
Citation 
 Permanent Link:
 http://ufdc.ufl.edu/AA00003782/00001
Material Information
 Title:
 Uniform stabilization of the EulerBernoulli equation with active Dirichlet and nonactive Neumann boundary feedback controls
 Creator:
 Bartolomeo, Jerry, 1960
 Publication Date:
 1988
 Language:
 English
 Physical Description:
 vi, 47 leaves : ; 28 cm.
Subjects
 Subjects / Keywords:
 Boundary conditions ( jstor )
Euler Bernoulli beam theory ( jstor ) Hilbert spaces ( jstor ) Mathematical theorems ( jstor ) Mathematics ( jstor ) Semigroups ( jstor ) Topological theorems ( jstor ) Topology ( jstor ) Vector fields ( jstor ) Wave equations ( jstor ) Bernoulli polynomials ( lcsh ) Boundary layer control ( lcsh ) Differential equations, Partial ( lcsh ) Hilbert space ( lcsh )
 Genre:
 bibliography ( marcgt )
theses ( marcgt ) nonfiction ( marcgt )
Notes
 Thesis:
 Thesis (Ph. D.)University of Florida, 1988.
 Bibliography:
 Includes bibliographical references.
 General Note:
 Typescript.
 General Note:
 Vita.
 Statement of Responsibility:
 by Jerry Bartolomeo.
Record Information
 Source Institution:
 University of Florida
 Holding Location:
 University of Florida
 Rights Management:
 Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for nonprofit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
 Resource Identifier:
 024531425 ( ALEPH )
AFL1799 ( NOTIS ) 19976728 ( OCLC )

Downloads 
This item has the following downloads:

Full Text 
UNIFORM STABILIZATION OF THE EULERBERNOULLI EQUATION
WITH ACTIVE DIRICHLET AND NONACTIVE NEUMANN
BOUNDARY FEEDBACK CONTROLS
By
JERRY BARTOLOMEO
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1988
ft E ER LIBRARIES
ACKNOWLEDGEMENTS
I express my sincere appreciation to my advisor Dr. Roberto Triggiani for his leadership and support
over the past four years and for directing this research project I would also like to thank the other
members of my committee, especially Dr. Irena Lasiecka who also played a major role in my education.
Also, I would like to thank my parents, James and Elisa Bartolomeo, and my wife Michelle, for their con
tinued encouragement. Finally, special thanks are due to John Holland who did an excellent job in prepar
ing this document.
TABLE OF CONTENTS
page
ACKNOW LEDGEMENTS ......................................................................................................... ii
ABSTRACT .................................................................................................................. .................... v
CHAPTERS
1 INTRODUCTION, PRELIMINARIES, STATEMENT OF MAIN RESULTS .............. 1
1.1 Introduction And Literature ........................................................... ..................... 1
1.2 Formulation of the Uniform Stabilization Problem and Main Statements ............... 3
Choice of Operators F and F 2 ................................................... ............... 5
Theorem 1.1 ....................................................................................................... 7
Theorem 1.2 ...................................................................................................... 8
Theorem 1.3 ............................................................................................................ 8
2 WELLPOSEDNESS AND STRONG STABILIZATION .......................................... 10
2.1 Preliminaries, Choice of Stabilizing Feedback ............................................ ....... 10
2.2 W ellPosedness and Semigroup Generation ............................................ ......... 14
Lemma 2.5 ......................................................................................................... 14
Theorem 2.6 ....................................................................................................... 15
Proof of Theorem 2.6 ......................................................................................... 16
Lemma 2.7 ......................................................................................................... 18
Proof of Lemma 2.7 ........................................................................................... 19
Corollary 2.8 ...................................................................................................... 20
Theorem 2.10 ..................................................................................................... 21
Proof of Theorem 2.10 ....................................................................................... 21
3 UNIFORM STABILIZATION ................................................................................... 22
3.1 Preliminaries, Change of Variables ....................................................................... 22
3.2 Uniform Stabilization ................................................................................................. 23
Theorem 3.1 ....................................................................................................... 23
Theorem 3.2 ....................................................................................................... 24
Proof of Theorem 3.2 (A multiplier approach) ...................................... .......... 24
Lemma 3.3 ......................................................................................................... 24
Lemma 3.4 ......................................................................................................... 26
Lemma 3.7 ......................................................................................................... 29
Lemma 3.8 ......................................................................................................... 30
Proof of Lemma 3.8 ................................................................................................. 31
 111ii
Lemm a 3.9 ............................................................................................................... 32
Lemm a 3.10 ....................................................................................................... 32
Lemm a 3.11 ....................................................................................................... 34
Proof of Theorem 3.1 ......................................................................................... 35
APPENDICES
A BASIC IDENTITIES .................................................................................................. 37
B TO HANDLE DIFFERENCE OF ENERGY TERM ...................................... ......... 39
C TO OBTAIN GENERAL IDENTITY ..................................................................... 40
REFERENCES ............................................................................................................................ 43
BIOGRAPHICAL SKETCH ....................................................................................................... 47
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy
UNIFORM STABILIZATION OF THE EULERBERNOULLI EQUATION
WITH ACTIVE DIRICHLET AND NONACTIVE NEUMANN
BOUNDARY FEEDBACK CONTROLS
By
Jerry Bartolomeo
December 1988
Chairman: Dr. Roberto Triggiani
Major Department Mathematics
Given Q an open bounded domain in IR" with sufficiently smooth boundary r, we consider the
nonhomogeneous EulerBeroulli equation in the solution w (t,x):
w + A2w =0 in Q = (0,o) x Q (a)
w(0,')= wo ; w',(O,)= w in Q (b)
wl =g e L2(j)=L2((0,oo);L2(L)) onE=(0,)o)xr (c) (1)
a =g2=0 on E (d)
We seek to express the nonzero control function g as as suitable linear feedback applied to the velo
city w, i.e., w 1= Fw,, such that Fw, E L2((0,oo); L2() ), and the corresponding closed loop system
obtained by using such a feedback in (1) generates a Cosemigroup which decays uniformly exponentially
to zero as to in the uniform topology of Z = [D(A"4)]' x [D(A1^)]':
II w(t),w(t) II Ce II [wo,wl i IIZ for all t >0 and some C, 8 > 0
Having identified the candidate Fw, = [A(A w,)], where A is the operator defined by
Af A2f; D(A)= feL2(): 2f eL2(), = = = 0 ,we prove two stabilization results,
the second of which (2) is the goal of this thesis and implies the first. However, we include both results to
illustrate a contrast in modern day control methods.
Specifically if the domain 0 satisfies a radial vector field assumption
(xxo) v y > 0 on r (3)
where v is the outward unit normal vector, we prove strong stabilization, i.e., solutions go to zero in the
strong topology of Z : lim II [w (t),w,(t)] II z = 0, by the use of a Hilbert space decomposition for contrac
tive semigroups. Finally, if 0 satisfies (3), we obtain the desired uniform stabilization (2) via a change of
variables followed by the use of multipliers.
CHAPTER 1
INTRODUCTION, PRELIMINARIES, STATEMENT OF MAIN RESULTS
1.1 Introduction And Literature
Let 0 be an open bounded domain in IR", n typically 2 2, with sufficiently smooth boundary r. In
Q we consider the EulerBernoulli mixed problem in w (tx) on an arbitrary time interval (0,T] :
w,, +Aw = 0 in (0,T]x Q (a)
w(O, x)= wo(x); w,(O, x)= wl(x) in Q (b)
w(t, a)= g (t, a) on (O,T] x F (c) (1.1)
aw
(t, a) = g2(t, a) on (0,T] x r (d)
with nonhomogeneous forcing terms (control functions g and g2 in the Dirichlet and Neumann boundary
conditions). In (1.1d), v denotes the unit normal to F pointed outward. Recently there has been a keen
resurgence of interest (e.g. [LL.1], [R.3] and references cited therein) in the theory of plate equations, of
which the EulerBernoulli equation (1.la) is a canonical model, presumably stimulated by two main
sources:
i) renewed studies in the dynamics, feasibility, and implementation of socalled large scale flexible
structures envisioned to be employed in space;
ii) recent mathematical advances in regularity theory of second order mixed hyperbolic problems
canonicallyy, the wave equation of both Dirichlet Type [L.8] [LT.8] [LT.9] [LLT.1] and of Neu
mann type [LT.8], [LT.9], [LT.10], [LT.13], [LT.14], [LT.15], [S.1]) with L2 boundary data.
In either case, a prime thrust of motivation has come from dynamical control studies, at either an
engineering or a theoretical level. With reference to the specific problem (1.1), we cite, [L.7], [LT.1],
[LT.2] for optimal regularity theory and exact controllability theory with respect to classes of interest for
the initial data [ wo,wl ] and for the boundary data {g1,g2}, which markedly improved upon prior litera
ture [LM]. Our interest in the present work is on the problem of boundary feedback uniform stabilization
1
for the dynamics (1.1) by explicit feedback operators, to be more properly defined below. Our results are
fully consistent with the corresponding exact controllability results [LT.l], [LT.2], not only with respect
to the function spaces for {gl,g21 and [w,w,] as mentioned above, but also with respect to the lack of
geometrical conditions on Q when both g and g2 are active, or else with respect to the presence of similar
geometrical conditions on 0 when only g 1 is active and g2 is taken g2 = 0. This consistency is most desir
able, and indeed has to be expected, in view of the known relationship between exact controllability and
uniform stabilization for time reversible dynamics such as (1.1) [R.1] [R.2]. We note, in passing, that uni
form stabilization of problem (1.1) by means of a feedback operator ( acting on [w,w,] ) which is defined in
terms of the algebraic Riccati operator (which arises in the study of the optimal quadratic cost problem on
(0,,) ) was already achieved in the abstract treatment of [FLT.1]. Mathematically the present work is
guided by and partially rests upon techniques developed in two main sources:
(i) the studies of exact controllability [LT.1], [LT.2] for problem (1.1);
(ii) the study of uniform stabilization of the wave equation with boundary feedback in the Dirichlet
boundary conditions [LT.4] and in the Neumann boundary conditions [T. 1].
Of course these studies have to be seen in the context of recent investigations including:
a) uniform stabilization of the wave equation with feedback in the Neumann boundary conditions
[C.l], [C.2], [L.4], [L.5], [LT.4. sect 4];
b) regularity theory for hyperbolic equations in [L.8], [LT.8], [LT.9], [LLT.1], as well as
corresponding exact controllability theory [L.6], [LT.12], [H.1], [T.2], [T.3];
c) exact controllability results for EulerBernoulli equations with different boundary conditions
[L.6], [L.7], [LT.6], [LL.1]; and, finally,
d) corresponding optimal quadratic cost problems [DLT.l], [LT.11 [FLT.1].
A point of view which we stress is the following: we choose g in 'open loop' form to be in
L2((0,T); L2(r)), T
wo = wl = g2 = 0 satisfies [w,w,] E C([0, T]; Z), where Z is the space identified in (1.7) below. This is
an optimal regularity result [L.7], [LT.2]. Next, we choose g2 such that the corresponding solution of
(1.1) with gl = o= = 0 also produces [w,w,] e C([O, T]; Z), again as an optimal regularity result
This leads to [w,w,] e Z and g2 L2( (0,T) ; H ()) [L.7], [LT.1] [LT.2].
In other words, only one choice is made, that g1 e L2( (0, T); L2() ); then, we work with data and
solutions in the corresponding optimal spaces. Our solution of the uniform stabilization problem below is
fully consistent with these 'open loop' considerations: uniform stabilization will be achieved in the space
Z with controls in feedback form gl E L2((0,oo) ; L2(0)) and g2 L2((0,oo); H'(T)), (see Theorem
1.2 below). However, the present thesis will treat only the case g I L2( (0,oo); L2(f) ) in feedback form
and g2 = 0, leaving the more general case to our successive effort [BT.1].
We finally point out that other uniform stabilization problems for plate equations have been/are being
considered following LagneseLions recent monograph [LL.1] on plates [L.1], [L.3]. In these works,
however, different boundary conditions occur, typically of higher order, e.g., Aw and (Aw) .As a conse
av
quence, uniform stabilization results are sought and obtained in higher topologies on 0 (so called
corresponding 'energy spaces'). The problem considered in this thesis with boundary conditions as in
(1.lc), (1.ld) of lower order have a natural (and optimal, in fact) setting in very low topologies on Q; see
the optimal space below. This produces additional mathematical difficulties. To overcome these obstacles,
it will be necessary to introduce a new variable, the variable p below in (3.3), which lifts the topologies on
Q to the level where the multiplier techniques which were successfully used in [LT.1], [LT.2] for the
corresponding exact controllability problem are applicable. A similar lifting was employed in the case of
the uniform stabilization problem for wave equations with Dirichlet feedback in the (low, but optimal)
topology L2(p) x H1 (Q) [LT.4].
1.2 Formulation of the Uniform Stabilization Problem and Main Statements
Throughout the paper we let < >n denote the L2(0)inner product with associated norm
I II n, and < >r denote the L2(T)inner product with associated norm II II r. In addition, L(f)
denotes the Hilbert space of all bounded linear operators on L2(fl) and L(r,T) denotes the Hilbert space
of all bounded linear operators from L2() into L2(0). Finally, L(X) will denote the Banach space of all
bounded linear operators on X.
We begin by letting A: D(A) c L2(f) L2(Q) be the positive, selfadjoint operator defined by
AfA2f; I(A)=H4 (Q)fH(Q) (1.2)
With the operator A defined as such, it then follows that [ LT.2, Appendix C ]
1D(A 4)=Hi (Q) {fe H (Q) :f =0 } (1.3)
D(A 3) = V fe H(Q) : flr = =0 (1.4)
where we use = to denote norm equivalence.
Thus, forf e D(A")
112
Ilfllo(A,)= IIA'flla = IlfllH.(O= { IVfl2dQ } (1.5)
where the last equivalence follows by Poincare inequality.
Similarly forf E ID(A)
1/2
llfll ( =I A3'f { = nV(Af/I2d } (1.6)
As suggested by [LT.1], [LT.2], our optimal space in which to study stabilization will be,
Z = H' (Q) x V' = [ID(A "')]' x [D(A 3)]' (1.7)
where denotes duality with respect to the L2(Q) topology. Next, let gl = g2= 0 in (1.1). Then, the
corresponding evolution of (1.1) is governed by the operator
A= IOA101 (1.8)
Ao= (A1 8)
which generates a strongly continuous unitary group ( on the space D(A 2)xL2(Q) with domain
1D(Ao) = (A)x D(Aa2) and hence ) on the space Z of our interest with domain
D)(Ao) = AD(A "4) x [1)(A 4)]' = H (0) x H' (f). We denote this unitary group by e At. Thus, the free
solutions of (1.1) with g = g2 = 0 are normpreserving in Z:
II[ w(t),w,(t)] I Z I leAt[ wo,wl i11 z= II [ wo,w1, i z, forall t E ?
With this wellknown result at hand, we can state the aim of the paper. Motivated by and consistent
with the function spaces in the optimal regularity and exact controllability theory of (1.1) [L.7], [LT.1],
[LT.2], we shall study the question of existence and construction of explicit boundary feedback operators
F1 and F2 based on the 'velocity' w,
F1 (w,) L((0,oo) ; L2()) (1.9)
F2(W,) E L2((0,o) ; H()) (1.10)
such that the boundary feedback functions
gl =F1 (w,), g2=F2 (w,) (1.11)
once inserted in (1.1 cd) produce a (feedback) Cosemigroup eAt which is exponentially stable in the uni
form operator norm L(Z) of the space Z in (1.7): namely, there exist constants M > 1,8 > 0 such that for
all t0
IleAt II L(Z)
We then say that such operators Fi and F2 uniformly (exponentially) stabilize the original norm
preserving (conservative) dynamics (1.1) with g = g2 = 0.
A weaker concept is that of 'strong stabilization', by which we seek operators Fi and F2 as in
(1.11) such that once inserted in (1.1 cd) produce a (feedback) Cosemigroup eA on Z which decays
strongly to zero.
IIeA II z ast o, forallzeZ (S.S)
We note that for compact semigroups such as those arising in parabolic equations on a bounded
domain Q, the concept of stability in the strong topology (even weak topology) is equivalent to the concept
of stability in the uniform topology. However, for second order hyperbolic problems and plate problems
such as (1.1), the two concepts of strong and uniform stability are distinct. For recent optimal results on
the lack of uniform stabilization, see [T.5], [T.6].
