Mech. , 24, 988. Parks, P. C. 1966, A stability criterion for panel flutter via the second method of Liapunov, AIAA J. , 4_, 175. Peczkowski, J. L. and Liu, R. W. 1967, A format method for generating Liapunov functions, J . Basic Eng. , Trans. ASME , ser. D, preprint no. 66-WA/Aut-17 . Rekasius, Z. V. and Gibson, J. E. 1962, Stability analysis of

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1.30 nonlinear control systems by second method of Liapunov, IRE Trans, on Automatic Control , 7_» 3. Schultz, D. G. and Gibson, J. E. 1962, The variable gradient method for generating Liapunov functions, Trans, of AIEE , 81, 203. Slobodkin, A. M. 1962, On the stability of equilibrium of conservative systems with an infinite number of degrees of freedom, Appl. Math. Mech. , 26 , 356. Szego, G. P. 1962, A contribution to Liapunov's second method: nonlinear autonomous systems, J. Basic Eng. , Trans. ASME , ser. D , 84, 571. Tauzsky, 0. 1961, A remark on a theorem of Liapunov, J. Math. Anal. Appl. , 2, 105. Walker, J. A. and Clark, L. G. 1967, An integral method for Liapunov function generation for nonautonomous systems, J . App 1 . Mech. , Trans. ASME , ser. E , 87 , 569. Wang, P. K. C. 1964, Control of distributed parameter systems, Advances in Control Systems , _1, 75. , 1965, Stability analysis of a simplified flexible vehicle via Liapunov's direct method, AIAA. J. , _3, 1764. Warden, R. B., Aris, R. and Amundson, N. 1964, An analysis of chemical reactor stability and control XIII, Ch.E.Sci. , 19 , 149. Zubov, V. I. 1961, The Methods of A.M. Liapunov and Their Application , AEC-tr-4439, Department of Commerce, Washington, D.C.

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1.31 HYDRODYNAMIC STABILITY Bell, E. T. 1955, Mathematics : The Queen of Science , Dover Pub., Inc., N.Y. Benard, H. 1901, Les tourbillons cellulaires dans une nappe liquide transportant de la chaleur par convection en regime permanent, Annales de Chemie et de Physique , 23 , 62. , 1927, Sur les tourbillons et la theorie de Rayleigh, Comp . Rend . , 185 , 1109. Benney, D. J. 1964, Finite amplitude effects in an unstable laminar boundary layer, Phys. Fluids , _7_, 319. Chandrasekar , S. 1961, Hvdrodvnamic and Hvdroma ,g.net 1 ic Stability. Clarendon Press, Oxford. Coles, D. 1965, Transition in circular Couette flow, J. Fluid Mech. , 21, 385. Conrad, P. W. and Criminale, N. 0. 1965, The stability of timedependent laminar flows: parallel flows, ZAMP , 16 , 233. Dolph, C. L. and Lewis, D. C. 1958, On the application of infinite systems of ordinary differential equations to perturbations of plane Poiseville flow, Quar. Appl. Math, 16 , 97. Eckhaus, W. 1965, Studies in Nonlinear Stability Theory , Springer Verlag, N.Y. Elder, J. 1960, An experimental investigation of turbulent spots and breakdown to turbulence, J . Fluid Mech. , _9, 235. Emmons, H. W. 1951, The laminar-turbulent transition in a boundary layer, J. Aero. Sci. , 18 , 490. Gill, A. E. 1965, A mechanism for instability of plane Couette flow and of Poiseuille flow in a pipe, J . Fluid Mech. , 21 , 503. Greenspan, H. P. and Benney, D. J. 1963, On shear-layer instability, breakdown, and transition, J . Fluid Mech. , 15 , 133. Hamel, G. 1911, Zum Turbulenzproblem, Nachr . Ges. Wiss . Gottingen, 261. Howard, L. N. 1963, Heat transport by turbulent convection, J. Fluid Mech. , 17, 405.

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1.32 Joseph, D. D. 1965, On the stability of the Boussinesq equations, Arch. Rat. Mech. Anal. , 9_, 59. Joseph, D. D. 1966, Subcritical convective instability, J. Fluid Mech. , 26, 753. Klebanoff, P. S., Tidstrom, K. D. and Sargent, L. M. 1962, The three-dimensional nature of bdy. layer instability, J. Fluid Mech. , 12, 1. Kovasznay, L., Kamoda, H., and Vasudeva, B. 1962, Detailed flow field in transition, Proc . 1962 Heat . Transf . Fluid Mech. Conf . , Stanford U. Press, Stanford, Calif. Lamb, H. 1945, Hydrodynamics , sixth edition, Dover Publications, N.Y. Lin, C. C. 1955, Theory of Hydrodynamic Stability , Cambridge Univ. Press, Cambridge. Lorentz, E. N. 1962, Simplified dynamic equations applied to rotatingbasin experiments, J Atmos . Sci . , 19 , 39. Malkus, W. and Veronis, G. 1958, Finite amplitude cellular convection, J. Fluid Mech., 4, 225. Meister, B. 1963, Die Anf angswertauf gabe fur die storungs differentialgleichungen des Taylorschen Stabilitatproblems , Arch. Rat. Mech. Anal. , 14 , 81. Miller, J. A. and Fejer, A. A. 1964, Transition phenomena in oscillating boundary layer flows, J. Fluid Mech., 18 , 438. Orr, W 1907, The stability or instability of motions of a liquid. Part II: a viscous liquid, Proc. Roy. Irish Acad. (A) , 27 , 69. Palm, E. 1960, On the tendency towards hexagonal cells in steady convection, J . Fluid Mech. , _8, 183. Palm, E. and Qiann, H. 1964, Contribution to the theory of cellular thermal convection, J . Fluid Mech. , 19 , 353. Rayleigh, Lord 1880, On the stability, or instability, of certain fluid motions, Scientific Papers, 1, 474, Cambridge Univ. Press, Cambridge. , 1887, On the stability, or instability, of certain fluid motion. II, Scientific Papers , 3_, 2,

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1.33 1895, On the stability, or instability, of certain fluid tions: III, Scientific Papers , 4_, 203, Cambridge Univ. mo Press, Cambridge. Reynolds, 0. 1895, On the dynamical theory of incompressible viscous fluids and the determination of the criterion, Phil. Trans. Roy. Soc. London (A) , 186 , 123. Sani, R. 1963, Ph.D. Dissertation , Univ. of Minn., Minneapolis, Minn. Schiilter, A., Lortz, D. and Busse, F. 1965, On the stability of finite amplitude convection, J. Fluid Mech. , 23 , 129. Segal, L. A. 1962, The nonlinear interaction of two disturbances in the thermal convection problem, J. Fluid Mech. , 14, 97. 1965a, The structure of nonlinear cellular solutions to the Boussinesq equations, J. Fluid Mech. , 21 , 345 1965b, The nonlinear interaction of a finite number of disturbances to a layer of fluid heated from below, J. Fluid Mech. , 21, 359. 1966, Nonlinear hydrodynamic stability theory and its application to thermal convection and curved flows, in Non-Equilibrium Thermodynamics, Variational Techniques, and Stability , Univ. of Chicago Press, Chicago, 111. Serrin, J. 1959, On the stability of viscous fluid motions, Arch. Rat. Mech. Anal. , _3> 1~ Sorger, P. 1966, Uber ein Variationsproblem aus der nichtlinearen Stabilitatstheorie zaher, inkompressibler Stromungen, ZAMP , 17, 201. Stuart, J. T. 1960a, On the nonlinear mechanics of wave disturbances in parallel flows I, J. Fluid Mech. . 18, 841. 1960b, Hydrodynamic Stability, Appl. Mech. Rev. , 523. Synge 1965, On the cellular patterns in thermal convection, J. Fluid Mech. , 18, 841. 1965, Hydrodynamic stability, Appl. Mech. Rev. , 18 , 523, J. L. 1938a, Hydrodynamic stability, Semi-centennial Publications of Amer. Math. Soc . 2 (Addresses), 227. 1938b, The stability of plane Poiseuille motion, Proc, Fifth Int er. Cong. Appl. Math. . 326, Cambridge, USA.

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1.34 Taylor, G. I. 1923, Stability of a viscous liquid contained between rotating cylinders, Phil. Trans. A , 223 , 289. Thomas, T. Y. 1943, On the uniform convergence of the solutions of the Navier-Stokes equations, Proc . Nat. Acad. Sci. USA , 29 , 243. Thompson, J. J. 1887a, Rectilinear motion of a viscous fluid between parallel planes, Mathematical and Physical Papers , k_, 321, Cambridge Univ. Press, Cambridge. , 1887b, Broad river flowing down an inclined plane bed, Mathematical and Physical Papers, 4_, 330, Cambridge Univ. Press, Cambridge. Tippleskirch, H. 1956, Uber Konvektionzellen, imbesondere im flussigen Schwefel, Beitr. Phys. frei Atmos . , 29 , 37. Tippleskirch, H. 1957, Uber die Benard-Stromung in Aerosolen, Beitr. Phys. frei Atmos. , 29 , 219. ft Velte, W. 1962, Uber ein Stabilitatskriterium der Hydrodynamik, Arch. Rat. Mech. Anal. , 9_, 9. Veronis, G. 1963, An analysis of wind-driven ocean circulation with a limited number of Fourier components, J . Atmos . Sci . , 20 , 277. Veronis, G. 1965, On finite amplitude instability in thermohaline convection, J . Marine Res . , 23 , 1. von Karman, Theodore 1924, Uber die Stabilitat der Laminarstromung und die Theorie der Turbulenz, Proc. First Inter. Cong. Appl. Mech. , 97, Delft. Watson, J. 1960, On the nonlinear mechanics of wave disturbances in stable and unstable flows II, J. Fluid Mech. , 9, 371.

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CHAPTER II THE GENERAL STABILITY PROBLEM II. 0. Prolegomena II. 0. a. Introduction Given a physical system, the selection of an appropriate state space and an operator equation are primarily questions of utility and of accuracy. As important as such questions are in formulating a mathematical model of a physical system, we shall defer them until Sections II. 2 and II. 3. The operator equation of evolution may be looked upon as an abstract operator that maps a given initial state £ into the sequence of states of the system; a sequence of states [s,i.HVso generated will be referred to as a state space trajectory. In actuality, however, if we write the operator equation in the form then the operator S< t > may be viewed as a temporal connection* that maps the state of the system at one instant of time, s^ , into the state of the system at the "next" instant of time, s. r.. This suggestive terminology for the right side of the equation of evolution will often be used in the sequel, but it must be kept in mind that *The interpretation given to the words "temporal connection" is analogous to the use of the words connection coefficient in differential geometry. The operator S<.>may also be looked upon as a map from the state space to the space of linear and continuous operators from the real line to the space state. II. 1

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II. 2 the temporal connection will be single-valued only if the boundary conditions are incorporated into S<.> — which means that the equation must be reinterpreted, say as an integral operator incorporating the boundary conditions — and if the problem were well-posed to begin with h so that there are single trajectories emanating from each element of the state space. It is often the situation that arbitrary initial states or arbitrary classes of initial states are to be considered, in which case we speak of the flow of a set of states from the initial set. The classical theory of stability is the study of the fate of a perturbation superimposed upon a given state space trajectory, or it may be the study of a state space trajectory acted upon by intermittently or continuously acting perturbations. Or, more generally, one can study a class of perturbations acting either once and for all, intermittently, or continuously but superimposed on either a state space trajectory or upon a flow of state space elements. If the evolution of the state of a system can be described accurately by a finite set of ordinary differential equations, then the state space is finite-dimensional. There are standard definitions of the -different types of stability in finite-dimensional state spaces (see, e.g., Kalman and Bertram, 1960; Hahn, 1963), If the changes of state of a system must be described by a system of partial differential equations to achieve the desired accuracy, then the state space is infinitedimensional. There also are available standard definitions of stability in infinite-dimensional state spaces (see, e.g., Zubov, 1961;

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II. 3 Hahn, 1963, 1967). In stability theory, as in any mathematical formulation of a specific physical system, the proper selection of a specific metric, or at least a specific topology is of utmost importance. This is especially true in stability theory for we are interested in the closeness of both trajectories and states. Conversely, the selection of a specific metric may limit a theoretical analysis needlessly. No specific metric or other type of topology is considered explicitly: we consider abstract spaces as state spaces and an intuitive (essentially topological) notion of stability and asymptotic stability.* Not content with this level of generality, we shall consider (see Sections II. 3. a., II. 3. b., II. 3. c, and II. 2) the formulation in the context of a general dynamical process — by which we mean the set of state space elements of a flow, the operator equation of evolution generating that flow, and the boundary and initial conditions delineating the succession of state space elements. By considering perturbations of dynamical processes as well as perturbations of basic states (please see Section 11,3), classical perturbation theory and stability theory under perturbations to both boundary conditions and initial conditions are brought under the same province as classical stability theory. There is, of course, a certain unity to be gained from this. It is to this and to the abstract formulation with an "arbitrary" set *In Appendix A, however, we do consider, for illustrative purposes, specific normed linear spaces appropriate to specific equations drawn from continuum mechanics, and we consider the physical meaning of different norms and their relations to different types of stability.

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II. 4 of general balance equations that we refer when we speak of a unified stability theory. II.Q.b. On the Scope of .the Theory Although the fate of a perturbation, loosely speaking, is our ultimate aim, it is first necessary to attempt to say what is meant by the terms "state space," "perturbation" and "relevant describing equations." Thus, with regard to the last, the derivation of a quite general form for the operator equations of evolution in terms of essentially three possible representations (integral equations, differential equations, difference equations and combinations thereof) is discussed. The two factors which are helpful in making possible the formulation of a unified approach to stability theory based upon the general balance equations are: (i) each of the principles of physics — that is, laws of our method of representation — has the form of this equation but with a different physical interpretation attached to the mathematical symbols in each principle, and (ii) the facility of Liapunov's direct method for dealing with classes of equations as well as with specific equations. In particular, we state at the onset that the mathematical terms of the general balance equations are to be interpreted within the context of continuum mechanics, thereby, also including mass point mechanics.* The unification of the various *Hamel (1908) has shown that when the motion of a body about and relative to the center of mass may be neglected, then the equations of continuum mechanics reduce to those of mass point mechanics. In his formulation it is stress rather than force which is of central importance. See also Noll (1959).

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II. 5 principles of continuum physics in the form of a single general balance equation emphasizes the now traditional, essentially axiomatic formulation of physics. In this way we are able to present a unified treatment of the class of stability problems associated with the objects and phenomena within the range of the physical theories referred to as continuum physics. Accordingly, subclasses of stability problems associated with physical systems which involve continuous media, classes of phenomena, classes of materials, and classes of state space trajectories may be delineated by the specification of a set of hypotheses. It is clear, moreover, that the selection of the relevant describing equations cannot be divorced from the selection of the state space, nor can the selection of a state be separated from the selection of a set of describing equations. The two are but different aspects of a whole, the description of the processes occurring in a given physical system. It is nevertheless useful to make the distinction, however useless it may seem, in formulating the problem, for it is the very essence of solving the problem. In other words, the stability problem is so formulated that its physical — and mathematical — complexity may be regulated by specifying the number and nature of the state variables of the system, by specifying the nature of all significant mechanisms for transport — the internal couplings in the system, by specifying the nature of volume interactions with the surroundings, and by specifying the nature of the interactions of the system with the continguous sur-

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II. 6 roundings. This formulation is accomplished by simultaneously considering a finite number of equations of the form of the general balance equation, by considering the basic state whose stability is being investigated as a solution to this set of equations, and finally by employing slight revisions of some familiar mathematical relations in order to obtain a general operator equation of evolution for the difference between a perturbed state and a basic state. II. 1. The Space of States Because we have chosen to interpret the terms in the general operator equation of evolution within the context of continuum mechanics, the state variables naturally appear as fields defined over a region of physical space. The values of these fields at a point in the region of physical space — that is, the physical properties such as temperature, velocity, stress, density, concentration, polarization density, etc. — describe, indicate, and provide some measure of the physical situation at that point. To be more precise, we have D efinition B.I.I : At an instant of time, the local state of a point in the region of physical space, X^~ , is said to be known if the values of an independent set of these physical properties are known. The members of this independent set are referred to as the local state variables. In a particular description of some physical phenomena, the meaning of an "independent" set is intimately linked to both the nature of the continuous medium occupying the region of

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II. 7 interest, *R , of physical space and the physical theory being used to describe the phenomena. Alternatively, the choice of what constitutes an independent set of these physical properties may be used to characterize the range of a proposed physical theory (cf . e.g., Toupin's theory of an elastic dielectric [Toupin, 1956]; Coleman and Noll's theory of thermostatics [Coleman and Noll, 1958]). We do not take up the general and profound question of what constitutes an independent set of state variables, although we do take it up peripherally as we consider specific (but still somewhat general) situations of the basic equations that describe them. Continuum mechanics is constructed, however, so as to deal directly with gross phenomena occurring in finite portions of physical space. For this reason it deals with fields whose independent variables are the points within the region of interest, f\ , and whose values are the various physical properties. Thus, the specifications of a field is actually a specification of a continuum of local properties. In view of this, we define, Definition II. 1.2 : At any instant of time, the global state of a region of physical space, ft , is said to be known if the field yielding the local state at each point of K is known. Now corresponding to local and global states, two types of state space may be constructed. In particular, the local state at some point, x c , in the region, 1^ , may be of particular interest. The instantaneous local state at that point, x , may be given as an N-tuple whose elements are the values of an independent set of the

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II. 8 physical properties. Some of these elements will be real numbers, some vectors, some dyadics, etc., but for convenience, this N-tuple may be represented as a point in an appropriate state space formed by taking (N) direct products of spaces to which each of the components of the N-tuple belong. In general, the local state at the point, x , will be different at different times. It would, therefore, be represented by different points in the local state space of the point, x , of |^ . The totality of all the local states associated with a point, x , of "Tx^ as time elapses is a curve in this local state space. This curve is called the trajectory of local states associated with the point, x , of 1\. . On the other hand, our interest may be in the instantaneous global state of a region, \\_ . In order to specify the global state, a continuum of these local states must be specified. The concept of a field, however, precisely specifies this global state by means of an N-tuple of fields — one element corresponding to each of the local state variables. Again, for convenience, one may look upon this N-tuple of fields as a point in an appropriate state space. This state space will now be formed, however, by (N) direct products of appropriate function spaces (or, in general, operator spaces). The choice of the particular types of function spaces involved in the (N) direct product operations depends upon the nature of the physical properties associated with the fields describing the system and involved in the N-tuple. Because the local states may vary with time, the global states of necessity will also vary with time. Different

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II. 9 global states associated with different times are represented by different points in this global state space. The totality of all these points as time is varied continuously is called the trajectory of global states . Any point on this curve, along with the associated instant of time, is referred to as an event . An event is thus specified by an (N+l)-tuple. Intimately connected with this concept of "state" is the concept of "system." The conceptual division of the universe into system and surroundings allows a discussion of process of primary concern without discussing all processes in the entire universe, a problem of trivially insurmountable proportions. Although this division of the universe is arbitrary, a system may only be isolated for study if the influence of the rest of the universe — the surroundings — may be adequately controlled or described or both. The intimate relation between the state of a system and the surroundings of a system thus occupies a central position in the construction of any meaningful theory. Another distinction which will be found convenient because we intend to deal with stability problems is that of environment of a system, by which we mean all alterable (i.e., controllable) external influences on the system. Suppose, for example, that a particular physical system has been designated for study. The surroundings are immediately fixed by this and an appropriate selection of the state variables. However, the environment is not automatically fixed. For example, one might only be interested in and able to

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11.10 control the effect of alterations of only a specific type of external influence (e.g., an external electric field or a condition at the physical boundary) , in which case only that external influence would constitute the environment. The environment is thus a controllable subset of the surroundings of a given system. Needless to say, fields that are parts of the environment in one class of systems may be parts of the system or nonenvironmental parts of the surroundings in other classes of systems. The choice, then, of what constitutes the environment is directly related to the question "stability with respect to what?" because it may specify some of the ways in which perturbations may be generated. Thus, as observers we are necessarily a part of the surroundings and, presumably, are the modus operandi behind alterations of the controllable external influences that can perturb the state of a system. In classical stability theory, no allowance is made for perturbing the system to its new state nor of permitting the system any "inertia" in that direction. One simply assumes that perturbations are achieved instantaneously and then the system ceases to be acted upon by the external influences that provided the original (step) change of state. The tacit assumption, of course, is that by considering all possible perturbations (at least of a given class) , one accounts for many of these effects. More generally, intermittent or continuous perturbations provide a more reasonable description of these effects. (A further discussion of perturbations arising from environmental influences is presented in an addendum at the end

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11.11 of this section). There are, however, uncontrollable external influences that can perturb that state. Worse, there are molecular sources providing for random fluctuations of the continuum (or macroscopic) variables, and these sources of perturbations lie outside the framework of the commonlyused theories. By the same token, there has been little study of the influence of macroscopic fields upon the microscopic processes, for example, upon the theory of fluctuations. Some usual choices for the environment are: (1) certain classes of boundary conditions (the specification of fields or of an operator equation for the fields on a mathematical surface or surfaces bounding the region f\ and representing the interaction of the system with its contiguous surroundings) ; (2) certain classes of body force fields (the specification of the interaction of system and external influences acting throughout the region K ). A variety of terms such as physical property, global state, event, trajectory, system, surroundings, and environment, have been introduced. They have been discussed separately, and the relationships between them have not been emphasized. To provide a degree of unity another concept must be introduced, that of a dynamical process. The dynamical process (DP) consists of all the information that is needed to transform the present state into the next state in a manner which conforms with physical reality.* Consequently, the particular *This is the ultimate criterion of any physical theory, and although it is universally recognized, it should nevertheless not go unstated.

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11.12 elements in a dynamical process depends upon the previous selection of (i) the state variables (ii) system and, (iii) the surroundings, including the environment. Although the choice of specific quantities for these three elements of a dynamical process should be made simultaneously, let us discuss separately how the "next" state is affected by them. A choice of the state variables (i.e. an independent set of fields) limits the class of possible "next" states in three ways. First, it indicates the types of physical phenomena which are included in the study (the range of a physical theory). Second, it suggests which of the principles of physics will supply an independent set of governing equations. Thus, if the state variables for a particular problem have been correctly chosen, the relevant principles of physics are precisely those sufficient to describe the evolution in time of these state variables.* The third effect precipitated by a choice of the state variables is a restriction it places upon the choice of an environment.** *As a simple illustration, the models of isothermal and aonisothermal flow of a single-component newtonian fluid may be considered. In the model of isothermal flow the Navier-Stokes equations are taken to be the describing equations. On the other hand, in the nonisothermal model these equations must be considered simultaneously with an equation governing the temperature field. **Again, the model of isothermal flow of an incompressible pure newtonian fluid may be used as an illustration. If the density and velocity fields are selected as the state variables, then it will not be consistent to choose for the environment — i.e., the alterable external influences — anything that may significantly alter the temperature field.

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11.13 Because the selection of a system and its environment are intimately related, the question of specifying the state variables as distinct from the environmental influences is quite delicate. It traditionally hinges more upon theoretical and experimental (or both) tractability than upon a bona fide effort at dealing with coupled fields. For example, suppose we have a newtonian fluid flowing, g subject to an electromagnetic, as well as a gravitational, field. The describing equations are the Navier-Stokes equations (with Maxwell stresses) and Maxwell equations for a flowing (continuous) medium. The relevant boundary conditions plus the (applied) body force fields constitute the surroundings, and the problem is then formulated. Unfortunately, we are not yet in a position to solve nonlinear partial differential equations, much less highly coupled, nonlinear sets of such equations. The uncoupling device is the following: assume electromagnetic fields that result as solutions of Maxwell's equations for certain similar geometries, etc., and use them as the entries in the Navier-Stokes equations. In the latter situation, then, electromagnetic fields are taken to be part of the environment, whereas in the former they are part of the system and are state variables. It is clear, then, that were we able to solve the full, coupled, set of equations, the state space trajectories would provide us a full description, but because we are mathematically inept, we must resort to a specification of certain of the state variables once and for all in order to find the approximate temporal and spatial variations of the remaining fields. Clearly,

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11.14 the selection of a "smaller" state space will strongly prejudice the accessible next states* of the system, simply by excluding the full range of values for those "state variables" that we have been forced to specify as part of the environment. The way in which the environment influences the next state may be loosely described as follows. In the absence of environmental influences and presuming uniqueness (uniqueness is assumed throughout the discussion; the argument may be generalized to nonunique situations) , there will be a single trajectory passing through a given state, with the "next" state being thus defined. For a given non-zero value of the environment, a different trajectory will in general pass through that state, and as the environment is varied, still other trajectories will result-, With each change of trajectory, there will be, of course, a change of "next" states. The class of possible "next" states is, of course, also affected by the past history of the system simply because we consider a state space trajectory as beginning somewhere, at some initial event. In general, if the system passed through different initial events in the past, they will occupy different current events at present and therefore have different future events. For certain classes of equations of evolution, however, the past history of a system assumes an even more significant role, in that a complete *To reiterate, the colloquialism "next" state is used solely for descriptive purposes. Time is taken throughout to be a continuous, not a quantified, independent variable.

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11.15 description requires not simply an initial event , but rather an initial history.* The equations of evolution appropriate for the model of a material with a memory, for example, requires an initial history. In closing, we should note that if, rather than proceeding from a set of hypotheses about the state variables of a system, the particular form of the principles of physics, and the environment, we had started from a particular physical situation and attempted to determine the elements of that particular dynamical process, then the following steps would have been taken: (1) A consideration of the class of physical systems and the range of the physical theory necessary to describe the events that can take place in the system. (2) A system which is consistent with (1), (3) An environment which is consistent with both (1) and (2) . (4) and based upon (1) (3), decide upon the minimum amount of information needed to predict subsequent events in a manner such that at least for a certain range of the variables (i.e kj level operation of the system) this prediction (i) exists, (ii) is unique, and (iii) depends continuously upon prescribed data.** The decision mentioned in (1) amounts to a specification of the type of phenomena under consideration and hence the relevant form *An example of this situation is a situation in which the relevant describing equations are differential equations with retarded arguments. The interested reader is referred to El'sgol' (1964 1966) for a more detailed discussion. ' **These three conditions are classically referred to as Hadamard s conditions for a well-posed problem. The interested reader may I*n ^rT 1 ' 3 t6Xt ln Partlal differen tial equations (e.g. Garabedlan, 1964; Courant and Hilbert, 1952).

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11.16 of the principles of physics involved as well as those principles to be used. Thus, from (1) and (2) one may find which and how many of the principles of physics lead to an independent set of equations of evolution (as well as the conditions imposed by the surroundings) . The part of the surroundings which will be — or may be — considered alterable is specified in (3). If the above steps are used to determine a dynamical process for phenomena associated with continuous media, the following considerations result: (1) the principles of physics provide the governing equations and their specific form. (2) the state variables provide, together with (1), the knowledge about an independent set of above equations. (3) the past history and the initial state provide initial data for equations of evolution. (4) the surroundings and environment provide the boundary conditions and body force terms for the equations. Alternatively, the above four points may be viewed as a -larification of points concerning the necessary elements of a set of hypotheses that delineate a class of stability problems associated with a particular dynamical process or a class of dynamical processes from a still larger class of problems.

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11.17 Addendum on the Generation of Perturbations through Environmental Influences The fate of perturbations superimposed upon solutions of relevant describing equations is indeed the main topic of stability analysis. However, another important question which naturally arises is: "how was the basic flow actually perturbed?" We, as observers of the system, are part of the surroundings. Consequently, in order to exert our influences upon the internal fields in the system — in the form of perturbations — we must be able to affect the interaction of the system with its surroundings. In this chapter interactions were divided into two classes: (i) volume or body interactions and influences, and (ii) surface or contact interactions and influences. They are, therefore, the means by which we may perturb the existing internal fields. As may be inferred from the above, this question "how" provides one link between the physics and the mathematics of hydrodynamic cability theory. The relationship and the inherent difficulties may be therefore illustrated most easily by the examination of a concrete example. Therefore, let us focus attention upon the idealization referred to as plane Couette flow. The corresponding physical situation is a fluid contained between two infinite parallel plates separated by a small gap (Figure II. 1). The basic velocity field is generated by translating the upper plate at a constant velocity, TJL , parallel ~.o itself,

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11.18 Suppose the region over which are stability question is posed is that contained within the dotted lines on Figure II. 1, The boundary of this region may be decomposed into the six parts which are also illustrated on the figure. Two of these surfaces, namely dl\ » and 0IX4, are at the interface between the fluid medium and the solid boundary. Another difference between « n, and This, in turn, alters the stress field along a K4. , which in turn affects the velocity field throughout K Consequently, the velocity field within IX has been perturbed by altering conditions — the strain field or the stress field — at one of the exposed surfaces. In this case it was the strain field along * which was directly altered, whereas a pressure pulse in the case of plane Poiseuille flow is an example of an alteration of the stress field.

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11.19 CJ C3

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11.20 Difficulties,, however, still remain in the details of our "almost" experiment. Essentially they have arisen because our system is: (i) an open system, and (ii) because we have direct control over only two of the surfaces, namely, d"K^ and dKi . The difficulty is "what are the conditions along d^ and ol^ ?" (Figure V.2). Althought the velocity field is specified directly on djv^and dy^A » all that is known about the velocity field on dIT and dlX?-, after the pulse is that it must satisfy the describing equations at each of their points. Because the basic velocity field does satisfy the describing equations, the special case might be thought of where the velocity field along d'KL and ovv, is the basic velocity field. For plane Couette motion, this can only be a "thought" experiment which may be imagined while for some other flows, such as plane Poiseuille flow, it might actually be obtained in practice if ^Iv* and a'K ? were chosen carefully. Only plane Couette flow has been discussed, and similar difficulties arise in analyzing other members of the class of parallel flows c For still more complicated flows the situation is still more complicated, for as one attempts to develop a tractable theoretical model that is useful in analyzing a given experiment, more and more discrepancies can arise. Thus, as we investigate the stability of even this simplest class of basic flows, it is well to keep in mind that the stability investigation is above all a "Gedanken" experiment.

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11.21 (a)+t
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11,22 II. 2. Constraints upon Possible States Of all the states, state space or trajectories, in a given state space, only a limited number yield identities upon insertion into the describing equations; that is, only a small subset of states or trajectories are solutions of the operator equations, A state that does satisfy the basic equations is termed an admissible state or an admissible trajectory , as it is a solution of the steady or insteady equations. The equations of evolution may thus be said to constrain the system, being in fact sometimes described as governing equations. From a consideration of the general form of a dynamical process associated with a continuous medium, two different classes of constraints may be distinguished, namely, internal and external constraints, of which the basic equations are of the former class and the body forces couples, and, the boundary conditions imposed on the system (reflecting the influence of the surroundings) are of the latter class. We hasten to add three points: first, that the division is artificial because the body forces and couples are present in the basic equations; second, because the boundary conditions are also in an integral formulation of the basic equations, and third, that our use of the terminology of internal and external constraints as distinct follows that of, for example, Noll (1959), Our usage of the qualifiers internal and external is, however, clear, and the notion of the term constraint is equally transparent, for constraints follow the class of admissible states and they serve to select the next state or states, depending upon whether or not uniqueness prevails,

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11.23 II. 2. a. Internal Constraints Within the class of internal constraints, there are also two major subdivisions: those arising from the fundamental principles of physics, by which we shall always mean classical physics, and those arising because of the nature of the material within the region T\ , of interest. In the first subdivision there are essentially six of these fundamental principles of physics: (1) Balance of Total Linear Momentum (2) Balance of Total Angular Momentum (3) Balance of Total Energy (4) Balance of Total Mass (5) Balance of Total Electric Charge (6) Balance of Magnetic Flux These principles — or statements — when expressed in a mathematical form yield the operator equations of evolution (in more standard but inverted terminology, the governing equations) which must be satisfied by the admissible states. The mathematical representations of *:hese principles may all be placed within the framework of the general equation of balance, i,e., of Accumulation 1 \ Net influx of the specified ? specified quantity I = / quantity through the surface! given control volume) (.bounding the control volume J Rate of a V in a given control volume! L bounding + j Rate of Generation of the specified quantity within f (II. 1) the control volume t Conversely, these principles provide the specific items to appear in the braces. In itself the equation of balance is nothing more than a

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11.24 bit of common sense; it is thus only a skeleton — the tlesh is supplied by the principles of physics.* To be embarrassingly specific, the principles are, in this context, what the flux of the specified quantity is and how the specified quantity may be generated within the volume. To reiterate, the mathematical representation of the principles of physics, in the form of equations of balance, restricts the class of all possible next states and delineates the class of admissible states. Among the various possible ways of interpreting this general equation of balance we have selected the viewpoint provided by continnum mechanics.** However, rather than the usual approach taken therein of dealing with body manifolds*** (see, e.g., Noll, 1959; Truesdell, 1960), we choose to focus our attention upon a certain region of physical space — the control volume, b. Furthermore, specific representations of the principles of physics are viewed as plausible postulates, rather than hard and fast laws, *In speaking of this equation, Truesdell (1960) says, "This statement is sufficiently general in itself; it doesn't say anything at all." **In that we will use its defining concept (to again borrow the words of C. Truesdell) of a generalization of the "stress hypothesis of Cauchy" — or in our terminology, "the equipollence hypothesis." ***In this way we have avoided any questions dealing with the "actual" structure of the body — that is, about the connections between "material points" within the medium — which are necessary hurdles to overcome in defining the mathematical operations integration over a set of material points. We feel that the approach we have taken is a more practical and a more convenient — and further, one that is still consistent with the fundamental goal of continuum mechanics — that is; to explain, represent, and predict gross phenomena in a macroscopic portion of physical space .