Choice of Operators Fi and F,
It is justified in Chapter 2 in the case of F1 and in [BT.1] in the general case that the following
choices of Fi and F2
g = F (w,)= k (x)G*Alaw, = k (x)GAAmw, =kI(x) A() ~ (1.12)
2 = F2(w,) = k2(x)A2GAAMw, = k2(x)A2A(Aw,) (1.13)
provide reasonable candidates for the uniform stabilization problem of (1.1), in the sense that the closed
loop feedback dynamics with (1.12) and (1.13) inserted in (1.lc) and (1.ld), respectively, is wellposed in
the semigroup sense in Z and the Znorm of all of its solutions originating in Z decreases as t + +0 ( this,
however, does not say that such Znorms decreases to zero as t > +o, (strong stabilization), let alone in
the uniform norm of L(Z)). To show this conclusion will be our major task in Chapter 3, at least in the
case where g2 0 at the outset. In (1.12), (1.13) we have that:
a) ki(x) = smooth functions on F, ki(x) > 0; (1.14)
b) A : (onto) isomorphism H'(F) > H''(), selfadjoint on L2() (1.15)
so that
1/2
lAgL'( 11= \\\\ g ; =[\r I Veg 8dr (1.16)
where VS denotes the tangential gradient on F;
c) The operators G* are the adjoints, in the sense that
< Gig, z > = < g, G*z >r, g L2(), z E L2() (1.17)
of the operators Gi defined by:
A2x=0 in Q (a)
Glg=x ifandonlyif xj =gl onF (b) (1.18)
*r
ax I
0 onFr (c)
A2y = 0 in Q (a)
G2g=y if and only if y =0 on (b) (1.19)
y = g2 onr (c)
av r
Elliptic theory [LM, Vol. 1] gives for any s E R
G 1 : continuous H'(I) H"'+ '"(f)
G2 :continuous H'() H'"2(Q)
Moreover, by Green's theorem it is proved that [LT.2, Lemma 2.0 and Lemma 4.0, respectively]
G*IAf = l fE D(A)
0V Ir
(1.20)
(1.21)
(1.22)
G;Af =(Af) fe D(A) (1.23)
Identities (1.22), (1.23) are used in the last step of (1.12), (1.13), respectively. Thus, the resulting candi
date feedback system, whose stability properties in Z we shall investigate is,
w, + A2w =0 in (0,oo) x Q = Q (a)
w(0,x) = wo(x) ; w,(0,x)= w1(x) in Q (b)
Sw =kx(x) ( on (0,o) x r = (c) (1.24)
dv 1x
SI=k2(x)A2A(A3aw)I on E (d)
Using the techniques of [T.4], problem (1.24) can be rewritten more conveniently in abstract form as
dt ;II=AI
A= 0 I
A= IA A[kxG1GIA1 +k2G2A2G2A1']
o)(A) = yeZ: AyEz
A more explicit description of D(A) will be given below. Our main results are as follows.
(1.25)
(1.26a)
(1.26b)
Theorem 1.1
(i) Wellposedness on Z
The operator A in (1.26) is dissipative on Z = [D(A' )]' x [1D(A 3)]', see (1.7), and satisfies here:
range (QA) = Z for X > 0. Thus, by LumerPhillips theorem [P], A generates a strongly continuous
contraction semigroup eAt on Z, and the resolvent operator R(X; A) is compact on Z, for ReX > 0.
Moreover 0 E p(A), the resolvent set of A.
(ii) Boundedness of feedback operators
For [ wo,w I ] Z, we have
A(A3w1)
k1G*A'aw,=k E L2((0,o) ; L2(P)) (1.27)
k2A2GA2w, = k A2A(Amw,) e L2( (0,*) ; H1 (T) (1.28)
more precisely
o IIG k Aw, Idt l [wo., Z (1.29)
Io I k2A2G A`w, rdt  [ wo.wl i II Z (1.30)
The proof of Theorem 1.1 will be given in Chapter 2, in the case of g2 0, and in [BT.1] in the general
case.
Theorem 1.2 ( Uniform stabilization on Z with both feedback operators in the absence of geometrical con
ditions on Q )
The following property holds for the feedback problem (1.24), or (1.25), (1.26): there are constants
M > 1 and 6 > 0 such that for all tr0
!w(t) o 0o
wII i(t) i AuI w IIz z Meu II lIIZ (1.31)
Theorem 1.3 ( Uniform stabilization on Z with only the first feedback operator g and g2 = 0, in the pres
ence of geometrical conditions on Q ).
Consider the feedback problem (1.1) with gl given by (1.12) while g2 = 0. Then there is a constant
8 > 0 such that the uniform decay (1.31) holds true, provided Q satisfies the following geometrical condi
tion (radial vector field assumption ):
there exists a pointxo E R" such that (xxo) v y > 0 on r (1.32)
Theorem 1.3 is the main result proved in this thesis. It may be extended to more general domains Q
which satisfy a weaker geometrical condition than (1.32), expressed in terms of a more general vector field
than the class of radial fields (xxo). This will be done in [BT.1]. Also Theorem 1.2the general result
with two feedback operators and no geometrical conditionwill likewise appear in [BT.1]. Instead, in
Chapter 2 of this thesis we shall also prove a strong stabilization result (Theorem 2.10) with g i as in (1.12),
( and k1=l ) and g2 = 0, under the same geometrical assumption (1.32). Though the strong stabilization
result of Theorem 2.10 in Chapter 2 is implied by the uniform stabilization result of Theorem 1.3 in
Chapter 3, we feel that its inclusion in this thesis is justified by the following considerations. It shows 'how
far' the purely operator techniquesbased on the operator model (1.25), (1.26) and the NagyFoiasFoguel
decomposition for contraction semigroupscan be carried out along the lines of arguments first used for
boundary control problems for second order hyperbolic equations in [LT. 16], [LT.17], [LT.4] and also in
[T.1] for different feedback operators. A new obstacle arises, however, in the case of plate problems
(fourth order in the space variable rather than second order in space as in hyperbolic problems) at the level
of excluding the presence of eigenvalues along the imaginary axis. This is Lemma 2.7, where only three
homogeneous boundary conditions (2.31bcd) are in place for the 'eigenproblem' (2.31a), as opposed to
the four homogeneous boundary conditions for a fourth order operator covered by standard theory. (In pre
vious arguments for second order hyperbolic problems as in [LT.16], [LT.17], [LT.4], [T.1], the
corresponding 'eigenproblem' has two homogeneous boundary conditions for a second order operator and
hence is covered by standard theory). A novelty is then that Lemma 2.7 uses a multiplier technique proof,
which is supported by the exact controllability problem [LT.1], [LT.2], as applied however to the (sta
tionary) elliptic problem this time. It is the multiplier technique that requires the geometrical condition
(1.32) in the proof of Lemma 2.7. Lemma 2.7 appears to be new in elliptic theory. This fact alone would
justify its inclusion here, even though Lemma 2.7 plays only a secondary role in the problem of stabiliza
tion considered here, as it leads only to strong stabilization (Theorem 2.10). It is through the more ela
borate and lengthier arguments of Chapter 3which are necessitated by use of the same multiplier applied
to the feedback dynamics (1.24)that we will eventually obtain the uniform stabilization result of Theorem
1.3.
CHAPTER 2
WELLPOSEDNESS AND STRONG STABILIZATION
2.1 Preliminaries, Choice of Stabilizing Feedback
Let Q be an open bounded domain in IR", n>2 with sufficiently smooth boundary r. Consider the
nonhomogeneous problem in the solution w (t,x):
wt+A2w=0 in Q = (0,o) x Q (a)
w(0O,)=Wo w,(0,)= W in Q (b)
w I=gl EL2(Y)=L2((0,_O);L2(I)) on =(0,_)xr (c) (2.1)
a =82=0 on E (d)
The goal of this chapter is to obtain strong stabilization of the system (2.1) via a closedloop feedback gI
based on the velocity w,. However, the optimal function space in which to work (a cross product space for
position and velocity) is obtained from the exact controllability result to be summarized below.
First, we define the positive, selfadjoint operator A: D)(A) c L2() + L2(() by
Af = A2f (2.2a)
D(A)= {fE L2(): A2fE L2'(), fr \= 0 = H 4() H () (2.2b)
Ir = v r
Since Q is bounded in R", then A has compact resolvent R( ; A). Also, if Af = 0 for fe D(A) then by
(A.O) and (2.1c,d) we have
O= < Af f > = < A(A), f >
=< Af, Af>n+ < (A f>rr
=< A, Af>n= II Afll
10
This implies f = 0, so therefore,
A' L(() (2.3)
Next, we let
V= { feH 3 ): : fi= f =0 (2.4a)
and consider the space Z = H' (Q) x V'. As shown in [LT.2], Z can be characterized by using equivalent
norms as
Z = [I(A 1)]' x [D(AW3)]' (2.4b)
where' denotes duality with respect to the L2(()topology.
The norms on these spaces are given by
(2.5)
Ilxll(Ai)= IIAx lln; IlxltII r((P = IIAxlln a,p >0 (2.5)
Below we state the regularity result as well as the exact controllability result.
Theorem 2.0 (Regularity) [L.71, rLT.21
Consider the problem (2.1) subject to [w0,w1] EZ, ge L2((0,T); L2(r)),
g2 E L2((0,T); Ht'()). Then the map {wo,w1,gl,g2) [w(t),w,(t)] e C([0,T] ;Z) is continuous
forany0
Theorem 2.1 (Exact Controllability) fLT.21
(i) Assume there exists a point xo E R?" such that (xxo) v 2 y > 0 on r where v is the unit out
ward normal vector. Let 0 < T < be arbitrary. If [wo,wl ] Z arbitrary, then there exists a suitable con
trol function gl e L2( (0,T); L2)), such that the corresponding solution of (2.1) with g2 0 satisfies
w(T,) = w,(T,) 0 and in addition [w,w,] e C( [0,T] ; Z)
(ii) The same conclusion holds true without geometrical conditions if g2 is taken within the class of
L2((0,T) ; H'(I)) controls. O
By time reversibility, we see that at any finite T the totality of all solution points (w(T),w,(T)) of
problem (1.1) with w0 = wl = 0 fills all of the space Z when either gl runs over all of L2(0(,T) ; L2() )
and g2=0 under geometrical conditions on Q, or else when (gl,g2) runs over all of
L2( (0,T); L2()) xL2( (0,T); H' () ) without geometrical conditions. Therefore, since the space of
exact controllability is the space of maximal regularity, we seek stabilization in exactly this space Z.
We define the "energy" E (t) for the dynamics (2.1) over the space Z = [D(A 1))]' x [I(A )]' by
I W(t)l 112 1 2 2
E (t)= Iw,(t)l = II [(A ) X wNtx )II [(A")]
= IA"w(t) II + IA34w,(t) I (2.6)
dE
Next we seek a candidate g which at least produces 0, i.e., energy "decrease." This does not,
dt
however, guarantee lim E (t)= 0 (which is precisely strong stability of (2.1)), let alone uniform stability.
Remark 2.2
Below we shall show wellposedness in Z, with g2 = 0. Then since w, e [ID(A )]', it follows that
A3w, = A 3A 4w, e D(A 3) a V. Therefore, A w, satisfies the required boundary conditions.
A3a2w =a(Asw,) = 0 (2.7)
IF rFV IF
By writing E(t)= < A"w A w >Q + < A4w,, A 4w, >0 and differentiating with respect to t
we have that
dE= < A'w, < 1/4A w, A4w, >a
2dt
by (2.1a)
= < w, AU2w > < Aw, Amw, >a = < w, A1aw, >Q < A(Aw), A3w, >0
by (A.O)
= < w A2w, >a < (Aw) A'w, >r < Aw, (A w,) >r + < Aw A(A32w) >0
by (2.7)
= < w A'w, >n < Aw, A(Amw,) >n
13
by (A.0)
= < w, Anw, >0 < A(A w,) >r < w, [A(Aawt)] >r + < w, A(Amw) >n
= < w, A > + < w, [A(Aw,)] >r < w, Aw, >a = < w [A(Aw)] >r
ov dv
Therefore, by selecting the simplest choice
wI =g = A[A(A2w,)] (2.8)
dE
We obtain =2 1 gi 1 0, our desired energy decrease. 0
Next we will show how our feedback can be expressed in terms of an operator (Green map) which
acts from boundary F to interior Q. Following [LT.1],[LT.2] we define G : L2(F) + L2() by
Ay =0 in Q (a)
G lg=y if and only if y =g on F (b) (2.9)
'I = 0 on r (c)
We quote the following Lemma which will be used below.
Lemma 2.3 [LT.21
Let G* :L2(p) * L2(r) denote the continuous operator defined by
< Gig v > = < g, Gvv >r, g e L2(), v e L2(), i.e., G\ is the adjoint of G. Then
G;Af= (Af) forofr D(A) (2.10)
Now using (2.8) and (2.10) we see that
w I= = [A(Aw,)] = GIA(Aww,)= G*Aa2w, (2.11)
Using elliptic theory [ LM, Vol I, p. 188 ] we have that for any s real
G : continuous H'(") ~ H'+ a(0) (2.12a)
and in particular for s = 0
G1 : continuous L2(r) Hm(0) (2.12b)
14
We also have that by duality on (2.12a) with s = 3/2 that
G : continuous H1(Q) H11(r) (2.13)
so that (2.12a), (2.13) imply
G1G : continuous D(A')= Hn(Q) H2(Q) (2.14)
to be used below (2.17b) in the description of the domain of the feedback generator.
2.2 WellPosedness and Semigroup Generation
First we want to introduce an abstract operator model for problem (2.1). According to [T.1], [T.4],
problem (2.1) with g 2 = 0 admits the following abstract versions:
as a second order equation
iw = A [ w G gl] = A [ w + GIG*A`aw] (2.15a)
or else as a first order system
 =Ali ; [w, w Z] Z =[D(A m)]' x [D(A )]" (2.15b)
0 1
where A= IA AG G*,AaI (2.16)
More explicitly if ye D(A) then we can write
.A Io 1 + GIG'A1 y2
Ay AOI il (2.17a)
Thus,)(A)= { [y y21E Z:y2 [)(AI )]andA [y, +G G*IALay2] [D(A4)]' ,i.e.,
yi + GGIAmy2 e 0D(A"4) = Hoi() which implies y, H1'() } (2.17b)
The operator A defined above is our candidate to be the generator of a feedback semigroup. The first step
in this direction is the following Lemma.
Lemma 2.5
The operator A is dissipative on Z = [D(A 14)]' x [ID(A 3)]'.
15
Proof
Let z E Z, then using below the skewadjointness of IA 01 we have for Z E I(A)
Re< Az, z >z=Re< A 0 21
+Re< AGIGA II21 I 21
=0< AGIG*A'Z2 zz2 >[(A")]' = < A4AG1G*A'^Z2, A~z2 >0
=II GA'2z2 II < 0 and dissipativity holds. 0
The above proof is a reformulation of our argument below (2.7). Now we come to our result on semigroup
generation.
Theorem 2.6
(i) The dissipative operator A in (2.16) also satisfies range ( I A) = Z on Z for X > 0.
Thus, by the LumerPhillips Theorem [P], A generates a Cosemigroup of contractions eAt on Z, t 2 0
and the solution of (2.1),(2.11) is given by
w(t wo ,1) WO
IW(t W W)I= eAt all t>0, [WO, w~ lZ (2.18a)
w,(t Wo, W) wi
and in fact
IIeAt:I Iz=E(t)=J { IA"wlI2+ IA"w,I2} dn (2.18b)
(ii) The resolvent operator R(X ; A) of A is given by
I V()I V()_'A_'
R(X; A)= vX)1 V(^ A11 (2.19a)
where V(X) = [I + XG1G IA1 + V2A ] (2.19b)
at least for all X satisfying ReX > 0 Moreover X = 0 belongs to the resolvent set of A and R(X; A) is
compact on Z.
16
(iii) If the domain l is such that there exists a point x0 E R" such that
(xxo)v>y>0 onr (2.20)
then R(1; A) is welldefined and compact on Z also on the imaginary axis and hence, for all X satisfying
ReX > 0. Thus, the spectrum (point) of A satisfies
a(A) c ( X: ReX < 0 (2.21)
Remark
A stronger result will follow below once we prove our uniform stabilization Theorem 1.3, that in fact
a(A)c ( {: ReX 8 < 0 ). 0
Proof of Theorem 2.6
Dissipativity of A on Z was already shown in Lemma 2.5. Next, fix X > 0 and let z E Z and we
want to solve (X A)y = z, i.e.,
Y1 Y2 = Z E [ID(A"4)]' (2.22a)
A(y1 + G1G Ayz2) + y2 = z2 [D)(A)]' (2.22b)
for y e D(A). We apply A1 to (2.22b), multiply (2.22a) by X and subtract to obtain:
V(X)y2 = A'z2 zI E [D(A"4)]' (2.23)
with V(X) defined in (2.19b).