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11,25 Thus, for purposes of description at the outset, we are interested in phenomena occurring in some compact region, b, of physical space. We adopt the viewpoint that the phenomena occurring fall within the range of physical theories referred to as continuum mechanics (so that the local physical properties may be specified by their associated physical fields) , and we assume that what we call an equipollence hypothesis may be used to characterize internal interactions* within the region of interest, b, Let us denote by lp a "typical"** physical field (of the per unit (J) volume nature), which is used to describe the phenomena of interest. Let a continuous medium,*** 15 , occupy the spatial region~BCE (physical space) at time t, and letoP denote the surface in CT coinciding with the elements of the material surface o\6 bounding the body O at time t (see Figure II. 3). Further, suppose that the influence of the universe exterior to UJ , which we will assume is also occupied by a continuous medium, upon the rate of change of the field H^ T) may be characterized by specifying: (i) a surface influence, JC7 ) of the exterior to 13 that 3 upon !-> by acting only upon (and being defined upon) a O , and li) a body influence that acts through ID (being defined per unit volume), { . B(J) *This is not to say that one must consider the media which are the subject of continuum mechanics to have no molecular structure. Rather one avoids the explicit use of any molecular structure they may have, taking it into account, loosely speaking, only implicitly through its macroscopic effects. To this end, molecular theories may and do play a qualitative role in continuum physics. **The quotes are to remind the reader that the principles of physics are unified by eq, (II. 1). ***By a continuous medium we mean any matter which, as far as the phenomena being considered are concerned , falls under the range of the physical theory called continuum mechanics (cf . , our previous remarks on the range of continuum mechanics above).

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11.26 \ / y 3b FIGURE 11.3, NOMENCLATURE

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11.27 Because at time t, ^Q occupies the region f> we may view J { as being defined on^b(T by 1 -W .,(* A) ,yt3B and -f as defined in D at the instant t. Further, suppose that at time t an observer selects an arbitrary control volume ( ]q , as indicated in Figure II. 3. This control volume, bounded by the mathematical surface ^ \o need not be contained entirely within, nor need it contain entirely within itself, the spatial region T*> ; but, again, the matter within is to be viewed as a continuous medium. Thus, the influence of the surroundings upon be'B may be characterized by: (i) a volume influence per unit volume, j^ , and (ii) a surface influence,^ defined over the surf ace ^e/B • This surface influence, i^ — unlike the surface influence, ^Cn' which is imposed by some external agent — reflects the internal interactions between the various material points.* In the special case of the balance of linear momentum this hypothesis is usually referred to as the stress hypothesis. In its general context, we refer to it as the equipollence hypothesis, for it asserts that -the .interaction of the material -points external to ^koB at time t upon those within C.^0 fc) VJOSnV) at time t is equipollent -as far as its contribution to the temporal rate of *It is of interest to note that Poincare in his writings on the philosophy of science argued against the use of the stress hypothesis for exactly this reason, He felt that the concepts of physics should all be real — that is, observable — and not owe their existence to a hypothesis which may not be directly tested. The reader xs referred to Synge (1960, pp. 4-5) and Truesdell and Toupin (1960, p. 229) references contained therein for further discussion of this* operational" philosophy of science and its ramifications.

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11,28 change of Y(t) in t©3 is concerned -to a field h .defined on the surfaced b©D . Further, it is assumed that this field is dependent* only upon the position on the surface, time, and tangent plane to the surface at that point, that is, L b(J) K(J)Cx ; -fc.vi) where x£^b©Band T£_ is the normal to this surface pointing away from 0©TS» By the use of the equipollence hypothesis again, this time to characterize the internal interaction of material points in !>©\o at time t upon material points of ~&pU> ,** we obtain the field K (-)defined on the surf ace 9bO~E> • Again, n n , >is assumed to depend only upon position on the surface, time, and the tangent plane to the surface at that point, that is, X "cdbno and ' vv is the normal to this surface wnere *We have followed the conventional presentation (e.g., Noll, 1959) by enumerating the dependence of In^ ^upon a single geometric property of the surface, the normal. This is often accurate assumption at physical boundaries, but the recent work of Scriven, Eliassen, and Murphy (Eliassen, 1963; Murphy, 1965; Murphy and Scriven, 1968) demonstrates that other geometric properties enter if the boundaries of the control volume are of high curvature. These points must therefore be kept in mind when selecting a control volume. **It should be mentioned that because of the conventions we have chosen, any internal interaction of matter exterior to IsflB — other than material points in "B at time t — upon material in bOB at time t must be accounted for in the volume interaction term. This, however, is consistent with the usual convention in electromagnetic theory (see, e.g., Elsasser, 1956; Sommerfeld, 1952; Toupin, 1956).

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11.29 pointing into B©k> • For convenience, we define the following fields, K^^ and l (X) so that equation (II. 1) when applied to the region, b subject to the above decompositions of influences and interactions becomes, at time t, bfUB \ (n.2) In words, the above equation and preceeding discussion may be collected in the form J of 'rhf AcC ^ lat i t° n ] |" Net influ * of f^O) 1 f Net increase of I of the quantity f (J, I = due to internal / + ?{ \j. to the , within the control j ^ interactions , sub} HSce sources /oiume b J ject to equipollencej due to surround-, (hypothesis J I ing actions upon' ( the body 3 Rate of Generation] • ^) of f^jj within b /due to volume influences Here, £ and x have been used to indicate dummy integration variable along the various surfaces (see, Figure II. 1) and \bj : element of volume in b, Ubj : element of surface area on 3L> \^BJ : element of surface area on^ .

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11.30 The term deserves further comment, for, although an analogous term commonly appears in the balances taken in electrostatics, it is scarcely seen in the balances taken in other fields of continuum mechanics. This term plays a versatile role in the balance equation, depending upon the physical situation at hand; at times it may represent the contribution from a flux which is presented at the boundary of B (possibly by the dictation of some external agent), represent the contribution from a surface source, or both of these. As an example of its representation as a surface source term we may imagine a mass transfer problem in which a chemical reaction is occurring only at the sur face of a catalyst particle and we have selected our control volume to include some of the catalyst as well as some of the surrounding fluid mixture. In this type of physical situation the term \ would represent the rate of generation of material within the control volume due to chemical reaction at the surface of the catalyst particles. On the other hand, physical situations where it represents the contribution from a prescribed flux field are most easily found in heat transfer problem, e.g., a solid whose surface is completely covered by heating wires. It is of importance to note that this term does not account for all the flux of heat across the

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11.31 surface of the solid — just that which is externally prescribed, If the volume and surface integrals of eq, (II, 2) are placed on different sides of the equation and s_ is brought within the inctt tegral, there results I* i h (IIi3) We now invoke a general form of the classical tetrahedron argument* whereby it is demonstrated that the internal interaction at time Z between a material point at >elo and the material points in b^B — as given by K-/*i"C'M — is the value of a linear operation on the vector l£ . The argument goes as follows. Suppose that the control volume is selected such that loO^-tO^ , the null set — that is, b is either completely within or completely external to B. Further, suppose that the control volume, b, is selected to be a small tetrahedron of volume, &0T b , and bounded by a surface of area, &o. ^ , which does not contain any points of ^B (see, Figure II. 4). Then the mean value theorem for volume integrals is invoked, lich leads to the equation At this juncture, the classical derivation divides this result by'&a.| j and takes the limit of both sides as Mr^O ; thereby, *The reader unfamiliar with the following development is referred to the discussion of the stress tensor as given by Truesdell and Toupin (1960); Eringen (1962), and Aris (1962).

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11.32 FIGURE II. 4. THE CLASSICAL TETRAHEDRON

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11.33 obtaining the classical result where has been used £&. b -*0 c. jv* ,x Jttt i H [ \ T1 ^ |X . e,^ n.„ 5b, where x = dummy integration variable indicating points on i th coordinate surface \qfyj element of surface area on side of tetrahedron which is perpendicular to i th direction C,;| = unit vector in i coordinate direction. In the limiting situation as the tetrahedron shrinks to a point (see, e.g., Cesari, 1956) it is assumed that

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11.34 or S-st-tttS where "H. denotes the normal to the slanted surface of the tetrahedron andldta lis an element of surface area on this face. If one substitutes these relations into eq.(ll.5), the following equation results At this point the classical derivation draws the implication from this equation that •tfViSt 5-[^Vrl^, + fe-feV^v^fol^t (II ' 6) *. where x^denotes some point within the infinitesmal volume. Now the quantity, Q) , is defined by 1Akl) $ ^•Ua rs _eJe.-Vi tv„ ^ -e,VVi <*. x -e ^ e (II ' 7) CT) lT| / / ^1-1 (J, * , K > -Z-^z which is independent of the

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11.35 mathematical surface — thus, loosely speaking, this relation separates the mathematical geometry (as embodied in "W , ) from the physics (as embodied in (&.*) . Because the £, 5 occurring in the definition of Q^i depend, at time T, only upon x , the following identifications* may be made: ^ ' ^* rO = Kti u * re -e '"j and By combining these identifications with eq. (II. 7), one is led to %, " ta,^%T. V^keV^' 6 ™' (II 9) at each instant t. If eq. (II. 9) is substituted into eq. (II. 8), one obtains the representation Ihis representation may be viewed in either of two ways.** It may be looked upon as defining a linear operator (b Cy x 'U ^ at the place x at time T which operates on n to yield the local influence n (xT'rt^, w (-n 3 j i~ *The reader looking for analogies is referred to similar identifications made in developing the concept of a stress tensor (see, e.g., Truesdell and Toupin, 1960; Eringen, 1962; Aris, 1962), **The reader is reminded of the distinctions made in finitedimensional and abstract spaces between the differential, the derivative, and the two possible interpretations of the derivative (please see Appendix B, section II. 1. a).

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11.36 which is a "2. -form in E in the mathematical theory of differential forms (see, e.g., Buck, 1956; Guggenheimer , 1963; Spivak, 1965; Willmore, 1959). An alternative interpretation is obtained by viewing it as a definition of a f ield Cp (. X'Vi-.j — that is, a global quantity — at time t associated with the quantity h._. whose values at a point X^.XB> are the linear operators yielding local values of the associated quantity , v\ .,. . In the classical context of Cauchy's law, where \ C. I-") assumes the role of HV^i and tCxx-n') assumes the role of \\ Vxi; ' yC\* e ^' (II»10) is referred to as the fundamental theorem of Cauchy which asserts the existence of a stress tensor field, ~T7.' *) » which characterizes the local stress vectors, tC^.T'. w. \ . In its more general context we refer to it as the generalized fundamental theorem of Cauchy , Within this general context, the primary utility of the generalized fundamental theorem of Cauchy ~ and therefore, also the classical theorem — becomes strikingly apparent. Because V\ _ is a 2-form in E the analogs of the classical integral theorems of vector analysis may be applied to it. The classical divergence theorem may be applied to the surface integral of n with an impact upon conW) tinuum mechanics at the deepest level. Before proceeding with the development, however, it is well to list the five key points which led to eq.(II.lO); (i) the form of the general balance equation, (ii) the equipollence hypothesis, (iii) the assumption that the value of r\ at a point on ot> depended only upon the point and the tangent plane to ^ at that point, (iv) the fact that control volume b

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11.37 b contained no points of the surface B where there was an imposed surface traction, and (v) the assumption that the passage to the limit in the generalized tetrahedron argument is a valid procedure. These points must be kept in mind as we now make use of the representation of In as a 2-form in E CJ) These points must be kept in mind as we now apply the volume to surface integral theorem* (see, e.g., Bergman and Schiffer, 1953; Buck, 1956; Kellogg, 1929; Sommerfeld, 1952) by selecting a control volume such that bfloB-^O 1 and by substituting eq . (11.10) into eg. (II.3)to arrive at Then applying the volume to surface integral theorem under suitable conditions on

(see, e.g., Kellogg, 1929) — for example, ^k'J) providing ^p is a continuously dif f erentiable field and do is a regular surface** — eq. (11.11) may be expressed solely in terms of volume integrals, that is jlVt P ^> C -^ f &'*^J*.^)]® OLIO On the other hand, if we impose weaker conditions on the field


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11.38 then a suitably revised form of the volume to surface integral theorem (see, e.g., Bergman and Schiffer, 1953, p. 363; Truesdell and Toupin, 1960, p. 526) leads to the equation, |J^ U 'M^s-* W)01 S r l$4,;^© (II ' 13) where (J? is the [c region of volume measure zero on which Cp is fT 1 ~ W discontinuous: h^ is the jump in the field mD , across 07 < "V\, is the outward normal to $7 , positive when pointing from © to (J) in relation to the convention chosen for , It should be noted that eq. (11.12) is a special case of eq. (11.13) when the field is continuous throughout b. We have, in deriving eq. (11.13), allowed the fields to be discontinuous on some surface or surfaces. Although it may be argued that these discontinuous fields are not in the true spirit of continuum mechanics, they are frequently convenient and sometimes necessary idealizations; indeed, they are necessary if one wishes to analyze phenomena such as shock waves within the realm of continuum mechanics. Furthermore, just as surfaces of discontinuity have sometimes proved convenient, so also have the idealizations of lines and even points of discontinuity.* We do, however, limit our development only to cases of surfaces of discontinuity. *Although these idealizations are abundantly used in the theory of electrostatics, they are also used in many other disciplines. In mass : transfer , for example, the model of diffusion from a point source to a moving stream is just such an idealization. Often these idealizations have experimental significance; the analysis of the example cited is used to determine "eddy dif fusivities (see, Sherwood and Pigford, 1952, pg. 42).

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11.39 The quantity $> may "be decomposed into two parts: a convective contribution due to a transport of the physical properties associated with material points as the material points themselves move across the surface db into b , and a diffusive contribution due to the transport , across db into ^ > of the values of the physical properties associated with material points which need not cross db , that is, $> ~ ~ $> 4
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11.40 the place x, and the particular physical property in the multilinear combination, * ~ P^X l S^ T £*,*1 (J}c If this relation — namely that P^WCX,^ is the linear operator °n t^j, whose value is the convective contribution mentioned above is substituted into eqs. (11.12) and (11.13), one obtains C&^*l tH 3-ttL^ (11.14) Jb and Because these equations have been developed for an arbitrary control volume, b, the familiar assumptions and arguments (see, e.g., Truesdell and Toupin, I960and similar arguments in the calculus of variations used to obtain the Euler-Lagrange equations, e.g., Courant and Hilbert, 1952) used to derive local relations (differential ations) from global relations (integral equations) may be applied to obtain the local relations (jt? 4 ^ + ^f^jf'SAiafULr OVxiTB (ii. at and 16) L^>ci p*^J O (11.17)

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11.41 Each of the fundamental principles of physics may be placed within the framework developed in eqs. (11.14) to (11.17) by different interpretations of the quantities M^ , Cb ,4• They are cataloged in Table (II. 1). For a given physical system, the number of equations of this form which constitute an independent set of equations of evolution depends upon the range of the physical theory, and both the range and nature of activity within the associated dynamical process. Thus, an explicit expression


PAGE 90

11.42 CO CJ M h cu o w w J Ph M O a M PS cu r-l
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11.43 3 a •H CD 03 X 3 3 iH <-\ CO a CD W IJH 3 \7 rH jU/ 1 4J u „ to o U X 4J 3 -H oi im .-I to 01 00 A! Q] l-i a x: •H 4-1 COl u oj 3 a) 3 u o o X a) a) -H x: x: U O 4J 5 4J o -w >n 3 4-1 O WS" b U •H 4J 3 o a; 3 lH o t3 3 a) to -. gpJH 60 (fl II 0J o 0--. K -H X> 01 4J X! I 4-1 3 Ci-i o _tO CJ 4-1 i Xl -H -H II 1-1 pcj -. x 3 4J U 01 4J 01 X U 13 >, 01 CO 4-1 tO o o •> 4-4 •a 4-1 U X 01 t( 60 3 PQ vH CO _1 m -a CJ H C >-i en tO 3 0) X H !>% 0> 60 M to o, OJ •> 60 OJ o 4J 01 4J u • tO ''"N S o 01 v£> 4-1 ON CO r-4. ! ;i I to 01 u U 0) &X X 4-1 01 o 2^ 01 u s o 01 4-4 60 to 01 3 u o O i-t c 3 o o U ON 4-1 o o 14-1 "3 X! U 60 t-l -H T3 01 3 3 •H 4-> 3 o u , b 0) 3 01 a «i-t 0' u jc 3 3 4JH O -H O m3> CM 00 I XI to 0) 1 Ct4 U 0) o 60 3 O 01 CO 3 o O -H •H 4.) £. X o w g 3 o 01 -H > 4J •H CJ U) (0 3 >-i y-i 4-1 H 3 Q M 0) 01 3 4J X5 O -H U ft IS 4-> 3 O 0) 3 "3 ^ , A 3 4J , O 0) + V4 ^1 3 >% 0) 4-1 -H 01 to O 3 tO 0) H T3 3U) 4J •H 3 13 01 •• >-l U 0) tfr\ u (D 3 a) 14-j o 01 01 O60 CO 3 01 V4 U 0) > to x: 3 CO 4-1 3 tH & to 3 O > o X! H U •rH I u 3 4-1 01 T3 01 rH 0) 01 60 •,3 4-1 Ujx: -h «?j-<0 to o ^ .. H co 3 M a) vil 60 u 01 3 0) ca cu •H 3 O r-l 4-1 3 P. co to CO 4-1 3 O S 3 ^ 4-1 <« -1 CO

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11.44 (ii) it restricts attention to "a specific ideal material or class of ideal materials; (iii) it introduces into the formulation various phenomenological coefficients as parameters describing the "level of operation" of a dynamical process within a given class of ideal materials. Effects (i) and (ii) are complementary in that an explicit H expression for j Cp V will exclude possible couplings between a set of fields SUJ r within the constitutive assumptions. To be sure, the velocity field is linearly coupled to all other fields by its appearance in the convective terms of the field equations of the dynamical process, so that it occupies regal position within the class of all physical fields, as does the density field. The three effects may be illustrated by considering the constitutive assumption for an incompressible newtonian fluid, namely, where ~\~ is the total stress dyadic, f-*is the viscosity coefficient, be I and L) is the deformation rate dyadic. This expression conveys the idea that the material depends upon the deformation rate and does so in a linear manner; therefore, we have focused attention upon a certain class of ideal materials — ideal because a specific real material may behave as indicated under a certain set of circumstances, and yet behave differently under, a different set of circumstances. Furthermore, if one places the restriction upon K*that it is a constant, then this class of ideal materials does not have the

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11.45 possibility of any explicit coupling between the other physical fields, e.g., temperature, composition -except, of course, through the coupling of the velocity field with all others in the remaining equations of balance. This parameter, u. may be used as an example to demonstrate the third effect, in that a specific value for it characterizes the strength of response to a certain stimulus — within the class of incompressible newtonian fluids. In some detail, if we considered two different members of this class, A and B, with their corresponding viscosity coefficients such that > r a > h Bj then they will have different values for the stress field for the same deformation rate field and vice versa; thus, the stress field may be viewed as the stimulus or the response and the formation rate as the converse. If the stress is viewed as the stimulus, although the mechanism for the response, as characterized by both the form of


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11.46 momentum, which then becomes the Navier-Stokes equation, which may be -made dimensionless by taking where u_ , \ , sr~ , -jt , a> , ~X are the dimensionless variables, thus giving rise to the dimensionless parameter , JJq > the Reynolds number. This parameter, a number, reflects "level of operation" of a material within the class of ideal materials mentioned above. It may be used to do much more than that. Indeed the Reynolds number may serve to indicate the "level of operation" of a system , an ability obtained by including characteristic geometrical , kinematical, as well as the physical properties, of the material. The reader wishing to pursue the subject of similarity and modelling in greater detail is referred to the delightful little book by Birkhoff (1960) and references therein. II. 2. c. External Constraints The interaction of the system with its surroundings comprises the external constraints. External constraints may be of three basic types: (i) those due to configuration or kinematical conditions (e.g., conditions on the location or movement of the bounding surfaces); (ii) those due to the interaction with or influence from contiguous surroundings, in the form of boundary conditions; (iii) the influence

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11.47 of distant surroundings upon the system, in the form of body forces acting throughout the system. The last have already been accounted for, having appeared as source terms in the general balance equations, and whereas boundary conditions may sometimes describe a genuine coupling between the system and its (contiguous) surroundings and sometimes not, body forces always influence the system, yet are themselves uninfluenced by the system and its behavior. The division into internal and external constraints is, to be sure, somewhat artificial, and depending upon whether the equations are formulated and considered in integral or differential form, it may seem more or less artificial, depending on one's taste. Thus, as a set of partial differential equations, the field equations require appropriate boundary conditions that appear separately, yet the body forces occur in the equations themselves. Conversely, in integral form the basic equations incorporate both external and internal constraints, both body forces and boundary conditions on an equal footing. As a final point with regard to all three kinds of external constraints that is perhaps worth recalling, they may each be subdivided into those conditions that are susceptible to control ("pure influence") and those that are not (an interaction, or coupling) . The boundary conditions may be made dimensionless in the same manner as the partial differential field equations. But whereas the dimensionless numbers that appeared there measured, or were at least indicative of, the level of operation of the system , the dimensionless parameters that arise from nondimensionalizing the boundary conditions describe the level of communication between system and surroundings.

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II. .48 II. 3. Perturbation Problems Associated with Dynamical Processes Because stability theory is the topic of this dissertation and because stability problems are concerned with the fate of perturbations, it is appropriate that the position occupied by stability* problems within the class of perturbation problems be considered. Moreover, the vantage point constructed in Section II. 1 — that is, the concepts of a state space and a dynamical process — will allow us to distinguish the classes of perturbation problems in a straightforward manner. Once the class of stability problems is clearly distinguished from other types of perturbation problems, we will concentrate upon a specific class of dynamical processes (in Section II. 4), those for which the equations of evolution are of the form of the general balance equation of Section II. 2. In this classification of perturbation problems associated with dynamical processes, it is advantageous to view the dynamical process as an operation, (DP), which transforms the present state of a system into a succeeding state, both states, of course, belonging to the state space associated with the system. Among all the elements of this state space, a particular set of elements may be distinguished, those that are transformed into themselves by the dynamical process — that is, they are the set of fixed points of the operation (DP) . Any element which belongs to this set is referred *By stability problems we mean not the classical problems of stability with respect to boundary perturbations, or with respect to perturbations in the body force field; rather we mean the stability of a basic state of a particular dynamical process (see, e.g., Lin, 1955).

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11.49 to as a basic state of that dynamical process. One type of perturbation problem is concerned with perturbations of the basic states of a specific dynamical process, while the other is concerned with perturbations of the operation — the dynamical process (DP) — itself. In the first type of perturbation problem, which we have taken as our prototype stability problem, is usually centered upon a particular basic state without regard for the method in which this basic state will be perturbed. On the other hand, in the second type of perturbation problem — that is, perturbation of the dynamical process — the method of perturbation is of primary concern. II. 3. a. Perturbation of the Dynamical Process Itself Although the main topic of this dissertation is with the perturbation of a basic state of a dynamical process, we would like to first briefly discuss the class of problems which involve the perturbation of the dynamical process itself. This class of perturbation problems, which includes perturbation of boundary conditions, perturbations of the shape of the physical boundaries, perturbations of the external influence fields, and perturbations of the operator in the constitutive assumption, is currently a topic of interest in both mathematics and engineering (see, e.g., Van Dyke, 1964; Seeger and Temple, 1965). In its broadest sense, the question raised in this cype of perturbation problem is: if one is given or can easily find the trace of a particular dynamical process, (DP), as t is varied in 1

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11.50 event space, then how may the trace of some other dynamical process, (DP ) — which is close in some sense to (DP ) — be expressed in terms of a modification of that of (DP ) . Perhaps the most straightforward modification one could imagine is a series expansion in terms of a scalar perturbation parameter. Indeed, this is the technique proposed by Poincare in 1892 — when such a modification was not so obvious. Essentially Poincare* 's method consists of expanding the dependent variables (e.g., the state variables) of the relevant governing equations in a power series in the perturbation parameter; substituting the series into the relevant equations; equating terms of similar powers in the perturbation parameter (powers of a scalar real variable are linearly independent); then solving the resulting system of equations successively. The equation corresponding to the zero power in the perturbation parameter is the governing equation for (DP ).. It is found that for a sizeable class of perturbation problems — referred to as regular perturbation problems — this technique provides a valid representation of the trace of (DP ) based upon that of (DP ) throughout physical space. However, there exists an important class of dynamical processes in mathematical physics for which this technique is not adequate. For example, if the governing equations are differential equations and the scalar parameter appears as a coefficient of the highest order derivative, then Poincare 's technique does not yield a valid representation throughout all of physical space. The most frequently used techniques of handling this type of perturbation problem

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11.51 for differential equations are: (i) Prandtl's method of inner and outer expansions, and (ii) Lighthill's method (see, e.g., Van Dyke, 1964). In Prandtl's method, series expansions are obtained that are for valid (approximate) representations of different (hypothetical) dynamical process occuring in different regions of physical space, and it is assumed that there is a region of physical space in which both expansions are valid and can be matched. The result, if successful, is a uniformly valid approximation to the trace of the actual dynamical process. Prandtl's original development was based solely on physical intuition and certain orderly arguments in the Navier-Stokes equation. It was half a century before any degree of mathematical vigor could be given to Prandtl's arguments, despite the pioneering efforts of Lagerstrom, Kaplun, and Cole (see, e.g., Lagerstrom and Cole, 1955; Kaplun and Lagerstrom, 1957; Kaplun, 195 4), many open questions remain. In Lighthill's technique, on the other hand, che dependent as well as the independent variables (e.g., positions in physical space) are expanded in terms of a perurbation series in a third set of variables. The choice of explicit relations for this third set of variables is subject to a set of guidelines that lead, under certain conditions, to a uniformly valid representation of (DP) . These guidelines provide a systematic scheme for finding a solution by a change of variable — as evidenced by the expansion of the independent variables as well as the dependent variables.

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11.52 However, the validity of using the various techniques of handling perturbation problems of this type rests upon the answer to a more basic question: "Is the problem correctly set (or wellposed)?" This question is a major consideration in determining the elements of a dynamical process as well as the validity of any of the above perturbation solution techniques. It is concerned with the validity of a mathematical representation and thus is inherently based upon the belief that if the mathematical representation of some stable (in the physical sense) physical phenomena is sufficiently well formulated, then the presence of small errors in the prescribed data should result in small changes in the solutions. On the other hand, if the physical situation was physically stable, yet the mathematical model did not behave in this manner, then one would conclude that the model was badly formulated. Consequently, if we wish to relate physical instability to the instability of a dynamical process, it is first necessary that the dynamical process provide a valid description of the physical situation. By valid, we mean in the sense that the dynamical process provides a mathematical formulation for which a solution (i) exists, (ii) is unique, and (iii) depends continuously upon prescribed data. In order that we may attach some physical significance to any stability analysis of a dynamical process, we must suppose that it is a valid representation of the physical situation.

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11.53 II. 3. b. The Stability Problem The class of stability problems is comprised of those perturbation problems concerned only with perturbations of basic states, Basically, one is given a certain set of information and one asks certain kinds of questions. Thus, one is generally given a basic state, s. of a specific dynamical process, (DP), and one considers perturbations at time t of the system to other states s in some -A neighborhood of s . There is presumed a single trace through s_ B and through each s in the prescribed neighborhood, The kinds of questions then asked are of the following sorts. Does the trace through a given state s at t become ar~1 ° bitrarily "far" from the trace through s ? For what neighborhood ? of states s of s is the answer to this question the same for every such state s ? "1 Does the trace through a given s at t fl remain within a prescribed neighborhood of the trace through £ ? For what set of states s^ is the answer to this question always the same? Does the trace through a given s at t Q approach the trace E s^ with sufficient elapse of time? For what neighborhood of £ do all such points s have traces tending toward the trace of s ? At what rates do such tendencies occur and how are they related to different neighborhoods of s ? In order to be somewhat more precise, we now consider a hypothetical stability problem and formulate it within the context of concepts defined in Section II. 1. We denote the set of global

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11.54 states associated with the system by J\ consisting of elements such that the relevant equations of evolution have meaning.* From among the elements of A consider the subset kJ consisting of all states DP belonging to J\ which satisfy all the requirements of the particular dynamical process (DP). Further, let us assume that the algebraic operations of addition and scalar multiplication may be introduced into J\ such that it is a linear algebraic system.** Also, suppose that a topology, {j , may be introduced in a manner such that the algebraic operations are continuous, so that A becomes a linear topological space, say \ J\ fc) 4( with the topology . A major advantage gained by having K J\ (j + • I a linear topological space is that a general stability theory may be based upon the stability of Q -the zero element of the space J\ . This happy circumstance arises because if we do have a linear topological space associated with the set J\ , then the stability problem associated with an arbitrary dynamical process (such that O C A ) ancl DP concerned with any of its basic states may be transformed into a -ability problem concerned with the element (J* in a straightforward xanner. To this end, let us consider 5. n ^ ^^r)T> to ^ e a basic state of (DP) whose stability is being investigated. That is, *For example, they are functions which are continuously dif f erentiable to some degree. However, the elements of are not required to satisfy any boundary conditions and any other constraints, **This is the same entity as the usual linear system or vector system. The adjective has been added to emphasize that it is an algebraic system at this point without a specific topology.

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11.55 let £ be a fixed point of the operator, (DP), therefore s 3 = (pr)<* b > (11.18) Further, let us consider another state s £ JO which is not the basic _ 1 ttp state of (DP), that is, X *«>»**;>} V^s,, y,,^^ ei.-i9> There are several tacit assumptions here, some of which are earlier stated explicitly. The statement that y
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11.56 as we limit ourselves to stability questions of the kind mentioned in the opening of this sub-section (II. 3. b) . If eq. (11.18) is subtracted from eq„ (11.19) — an operation which is defined and continuous since the elements belong to J\ there then results K-Wo^j,**, ,,**>. (11.20) Because we have taken s to be invariant under (DP), eq. (11.20) may be rewritten as y* = (3>p*)< ? v> (n.21) where y*i?tie-*Al a-.js-a,, „ i^ p \ and Thus, if the operator (DP) is continuous, then by setting s_ — O i.e., s_ s s_ ) on the right-hand side of eq . (11,21) we find (PP*)<< >] = toP*) -_5BA or Q A = ^KQ^ (n.22) Eq. (11.22) states that Q is a fixed point of (DP*) , that is, a basic

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11,57 state of (DP*) . Hence we have transformed the stability problem associated with (DP) and £ to a stability problem associated with (DP*) and (\ 3 There are, in addition, several other interesting points that arise in considering the above transformation. In the first place, by inspection it is clear that if (DP) is an additive operator, then (DP*) and (DP) are equal; thus, for linear dynamical processes, (DP*) and (DP) are necessarily the same operators. We say that a dynamical process is linear if and if the formulation is differential, we speak of a linear field operator in the case of linear field equations and linear boundary operators in the case of linear boundary conditions. With this nomenclature in mind, then, we can say in the second place that if the boundary operators in (DP) are linear, then those in (DP*) are homogenous, regardless of the possible nonlinearity of field operator, and conversely if the field operators for (DP) are linear, then those tor (DP*) are homogeneous, regardless of the possible nonlinearity of che boundary operator. In the third place, eq. (11,22) indicates that (DP*) is an operator for which nonlinear eigenvalve problems and questions of bifurcation points may be phrased meaningfully, as well as questions about fixed points of the operator (see, e.g., Krasnosel'skii, 1964). The interested reader is referred, for example, to the article by Ukhovskii and Iudovich (1963) in which the theory of bifurcation points is applied to the Benard stability problem.