We next note that V(X) is boundedly invertible on [1D(A 4)]' since equivalently
A4V(X)A = = I + AG ,1GA~' + 2A' is boundedly invertible on L2(Q) (being selfadjoint, strictly
positive on L2() ) with inverse
A ' V1 (X)A/4 e L(Q) (2.24)
Thus, from (2.23)
Y2 = V1'() (A'z2 1) e [(D(A 14)]' (2.25)
which then inserted in (2.22a) yields
Y = zi L 1 + V' 2 (2.26)
X*
17
Then (2.19a) follows from (2.25) and (2.26). Note that from (2.22b) and (2.25) that
Yi + G1GAAl2y2 =Azz XA'y2 E D(A'M) (2.27)
So that recalling (2.17b) we see that from (2.25) and (2.27) it is verified that y e D(A). The compactness
of R(X; A) on Z is readily seen from (2.19a) to be equivalent to compactness on L2(f) of the following
operators:
A"4(I V (X) )A 1 (2.28a)
A "4 V1 ()A14 = A" V1 (X)A '1A/2 (2.28b)
A 34V1 ()A 1' = A 1A1 V'1 (XA 1 (2.28c)
A4 V1 (X)A 14 = A UAl'V1 (X)A A l/2 (2.28d)
First, compactness of the operators (2.28bcd) on L2(0) is plain from (2.24) and A a a> 0 being
compact on L2(9). For (2.28a) apply V1 (X) on (2.19b) so that
I = V'1() + XV'(X)G G*A2 + 2V1 ()A1
and then
A"4 [I V1 () ] A 4 = A 4V1 (X)G iGIA"4 + X2A 1 ()A3/4
= a '"V (X)A A'G G*IA '4 + 2A4V (X)A 14A1
which is compact on L2() by (2.34) since A'4G I GIA"4 E L(a) D
To complete the proof of Theorem 2.6, we must show that o(A) does not contain any points on the
imaginary axis (we already know that o(A) does not contain points in ( ReX > 0 ) since A is the genera
tor of a contraction semigroup).
Thus, we need to show that
V(X)' E L( [fl(A" )]') for = ir, re BR, r 0 (2.29)
To this end let x e [ID(A l4)]' and suppose V(X)x = 0 for = ir. Then from (2.19b),
0= < V(A)x, x >[D(A'1)]
= < x, x >[ID(A")]' +ir< G1GAAlx, x >[D(A")]' r2< A1x, x >[ID(A14)]'
= < Aax x >O +ir II G;Amx 1 r2< Ax x >0 (2.30)
Since the middle term in (2.30) is purely imaginary we must have that via (2.10)
G IA'txx =GA [A a3x2] = [AAA3x)lr=0 (2.31)
Ir 0IV
18
Also, we have that by (2.30), A'2x = r2AMx, i.e.,
Ax = r2x (2.32)
which means that x must be an eigenvector of A say x = e, with eigenvalue r2. Therefore, since
e E D(A) we have that it satisfies the two zero boundary conditions associated with 1D(A) ( see (2.2b)), as
well as (2.31). Therefore, the following Lemma will complete the proof of Theorem 2.6.
Lemma 2.7
Let X = r2 > 0 and suppose Q satisfies the radial vector field assumption (2.20). Then the problem
A2O = 0 .in Q (a)
= =0 on r (b)
Sl(2.33)
=0 on r (c)
Ov IF
 0 on r (d)
has only the trivial solution a 0.
Notes
1. Since A3x = A3e, = rMe, ; (2.33d) follows from (2.31).
2. The above Lemma is not covered by standard elliptic theory since only three boundary conditions,
instead of four, are involved for the fourthorder elliptic operator in (2.33).
3. Recall that as given in Appendix C, if h(x) = (xxo) then
H(x) = 1 (identity matrix) (2.34a)
divh = n = dim Q (2.34b)
4. If 0 is a solution of (2.33) then it is immediate that via multiplication by AO and J dLr we have
J IV(AO)12dl =X I VIl2d (2.35)
19
Proof of Lemma 2.7
Multiply both sides of (2.33a) by h V(AO) and integrate dQ.
Left hand side: We have by (A.8), (2.33d), (2.34ab), (2.35)
IA(A4)h V(A<)dL2= I V(A4)12h vdr + ( 1)Xj I Vo I2d (2.36)
Right hand side:
X nOh V(A)df
by (A.5)
= F A h vdr f, AOdiv(Oh)dQ
by (A.3),(2.33b),(2.34b)
= XJ Ah Vdf nx Afmd
by (A.8),(2.34a,b),(A.1)
=Fr h *V dvF+ fIVOIh vdr+f IVOIzd J IVni2dQ
S*Jrv 2 'n 2 fn
nkfJ Tdr+n X0j IVO 2d
by(2.33bc)
=J IVI2hvdr+( +1) I VI2d2
Butsinceol = 0wehave I V I = I =0. Therefore,
1 Oh V(AO)df = ( + 1) X IV,122d (2.37)
Sfa 2 kZ
Setting (2.36) = (2.37) and simplifying we get
I I V(A*) 12h vdr= 21 ,V I2dr (2.38)
2 Jk a&
Now if we analyze (2.38) we see that by (2.32) the left hand side is nonpositive and the right hand
side is nonnegative, therefore both sides must equal zero. This then implies VO 0 a.e. and hence, 0 = c
a.e. But we have that 0 eigenfunction implies 0 smooth, and together with 0 we have 0. There
fore, Lemma 2.7 and hence Theorem 2.6 are proved. 0
Now that we have proven that A generates a Cosemigroup of contractions eAt on Z, it follows that
20
E (t) E (0) for t 0 (2.39)
This fact will be used crucially below. The next corollary is a consequence of the dissipative feedback per
turbation on the boundary.
Corollary 2.8
By choosing w =gl = GiA aw, it follows that G*A2w, E L2((O,oo); L2(r)) and in fact
I GIAm = I GA dt 5 E(0) (2.40)
for all initial conditions [wo,w ] e Z.
Proof of Corollary 2.8
Let [ wo,w 1 e ID(A) and recall for convenience
iw(t) t) 2 At Wi 2
E(t)= 11 Iw, () Iz= IlIeA Iz for t>0 (2.41)
Now d E(t)= 1 d < eAtlw A
2 di 2 di WI W1
Iww)Iw( w,(t)
=Re< AAtwj, e AjI:wI >Z=R>< A W,(1)1 ,,,,(t)W >Z
via the proof of Lemma 2.5
= II G*Aaw, I2 0 (2.42)
Remark 2.9
We see that (2.42) shows that such a choice of g does lead to an energy decrease as was demon
strated in another way (using Green's formula) in Remark 2.2. 0
Continuing the proof now we integrtee dt both sides to obtain:
,io
21
1iM T d 1 'd (t)dt
f i 11 A'2w, II dt E()dt lim E(t)d
2Tf o d t 2T, dt
1 1
E (0) lim E(T) < E (0)
2 2To
where in the last inequality we used the contraction of the semigroup, i.e., (2.39). Extension by continuity
yields (2.38) for all [wo,w ] e Z. [
Theorem 2.10
Let the radial vector field assumption (2.20) on Q hold. Then for any [wo,W ] E Z we have that
w(t, w0, wl) 2 I wol 2
E(t)= II w(t, w z= le I lz 0 ast (2.43)
wt(t, w O WO) 1
Proof of Theorem 2.10
The above result follows by appealing to the NagyFoiasFogel decomposition theory [L]. Since
eAt is a Cocontraction semigroup by Theorem 2.6, the Hilbert space Z can be decomposed in a unique
way into the orthogonal sum;
Z= Z.eZ, (2.44)
where both Z,, and Z. are reducing subspaces for e At and its adjoint.
It is also true that
(i) on Z,,, eAt is completely nonunitary and weakly stable
(ii) on Z,, e is a Counitary group.
In our case, Z= 0 1, the trivial subspace, because otherwise an application of Stone's theorem
[P] would guarantee at least one eigenvalue of A on the imaginary axis, but this is clearly false due to
Theorem 2.6. Hence Z Z, and therefore eAt is weakly stable on Z. However, since A has compact
resolvent, it follows that eAt is stable in the strong topology of Z[B]. Therefore, eAtz > 0 as t +* for
all z e Z and strong stability is verified. O
CHAPTER 3
UNIFORM STABILIZATION
3.1 Preliminaries, Change of Variables
Recall our feedback system
w,+ A2w = 0 in Q = (0,) x Q (a)
w(0,)=wo w,(0,)=wi in (b)
w =,= [A(Amw,)] on Z = (0,o) x r (c) (3.0)
=I g,=0 on Z (d)
and the corresponding "energy" E (t) defined by the squared norm of the semigroup
E(t)=E(w,t)= IleAtl l I= II l) = Il AW I + IIA A^w, 12 (3.1)
We want to show that under suitable assumptions on Q the energy E (t) decays uniformly, exponen
tially to zero. More precisely there exists constants C,8 >0 such that for any
[ wo,w ] E Z = [ID(A"4)]' x [D(A3')]' the corresponding solution of (3.0) satisfies
E(t) Ce4tE(O) for all t0 (3.2)
The proof of (3.2) will require a different approach than was used in Chapter 2 to achieve strong sta
bility. The proof will involve multipliers, but before we can proceed a change of variables must be ini
tiated. Let
p(t) =A'w,(t) for [ wo,w ] e Z (3.3)
Then since w,() e [D)(A3/4)]' i.e. Awt(t) e Lf 2), it follows that
A p (t)= AwAmw,(t)= A'3w,(t)e L2(Q). Thus
22
23
p(t) ID(A")= {fE H3(Q):f'r= =01
So in particular
p\ =0 (a)
(3.4)
=1 0 (b)
Now recalling (2.15) we have using (3.3)
p, =Asaw, =Aa[Aw AG 1iGA2w,] (3.5)
= Araw AaG1GGA1aw, (3.6)
Differentiating once more in time and using A'aw, = Ap and A2wa = Ap, we get
P. = ApAaG 1G 'Ap, (3.6)
and hence
p,+A2p = AaG IG'Ap, in Q (a)
p(0,.)=po =AMW, p,(0,)=pl =AlawoA1naG 1GAl'wl in Q (b)
SI =0 on (c) (3.7)
l =0 on E (d)
3.2 Uniform Stabilization
Our main result referred to in Chapter 1 as Theorem 1.3 is as follows:
Theorem 3.1
Assume that there exists a point x0 E IR so that the radial vector field defined by h(x) = (xxo)
satisfies
h(x)v>y>0 onr (3.8)
where v is the unit outward normal, then there exists positive constants C, 8 such that
E (t) < CeEE (0) for all t 0 (3.9)
The proof of Theorem 3.1 will follow directly from the next theorem, by taking lim in (3.10) and invoking
04
Datko's theorem [D.1].
24
Theorem 3.2
Under the same assumption (3.8) as in the above theorem, there exists a constant K such that for all
initial data [wo,w] e Z ( therefore yielding E (0) < +oo ) and all 0 < P <1 it follows that
w 2 IWo 2
e E(t)dtW=e(t) II = II eAtw II dt
Proof of Theorem 3.2 (A multiplier approach)
First, we take initial data smooth [wo,wI ] eD(A) (see 2.17b), which then guarantees
[w(t),w,(t)] e C([0,T]; D(A)) for all T < oo and find the desired estimate with K independent of
[ wo,wl ] and of p. Then extension by continuity yields (3.10) for all initial data [wo,W1] e Z.
Now we recall our starting identity which we derived in the appendices via the multipliers
e0th V(Ap) and e2tApdivh (recall (C.8)):
Je2pt (L)h V(Ap)d ' e2t I V(Ap) I2h vdE+ n e2t ApdL
= J eP I Vp, 12dQ + J e2 I V(Ap) 12dQ 2p e2lptp,h V(Ap)dQ
n BQel2p,ApdQ Qe AaGiGAl p,h V(Ap)dQ
n 2Qe*AmGiGIApApdQ lim e2< p, hV(Ap) >Q
2 Q TL. 0i T
n lim et< p,, Ap > T (3.11a)
2 rTL To
= e2tIVp 12dQ+ Qe2tIV(Ap)12dQQ1 Q2 Q3 L1 L2 (3.11b)
where the Qj, i = 1,2,3,4 are the corresponding integrals over Q and the Li, i = 1,2 are the corresponding
limit terms. The first two integrals are most important as we shall see below. Next we prove:
Lemma 3.3
There exists a constant C such that
II Ap 112 C II A4p i12 (3.12)
25
Proof
2 2
First by trace theory [LM] II Ap II, < C II Ap II Hm(a)
2
< C IIAp IIH 'l) since Hl'() isa stronger topology
= C Ap + C Ij V(Ap) 1 by definition
5 C II Ap ii2 + CCp3 II A"p II2 by norm equivalaence
We continue the proof of Theorem 3.2 which will be divide into seven steps.
Step 1: Absorption of the boundary terms in (3.11). Let M = max I h12 and choose el > 0, E2 > 0
such that
eIM, < 2 (3.13a)
2 n < al (3.13b)
2
where C1 as in Lemma 3.3 and ac to be given below.
Before proceeding, note that the following inequality will be used extensively below:
For any e > 0 ; 2ab < ea2 + lb2 (*)
Next we operate on the left hand side (L.H.S) of equation (3.11) by using (*) with E1 and C2 as well
as the assumption on h v to obtain:
e 2t a A) h V(Ap)dl f e2 I V(iAp) 12h vd + fe2 Apd
f _2~ 2 a() 2d+eMbf e2 lV(Ap)12dZ e2 IV(Ap)12dI
le < v 1 2z
+ n e 2 I a 'p) 12dl+ CE2 ez I,11 1 Ap d
262 f < v 2 ,
+( +E2 )E()+(e1Mhb fe IV(Ap)l'dE+ 2 en A31
where the E(0) term follows by using (2.40) and we have also used Lemma 3.3 for the Ap term.
Now we can drop the middle term because by (3.13a) e M 1 < 0 and use the fact that
II Ap II = I A'w, II2 < E(t) to obtain:
26
(L.H.S.) of (3.11) < K1E(0) + ' eWE(t)dt (3.14)
2 n
where K = + 
el 2e2
Finally, utilizing (3.14) in (3.11) and isolating the important terms we obtain:
Qe2t IVp1dQ + Qe2t IV(Ap)12dQ K E (O) + 2nC 2eE (t)dt
+Q1+Q2+Q3+Q4+L1 +L2 (3.15)
Ste 2: Isolating the energy integral oetE (t)dt. First since p, E D(A w) there exists a constant
Cpi > 0 such that
II Vpll" I Cp II A "p, ll [LT.2] (3.16)
Similarly p E D(A 4) implies the existence of Cp3 > 0 such that
 IV(Ap) 11 >Cp3 JJAp [LT.2] (3.17)
Next using (3.15) and (3.16) we have
II IV 112 Cpl IIA1p, 1
=Cpj IIA"w II2 + 2CppRe< A"w A'GIGlAuZw, > a+Cpl I A LGIG A2w, 112
Now to bound below we use 2ab ea2 b2 on the middle term and taking e = we obtain
II jVp, i 2 IIA'4w II Cp II A"GiG A'w, 112 (3.18)
Also it is immediate that
l (Ap) Cp3  A4p 12 = Cp3 A'w, i (3.19)
Putting the pieces together in a Lemma:
Lemma 3.4
Let a =min {C1 ,Cp3 } (3.20)
2
and K2=Cpj II A'G, II L(r.a) (3.21)
Then e2P I Vp, 12dQ + Qe2 IV(Ap)I2dQ >a e0E(t)dt K2E(O) (3.22)
27
Proof
Q eI V(Ap)12dQ + QeIVp,12dQ
by (3.18), (3.19)
SCps Je d + e1 A, w I2 dt Cp1 oe II A1 GG*iAU2w, II2dt
>,afe 20t[ IlA A4wI 2+ hiA"w, 2II dtCpl JiAUGi [I r.)l II GA'2w, Idt
2
>by(2.40) aiJoeP E(t)dt Cp IIA 'G1 I( Lr.n)E(0). O
Collecting our results so far we have via (3.13b), (3.15), (3.22) that
ac2Je2PtE(t)dt 5K3E(0)+i+LZ2+Ql+Q2+Q3+Q4 (3.23)
where
K3 =K + K2 (3.24a)
and
nC1
a2 = al E2 > 0 (3.24b)
Ste 3 : Handling the limit terms arising from integration by parts in time.
Remark 3.5
The following will be used in evaluating the limit terms in (3.23). Since we are taking initial condi
tions [ wo,wi ] e D(A) we have by the regularity theory [LT.5], [w(t), w,(t)] e C([0,T] ; D(A)), i.e.,
w(t) e C([0,T]; H(K)) and w,(t) e C([0,T]; [D(A4)]).
Computing II I I A= wII Hllt I A)
= IIAeAtIlwlI IZ= IleAtAlwI Ilz IIAIwI ilz (3.25)
where we have used 0 E p(A) quoted in Theorem 2.6. Hence we have that
2Ilwl + I '" 0 ( ) W
I1 w 11 H(Q) + IIA" 11 A[ II Z forallt0 (3.26)
28
Now letting M = I h 12 we have
lim e21 < p,(T), h V(Ap(T)) >a lim e2 I p,(T) 2 + M 11 A31'p(T 112
by (3.5), (3.9)
m el T 2 II A L(4 IIAw(T) IL
2 12
+2I A'GiG*A'" II L(o) II Awr(T) II + ME(0)
by (3.26)
< lim eA CE(0) + 2 11 AaGIGA II A 1, z =0 (3.27)
A similar computation shows that lim e'T< p,(T), Ap(T) >0a =0 (3.28)
IT  I
Hence we have thatL1 +L2=< pi h V(Apo) >n "n< Pi Ao >0
2
< A"lwo +Al'GiGAAl'aw, h V(Apo) >a+ < A Awo +AGG w, Apo >n (3.29)
Ste4 : Handling the terms premultiplied by 0, i.e. the terms Q and Q2. First
Q I =2Pj ep,h V(Ap)dQl
< PjIe* I P, 12 + 1 Mhf et II V(AP) I dt
by (3.5)
<23 II Al' II L(n)Je I2I AII Aw 12dt
+ 2 II AG, II L(r. o)Jfe II G Ar"w, 2d + MACP3 e2 II A4w, I\dt
2 2
< 2( 11 A"4 II L()+MhCp3)0 e2 E(t)dt + II AmGI II (r,a) E(0)
(where we have used the facts that I A "w II 5E(t) and similarly
29
II A3w, 112 E(t) as well as (2.40))
2 2
<[ 1 A1II a)+MhCp3+ IA aG 1 ILr.Ia) E(0)=K4E(0) (3.30)
where we also have used E (t) E (0) for all t 2 0 (contraction of the semigroup) and
fedt = (3.31)
o 20
A similar computation shows that there exists a constant Ks (independent of 3 ) such that
IQ21 Ks E(0) (3.32)
Thus we arrive at
Lemma 3.7
Let K6 = K3 + K4 + Ks (3.33)
Then a2foe" E(t)dt a
+ n< A'wo +AIRGIGGA12wl Apo >n (3.34)
2
Proof
Follows immediately by using (3.29), (3.30), (3.32) in (3.23). 0
Now having dispersed with the "lower order" terms, our task is to absorb the more difficult interior
integrals.
Step : The term Q3. We apply integration by parts in time with dv=A 'G1G*A'w,,
u = e2th V(Ap) to obtain
Q T+[L 0
+ e2 eaGIGAGaw,,h V(Ap)dQ = lim [ep < A2G I GALaw, h V(Ap) >n (3.
+2a2J e A AaG1G*IA 'awh V(Ap)dQ JeAaG1Ga AI wh V(Ap,)dQ (3.35)
30
Part A
Since A'G,G*Aaw,(T) I < II A'GG*IA^ 1 L()  IIA'w,(T)
by(3.26) < IIAlGiG*IA"M IL(n)2 IIAJw 0
an argument exactly like the one used to obtain (3.27) yields:
lime2PT< Ai`G GI*GA"aw,(T), h V(Ap(T)) >Q = 0 (3.36)
T4
Thus we only have a contribution from t = 0, i.e.,
lim [e2< AmGIGIA w,, h V(Ap) >] =< A2GGIA'2aw, h V(Apo) >0 (3.37)
T+L 00
Part B
2PJ e2 < AIGIG A"2w h V(Ap) >ndt
< 2I e2Mt A1G1G*GAaw, II Idt + 2P3M o et II V(Ap) Idt
<(2 1 A"2G1 I 12 +MhCp3)E(0) (3.38)
where we have used (2.39), (2.40) and (3.31)
Part C The following Lemma will enable us to handle the remaining term and hence complete Step 5.
Lemma 3.8
For h E [C2(C)]" we have that there exists a constant CA such that
II A2(h V(Ap,)) I1 CI,  Ap, 1 (3.39a)
Remark
Since D(A 1')= H2() (see [LT.2, Appendix C]) so that [ID(A1')] '=H2() (with equivalent
norms) we have
II A'a(h V(Ap,)) I = II h V(Ap,) II [D(A')]' = II h V(Ap,) II H(n) (3.39b)
Claim Let z E H2(0), h e C2(0). Then hz e H2(4).
Proof of claim (See also [LM, Vol I, Theorem 7.3, p31])
By assumption o= zgd< *a ;gE HI () and we want to show that
By~f as0plo
31
< hz g>=J zhgd < ; ge HO(P).
Thus it suffices to show hg e H2(2), if g e H2(). Clearly hg E H2(0). So we must show
hg r h r= 0
IF r
buthg I=h g = 0 since g E H2().
Also = V(hg) v = { [(hg), (hg), ,(hg) [V ,2, ,
= { [hg + hg,]vi + [hg + hgV2 + + [hg +hg }v. = 0
since g H (f) implies g =gx,1 =gIx, =. = =o 0
*r F r r
Proof of Lemma 3.8
Using the above claim in (3.39b) we have
II A (h V(Ap,)) I = II h V(Ap,)) II Ha)
< M, I V(Ap,) II (Q)
5 CM, Cp1 II I Vp, II by Poincare Inequality
< C1, IIA'p, II a via norm equivalence 0
Continuing Part C, choose C3 > 0 such that 2ChE3 < Oa. (3.40)
Now we compute IJQ e2AA'G GA2w, h V(Ap,)dQ
< ,e2jt I< GIGAa2w,, Aa(h V(Ap,)) >n dt
by (3.39a)
5< JGIIGL(, 11G1 GIIG1A 2w, Ildt +C 0 e2 11 A"4p, IL dt
32
by (2.40),(*)
< II G L f A)E(0)+ l AGGA"2w, 112dt +2CAJ e2X 11AI'w dt
< II G lLrn) + 3 IA "G1 II E (0)+2Che3J etE(t)dt
= KgE(O)+ 2ChSe2 E(t)dt (3.41)
Again organizing our results we have
Lemma 3.9
Let K9 = K6 + K7 + Ks (3.42)
and a3 = a2 2ChE3 > 0 (3.43)
then a3j eE(t)dt <5KE(0) +Q4 + < Aawo h V(Apo) >
+ n < Awo +A2G1GiAlzwl Apo > (3.44)
2
Proof
The proof follows directly by utilizing (3.36), (3.38) and (3.41) in (3.34) and taking advantage of the
cancellation of terms
< Aa2wo +AaGjG*A`awl h V(Apo) >a < AflGiGIAawi h V(Apo) >a
= < AwWo h V(Apo) >n 1
Before beginning Step 6, the following Lemma will be needed.
Lemma 3.10
The L2norm of Ap is bounded by the initial energy E (0), i.e.,
11 AP 112= I I A 1II2 = 1 AI A1^4A3p I2 < IA' I12 1 IAL p I(2
= IIA'4I2 IAw, ll < 1IA' 112\ )E(t),5 AI4 11 E(0) (3.45)
L(Q) a L(QL
33
Proof
IIA'p 12 = < Ap p >0= < A(Ap), p >n
by(A.0) = < Ap, Ap >Q+J pdr Apa dr
by(3.4a,b) = < Ap, Ap >a= II Ap I 0
Ste 6 : The term Q4. As in Step 5, we begin with integration by parts in time with
dv =AflG1G Alawu, u = e2Ap to obtain:
nf e A2GiGGAwuApdQ = n lim e2P < A"2GG A a1, Ap > T
2 Q 2T,[ Jo
+n pfeA2GIGA2wApdQ e2PAt2GiG*A2wsApdQ (3.46)
Now we begin a three part process as in step 5.
Part A Using (3.26) and (3.45) it follows directly that
n lim e2T< A2G I GA'aw,(T), Ap(T) >a < nC lim e2 = 0 (3.47)
2rT4 2 Tr
So again we have only a contribution from t = 0 i.e.
2 lim e2P< A'aG,GiA1w, Ap >] T=n< A2Gr1GAawl Apo >n (3.48)
24 o 2
Part B
nPf e2t< AaG 1G;Aaw,, Ap >adt
nI n AIGI A12G1 IIL 11e GIA"w, 112dt +nl 1 e II A 11
by (2.40), (3.32), (3.45) n IIAG I L2 ) + 11lA1'4 12 E (0)= KoE(0) (3.49)
Part C For C2 as given below, choose E4>0 such that
E4nC2 IA" L(n) < a (3.50)
L(Q) < Q3
Qe2P< A'2G1lGiA2w,, Ap,>QdQ
5 nJe < GiG'1A'2w, A'2(AP) >.IdQ
< e2 II GIGIA"w, 11 dQ + E4n T II A'aAp,) 12dQ (3.51)
34
Examine the integral on the right in (3.51)
Using the facts that Da(A la)= H2 () [LT.2, Appendix C] (3.52a)
and II AfllH2f )
We have II A Ap, 1 = I Ap, II2 = I Ap ff2()]
= 1 II 11 II ~, 1 l2 C21I A"I ) IIA1MP, 11 (3.53)
so continuing(3.51) < IIG 11I L(.l)E(0)+ 4 2 IA'4 I2L(n) oe2 IIA"p, la dt
F4 L(r, 1) 22
Si IG, II E(0)+ e4nC2 I A" 1 L( eE(t)dt
2F4 L(rQL)
+ E4nC2 IIA'G1 I L(r, )E (0)
= KIE(0)+ E4nC2 IIA"4 II eE (t)dt (3.54)
Lemma 3.11
Let K 2= K,+Ki+KI (3.55)
and a4 =a3 4nC2 i A'4 I1(1 ) > 0 (3.56)
Then o(4 eE(t)dt < K2E(0) + < A'wo, h *V(Apo) >0
+ < Al"wo, APo >n (3.57)
2
Proof
The proof follows again directly using (3.48), (3.49), (3.54) and again utilizing the cancellation of
terms:
n < A"2Wo +A_2GiGA"aw, Apo >o n< A'G,1GA"'wl Apo >n
2 2
n < A2wo, Apo >0a
2
Ste_ 7 : The initial conditions.
First, 1< A^'wo, h V(Apo) >na< II A'o n2 +MACp3 II A3wl 1
<[ lA"4 IIL( +MACp] E(0)= KE(O) (3.58)
35
Second, I < A2wo, Apo >nl < [ HA12wo 11 + IIApo0 11]
<5 A 1IM E2(0)+ IIA 1 ) E(0)] =n AA IL()E(O)=K14E(0) (3.59)
where we have applied Lemma 3.10 for the Apo term.
Finally using (3.58), (3.59) in (3.57) we have
a4 e E (t)dt 5 KisE (0) (3.60)
where K1 = K2 + K13 +K 14 (3.61)
K15
Dividing (3.60) by o4 > 0 and letting K = Theorem 3.2 is proved. O
04
Now the proof of the main result (Theorem 3.1) will follow easily from the below quoted result
Theorem (Datko) [B. page 1761
Suppose T(t) is a semigroup on a Banach space X. If for all x E X
Jf011T (t)x IX dt <
then the semigroup is uniformly (exponentially) stable, i.e., there exists constants C,8 > 0 such that
IIT(t) L(x) Ce forall t> 0 O
Proof of Theorem 3.1
We have seen that JeE (t)dt < K E (0) where K is a constant independent of ~ e (0,1] and of
[ wo,w1 ] D(A). Thus taking the lim we get
oE (t)dt
But E(t) is simply the squared norm of the semigroup applied to any initial conditions [ wo,wl ] e 2D(A)
and hence by continuity all [ wowi ] e Z, i.e,
substituting E(t)= IleAw l w IZ into (3.62),
weget oII eAl l Ilzd1 KE(0)
36
Thus by Datko's Theorem there exists C 1,81 > 0 such that
II eAt IIL(Z)> Clekt and hence II eAt IL(Z) < Ce2
therefore by taking C = C2 and 8 = 28 we have
whc is t Au uO 1 r
E(t)= (leAlsj 1t IIu LZ) II j. II rCe E(0)
which is exactly our uniform stabilization result D
APPENDIX A
BASIC IDENTITIES
Let g be scalar functions, and h (x) e C2() vector field. The following identities will be needed:
Green's Second Theorem Afgd2 Afd = f Jgdr f fdr (A.O)
Green's First Theorem Afgd = f gdf Vf Vf Vgd (A.1)
Divergence Theorem f divhdK2 = frh vdr (A.2)
div(fh) = h Vf + fdivh (A.3)
fh Vf= h V(f2) (A.4)
J h Vfd =fJ (divfh fdivh)d2= Jrfh vdr Qfdivhdi (A.5)
Vf V(h Vf)= HVf Vf+ h. V(IVf12) (A.6)
ah1 ah1
ax1 ax,
where H(x)=
ah, ah,
ax 1 ax,
Vf V(fdivh) =fV(divh) Vf+ I Vfl2divh (A.7)
JA fh.Vfda=f ,h. Vfdr1 IVfl2h vdr
HVf VfdM + 1 Vf I2divhd (A.8)
37
38
Proof of (A.8):
Multiply Af by h Vf, integrate J dQ, and successively apply (A.1), (A.6), and (A.5).
SAf Vfd = h Vfdfr jVf V(h Vf)d
=fr h Vfdf HVf VfdQ fh V(IVf I2)d(
Ir *Vfdr HJVf.Vfdn rIVf12h vdr+ IVfI2divhdo
Another useful identity arises by applying (A.1), (A.7) to get:
f Affdivhd(= FfdivhdFr Vf V(fdivh)dQ
= fraydivhdr fV(divh) Vfd f I Vf lI divhd
Nrov l'Q f' Q
(A.9)
APPENDIX B
TO HANDLE DIFFERENCE OF ENERGY TERM
We multiply both sides of our equation pa + A2p = A'2G 1GAp, by e2tApdivh with 0
integrate both sides over Q: JdQ = Jf dtdl .
p. term: First integrate by parts in time, and then apply (A.9) to get:
f Pe2ApdivhdtdQ = lim e2 < p,, Apdivh >
+ 2p e2ftAppdivhdQ e2stAppdivhdQ
= lim e2< p,. Apdivh >] + 2 eWAppdivhdQ
rTL 0o Q
I f, 2 p,divhdE+ e2tp,V(divh) Vp,dQ + e2Vp, 2divhdQ (B.1)
A2p term: First apply (A.1) and then (A.7) to get:
JIe2A(Ap)ApdivhdQ = e2t~) ApdivhdE Q e2V(Ap) *V(Apdivh)dQ
= fe2pr '(MAvApdivhdE eApV(divh) V(Ap)dQ f e I V(Ap) 12divhdQ (B.2)
1
Now isolating difference of energy term and multiplying by we get
21 e2 I Vp.l 2 V(Ap)12] divhdQ =lim[e2pt< pr, Apdivh >n]
PQe tAppdivhdQ + e2t pdivhd 1 e 'pV(divh) Vp,dQ
e2Ap l) ApdivhdZ + Q e2tApV(divh) Vp,dQ
S2 Qe GGA pdivhdQ B.3)2
1l eVAv2GGApApdivhdQ (B.3)
2Q
APPENDIX C
TO OBTAIN GENERAL IDENTITY
We multiply our equation p,,+A2p=AaG1GiGAp, by e2th V(Ap) with 0
integrate both sides over Q as in Appendix B.
p,, term: First integrate by parts in time, then successively apply (A.5) and (A.3) to get:
ep,,eh V(Ap)dtd = lim e t,< p, hV(Ap) > T
+ 2pQ etp,h V(Ap)dQ Qe~'p,h V(Ap,)dQ
= lim e2t< p,, hV(Ap) >0 + 20p e p,h V(Ap)dQ
Je 2Apph vd + QeXApdiv(ph)dQ
= lim e2< p,, hV(Ap) >o] +2o e ph V(Ap)dQe2tApph *vdZ
TL o 0 Q 
+ fe2PtAph Vp,dQ + ae2AppdivhdQ (C.1)
Next, we will apply (A.8) to J e2Ap,h Vp,dQ and (A.9) to IQetApp,divhdQ simultaneously to
obtain:
2ofPce20 h V(Ap)dtdG
= lim eW2< p,. hV(Ap) > + 2 2pfQetp,h V(Ap)dQ Ie2tApp,h vdE
+ e P'hVp,dZ J et IVp,12h vdEIQe2P tHVp 'Vp,dQ + J e~ lVp 12divhdQ
+ f e2 p, divhdY e2fp,V(divh) VpdQ e2t I Vp, 12divhdQ (C.2)
W Q 0
40
41
A2p term: Applying (A.8) we get
J e A(Ap)h V(Ap)dQ = e2t h V(Ap)dE
eZ I V(Ap) I 2h vdE Je2tHV(Ap) V(Ap)dQ + 2Je I V(Ap) IdivhdQ (C.3)
Now we put all boundary terms on the left side, all interior terms on the right side and simplify to obtain:
e Apph vd.+ e h Vpd etIVp, 12h vdW
fE fv Y. 2y e
+ e2 pt p,divhd + f e2a(P) h V(p)d 1Je I V(Ap) 1 h vdW
N2 If av 2
= lim e2t< p,, hV(Ap) >a 2 e2p,h V(Ap)dQ
rL o 0 Q
+ IQe2PHV(Ap) V(Ap)dQ + f e HVp, VpdQ
+ Jep, V(divh) Vp,dQ + e2 [ I Vp, 12 I V(Ap) 12] divhdQ
fe2W Al2G 1G*Ap,h h V(Ap)dQ (C.4)
Before proceeding we make some simplifications which arise due to the boundary conditions.