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11.58 II. 4. The General Stability Problem, for. a.Dynamical Process Associated with a Continuous Medium In order to place the above formulation of the general stability problem in a still more concrete setting, let us consider a specific class of dynamical processes, namely, dynamical processes whose operator equations are of the form of the general balance equation of Section II. 2 and to physical systems which are contained within the domain of the theory of continuum mechanics. With this restriction in mind, the state, s_, is now generally specified by (N) fields of the physical properties — the vL) V in the general balance equations — that is,* K^j >%.} Also in the balance equation for each kp , we have a term for the flux field, Cp and the source field, 4^* , Thus, the equations of evolution for a particular subclass of this class of dynamical processes is readily obtained by specifying particular relations between the flux and source fields and the fields \(\) > and also operations upon the V^ 3 ^J> . 5. We will assume that such relations are given a priori , denoting them by i) Cf") and 4. For ease of mathematical manipulations, it will be convenient to decompose the
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11.59 which remains in the absence of gradients of the 4> ^(denoted by , which we will denote by the symbol d) With these considerations, we may now relate the general formulation of Section (II. 3. b) to the formulation for the specific class of dynamical processes of this section. If, for concreteness, we consider the form of the operator equations of evolution to be of the differential form of the general balance equations, then the set J\ consists of the cartesian product of the (N) classes of functions such that the differential equations for each of the Vp 5 are meaningful — that is, sufficiently dif f erentiable. Further, for concreteness, suppose that the class of functions for each *\> is such that a normed topology and the operations of addition and scalar multiplication may be introduced into the class in a manner that leads to a normed linear function space corresponding to that kjj . Then the cartesian product of a finite number of normed linear spaces, each of which is a function space, may be considered as a normed linear space with the topology, G , induced by and the usual definitions of addition and scalar multiplication (cf . , e.g., Dieudonne 1960, p,99, for normed linear spaces; Edwards 1965, p<59, for the more general case of linear topological spaces) for products of linear algebraic systems, The result of this construction is the linear topological space /j6 ( i -S mentioned in the previous section. From among the elements of this space we select those which

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II 60 satisfy the requirements of a dynamical process, such as: the equations of evolution, the boundary conditions, and other additional constraints such as some of the components of s_ being solenoidal. Such elements will compose the subset* >0 for that particular dynamical process, A basic DP state, S B £. -O^p > will then be a solution to the steady state form of the operator equations of evolution which satisfies the other constraints of the particular dynamical process. The collection of relevant equations of evolution, each of the general balance form of eq . (11.16) may be written compactly as where we have defined the (N)-tuples of field as K^-v) * $frMt>, , i.^A^vT) It is easily seen that both the mathematical and physical complexity of the system, which this equation partially represents, is regulated by the selection of the relevant physical properties and explicit expressions *This subset with the topology U will not, in general, be a linear topological space itself unless the subset io$p is a linear algebraic system in itself. However, since C^^pis a subset of J\ , from the vantage point provided by|/\g^>\ the algebraic operations will still be defined and continuous as elements of the space J^ .

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11.61 for <£> S and -f '$ — but they are all unified by this general equate) CTD tion. In arriving at this equation from (N) equations of the form of eq. (11.16), we have employed the balance of total mass in the form } In order to correspond to the formulation of section (II. 2. b), we now wish to form the anolog of the operator (DP*) in eq, (11.22), at least as far as the differential equations in (DP*) are concerned. This may be done by noting that the basic state is a solution to the steadystate form of eq. (11,23), that is, Because, s , is supposed to be known a priori — it is the basic state — fc whose stability is being investigated — the above equation is really an algebraic identity , Also, we know that in order for any perturbed state to be admissible, s , , it must satisfy the equation of evolution in the form of eq. (11.23), thus supposing that we have the same fluid as in eq. (11,24). The operator equation part of (DP*) may now be formed by subtracting eq . (11,24) from eq, (11.25) and defining the difference state, s , by — p S » S 5*-S B

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11.62 to arrive at — after some simple manipulations — the equation (I) (ID (III) (IV) (V) (VI) (11,26) (VIII) where we have used the nomenclature § d C ^6 J bCs^5 B V-')= ^.^V.}-^ ^5b,B
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11.63 velocity fields. Term (V) represents a contribution from quantities which appear regardless of the vanishing of the gradients of the physical property fields, and finally term (VII) represents a rate of generation or source for the difference state. Similar to the usefulness and importance of the concept of energy in the explanation of transient phenomena, it is reasonable to suppose that the variance of the difference state, defined by "V<£/h* Sp^-^V^^v^-tw +114W11* .fc will be useful and important in the present context. An equation which governs the temporal rate of increase of this local variance may be obT tained by multiplying eq . (11.26) by • 5 and rearranging to arrive at D the result, (A) (B) (C) (D) (E) (F) (G) +(H<^-£ + ^) (H) (I) -^<^| T a + ?T "? D (H.27) (J) (K) Again, each of the terms of this equation may be interpreted as contributions to the temporal rate of increase of variance. Term

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11.64 (A) provides a measure of the temporal rate of increase of variance per unit volume. On the other hand, term (B) measures the rate of increase of variance per unit volume due to the coupling arising because of the difference between the perturbed and basic velocity fields. It is basically of the nature of a convective term. The rate of increase of variance due to the coupling of various velocity fields and the fields of the physical properties are given by terms such as (C) , (D) , (E) , (F) and (G) . Again (K) will correspond to a source term — now a source of variance due to external influences „ Terms (I) and (J) have very interesting and important interpretations. Term (I) gives rise to terms which are concerned with the internal redistribution of variance per unit volume while the term (J) gives rise to terms which are concerned with the dissipation of variance per unit volume — a characteristic of all physical systems. These latter interpretations may be more easily seen for the special situation where each flux, , in the set of fluxes,


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11.65 If this relation is substituted into terms (I) and (J) of eq . (11,27), one obtains and — d Eq. (11.29) has the form of an internal redistribution of variance with the V being the phenomenological coefficients and the gradient of variance as the driving force. Furthermore, because each of the phenomenological coefficients are greater than zero, the right-hand side of eq. (11.30) is negative definite — thus indicating that it is a rate of dissipation of variance. The variance we have defined so far, ~\J , is a local measure. Because we wish to consider regions of physical space, y\. , rather chan only points of r^ we would also like to have some global measure of variance. One possible way of defining the "global variance," V , jy summing the local variance over all points of "p^ , that is V*)?*sl "Ll^ f 3.-SL> (11,31) An expression for the temporal rate of increase of global variance may easily be derived by integrating equation (11.27), using the volume to surface integral theorem, and also applying the boundary conditions. Ihe boundary conditions will be of the form C<5 D > = Q

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11.66 if the boundary operators are linear and the dynamical process itself is not being perturbed. Again, if (2
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11.67 density variations are neglected except in the source term — simplify this equation somewhat. It should be noted that both Serrin (1959)'s and Joseph (1965) 's formulations are special cases of equation (11.34). In a manner similar to that of Serrin (1959) and Joseph (1965), the terms appearing on the right hand side of eq . (11,34) may be estimated by means of some standard inequalities. The relevant inequalities which are derived in a like manner to Serrin 's and Joseph's derivations are: (1) By the Schwarz inequality we have II « < f F ' s -, 7 >I^I' 4 V" % (11.35) where ^ j * ^p.p^ V* =i;x t b p i,;> and* M v, „ \ |I«'^>|S N*5 /l I* Ov 1 *36) where N = number of state variables 5 l) *This estimate is cruder than the analog in Joseph (1965), but because a more general case is being considered this is to be expected,

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11.68 (2) The analog to Poincare's inequality, as used by Serrin (1959) and Joseph (1965) may be arrived at from 5 Me +^ fe io<^-a^ «^<*o7>i> p. -lUinv^i^j/j/ (11.37) where h is an arbitrary continuous vector field defined in "H and €: is a real number. The relevance of this latter inequality becomes apparent only when one substitutes a constitutive relation such as that preceeding eq. (11.29) — that is, the fluxes composing (D are linear operations in the gradients of the associated physical constructs and there exists no coupling between the fluxes in this term (the latter condition, is, of course, consistent with the Boussinesq approximation). With this in mind, let us consider the following hypothetical mathematical model for the purpose of illustration, namely, (i) the boundary conditions for the dynamical process under investigation are not perturbed, that is, s d| Q (ii) the constitutive relations are as mentioned in the previous paragraph, that is, such that (£ , is of the form where 7 is a diagonal (N x N) matrix;

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11.69 (iii) the conditions are such that the Boussinesq approximation is appropriate; (iv) the static contributions to the flux fields, as accounted for by 95 > are such that o< 1 -lp 1 and *t =0('l=2) where h is the hydrostatic pressure field; (v) the fluid medium being considered is incompressible. Under these assumptions, eq . (11,34) reduces in a straightforward manner to + ^ R (11.38) Now the main objective in the following manipulations will be to obtain as many estimates as possible explicitly in terms of V or Y . In order to meet this objective it will be necessary to compare, in the present context, quantities which are of different dimensions . In view of this let us now suppose that the original equations as well as the boundary conditions have been placed in dimensionless form;* the resulting *We do not wish to imply that there is no subtlety involved in placing the governing equations in dimensionless form; this is not the case as Sani (1963) has amply illustrated by showing how the properties of the operators in the governing equations depend upon the method in which the equations are made dimensionless. However, because we have taken the viewpoint that the governing equations are given a priori , the manner in which they have been made dimensionless is not of central Importance for our development.

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11.70 form of our equations in that context are easily obtained by setting the density o = 1, the diagonal elements in V equal to corresponding dimensionless ratios (e.ftu. —» j_ ) and possibly the multiplication of the 1 % usual components of the source term, \~ , by some dimensionless numerical factors. With this new interpretation of the symbols appearing in the equations — but the old symbols — the use of the estimates given in eqs. (11.35) to (11.37) subject to the homogeneous boundary conditions of our model on eq. (11.38) leads to the estimate, + i 1+6' F^U 2 -^)^] ^J^^ (II ' 39) Our objective to obtain all estimates in terms of V and A7 has been attained in all but the last two terms. Both these terms may be expressed in a more tractable form by defining two new operators. First we consider the term and we define a diagonal matrix, 1 , such that the element corresponding to a vector field component in S is the real number one and the diagonal element corresponding to a scalar field component of A* is the unity dyadic. As a result of this definition of \_ it follows ,* that matrix multiplication of ( 'SL "S ) with \ will produce a matrix ~X> — t> —

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11.71 whose diagonal elements are second order tensors. For example, if then -oVi = t^&^T^ 1 O L — V v + T T \ a* With the use of J. , the following equation is easily verified by direct computation, namely ^!l>^I^Ul>i D ^^ (H.40) 'tr Likewise, for the term, \W^^<5,yyi> the definition of a diagonal matrix, 1. , each diagonal element of which is the unit dyadic allows us to rearrange this term into the re convenient form I K
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11.72 If equations (11.40) and (11.41) are substituted in eq . (11.39), then one obtains 1, t ^ % &fckv>* nV^'Xu " &#-[V> < 5.5.ffcll^VK^i'KVK^ (11.42) where From this equation, the role played by the arbitrary vector field, r\ , in estimating quantities in terms of "V (or ~\J because the quantities are dimensionless) is evident. We become closer to our final objective if h is chosen such that (i) HM^-^.V* = p = ^ ere Is no sum on the \ in With these restrictions upon h , we then find that where there Is no sum on the Jp in ^\ .* (il)s Ks v i^<«v o + ij<, 1F o> V, + ^i leGA n U+e> 1+e' y 2 ^\^--[u,i.v n<^>n> v 7^^< tt ^c> (11,43) *These two restrictions on h imposed the compatability condition that Hhl|' t =<1-+M*HV>0^iA Note" that if X=0 Vx£H then ^IV 2 "? ^ •

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11.73 where A "iK\^vvv^. If we now restrict ourselves to q. = O » then the last term of equation (11.43) always makes a negative contribution therefore, Our objective of obtaining all estimates in terms of V can finally be reached if we impose the conditions that all vector field components of 3_are solenoidal and that X C x) + X Cx) •+• > 5 (v^ = O ; Vx[^ . The latter condition implies that because of the other conditions imposed upon n . It may be seen that this last term always makes a negative contribution — under the above conditions — by considering these conditions a typical vector field component of S .j. . say &^ , and its contribution Corresponding to a scalar field component, say \j , the contribution is i^^A>l\
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11.74 Consequently, subject to the imposed conditions mentioned above we have reached our objective, namely, 1 L<-\7> % N Vtr-Va-r Vz ^V^'V^v 4 ^ 4.r^_ _ 2eGA ~K 7 (11.45) In summary, the conditions we have imposed to obtain this equation are: (1) on the model — each vector field component of 5r> must De solenoidal, 3 — (j , linear constitutive relation with no coupling of flux fields, the Boussinesq approximation is appropriate, no contribution from static part of flux fields; (2) on the constan t ^ and the vector field Q. 9 \^)<-> 2 W+\W^0lVu^ and fc%0.

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BIBLIOGRAPHY REFERENCES FOR CHAPTER II Aris, R. 1962, Vectors, Tensors, and the Basic Equations of Fluid Mechanics , Englewood Cliffs, N. J. Bergman, S. and Schiffer, M. 1953, Kernal Functions and Elliptic Differential Equations in Mathematical Physics , Academic Press, Inc. , N. Y. Birkhoff, G. 1960, Hydrodynamics, a study in faet, logic, and. similitude , Princeton Univ. Press, Princeton, N. J. Buck, R. C. 1956, Advanced Calculus , McGraw-Hill Book Co., N. Y. Cesari, L. 1956, Surface Area , Princeton Univ. Press, Princeton, N. J, Clement, P. R. and Johnson, W. C. 1960, Electrical Engineering Science , McGraw-Hill Book Co., N. Y. Coleman, B. and Noll, W. 1958, On the ftiermostatics of continuous media, Arch. Rat. Meeh. Anal<, 4 , 97. Courant, R. and Hilbert, D. 1952, Methods of Mathematical Physics, I , Interscience Publishers, N. Y. Dieudonne, J. 1960, Foundations of Modern -Analysis , Academic Press, Inc., N. Y. Edwards, R. E. 1965, Functional Analysis , Holt, Rhinehart, and Winston, Inc., N. Y. Eliassen, J. D. 1963, Ph.D. Dissertation , Univ^ of Minn., Minneapolis, Minn. Ellasser, W. M. 1956, Hydromagnetic dynamo theory, Rev . Mod . Phys, 28, 135 El'sgol'c, L. E. 1964, Qualitative Methods in-Mathematical Analysis , American Mathematical Society, Providence, R. I. El'sgol'c, L. E. 1966, Differential Equationswith. Deviating. Arguments , Holden-Day, . Inc. , San Francisco, Calif. Eringen, A, C1962, Nonlinear Theory of Continuous Media , McGrawHill Book Co. , N. Y. 11.75

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11.76 Flanders, H. 1963, Differential Forms , Acedemic Press, Inc., N. Y-. Freudenthal, H. 1963, The Concepts and Role of a Model in Mathematics and Natural and Social Sciences , Gordon and Breach, Inc., N. Y, Garabedian, P. 1964, Partial Differential Equations , J. Wiley and Co., N. Y. Guggenheimer, H. W. 1963, Differential Geometry , McGraw-Hill Book Co., N. Y. Hahn, W. 1963, Theory and Application of Liapunov's Direct Method , Prentice-Hall, Inc., Englewood Cliffs, N. J. Hahn, W. 1967, Stability of Motion , Springer Verlag, Inc., N, Y. Hamel, G. 1908, Uber der Grundlagen der Mechanik, Math. Ann., 66 , 350. Joseph, D. D. 1965, On the stability of Boussinesq equations, Arch. Rat. Mech. Anal, , 9, 59. Kalman, R. E. and Bertram, J. E. 1960, Control analysis and design via the second method of Liapunov, J. Basic Eng., Trans. ASME, ser. D, 82 , 371. Kaplun , S. 1954, The role of coordinate systems in boundary-layer theory, ZAMP, 5 , 111. Kaplun, S. and Lagerstrom, P. 1957, Asymptotic behavior of NavierStokes solutions for small Reynolds numbers, J. Math, Mech. , 6, 585. Kellogg, 0. D. 1929, Foundations of Potential Theory , Fredrick Ungar Pub. Co., N. Y. Krasnoselskii, M. A. 1964, Topological Methods in the Theory of Non linear Integral Equations , the Macmillian Co., N. Y. Lagerstrom, P. A. and Cole, J. D, 1955, Examples illustrating expansion procedures for the Navier-Stokes equations, J. Rat. Mech. Anal. , 4 , 817. Lin, C. C. 1955, The Theory of Hydrodynamic Stability , Cambridge Univ. Press, Cambridge. Maxwell, J. C. 1892, A Treatise on Electricity and Magnetism , third edition, Oxford Univ. Press, Oxford. Murphy, C. L. 1965, Ph.D. Dissertation , Univ. of Minn., Minneapolis Minn.

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11.77 Murphy, C. L. and Scriven, L. E. 1968, Low tension and highly curved interfaces, paper presented at A.I.Ch.E. Tampa Meeting . Noll, W. 1959, The foundations of classical mechanics in light of recent advances in continuum mechanics, in The Axiomatic Method , North Holland Pub. Co., Amsterdam. Sani, R. 1963, Convective instability, Ph.D. Dissertation, Univ. of Minn., Minneapolis, Minn. Seeger, R. and Temple, G. 1965, Research Frontiers in Fluid Dynamics , Interscience Pub., N. Y. Serrin, J. 1959, On the stability of viscous fluid motions, Arch . Rat. Mech. Anal., 3 , 1 . Sherwood, T. K. and Pigford, R. L. 1952, Absorption and Extraction , McGraw-Hill Book Co., N. Y. Shilov, G. E. and Gurevich, B. L. 1966, Integral, Measure, and Derivative: A Unified Approach , Prentice-Hall, Inc., Englewood Cliffs, N. J. Sommerfeld, A. 1952, Electrodynamics , John Wiley and Co., N. Y. Spivak, M. 1965, Calculus on Manifolds , Ws Ai Benjamin, Inc., N. Y. Synge, J. L. 1960, Classical mechanics, in Handbuch der Physik , III/l , Springer Verlag, Berlin. Toupin, R. A. 1956, The elastic dielectric, J. Rat. Mech. Anal., 5 , 849. Truesdell, C. A. 1960, The Principles of -Continuum-Mechanics , Colloquium Lectures in Pure and Applied Science, no. 5, Socony Mobil Oil Co., Dallas, Tex. Truesdell, C. A. and Toupin, R. A. I960, The classical field theories, in Handbuch der Physik, III/l , Springer Verlag, Berlin. Ukhovskii, M. R. and Iudovich, V. I. 1963, On the equations of steadystate convection, Appl. Math. Mech. , 27, 295. Van Dyke, M. 1964, Perturbation Methods in Fluid Mechanics , Academic Press, Inc., N. Y. Willmore, T. J. 1959, An introduction t o Differential Geometry , Oxford Univ. Press, London. Zubov, V. I. 1961, The Methods of A. M. Liapunov and their Application , ACE-tr-4439, Department of Commerce, Washington, D.C.

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CHAPTER III POSITIVE CONES AND LIAPUNOV OPERATORS III.l. Introduction The history of mathematics abounds with examples of simple concepts masquerading, unnoticed, in different guises. Yet they are always found out, and mathematics is the richer for discoveries of a unifying threads in seemingly disparate disciplines. The distillation of a concept to its essence is generally termed an abstract formulation, for then it is longer bounded to its pedestrian surroundings of one application but rather stands at a distance overviewing many possible applications. So it turns out to be in the case of Liapunov's direct method. There are two basic features in Liapunov's method: (i) the positive-definite functional whose domain space is the state space for a given system and whose range space is the nonnegative portion of the real line, the latter of which is used to provide an ordering of elements in the state space, and (ii) the derivative of that functional, which, with requisite properties, may be used to achieve an ordering of the succession of elements along admissible trajectories in the state space. Because Liapunov's original formulation (see, e.g., Hahn, 1963, 1967; We will later (see Section III. 3) attach adjectives to the word "ordering" to somewhat qualify this property. At this point it is sufficient, however, to realize that with its usual ordering relation, the set of all real numbers is a totally ordered linear space. III.l

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III. 2 Lasalle and Lefschetz, 1961; Lefschetz, 1963; Liapunov, 1966) was concerned with stability problems involving ordinary differential equations and finite-dimensional state spaces, there are at least four avenues of generalization. Of these, the following three have already been pursued: (i) the state space the domain of the Liapunov functional may be generalized, for example, to infinite dimensional linear spaces and to abstract metric spaces (see, e.g., Hahn, 1963, 1967; Zubov, 1961), thereby permitting a wider class of problems to be analyzed by Liapunov ' s direct method, (ii) Liapunov 's definitions of stability and asymptotic stability may be modified and refined, for example, to concepts such as uniform stability, equi-asymptotic stability, stability in the large, etc. (see, e.g., Hahn, 1967; Kalman and Bertram, 1960), and (iii) the form of the operator equations to be analyzed may be extended, for example, to equations involving differentials rather than derivatives (see, Yorke, 1968, and references therein). In each of the above avenues of generalization, however, the range of the Liapunov functional is the nonnegative portion of the real line, and the values taken there by the Liapunov functional are used to induce an ordering of the state space. Now there is nothing, other than convenience and familiarity, that singles out the nonnegative portion of the real line for this role; indeed, because we will find that a partial ordering of the state space induced by the values of a Liapunov operator is adequate for the purposes of the direct method, any positive cone having certain essential properties of the nonnegative portion of the real line will do equally well. This partial, rather than a total, ordering is adequate precisely because Liapunov' s theorems provide

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III. 3 sufficient and not necessary conditions for stability. Thus, the use, rather than just the recognition, of this fact leads us to our generalizations of Liapunov's basic theorems by means of the concepts of admissible trajectories (Section III. 3) and Liapunov operators, as opposed to Liapunov functionals (Section III. 2) Because the sufficient nature of Liapunov's method permeates the subject, it has been recognized for some time that only a few systems can be said to have all their trajectories possessing a given property say, being asymptotically stable and consequently conditional stability has come more into prominence as a means of considering classes of trajectories possessing a given property (see, e.g., Massera and Schaffer, 1966). It is not an unhappy circumstance, then, when the generalization of the direct method permits us to consider only admissible trajectories, for this leads to stronger mathematical results and more realistic physical conclusions. More importantly from a theoretical point of view, this avenue of generalization brings out the centralized role played by the Liapunov operator. In previous works dealing with Liapunov's direct method (see, e.g., Hahn, 1963, 1967; Kalman and Bertram, 1960; Zubov, 1961, and references therein), the main role played by Liapunov functional has been to provide a means of estimating the norm, and in this sense to order the state space. This, however, is not a natural role for the Liapunov functional, for it generates a great deal of extracurricular activity in the proofs of the theorems, and, in effect, almost hopelessly entangles the concepts of a norm and a Liapunov functional. Our theorems show this to be quite unnecessary; the actual

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III. 4 role of the Liapunov operator to induce a partial ordering in the state space is thereby considerably clarified. An additional benefit arising from this process of clarification is that the class of Liapunov operators is significantly enlarged, thereby, significantly increasing the number of ways that the state space may be partially ordered. If the operator equations are ordinary differential equations and the state space concomitantly finitedimensional, the class of Liapunov functionals may well be adequate in stability analyses. However, for the more general operator equations with state spaces that are accordingly more general, the distinctions between elements and also trajectories will be more subtle. Naturally, it will become more essential to discriminate between different classes of elements as well as different classes of trajectories, and consequently, the more desirable it becomes to have more sophisticated and delicate means of ordering the state space than simply that induced by the positive cone of the nonnegative portion of the real line. III. 2. Preliminaries: Relevant Definitions, Properties, and Concepts The proofs of the classical stability theorems of Liapunov rely heavily upon the state space being endowed with the properties of a linear topological space. The space, in other words, is not a linear space with only the algebraic operations as structure, nor is it an arbitrary topological space. Within the framework of linear topological spaces, the topological structure may be introduced arbitrarily; in particular, topology need not be that induced by

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III. 5 operators into the real line functionals as is the case, for example with normed linear spaces. Moreover, in the proofs of the classical theorems of Liapunov's method, it is important that the range space of the Liapunov functional the nonnegative portion of the real line have the properties of an ordered set as well as those of a linear space. The relevant abstraction of the range space for our generalization is therefore that of a partially ordered, linear topological space (see, e.g., Hille and Phillips, 1955), that is, Definition III. 2.1. The five-tuple I X / + . —
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III. 6 In general, there are a variety of ways of introducing an ordering relation into an abstract space. However, because we wish to generalize Liapunov's direct method, we focus upon an ordering induced by the specification of a closed positive cone, "K , in li the natural abstraction of the nonnegative portion of the real line. By a (closed) positive cone (cf., Hille and Phillips, 1955; Krasnoselskii, 1964a) we mean: Definitio n III. 2. 2. If 7J is a linear topological space, then a subset of f\ is said to be a (closed) positive cone in li if the following are satisfied: JkJ j^ is a (closed) subset of 7J , g[ if i^ ; V£ jS then *u+ (5v £^< ; V^^O ; fi>0 ; o< R E F JK^j if ^^ and-l^X thenUSO,. Given a positive cone in a linear topological space, we may induce a partial ordering of the linear topological space based upon it by defining (--( ) such that Because of postulate |gj , the above ordering relation is continuous in the sense that if ^]_^w , [y k ] _^ v .andu^v^ for each k, then u -< v (see, e.g., Kantorovich and Akilov, 1964). In the nomenclature of Edwards (1965) our positive cones are pointed, closed, convex, and salient. Incidentally, Edwards (1965) as well as several other authors (see, e.g., Hille and Phillips, 1955; Krasnoselskii, 1964b), do not explicitly attach the adjective positive to their definitions of cones.

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III. 7 If the ordering is induced by a specific positive cone, then the positive elements, that is, are exactly the members of that positive cone. On the other hand, if in a given linear topological space we single out a class of elements, W > as being the set of all positive elements, and if we define x— < y to signify that Cy ~ *")£ 'P* » we have the only structure of a partially ordered linear topological space in CJ for which t is precisely the set of positive elements. Again going back to the classical theorems of Liapunov and it his positive-definite functionals the Liapunov functionals utility of an operator mapping into a positive cone rather than the nonnegative portion of the real line is not unexpected. However, because we wish also to satisfy postulate |K 1 , we first define: Definition III. 2. 3. An operator vV < ^"^"3£"*' ^4 ^ s called closed if its graph is closed, that is, if X^£ Jt. y ~^_ — »• y and •j — y o .We now can define a positive-semidef inite operator as: Definition III. 2. 4. A closed operator , \AJ<^ "> : _^C — * H > wi H De called positive-semidef inite if it satisfies: (i) "V0= O^; and (ii) O y -<"U)<^>jV^ e ^ J **°3C The terminology in the literature on Liapunov 's direct method is somewhat confusing at this point. Some authors require that the functionals derivative have certain desired properties before it is referred to as a "Liapunov functional"; however, we make no such requirement on the operator's differential before it is called a Liapunov operator.

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III. 8 It is called positive definite if (ii) is replaced by (ii') 10 <>> £ ^ O0-u ; Vx£3£ >*O x in which J\ is a positive cone in L| . For brevity, in the following condition (ii') is denoted by Just as in the case of classicial Liapunov functionals, we will have need of counterparts of these positive-semidef inite and definite operators, namely, Definition III. 2.5. A closed operator , ~V\]< ."> '. 3£ — => 7_\ will be called negative-semidef inite if it satisfies: (i) ^-Ox, and (ii)-"W<^x> >-0 Vx ale^ >^O x . A negative-semidef inite operator will be called negative-definite if (ii) is replaced by di') o^-~V0 <*> ; V > £* j x + O* From the standpoint of generalizing Liapunov 's theorems, the most important property possessed by these definite operators may be referred to as the separation property. This separation property may be characterized as follows: given any nonzero element, P , of a positive cone j( , all the elements of the domain whose corresponding values in the range space are comparable with P may be classified into three sets, namely,

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III. 9 In the sequel, this property is used not only to separate comparable (in the sense of the ordering induced by W<.>) states within the state space, but also to characterize the fate of a succession of states that is, a trajectory in the state space that satisfies the operator equation of evolution. Now this task of characterizing trajectories may be easily accomplished, if, in addition to satisfying the operator equation of evolution, each succeeding state along a trajectory is comparable with the immediately preceeding state in the sense of Vv^*^ ; each such trajectory will be called an admissible trajectory with respect to that W < •"? , and the set of all such trajectories is called the set, or class, of admissible trajectories with respect to "V\/^»"> . When combined with the inherently sufficient nature of Liapunov's theorems, the significance of this set will become apparent; however, until we discuss the generalizations of the theorems themselves we will employ this separation property primarily in generalizing and placing on a more geometrically and physically intuitive level the classical notions of stability and asymptotic stability, as well as the complementary notions of instability and unbounded instability. In particular, the classical definition of stability is replaced by Definition III. 2. 6. The origin, CL , of the state space 5> is called a stable basic state of a dynamical process relative to the Liapunov operator \K '^ and to the set Cq,*J %Z-S I v — < ^ J if any admissible trajectory passing through a state 5 £ (^p d (A, never reaches a state 3^3^ p ^ { I £ S* ("V f ^

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III. 10 and the notion analogous to the classical concept of asymptotic stability is: Definition III. 2. 7. The origin, Q^ , of the state space S is called an asymptotically stable basic state of a dynamical process relative to the Liapunov operator V< > and the set ^p of Definition III. 2. 7 if any admissible trajectory passing through a state S £ (^> © d &>o is stable and if, in addition, it will eventually reach the basic state VJ^i . These notions of stability differ from the more classical definitions in two important ways other than that a specific topology is not explicitly mentioned. First, both of these definitions include a phrase "relative to the Liapunov operator ~\7< >> and the set Q~) p ," which corresponds to the more natural condition of being relative to a class of disturbances in the sense of the Liapunov operator, rather than with respect to a possibly unphysical as well as mathematically restrictive neighborhood of the origin, say, in the sense of a norm. However, these two definitions differ in a manner more subtle than might, at first, be apparent. It is more than simply a difference between asymptotic stability and stability. In particular, if a given Liapunov operator is used to conclude that a certain admissible (in the sense of the given "V< "> ) trajectory is asymptotically stable, then this trajectory is asymptotically stable whether or not it is admissible for a different Liapunov operator. On the other hand, a certain trajectory may be stable relative to a given Liapunov operator but the trajectory can lose this

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III. 11 property upon changing Liapunov operators. We are thus led to refer to stability as a relative concept, and asymptotic stability as an absolute concept. With this convention, we can say that the notion of instability as defined by: Definition III. 2. 8. The origin, O^ . of the state space b of a dynamical process is called an unstable basic state of the dynamical process relative to the Lipaunov operator ~V^ •"> and to the set Qo t ) 5. £ S I "V < ^' 5 '> ^~ P 1 i f an y admissible trajectory passing through a state S£ (~> © o^D will never reach a state belonging to the set °&p f is also relative. However, the notion of unbounded instability, the complement of asymptotic stability, as defined by: Definition III. 2. 9. The origin, O 3 , of the state space O of a dynamical process is called an unboundedly unstable basic state relative to the set (2>p = i^^ j V>-fj if any admissible trajectory passing through a state s£ (.-^ p ©^G-b will eventually penetrate r f ^ ** the outer boundary of every possible set of the form \D n p2 — |5SO [(^"V^-^J where a is fixed and p varies throughout the positive cone in (J in such a manner as to satisfy the inequality yet retain comparability i may be regarded as absolute. The central role of the Liapunov operator is emphasized by these definitions, for they make full use of the operators as a means of inducing a partial ordering in the state space, This is in

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III. 12 contrast to the classical definitions and theorems in which the Liapunov functional plays an essentially ancillary role by providing a partial ordering that is used only for purposes of comparison with another ordering, say that induced by a norm. Moreover, although the introduction of a norm into the state space gives the appearance of transforming our "relative" statements into "absolute" results, such conclusions are specious until largest domains of stability, etc., are obtained, and these may be more naturally sought in the sense of a Liapunov operator than in norm. III. 3. Liapunov 's Stability Theorems and Positive Cones The fundamental principle underlying the utility and versatility of our generalization of Liapunov 's direct method is the partial ordering of the state space as induced by a positive-definite operator ~V< ."> mapping into a positive cone. The power of the generalization arises because of this ordering of states and, more important, because of the characterization of the fate of admissible trajectories that is obtained without an explicit knowledge of the solutions of the operator equations of evolution; this ordering is obtained merely by the intelligent use of information about, and properties of, Liapunov operators and their differentials or derivatives, as the case may be. In some detail, because the differential evaluated at a state on an admissible trajectory in a direction specified by the operator equation of evolution may be interpreted as a linear approximation to the change in values of the Liapunov operator upon proceeding along the admissible trajectory, the differential may serve as an

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III. 13 indicator of the behavior of the trajectory through additional knowledge of the nature of the Liapunov operator. Loosely speaking, the Liapunov operator characterizes "where a dynamical process is" in relation to L)^ , and the differential evaluated along an admissible trajectory characterizes "where the dynamic process is going." With this overview of the roles played by the Liapunov operator and its differential, the first of generalizations of Liapunov 's main stability theorems may be now formulated. Theorem III. 3.1. Consider a dynamical process with its evolution in a state space 3 characterized by the field, 5 5 , and with the state (j^i as a basic state. Then for the basic state O^ of this dynamical process to be stable relative to a positive-definite operator, ~V<.> •* ^ — * "M » with both S and J\ linear topological spaces, and relative to the set QL £ i S£^ [V<' s ->- J> ic is sufficient that there exist such a "V<,."> which is (F)-dif f erentiable and such that its (F)-dif ferential is a negative-semidef inite operator along all admissible trajectories passing through states belonging to Cop ; in symbols, - VO^Vst 6 f Proof: From the hypotheses of the theorem and the postulates of a partially ordered linear topological space, we have cTv^sr^o^v^fc . (III>1) The reader is referred to Appendix B and references therein for a definition and discussion of the (F)-dif ferential .