Since p = = 0 it is immediate that
a 0v X
p, =a =0 (C.5)
Also VpFiand Vplr imply I Vp I = I I 0 on
ap,
and IVp,I= II 0 on (C.6)
And by utilizing h (x) = (xxo) the radial vector field, we have that
H (x)= 1 identity matrix
divh = n = dim t (C.7)
V(divh) 0
42
Now inserting (B.3) for the difference of energy term in (C.4) and utilizing the simplifications (C.5), (C.6)
and (C.7), we finally arrive at the desired identity:
,e2~P h V(Ap)dE f e I V(Ap) I2h vdC + ,e2ze pd
= i e2Ee < p,, hV(Ap) >a lim [e2 p,, Ap >O]
T,L o0 2 TJL 0
2p e p,h V(Ap)dQ n 3Q e p,4p dQ
+ e IVp,12dQ +f e2 t IV p) 2dQ
fe2A l'2CG Ap,h V(Ap)dQ n feAAGiG GArApdQ (C.8)
Q2
REFERENCES
[B] A.V. Balakrishnan, "Applied Funtional Analysis," 2nd edition, SpringerVerlag, New
York/Berlin, (1981).
[BT.1] J. Bartolomeo and R. Triggiani, Uniform stabilization of the EulerBemoulli Equation With
Dirichlet and Neumann Boundary Feedback, Report, Dept. of Applied Mathematics, Univer
sity of Virginia, 1988.
[C.1] G. Chen, Energy Decay Estimates and Exact Controllability of the Wave Equation in a
Bounded Domain, Journal de Mathematiques Pures et Appliquees (9) 58 (1979), 249274.
[C.2] G. Chen, A Note on the Boundary Stabilization of the Wave Equation, SIAM J. Control 19
(1981), 106113.
[D.1] I. Datko, Extending a Theorem of Liapunov to Hilbert Spaces, J. Mathem. Anal. and Applic.,
32 (1970), 610613.
[DLT] G. DaPrato, I. Lasiecka and R. Triggiani, A Direct Study of the Riccati Equation Arising in
Hyperbolic Boundary Control Problems, J. Differential Equations 64 (1986), 2647.
[FLT] F. Flandoli, I. Lasiecka and R. Triggiani, Algebraic Riccati Equations With Nonsmoothing
Observation Arising in Hyperbolic and EulerBernoulli Equations, Annali di Matematica Pura
e Applicata, to appear.
[H.1] L. F. Ho, Observabilite' Frontiere de L'equation des Ondes, CRAS, 302 (1986), 443446.
[L] N. Levan, The Stabilization Problem: A Hilbert Space Operator Decomposition Approach,
IEEE Trans Circuits and Systems AS2519 (1978), 721727.
[L.1] J. Lagnese, a paper presented at the International Workshop held in Vorau, Austria, July 10
16, 1988.
[L.2] J. Lagnese, "Boundary Stabilization of Thin Elastic Plates", to appear.
[L.3] J. Lagnese, Uniform Boundary Stabilization of Homogeneous, Isotropic Plates in "Lecture
Notes in Control Science #102", SpringerVerlag, New York, pp. 204215, 1987 Proceedings
of the 1986 Vorau Conference on Distributed Parameter Systems.
43
44
[L.4] J. Lagnese, Decay of Solutions of Wave Equations in a Bounded Region with Boundary Dissi
pation, J. Differential Equations 50 (1983), 163182.
[L.5] J. Lagnese, A Note on Boundary Stabilization of Wave Equations, SIAM J. Control, to appear.
[L.6] J. L. Lions, Exact Controllability, Stabilization and Perturbations, SIAM Review, March 1988.
[L.7] J. L. Lions, A Resultat de Regularite (paper dedicated to S. Mizohata), "Current Topics on Par
tial Differential Equations," Kinikuniya Company, Tokyo, 1986.
[L.8] J. L. Lions, "Controle des Systemes Distribues Singuliers", Ganthier Villars, Paris, 1988.
[LM] J. L. Lions and E. Magenes, "Nonhomogeneous Boundary Value Problems and Applications,"
Vols. I, II, SpringerVerlag, BerlinHeidelberg, New York, 1972.
[LL.1] J. Lagnese and J. L. Lions, "Modeling, Analysis and Control of Thin Plates," Masson, Paris,
1988.
[LLT.1] I. Lasiecka, JL. Lions and R. Triggiani, Nonhomogeneous Boundary Value Problems for
Second Order Hyperbolic Operators, Journal de Mathematiques Pures et Appliquees 65
(1986), 149192.
[LT.1] I. Lasiecka and R. Triggiani, Exact Controllability of the EulerBernoulli Equation with
L2(E)control Only in the Dirichlet Boundary Conditions, Atti della Accademia Nazionale dei
Lincei, Rendiconti Classe di Scienzefisiche, 81 (August 1987).
[LT.2] I. Lasiecka and R. Triggiani, Exact Controllability of the EulerBernoulli Equation with Con
trols in the Dirichlet and Neumann Boundary Conditions: a Nonconservative Case, SIAM J.
Control and Optimization, to appear.
[LT.3] I. Lasiecka and R. Triggiani, A Direct Approach to Exact Controllability for the Wave Equa
tion with Neumann Boundary Control and to an EulerBeroulli Equation, Proceedings 26th
IEEE Conference, pp. 529534, Los Angeles, December 1987.
[LT.4] I. Lasiecka and R. Triggiani, Uniform Exponential Energy Decay of the Wave Equation in a
Bounded Region with L2(0,oo; L2())feedback Control in the Dirichlet Boundary Conditions,
J. Diff. Eqts. 66 (1987), 340390.
[LT.5] I. Lasiecka and R. Triggiani, Regularity Theory for a Class of Nonhomogeneous Euler
Bernoulli Equations: a Cosine Operator Approach, Bollettino union matematica Italiana, (7)
2B, December 1988.
[LT.6] I. Lasiecka and R. Triggiani, Exact Controllability of the EulerBernoulli Equation with Boun
dary Controls for Displacement and Moments, J. Mathem. Analysis and Applic., to appear.
[LT.7] I. Lasiecka and R. Triggiani, Uniform Exponential Energy Decay of the EulerBernoulli Equa
tion on a Bounded Region with Boundary Feedback Acting on the Bending Moment, to
appear.
45
[LT.8] I. Lasiecka and R. Triggiani, A Cosine Operator Approach to Modeling L2(0,T;LZ2(r))
boundary Input Hyberbolic Equations, Applied Mathem. and Optimiz. 7 (1981), 3593.
[LT.9] I. Lasiecka and R. Triggiani, Regularity of hyperbolic equations under L2(0,T ;L2(T))
Dirichelt Boundary terms, Applied Mathem. and Optimiz. 10 (1983), 275286.
[LT.10] I. Lasiecka and R. Triggiani, A Lifting Theorem for the Time Regularity of Solutions to
Abstract Equations with Unbounded Operators and Apllications Through Hyperbolic Equa
tions, Proceedings American Mathematical Society, to appear.
[LT.11] I. Lasiecka and R. Triggiani, Riccati Equations for Hyperbolic Partial Differential Equations
with L2(0,T ; L2())Dirichlet Boundary Terms, SIAM J. Control and Optimiz. (5) 24 (1986),
884926.
[LT.12] I. Lasiecka and R. Triggiani, Exact Controllabilty for the Wave Equation with Neumann Boun
dary Control, Appl. Math. and Optimiz., to appear.
[LT.13] I. Lasiecka and R. Triggiani Sharp Regularity Theory for Second Order Hyperbolic Equations
of Neumann Type. Part I: L Nonhomogenous data, to appear.
[LT.14] I. Lasiecka and R. Triggiani, Sharp Regularity Theory for Second Order Hyperbolic Equations
of Neumann Type. Part II: The General Boundary Data, to appear.
[LT.15] I. Lasiecka and R. Triggiani, Trace Regularity of the Solutions of the Wave Equations with
Homogeneous Neumann Boundary Conditions and Compactly Supported Data, J. Mathem.
Anal. and Applic., to appear.
[LT.16] I. Lasiecka and R. Triggiani, Nondissipative Boundary Stabilization of the Wave Equation via
Boundary Observation Journal de Mathematiques Pures et Appliquees 63 (1984), 5980.
[LT.17] I. Lasiecka and R. Triggiani, Dirichlet Boundary Stabilization of the Wave Equation via Boun
dary Observation J. Mathem. Anal. and Applic., 87 (1983), 112130.
[P] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,"
SpringerVerlag, New York, (1983).
[R.1] D.L. Russell, Exact Boundary Value Controllability Theorems for Wave and Heat Processes in
Star Complemented Regions, in "Differential Games in Control Theory," Dekker, New York,
1974.
[R.2] D.L. Russell, A Unified Boundary Controllability Theory for Hyperbolic Partial Differential
Equations, Stud. Appl. Math. 3 (1973), 189211.
[R.3] D.L. Russell, "Mathematical Models for the Elastic Beam and Their Controltheoretic Implica
tions; in Semigroups Theory and Applications, Pitman Research Notes in Mathematics 152,
1986.
[S.1] W. Symes, A Trace Theorem for Solutions of Wave Equations, Mathematical Methods in the
Applied Sciences 5 (1983), 131152.
46
[T.1] R. Triggiani, Wave Equation on a Bounded Domain with Boundary Dissipation: an Operator
Approach, J. Mathem. Anal. and Applic., to appear.
[T.2] R. Triggiani, "Exact Controllability of Wave and EulerBeroulli Equations in the Presence of
Damping," Proceedings of International Conference on Differential Equations held in
Columbus, Ohio, March 2115, 1988.
[T.3] R. Triggiani, "Exact Controllability on L2(x) x H () for the Wave Equation with Dirichlet
Control Acting on a Portion of the Boundary, and Related Problems", Applied Math. and Opt.
18 (1988), 241277.
[T.4] R. Triggiani, A Cosine Operator Approach to Modeling L2(O,T; L2(C))boundary Input Prob
lems for Hyperbolic Systems, pp. 380390, SpringerVerlag, 1978. Proceedings 8th IFIP
Conference, University of Wurzburg, W. Germany, 1977.
[T.5] R. Triggiani, Lack of Uniform Stabilization for Noncontractive Semigroups, Proceedings
Amer. Mathem. Soc., to appear.
[T.6] R. Triggiani, Finite Rank, Relatively Bounded Perturbations of Semigroup Generators. Part III:
A Sharp Result of the Lack of Uniform Stabilization, Proceedings of First Conference on
Communication and Control Theory, Washington, D.C., June 1987, to appear.
[T.7] R. Triggiani, On the Stabilization Problem in Banach Space, J. Mathem. Anal. and Applic., 52
(1975), 383403.
BIOGRAPHICAL SKETCH
Jerry Bartolomeo was born June 17, 1960 in Brooklyn, New York. In 1972, he moved to Home
stead, Florida, and graduated from South Dade High School in June 1978. After entering the University of
Florida in Fall 1978, he received his B.A. in mathematics in 1981, and then his M.S. in mathematics in
1984. While working on his Ph.D., he spent a year teaching at the University of Virginia, School of
Engineering. He is currently living with his wife and son in Miami, Florida, and working as an instructor
at Nova University in Ft. Lauderdale, Florida. Upon receiving his Ph.D. in mathematics, he plans to con
tinue working in the area of Boundary Control for Partial Differential Equations and continue his involve
ment in program development at Nova.
47
I certify that I have read this study and that in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor
of Philosophy.
Roberto Trigglani, Chair
Professor of Mathematics
I certify that I have read this study and that in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor
of Philosophy.
Irena Lasiecka
Professor of Mathematics
I certify that I have read this study and that in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor
of Philosophy.
Jorg6i nez L
Professor of Mathematics
I certify that I have read this study and that in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor
of Philosophy.
Murali Rao
Professor of Mathematics
I certify that I have read this study and that in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor
ofPhilosophy.
Carmen Lanciani
Professor of Zoology
This dissertation was submitted to the Graduate Faculty Department ( OR School ) of ... in the
College of .. and to the Graduate School and was accepted as partial fulfillment of the requirements fro
the degree of Doctor of Philosophy.