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III. 14 we now select an arbitrary nonzero element, 5, , of ^3„ such that or, equivalently, "V , to the basic state or not. In other words we are interested in knowing if all succeeding states along this admissible trajectory remain within the set Co <£> ^^p To this end the approximate property of the Frechet differential (see, e.g., Vainberg, 1964; or Appendix B) may be employed in order to obtain the estimate A7 J\7 « S\l< l<} Vs> (III>3) Because the field dS is characteristic of "how the given dynamical process evolves in time," the quantity s, -t-cS may be interpreted as an approximation to the succeeding state along the admissible trajectory, and the approximation may be made as accurate as desired simply by letting the time interval become sufficiently small. We therefore use the following nomenclature: 5, = 5. + Ss ~t > "t and \t~T
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III. 15 \7 -"V<5« o ? = J"V (in. 4) relations (III.l) and (III. 2) and substituted into (III. 4), and 5, £ G © ^^o is utilized, then because U is a partially "t 6 r r —J ordered linear topological space, there results V< s t > -4 ?o + Ov, f (in. 5) and by construction we know that a — <1 f> . Because S. and 'twere selected arbitrarily we conclude from relation (III. 5) that no state along an admissible trajectory passing through a state belonging to the set C3 p ^>^C*3p ever attains a state belonging to the o&p . In addition to satisfying our formal definition of stability (Definition III. 2. 6), it corresponds to our intuitive definition of stability in the Liapunov operator relative to (^> p . We preface the presentation of the theorem concerned with sufficient conditions for asymptotic stability with a simple lemma which will be helpful in the final steps of the proof of that theorem. Lemma 1 1 1. 3.1. Consider LA to be a linear topological space over the field "fl^ and J^ to be a positive cone in that space. Then for each interior point of that cone, O £ j^ , there exists a region lZi o in the vicinity of the origin, On , such that each element of o is comparable V Proof: Consider an arbitrary interior point of the positive cone, say From the definition of an interior point and the properties of a linear

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III. 16 topological space, we have that for each interior point, n , of the set, 7\^j , there exists a neighborhood, )\J q , of the fundamental system of neighborhoods of O , \J , such that However, each neighborhood belonging to U o must be a balanced set (see, e.g., Kantorovich and Akilov, 1964, p. 392), that is x>M a CM Q ) v 1x1 = 1 Therefore the sets and since XJI.c)f 0> -XJI.OI. . Let us now take see whether all the elements of this are comparable to n . Therefore we select an arbitrary element £ € £ and form the element of Cj . Because f^c^as selected above, it may be represented in the form Thus the element ia. may be written as Consequently

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III. 17 and because we have the result that Thus *7 is comparable to each element of ^ and, in fact, exceeds each element of Cj in the sense of the induced ordering. Let us now consider the analog to Liapunov's other main stability theorem having to do specifically with sufficient conditions for asymptotic stability, that is, Theorem III. 3. 2. Consider a dynamical process with its evolution m a state space 3 characterized by the field, J3 » and with the state ^3 as a basic state. Then for the basic state CL of this dynamical process to be asymptotically stable relative to a positivedefinite operator, \7< •> '. 3 — * \] with both 5 and t\ being linear topological spaces, and relative to the set Qr>„ = { \£ o | ~S] < *>_> — <, n V , it is sufficient that there exists such a ~SJ < •"> , that is (F)diff erentiable and such that its (F)-dif f erential is a negativedefinite operator along all admissible trajectories passing through states belonging to £?_ : in symbols JV = O^ and Proof: Because a negative-definite operator is also negative-

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III.lt semidef inite, the conditions of Theorem 1 are also satisfied by the hypotheses of this theorem, and therefore O. is a stable basic state relative to Gd . Mathematically stated, we have if ti £. Qi> @ "^fep and the trajectory from 3/ to 5 , " t p r -T -<> — t characterized by 35 , is admissible. Let us now select an arbitrary state Sz 2 C~> ©^(ife "*0 P I such that the trajectory through 5+ aC "t is admissible, in which case we have "V<^> =^ o ^J^ ^t"V<5f> (III. 6) The negative-def initeness of the Frechet differential along admissible trajectories passing through states within Q~> p implies by the following argument that succeeding admissible states are successively closer to \J, in the sense of the ordering induced by v<.*>. We may look upon 5, as belonging to both and Furthermore, by construction we have The approximation principle of (F)-dif f erentials is now employed to estimate whether or not the succeeding state on the admissible trajectory through 5, is closer to in the sense of "V<,> > and we obtain the estimate

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III. 19 Now because the field, ^S , is characteristic of how the dynamical process evolves in time, we can rewrite relation (III. 8) as in which the error in the estimate may be made less than any arbitrary tolerance by selecting simply Qfc ~t *) sufficiently small. However, because of the hypothesis and our construction, we have and thus, -^V' *!>-\-4°m ; t i £"v<^. (III,10) 6 Relation (III. 10) is now substituted into relation (III. 9) and by recalling that M is a linear topological space, we obtain "V ~fo-"^ "
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III. 20 for t~t' -tc) sufficiently small. We could now repeat this construction algorithm by starting at i! at S^ , and employing the hypothesis of a negative-definite differential along an admissible trajectory to show that at t">t' Ct"-t') sufficiently small, the succeeding admissible state, 5 , is again 'closer" to O^ than S^, in that In fact, this construction algorithm may be carried out ad infinitum, always with the same conclusion, namely, that succeeding states along admissible trajectories are always closer in the sense of the induced ordering to O. than the preceeding states. In this way a generalized sequence* of elements belonging to a topological space may be constructed by following the values of ~V<-> along an admissible trajectory passing through an 3, £ £ ~ 3& D at time ^ ' that ls a sec l uence {V J i^t^o] (III. 12) with the property that ^ -^~V ; Y-t">i'Si D . cm. is) Because the algorithm may be performed for each 5 &&„&!}& which has an admissible trajectory passing through it, and each member of a class of such generalized sequences may be considered in precisely That this may be considered a generalized sequence (see, e.g., Kantorovich and Akilov, 1964, p. 34; Edwards, 1965, p. 14) follows from the fact that "V£ IJ (a topological space by hypothesis) for each -fc = , and the set {{^|t=o} is a directed set (see, e.g., Kantorovich and Akilov, 1964, p. 35; Edwards, 1965, p. 7) and because IF ^ , is postulated as an Archimedian ordered field (see, e.g., Dieudonne, 1960, p. 16).

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III. 21 the same manner by dealing only with the characteristic properties as given by relations (III .12) and (III. 13). But by our definition of positive-definite operators (Section III. 2), V\ o / is closed, and every convergent generalized sequence in V\S/must converge to an element of V\o/ (see, e.g., Kantorovich and Akilov, 1964, p. 45; Edwards, 1965, p. 14). From a consideration of relations (III. 12), (III. 13), and our algorithm for constructing these generalized sequences in terms of succeeding states along admissible trajectories, if we have given a comparable nonzero interior element of the cone, say h , that we can continue this procedure until a time K^) is reached such that for all we have (III. 14) The reason for this, to be intentionally redundant, is the negative-def initeness of the differential, which insures that the generalized sequence will continually get closer and closer to O., , and the sequence must eventually reach a state that is both comparable and "less than" V) by Lemma (III. 3.1) because *9 is an interior point and the trajectory is admissible. This is, therefore, true for each member of the class of generalized sequences starting at different 5 £. G> <~^ d(r> p at t and following admissible trajectories. If we wish, therefore, to consider collectively this class of admissible trajectories, then I will

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III. 22 generally depend upon 5. as well as r) . For this class of generalized sequences, therefore, we can say that given a nonzero interior point vy of ~V<5>, there exists a ^(^ rj) such that for all 1 = " il ±aJ rp we have ~V -<1« (III. 14) Because relation (III. 14) holds for an arbitrary comparable nonzero interior element, n— <1 p , and also because of the negative-def initeness of » it follows that for each subset of XX of the form there exists a number .^(.^ n) , such that for all ~t I C? t n) the elements of the characteristic generalized sequence belong to the above set. However, because VI is arbitrary and this progression towards O^ terminates only if «=l"V<. ; ' o"s> = On , we say that CL is the limit of the characteristic generalized sequence; thus, C> is the limit for each member of this class. We write, therefore, L'rvitV<^>j ~ °Ll as t grows (III. 15) Upon recalling that ~V<."> was (F)-dif f erentiable throughout ^ , which implies that ~V^.~> is continuous in o , and because If we could find a single time T^S, n") , for which relation (III. 14) holds for all members of this class, "then we would by analogy to classical definitions refer to the resulting asymptotic stability as "equiasymptotic stability in the Liapunov operator."

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III. 23 "VC^ is positive-definite we have that relation (III. 15) implies Li.*Uti = 3 ^ t grows (HI. 16) Therefore, under the hypotheses of the theorem, the succession of states along admissible trajectories tends to the origin, U^ . Because proofs of the generalized theorems for instability and unbounded instability are so akin to the two proofs already presented, with only evident changes in the senses of the various inequalities, we clutter the presentation no further and only present statements of the theorems: Theorem III. 3. 3 Consider a dynamical process whose evolution in state space is characterized by the field, o5 , and which has the state U^, as a basic state. Then for O. to be an unstable basic state of the dynamical process relative to a positive-definite operator ,y<>'?\ S~*1-\ > with both J and 1J being linear topological spaces and relative to (^* = {^ &S |'V>-p? , it is sufficient that there exists such a ~V<.>, which is (F)-differentiable and such that the (F)-dif f erential is a positive semi-definite operator along all admissible trajectories originating at states belonging to VOp • Theorem III. 3. 4 Consider a dynamical process whose evolution in state space is characterized by the field, S3 » and which has M^ as a basic state. Then for 0. of this dynamical process to be an unboundedly unstable basic state relative to a positive definite operator ,v<.>:o& lj with both S> and CjJ being linear topological spaces, and relative to (~? , it is sufficient that there exists such a V ^ • > which is

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III. 24 (F)-dif f erentiable and such that its (F)-dif f erential is positive definite along all admissible trajectories passing through states belonging to vd III. 4 Concluding Remarks The statements of the four theorems provides formal recognition of the essence of the generalization of Liapunov's direct method, namely, from a Liapunov operator with the nonnegative portion of the real line to a positive cone in a linear topological space. The extent of one's grasp of the essence of the generalization comes only upon considering their proofs, for only then is the impact of the extension from a totally to a partially ordered space felt. Only then, for example, does one understand the notion of an admissible trajectory. Only then does one see clearly how firmly rooted are the stability and instability theorems in their particular Liapunov operators, nor would one have readily anticipated this relativity of even these conditional results. And only then does one begin to sense the role of the element p selected in the positive cone J\^ , for it is only then that one can wonder about the complexion of the set of points vE) p in the state (domain) space that are mapped into J).. , yet comparable to p Precisely because of the several facets of the generalized direct method, however, there is a considerable potential for application of the method, perhaps to problems hitherto not amenable to the classical method of Liapunov and probably to more highly refined results. Certainly there is the enlarged class of Liapunov operators to commend it, even should the more subtle distinctions of more sophis-

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III. 25 ticated spaces be only partially realized. The emphasis on "stability in the Liapunov operator," without either the obfuscating or detrimental introduction of a norm or even an a priori topology is laudable, if momentarily unfamiliar, for the class of disturbances to be analyzed, say for stability, are precisely those that will b_e stable, provided the hypotheses of the theorem are met. And the realization that conditional stability is necessitated here is a desirable one to make in general because of the more subtle and mathematically precise results and the more scientifically powerful and appealing conclusions. An increase in versatility generally places a greater burden upon the user, and that is the case with the generalized direct method. The central difficulty is the fact that not only is <-J a partially ordered space because of the positive cone Jiu , but J)^. itself is only partially ordered. The nature of the set of elements of J\j. comparable to a given element p ^-^i_i » an< ^ ^he set of elements in the domain space of the Liapunov operator that map into them is quintessential, for one wishes to use values of the operator to "separate" the state space. But by varying p , one can hope to accomplish this, thereby bringing about an enlargement of the class of admissible trajectories. In addition, one can vary J\), within U , as well as varying P for each j)-^ . Moreover, the range space U can be changed as well as the M u '$ in IX and the p'5 in the jr^ . And finally, the Liapunov operators could be changed for each J\^ in each U , with p being varied in any event.

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BIBLIOGRAPHY REFERENCES FOR CHAPTER III Dieudonne, J. 1960, Foundations of Modern Analysis , Academic Press, Inc., N.Y. Edwards, R. E, 1965, Functional Analysis , Holt, Rhinehart, and Winston, Inc. , N.Y. Hahn, W. 1963, Theory and Applications of Liapunov' s Direct Method , Prentice-Hall, Inc., Englewood Cliffs, N.J. Hahn, W. 1967, Stability of Motion , Springer Verlag, Berlin. Hille, E. and Phillips, R. 1955, Functional Analysis and Semi -Groups , Amer . Math. Soc. Coll. Publications, vol. 31, Waverly Press, Inc., Baltimore, Md . Joseph, D. D. 1965, On the stability of the Boussinesq equations, Arch. Rat. Mech. Anal. , 9, 59. Kalman, R. E. and Bertram, J. E. 1960, Control analysis and design via the second method of Liapunov, J. Basic Eng. , Trans. ASME , ser. D, 89. Kantorovich, L. V. and Akilov, G. P. 1964, Functional Analysis in Normed Spaces , The Macmillian Co., N.Y. Kelley, J. 1955, Topology, Van Nostrand Co., N.Y. Krasnoselskii , M. A. 1964a, Positive Solutions of Operator Equations , P. Noordhoff Ltd., Groningen, The Netherlands. Krasnoselskii, M. A. 1964b, Topological Methods in the Theory of Nonlinear Integral Equations , The Macmillian Co., N.Y. Krasovskii, N. N. 1963, Stability of Motion , Stanford Univ. Press, Stanford, Calif. Lasalle, J. P. and Lefschetz, S. 1961, Stability by Liapunov's Direct Method with Applications, Academic Press, N.Y. Lefschetz, S. 1963, Differential Equations: Geometric Theory , 2nd edition, Interscience Pub., Inc., N.Y. Liapunov, A. M. 1967, General Problem of Stability of Motion , Academic Press, N.Y, III. 26

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II 1. 27 Massera, J. L. and Schaffer, J. J„ 1967, Lin ear Differential Equations and Functional Analysis , Academic Press, Inc., N.Y. Serrin, J, 1959, On the stability of viscous motions, Arch, Rat. Mech. Anal. , 3, 1. Simmons, G, F, 1963, Introduction to Modern Analysis , McGraw-Hill Book Co. , N.Y. Vainberg, M. M. 1964, Variational Methods for the Study of Nonlinear Operators , Holden-Day, Inc., San Francisco, Calif. Yorke , J. A1968, Extending Liapunov' s second method to non-Lipshitz Liapunov functions, Bull. Am. Math. Soc , 74 , 322. Zubov, V. I. 1961, The Methods of A. M. Liapunov and Their Application , AEC-tr-4439, Department of Commerce, Washington, D.C.

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CHAPTER IV QUADRATIC OPERATORS AND LIAPUNOV OPERATORS IV. 1. Introduction There exists a rich and useful relationship between abstract quadratic operators and Liapunov operators, similar to that between abstract quadratic forms and Liapunov functionals. In particular, we note that as each classical quadratic form has a bilinear form associated with it, so also does each quadratic operator have an associated bilinear operator. More important, perhaps, from the standpoint of utilizing the theory of abstract quadratic operators to generate Liapunov operators for the generalized direct method (see, e.g., Theorems III. 3,1-4) is the happy circumstance that the Frechet differential of a quadratic operator may be expressed in terms of its associated bilinear operator, With the Liapunov operator and its differential represented in terms of a quadratic operator and its associated bilinear operator, the inequalities for the Liapunov operator in the basic theorem (s) become conditions for a quadratic operator to be a Liapunov operator . Quadratic operators and their associated bilinear operators, as well as the special cases relevant to our discussion of positive (and negative) definite and semidefinite quadratic operators, are defined in the second section. After the conditions to be satisfied by a Liapunov operator in the various stability theorems of Chapter III are recalled in the third section, we develop sufficient conditions in the form of equations for a quadratic form to be a Liapunov operator for a quite general form of an operator equation of evolution. Moreover, we pursue IV. 1

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IV. 2 an obvious generalization as well as consider the noteworthy special case of a general linear dynamical process. In the final section we take up some special cases of the more general treatment of the previous sections, thereby arriving readily at analogs of classical results, for example, the results of Krasovskii. IV, 2. Definitions and Preliminaries We begin with the simplest method of construction, that of the quadratic form, an abstraction of the essentially trivial observation that the square of a real number is positive unless the number is zero, and if the square of a real number is zero, the number is zero. In a Hilbert space, in fact, the functional defined as the inner product of an element with itself is also positive-valued for all elements, being zero if and only if the element is the null element; in symbols this may be stated as (i) cxlx") >0 <=5> x *0 (ii)(x |xl =0 <^> xO Now the nature of the squaring operation on the real line is unaffected by multiplication by a positive scalar, and it is well known that the essential properties may be extended to positive operators, say A, in a Hilbert space, so that (i) (x I Ax) >0 <=> x^O (ii) (x f A*) =0 x-O By the same token, it is also possible to define a positive operator A that maps a Banach space 3£. into its dual 3£ with the property that

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IV. 3 (i) [ X; Ax7>0 <=^ x*0 (ii) [x, Axl^ O <=> x-=0 with Q-3 the bilinear functional between the space and its dual (see any standard reference). Other natural generalizations may be made, but these suffice for our purposes. The basic idea in abstracting the squaring property is clear. We begin with the necessary definitions and certain properties required for the subsequent development. ee Definition IV. 2.1. Suppose that jl , jL , and M are any thr linear spaces over the same field 7 1 , and consider an operator 4? is called a bilinear operator if it satisfies the conditions, and it is called a bilinear functional, or form, if L — Ik*. Definition IV. 2, 2. If 3c and \A are linear topological spaces over the field of real numbers, if?. , then is called a quadratic operator from x to U if it satisfies the conditions (*) |J ; Vx£3£ and X c R^ p<. -"^ will be called the bilinear operator associated with

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IV. 4 the quadratic operator \7 . From this definition, (a) "\7= Ou> (b) "V<-*> = "N7<>0> and (c) ^7 1 P><^,>> are easily verified. From property (c) , we see that we may look upon the quadratic operator as generating an associated bilinear operator or, equivalently , as a bilinear operator generating a quadratic operator. This relationship is further emphasized by the following theorem. Theorem IV „ 2,1. Every continuous quadratic operator mapping one Banach space, say T5v , into another, iJ^i , has an (F)-dif f erential given by clV<*'Vi>(?><*, Vf> (iv. l) or, in terms of an (F)-derivative , o\^ s fi><£ A; VO (iv. 2) in which ° indicates that the derivative evaluated at £. is a linear operator. Proof: From the definition of a quadratic operator, we have and thus, if it may be shown that 6* /5<^ ; V^ will be the Frechet differential of ~\7<.> at the point ^^ in the direction V\ . With this in mind, recall that

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IV 5 But ^ • ; >~^ is a continuous bilinear form, and hence we have We then have hr-^A II MU. i Ihl^O II In immediately, and the theorem is proved The (F)-dif f erential of a quadratic operator, then, clearly has a representation in terms of its associated bilinear operator, and therefore if one were seeking Liapunov operators that were also quadratic operators, a representation for the (F)-dif ferential of the latter is immediately available for comparison with the appropriate inequalities in the hypotheses of the theorems of the generalized direct method. Before use can be made of the representation in this context, however positive-semidef inite and positive-definite quadratic operators must first be defined Definition IV , 2 3 If '(3-r an d ^?j are Banach spaces over the field of real numbers [j^ , and if J ^)x . contains a positive cone, j\^, then a quadratic operator, ~V<"> ' p^. — * 'fe > will be called a positivesemidefinite quadratic operator if (i) ^7<~> E ^ ; V x £ 12^ Definition IV. 2 4. A positive-semidef imte operator will be called See Chapter III for definitions

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IV 6 positive-definite if, instead of (i) , it satisfies These two definitions serve as bases for the definitions of negativesemidefinite and negative-definite quadratic operators. Definition IV„2,5. A quadratic operator, V< .'? , will be called a negative-semidef inite quadratic operator if (-1) "V< •> is a positive-semidef inite quadratic operator, Definition IV» 2 6 A quadratic operator, ~V< .> , will be called negative-definite if (-1)~V<.> is a positive-definite quadratic operator. The analogs of a positive-definite and positive-semidef inite matrices that is, positive definite and semi-definite operators with respect to a bilinear operator are, Definition IV, 2 7 Suppose that 9C , }£ , and "Lj are linear spaces over the same field ~p , with j\^ a positive cone in U , and suppose that A , £• *£c L9£,| ^ 3£ z J „ Then an operator A^ is called positive-definite or posit ive-semidef inite relative to a bilinear operator <^p '• It. ® 36 "~* T--J if the quadratic form associated with 4>< \ 1 } A,(* 2 )*> » that 1S > 4><* 7 ,Ax(>b> » is accordingly posit definite or positive-semidef inite . We may also define, by analogy, negative-definite and semidefinite operators^ Definition IV, 2 8 An operator Aj^ £ iL c tj3£ 35^*3 is called negativedefinite or negative semidef inite accordingly as (— O C A ,^) is a lve-

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IV ; positive-definite and positive-semidef ini' e IV 3 The Method of Squares We restrict ourselves, unnecessarily but for ease of presentation in the preliminary report, to Banach spaces. In particular, the state space o of the dynamical process l = f^,-t"i (17.3) is taken to be a Banach space. Equation (IVo3) is, of course, the operator equation of evolution the law of correspondence between an event, (5 "t ") and the quantity s describing the dynamical process, with -jl.,,*) an operator mapping an event, (js ; "t) into the instantaneous temporal rate of change of the state of the system, 5 If s(t) is interpreted as a continuous derivative, then the operator, 4(. . ^ > evidently acts from *3<2) fi? + to //_ £ ft^_ .0 J , the space of linear and continuous operators from lR + to -> , with lR, LO ; 00 ) Although the operator -f-d '*) is not, in general, linear, each of the results of this chapter may be carried further for the special class of linear dynamical processes for which -f-G..} may be represented in the form, fav = A,rt)-i (IV>4) in which Aft) Z IL^l^j^^ and e is used to signify that A,(r) is a linear operator If the definitions of Section IV 2 are combined with the

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IV. 8 hypotheses of Theorems III. 3.1 and III. 3. 2, the following conditions are placed upon the quadratic form v<\> in terms of its associated bilinear operator /3^> : (1) for stability 2' ; **j J V *6 f J "^ ^-P*. (IV. 5S) cW-(?>-<0* Vs.L& f (IV ' 6S) (2) for asymptotic stability V<^>if><3 ; S>l>-0^ ; Vs£& p ^4=0^ (IV.5AS) <*V-= (3 -^O^'jVse^C t (iv.6as) and and in which dt is a positive infinitesimal increment in time and Cr?p is as defined in Chapter III. For the widest generality, yet still retaining utility, we make the trivial observation that if /S< 1; ."> is bilinear operator associated with a quadratic operator, then for a fixed operator is a quadratic operator. With this observation and a generalization of the chain rule of differentiation, equations (IV.5AS) and (IV.6AS) may be written _ \ "V = £^<^ J ^> t>-o^ J Vs £ e ? ,i40^ e (IV. 7) and = ^?M'.-f <*>«"* ° M )Vlt + (IV . 8) For brevity, we have not and will not in the sequel explicitly write out the corresponding relations for a Liapunov operator insuring instability or unbounded instability. They may always be easily obtained from those given merely by reversing the sense of the inequalities in relations (IV. 6S) and (IV.6AS),

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IV 9 in which S.C.") is a specific (F)-dif ferentiable operator from S to S , and <3/(..)is, of course-, its (F)-dif f erential at s The operator qt.) is selected (beforehand) in a given situation based upon its desirable properties for that situation. In the sequel, the inequalities (IV, 7) and (IV08) are taken as the starting point, rather than relations (IV 5) and (IV. 6), for they may always be regenerated by selecting d.(0 as the identity operation thus, tf'c.) becomes the identity operator, In this preliminary investigation we do not seek a general representation for a positive-definite quadratic operator in an abstract space, nor do we seek a class of dynamical processes for which sufficient conditions may be obtained for the existence of such operators to be Liapunov operators. Nor do we take up the natural concrete spaces in which such representations would be relatively straightforward. Instead, we suppose that positive-definite quadratic operators are known, say in terms of their associated bilinear operators, T[~,(, ,") > we make the same obvious generalization to operators £> t • ) : f5^ — "* 13^ that we made for operators q<0 : fo^ — > *&%_ , and by Identifying j. /?/ -^ with, say, T\ < -> in the following fashion, and where JT TI Z £TP C = [7T <. , . > is bilinear , 1\ \ ft^® fa^^l JT0 *~Oq\ We note that if we had a specific /5<._,.> S. lf\, and, relative to that ^< . ."> a linear positive-definite operator A* , then the operator

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IV. 10 &»<•,,.> »|S<.,A,c. ) "> is again a member of ll c as Is suggested by the nomenclature, The corresponding equations for this generalization (or special case, depending upon the point of view taken) follow merely by the substitution of this identification into equations (IV 9) and (IV, 10)., We therefore do not explicitly write out these equations for the case in what follows Now in order to place equations (IV, 9) and (IV, 10) in a more explicit and useful form, one should actually have a general representation for a bilinear operator from \b^® i-^" *" f^u an d the members of ]f^ as well as specific selections of the operators pc.5 and a(») Also there are at least two alternative routes which may be selected in making use of these relations corresponding to assuming a general form for the Liapunov operator or its Frechet differential, Within the context of the nomenclature of this chapter, these two approaches may be phrased as follows: (A,l) start with a general form for ^<.^»"> by selecting a priori 7T„ <,,.>,(}(.) ; <^U ) " thus insuring that it is a member of ]• and then check to see whether the #<, c "> determined will satisfy equation (IV, 10)(Ao2) start with a general form for /?<, > by selecting 7f 2 < ,, .> } \)(.,) ; «(>) a priori and then seeing whether a specific A< -> may be determined which belongs to (P The utility of one approach over the other depends upon the particular situation at hand, and the distinction between them will become clearer

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IV. 11 rr> W in the sequel „ IV, 4. The Method of Squares for Simple Bilinear Operators Perhaps the simplest representation of a bilinear operator useful in a generalization of the method of squares is that of a multiplication operation, [3 , defined as follows (see, e.g., Greub, 1967; Liusternik and Sobelev, 1961): Definition IV „ 4.1, Suppose that 3£. and Lj are linear topological spaces Then a multiplication operation El from 9£<2)36. — * \Jt must satisfy The first two axioms insure bilinearity whereas the third insures that the multiplication is a continuous operation. In addition to these properties, we require in view of our application-orientation for the method of squares that LI be partially ordered by a positive cone, 4^ , and that the multiplication operation also satisfy the condition Xt3x >-Ou ; V xtX ©Q^ Therefore we define formally the notion of positive multiplication as: Definition IV. 4,2, A multiplication operation £1 from X®X-^^J\ withbt Because Greub (1967) is concerned only with the algebraic aspects of a multiplication operation, whi ch he refers to as a tensor product, he does not require axiom 1 M„ I and 3E. and 7J are linear spaces without a topology.

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IV, 12 partially ordered by a positive cone d\ ^ , will be called positive if the condition M Al \ , VxtXeO^ in addition to the axioms |M 11 th of Definition IV ,4,1. A particularly simple example of a multiplication operation (according to Definition IV, 4 1) which is not a positive multiplication is the vector cross product in [R, On the other hand, given a continuous bilinear operator, C{)<.^. *> , between two linear topological spaces 36 and X^ into a third LA and an A.StLJLX. X,! which is a positivedefinite operator relative to (please recall Definition IV, 2 ,7), then the operation from ?£® 36 — *»<-A defined by actually defines a positive multiplication operation. Therefore, in the sequel we may interpret, if we wish, ( ,) Q C . ) as a short hand for where ob and a\ . are as discussed above. Consequently we do not explicitly take up this generalization We now consider more specific cases than the linear topological ones mentioned above but with positive multiplication operations defined on them, In particular we take up (a) a Banach space with a general positive multiplication operation, (b) a Hilbert space with the inner product as the positive multiplication operation, and finally, for reasons of comparison with more familiar results, we consider (c) an N-dimensional Euclidean space with the usual scalar product as the positive multiplication operation

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IV. 13 IV.A.a, Banach Space with a Positive Multiplication. Let |3^ denote a Banach space over the field, 71 , and suppose that a positive multiplication operation, d , is defined between the elements of 'uL and a Banach space 15^. that is partially ordered by a positive cone T\u, with respect to which the norm is monotomic. It then follows from the definition of a positive multiplication operation that for any fixed continuous linear operator, L'. u)x~^ !%£. » the operator p^..,,"7 defined by is a bilinear operator, in which « is used only to indicate that I— is a continuous linear operator. If expression (IV. 11) is substituted into relations (IV. 9) and (IV. 10), we obtain i ^(1M3 !_• ^\~) ^ j>Cl)Q L,, «j?Cl^ (IV. 12) and ^WQCL-^-fcs^V -C^oal^cr>) (IV. 13) There is a strong temptation to use, because of the discussions following equation (IV. 10) and Definition IV. 4. 2, the notation l_ for a continuous linear operator that is positive-definite relative to the multiplication operation Q However, rather than clutter our equations with complicated nomenclature where it is unnecessary we simply use the symbol L with this understanding Of course, various particular forms of equations (IV. 12) and (IV13) are possible for particular choices of qt.) , be,) , J_ and L_ 2 , and we seek an t_ €. IP such that equations (IV, 12, 13) are satisfied. For example, if P = q_ then equation (IV. 13) leads to

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IV-14 L 6 V^fc^-t*) = L z ° = ^.C5.)H {1L. { o a ir ^ (IV. 17) is a Liapunov operator sufficient to insure asymptotic stability By selecting a. ti •> -=|hs_-) = 1 £ % and following the approach (A2) , another interesting form of the equation (IV ,.13) is obtained, Because of the nature of our multiplication operation the unity operator A always belongs to this class of operators.

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IV. 15 •-"i/L °£ C ^ = "" L ^°^ ; V-5&6 (IV. 18) Again an operator l_ satisfying equation (IV. 18) must be verified to satisfy a relation such as equation (IV. 15) before it is a Liapunov operator. Still another interesting form of equations (IV. 9) and (IV. 10) may be arrived at by choosing y 3 4and by using approach (A2) . Upon making these substitutions in equation (IV. 13), the following operator equation is obtained L-^Cs^vfc^-t^-L^-fcs^ vu6 f (iv. 19) where £ (s "f) is t ^ ie (F) -derivative o-f-f (,.,') with respect to the first variable. Again, the solution, l_ , of this equation does not specify an associated Liapunov operator unless a condition similar to equation (IV. 15) is satisfied. Equation (IV, 19) assumes an especially simple form if only the subclass of linear dynamical systems is considered: L-A^l" Af-tl-5 = -L L °A^)°£ (IV. 20a) One next must see if such an L is a positive operator, relative to Q An alternative procedure, of course, is to see if L— ~2-\-\ » obtained as a possible solution of [i tyfl-s] ® [l A f 1+1.3] =[tyfl.s] Q [ Ll . A,(i).s] is also such that it satisfies The simplicity of the above operator equations, (IV. 14),

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IV16 (IV. 18), (IV. 19), and (IV. 20) may be deceiving for they can be carried no further without a specific definition of the positive multiplication operation, EJ , being made. IV. 4. b. Hilbert Space As a concrete example of a Banach space with a positive multiplication operation, we consider a Hilbert space with its inner product, denoted by j-\ and (>ly)(for "*,y £ \H ), respectively. We shall have need of the well-known fact that any continuous linear functional HC9.(K)| L • 4ty{) 4-C L ° 4.^) I ^.(.x^ (IV. 22) in which
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IV. 17 and h [((Ki»lP r |,(s)tCP.^i^]civ . 24) in which r> TD3 'Dc )l rU 4il I tne associated bilinear form? % fl L V 1 c ' ' belongs to (f>. ] If the transposed operator is used, (IV. 23) and (IV. 24) become and (^l)|L^;»fc^))+(L^;of(s ; i)|^)^0(^ll^4-? T )4^^Vs£^ (IV. 26) respectively. If, in addition, we use the symmetry of the inner product and at the same time select jx.)=a(.), then a solution |_ of 1Lc^o|(s^)--C? 2 4-?7)^Cs) (IV . 27) will also solve (IV. 26), and if L is positive»def inite as well, then it may be used as a Liapunov operator. Thus, if q.V.0 is a continuous operator and if L is such that i( f j|L T )>0, ;V5 ^^ s+ ^ (IV>28) and (IV. 29) 2 can serve as a Liapunov operator, or functional, in this case, where (IV. 29) is automatically satisfied if q(,) is chosen so that

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IV. 18 and Let us consider another form of equation (IV. 27) obtained from the choices £>(,)f (•;.") -f5 ; t)=-(?, + ^ T )of ( s ; t) (IV. 30) an operator equation for |__ . Again -J(s t) is the (F)-derivative of J-( . , ") with respect to the first variable evaluated at the event ^S."fc) . In general, the operator j(, t) is nonlinear and therefore a different member of lL c Ln ; H J for each 5 E. \\ . Consequently, to proceed further with equation (IV. 30) a particular dynamical process must be specified. Alternatively, we might focus attention upon the class of linear dynamical processes, in which case we have. "f, <-S/t) = A(t)°5 (IV. 31) that is, the same member of IL^jHJfor all 5 . For this class of linear dynamical processes, equation (IV. 30) may be written as ZL«A^= -tP^+^j (IV. 32) which does not contain the variable 3 . If an inverse operation A. (f) is defined for all "t then equation (IV. 32) may be solved for L to yield L— i(P 2 + P>A~ 4 <« (IV .33) In particular, if "P={ (a valid choice as is seen from the axioms of an inner product), there results

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IV. 19 L-A'Vt) (IV. 34) For the functional Y^> = CA(i)os|(-A 1 (t))=A(i1<=i)-CA(i>sU) to be a Liapunov operator insuring asymptotic stability it must be verified that -(Arsis') > ; Vs_£6 p ( IV -35) for all "t . If the linear operator A(rO has a spectral decomposition, (IV. 35) for example, equation (IV. 35) is satisfied if all the eigenvalues J\\i) are negative for all L . For A constant, the obvious conclusion follows. An interesting form of equation (IV. 24) results if it is examined locally, that is, at a fixed S.Sh and the famous theorem on a sufficient number of linear functionals in a Banach space is utilized: namely, for any fixed nonzero element of n there exists a linear continuous functional which takes on a preassigned value at that element. In particular, consider the linear continuous functional (* 1 ^(yj) in which >u(,) is a mixed operator from j-1 to *Vi and y is a fixed nonzero element of \-\ , and denote its value at x. Z. \\ >< i O ^Y /I 1H (*i I >^(yV) . From the theorem mentioned above, we know that there exists a linear continuous functional, a (.") , such that h A * -\7 n.b. Not simply nonpositive, but negative. Otherwise Vcan vanish without "5 vanishing.

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IV. 20 VV 1 = c V~*yfi However, because we are dealing with Hilbert spaces, the linear functional ci ( ") , may be represented in terms of the inner product and an element "X cm uniquely determined by the functional, q (.,*) that is, Therefore for a fixed x and y belonging to / >4 , we have The element of 'j-jv* which satisfies this relation varies as both Ya and the value of Wi(,)at y are varied. At a fixed X. and y, we write in which ivi y(,) belongs to LL CH W 3 because / C is a linear continuous functional in > for a fixed y. It will, however, need to be a different element of ^ c T / H.'fJj for each yt j| unless, of course, "^id) is a linear operator, M(,) , is which case y.. may be characterized once and for all by another linear operator, jVJMi), such that Xf = m cx^ = MT, Xl ; v yzl4^ that is, the familiar adjoint operator. We now utilize these ideas by selecting a ( >)yd ; "t") , be )— 1.., <>(.')> and evaluating the equation at s=^ , to arrive at where -f^-^ f( . ; "t) and at a particular 5 1 the operation -h c V ma y be defined as above.

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IV. 21 Thus, it follows from equation (IV. 36) that each ^ £ Cx, the relation ^(^WM^^LM^r^ -^^ e 5i (iv.37) must be satisifed. The above operator, L , must also satisfy the relation for each s . These last two relations have been developed by local considerations and should therefore be treated as such. In obtaining the equations, (IV, 27) (IV. 38), approach (A2) has been used exclusively. If approach (Al) had been used, an Ltlfy would have been chosen initially and VC^\L-^) (IV. 39) with fl(,) such that qts^Qq implies ^ Q^ » would be tried as a possible Liapunov operator. From the general representation of the (F)-differential of "VOand the second requirement of a Liapunov operator expressed by the appropriate specialization of equation (IV .19), this l_ must also satisfy C^lL°q/ s 4(i,±)U(l°^°fts^^%^ -A O^ (iv. 40) if asymptotic stability is to be insured. IV. 4. c. N-dimensionai Hilbert Space In this final subsection, we consider the especially happy situation of a finite-dimensional Hilbert space, and we both specialize and utilize the results of the preceeding subsections to obtain the standard

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IV -22 methods used in treating systems of ordinary differential equations, those of Krasovskii (cf. Kalman and Bertram, 1960; Krasovskii, 1963). Not inconseque ially, Krasovskii's method, which is thus subsumed as an especially simple special case, provides a basepoint with which to compare and to view the general framework erected. In j-1 ( (R M with the standard inner product), /i_ C C 4^,143 becomes the set of all linear operators on j~i M , which is isomorphic to the set if all NxN matrices; we also represent elements of j-T/sj as N-tuples because of another such (quite well-known) isomorphism, if thus becomes the set of all positive definite NxN matrices, the adjoint operator becomes the transpose, the inner product is that of matrix analysis, and the operation ° becomes that of matrix multiplication. Finally Q is now to be interpreted as the Jacobian associated with the transformation * With these interpretations understood, equations (IV. 23) and (IV. 24), which, in turn, were derived from equations (IV. 9) and (IV. 10), become V^i-l ^+[L«j
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IV, 23 where I , P Z W., (now the class of "positive definite" matrices) If we use the properties of the transpose operation with respect to this particular inner product, the above equations become ^^U+C]a^)-V T ^5^V CL1 ; VstG r (IV,43) and fC£, L J^J £ C s p 4 fcfl 3^J £ f> = M£ + P^ W ; VT26/^ . 44) If the particular selection bso. is made analogously to equations (IV. 14) and (IV. 27) one obtains from equation' (IV. 44), If approach (A2) is used, this relation may be viewed as an operator equation for 1_ by considering (i) V* to be a chosen "positive definite" matrix, (ii) CK.) to be a chosen vector valued function, (iii) t(';>") as characterizing a specific dynamical process or class of dynamical processes. A solution, 1_ , to this equation will not be associated with a Liapunov operator for a specific dynamical process as characterized by +-(., .) unless it also satisfies jm L ^ y O, Vs L6 f ; s+ o^ (IV . 46) and q (O) U(o) = O As a special case of equation (IV. 45), one may choose q( ,)=+-(. -f) and if there is an L. such thac

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IV, 24 2-WL C *5 ~ ~^?z +E T ) , YS£ If L_ and R belong to the subclass of symmetric matrices in 11^. , then equation (IV. 48) may be written which, if considered pointwise, is essentially the relation used in Krasovskii's well-known theorem (cf., Krasovskii, 1963) the search now being for a V at each 3^ „ Furthermore, we may utilize this equation by selecting an L a priori and investigating the region in the neighborhood of each 5, to determine over what region the expression on the left side of (IV. 49) remains negative-definiteOne should notice that with these restrictions placed upon L , conditions such as (IV. 46) are automatically satisfied, and in particular if L — ^y_ is selected, the the other standard special case of Krasovskii is retrieved. Similar to the situation when general Hilbert Spaces were considered, more explicit equations may be obtained if the class of linear dynamical processes are considered. In that case, from equation (IV. 47) or equation (IV. 32) there follows

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IV. 25 1 L A (« = -(&+ FJ) (iv. so) in which L } (\[\) P, are now matrices. If a matrix A (t) exists for all T , then an explicit solution for L may be obtained as L '-iCPz + Pj)^ i [s T Mi)(P z + Pj)^] (IV. 52) is a Liapunov operator sufficient for asymptotic stability. A somewhat trivial, but nonetheless interesting, result may be derived from equations (IV. 51) and (IV ,52) if attention is limited to the class of asymptotically stable linear dynamical processes. These processes are characterized by matrices, A (-f) , having eigenvalues with negative real parts." If the matrix MtJ can be transformed to a diagonal form Dit) , then each of the components along the diagonal of £)(t) is negative. From equation (IV. 51) with the choice p^ = x E fKj , it is found that L g" V) Because T) {{) has as its components the reciprocal of the respective components of D/$ L is also a diagonal matrix, each of whose components are positive. Therefore a relation similar to (IV. 46) is identically satisfied and is one possible Liapunov operator „ Again, not just nonpositive real parts,

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IV -26 In this chapter, various operator equations have been proposed which if solved would yield a Liapunov operator which was concomitantly a positive-definite quadratic formAny proposed operator equation naturally gives rise to two questions of practical and mathematical significance: (i) Does a solution exist?, and (ii) Is the solution unique? With regard to the latter, it is clear that if a Liapunov operator exists, it is not unique, so that only the former is nontrivial . Because our equations are only for a subclass of Liapunov operators those which may be represented as positive-definite quadratic forms the existence or nonexistence of solutions to these operator equations has meaning only with respect to this particular subclass of Liapunov operators. At any rate, however, the investigation of the existence of solutions and the explicit forms of these equations for specific stability problems, await explicit expressions for -p( « } . ) characterizing the class of dynamical processes. The following chapter is devoted to obtaining just such explicit expressions for a special subclass of problems formulated in Chapter II in a quite general manner.

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BIBLIOGRAPHY REFERENCES FOR CHAPTER IV Greub, W.^H. 1967 /MuUiUnea^Algebra, Springer-Verlag, N.Y. , Inc., Hahn ' W ' 1%7 ' Stability of Motion, Springer-Verlag, Berlin. Kalman, RE. and Bertram, J. E. 1960, Control analysis and design via ^second ^ethod of Liapunov, J^gasic En fi f, Trans. ASME? Krasovskii,^. N. 1963, Stability of Motic^, Stanford U. Press, Stanford, Liusternik, L. A. and Sobelev. V J ]9ft1 pi omm * * t, Fred. Ungar Pub. Co.! N.Y. ' Elements ° f Function 1 An»l™<, , IV. 27

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CHAPTER V DIFFERENCE EQUATIONS FOR A CLASS OF BASIC FLOWS The basic aim of the theory of hydrodynamic stability, which until recent years was synonymous with linearized hydrodynamic stability theory, is the prediction of the onset of instabilities. The form of the results of classical linearized hydrodynamic stability analysis are usually a set (or possibly one) of critical values of relevant dimensionless parameters; if these critical values are exceeded, one may expect the onset of instabilities in a hitherto stable system. With this in mind, we intend to construct the machinery required to utilize the formalism of the preceeding chapters in order to pave the way for obtaining analogous results. Thus, explicit expressions for classes of operators f(.,.) — that is, for a class of basic flows of several classes of fluids — are developed in this chapter. The materials of construction for our machinery will be the concepts and equations developed in Chapters II-IV. The welding of the abstraction of a class of mathematical models of flows (as possible solutions of the describing equations) to classes of mathematical models of fluid behavior (as classes of constitutive relations) results in machinery which is both useful and versatile. The utility of the machinery is demonstrated, if briefly, by considering specialized dynamical processes, the global state of which is specified only by the velocity field. In effect, therefore, we have also limited ourselves to situations where the describing equations V.l

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V.2 and the boundary conditions are sufficient to determine the pressure field from the velocity field. Furthermore, the versatility of the machinery is demonstrated by the wide variety of formulations which may be placed within the formalism that has been developed, even for this class of specialized dynamical processes. V.l. A Class of Basic Flows Until a basic state is at hand, it is a vacuous exercise to investigate its stability. The obvious first step in developing explicit expressions for the operator, f (.,.), then, is to define a class of basic states. To make efficient use of the power of Liapunov's direct method, and in particular its facility for dealing with classes of problems, the essential qualities of the classical solutions to the Navier-Stokes equations are abstracted. We restrict ourselves to an admittedly small class of flows called parallel flows. This class of basic flows is characterized by a time-independent velocity field, v(.), such that in some orthogonal coordinate system it has the contravariant components, {^v^HtO, vcx^oj (v.l) Included within this class of flows are (i) Poiseuille flow between parallel planes, (ii) Poiseuille flow in tubes of circular crosssection, (iii) Poiseuille flow in annuli between concentric tubes of * The class of parallel flows is not equivalent to the class of viscometric flows of Coleman, Markovitz, and Noll (1966). In their terminology, parallel flows are a subclass of curvilinear flows which are in turn a subclass of viscometric flows.

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V.3 circular cross-section, (iv) Couette flow within a rotating cylinder, (v) Couette flow between rotating concentric cylinders of circular cross-section, and (vi) (plane) Couette flow. Rather than view this class of velocity fields as possible solutions to the Navier-Stokes equations, they will be viewed as possible solutions to the balance of linear momentum as expressed by Cauchy s First Law. The orthogonal coordinate systems which may be used to obtain the contravariant components in the form of equation (V.l) are restricted to be such that the local geometry does not vary in the direction of flow, i.e., J'i " 'Jl^**'*^ (V.2) From this requirement and equation (V.l), it immediately follows that motion within the class of parallel flows is isochoric It is then only a small price to pay and a considerable benefit accrues if we restrict ourselves to incompressible fluids, as we shall, in order to have the balance of total mass equation identically satisfied. With the velocity fields as given by equation (V.l) and the characteristics of the coordinate systems as given by equation (V.2), various useful quantities may be directly calculated. For convenience in the following sections these calculations are summarized in Table V.l. J. L. Ericksen (1956) has considered the question as to when this class of velocity fields are "dynamically possible" from this more general viewpoint. By dynamically possible he means that the balance of linear momentum does not over-determine the velocity field.

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V.4 J < CO O fe

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V.5 T3 CD 3 rt •H u c o e> w 1-1 M <; C o H CO to CU S-i p. X w M

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V.6 V.2. Equations Describing Velocity and Vorticity of Difference Motions The equations to be developed in this section are a special case of those of Chapter II for which s^— | v> and the basic equation is the balance of linear momentum in the form of Cauchy s First Law. Although most of the steps performed in this chapter have analogs in the formulation presented in Chapter II, it is instructive to repeat them in detail for this special case because: (1) it is a concrete example of the steps previously performed in an abstract form, (ii) it serves to illustrate the inclusiveness and usefulness of the abstract formulation of Chapter II, and (iii) it indicates the wide variety of possible stability problems which are still included under this very special instance of the abstract formulation. Thus, at the expense of being repetitious, we consider a situation in which the state variable is a solenoidal velocity field and the basic equation is Cauchy's First Law. In order for a velocity F ield to be admissible — as both the disturbed and basic velocity field mast be — the balance of linear momentum in the following form must be identically satisfied, * For emphasis, we again note that this places a restriction upon the class of possible field equations and boundary conditions because we wish to be able to determine the pressure field from a knowledge of the velocity field, thereby having p as a dependent (and therefore nonstate) variable.

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V.7 for any point (x,t) of space-time. It is convenient to decompose the stress tensor field T(x,t) into (i) the dynamic pressure f ield, ~h (> "(f) , which reduces to the hydrostatic pressure in the absence of bulk motion, (ii) the extra stress field j3(x,t), which is present in any bulk motion and which vanishes if there is no bulk motion. Upon inserting the decomposition into the balance equation above, the following relation is obtained: It should be noted that different classes of fluids with different velocity fields may have the same extra stress field, and in this respect they might therefore be indistinguishableThis is seen from jfy 4-)r O Xv D A( v ) ) ' S £-\ admissible constitutive relations C vhich is nothing more than a statement that the stress in this particular class of fluids at any place and at any time depends only upon the velocity field through certain more or less standard operations upon it. The specification of the form of the relation S (.,.,.., ) tells us how it depends upon these quantities * Consequently, the form of the operator J3 and its arguments are the elements which characterize and distinguish different classes of materials, e.g., newtonian fluids, stokesian fluids, and viscoelastic materials.

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V.8 With these preliminaries, the equation describing the difference velocity, u, between the disturbed velocity field, v , and the basic velocity field, v, may be developed. Both the disturbed and the basic velocity field satisfy the balance of linear momentum, ^v and The form of S <•> is the same for both equations because we are investigating the effect of perturbations of the time-invariant basic velocity field for a given fluid. By subtracting equation (V.4) from equation (V.3), by rearranging, by adding the restriction to incompressible fluids, and by defining U^:V. v one obtains (V.5) In obtaining this equation of course, the linearity of the operations Nf°(..} ^C. 1 ) and — C ,N ) has been used. In order to express equation (V.5) in a more compact form, the operation, J5, is defined by *The requirements defining the calss of admissible constitutive relations is presently an open research question (see, e.g., Truesdell and Toupin, 1960; Truesdell, 1960; Eringen, 1962, and Noll, 1959).

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V.9 ^vt./rv^.^jv^Cv^^^^v*,..^^..^^,.^^,,) (V.6) Equation (V.5) then becomes iu. * t5t + **v«i + .Y»7u+u.yv--wj 3t -^ + V*3, , tf (f | .|!) (V.7) This equation describes the velocity field of the difference motion. It is a fundamental equation from which many other useful relationships may be derived, such as: (1) the equation describing the local variance of the difference velocity field (called the kinetic energy of the difference motion), which is obtained by taking scalar product of u and equation (V.7), that is f ftfctf m m + rKiify s «&,] (V.8) where LA — LAo Uand D -— Nv+^V = the deformation rate ten ^ -^ of basxc velocity field, ensor (2) the equation describing the vorticity field (an example of the operator q(.) of the preceding chapter) of the difference motion, which may be obtained by operating on both sides of equation X (V.7) withy^.), that is,

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V.10 *f 2 *<£*-£> " (V.9) where \aTV * u. = vorticity field of difference motion and -£~V * V = vorticity field of basic motion, and, finally, (3) the equation describing the local variance of the vorticity field associated with the difference motion, which may be derived by taking the scalar product of w with equation (V.9), that is +^4~Y*c^-f)j (v 10) where £> y u 4 ^ I ^a 4C\}u') 1 \} ' 0ther useful forms °f the last two equations may be obtained from the use of the identity X v r V^(7«S^=VDepending upon various properties associated with the constitutive operation, S , and properties of its arguments, equations (V.7), (V.8), (V.9), and (V.10) may be placed in more explicit forms. For example, ... * (i) if S is symmetric, then S is symmetric (ii) if S is a linear operation in all its arguments, then

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V.ll y. and -YxAfe),...> and (iii) if _S, is a linear operation and all its arguments are linear operations, then x .* The forms which these equations take for various properties of the operator S(.) are displayed in Table V.3. The key is given in Table V.2. The equations developed so far in this chapter have been of the differential type that is, giving information at individual points in our region of interest, \\ , of physical space. Alternatively, these equations may be cast in an integral formulation, thereby defining types of global properties of the system which may be viewed as coarse-grained averages of this local information; they are not to be confused with the notion of the global state of the system. a sense, therefore, some of the local information is bartered for an ability to deal with the region of interest as a whole. Other immediate advantages of an integral formulation are: (1) the integral operation is bounded and continuous by definition (cf , Shilov and Gurevich, 1966) , and (2) the boundary conditions which naturally go with the differential equation formulation may be introduced directly into the basic equations. This boundary-condition advantage depends upon the existence of

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V.12

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V.13 TABLE V.2 (Continued) © r R <©> © i„<©> 3> I^ 9 I^ R <^o KrY > d£ ® **<*•£ -I > ® T *<^*L > 3R' -'an ^ I R <©> © I R <©> © I R <@> ® T R <^-({,-i)> la) i R ^R

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V.14 TABLE V.2 (Continued) © r'UV» + ±_V*p] i 22 © i b < f <*V[v v * + ^ \u^]> x ^r4^U> I^V^

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o •H 4J QJ V.15 u CD a o u go co cu •H M o CO$J 0) CO CO a) S co oo a) a) CO u to a) C to 05 •ri a 0 •u ri tO to u QJ W t-{ (X U COil O -H a] •H a) to co h-J CO Z o M I w § H h-l BJ u CO UJ Q Pn O O w CO c o to 3 a w * <7< S f3\ i U ^ & /-o B8 »o + ley -t^ © © I) % tV'A'

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V.16 u QJ PO u
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V.17 a volume surface integral theorem of the form where I D is the integral operation associated with a volume measure defined on "R^ and L. is an integral operation associated with a surface measure defined on <^T^ related to the volume measure such that the integral theorem is valid (cf . Chapter II and Appendix B) . The results of operating on both sides of equations (V.7), (V.8), (V.9), and (V.10), making use of the above integral theorem whereever possible, and considering various possible boundary conditions and properties of the constitutive operator are summarized in Tables (V.4), (V.5), (V.6), and (V.7) where the key is given in Table V.2. V.3. Form of Governing Equations for Particular Classes of Fluids The equations of the preceding section assumed no particular rheological model for the fluid; rather they were based upon specific properties of a constitutive operator. In this section the forms assumed by these equations for familiar models of fluid behavior will be developed based upon combinations of properties possessed by a particular _S_ <^. The classes of fluid models to be considered are: (i) newtonian, (ii) stokesian, (iii) linear finite viscoelastic, and (iv) simple fluids.

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V.18 <: CO O l-l H H M C_> O ,-! W > o co O M 3 o 3 o w H 3 0) O U &4 m e o CO 3 w -e-i Q< ii 4-1 II CJ -H en i-l a o cd cr 0) CO -l cd CD s cd en 3 e 3 cd CO 4-1 a cd 3 o cd M cfl O 0) w cd 0) a. o A T^U

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V.19 u-i W -J 2: o M H w H M u M H 8 > O En o M u o M P* CO a o cO 3 T W /ff^ + l) H Ql Qi. dl __ ; 5 >> o M + ll N/ M

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V.20 vD eq w u 2: 3 H M u o -I w > a! o o o 3 o H 3 u CU a. o u a o cO 3 a* W /^, T> 3$ a . ,-T> ^i

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V.21 u u u 0) ex o u P-, C> 01 iJ

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V.22 V.3.a Newtonian Fluids For the class of newtonian fluids the constitutive operator, S^<\) , is defined by where J-)„, . is the symmetric part of V V . This particular S«^' ) nas the following properties: (i) it is a symmetric dyadic field , (ii) it is a linear operation in its one argument, ^D vy/ » (iii) its one argument "D_ v is also a linear operator in v. If these three properties are used in conjunction with Tables V.3 to V.7 and the key in Table V.2, the following equations for this class of fluids may be written by inspection: (N.l) Difference Velocity (differential formulation) f H © © © -S> +®+ 2 ^°l^ (V U) where the last term may be alternatively written as UL ^ tir , (N.2) Difference Vorticity (differential formulation) f&= ©+ © + @+®+®+©+2hZ a Rv»r (V.12) where the last term may be alternatively written as LiAw , (N.3) Difference Velocity Variance (differential formulation)

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V.23 (N.4) Difference Vorticity Variance (differential formulation) f^V 2 >--@-@)+@^®-@ + ©-2^^g v ^ (vi4) (N.5) Difference Velocity (integral formulation) l R = -@h©-@-e>© + 2 t x hR<^°l„ I ) (v#15) (N.6) Difference Vorticity (integral formulation) (N.7) Difference Velocity Variance (integral formulation) x^i(yu*);> , _@) -@ -©-© +@> + ^ftfe^ls^ii )l >-I„^?^0 (V.17 \D f\ ^_ — / <)R (N.8) Difference Vorticity Variance (integral formulation) The viscous character of the fluid manifests itself explicitly in equations (V.17) and (V.18) in two places, namely, in a surface integral term and in a volume integral term. The integrand in the volume integrals is always positive, and therefore this integral term always contributes to a damping out of any disturbance. On the other hand the surface integral terms do not have

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V.2A any such convenient property, as well they should not, for otherwise physical flows would have an entirely "unphysical" character, however mathematically facile they might be. Consequently, within the bulk of the fluid viscous forces do tend to damp disturbances while at the boundaries of y\ , viscous forces may actually cause an instability, the reasoning which invalidated Rayleigh's (1912) argument about the centrality of an inviscid stability analysis. Another conclusion may be drawn from equation (V.17), namely, that it reduces to the basic equation of Serrin's (1959) analysis. Indeed, if the stability problem where +-^ s-Cand \x | = Q , then equation (V.17) reduces to The use of the volume surface integral theorem and the condition IX \-0 leads directly to Serrin's fundamental equation. V.3.b. Stokesian Fluids with Constant Coefficients For the class of stokesian fluids with constant coefficients, the operator S, <"•> is defined by |1
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V.25 (iii) its only argument Dy V is a symmetric dyadic field and a linear operation in V . From these properties of S ^-) and its argument, the following relationships may be easily derived, and If these relationships are used in conjunction with Tables V,3 to V.7 and the key given in Table V.2, the following equations may be written by inspection, namely, (5.1) Difference Velocity (differential formulation) + oC z ^^u°^w^ v ,^ v J (V ' 19) (5.2) Difference Vorticity (differential formulation) ^^Cy^P °D +£> "D -f-"D„ * x ,+ ^ ESva g^ + ^^°5^ + gw-g v J5

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V.26 (V.21) (S.4) Difference Vorticity Variance (differential formulation) 2 ^^^W *£w^>^ CV.22) (S.5) Difference Velocity (integral formulation) (S.6) Difference Vorticity (integral formulation) + ^^l> + ^^B^-i^fiU+SuJ W J)(V . 24) (S.7) Difference Velocity Variance (integral formulation) r *<& W»-© -©-© -©+@^ R

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V.27 (S.8) Difference Vorticity Variance (integral formulation) These equations are obviously much more complex than those for the newtonian fluid, as one would expect since the constitutive operator is now nonlinear. However, one of the terms makes this ^-f^*^ nonlinear does not affect the stability namely the term with the coefficient Cx C . Because of existing evidence (cf. Truesdell, 1952) that °^ z is always greater than or equal to zero, the other nonlinear term will tend to both damp and amplify disturbances. Unlike the ^^ term, however, it will have this dual effect both within the bulk of the fluid and on the boundary of +"S . Therefore, it is the net effect within the bulk versus the net effect on the boundary rather than just the net effect in the bulk versus on the boundary which must be considered in relating the effect of a non-linear fluid of this nature. In short, in a stokesian fluid there are more ways for an instability to grow and be maintained, and there are more ways for it to be damped as well, so that questions of stability for stokesian fluids are much more

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V.28 complicated for their viscous counterparts, nawtonian fluids. V.3.c Finite Linear Viscoelastic Fluids In a delightfully lucid paper, Coleman and Noll (1961) have provided a firmer foundation for the linear theories of viscoelasticity. Basically, they attempt to provide a proper mathematical of the physically appealing (or intuitive) model of a linear viscoelastic fluid. Their approach is solidly centered upon three notions: (i) the decaying "memory" of the material for past states, and thus, the tendency of the material to be more influenced by its immediate past, (ii) a Hilberc space of histories in which the "closeness" of two histories is measured (by means of a norm) in terms of the values of a memory functional which is based on (i) , and (iii) the smoothness of the constitutive functionals (whose value at any time t is the stress dyadic field) as evidenced by the requirement that these constitutive operators be Frechet-dif f erentiable in the histories. Much as the constitutive assumption of a newtonian fluid may be viewed either as a first order approximation to that of a stokesian fluid or as a definition of a class of ideal materials, so also may the theory of finite linear viscoelasticity be viewed either as a first-order approximation to the theory of simple fluids or as the definition of a class of ideal materials. In particular, by allowing for finite deformations which nevertheless occur slowly (and thus, are small in the sense of the norm mentioned above), yet not restricting themselves both to slow motions and

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V.29 infinitesimal deformations, they arrived at a first order approximation to a simple fluid which may alternatively be viewed as a definition of a class of ideal materials finite linear viscoelastic fluids. The theory of this class of materials based upon this approximation contains the classical infinitesimal theory of viscoelasticity as a subclass, while at the same time being contained in the more general theory of simple fluids. A finite linear viscoelastic fluid is defined as one for which the constitutive operator is of the form o in which tt-t-'s) ± s called the influence function which gives added weight to the more recent history, as measured by "|_w-t ; '£?) ~ J in which C , (_X;-t -S ) is the right Cauchy-Green finite deformation dyadic (with the unit dyadic subtracted, it is called the right Cauchy-Green strain dyadic) . With this brief discussion of the model, let us now list some of the properties of the constitutive operator o j , namely, (i) it is a symmetric dyadic field because bothQt an d 1 are symmetric tensor fields, (ii) it is a continuous linear operation in the space of all histories ^--t C x ; "^ _ s y tt. because it is an integral operation, (iii) its argument Q t -3= is a nonlinear operation in the relative deformation dyadic, p \/ h -h ^ n wn: >c h the