December 1988
Dean, College of Mathematics
Dean, Graduate School
UNIVERSITY OF FLORIDA
3 11111111122 011118 IIII 4 0259II III l I111
3 1262 08554 0259

Full Text 
xml version 1.0 encoding UTF8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd
INGEST IEID EUR465RI4_SEADPZ INGEST_TIME 20120207T15:10:05Z PACKAGE AA00003782_00001
AGREEMENT_INFO ACCOUNT UF PROJECT UFDC
FILES
PAGE 1
81,)250 67$%,/,=$7,21 2) 7+( (8/(5%(5128//, (48$7,21 :,7+ $&7,9( ',5,&+/(7 $1' 121$&7,9( 1(80$11 %281'$5< )(('%$&. &21752/6 %\ (55< %$572/20(2 $ ',66(57$7,21 35(6(17(' 72 7+( *5$'8$7( 6&+22/ 2) 7+( 81,9(56,7< 2) )/25,'$ ,1 3$57,$/ )8/),//0(17 2) 7+( 5(48,5(0(176 )25 7+( '(*5(( 2) '2&725 2) 3+,/2623+< 81,9(56,7< 2) )/25,'$ I_ '( ) /,%5$5,(6
PAGE 2
$&.12:/('*(0(176 H[SUHVV P\ VLQFHUH DSSUHFLDWLRQ WR P\ DGYLVRU 'U 5REHUWR 7ULJJLDQL IRU KLV OHDGHUVKLS DQG VXSSRUW RYHU WKH SDVW IRXU \HDUV DQG IRU GLUHFWLQJ WKLV UHVHDUFK SURMHFW ZRXOG DOVR OLNH WR WKDQN WKH RWKHU PHPEHUV RI P\ FRPPLWWHH HVSHFLDOO\ 'U ,UHQD /DVLHFND ZKR DOVR SOD\HG D PDMRU UROH LQ P\ HGXFDWLRQ $OVR ZRXOG OLNH WR WKDQN P\ SDUHQWV DPHV DQG (OLVD %DUWRORPHR DQG P\ ZLIH 0LFKHOOH IRU WKHLU FRQn WLQXHG HQFRXUDJHPHQW )LQDOO\ VSHFLDO WKDQNV DUH GXH WR RKQ +ROODQG ZKR GLG DQ H[FHOOHQW MRE LQ SUHSDUn LQJ WKLV GRFXPHQW X
PAGE 3
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f /HPPD /HPPD /HPPD /HPPD 3URRI RI /HPPD LLL
PAGE 4
/HPPD /HPPD /HPPD 3URRI RI 7KHRUHP $33(1',&(6 $ %$6,& ,'(17,7,(6 % 72 +$1'/( ',))(5(1&( 2) (1(5*< 7(50 & 72 2%7$,1 *(1(5$/ ,'(17,7< 5()(5(1&(6 %,2*5$3+,&$/ 6.(7&+ LY
PAGE 5
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f Zf $Z LQ 4 rrf [ Â Df Zf Z Zf Z LQ e Ef n Z, r /,f /RRf/Uff RQ ( rrf [ 7 Ff Âf WUOV A R rQ] Z :H VHHN WR H[SUHVV WKH QRQ]HUR FRQWURO IXQFWLRQ J M DV DV VXLWDEOH OLQHDU IHHGEDFN DSSOLHG WR WKH YHORn FLW\ Zf LH Z )Zf VXFK WKDW )Z H / rrf /7ff DQG WKH FRUUHVSRQGLQJ FORVHG ORRS V\VWHP REWDLQHG E\ XVLQJ VXFK D IHHGEDFN LQ f JHQHUDWHV D &VHPLJURXS ZKLFK GHFD\V XQLIRUPO\ H[SRQHQWLDOO\ WR ]HUR DV LQ WKH XQLIRUP WRSRORJ\ RI = >,'$9Lf< [ _'$nf@n __ ZfZf ,, ] A &H__ > KnTM+nL @ ,, ] IRU DOO DQG VRPH & f Y
PAGE 6
+DYLQJ LGHQWLILHG WKH FDQGLGDWH )Z fÂ§>$$ Zf@ ZKHUH $ LV WKH RSHUDWRU GHILQHG E\ $I $ ,'$f ^ IH /4f $H /4f M e_ ` ZH SURYH WZR VWDELOL]DWLRQ UHVXOWV WKH VHFRQG RI ZKLFK f LV WKH JRDO RI WKLV WKHVLV DQG LPSOLHV WKH ILUVW +RZHYHU ZH LQFOXGH ERWK UHVXOWV WR LOOXVWUDWH D FRQWUDVW LQ PRGHP GD\ FRQWURO PHWKRGV 6SHFLILFDOO\ LI WKH GRPDLQ e VDWLVILHV D UDGLDO YHFWRU ILHOG DVVXPSWLRQ MFfÂ§&f f Y \ RQ U f ZKHUH Y LV WKH RXWZDUG XQLW QRUPDO YHFWRU ZH SURYH VWURQJ VWDELOL]DWLRQ LH VROXWLRQV JR WR ]HUR LQ WKH VWURQJ WRSRORJ\ RI = OLP __ >ZIfZ@ ,, ; E\ WKH XVH RI D +LOEHUW VSDFH GHFRPSRVLWLRQ IRU FRQWUDF ÂfÂ§!f WLYH VHPLJURXSV )LQDOO\ LI 4 VDWLVILHV f ZH REWDLQ WKH GHVLUHG XQLIRUP VWDELOL]DWLRQ f YLD D FKDQJH RI YDULDEOHV IROORZHG E\ WKH XVH RI PXOWLSOLHUV YL
PAGE 7
&+$37(5 ,1752'8&7,21 35(/,0,1$5,(6 67$7(0(17 2) 0$,1 5(68/76 /, ,QWURGXFWLRQ $QG /LWHUDWXUH /HW 4 EH DQ RSHQ ERXQGHG GRPDLQ LQ ,5Q Q W\SLFDOO\ ZLWK VXIILFLHQWO\ VPRRWK ERXQGDU\ 7 ,Q e ZH FRQVLGHU WKH (XOHU%HPRXOOL PL[HG SUREOHP LQ ZW[f RQ DQ DUELWUDU\ WLPH LQWHUYDO n@ Zf $Z LQ 7@ [ IL Df Z [f Z[f Z [f :L [f LQ 4 Ef ZWDf JGWRf RQ 7@ [ U Ff _ADf JL FWf RQ RQ ; U Gf ZLWK QRQKRPRJHQHRXV IRUFLQJ WHUPV FRQWURO IXQFWLRQV J? DQG J LQ WKH 'LULFKOHW DQG 1HXPDQQ ERXQGDU\ FRQGLWLRQVf ,Q OOGf Y GHQRWHV WKH XQLW QRUPDO WR 7 SRLQWHG RXWZDUG 5HFHQWO\ WKHUH KDV EHHQ D NHHQ UHVXUJHQFH RI LQWHUHVW HJ >//O@ >5@ DQG UHIHUHQFHV FLWHG WKHUHLQf LQ WKH WKHRU\ RI SODWH HTXDWLRQV RI ZKLFK WKH (XOHU%HPRXOOL HTXDWLRQ Df LV D FDQRQLFDO PRGHO SUHVXPDEO\ VWLPXODWHG E\ WZR PDLQ VRXUFHV Lf UHQHZHG VWXGLHV LQ WKH G\QDPLFV IHDVLELOLW\ DQG LPSOHPHQWDWLRQ RI VRFDOOHG ODUJH VFDOH IOH[LEOH VWUXFWXUHV HQYLVLRQHG WR EH HPSOR\HG LQ VSDFH LLf UHFHQW PDWKHPDWLFDO DGYDQFHV LQ UHJXODULW\ WKHRU\ RI VHFRQG RUGHU PL[HG K\SHUEROLF SUREOHPV FDQRQLFDOO\ WKH ZDYH HTXDWLRQ RI ERWK 'LULFKOHW 7\SH >/@ >/7@ >/7@ >//7O@ DQG RI 1HXn PDQQ W\SH >/7@ >/7@ >/7@ >/7@ >/7@ >/7@ >6O@f ZLWK/ ERXQGDU\ GDWD ,Q HLWKHU FDVH D SULPH WKUXVW RI PRWLYDWLRQ KDV FRPH IURP G\QDPLFDO FRQWURO VWXGLHV DW HLWKHU DQ HQJLQHHULQJ RU D WKHRUHWLFDO OHYHO :LWK UHIHUHQFH WR WKH VSHFLILF SUREOHP f ZH FLWH >/@ >/7O@ >/7@ IRU RSWLPDO UHJXODULW\ WKHRU\ DQG H[DFW FRQWUROODELOLW\ WKHRU\ ZLWK UHVSHFW WR FODVVHV RI LQWHUHVW IRU WKH LQLWLDO GDWD > ZnR:@ @ DQG IRU WKH ERXQGDU\ GDWD >JLJ@ ZKLFK PDUNHGO\ LPSURYHG XSRQ SULRU OLWHUDn WXUH >/0@ 2XU LQWHUHVW LQ WKH SUHVHQW ZRUN LV RQ WKH SUREOHP RI ERXQGDU\ IHHGEDFN XQLIRUP VWDELOL]DWLRQ
PAGE 8
IRU WKH G\QDPLFV f E\ H[SOLFLW IHHGEDFN RSHUDWRUV WR EH PRUH SURSHUO\ GHILQHG EHORZ 2XU UHVXOWV DUH IXOO\ FRQVLVWHQW ZLWK WKH FRUUHVSRQGLQJ H[DFW FRQWUROODELOLW\ UHVXOWV >/7O@ >/7@ QRW RQO\ ZLWK UHVSHFW WR WKH IXQFWLRQ VSDFHV IRU >J?J@ DQG >ZZ@ DV PHQWLRQHG DERYH EXW DOVR ZLWK UHVSHFW WR WKH ODFN RI JHRPHWULFDO FRQGLWLRQV RQ e ZKHQ ERWK J M DQG J DUH DFWLYH RU HOVH ZLWK UHVSHFW WR WKH SUHVHQFH RI VLPLODU JHRPHWULFDO FRQGLWLRQV RQ Â ZKHQ RQO\ J W LV DFWLYH DQG J LV WDNHQ J 7KLV FRQVLVWHQF\ LV PRVW GHVLUn DEOH DQG LQGHHG KDV WR EH H[SHFWHG LQ YLHZ RI WKH NQRZQ UHODWLRQVKLS EHWZHHQ H[DFW FRQWUROODELOLW\ DQG XQLIRUP VWDELOL]DWLRQ IRU WLPH UHYHUVLEOH G\QDPLFV VXFK DV f >5O@ >5@ :H QRWH LQ SDVVLQJ WKDW XQLn IRUP VWDELOL]DWLRQ RI SUREOHP f E\ PHDQV RI D IHHGEDFN RSHUDWRU DFWLQJ RQ >ZZ@f ZKLFK LV GHILQHG LQ WHUPV RI WKH DOJHEUDLF 5LFFDWL RSHUDWRU ZKLFK DULVHV LQ WKH VWXG\ RI WKH RSWLPDO TXDGUDWLF FRVW SUREOHP RQ rrff ZDV DOUHDG\ DFKLHYHG LQ WKH DEVWUDFW WUHDWPHQW RI >)/7O@ 0DWKHPDWLFDOO\ WKH SUHVHQW ZRUN LV JXLGHG E\ DQG SDUWLDOO\ UHVWV XSRQ WHFKQLTXHV GHYHORSHG LQ WZR PDLQ VRXUFHV Lf WKH VWXGLHV RI H[DFW FRQWUROODELOLW\ >/7O@ >/7@ IRU SUREOHP f LLf WKH VWXG\ RI XQLIRUP VWDELOL]DWLRQ RI WKH ZDYH HTXDWLRQ ZLWK ERXQGDU\ IHHGEDFN LQ WKH 'LULFKOHW ERXQGDU\ FRQGLWLRQV >/7@ DQG LQ WKH 1HXPDQQ ERXQGDU\ FRQGLWLRQV >7O@ 2I FRXUVH WKHVH VWXGLHV KDYH WR EH VHHQ LQ WKH FRQWH[W RI UHFHQW LQYHVWLJDWLRQV LQFOXGLQJ Df XQLIRUP VWDELOL]DWLRQ RI WKH ZDYH HTXDWLRQ ZLWK IHHGEDFN LQ WKH 1HXPDQQ ERXQGDU\ FRQGLWLRQV >&O@ >&@ >/@ >/@ >/7 VHFW @ Ef UHJXODULW\ WKHRU\ IRU K\SHUEROLF HTXDWLRQV LQ >/@ >/7@ >/7@ >//7O@ DV ZHOO DV FRUUHVSRQGLQJ H[DFW FRQWUROODELOLW\ WKHRU\ >/@ >/7O@ >+O@ >7@ >7@ Ff H[DFW FRQWUROODELOLW\ UHVXOWV IRU (XOHU%HPRXOOL HTXDWLRQV ZLWK GLIIHUHQW ERXQGDU\ FRQGLWLRQV >/@ >/@ >/7@ >//O@ DQG ILQDOO\ Gf FRUUHVSRQGLQJ RSWLPDO TXDGUDWLF FRVW SUREOHPV >'/7O@ >/7O @ >)/7O@ $ SRLQW RI YLHZ ZKLFK ZH VWUHVV LV WKH IROORZLQJ ZH FKRRVH J LQ fRSHQ ORRSf IRUP WR EH LQ /f/2f 7 7KLV GHWHUPLQHV WKDW WKH FRUUHVSRQGLQJ VROXWLRQ RI SUREOHP f ZLWK Z +n J VDWLVILHV >ZZ@ H &> 7@ =f ZKHUH = LV WKH VSDFH LGHQWLILHG LQ f EHORZ 7KLV LV DQ RSWLPDO UHJXODULW\ UHVXOW >/@ >/7@ 1H[W ZH FKRRVH J VXFK WKDW WKH FRUUHVSRQGLQJ VROXWLRQ RI
PAGE 9
f ZLWK JL Z :M DOVR SURGXFHV >ZZ@ J &> @ =f DJDLQ DV DQ RSWLPDO UHJXODULW\ UHVXOW 7KLV OHDGV WR >ZZ@ H = DQG J H / f BUff >/@ >/7O@ >/7@ ,Q RWKHU ZRUGV RQO\ RQH FKRLFH LV PDGH WKDW JM J / f /9ff WKHQ ZH ZRUN ZLWK GDWD DQG VROXWLRQV LQ WKH FRUUHVSRQGLQJ RSWLPDO VSDFHV 2XU VROXWLRQ RI WKH XQLIRUP VWDELOL]DWLRQ SUREOHP EHORZ LV IXOO\ FRQVLVWHQW ZLWK WKHVH fRSHQ ORRSf FRQVLGHUDWLRQV XQLIRUP VWDELOL]DWLRQ ZLOO EH DFKLHYHG LQ WKH VSDFH = ZLWK FRQWUROV LQ IHHGEDFN IRUP J[ J / rrf /^9ff DQG J H / rrf fnUff VHH 7KHRUHP EHORZf +RZHYHU WKH SUHVHQW WKHVLV ZLOO WUHDW RQO\ WKH FDVH JL J / rrf /7ff LQ IHHGEDFN IRUP DQG J OHDYLQJ WKH PRUH JHQHUDO FDVH WR RXU VXFFHVVLYH HIIRUW >%7O@ :H ILQDOO\ SRLQW RXW WKDW RWKHU XQLIRUP VWDELOL]DWLRQ SUREOHPV IRU SODWH HTXDWLRQV KDYH EHHQDUH EHLQJ FRQVLGHUHG IROORZLQJ /DJQHVH/LRQV UHFHQW PRQRJUDSK >//O@ RQ SODWHV >/O@ >/@ ,Q WKHVH ZRUNV KRZHYHU GLIIHUHQW ERXQGDU\ FRQGLWLRQV RFFXU W\SLFDOO\ RI KLJKHU RUGHU HJ $Z DQG $V D FRQVH RY TXHQFH XQLIRUP VWDELOL]DWLRQ UHVXOWV DUH VRXJKW DQG REWDLQHG LQ KLJKHU WRSRORJLHV RQ e VR FDOOHG FRUUHVSRQGLQJ fHQHUJ\ VSDFHVff 7KH SUREOHP FRQVLGHUHG LQ WKLV WKHVLV ZLWK ERXQGDU\ FRQGLWLRQV DV LQ Ff OOGf RI ORZHU RUGHU KDYH D QDWXUDO DQG RSWLPDO LQ IDFWf VHWWLQJ LQ YHU\ ORZ WRSRORJLHV RQ e VHH WKH RSWLPDO VSDFH EHORZ 7KLV SURGXFHV DGGLWLRQDO PDWKHPDWLFDO GLIILFXOWLHV 7R RYHUFRPH WKHVH REVWDFOHV LW ZLOO EH QHFHVVDU\ WR LQWURGXFH D QHZ YDULDEOH WKH YDULDEOH S EHORZ LQ f ZKLFK OLIWV WKH WRSRORJLHV RQ e WR WKH OHYHO ZKHUH WKH PXOWLSOLHU WHFKQLTXHV ZKLFK ZHUH VXFFHVVIXOO\ XVHG LQ >/7O@ >/7@ IRU WKH FRUUHVSRQGLQJ H[DFW FRQWUROODELOLW\ SUREOHP DUH DSSOLFDEOH $ VLPLODU OLIWLQJ ZDV HPSOR\HG LQ WKH FDVH RI WKH XQLIRUP VWDELOL]DWLRQ SUREOHP IRU ZDYH HTXDWLRQV ZLWK 'LULFKOHW IHHGEDFN LQ WKH ORZ EXW RSWLPDOf WRSRORJ\ /ef [ U ef >/7@ )RUPXODWLRQ RI WKH 8QLIRUP 6WDELOL]DWLRQ 3UREOHP DQG 0DLQ 6WDWHPHQWV 7KURXJKRXW WKH SDSHU ZH OHW !Q GHQRWH WKH /efLQQHU SURGXFW ZLWK DVVRFLDWHG QRUP __ __ T DQG !U GHQRWH WKH /,!LQQHU SURGXFW ZLWK DVVRFLDWHG QRUP __ __ U ,Q DGGLWLRQ /ef GHQRWHV WKH +LOEHUW VSDFH RI DOO ERXQGHG OLQHDU RSHUDWRUV RQ /4f DQG /7ef GHQRWHV WKH +LOEHUW VSDFH RI DOO ERXQGHG OLQHDU RSHUDWRUV IURP /7f LQWR /4f )LQDOO\ /;f ZLOO GHQRWH WKH %DQDFK VSDFH RI DOO ERXQGHG OLQHDU RSHUDWRUV RQ ;
PAGE 10
:H EHJLQ E\ OHWWLQJ $ ,'$f F Â4f ÂOf EH WKH SRVLWLYH VHOIDGMRLQW RSHUDWRU GHILQHG E\ $I $ '$f +?4fQ+O4f f :LWK WKH RSHUDWRU $ GHILQHG DV VXFK LW WKHQ IROORZV WKDW > /7 $SSHQGL[ & @ ='$0f +T4f ^IH+n4f ` f ,'$r1f 9 ^4f _U e_U ` f ZKHUH ZH XVH WR GHQRWH QRUP HTXLYDOHQFH 7KXV IRU IH '$9rf OOOO rff ,,An9,/ ,, ,, +n&Of a ^ 4,9_G4 ` 2f ZKHUH WKH ODVW HTXLYDOHQFH IROORZV E\ 3RLQFDUH LQHTXDOLW\ 6LPLODUO\ IRUH k$f ??I??P}$f ,, $n_ T ^ 9$,Ge ` f $V VXJJHVWHG E\ >/7O@ >/7@ RXU RSWLPDO VSDFH LQ ZKLFK WR VWXG\ VWDELOL]DWLRQ ZLOO EH = +aO ILf [9n >'$ rf@n [ >='$ ff@n f ZKHUH n GHQRWHV GXDOLW\ ZLWK UHVSHFW WR WKH /^4f WRSRORJ\ 1H[W OHW JL J LQ f 7KHQ WKH FRUUHVSRQGLQJ HYROXWLRQ RI f LV JRYHUQHG E\ WKH RSHUDWRU ZKLFK JHQHUDWHV D VWURQJO\ FRQWLQXRXV XQLWDU\ JURXS RQ WKH VSDFH ,'$LDf[/4f ZLWK GRPDLQ ='$f '$f['$nf DQG KHQFH f RQ WKH VSDFH = RI RXU LQWHUHVW ZLWK GRPDLQ ='$Rf ,' $,f [ >='$f@n +T [ Âf :H GHQRWH WKLV XQLWDU\ JURXS E\ H 7KXV WKH IUHH VROXWLRQV RI f ZLWK JM J DUH QRUPSUHVHUYLQJ LQ = ,, >ZUfZ@ ,, = ,, HAf? ZZ @ __ ] ,, >ZZL @ __] IRU DOO f 5 :LWK WKLV ZHOONQRZQ UHVXOW DW KDQG ZH FDQ VWDWH WKH DLP RI WKH SDSHU 0RWLYDWHG E\ DQG FRQVLVWHQW ZLWK WKH IXQFWLRQ VSDFHV LQ WKH RSWLPDO UHJXODULW\ DQG H[DFW FRQWUROODELOLW\ WKHRU\ RI f >/@ >/7O@ >/7@ ZH VKDOO VWXG\ WKH TXHVWLRQ RI H[LVWHQFH DQG FRQVWUXFWLRQ RI H[SOLFLW ERXQGDU\ IHHGEDFN RSHUDWRUV ) [ DQG ) EDVHG RQ WKH fYHORFLW\f Z
PAGE 11
f f )AZH/A2f /f )ZfH/af cnn2f VXFK WKDW WKH ERXQGDU\ IHHGEDFN IXQFWLRQV rL ALZ}f JL )\Yf f RQFH LQVHUWHG LQ FGf SURGXFH D IHHGEDFNf &VHPLJURXS HA ZKLFK LV H[SRQHQWLDOO\ VWDEOH LQ WKH XQLn IRUP RSHUDWRU QRUP /=f RI WKH VSDFH = LQ f QDPHO\ WKHUH H[LVW FRQVWDQWV 0 ! VXFK WKDW IRU DOO W! ,8Dn,,O]Vmf XVf :H WKHQ VD\ WKDW VXFK RSHUDWRUV ) ? DQG ) XQLIRUPO\ H[SRQHQWLDOO\f VWDELOL]H WKH RULJLQDO QRUPn SUHVHUYLQJ FRQVHUYDWLYHf G\QDPLFV f ZLWK JM J $ ZHDNHU FRQFHSW LV WKDW RI fVWURQJ VWDELOL]DWLRQf E\ ZKLFK ZH VHHN RSHUDWRUV )W DQG ) DV LQ f VXFK WKDW RQFH LQVHUWHG LQ FGf SURGXFH D IHHGEDFNf &VHPLJURXS HAf RQ = ZKLFK GHFD\V VWURQJO\ WR ]HUR ??HA __ ] fÂ§! DVW fÂ§IRU DOO ] J = 66f :H QRWH WKDW IRU FRPSDFW VHPLJURXSV VXFK DV WKRVH DULVLQJ LQ SDUDEROLF HTXDWLRQV RQ D ERXQGHG GRPDLQ e WKH FRQFHSW RI VWDELOLW\ LQ WKH VWURQJ WRSRORJ\ HYHQ ZHDN WRSRORJ\f LV HTXLYDOHQW WR WKH FRQFHSW RI VWDELOLW\ LQ WKH XQLIRUP WRSRORJ\ +RZHYHU IRU VHFRQG RUGHU K\SHUEROLF SUREOHPV DQG SODWH SUREOHPV VXFK DV f WKH WZR FRQFHSWV RI VWURQJ DQG XQLIRUP VWDELOLW\ DUH GLVWLQFW )RU UHFHQW RSWLPDO UHVXOWV RQ WKH ODFN RI XQLIRUP VWDELOL]DWLRQ VHH >7@ >7@ &KRLFH RI 2SHUDWRUV ) DQG )L ,W LV MXVWLILHG LQ &KDSWHU LQ WKH FDVH RI ) [ DQG LQ >%7O@ LQ WKH JHQHUDO FDVH WKDW WKH IROORZLQJ FKRLFHV RI ) [ DQG )
PAGE 12
J[ )OZOf NO[f*?$OOZO NOFf*?$$QZ NO[ffÂ§/A/_ f J 3Acf N[fK*n$$nnUOZW Â[f$$?ZKnf_ f Â‘O SURYLGH UHDVRQDEOH FDQGLGDWHV IRU WKH XQLIRUP VWDELOL]DWLRQ SUREOHP RI f LQ WKH VHQVH WKDW WKH FORVHG ORRS IHHGEDFN G\QDPLFV ZLWK f DQG f LQVHUWHG LQ Ff DQG OOGf UHVSHFWLYHO\ LV ZHOOSRVHG LQ WKH VHPLJURXS VHQVH LQ = DQG WKH =QRUP RI DOO RI LWV VROXWLRQV RULJLQDWLQJ LQ = GHFUHDVHV DV W } rr WKLV KRZHYHU GRHV QRW VD\ WKDW VXFK =QRUPV GHFUHDVHV WR ]HUR DV W fÂ§! N} VWURQJ VWDELOL]DWLRQf OHW DORQH LQ WKH XQLIRUP QRUP RI /=f f 7R VKRZ WKLV FRQFOXVLRQ ZLOO EH RXU PDMRU WDVN LQ &KDSWHU DW OHDVW LQ WKH FDVH ZKHUH J DW WKH RXWVHW ,Q f f ZH KDYH WKDW Df NL[f VPRRWK IXQFWLRQV RQ 7 Nc[f f Ef $ RQWRf LVRPRUSKLVP +67f } +V f7f VHOIDGMRLQW RQ /' f VR WKDW I ,, $GL $Q OOGO Q?U! ^UOYMmOGU f ZKHUH 9 GHQRWHV WKH WDQJHQWLDO JUDGLHQW RQ 7 Ff 7KH RSHUDWRUV *@ DUH WKH DGMRLQWV LQ WKH VHQVH WKDW *cJ ] !D J *8 !U r /' ] H /4f f RI WKH RSHUDWRUV *W GHILQHG E\ $[ LQ 4 Df *LJ [ LI DQG RQO\ LI MF, J@ RQ 7 fU &Ef f A RQ7 / Y OU Ff $\ fÂ§ LQ Â Df J aa \ LI DQG RQO\ LI \ RQ 7 Â‘U Ef f WUOUr6QU Ff
PAGE 13
(OOLSWLF WKHRU\ >/0 9RO @ JLYHV IRU DQ\ V H 5 *[ FRQWLQXRXV +67f }I 4f f FRQWLQXRXV +67f !ILf f 0RUHRYHU E\ *UHHQfV WKHRUHP LW LV SURYHG WKDW >/7 /HPPD DQG /HPPD UHVSHFWLYHO\@ *?$I IH'$f f *$I $f, J'$f f fU ,GHQWLWLHV f f DUH XVHG LQ WKH ODVW VWHS RI f f UHVSHFWLYHO\ 7KXV WKH UHVXOWLQJ FDQGLn GDWH IHHGEDFN V\VWHP ZKRVH VWDELOLW\ SURSHUWLHV LQ = ZH VKDOO LQYHVWLJDWH LV Zf $Z LQ RRf [ 4 4 Df Z[f Z[f Z[f :M&[f LQ 4 Ef $$ZZf 6Y RQ rrf [ 7 = Ff f RQ Â‘Gf 8VLQJ WKH WHFKQLTXHV RI >7@ SUREOHP f FDQ EH UHZULWWHQ PRUH FRQYHQLHQWO\ LQ DEVWUDFW IRUP DV f r Df $ $>N*[*?$P N*.*$nD`? '$f ^ \ H = $\ H= ` Ef $ PRUH H[SOLFLW GHVFULSWLRQ RI '$f ZLOO EH JLYHQ EHORZ 2XU PDLQ UHVXOWV DUH DV IROORZV 7KHRUHP /, Lf :HOOSRVHGQHVV RQ = 7KH RSHUDWRU $ LQ f LV GLVVLSDWLYH RQ = >,'$f@n [ >,'$f@n VHH f DQG VDWLVILHV KHUH UDQJH ;, $f = IRU ; 7KXV E\ /XPHU3KLOOLSV WKHRUHP >3@ $ JHQHUDWHV D VWURQJO\ FRQWLQXRXV FRQWUDFWLRQ VHPLJURXS H RQ = DQG WKH UHVROYHQW RSHUDWRU "; $f LV FRPSDFW RQ = IRU 5H;
PAGE 14
0RUHRYHU H S$f WKH UHVROYHQW VHW RI $ LLf %RXQGHGQHVV RI IHHGEDFN RSHUDWRUV )RU > YYR+fM @ H = ZH KDYH PRUH SUHFLVHO\ $$ Zf N[*[$LDZ N[fÂ§AfÂ§ H / rrf /7ff Y N.*?$aPZ N$Â$$LDZWf H /? rrf +a?7ff f f ??N[*?$PZW ?"7GW __ >KnRKnL @ __ ] f 4 ,, N$*r$aPZW __UGW __ > ZZL @ __ ] f 7KH SURRI RI 7KHRUHP ZLOO EH JLYHQ LQ &KDSWHU LQ WKH FDVH RI J DQG LQ >%7O@ LQ WKH JHQHUDO FDVH 7KHRUHP 8QLIRUP VWDELOL]DWLRQ RQ = ZLWK ERWK IHHGEDFN RSHUDWRUV LQ WKH DEVHQFH RI JHRPHWULFDO FRQn GLWLRQV RQ ef 7KH IROORZLQJ SURSHUW\ KROGV IRU WKH IHHGEDFN SUREOHP f RU f f WKHUH DUH FRQVWDQWV 0 DQG VXFK WKDW IRU DOO W! ,, _}_ 8RO ,, ] f 7KHRUHP 8QLIRUP VWDELOL]DWLRQ RQ = ZLWK RQO\ WKH ILUVW IHHGEDFN RSHUDWRU J[ DQG J LQ WKH SUHVn HQFH RI JHRPHWULFDO FRQGLWLRQV RQ 4f &RQVLGHU WKH IHHGEDFN SUREOHP f ZLWK J[ JLYHQ E\ f ZKLOH J 7KHQ WKHUH LV D FRQVWDQW VXFK WKDW WKH XQLIRUP GHFD\ f KROGV WUXH SURYLGHG 4 VDWLVILHV WKH IROORZLQJ JHRPHWULFDO FRQGLn WLRQ UDGLDO YHFWRU ILHOG DVVXPSWLRQf WKHUH H[LVWV D SRLQW[ H 5Q VXFK WKDW [[f f9 !\!RQ 7 f 7KHRUHP LV WKH PDLQ UHVXOW SURYHG LQ WKLV WKHVLV ,W PD\ EH H[WHQGHG WR PRUH JHQHUDO GRPDLQV e ZKLFK VDWLVI\ D ZHDNHU JHRPHWULFDO FRQGLWLRQ WKDQ f H[SUHVVHG LQ WHUPV RI D PRUH JHQHUDO YHFWRU ILHOG WKDQ WKH FODVV RI UDGLDO ILHOGV MF[f 7KLV ZLOO EH GRQH LQ >%7O@ $OVR 7KHRUHP WKH JHQHUDO UHVXOW
PAGE 15
ZLWK WZR IHHGEDFN RSHUDWRUV DQG QR JHRPHWULFDO FRQGLWLRQZLOO OLNHZLVH DSSHDU LQ >%7O@ ,QVWHDG LQ &KDSWHU RI WKLV WKHVLV ZH VKDOO DOVR SURYH D VWURQJ VWDELOL]DWLRQ UHVXOW 7KHRUHP f ZLWK JM DV LQ f DQG Â f DQG J XQGHU WKH VDPH JHRPHWULFDO DVVXPSWLRQ f 7KRXJK WKH VWURQJ VWDELOL]DWLRQ UHVXOW RI 7KHRUHP LQ &KDSWHU LV LPSOLHG E\ WKH XQLIRUP VWDELOL]DWLRQ UHVXOW RI 7KHRUHP LQ &KDSWHU ZH IHHO WKDW LWV LQFOXVLRQ LQ WKLV WKHVLV LV MXVWLILHG E\ WKH IROORZLQJ FRQVLGHUDWLRQV ,W VKRZV fKRZ IDUf WKH SXUHO\ RSHUDWRU WHFKQLTXHVEDVHG RQ WKH RSHUDWRU PRGHO f f DQG WKH 1DJ\)RLDV)RJXHO GHFRPSRVLWLRQ IRU FRQWUDFWLRQ VHPLJURXSVFDQ EH FDUULHG RXW DORQJ WKH OLQHV RI DUJXPHQWV ILUVW XVHG IRU ERXQGDU\ FRQWURO SUREOHPV IRU VHFRQG RUGHU K\SHUEROLF HTXDWLRQV LQ >/7@ >/7@ >/7@ DQG DOVR LQ >7O@ IRU GLIIHUHQW IHHGEDFN RSHUDWRUV $ QHZ REVWDFOH DULVHV KRZHYHU LQ WKH FDVH RI SODWH SUREOHPV IRXUWK RUGHU LQ WKH VSDFH YDULDEOH UDWKHU WKDQ VHFRQG RUGHU LQ VSDFH DV LQ K\SHUEROLF SUREOHPVf DW WKH OHYHO RI H[FOXGLQJ WKH SUHVHQFH RI HLJHQYDOXHV DORQJ WKH LPDJLQDU\ D[LV 7KLV LV /HPPD ZKHUH RQO\ WKUHH KRPRJHQHRXV ERXQGDU\ FRQGLWLRQV EFGf DUH LQ SODFH IRU WKH fHLJHQSUREOHPf Df DV RSSRVHG WR WKH IRXU KRPRJHQHRXV ERXQGDU\ FRQGLWLRQV IRU D IRXUWK RUGHU RSHUDWRU FRYHUHG E\ VWDQGDUG WKHRU\ ,Q SUHn YLRXV DUJXPHQWV IRU VHFRQG RUGHU K\SHUEROLF SUREOHPV DV LQ >/7@ >/7@ >/7@ >7O@ WKH FRUUHVSRQGLQJ fHLJHQSUREOHPf KDV WZR KRPRJHQHRXV ERXQGDU\ FRQGLWLRQV IRU D VHFRQG RUGHU RSHUDWRU DQG KHQFH LV FRYHUHG E\ VWDQGDUG WKHRU\f $ QRYHOW\ LV WKHQ WKDW /HPPD XVHV D PXOWLSOLHU WHFKQLTXH SURRI ZKLFK LV VXSSRUWHG E\ WKH H[DFW FRQWUROODELOLW\ SUREOHP >/7O@ >/7@ DV DSSOLHG KRZHYHU WR WKH VWDn WLRQDU\f HOOLSWLF SUREOHP WKLV WLPH ,W LV WKH PXOWLSOLHU WHFKQLTXH WKDW UHTXLUHV WKH JHRPHWULFDO FRQGLWLRQ f LQ WKH SURRI RI /HPPD /HPPD DSSHDUV WR EH QHZ LQ HOOLSWLF WKHRU\ 7KLV IDFW DORQH ZRXOG MXVWLI\ LWV LQFOXVLRQ KHUH HYHQ WKRXJK /HPPD SOD\V RQO\ D VHFRQGDU\ UROH LQ WKH SUREOHP RI VWDELOL]Dn WLRQ FRQVLGHUHG KHUH DV LW OHDGV RQO\ WR VWURQJ VWDELOL]DWLRQ 7KHRUHP f ,W LV WKURXJK WKH PRUH HODn ERUDWH DQG OHQJWKLHU DUJXPHQWV RI &KDSWHU ZKLFK DUH QHFHVVLWDWHG E\ XVH RI WKH VDPH PXOWLSOLHU DSSOLHG WR WKH IHHGEDFN G\QDPLFV fWKDW ZH ZLOO HYHQWXDOO\ REWDLQ WKH XQLIRUP VWDELOL]DWLRQ UHVXOW RI 7KHRUHP