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V.30 gradient is with respect to the location of material points in the reference configuration taken to be the current configuration for the relative deformation gradient: From property (ii) above, it is easily seen that 4-* g7 < y t t &) = ^ -f >= f /xGOfeOj t-fi)J^. tS )J@ where JCX;^-s;A d. CK",*"^-* _ -t has been introduced for the sake of compactness. If this relationship is used in conjunction with Tables V.3 to V.7 and the key given in Table V.2, the following equations may be written by inspection: (FL.l) Difference Velocity (differential formulation) (FL.2) Difference Vorticity (differential formulation) dt ° (V.28) (FL.3) Difference Velocity Variance (differential formulation) "&A V ^; -© © + ^ +<© -€) +@ T-v&fe, ?yyc*$&pafc&&-2)]B (v ' 29)

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V.31 (FL.4) Difference Vorticity Variance (differential formulation) _^ (FL.5) Difference Velocity (integral formulation (V.30) + VT: T\. (V.31) (FL.6) Difference Vorticity (integral formulation) (V.32) (FL.7) Difference Velocity Variance (integral formulation) -p°° _ -nr^ (V.34)

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V.32 These equations illustrate how fluids with a memory even a linear memory add a new dimension to stability theory. In addition to the consequences of having a "nonlinear" fluid at each instant of time, the consequences of what the perturbation did in the past at certain places in physical space are directly felt at any present instant. Consequently, the history of the perturbed flow assumes a position of equal stature and importance to the magnitude of the perturbed flow. V.3.d. Simple Fluids Many models of fluids exhibiting "memory" effects have been proposed, the most encompassing one which is yet quite simple conceptually is that of a simple fluid (see, e.g., Coleman, Markovitz, Noll, 1966). A simple fluid depends only upon the history of the strain in a small neighborhood of each material point, although its "memory" can extend backwards indefinitely. Thus, neighborhoods of different material points differ in their response only in so far as their histories may differ. Only incompressible simple fluids have been dealt with in any depth, but as we solve no explicit problems or examine no explicit stability questions, it behooves us, without cost, to remain fully general for possible future usefulness. For an incompressible simple fluid, one can say that if the strain of all past configurations relative to the present one are known, then the extra stress, j , is essentially determined . The constitutive operator for a simple fluid is written in

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V.33 the form S~c© in which ^hj <^«> is a functional over the space of all possible right Cauchy-Green deformation dyadics at (t s) with respect to the configuration at t. Furthermore Jj K})> is supposed to be a symmetric dyadic field and to depend continuously upon the past history of the right Cauchy-Green deformation dyadic. We note also that the right Cauchy-Green dyadic is itself a symmetric dyadic field. If these properties are used in conjunction with Tables V.3 to V.7 and the key given in Table V.2, the following equations may be written by inspection: (SF.l) Difference Velocity (differential formulation) (SF.2) Difference Vorticity (differential formulation) 4 f jf -(§>©+© +@ + ©^©4(jM <&)}7^%}] T (v . 36) (SF.3) Difference Velocity Variance (differential formulation) (V.37)

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V.34 (SF.4) Difference Vorticity Variance (differential formulation) -t*Di< &&)-£<<* ^>llV^ (V . 38) (SF.5) Difference Velocity (integral formulation) + X lR<^° ClK^O#>^<£ C ^/> (V-39) (SF.6) Difference Vorticity (integral formulation) <7"fc (SF.7) Difference Velocity Variance (integral formulation) (SF.8) Difference Vorticity Variance (integral formulation) + i °ff^61^ ^> f -j&<€c]J : Vtr) (V.42) Upon reviewing the equations of this section, it becomes apparent that the more restrictive the class of materials being

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V.35 considered, the more concrete and explicit are the forms assumed by the basic equations, in the obvious sense that classes of operators become specific operators: nonlinear operators become linear operators, etc.; for example, the operator ZJ^'s for simple fluids becomes an integral operator for finite linear viscoelastic fluids. Of course, more restrictive results also accompany more restrictive classes of fluids. Similarly, although the equations apply to all solenoidal basic velocity fields, advantages may also be gained by the consideration of only a special class of basic flows, namely, the class of parallel flows. V.4. Equations Governing Difference Fields for Parallel Flows The class of parallel flows embraces a sufficiently wide range of stability problems to be of interest, yet the governing equations are significantly more tractable than the equations for more general basic flows. The governing equations for this class of flows may be derived by substituting the quantities of Table V.l directly into the equations of the previous section. Once these general governing equations have been derived, this is a straightforward substitution; consequently, we will only explicitly present the differential formulation for the difference velocity and the integral formulation for the difference velocity variance and difference vorticity variance. Again, the explicit formulations will be derived for the four classes of fluids: (i) newtonian,

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V.36 (ii) stokesian, (iii) finite linear viscoelastic, and (iv) simple. V.4.a. Newtonian Fluids Upon using the equations of section (V.3.a.) and Table (V.l), the following equations may be directly derived, (N.l) Difference Velocity (differential formulation) (N.2) Difference Velocity Variance (differential formula(N.3) Difference Vorticity Variance (differential formulation) (N.4) Difference Velocity (integral formulation) (N. ) Difference Velocity Variance (integral formulation)

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V.37 X. (N.8) Difference Vorticity Variance (integral formulation) V.4.b. Stokesian Fluids a Jpu > vu. This following equation may be derived from Table (V.l) and equations (V.19), (V.25), and (V.26). . (S.l) Difference Velocity (differential formulation) (S.7) Difference Velocity Variance (integral formulation) Vf^c4^;>-x R -x R /r > -x,<%4 cgwj va ^>

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V.38 V.4.c. Finite Linear Viscoelastic Fluids The following equations may be derived for Table V.l and equations (V.27), (V.33), and (V.34). It will also be convenient to define the following quantities A 4^= and °1 o H f °1 3 ! ! °M 3 Then one obtains (FL.l) Difference Velocity (differential formulation) (FL.7) Difference Velocity Variance (integral formulation)

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V.39 (FL.8) Difference Vorticity (integral formulation) <>£ -4-sj ^fftih 1 where in the above equations r 2 fc "7 4 -V'C^J^^XX; J o v / ^7

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V.40 and V.4.d. Simple Fluids " From Table (V.l) and equations (V.35), (V.41), and (V.42), the following equations may be derived. (SF.l) Difference Velocity (Differential formulation) (SF.7) Difference Velocity Variance (integral formulation) (SF.8) Difference Vorticity Variance (integral formulation) +a^3) -(ik) + (f 27 ) -(^) ~(p1) Mfeo, These equations applying to simple fluids may alternately be written in terms of the material functions X. CK),
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V.41 -f«i*(t-s^C«)^fr#5c*)
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BIBLIOGRAPHY Coleman, B., Markovitz, H., and Noll, W. I966, Viscometric Flows of Non -Newtonian Fluids , Springer Verlag, Inc., N.Y. Coleman, B. and Noll, W. 1961, Foundations of linear viscoelasticity, Rev. Mod. Phys. , 33, 239. Ericksen, J. L. 1956, Overdetermination of the speed in rectilinear motion of non-newtonian fluids, Quar . Appl . Math . , 1^ , 318. Eringen, A. C. 1962, Nonlinear Theory of Continuous Media , McGrawHill Co., N.Y. Joseph, D. D. 19655 On the stability of the Boussinesq equations, Arch. Rat. Mech. Anal. , 9, 59. Noll, W. 1959? The foundations of classical mechanics in light of recent advances in continuum mechanics, in Axiomatic Methods , Nordhoff, Ltd., Delft, Netherlands. Rayleigh, Lord 1912, Further remarks on the stability of viscous fluid motions, Scientific Papers , 6, 226. Serrin, J. 1959? On the stability of viscous fluid motions, Arch. Rat. Mech. Anal. , 3, 1. Truesdell, C. A. 1952, The mechanical foundations of elasticity and fluid mechanics, J. Rat. Mech. Anal. , 1, 1. Truesdell, C. A. i960, The Principles of Continuum Mechanics , Colloquium Lectures in Pure and Applied Science, no. 5? Socony Mobil Oil Co., Dallas, Tex. Truesdell, C. A. and Toupin, R. A. i960, Classical field theories, in Handbuck der Physik , IIl/l , Springer Verlag, Berlin. V.U2

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APPENDICES

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APPENDIX A ON THE PHYSICAL INTERPRETATIONS OF MATHEMATICAL STABILITY The centrality of an appropriate selection of both the state space and the topology defined on that state space cannot be overemphasized. The selection of a set of instantaneous states the state space involves more than just a decision as to the types of mathematical descriptions of the state of a physical system which are admissible; it involves also a decision as to the mechanisms that are significant in the physical system under study. Without a topology on this state space e.g., one induced by a concept of "closeness" as defined by a metric it is an amorphous set of rather limited utility in analysis. Thus, the selection of a topology is also more than just a matter of an acceptable physical notion of "closeness" or, on the other hand, only an acceptable mathematical definition; it is both, and, moreover, the selection of a particular state space and a particular topology should always be done so that the combination of the two has properties which are useful in analysis, e.g., continuity of algebraic operations, completeness, etc. Likewise, the delicate and intimate relationship between the particular selection of this combination and the physical interpretations of the various mathematical notions of stability also can not be overemphasized. In this brief appendix we intend to comment on this relationship in three ways: (i) by providing concrete illustrations (see, Table A.l) of the general comments made in Section II. 3 about the roles played by the state space, the topology, A.l

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A. 2 w J M <3 H H pq < H CO Pn O CO H PH W U 3 o c_> Q CO W l-l O O o P-, o H CO w c_> o P-1 CO w H a o u M

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A. 3 OJ CJ a c o o <4-l o en u o

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A. 4 o» 3 3 3 o u w J CD u s 0) oo M 0) o o to s o Pn a)

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A. 5 T3 QJ 3 C •rl 4-1 C o u w < H 04 o c qj ac u •v > c o u

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A. 6 01 o e 0) M U QJ > c o u x) m 0) o 3 co •h g 4-1 r-4 c o O Pn u w J QJ 60 I >-4 CD cd rH iJ W 01 rH X rH c c M 4-4 QJ = 06 y-i i

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A. 7 x) OJ 3 3 H 4-1 c o u 4-1

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A. 8 TJ OJ 2) 3 c o w r-1 < QJ O c 0) 00 H 0) > c o o l+H o CO e o QJ oo i H QJ cfl rH CD rH 42 .h 4-i 3 c M-l I O 0) rH : w •H 3 H CT rH W 3 = S

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A. 9 CD cj c CD M Q) > c o T3 0) U-i 3 O C O fn w XI H tu

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A. 10 -a 3 3 C c o o w 1-1 m c o u MH o CO £ u o Pn 0)

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A. 11 and the numerous mathematical notions of stability for particular function spaces, (ii) by accentuating the close relationship between the selection of a state space with a topology and the corresponding physical interpretations ascribed to various mathematical notions of stability by considering the meaning of various concepts of mathematical stability for the same metric, and (iii) by accentuating the manner in which physical intuition may be used in a stability analysis by indicating how various physical interpretations may be ascribed to the same mathematical notion of stability for the same physical system thus, loosely speaking, illustrating how physical intuition is translated into mathematical language. These latter two points are considered, naturally, at one and the same time concomitantly. A concrete discussion necessarily involves explicitly-defined topologies usually induced by a norm as well as a particular class of physical systems. For purposes of illustration, let us consider the class of physical systems whose state is specified by a solenoidal velocity field, that is, Before any physical interpretations may be given, however, we must also specify a topology and the set of instantaneous states as well as the state variables that is, the function space to which v(,.) belongs. This involves, of course, the specification of the set of functions to which y^. ) is a member as well as a topology on that set. In selecting a function space, a basic consideration is the form of

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A. 12 the general balance equation being used (that is, the integral, differential, or difference equation formulation). Thus, for the sake of brevity, we have decided to select only a generalization of the familiar set (£_, L'f?. J with, however, a variety of possible normed topologies. We begin by considering V(.) belonging to (C f_^~| . which we define as the set of all continuous functions from j^C ~E t0 C. , whose component functions have continuous derivatives up to the second order. We now consider possible norms over this set and the resulting physical interpretations: (A) let VC.1 £ C [ 1?1 with the norm defined as || VCMI = ^>4>{ 'Ifc^U-g } (A.l) where Wv^f v W „ V0 With this norm, the stability of the null state means that if the magnitude difference velocity at each point of f\ is sufficiently small at some initial instant, then the maximum magnitude of the difference velocity will always remain bounded by some finite real number. On the other hand, asymptotic stability with respect to some set of '.litial disturbances means that not only does the maximum magnitude of the difference velocity corresponding to each member of their initial set remain bounded by a finite real number, but it actually tends to zero as time increases indefinitely; thus, we also have that v(.i— >-Q Thus, each of the component functions (with respect to some is) belongs to the space (C^C 12"] of Table A.l.

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A. 13 as time increases indefinitely for each of these difference velocity fields. That the null state of some physical system is equiasymptotically stable may then be interpreted as the quality that for a set initial disturbances (a set of functions such that the magnitude of the difference velocity at all points of "K is l ess than some finite real number) the corresponding difference velocity fields will, after the time interval X 4I , all conform to one mode of approach to the null state that is, approach the null state uniformly. The "in the large" types of stabilities mean that we have the pertinent type of stability when the initial disturbance is any element of
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A. 14 that is, on the quantities ] XD<*,ty • y^*,-0 and Thus, if a basic state of a physical system possesses one of the types of stability with respect to the norm defined in equation (A.l), then it does so with no restriction that the global kinetic energy or global stress power be initially small . It will, therefore, be of interest to consider interpretations with a norm defined over the same set of functions but with a term involving the gradient of the initial difference velocity field. In particular, let us now define the norm by where lNvO0>f= NWkxIvW) Thus, the stability of a basic state means that if the magnitude of both the initial difference velocity and the gradient of initial difference velocity at each point of \\_^ is sufficiently small, then the sum of these quantities is always bounded by some finite real number that is, both the quantities must always remain bounded. On the other hand, if the basic state is asymptotically stable then the sum of these quantities actually approaches zero as time grows indefinitely. Because both the magnitudes are positive, the latter implies that not

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A. 15 only does the difference velocity field tends to the zero function but so also does the gradient of the difference velocity field. The significance of this may be seen by noting that Mv is often the most important kinematical variable for a fluid in fact, for a newtonian fluid the quantity is the local viscous dissipation up to a multiplicative factor. Although a small norm as defined by equation (A. 2) does not necessarily imply a small global stress power, for many classes of fluids it does imply that the global stress power is finite (if the region of interest,^ , is compact). In fact it does imply for asymptotic stability that system eventually reaches a state where the global stress power of the difference velocity field is zero. However, we may also define a norm over the class of functions (L L'fH 2 '^ ^ ^ytV (a. 3) ft Thus, for many classes of fluids (e.g., newtonian fluids) a small norm as defined by equation (A. 3) implies a small global stress power. Thus, stability of a basic state with respect to this norm is to be interpreted as: if the value of the averaged magnitude of the initial velocity gradient field is sufficiently small (small deformation rates) then the difference velocity field will always be such that the averaged magnitude of its gradient is bounded by a finite number. Asymptotic

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A. 16 stability of a basic state requires, in addition, that this averaged magnitude actually approaches zero as time grows indefinitely; this is not to say that the perturbed velocity field actually approaches the basic velocity field but rather that it approaches some velocity field where the averaged magnitude of its associated gradient field is the same as averaged magnitude of the gradient of the basic velocity field. With this norm, the property that the null state is equiasymptotically stable with respect to some set of initial disturbances (a set such that the averaged magnitude of the velocity gradient field is sufficiently close to that of the basic flow) means that all the corresponding difference velocity fields will, after some time interval "t+ I , all conform to one mode of approach to the equivalence class of velocity fields which have the same averaged magnitude of gradient field as the basic velocity field that is, a uniform convergence to this class. If the global kinetic energy rather than the averaged velocity gradient field is of particular significance, then it may be appropriate to define a norm as over the set of functions (C C^l • Then the various types of stability will have interpretations analogous to those of the previous paragraph with the words "averaged magnitude of the velocity gradient" replaced by "global kinetic energy." Furthermore, if we had taken the original set of functions as ~fl ["\£l (the set of all vector-valued functions whose component functions and all their

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A. 17 derivatives to second order are Lebesgue square integrable) instead of
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APPENDIX B ON THE CALCULUS IN ABSTRACT SPACES In an effort to formulate a unified approach to stability theory, it is not simply convenient to employ a function space to represent the state of a physical system, it is necessary if the system has spatial as well as temporal variations (Zubov, 1961; Hahn, 1963, 1967). Thus, if the instantaneous state of a physical system may not be characterized by a set of numbers (e.g., a point in a finite-dimensional Euclidean space, E ), but rather is characterized 3 by a set of functions defined over a region of physical space (an E ) , then the instantaneous state must be represented as a vector-valued function defined over a region of physical space, that is, as a point in an appropriate function space. In extending the concept of a Liapunov functional from finite-dimensional state spaces (E 's) to infinite-dimensional spaces (e.g., function spaces; see, e.g. Chapter III; Zubov, 1961; Krasovskii, 1963) there seems to be only a slight conceptual change. In extending Liapunov' s method itself, however, it becomes apparent that classical differentiation operations do not suffice because the independent variables of the Liapunov functional the state variables are no longer elements of a finite-dimensional space. Rather they are elements of an infinite-dimensional space. Thus there arises, in the present context, the need for the notion of a differential of an abstract operator abstract in the sense that the * We reiterate that a theory may be a unified one without it being a unifying one (cf. our previous remarks in Chapter I). B.l

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B.2 range as well as the domain of the operator are abstract spaces. Likewise, the solution of operator equations for Liapunov operators defined on an abstract space, as discussed in Chapter IV, will depend upon a knowledge of the various methods of solving operator equations; each method, in turn, is based upon the calculus in abstract spaces. By the calculus in abstract spaces we mean a generalization of the classical calculus from the space of real numbers or from finitedimensional Euclidean spaces (E n 's) to abstract, in general, infinitedimensional, spaces. Such a generalization can take either of two tacks: (i) one may take the classical definition but generalize the class of objects to which the definition is applied, or (ii) one may focus upon certain desirable properties following from the classical definition and seek to construct a definition for the more general class of objects which retains these desired properties. In the first approach, one invariably sacrifices some of the properties similar to those following directly from the definition in the classical case, but, on the other hand, one insures that the definition will reduce to the classical one if the class of objects is that of classical analysis. In the second approach, which is often called axiomatic, one begins with the abstraction of a number of desirable properties and then attempts to formulate a definition for the more general class of objects which possesses the abstracted properties. There is, loosely speaking, a restriction in the axiomatic approach in that the generalized definition reduces to the classical definition if the class of objects is that of classical analysis.

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B.3 B.l. Differentiation Classical analysis emerged as a branch of mathematics with the introduction of the concepts of differentiation and integration both of which are examples of limit processes. One would therefore expect that modern analysis, in an effort to lay bare the inner foundations and the underlying ideas behind the techniques of classical analysis, would be intimately concerned with the generalization of the concepts of a limit and of limiting operations and indeed it is. It is thus appropriate to preface our discussion of abstract differentiation, integration, and methods of solution of operator equations by a discussion of limits, convergence and continuity. Generally speaking, three modes of convergence of elements may be distinguished, namely, (i) weak convergence, (ii) strong convergence, and (iii) uniform convergence. Following Vainberg (1964) the definitions will be stated for situations where normed spaces are involved, although analogous definitions exist for more general abstract spaces. The concept of weak convergence of elements of a normed linear space,3t , is most naturally defined by means of a definition of the weak convergence of a sequence of linear functionals, {f } , belonging to the algebraic dual space ,3E. (the space of all linear functionals on 3c ) . Definition B.l.l A sequence of linear functionals, {f ], where f^e$L, is said to be weakly convergent to the functional, f, if for every

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B.4 e>03n (e,x) such that If (x) f(x)|n (x,e), for each o n = o xe3£. As an aside, from this definition, the concept of the weak convergence of elements of a normed space is obtained by considering the weak convergence of a sequence of functionals, {g }, defined by g n (f)A_f(x n ),fe3€* The weak convergence of the functionals defined in the above manner implies that |f(x ) f(x)|n (x,e) •fe3E* n =o The concept of weak convergence of a sequence of linear functionals and, therefore, also that of weak convergence of a sequence of elements employs the familiar definition of convergence in the set of real numbers in terms of the absolute value, whereas the concept of strong convergence is phrased in terms of a generalization of the absolute value namely, the norm. The definition of strong convergence is as follows: Definition B.1.2 A sequence of operators, {t } where T :3C~ *AJ an ^ ,\X are normed linear spaces, is said to strongly convergent to an operator, T if given an e>0 3 an n (x,e) such that o IT T | n (e,x) for each xe3£ n ' >m = o The similarity of the above definitions to pointwise convergence in classical analysis is evident, and the next higher level of generalization, to uniform convergence, is an equally logical step.

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B.5 Definition B.1.3 A sequence of operators, JT }, where T :^£ -"-}_] and x. M are normed spaces, is said to be uniformly convergent to an operator, T, if given an e>0a an n (e) such that o I It T I n (e) for all xe3£ n 'u = o The difference between the last two definitions is precisely that of the classical calculus: the n in Definition B.1.3 is universal: o that is, it is independent of any particular x. It follows therefore that the last definition may alternatively be stated as, given an e>0 there exists an n (e) such that o T " T l I r-v >.i n ( £ ) n [% ,\j] = o where |T|| ASupt ||T|| } L3t ' l i J -|ixii n in the definitions as is shown below: — o Definition B.1.4 An operator, T, is said to be weakly continuous at x e 3c. if , for any sequence x -> x (weakly) the sequence JT} x -»• x (weakly) is required in the definitions because otherwise weak continuity might be stronger that strong continuity (see e.g., Vainberg, 1964).

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B.6 converges weakly to T. Definition B.1.5 An operator, T, is said to be strongly continuous at x elf if, for any sequence x * x (weakly), the sequence {T} o no l n converges strongly to T. And finally, Definition B.1.6 An operator, T, is said to be uniformly continuous on Jk. if, given an e>0, there exists a 6(e)>0 such that I |x x I I <5 (e) implies |T T I L. for all xe>C. II n ' ' ' ' n 'y There are a number of theorems relating these concepts of continuity but rather than dwell on these", we proceed to a discussion of abstract differentiation. For reasons of comparison between the two approaches to generalization, let us recall some of the more important properties of the classical differential: (i) the differential, 6f(x;6x), is a first order approximation to f(x + 6x) f (x) as 6x-*0, (ii) the differential is continuous in 6x, (iii) the differential is linear in 6x, (iv) the differential satisfies a rule for composite differentiation: thus, if g(x) = f((x)) then 6g(x;6x) = 6f(y;6y) * The interested reader is referred to Vainberg (1964) for a summary of these relations.

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B.7 where y = (j>(x),6y = 6<()(x;6x), (v) the differential is such that if a function is differentiable at a point, then the function is continuous at that point. B.l.a Some Concepts of Abstract Differentiation With these five properties at hand, let us take the first approach to generalizing a classical differential and consider which properties are lost in doing so. As a starting point, let us use the definition of a directional differential (cf. Buck, 1956, p. 180) as a basis and by analogy define: Definition B.1.7 The Gateaux differential (or, for brevity, the (G) -differential) of the operator FiK+liat x in the direction ( with the increment) h is given by or 6FA o = [F F] (B.l) a o o i'm a+0 if this limit exists for all heX both ck and|i are considered to be normed 1: with 3£ ) aormed linear spaces and x ,he± while ae ji (the field associated ° 3t Graves (193 5) has pointed out that an equivalent way of expressing Definition B.1.7 is 6F 4~ {F } I (B.2) o da L o ' ' „ a=0 Equation (B.2) is the form usually encountered in the calculus of

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B.8 variations, and in fact, is the usual starting point for the famous Euler-Lagrange equations. Because of this link with the calculus of variations, the (G)-dif f erential is often referred to as the first variation of F at the point x in the direction h. Despite this correspondence with classical analysis, the (G)-differential does not provide us with a generalization possessing all the desirable properties of the classical differential. This should not be totally unexpected, for, even in the case of functions of a finite number of variables, property (v) is not satisfied without additional conditions (cf., e.g., Buck, 1956, p. 184). It is found, moreover, that none of the desirable properties follow directly from Definition B.1.7 in the general case. If the (G)-differential does possess property (iii) that is, if it is linear in h then a (G) -derivative may be defined by the relation ^=-\ a<5f ox 1 = o X o 6F in which the x in the denominator of — serves to indicate a ox ' X o differentiation with respect to that dummy variable. Thus, the derivative of F at x , -7— | , is an element of the space |jj3£ ,1J] that x The conditions for this are that F possess a (G)-diff erential at x such that it is continuous in the first argument at x o o uniformly in h.

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B.9 is, the space of all linear operators fromx. to\J which associates an element of the range space of F,\l, with each element of the domain space of F,3cAlternatively, it may be viewed as the valu e of an operator which associates an element of li_[3£.AJ] with a fixed x in the domain.* It is found that the (G) -derivative does not possess the desirable properties associated with the classical derivative a natural consequence of the constructive approach to generalization. On the other hand, if the second, or axiomatic, approach to generalization is taken, the abstract differential possesses all the desirable properties (i) to (v) . The abstract differential corresponding to this approach is referred to as the Frechet differential (or (F) -differential) and is defined as follows: Definition B.1.8 If at a point x elf we have F = H + w , where H o o o o o is linear and continuous in h, and if w< x ,h> 2 ^L= o INI „_im | jh| | then dF^x ;h>AH is called the (F)-dif f erential of the o = o operator F at x e3£ and w is called the remainder. Alternatively, ft The distinction between a derivative and a differential was made by Zorn (1946)

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B.10 the definition of the (F)-dif f erential may be written explicitly in terms of F F dF | L = 0, (B.3) o . o o jj lm | |h| | + given the existence of a continuous linear operator, dF over the increment he9£ . If the (G)-diff erential is linear in h and uniformly bounded for all h, then it is equal to the (F)-diff erential (cf. Vainberg, 1964). Another indication of the relation between these two types of differentials was provided by Zorn (1946) who showed that if the (G)-diff erential is homogeneous of degree one in h and is bounded in h, then 6F = dF o o when the spaces are normed linear spaces. Still more insight into the relationship between these differentials was provided by Graves (1935), who had shown earlier that the combination of equation (B.3) and the linearity of dF in h implies that dF is equivaJ o o lent to the quantity defined in equation (B.2) An (F) -derivative may be defined, in a method analogous to the "derivation" of the (G) -derivative from the (G)-diff erential: if dF is linear and uniformly continuous in h by the relation o 4^1 AdF dx 1 = o x o where — | is the value of an operation f rom J^ to |T [3£ ,\J] x o

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B.ll (the space of all linear and continuous operations from at*" (J) and, again, x indicates the dummy variable. The Frechet derivative, because of the requirement of uniform continuity in h, may be thought of as an all directional derivative that is, a generalization of gradient operation of classical analysis. Rothe (1953) has defined a gradient operation for functionals defined on Banach spaces by: Definition B.1.8 (Rothe, 1953) Let ^ be a Banach space and f<°> be a functional defined on 3£ • Suppose that the (F)-dif f erential of f is continuous in h; then df induces a bounded linear operation f rom 3c > 3t (algebraic dual space of^O for each x e9£. This induced linear bounded operation is referred to as the gradient of f at x . In a Hilbert space every continuous linear functional may be represented as an inner product, and hence the above definition may be expressed in a more familiar form, namely (G |h)Adf o ' = o where G^x > is the gradient operation on f at x . This concept of gradient operator has been extensively employed both in the method of steepest descent and in the existence theorems for certain functional equations (cf. Vainberg, 1964; Rothe, 1937, 1946, 1948, 1953) . The latter use arises from the natural generalization of potential functions to potential functionals. Thus, in classical vector analysis the well-known necessary and sufficient condition

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B.12 that a vector field, v, admit a potential is that the field be irrotational, (Curl \v\ = 0) , a condition that may be phrased X equivalently by requiring that Vv_ be a symmetric dyadic field. The necessary and sufficient condition in a Hilbert space is similar to the latter form of the condition in classical vector analysis; namely, if G<.> is an operator on a Hilbert space having an (F)differential which is continuous in its first argument and having the property that (dG|k) = (h|dG) then G<.> is the gradient of some functional (cf. Nashed, 1965). Further generalizations (in the sense of more general abstract spaces) of the operations presented here may be found in the literature, much of which appears in the bibliography. Those mentioned above have been the most widely used, but before closing our discussion of differentiation some of the other generalizations that have appeared are briefly mentioned. Michal (1939) defined a differential on a linear topological space, the topological groups of which are abelian with respect to the operations of addition and scalar multiplication. His differential, referred to as the Michal differential, reduces to the (F)differential when the particular linear topological space is a Banach space. This is, however, to be expected because Michal abstracted the approximation property (i) of differentials in his definition. Millsaps (1942, 1943) also abstracted the approximation principle, but he allowed the topological space to have non-abelian group

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B.13 operations. Again, by the abstraction of the approximation principle, Lasalle (1941) defined a differential for a linear algebraic system whose field was an arbitrary partially ordered commutative division ring rather than the usual field of real numbers. His generalization, unlike the other generalizations, is phrased in terms of pseudonorms . B.l.b Historical Note Abstract calculus as a distinct subject seems to have had its origin in Frechet's thesis in 1906 (cf, Frechet, 1906). His motivation most likely came from Volterra's concept of the gradient of a functional (see e.g., Volterra, 1959; Courant and Hilbert, 1952) as used in the calculus of variations. Later, Frechet (1910) used his notion of abstract differentiation to construct an abstract power series expansion. In applying Frechet's concepts to the construction of a theory of analytic functionals and functions of an infinitely many unknowns, Gateaux (1913) found that he could weaken the (F)-dif f erential somewhat and nevertheless obtain many significant . esults. From this period until the 1930 's this phase of abstract calculus underwent fewer developments while attention was focused on another phase-abstract integration. But, towards the latter part of the thirties, Michal and his students revived the prior interest in differentiation. Between 1936 to 1939, Michal developed the (M)differential and a theory of geometry in abstract spaces (see, e.g., Michal, 1939) the latter as a consequence of, or in conjunction with, his simultaneous interest in tensor analysis. During this

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B.14 period and the early forties, he published papers with Paxson on abstract differential, (Michal and Paxson, 1936) with Elcomn on a similar subject (Elconin and Michal, 1937), and with Clifford on a theory of analytic functions based upon a (G) -differential (Michal and Clifford, 1933). During the same period, Hyers (1941) published a paper which contained a generalization of the (F) -derivative and Zorn (1946) investigated the (F)-dif f erential and the generalization of the Taylor expansion. In the late forties and early fifties, Rothe (1937, 1946, 1948, 1953) and Vainberg (1964) began applying the concept of a gradient operator to Hammerstem 1 s nonlinear integral equation, Meanwhile, Cronin (1950) related the existence of critical points of a gradient operation to the existence of multiple solutions of elliptic differential equations. B.l.c Examples of Abstract Differentiation Example 1 . An illustration of a (G)-diff erential, a (F)-dlff erential, and the relationship of the definition of a norm. Consider the functional arising in the "simplest" problem in the Calculus of Variations, namely, b J = F(t,x,x')ftj (B.4) J a where F(.,.,.) is a continuous function of three variables and It is interesting to note that Liapunov also did some work in the early 1900 's on topological properties of integral equations, Liapunov (1906) .

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B.15 J thus denotes the value, a number, that the operator J<. assigns to a function x(.) belonging to C [a,b] , If one calculates the (G)-dif f erential of J<.> by equations (B.l) or (B.2), the following well-known result, in the notation of the calculus of variations, is obtained b J = J [g 6x + ||r 6x'] Q . (B.5) a Regardless of how the norm of the domain space of J<.> is chosen, this operation 6J has the properties (Courant and Hilbert, 1952) SJ = a6J because (6x') = (6x) ' and 6J = 6J + 6J because (6 X;L + 6x 2 )' = 6x| + 6x' 2 Therefore, 6j is an additive and homogeneous functional in 6x. If the chosen domain of J<.> is normed by the Chebyshev metric i.e., | |x| |A_Sup{ |x(t)} (.B.6) Ta,b] then 6J is not continuous in 6x because no restriction is placed upon the closeness of 6x' (cf., strong variations in the calculus of variations), Consequently, J-.> has a (G)-diff erential but no (F)-diff erential under the norm defined in equation (B.6),

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B.16 This is easily seen from the inequality 6J 6J is then continuous in 6x (corresponding to variations in the calculus of variations) . Both of the above norms are valid norms for the collection C [a,b], but only one leads to the existence of an (F) -differential. This simple example has another interesting 2 1 aspect. If C [a,b] rather than C [a,b] is chosen as the class of objects in the domain space of J<.>, integration by parts, b b b r h 3F . ,r-, 8F . d f dF \ t—1 jp6x ' = *? 6x " dTV a^") 6x W ' 5x a am may be used; and if the restriction that 6x(a) = 6x(b) = is imposed, then we obtain 6j^x;6x> = dJ in the norm defined by equation (B.6). Consequently, the (G)-diff erential may be changed to an (F)-dif ferential if the class of objects in the domain of J< . > is restricted to twice continuously dif f erentiable functions and the variations vanish at the end points.

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B.17 Example 2 . Reduction of an abstract derivative to the classical definition. In order to examine the meaning of the definition in equation (B.l), let X = IK * Then, F is an operation from n\ to some other space (J • Now if 6F is linear in 6x, then the (G) -derivative, 6F 'x , is a linear operator defined by, -r-\ < = 6F dx'x o If an arbitrary member of f]\ is denoted by x, then the general form of a member of |j_[ fj^ ,M] , the space of linear, bounded operators from ]f\ to U, (to which the derivative belongs) is yx, and in particular 6F, dx 'x = y o -x;y o e1/,xe3£. (B.9) 6F holds for the Gateaux derivative. If the operator -r-\ is normed dx 'x o in the usual manner then l | 6F , , \-r~\ = y *x = x y ll dx'x o "y "y o My MI-o" L j 6F, i i i i i i Hence the result hr~ 1 1 r n-il i i i y ..is obtained. If one 1 'dx'x o ' ' HUR ,(J ] '-'o ' 'VJ combines the general form given in equation (B.9) with the definition given in equation (B.l), one obtains F
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B.18 may be written as ill dx ' x = y~ = F F o o Ax lm Ax -> Now if Ax is identified with Ax, then the familiar definition 6F, dx 'x F F o o Ax (B.ll) _ — 1m Ax -> is obtained. Example 3. Three simple illustrations of gradient operations in Hilbert spaces . ii i i 2 (i) Calculation of Grad{||x|| } in a real Hilbert space. From the definition of a gradient operation (G|h) ' [f f is the gradient of f at x. Consequently, the first step is to form x + thl | 2 I Ixl I = 2t(x|h) + t 2 (h|h) From equation (B.12), one thus obtains G = Grad{ I Ixl I } = 2x x i i i i (B.13) (B.14) (ii) Calculation of Grad{ | |x| |}in a real Kilbert space. Because of the equality x + th| ' 1 x + thl I + x + th

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B.19 x + th t im t * ° 2t(x|h) + t 2 (h]h) t< | [x + th| | + | |x|.|) . im = (liVrr |h ) or G<| |x| |> = Gradl | |x| |} = Tl^TT (B.15) x I i x l I (iii) Calculation of Gradt (— A|x + th) -| (A^x>|x) = -| t([A + A*]-x>|h) + | t 2 (A^h>|h) is formed, then divided by t, and then the limit is taken as t * 0; that is, ^ {-| (A^x + th jx + th) -| (A|x)} = -| ([A + A*] |h) — im t * is calculated. Consequently, from the definition of the gradient operation one has G<-| (Ax|x)^ = Grad{-| (Ax|x)} = -| [A + A*]. (B.i6) In words, the gradient of one-half the quadratic form associated with the bounded, linear operator, A, evaluated at x is equal to the symmetric part of the operator acting upon x. In the special case where A is symmetric (or self-adjoint), this result may be written as Grad{ T (Ax | x) } = A-x> . x l

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B.20 Some simple illustrations of the use of the result given in equation (B.16) may be obtained, e.g., (iiia) Let x be a column matrix (say x) in an N-dimensional space, let A = A be an N x N matrix, and let (x|y) be the ordinary T scalar product (i.e., x «y) in a real, finite-dimensional Hilbert space. Then equation (B.16) becomes N N , N Grad{^ , v , V A. .x x. } = -r [ (A , + A. .)x, (iiib) Let x be a vector-valued function, say v(„), over a region R of physical space (an E ), let A be a constant dyadic over R, and let the inner product of two elements, v(.) and u(.), be defined as f (vlu)A vou rr~| j R where I r| denotes an element of volume of R. Then equation voA.v ( \ = ^r (A + A T )*v 2_ Vx*(B.16) gives Grad{^ v z R and, in particular, if A is the unit dyadic, 1, then the result is 1 r Grad{— v c v Tx\ } = lov = v x 2 j R (iiic) Let x be a dyadic-valued function, say D(.), over 3 RCE , let A = A be a constant tetradic, and let the inner product of two elements, D(.) and E(.), be defined as

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B.21 (D D|E)A D$E fr] Then equation (B.16) becomes Grad{y D (A8D) 8 D T frl = i (A + A T )8D St st s: irJ 2 <~ sr sr R T T where A is defined in terms of tensor components as A,., ,. = A. „, „ , tr Ikji ljkl' which is the transpose of A with respect to the inner product defined above, As a final illustration, let us consider a case of interest in the theory of integral equations that is 2 (iiid) Let x be cpeL [0,1] (the space of a Lebesgue square 1 integrable functions) let A = Ko [ . ]_A j K(x,y)[.] fy] , where K(.,y) 11 f f 2 is a continuous function of both variables and j J K (x,y) Jjc] [y] 6 I and let the inner product be defined by (i^)_A | \ [ T In the special case where K Is symmetric (or self-adjoint) that is K(x,y) = K(y,x) we have 1 -r r , Gradl^ (K c $|$)} = K o = I K(x,y)
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B.22 B.2. Abstract Integration These examples and those of the preceding subsections serve to illustrate the similarity that often exists between classical results and results for abstract spaces. This similarity of results is true for integral as well as for differential operations. It is also true for the theorem which is the connecting link between these two types of operations, namely, the fundamental theorem of integral calculus i r f(b) f(a) = | df J a The validity of any generalization of this theorem depends upon the b generalizations of the operations, (.) and d^) and upon the j connection between them. This theorem finds widespread use in classical analysis and in particular in the classical proof of Liapunov's theorems which suggests that the generalization would be an equally valuable theorem in abstract analysis. Even without a generalization of the fundamental theorem of integral calculus, however, the integral operation would be more than useful even in classical analysis. The notion that an integral operation permits us to find out something about the domain space based upon values of the operation in the range space has become most important in modern mathematics, as is evidenced by the rapid growth of the theory of distributions. In this particular theory, relations are characterized by the values which they assume when acting upon the elements of a certain function space the space of

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B.23 testing functions (see, eg,, Zemanian, 1965). In the sense that these distributions are continuous, linear functionals over this space of testing functions, they are an abstraction of an integral operation; but even more that this, the various operations upon distributions e.g., differentiation are based upon resales for the classical integral with the test function as the kernel. The two approaches to generalizing classical definitions are especially evident in integration theory, as is readily seen by comparing the approaches of Lebesgue and Danielle Lebesgue (see, e.g., Lebesgue, 1928) constructed an integration process for which one could interchange limiting operations, eg., b b f r Limit I x (t) t n n im a a n -* °° Daniell (1918), on the other hand, took the axiomatic approach and started with certain basic postulates which the generalized integral operation must satisfy on an unspecified domain; then he considered how the domain should be extended from that of classical calculus in a manner which preserved the validity of the basic postulates. Although Lebesgue' s approach to integration theory was constructive, his approach to measure theory, leading to an integral operation, was axiomatic. If our main interest is abstracting an integral operation, this approach has the disadvantage of having to construct a measure theory before proceeding to an integral operation if this extension may be made. Daniell 's approach, on the other hand,

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B.24 was axiomatic with respect to the integral operation each integral operation having an associated measure theory. The direct extension of the Riemann integral by generalizing the class of objects in the definitions is unsatisfactory in modern analysis for the following reasons: (i) the class of Riemannintegrable functions is a very restricted class of functions (e,g,, the Cauchy Convergence Test for "convergence in integral" is invalid) , and (ii) it may only be applied to functions of a finite number of independent variables, whereas in modern analysis one must often deal with operators on function spaces. The Riemann integral may, nevertheless, be used to construct a list of desirable properties which may, in turn, be abstracted as postulates for a generalized integration operation, (.), namely b r a D l if f ( o )>= and b^_a , then b f >0, in which >is a partial ordering symbol; that is, (.) i .s j a a a positive operator, DJ for all f-,^ belonging to the domain of definition, the operation is additive; that is, b b b r i (f n + f„) = I f, + | f„ , '1 V J 1 J 2 a a a |DJ if if n ) is such that f k+1 (s)>f fc (s) for all k and s and if Dieudonne (1960) has gone so far as to say that were it not for the fact that Riemann' s name is attached to it, this integral operation would have been abandoned long ago,

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f -» f then Limit n j n * =° a B.25 b r f = I f * In words, if 1 f i is a monotone n n of integrals increasing sequence of functions which approach f, then the sequence f f tends to f. Some other properties are , a a 1 = 1 for all a, b, and c, one has b c a f r r f + | f + | f = o, j I j a b c and \l I for all a, b, and h, one has LSI b b+h f(s) f(s-h) J j a a+h The first three were taken as postulates by Danieil, while the Lebesgue integral had in addition, the last three properties. Among the benefits reaped from having an operation endowed with the postulates of Danieil, the following may be listed: (1) Postulate Id j of Danieil, which endows the integral operation with the features of a positive operator, allows the use of this operation to derive many other equations e,g., the use of the classical integral in the calculus of variations permits the derivation of the Euler-Lagrange equations. (2) Postulate 'TdJ of Danieil that is, that the integral operation is a linear operation makes applicable a large number of

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B.26 the developments and theorems in the vast field of linear operator theory. (3) The combination of the above postulates that is, the integral operation is positive and linear implies that it is also isotone or order preserving. The major consequence ot this is that a known ordering in the range space of the integral operation may be used to induce an ordering in the domain space. (4) Postulate [dTI of Daniell, which endows the integral operation with the property of lower semi-continuity (see, e,g„, Edwards, 1966), is the essence of the extension of the domain space of "simple functions" made by Daniell, This property insures us that an analysis in the range space is an "almost" faithful reflection of the situation in the domain space. The "almost" is used here not in the sense of the "almost everywhere" of integration theory, but rather in the sense that an operation should be continuous in order to be truly relation-preserving. In addition to possessing the above properties (the postulates of Daniell generalized integral), the Lebesgue integral also possesses the properties [17] to TT1 of the Riemann integral. From JLJJ the Lebesgue integral, similar to the Riemann integral, has the properties that (i)

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B.27

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B.28 generalization of Riemann's integral is to abstract the range space of the class of objects over which the integral operates. Graves (1927) chose a natural abstraction of the real line, namely a Banach space. It inherited, however, the ills of its father in classical analysis the Riemann integral as far as the Cauchy Convergence Test was concerned. The next logical step was to define an analog to Lebesgue's integral for functions of a finite number of independent variables. This was done by Frechet (1915) at Radon's suggestion by introducing the concept of a positive, completely additive set function rather than the obvious generalization to an n-dimensional measure, Frechet 's work, in turn, led to generalizations of the Stieltjes integral, in that an analog to the formula for integration by parts becomes a basic postulate, Bochner (1933), following in the natural succession of generalizations, defined an integration process analogous to Lebesgue's over operators whose range was a Banach space and whose domain was a finite dimensional set. Dunford (1938), then extended this definition to measures of the Borel type, which includes measures of the Lebesgue type, Gowurin (1936) then made the generalization of Graves' integral analogous to the earlier generalization of Riemann's integral; namely, he considered a Riemann type of integration process defined on operators, the domains and ranges of which were Banach spaces. Price (1940) made a Lebesgue type of extension of Gowurin ' s integral, which was then followed by the extension of Gelfand (1938) and

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B.29 Pettis (1938). Their extension was in the analogous spirit of Dunford's extension of Lebesgue integration. B.2.b. Daniell's Theory of Integration Rather than this constructive approach, Daniell took an axiomatic approach and went through the details of che generalization once. He recognized that Lebesgue 's generalization of Riemann's process may be viewed as an extension of Riemann integration over a large class of functions, called elementary functions, which satisfy certain axioms when operated upon by an elementary operation that is positive, additive, and upper semi-continuous by the postulates is: D 3 ; the explicit expression of postulate nO depends upon the topology of the set of elementary functions and the topology of the range space of the range of the elementary operation. Armed with this set of axioms, Daniell developed a general theory of the integral. Phillips (1940) mentions that all the previously mentioned integrals may be placed within the framework constructed by Daniell with proper choices of the four degrees of freedom in Daniell's formulation, namely, (i) the set of elementary f unctions , X > (ii) the elementary operator, I<-,>, (iii) the topology of 36 , (iv) the topology of the range space of 1^.^Briefly what Daniell did was to extend the domain of the elementary operation, !<•>, from the set of elementary functions,^,

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B.30 to the set theoretic union of that set and all those functions which are limits of all possible non-decreasing sequences of members of 3£ . The value of the elementary operation for members of this extension are defined such that it is continuous. The final extension of the domain of the elementary operation is to all those elements which may be obtained as the difference between non-decreasing sequences of elements of 36, Again the value of I<*> for these elements is defined to insure continuity. In addition to showing this general framework of Daniell, Shilov and Gurevich (.1966) work through the details in several specific cases. These are summarized in Table B.l. Although in each of their examples the elementary functions are real, numerically valued functions and the elementary operator is a functional, extensions to situations where the range space of both the elementary functions and the elementary operation is a lattice or a partially ordered linear algebraic system is straightforward . If some of the other integration theories are to be submerged in the framework of Daniell, then this type of an extension is necessary, An integration operation defined over elements other than real-valued functions is of importance as a possible technique for solving operator equations containing abstract derivatives and operators. This solution might be either by direct integration or in transforming inherently unbounded operators into bounded operators

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B.31 «0 C H •H cfl •u u •-i at 3 cu CO 4J 0) c
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B.32 so that various iterative and approximate solution methods may be applied, much as is done when Picard's method is applied to ordinary differential equations. B.3. Methods of Solution of Operator Equations One of the important advantages gained when a problem is formulated in the abstract language of modern mathematics is that analogies between seemingly diverse problems will generally become apparent. Even more important to us, however, is the realization that methods of solution known for one type of problem may be analogously applied to another type of problem when they are both viewed as special cases of the same operator equation in abstract spaces . Indeed, the methods of solution of operator equations mentioned below had their origins in diverse disciplines in mathematics. For example, the Newton-Raphson-Kantorovich method generalized the NewtonRaphson method for finding roots of an algebraic equation, Picard"s method was originally used to solve ordinary differential equations, and Ritz's method was originally used to find the extremal function of some functional. However, rather than continue with these examples, let us examine some of these methods. B.3.a. The Method of Contracting Operators" This method follows as a result of the constructive proof of the *A operator P< . > is referred to as contracting in fi if p(P, P) <_ ap(x, Xl ) where x.x^fi and a belongs to the interval (0,1) and is independent of the elements x and x^

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B.33 Banach-Cacciapoli Theorem (cf . Kantorovich and Akilov, 1964, p. 687) for the existence and uniqueness of a solution, x , to the equation x = P The theorem statement is as follows, Theorem B.3.A. If P<.> in the above equation is a contracting operator in ftCX, a unique solution, x^, of the equation exists in ft. The solution, x^, ma} 7 be obtained as the limit of a sequence, (x, ) , where x k+i p< v ; k = °« 1 » and x is any given element of ft. The rate of convergence of {x } to the solution is given by n where a is the contraction coefficient. The method of solution offered by the constructive proof of this theorem is one of the most general techniques for solving operator equations. In applying this method, moreover, there is considerable freedom in satisfying the hypotheses of the theorem. Some of the degrees of freedom are: (i) there is usually more than one way to formulate any Janos (1967) has proved a converse to the above theorem, provided the space is a compact, metrizable topological space and P .> is such that A P n = la} a fixed point. n=l

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B.34 particular problem as a search for a fixed point, (ii) because there is usually more than one way to introduce a metric into any collection of objects it is not unlikely that under some metric the operator will be a contraction while for others it will not, (iii) it is possible for an operator to be a contraction in one subset Q of X and not in another. B.3.b. Implicit Function Technique Although this technique may be looked upon as one of the many variations of the above method, it deserves separate consideration because of its utility in modern analysis. Hildebrand and Graves (1927) applied a form of the implicit function theorem to investigate existence and uniqueness of solutions of various operator equations. The idea behind this method is very simple, namely, the introduction of a number of parameters into the problem such that an intelligent choice of the values of these parameters will make the relevant operator a contracting operator. As an illustration of the technique consider the inhomogeneous operator equation y = T^x> (B.18) where y is given beforehand. By rearranging and defining an operator $^.,.^ , this equation becomes *^y,x> = (B.19) where $Ay T .

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B.35 Now the solution x of equation (B.18) is also the solution x A = f(y) of equation (B.19), for if x^ is such that y = t then $ = 4> = y T = y T = y y = Let us now operate upon $ with some operator V which is such that: (i) V<0> vanishes, and (ii) •<$*,.>> is bounded operation. If equation (B.19) is operated on by •<•> and multiplied through by some constant, K, then it becomes K^$> = The solution x A of equation (B.18) is also a solution to this equation because YLV«b is introduced by z = P> (B.20) then it is seen that, x A , the solution of equation (B.18), is also a fixed point of the operator P<.> in equation (B.20), that is P | = x^ x=x The difference in the second formulation (i.e., equation (B.20))

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B.36 between the two formulations is that there exists two quantities, K and 4 1 , which may be adjusted in an attempt to obtain a contracting operator in the equation, and if one is, K and y may also generally be adjusted to increase the rate of convergence of the iterations. B.3.c. Newton-Raphson-Kantorovich Method Newton's method (see e.g., Newton Ostrowski (1966): Stiefel (1965); Scarborough (1958)), as applied to algebraic equations, is a successive approximation technique in distinction to the iterative technique of contracting operators for finding the roots of an equation, P(x) = 0. The method is often referred to as the method of tangents, for its essence lies in the successive solution of a linear equation tangent to the equation P(x) = at each successive step. Figure (B.l) graphically illustrates the technique as applied to a single algebraic equation in cases where it does and does not converge. It is of interest to notice that the convexity of the graph of the equation P(x) = in relation to the yaxis is an ideal situation for the convergence of the method at least for algebraic equations. Fine (1916) and later Raphson made the natural observation that if an ordinary derivative is reinterpreted as the gradient of a function or the Jacobian matrix respectively, then there is nothing to prevent the method from being generalized to equations in finite-dimensional spaces. Kantorovich (1948a, 1948b, 1948c), in turn, using a still more general but analogous notion of a derivative, the Frechet derivative, further generalized

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B.37 GRAPH OF f(x)--0 GRAPH OF LINEAR EQUATION TANGENT TO f(x) AT f(x ) -^-x x O X 1 X T X* (a) CONVERGENT SUCCESSIVE APPROXIMATIONS (X ) (b) NON-CONVERGENT SUCCESSIVE APPROXIMATIONS FIGURE B.I. NEWTON'S METHOD FOR ROOTS OF AN ALGEBRAIC EQUATION

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B.38 the method to operator equations in infinite-dimensional spaces, or, more generally, to abstract spaces. Since Kantorovich's pioneering efforts following World War II, there have been numerous reformulations, extensions, and modifications of the Newton-Kantorovich technique. Of these, we mention only Altman (1957, 1961); Bar tie (1955); Hart and Motzkin (1956); Mysovskikh (1954); Schroder (1957); Slugin (1958); Zaazik (1960); the reader is referred to Kantorovich and Akilov (1964), and Moore (1964) for a more extensive bibliography of recent results. Of these results we state, without proof, a theorem due to Bartle (1955) . Theorem B.3.b: Let T<.> be a continuously (F)-dif f erentiable operator mapping the sphere S (x„ ,a)G£ititolj . Further, let ?1 = T '(* ) dx o x o have a bounded inverse i.e., | | (T ' ) _1 1 | <\ . Let there be a positive number, 5, such that ||x x q | | < 6 implies i I 1 T' T' < Ty 11 O ^Js and let T be bounded as follows: o T< x > h T\ in which gAMin{l,a,6(x o , -^ ) }

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B.39 Then the sequence, {x } , with x = x (T'(x )) -1 T n n-1 n-1 n-1 converges to the solution x^ of the equation T = Moreover, x^ is the only solution in the neighborhood S(x ,3), and the rate of convergence to x^ is given by j | x x | | < 2 $ . n x The number and stringency of the hypotheses in the theorem are necessary to guarantee uniqueness and convergence based upon an initial estimate x ; although, in practice, rather than checking the hypothesis it is usually simpler to choose an x , begin the successive approximations, and determine whether the successive approximations seem to be converging. The nature of the hypotheses of this theorem is somewhat analogous to those of Newton's method for algebraic equations. This, and also the analogous formula for the successive approximations, suggests that the difficulties which plague Newton's method are likely to arise also in the NewtonRaphson-Kantorovich method. Indeed this is the case, and, quite naturally, the techniques developed to circumvent these difficulties in the Newton-Raphson-Kantorovich method are reminiscent of those developed for similar difficulties in the Newton and Newton-Raphson methods. For example, a modification of the Newton-Raphson-Kantorovich

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B.40 method which replaces (T') by (T') is used to reduce n o computation time, as a similar modification of Newton's method. Similar to a modification of Newton's method when a derivative is small or zero, Altman (1957) has modified the usual successive approximations to the form l|T|| X n+1 = X n " 2 M Q,, Q< V I n | where QA(T') T^x > , (*) indicating the adjoint operation when the method is besieged by the difficulty that (T')~ n does not exist. Perhaps this abstract method of solution serves better to illustrate the benefits reaped from a abstract approach than the other methods to be mentioned because of the familiarity of its motivation, the directness of its generalization, the simplicity of the technique, and the universality of its possible application. Indeed, Kantorovich started with a method familiar to any first year calculus student; abstracted the essential properties of the operations involved in the method while retaining the basic idea or motivation behind the method; thereby, developing a simple successive approximation technique of universal in the sense of amenable concrete types of equations application. His application of Newton's basic idea to the study of operator equations of the form P(x) =

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B.41 where x and are elements of Banach spaces. The explicit form of the equation used in the successive approximation steps will depend upon the concrete nature of the operator P<.> that is, algebraic, differential, integral and concrete nature of the elements of the Banach spaces between which P<.> maps; the conditions for the convergence of the successive approximations assuming the nature of the operator is fixed will depend upon how the Banach spaces are normed. The possible applications of this method, therefore, stretches the limits of one's imagination, For example, Kantorovich and Akilov (1964) apply the method to the concrete operator equations such as: (i) a finite system of algebraic equations, (ii) an integral equation of Hammerstein's type, (iii) an ordinary differential equation, and (iv) a partial differential equation. Along with the study of each concrete operator with its own interpretation of the meaning of the Frechet derivative they consider the convergence conditions for two different types of possible norms on the set of elements which are transformed by the particular operator which, in turn, leads naturally to two different explicit sets of convergence conditions. The recent literature abounds with other applications of the Newton-Raphson-Kantorovich method, of which we only mention: Since Kantorovich s early work on the above method its applicability has been extended to complex linear topological spaces (see Hirasawa (1954)) and to partially ordered linear systems (see, e.g., Kantorovich (1948c); Schroder (1957)).

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B.42 Mysovskikh (1954); Collatz (1953, 1954, 1966); Koshelev (1954); Chandrasekar and Breen (1947); Radbill (1964); Bellman and Kalaha (1964). B.3.d. Method of Steepest Descent The method of steepest descent, as the Newton-RaphsonKantorovich method, had its origin in the methods of solution, namely as a successive approximation technique for finding the extrema of a function. Likewise, in the abstract method of steepest descent (see, e.g., Mikhlin (1964), Kantorovich and Akilov (1964)), the basic goal is to minimize the value of a functional that is, a variational method. To this end, the method involves successive steps in the direction of most rapid decrease in the value of the functional as obtained from the abstract derivative. Therefore to apply this method to find a solution of an operator equation, one must be able to construct a functional whose extrema are the solutions to the operator equation (for linear, positive, self adjoint operators in Hilbert spaces (see, Kantorovich and Akilov (1964), p. 604). Thus, a central issue in the consideration of the technique is a statement of the conditions under which an operator equation may be reformulated as a variational problem. The following theorem, referred to as Kerner's symmetry condition by Nashed (1965), is In their generality, variational problems have meaning for any type of operator whose range is a lattice, or some other partially ordered space, e.g. the positive operators defined in Chapter III.

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B.43 such a statement for Hilbert spaces: Theorem B.3.c: Let T<.> be an operator from a Hilbert space, Jrf , into itself, and have a (G) -differential, bounded in its second argument, at each xe[the sphere S(x ,r)}. Further, at each xeS(x ,r), let the functional (6T|k) be continuous in x for each h,kej-|. Then for T< . >to be the gradient of some functional t< , > , i.e. Grad t< .> = T<.>, defined on S (x ,r) , it is necessary and sufficient that the bilinear functional (6T |k) by symmetric in h and k for each xeS(x ,r) . That is, the following condition must hold: (6T|k) = (6T|h) Therefore, given that an operator, T^.>, satisfies the conditions of the above theorem, it is thus clear that the operator equation of the form T-x> = (B.21) is precisely the condition that the associated functional, t<.>, assumes a critical or extremal value. The latter problem that of finding the element which extremizes a functional is amenable to attack by the method of steepest descent. The successive approximations used in the method are of the form x , .. = x + a z n+1 n n n For applications of this type of reasoning for the study of topological properties of operator equation the reader is referred to Krasnosel'skii (1964); Vainberg (1964); Rothe (1953).

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in B.44 which z is the direction of greatest decrease of the functional n t<.> from the point x , and in which a is the distance traversed n n in that direction. In deciding upon the particulars of the successive approximation scheme, the following theorem (see, e.g., Nashed, 1965; Mikhlin, 1964) is both typical and useful. Theorem B.3.d: Let {U } be a sequence of self-adjoint, positive n definite, bounded linear operators defined on a real Hilbert space, j-4 , with the inner product, (. 1 .). In particular (U |h)^m(hih), m>0, he 44 n = and therefore let the metric, p (.,.), be defined for the n-th iteration as p (x,y) = (U n |x y) If the operator equation is as equation (B.21); the operator satisfies the conditions of Theorem B.3.c; x is the initial estimate, then the direction of steepest descent of t<>> from x q is given by z = -U~ T , o o o and in general z = -U T n n n In this case the successive approximations are of the form x , = x a U T n+1 n n n n There are, moreover, several standard ways of choosing the a in these successive approximations, the most popular of which are:

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B.45 (i) the optimum gradient method a is chosen to minimize the functional t>, i.e. the smallest positive n n n n 3 1 root of — = (this is the technique suggested by Kantorovich da n and Akilov 1964, p. 603), (ii) the approximate optimum gradient method a is chosen to minimize I |T a 6T> I i 11 n n n n n ' ' The above terms are the first two terms of a Taylor series expansion of T about x in the direction of steepest descent, n+1 n (iii) the sequential descent method a is chosen by n methods (i) or (ii) above and then U is adjusted to speed convergence, B.3.e, Method of Weighted Residuals Although the method of weighted residuals, similar to the method of steepest descent, had an origin in variational problems, they were variational problems connected with the calculus of variations rather than those connected with functions. The basic objective of the method is to obtain an approximate solution to the operator equation T = by setting a finite number, N, of weighted averages A In the following we restrict our discussion to operator equations in Hilbert spaces.

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B.46 * (w. |T) = * where x is some general form containing (N) degrees of freedom for the approximate solution. These degrees of freedom are then determined such that the above equation is satisfied for each j . The w.'s are referred to as the weight operators and depending upon the choice of them, various analogs to the classical methods of (i) the collocation method, (ii) the subdomain method, (iii) the method of minimum mean square residuals, (iv) Galerkin's method, and (v) the method of moments £s is pointed out by Finlayson and Scriven } 1966, for the special case of differential equations) are obtained. As in the case of differential equations, one often finds that a solution to an operator equation in addition to satisfying T = must also satisfy auxiliary constraints such as boundary, and, or side conditions, etc. If this is the situation then it is advantageous to choose this approximate form such that these auxiliary conditions are satisfied regardless of how the (N) degrees of freedom are chosen, In order to place these words in a more precise mathematical framework, let us suppose for concreteness that we wish to find a solution, x, such that T = and such that it also satisfied the auxiliary condition * = y (B.24) If this operator $<.> is additive and an 3c can be found such that

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B.47 $ ~ y, then one could select, x , as * — . — * X = X + X (B.25) _* where x belongs to the null space of the operator $<•>. If, on the other hand, T<.> were additive and we knew a representation for elements of its null space, then the roles of T<.> and $<•>, may be reversed; thereby, we obtain the analog to what Finlayson and Scriven (1966) refer to as boundary methods. We have proposed operator equations for Liapunov operators in Chapter IV and these are some standard techniques of finding solutions to operator equations. It is likely, therefore, that they will be useful in solving the operator equations for Liapunov operators.

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BIBLIOGRAPHY Altman, M. 1957, Concerning approximate solutions of non-linear functional equations, Bull. Acad. Polon. Sci . Ser . Sci. Math. Astronom. Phys. , _5, 461. , 1961, Concerning the method of tangent hyperbolas for operator equations, Bull. Acad. Polon. Sci. Ser . Sci. Math. Astronom. Phys. , 9_, 633. 1966, A generalized gradient method for minimizing a functional on a Banach space, Mathematica , 9_, 15. Summary in Math. Rev. , 35, 218. Anselone, P. M. and Moore, R. H. , 1966, An extension of the NewtonKantorovich method for solving nonlinear equations with an application to elasticity, J. Math. Anal. Appl. , 13 , 476. Summary in Math. Rev. , 32, 1393. Bartle, R. G. 1955, Newton's method in Banach spaces, Proc. Amer. Math. Soc , 6, 827. Bastiani, A. 1964, Applications dif f erentiables et varietes differentiables de dimension infinie, J . Anal. Math. , 13 , 1. Summary in Math. Rev. , 31 , 278. Bellman, R. and Kalaba, R. 1964, Quasilinearization, Elsevier Pub. Co., San Francisco, Cal. Bochner, S. 1933, Integration von Funktionen deren Werte die Elemente eines Vektorraumes sind, Fund. Math. , 20 , 262. Buck, R. C. 1956, Advanced Calculus , McGraw-Hill Book Co., N.Y. Chandrasekar , S. and Breen, F. 1947, On the radiative equilibrium of a stellar atmosphere, Astrophys. Jour. , 106 , 143. Collatz, L. 1953, Einige Anwendungen funktional analytischen Methoden in der praktischen Analysis, ZAMP , 4_, 327. , 1954, Das vereinfachte Newtonsche Verfahren bei nichtlinearen Randwertauf gaben, Arch. Math. , _5, 99. 1966, Functional Analysis and Numerical Mathematics , Courant, R. and Hilbert, D. 1952, Methods of Mathematical Physics , Vol. I. , Academic Press, Inc., N.Y. Cronin, J. 1950, The existence of multiple solutions of elliptic differential equations, Trans. Am. Math. Soc , 68 , 105. Daniell, P. J. 1918, A general form of integral, Ann. Math. , 19 , 279. B.48

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B.49 Dieudonne, J. 1960, Foundations of Modern Analysis, Academic Press, Inc. N.Y. Dunford, N. 1938, Uniformity in linear spaces, Trans . Am. Math. Soc . , 44_, 305. Elconin, V. and Michal, A. 1937, Completely integrable differential equations in abstract spaces, Acta Math. , 68 , 71. Fine, H. B. 1916, On Newton's method of approximation, Proc. Nat. Sci. , 2_, 546. Finlayson, B. and Scriven, L. 1966, The method of weighted residuals, Appl. Mech. Rev. , 19 , 735. Frechet, M. 1906, Sur quelques points du calcul fonctionel, Rend. Math, di Palermo , 22 , 1. , 1910, Sur les f onctionelles continues, Ann. Ecol. Norm. , 27, 193. 1915, Sur l'integrale dune fonctionelle entendue a une ensemble abstrait, Bull, de la Societe Math, de France , 43 , 248. Gateaux, R. 1913, Sur les f onctionelles continues et les fonctionelles analytiques, Comp . Rend. , 157 , 325. Gelfand, I. M. 1938, Abstrakte Funktionen und lineare Operatoren, Math. Sbor. , 4_, 235. Gowurin, M. K. 1936, Uber die Stieltjesschen Integrationen abstrakten Funktionen, Fund. Math. , 27 . 254. Graves, L. M. 1927, Riemann integration and Taylor's theorem in general analysis, Trans. Am. Math. Soc , 29 , 163. , 1935, Topics in functional analysis, Bull. Am. Math. Soc , 41 , 641. Hahn, W. 1963, Theory and Applications of Liapunov's Direct Method , Prentice-Hall, Inc., Englewood Cliffs, N.J. , 1967, Stability of Motion , Springer Verlag, Berlin. Hart, W. and Motzkin, T. 1956, A composite Newton-Raphson gradient method for the solution of systems of equation, Pac J . Math. , 6., 691. Hildebrand, T. H. and Graves, L. M. 1927, Implicit functions and their differentials in general analysis, Trans. Am. Math. Soc , 29, 127. Hille, E. and Phillips, R. 1955, Functional Analysis and Semi-Groups , Amer. Math. Soc. Colloquium Publication, vol. 31, Waverly Press, Inc., Baltimore, Md.

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B.50 Hirasawa, Y. 1954, Newton's method in complex linear topological spaces, Comm. Math. Univ. St. Paul , 3_, 15. Hyers, D. H. 1941, A generalization of the (F)-dif f erential, Proc . Nat. Acad. Sci. , 27 , 315. Janos, L. 1967, A converse of Banach's contraction theorem, Proc. Am. Math. Soc , 18 , 287. Kantorovich, L. V. 1939, The method of successive approximations for functional equations, Acta Math. , 71 , 63. , 1948a, On Newton's method for functional equations, Dokl. Acad. Nauk, SSSR , _59, 1297. 1948b, On a general theory of approximation methods of analysis, Dokl. Acad. Nauk. SSSR , 60, 957. 1948c, Functional analysis and applied mathematics, Uspehi. Mat. Nauk. , 3, 89. and Akilov, G. P. 1964, Functional Analysis in Normed Spaces , The Macmillan Co., N.Y. and Krylov, V. I. 1958, Approximate Methods of Higher Analysis , Interscience Publishers, Inc., N.Y. Koshelev, A. I. 1954, The existence of a generalized solution of the elastic-plastic problem of torsion, Dokl. Acad. Nauk. SSSR , 99, 357. Krasnolsel' skii, M. A. 1964, Topological Methods in the Theory of Nonlinear Integral Equations , The Macmillan Co., N.Y. Kwapisz, M. 1967, On the approximate solution of abstract equations, Ann. Polonam. Math. , 19 , 47. Summary in Math. Rev. , 35 , 222. Lasalle, J. P. 1941, Pseudo-normed linear spaces, Duke Math. J . , 8_, 131, Lebesgue, H. 1928, Lecons sur 1 ' integration , in Gauthier-Villars , Paris . Liapunov, A. M. 1906, Sur les figures d'equilibre peu differentes des ellipsoides d'une masse liquide homgene donee d'une mouvement du rotation. Premiere partie. Etude generale du probleme, Zap. Acad. Nauk., St. Petersburg , 1, 1. McShane, E. J. 1965, Integration in linear spaces, Arch. Rat. Mech. Anal. , 18, 403. Summary in Math. Rev. , 31 , 56.

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B.51 Michal, A. D. 1939, General differential geometry and related topics, Bull. Am. Math. Soc . , 45 , 529. , and Clifford, M. 1933, Fonctione analytiques implictes dans des espaces vectoriels abstraits, Comp . Rend. Acad. Sci , Paris , 197 , 735. and Paxson, E. W. 1936, La dif f erentielle clans les espaces abstraits lineares avec une topologie, Comp . Rend . Acad. Sci. Paris , 202 , 1741. Millsaps, K. 1942, Differential calculus in topological groups, I, Rev, de Cien. , 44 , 485. Millsaps, K. 1943, Differential calculus in topological groups, II, Rev, de Cien. , 45 , 45. Mikhlin, S. G. 1964, Variational Methods in Mathematical Physics , The Macmillan Co., N.Y. Moore, R. H. 1964, Newton's method and variations, in Nonlinear Integral Equations , ed . P. M. Anselone, Univ. of Wisconsin Press, Madison, Wis. Mysovskikh, I. P. 1954, On a boundary problem for the equation Au = k(x,y)u , Dokl. Acad. Nauk. SSSR, 94, 995. Nashed, M. Z. 1965, On general iterative methods for the solution of a class of nonlinear operator equations, Math. Comp., 19, 14. Nemyetskii, V. V. 1936, Fixed point methods in analysis, Uspehi . Mat. Nauk., 1, 1. Ostrowski, A. 1966, Solution to Equations and Systems of Equations , Academic Press, Inc., N.Y. , 1967, The round-off stability of iterations, Z.A.M.M. , 47 , 77. Summary in Math. Rev. , 35 , 402. Pereyra, V. 1967 , Iterative methods for solving nonlinear least squares problems, SIAM J. Num. Anal. , 4_, 27. Summary in Math. Rev., 35 , 1402. Pettis, B. J. 1938, On integration in vector spaces, Trans. Am. Math. Soc, 44, 277.

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B.52 Phillips, R. S. 1940, Integration in a convex linear topological space, Trans. Am. Math. Soc , 47 , 114. Price, G. B. 1940, The theory of integration, Trans. Am. Math. Soc , _47, 1. Radbill, J. R. 1964, Application of quasilinearization to the boundary layer layer equations, AIAA. J. , 2 , 8. Rothe, E. 1937, Zur Theorie der topologischen Ordnung und der Vektorf elder in Banachschen Raumes, Composito Mathematica , 5, 177. , 1946, Gradient mappings in Hilbert space, Ann, of Math. , 47_, 580. 1948, Gradient mappings and extrema in Banach spaces, Duke J. of Math. , 15, 421. 1953, Gradient mappings, Bull. Am. Math, Soc , 59, 5. Scarborough, D. 1958, Numerical Mathematical Analysis, Johns Hopkins Press, Baltimore, Md. Schroder, J. 1957, Uber das Newtonsche Verfahven, Arch. Rat. Mech. Anal. , 1 , 154. Sen, R. N., A modification of the Newton-Kantorovich method, Mathematica , 8, 155. Summary in Math . Rev . , 35 250. Shilov, G. E. and Gurevich, B. L., 1966, Integral, Measure, and Derivative: a Unified Approach , Prentice-Hall, Inc. Englewood Cliffs, N.J. Slugin, S. N. 1958, On the theory of Newton's method and Chaplygin's Method, Dokl. Acad. Naul. SSSR , 120 , 472. Stiefel, H. 1965, Einfuhrung in der Numerische Mathematik , Tenbner von Stuttgard, Germany. Vainberg, M. M. 1964, Variational Methods for the Study of Nonlinear Operators , Holden-Day, Inc., San Francisco, Cal . Volterra, V. 1959, Theory of Functional and of Integral and IntegroDifferential Equations , Dover Publications, N.Y.

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B.53 Zaazik, Y. J. 1960, On the approximate solution of nonlinear operator equations by iterative methods, Am. Math. Soc. Tranl . , ser. 2 , 16, 410. Zorn, M. A. 1946, Derivatives and Frechet differentials, Bull. Am. Math. Soc, 52, 133.

PAGE 293

BIOGRAPHICAL SKETCH John Paul San Giovanni, the youngest son of Carlo and Maria, was born on 26 June 1941 in Rye, New York. He graduated from Archbishop Stepinac High School in June, 1959, and began his university studies at Manhattan College the following September. He received his B.Ch.E. degree in June, 1963, and was married to the former Marie Santoro later the same month. In July, 1963, he entered graduate school at Iowa State University and received his M.S. degree in February, 1965, under the direction of Dr. L. E. Burkhardt with a thesis entitled "The Calculation of Profiles and Surface Areas of Slowly Expanding Interfaces." During this period, in October, 1964, his oldest son John Paul, Jr. was born. In March, 1965, he entered the University of Florida where Dr. R. W. Fahien one of his professors at Iowa State became Chairman of his Supervisory Committee. In June, 1966, his son, Thomas, was born. Dr. X. B. Reed became the Co-chairman of his Supervisory Committee in December of the same year. John Paul San Giovanni is a member of the Phi Kappa Phi Honor Society and an associate member of the American Institute of Chemical Engineers. B.54

PAGE 294

This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved bv all members of that committee. It wa iitl d to the P '.n r,f the College of Engineering and to the Graduate Council, and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy, March 1969 (jjL*Zt£< Dean, Co] lege of Engineer/ng D an, Graduate School Supervisory Committee: 5?T

PAGE 295

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onliapunovsdirec00sangrich_Page_227.pro
R228 41106
onliapunovsdirec00sangrich_Page_228.pro
R229 51142
onliapunovsdirec00sangrich_Page_229.pro
R230 37828
onliapunovsdirec00sangrich_Page_230.pro
R231 33516
onliapunovsdirec00sangrich_Page_231.pro
R232 22031
onliapunovsdirec00sangrich_Page_232.pro
R233 39037
onliapunovsdirec00sangrich_Page_233.pro
R234 37788
onliapunovsdirec00sangrich_Page_234.pro
R235 40718
onliapunovsdirec00sangrich_Page_235.pro
R236 29147
onliapunovsdirec00sangrich_Page_236.pro
R237 35645
onliapunovsdirec00sangrich_Page_237.pro
R238 39502
onliapunovsdirec00sangrich_Page_238.pro
R239 6840
onliapunovsdirec00sangrich_Page_239.pro
R240 40950
onliapunovsdirec00sangrich_Page_240.pro
R241 43249
onliapunovsdirec00sangrich_Page_241.pro
R242 37172
onliapunovsdirec00sangrich_Page_242.pro
R243 33915
onliapunovsdirec00sangrich_Page_243.pro
R244 30970
onliapunovsdirec00sangrich_Page_244.pro
R245 27985
onliapunovsdirec00sangrich_Page_245.pro
R246 28412
onliapunovsdirec00sangrich_Page_246.pro
R247 34107
onliapunovsdirec00sangrich_Page_247.pro
R248 30228
onliapunovsdirec00sangrich_Page_248.pro
R249 31422
onliapunovsdirec00sangrich_Page_249.pro
R250 36491
onliapunovsdirec00sangrich_Page_250.pro
R251 39486
onliapunovsdirec00sangrich_Page_251.pro
R252 42062
onliapunovsdirec00sangrich_Page_252.pro
R253 35099
onliapunovsdirec00sangrich_Page_253.pro
R254 27410
onliapunovsdirec00sangrich_Page_254.pro
R255 33740
onliapunovsdirec00sangrich_Page_255.pro
R256 29098
onliapunovsdirec00sangrich_Page_256.pro
R257 22618
onliapunovsdirec00sangrich_Page_257.pro
R258 24162
onliapunovsdirec00sangrich_Page_258.pro
R259 27571
onliapunovsdirec00sangrich_Page_259.pro
R260 24134
onliapunovsdirec00sangrich_Page_260.pro
R261 37350
onliapunovsdirec00sangrich_Page_261.pro
R262 39068
onliapunovsdirec00sangrich_Page_262.pro
R263 36609
onliapunovsdirec00sangrich_Page_263.pro
R264 27409
onliapunovsdirec00sangrich_Page_264.pro
R265 32604
onliapunovsdirec00sangrich_Page_265.pro
R266 36399
onliapunovsdirec00sangrich_Page_266.pro
R267 40613
onliapunovsdirec00sangrich_Page_267.pro
R268 35395
onliapunovsdirec00sangrich_Page_268.pro
R269 37959
onliapunovsdirec00sangrich_Page_269.pro
R270 36673
onliapunovsdirec00sangrich_Page_270.pro
R271 37840
onliapunovsdirec00sangrich_Page_271.pro
R272 28819
onliapunovsdirec00sangrich_Page_272.pro
R273 31895
onliapunovsdirec00sangrich_Page_273.pro
R274 21498
onliapunovsdirec00sangrich_Page_274.pro
R275 40639
onliapunovsdirec00sangrich_Page_275.pro
R276 7744
onliapunovsdirec00sangrich_Page_276.pro
R277 26635
onliapunovsdirec00sangrich_Page_277.pro
R278 32738
onliapunovsdirec00sangrich_Page_278.pro
R279 33499
onliapunovsdirec00sangrich_Page_279.pro
R280 40833
onliapunovsdirec00sangrich_Page_280.pro
R281 38533
onliapunovsdirec00sangrich_Page_281.pro
R282 33416
onliapunovsdirec00sangrich_Page_282.pro
R283 29751
onliapunovsdirec00sangrich_Page_283.pro
R284 29722
onliapunovsdirec00sangrich_Page_284.pro
R285 34644
onliapunovsdirec00sangrich_Page_285.pro
R286 16445
onliapunovsdirec00sangrich_Page_286.pro
R287 48716
onliapunovsdirec00sangrich_Page_287.pro
R288 47193
onliapunovsdirec00sangrich_Page_288.pro
R289 45380
onliapunovsdirec00sangrich_Page_289.pro
R290 40041
onliapunovsdirec00sangrich_Page_290.pro
R291 38874
onliapunovsdirec00sangrich_Page_291.pro
R292 6223
onliapunovsdirec00sangrich_Page_292.pro
R293 29922
onliapunovsdirec00sangrich_Page_293.pro
R294 12512
onliapunovsdirec00sangrich_Page_294.pro
T1 textplain 474
onliapunovsdirec00sangrich_Page_001.txt
T3 125
onliapunovsdirec00sangrich_Page_003.txt
T4 134
onliapunovsdirec00sangrich_Page_004.txt
T5 708
onliapunovsdirec00sangrich_Page_005.txt
T6 830
onliapunovsdirec00sangrich_Page_006.txt
T7 1870
onliapunovsdirec00sangrich_Page_007.txt
T8 2062
onliapunovsdirec00sangrich_Page_008.txt
T9 1865
onliapunovsdirec00sangrich_Page_009.txt
T10 740
onliapunovsdirec00sangrich_Page_010.txt
T11 953
onliapunovsdirec00sangrich_Page_011.txt
T12 629
onliapunovsdirec00sangrich_Page_012.txt
T13 1469
onliapunovsdirec00sangrich_Page_013.txt
T14 1473
onliapunovsdirec00sangrich_Page_014.txt
T15 1638
onliapunovsdirec00sangrich_Page_015.txt
T16 1655
onliapunovsdirec00sangrich_Page_016.txt
T17 1712
onliapunovsdirec00sangrich_Page_017.txt
T18 1605
onliapunovsdirec00sangrich_Page_018.txt
T19 1736
onliapunovsdirec00sangrich_Page_019.txt
T20 1754
onliapunovsdirec00sangrich_Page_020.txt
T21 1662
onliapunovsdirec00sangrich_Page_021.txt
T22 1698
onliapunovsdirec00sangrich_Page_022.txt
T23 1670
onliapunovsdirec00sangrich_Page_023.txt
T24 1415
onliapunovsdirec00sangrich_Page_024.txt
T25 1452
onliapunovsdirec00sangrich_Page_025.txt
T26 1707
onliapunovsdirec00sangrich_Page_026.txt
T27 1735
onliapunovsdirec00sangrich_Page_027.txt
T28 1613
onliapunovsdirec00sangrich_Page_028.txt
T29 1621
onliapunovsdirec00sangrich_Page_029.txt
T30 1666
onliapunovsdirec00sangrich_Page_030.txt
T31 1674
onliapunovsdirec00sangrich_Page_031.txt
T32 1623
onliapunovsdirec00sangrich_Page_032.txt
T33 370
onliapunovsdirec00sangrich_Page_033.txt
T34 1604
onliapunovsdirec00sangrich_Page_034.txt
T35 1722
onliapunovsdirec00sangrich_Page_035.txt
T36
onliapunovsdirec00sangrich_Page_036.txt
T37 1778
onliapunovsdirec00sangrich_Page_037.txt
T38 1659
onliapunovsdirec00sangrich_Page_038.txt
T39 1653
onliapunovsdirec00sangrich_Page_039.txt
T40 1776
onliapunovsdirec00sangrich_Page_040.txt
T41 11
onliapunovsdirec00sangrich_Page_041.txt
T42 1901
onliapunovsdirec00sangrich_Page_042.txt
T43 1997
onliapunovsdirec00sangrich_Page_043.txt
T44 1414
onliapunovsdirec00sangrich_Page_044.txt
T45 1872
onliapunovsdirec00sangrich_Page_045.txt
T46 1942
onliapunovsdirec00sangrich_Page_046.txt
T47 2157
onliapunovsdirec00sangrich_Page_047.txt
T48 1536
onliapunovsdirec00sangrich_Page_048.txt
T49 1535
onliapunovsdirec00sangrich_Page_049.txt
T50 1756
onliapunovsdirec00sangrich_Page_050.txt
T51 1838
onliapunovsdirec00sangrich_Page_051.txt
T52 1774
onliapunovsdirec00sangrich_Page_052.txt
T53 1718
onliapunovsdirec00sangrich_Page_053.txt
T54 1626
onliapunovsdirec00sangrich_Page_054.txt
T55 1725
onliapunovsdirec00sangrich_Page_055.txt
T56 1755
onliapunovsdirec00sangrich_Page_056.txt
T57 1700
onliapunovsdirec00sangrich_Page_057.txt
T58 1741
onliapunovsdirec00sangrich_Page_058.txt
T59 1706
onliapunovsdirec00sangrich_Page_059.txt
T60 1886
onliapunovsdirec00sangrich_Page_060.txt
T61
onliapunovsdirec00sangrich_Page_061.txt
T62
onliapunovsdirec00sangrich_Page_062.txt
T63 1958
onliapunovsdirec00sangrich_Page_063.txt
T64 1368
onliapunovsdirec00sangrich_Page_064.txt
T65 1609
onliapunovsdirec00sangrich_Page_065.txt
T66 1810
onliapunovsdirec00sangrich_Page_066.txt
T67 16
onliapunovsdirec00sangrich_Page_067.txt
T68 1617
onliapunovsdirec00sangrich_Page_068.txt
T69 143
onliapunovsdirec00sangrich_Page_069.txt
T70 1773
onliapunovsdirec00sangrich_Page_070.txt
T71 1585
onliapunovsdirec00sangrich_Page_071.txt
T72 2053
onliapunovsdirec00sangrich_Page_072.txt
T73 2187
onliapunovsdirec00sangrich_Page_073.txt
T74 100
onliapunovsdirec00sangrich_Page_074.txt
T75 1943
onliapunovsdirec00sangrich_Page_075.txt
T76 1808
onliapunovsdirec00sangrich_Page_076.txt
T77 1330
onliapunovsdirec00sangrich_Page_077.txt
T78 1488
onliapunovsdirec00sangrich_Page_078.txt
T79 1525
onliapunovsdirec00sangrich_Page_079.txt
T80 65
onliapunovsdirec00sangrich_Page_080.txt
T81 853
onliapunovsdirec00sangrich_Page_081.txt
T82 1030
onliapunovsdirec00sangrich_Page_082.txt
T83 1281
onliapunovsdirec00sangrich_Page_083.txt
T84 1731
onliapunovsdirec00sangrich_Page_084.txt
T85 1616
onliapunovsdirec00sangrich_Page_085.txt
T86 1794
onliapunovsdirec00sangrich_Page_086.txt
T87 1694
onliapunovsdirec00sangrich_Page_087.txt
T88 1015
onliapunovsdirec00sangrich_Page_088.txt
T89 1549
onliapunovsdirec00sangrich_Page_089.txt
T90 1021
onliapunovsdirec00sangrich_Page_090.txt
T91 1826
onliapunovsdirec00sangrich_Page_091.txt
T92 1574
onliapunovsdirec00sangrich_Page_092.txt
T93 1569
onliapunovsdirec00sangrich_Page_093.txt
T94 1402
onliapunovsdirec00sangrich_Page_094.txt
T95 1847
onliapunovsdirec00sangrich_Page_095.txt
T96 1834
onliapunovsdirec00sangrich_Page_096.txt
T97
onliapunovsdirec00sangrich_Page_097.txt
T98 1763
onliapunovsdirec00sangrich_Page_098.txt
T99 1680
onliapunovsdirec00sangrich_Page_099.txt
T100 1586
onliapunovsdirec00sangrich_Page_100.txt
T101 1685
onliapunovsdirec00sangrich_Page_101.txt
T102 1843
onliapunovsdirec00sangrich_Page_102.txt
T103 1570
onliapunovsdirec00sangrich_Page_103.txt
T104 902
onliapunovsdirec00sangrich_Page_104.txt
T105
onliapunovsdirec00sangrich_Page_105.txt
T106 1668
onliapunovsdirec00sangrich_Page_106.txt
T107 1654
onliapunovsdirec00sangrich_Page_107.txt
T108
onliapunovsdirec00sangrich_Page_108.txt
T109 1266
onliapunovsdirec00sangrich_Page_109.txt
T110 995
onliapunovsdirec00sangrich_Page_110.txt
T111 1124
onliapunovsdirec00sangrich_Page_111.txt
T112 1441
onliapunovsdirec00sangrich_Page_112.txt
T113 1309
onliapunovsdirec00sangrich_Page_113.txt
T114 929
onliapunovsdirec00sangrich_Page_114.txt
T115 1000
onliapunovsdirec00sangrich_Page_115.txt
T116 1254
onliapunovsdirec00sangrich_Page_116.txt
T117 1600
onliapunovsdirec00sangrich_Page_117.txt
T118 1305
onliapunovsdirec00sangrich_Page_118.txt
T119 649
onliapunovsdirec00sangrich_Page_119.txt
T120 796
onliapunovsdirec00sangrich_Page_120.txt
T121 938
onliapunovsdirec00sangrich_Page_121.txt
T122 752
onliapunovsdirec00sangrich_Page_122.txt
T123 1733
onliapunovsdirec00sangrich_Page_123.txt
T124 1960
onliapunovsdirec00sangrich_Page_124.txt
T125 2081
onliapunovsdirec00sangrich_Page_125.txt
T126
onliapunovsdirec00sangrich_Page_126.txt
T127 1905
onliapunovsdirec00sangrich_Page_127.txt
T128 1833
onliapunovsdirec00sangrich_Page_128.txt
T129 1761
onliapunovsdirec00sangrich_Page_129.txt
T130 1695
onliapunovsdirec00sangrich_Page_130.txt
T131 1597
onliapunovsdirec00sangrich_Page_131.txt
T132 1610
onliapunovsdirec00sangrich_Page_132.txt
T133 1276
onliapunovsdirec00sangrich_Page_133.txt
T134
onliapunovsdirec00sangrich_Page_134.txt
T135
onliapunovsdirec00sangrich_Page_135.txt
T136
onliapunovsdirec00sangrich_Page_136.txt
T137
onliapunovsdirec00sangrich_Page_137.txt
T138 1515
onliapunovsdirec00sangrich_Page_138.txt
T139 1344
onliapunovsdirec00sangrich_Page_139.txt
T140 1537
onliapunovsdirec00sangrich_Page_140.txt
T141 761
onliapunovsdirec00sangrich_Page_141.txt
T142 1162
onliapunovsdirec00sangrich_Page_142.txt
T143 1275
onliapunovsdirec00sangrich_Page_143.txt
T144 1081
onliapunovsdirec00sangrich_Page_144.txt
T145 1775
onliapunovsdirec00sangrich_Page_145.txt
T146 1529
onliapunovsdirec00sangrich_Page_146.txt
T147 1507
onliapunovsdirec00sangrich_Page_147.txt
T148 1690
onliapunovsdirec00sangrich_Page_148.txt
T149 1745
onliapunovsdirec00sangrich_Page_149.txt
T150
onliapunovsdirec00sangrich_Page_150.txt
T151 1793
onliapunovsdirec00sangrich_Page_151.txt
T152 738
onliapunovsdirec00sangrich_Page_152.txt
T153 1790
onliapunovsdirec00sangrich_Page_153.txt
T154 1502
onliapunovsdirec00sangrich_Page_154.txt
T155 1260
onliapunovsdirec00sangrich_Page_155.txt
T156 1155
onliapunovsdirec00sangrich_Page_156.txt
T157 1282
onliapunovsdirec00sangrich_Page_157.txt
T158 1450
onliapunovsdirec00sangrich_Page_158.txt
T159 1455
onliapunovsdirec00sangrich_Page_159.txt
T160 1373
onliapunovsdirec00sangrich_Page_160.txt
T161 1631
onliapunovsdirec00sangrich_Page_161.txt
T162
onliapunovsdirec00sangrich_Page_162.txt
T163 1149
onliapunovsdirec00sangrich_Page_163.txt
T164 1501
onliapunovsdirec00sangrich_Page_164.txt
T165 1607
onliapunovsdirec00sangrich_Page_165.txt
T166 1319
onliapunovsdirec00sangrich_Page_166.txt
T167 1449
onliapunovsdirec00sangrich_Page_167.txt
T168 1251
onliapunovsdirec00sangrich_Page_168.txt
T169 1209
onliapunovsdirec00sangrich_Page_169.txt
T170
onliapunovsdirec00sangrich_Page_170.txt
T171 1322
onliapunovsdirec00sangrich_Page_171.txt
T172 1122
onliapunovsdirec00sangrich_Page_172.txt
T173 1245
onliapunovsdirec00sangrich_Page_173.txt
T174 1462
onliapunovsdirec00sangrich_Page_174.txt
T175 1249
onliapunovsdirec00sangrich_Page_175.txt
T176 1544
onliapunovsdirec00sangrich_Page_176.txt
T177 1396
onliapunovsdirec00sangrich_Page_177.txt
T178 1272
onliapunovsdirec00sangrich_Page_178.txt
T179 662
onliapunovsdirec00sangrich_Page_179.txt
T180 1669
onliapunovsdirec00sangrich_Page_180.txt
T181
onliapunovsdirec00sangrich_Page_181.txt
T182 1673
onliapunovsdirec00sangrich_Page_182.txt
T183 659
onliapunovsdirec00sangrich_Page_183.txt
T184 767
onliapunovsdirec00sangrich_Page_184.txt
T185 1608
onliapunovsdirec00sangrich_Page_185.txt
T186 1422
onliapunovsdirec00sangrich_Page_186.txt
T187 1335
onliapunovsdirec00sangrich_Page_187.txt
T188 1168
onliapunovsdirec00sangrich_Page_188.txt
T189 1114
onliapunovsdirec00sangrich_Page_189.txt
T190 1416
onliapunovsdirec00sangrich_Page_190.txt
T191 301
onliapunovsdirec00sangrich_Page_191.txt
T192 386
onliapunovsdirec00sangrich_Page_192.txt
T193 188
onliapunovsdirec00sangrich_Page_193.txt
T194 1210
onliapunovsdirec00sangrich_Page_194.txt
T195 1025
onliapunovsdirec00sangrich_Page_195.txt
T196
onliapunovsdirec00sangrich_Page_196.txt
T197 566
onliapunovsdirec00sangrich_Page_197.txt
T198 1211
onliapunovsdirec00sangrich_Page_198.txt
T199 1273
onliapunovsdirec00sangrich_Page_199.txt
T200 646
onliapunovsdirec00sangrich_Page_200.txt
T201 1197
onliapunovsdirec00sangrich_Page_201.txt
T202 1244
onliapunovsdirec00sangrich_Page_202.txt
T203 1338
onliapunovsdirec00sangrich_Page_203.txt
T204 871
onliapunovsdirec00sangrich_Page_204.txt
T205 846
onliapunovsdirec00sangrich_Page_205.txt
T206 1333
onliapunovsdirec00sangrich_Page_206.txt
T207 1656
onliapunovsdirec00sangrich_Page_207.txt
T208 1524
onliapunovsdirec00sangrich_Page_208.txt
T209 954
onliapunovsdirec00sangrich_Page_209.txt
T210 528
onliapunovsdirec00sangrich_Page_210.txt
T211 1596
onliapunovsdirec00sangrich_Page_211.txt
T212
onliapunovsdirec00sangrich_Page_212.txt
T213 827
onliapunovsdirec00sangrich_Page_213.txt
T214 1615
onliapunovsdirec00sangrich_Page_214.txt
T215 746
onliapunovsdirec00sangrich_Page_215.txt
T216 1055
onliapunovsdirec00sangrich_Page_216.txt
T217 711
onliapunovsdirec00sangrich_Page_217.txt
T218 469
onliapunovsdirec00sangrich_Page_218.txt
T219 717
onliapunovsdirec00sangrich_Page_219.txt
T220 267
onliapunovsdirec00sangrich_Page_220.txt
T221 1484
onliapunovsdirec00sangrich_Page_221.txt
T222 89
onliapunovsdirec00sangrich_Page_222.txt
T223 1737
onliapunovsdirec00sangrich_Page_223.txt
T224 1748
onliapunovsdirec00sangrich_Page_224.txt
T225 1063
onliapunovsdirec00sangrich_Page_225.txt
T226 1011
onliapunovsdirec00sangrich_Page_226.txt
T227 1320
onliapunovsdirec00sangrich_Page_227.txt
T228 2172
onliapunovsdirec00sangrich_Page_228.txt
T229 2737
onliapunovsdirec00sangrich_Page_229.txt
T230 2359
onliapunovsdirec00sangrich_Page_230.txt
T231 2003
onliapunovsdirec00sangrich_Page_231.txt
T232 1108
onliapunovsdirec00sangrich_Page_232.txt
T233 1624
onliapunovsdirec00sangrich_Page_233.txt
T234 1592
onliapunovsdirec00sangrich_Page_234.txt
T235 1717
onliapunovsdirec00sangrich_Page_235.txt
T236 1220
onliapunovsdirec00sangrich_Page_236.txt
T237 1493
onliapunovsdirec00sangrich_Page_237.txt
T238 1642
onliapunovsdirec00sangrich_Page_238.txt
T239 316
onliapunovsdirec00sangrich_Page_239.txt
T240
onliapunovsdirec00sangrich_Page_240.txt
T241 1804
onliapunovsdirec00sangrich_Page_241.txt
T242
onliapunovsdirec00sangrich_Page_242.txt
T243 1531
onliapunovsdirec00sangrich_Page_243.txt
T244 1505
onliapunovsdirec00sangrich_Page_244.txt
T245 1279
onliapunovsdirec00sangrich_Page_245.txt
T246
onliapunovsdirec00sangrich_Page_246.txt
T247 1805
onliapunovsdirec00sangrich_Page_247.txt
T248 1365
onliapunovsdirec00sangrich_Page_248.txt
T249
onliapunovsdirec00sangrich_Page_249.txt
T250 1561
onliapunovsdirec00sangrich_Page_250.txt
T251 1688
onliapunovsdirec00sangrich_Page_251.txt
T252 1671
onliapunovsdirec00sangrich_Page_252.txt
T253 1550
onliapunovsdirec00sangrich_Page_253.txt
T254 1299
onliapunovsdirec00sangrich_Page_254.txt
T255 1703
onliapunovsdirec00sangrich_Page_255.txt
T256
onliapunovsdirec00sangrich_Page_256.txt
T257 1528
onliapunovsdirec00sangrich_Page_257.txt
T258 1259
onliapunovsdirec00sangrich_Page_258.txt
T259 1359
onliapunovsdirec00sangrich_Page_259.txt
T260 1447
onliapunovsdirec00sangrich_Page_260.txt
T261
onliapunovsdirec00sangrich_Page_261.txt
T262 1732
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T263 1697
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P222 Appendix A 222
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