PAGE 16
&+$37(5 :(//326('1(66 $1' 67521* 67$%,/,=$7,21 3UHOLPLQDULHV &KRLFH RI 6WDELOL]LQJ )HHGEDFN /HW 4 EH DQ RSHQ ERXQGHG GRPDLQ LQ Q! ZLWK VXIILFLHQWO\ VPRRWK ERXQGDU\ 7 &RQVLGHU WKH QRQKRPRJHQHRXV SUREOHP LQ WKH VROXWLRQ Z W[f Zf $Z LQ 4 rRf [ 4 Df Zf Z Zf Z LQ LL Ef Z_ JOH//f / RRf/Qf Â‘O RQ e RRf [ 7 Ff f f RQ ( Gf 7KH JRDO RI WKLV FKDSWHU LV WR REWDLQ VWURQJ VWDELOL]DWLRQ RI WKH V\VWHP f YLD D FORVHGORRS IHHGEDFN J M EDVHG RQ WKH YHORFLW\ Z +RZHYHU WKH RSWLPDO IXQFWLRQ VSDFH LQ ZKLFK WR ZRUN D FURVV SURGXFW VSDFH IRU SRVLWLRQ DQG YHORFLW\f LV REWDLQHG IURP WKH H[DFW FRQWUROODELOLW\ UHVXOW WR EH VXPPDUL]HG EHORZ )LUVW ZH GHILQH WKH SRVLWLYH VHOIDGMRLQW RSHUDWRU $ ,'$f F /LOf fÂ§! /4f E\ $I $ Df '$f ^H/ef$H/4f _A e_ R` V 2i4f Ef 6LQFH 4 LV ERXQGHG LQ 5Q WKHQ $ KDV FRPSDFW UHVROYHQW 5? $f $OVR LI $I IRU H '^$f WKHQ E\ $2f DQG FGf ZH KDYH $I I!Q $$f I!D $ $I !T I!U $I !U $I $!T __$__Â
PAGE 17
7KLV LPSOLHV VR WKHUHIRUH $a[ H /4f f 1H[W ZH OHW DQG FRQVLGHU WKH VSDFH = + nLf [ 9 $V VKRZQ LQ >/7@ = FDQ EH FKDUDFWHUL]HG E\ XVLQJ HTXLYDOHQW QRUPV DV = >,'$Pf< [ >'$nf@n ZKHUH n GHQRWHV GXDOLW\ ZLWK UHVSHFW WR WKH /LfWRSRORJ\ Ef 7KH QRUPV RQ WKHVH VSDFHV DUH JLYHQ E\ f ,0, e!rff ,8frOOQ ,8 ,, >'D3f@n ,8 0LQ DS! f %HORZ ZH VWDWH WKH UHJXODULW\ UHVXOW DV ZHOO DV WKH H[DFW FRQWUROODELOLW\ UHVXOW 7KHRUHP 5HJXODULW\f >/ I/7 &RQVLGHU WKH SUREOHP f VXEMHFW WR >Z ZH= J[ H / f /7ff J H /f Uff 7KHQ WKH PDS >ZZcJLJ@ >ZfZf@ H &>7@ =f LV FRQWLQXRXV IRU DQ\ 7 rr Â’ 7KHRUHP ([DFW &RQWUROODELOLW\f I/7 Lf $VVXPH WKHUH H[LVWV D SRLQW [ H 5Q VXFK WKDW [[f f Y \ RQ 7 ZKHUH Y LV WKH XQLW RXWn ZDUG QRUPDO YHFWRU /HW 7 rr EH DUELWUDU\ ,I >ZRA @ H = DUELWUDU\ WKHQ WKHUH H[LVWV D VXLWDEOH FRQn WURO IXQFWLRQ JO H / 7f /)ff VXFK WKDW WKH FRUUHVSRQGLQJ VROXWLRQ RI f ZLWK J VDWLVILHV Z7f Z7f DQG LQ DGGLWLRQ >ZZ@ H & >@ = f LLf 7KH VDPH FRQFOXVLRQ KROGV WUXH ZLWKRXW JHRPHWULFDO FRQGLWLRQV LI J LV WDNHQ ZLWKLQ WKH FODVV RI / ff2f FRQWUROV Â’ %\ WLPH UHYHUVLELOLW\ ZH VHH WKDW DW DQ\ ILQLWH 7 WKH WRWDOLW\ RI DOO VROXWLRQ SRLQWV >Z7fZ7f` RI SUREOHP f ZLWK Z :M ILOOV DOO RI WKH VSDFH = ZKHQ HLWKHU UXQV RYHU DOO RI / 7f /7ff
PAGE 18
DQG J XQGHU JHRPHWULFDO FRQGLWLRQV RQ 4 RU HOVH ZKHQ OJ?JM` QLQV RYHU DOO RI /^ 7f /7ff [L 7f B7ff ZLWKRXW JHRPHWULFDO FRQGLWLRQV 7KHUHIRUH VLQFH WKH VSDFH RI H[DFW FRQWUROODELOLW\ LV WKH VSDFH RI PD[LPDO UHJXODULW\ ZH VHHN VWDELOL]DWLRQ LQ H[DFWO\ WKLV VSDFH = :H GHILQH WKH HQHUJ\ (Wf IRU WKH G\QDPLFV f RYHU WKH VSDFH = >.f^$,f@n [ >'$:f@n E\ ,: Wf R ZLfO + + >c'$nLf< ; ,, :f0 ,, >k$rf@n ,8ZOÂ __$fZf__T f G( 1H[W ZH VHHN D FDQGLGDWH JL ZKLFK DW OHDVW SURGXFHV fÂ§fÂ§ LH HQHUJ\ GHFUHDVH 7KLV GRHV QRW DW KRZHYHU JXDUDQWHH OLP (Wf ZKLFK LV SUHFLVHO\ VWURQJ VWDELOLW\ RI ff OHW DORQH XQLIRUP VWDELOLW\ 5HPDUN %HORZ ZH VKDOO VKRZ ZHOOSRVHGQHVV LQ = ZLWK J 7KHQ VLQFH Z H >'$f@n LW IROORZV WKDW $ aYOZ $ $ Z H ,' $f 9 7KHUHIRUH $ Z VDWLVILHV WKH UHTXLUHG ERXQGDU\ FRQGLWLRQV $n Z f %\ ZULWLQJ (Wf $ ,Z $ Z !D $ Z $ 0Z !Q DQG GLIIHUHQWLDWLQJ ZLWK UHVSHFW WR W ZH KDYH WKDW $fZ $Z !Q $Zf $ ZZ !Q DL E\ Df Z $BZ !D $Z $f0Z !Q Z $nPZ !D $$Zf $nPZ !D E\ $2f Z $aOOZc !Q AI$Zf $BZ !U $Z A$BZf !U $Z $$aLDZWf !Q GY RY E\ f Z $ LDZ !T $Z $$ PZf !Q
PAGE 19
E\ $2f Z $a,Z !D $$nKnf !U a Z A>$$ K!f@ !U Z $$ PZWf !D Z $ LQZ !T Z A>$$ :Zf@ !U Z $ PZW !D Z A>$$aZf@ !U GY GY 7KHUHIRUH E\ VHOHFWLQJ WKH VLPSOHVW FKRLFH Z? mL fÂ§ >$$BDZf@ RY f :H REWDLQ fÂ§ J M __ RXU GHVLUHG HQHUJ\ GHFUHDVH Â’ DW 1H[W ZH ZLOO VKRZ KRZ RXU IHHGEDFN FDQ EH H[SUHVVHG LQ WHUPV RI DQ RSHUDWRU *UHHQ PDSf ZKLFK DFWV IURP ERXQGDU\ ) WR LQWHULRU 4 )ROORZLQJ >/7@>/7@ ZH GHILQH /)f fÂ§! /ef E\ e LQ 4 Df *LJ \ LI DQG RQO\ LI f \ J Â‘U RQ U Ef f U L RQ U Ff :H TXRWH WKH IROORZLQJ /HPPD ZKLFK ZLOO EH XVHG EHORZ /HPPD Â/7 /HW *M /4f }/Uf GHQRWH WKH FRQWLQXRXV RSHUDWRU GHILQHG E\ *LJ Y!4 J *MY !U J H /)f Y H /4f LH *? LV WKH DGMRLQW RI *? 7KHQ *OL _M$f_ IRUH ='$f f 1RZ XVLQJ f DQG f ZH VHH WKDW Z_ JO aA>$$PZOf@ *nO$$rZf *?$O:O f 8VLQJ HOOLSWLF WKHRU\ > /0 9RO S @ ZH KDYH WKDW IRU DQ\ V UHDO *L FRQWLQXRXV +67f + 4f Df DQG LQ SDUWLFXODU IRU V *@ FRQWLQXRXV /If :4f Ef
PAGE 20
:H DOVR KDYH WKDW E\ GXDOLW\ RQ Df ZLWK V WKDW *? FRQWLQXRXV ÂÂf!Uf f VR WKDW Df f LPSO\ **O FRQWLQXRXV e!$ rrf Â 4f !4f f WR EH XVHG EHORZ Ef LQ WKH GHVFULSWLRQ RI WKH GRPDLQ RI WKH IHHGEDFN JHQHUDWRU :HOO3RVHGQHVV DQG 6HPLJURXS *HQHUDWLRQ )LUVW ZH ZDQW WR LQWURGXFH DQ DEVWUDFW RSHUDWRU PRGHO IRU SUREOHP f $FFRUGLQJ WR >7O@ >7@ SUREOHP f ZLWK J DGPLWV WKH IROORZLQJ DEVWUDFW YHUVLRQV DV D VHFRQG RUGHU HTXDWLRQ Z $ >Z*MJA$ >Z *?*?$aPZ@ Df RU HOVH DV D ILUVW RUGHU V\VWHP L8$8 >Z YY@ H = >M'$ 0f@n [ >'$nf@n Ef r Â‘ ZKHUH $ O$ $*M*O$ PO f (!$f WKHQ ZH FDQ ZULWH OOA *?*?$a[D\L $\ $ R_ ,, Df 7KXV =' $f ^ >\ \@H=\H >='$f@n DQG$ >\ *O*nO$aLD\@ H >k$rf@n LHf \ *;*?$P\ H '$f Â4f ZKLFK LPSOLHV\ H n4f` Ef 7KH RSHUDWRU $ GHILQHG DERYH LV RXU FDQGLGDWH WR EH WKH JHQHUDWRU RI D IHHGEDFN VHPLJURXS 7KH ILUVW VWHS LQ WKLV GLUHFWLRQ LV WKH IROORZLQJ /HPPD /HPPD 7KH RSHUDWRU $ LV GLVVLSDWLYH RQ = >'$f@n [ >,'$9Lf@n
PAGE 21
3URRI /HW ] H = WKHQ XVLQJ EHORZ WKH VNHZDGMRLQWQHVV RI T ZH rLDYH IrU p $f 5H $] ]!] 5H _Bf n_ ?>>? e_ != O_=nO _=f_ 5H OR $*;*?$, 8, f 8O != $*[*?$ P] ] !^Q;$\f`n $r1$*[*?$P] $ar1] !D __ *?$nP] __ DQG GLVVLSDWLYLW\ KROGV Â’ 7KH DERYH SURRI LV D UHIRUPXODWLRQ RI RXU DUJXPHQW EHORZ f 1RZ ZH FRPH WR RXU UHVXOW RQ VHPLJURXS JHQHUDWLRQ 7KHRUHP Lf 7KH GLVVLSDWLYH RSHUDWRU $ LQ f DOVR VDWLVILHV UDQJH ;, $f = RQ = IRU ; 7KXV E\ WKH /XPHU3KLOOLSV 7KHRUHP >3@ $ JHQHUDWHV D &VHPLJURXS RI FRQWUDFWLRQV HAf RQ = W DQG WKH VROXWLRQ RI ff LV JLYHQ E\ DQG LQ IDFW _: :R :2 : ?O H 0 DOO >Z Z@H= ,ZW Z ZfO _ZL A LLf 7KH UHVROYHQW RSHUDWRU 5 ; $f RI $ LV JLYHQ E\ L YAU 5;$f 3;f@$ 8L Y;f [SAUn$ ZKHUH 9;f > ;**r$a ;$n Df __ H$_ r_ __ ] (Wf I I ,$ ZOO$ AZ` G4 Ef _:O_ 4 Df Ef DW OHDVW IRU DOO ; VDWLVI\LQJ 5H; 0RUHRYHU ; EHORQJV WR WKH UHVROYHQW VHW RI $ DQG 5; $f LV FRPSDFW RQ =
PAGE 22
LLLf ,I WKH GRPDLQ e LV VXFK WKDW WKHUH H[LVWV D SRLQW [ J c5Q VXFK WKDW MFMFRf f Y \ RQ 7 f WKHQ 5 ; $f LV ZHOOGHILQHG DQG FRPSDFW RQ = DOVR RQ WKH LPDJLQDU\ D[LV DQG KHQFH IRU DOO ; VDWLVI\LQJ 5H; 7KXV WKH VSHFWUXP SRLQWf RI $ VDWLVILHV D$f F ^ ; 5H; ` f 5HPDUN $ VWURQJHU UHVXOW ZLOO IROORZ EHORZ RQFH ZH SURYH RXU XQLIRUP VWDELOL]DWLRQ 7KHRUHP WKDW LQ IDFW R$fF ^ ;5H; ` Â’ 3URRI RI 7KHRUHP 'LVVLSDWLYLW\ RI $ RQ = ZDV DOUHDG\ VKRZQ LQ /HPPD 1H[W IL[ ; DQG OHW ] H = DQG ZH ZDQW WR VROYH ;, $f\ ] LH ;\O\ ]H>O'$nmf@n Df $\ *O*n$aOO\f ;\ ] H >k$rf@n Ef IRU \ H ,'^$f :H DSSO\ $ WR Ef PXOWLSO\ Df E\ ; DQG VXEWUDFW WR REWDLQ 9;f\ ;$f] ] J >(!$0f@n f ZLWK 9;f GHILQHG LQ Ef :H QH[W QRWH WKDW /$f LV ERXQGHGO\ LQYHUWLEOH RQ >,'$f@n VLQFH HTXLYDOHQWO\ $f0;f$n ;$a:*?*?$a[Lr ;$aO LV ERXQGHGO\ LQYHUWLEOH RQ /Lf EHLQJ VHOIDGMRLQW VWULFWO\ SRVLWLYH RQ /]eff ZLWK LQYHUVH $LD\LA$r H /4f 7KXV IURP f \ .n;f;$]]fH >='$f@n ZKLFK WKHQ LQVHUWHG LQ Df \LHOGV U ,9a?;f f f \? ]O9?;f$O] f
PAGE 23
7KHQ Df IROORZV IURP f DQG f 1RWH WKDW IURP Ef DQG f WKDW \O**?$0\ $]8\H '$8rf f 6R WKDW UHFDOOLQJ Ef ZH VHH WKDW IURP f DQG f LW LV YHULILHG WKDW \ J ='$f 7KH FRPSDFWQHVV RI 5; $f RQ = LV UHDGLO\ VHHQ IURP Df WR EH HTXLYDOHQW WR FRPSDFWQHVV RQ /4f RI WKH IROORZLQJ RSHUDWRUV $ 9aO ;f f$ Df $\L9a^Nf$Yr $Yr9Nf$P$n Ef $ \LYa[ Nf$ $ P$ nff Nf$ Ff $ frr 9 ;f$ n $ B$ 9 ;f$ $ aP Gf )LUVW FRPSDFWQHVV RI WKH RSHUDWRUV EFGf RQ /4f LV SODLQ IURP f DQG $ f D EHLQJ FRPSDFW RQ /LOf )RU Df DSSO\ 9aO ;f RQ Ef VR WKDW 9aO ;f ;9 ;f* *M $ ;9a[ ;f$ DQG WKHQ $r > )B $f@ $ ;$)B ;f*A*?$ ; $)B ;f$n ;$ M ;f$ $ BP* L r $ Bn ;$);f$$B ZKLFK LV FRPSDFW RQ /4f E\ f VLQFH $B* M *r$ J /LOf Â’ 7R FRPSOHWH WKH SURRI RI 7KHRUHP ZH PXVW VKRZ WKDW D$f GRHV QRW FRQWDLQ DQ\ SRLQWV RQ WKH LPDJLQDU\ D[LV ZH DOUHDG\ NQRZ WKDW D$f GRHV QRW FRQWDLQ SRLQWV LQ ^ 5H; ` VLQFH $ LV WKH JHQHUDn WRU RI D FRQWUDFWLRQ VHPLJURXSf 7KXV ZH QHHG WR VKRZ WKDW 9;f H / >='$f@nf IRU ; LU U H ,5 U r f 7R WKLV HQG OHW [ J >,'$f@n DQG VXSSRVH 9;f[ IRU ; LU 7KHQ IURP Ef );f[ [ !>'$,0f@n [ [ !>='$f@n LU *[*?$aP[ [ !>='fÂ§ U $a[[ [ !>'$:f@f $aP[ [ !Q LU __ *r$f[ __A U $aP[ [ !D 6LQFH WKH PLGGOH WHUP LQ f LV SXUHO\ LPDJLQDU\ ZH PXVW KDYH WKDW YLD f f
PAGE 24
$OVR ZH KDYH WKDW E\ f $ P[ U$ P[ LH $[ U[ f ZKLFK PHDQV WKDW [ PXVW EH DQ HLJHQYHFWRU RI $ VD\ [ HQ ZLWK HLJHQYDOXH U 7KHUHIRUH VLQFH HQ H ,' $f ZH KDYH WKDW LW VDWLVILHV WKH WZR ]HUR ERXQGDU\ FRQGLWLRQV DVVRFLDWHG ZLWK ,'^$f VHH Eff DV ZHOO DV f 7KHUHIRUH WKH IROORZLQJ /HPPD ZLOO FRPSOHWH WKH SURRI RI 7KHRUHP /HPPD /HW ; U DQG VXSSRVH 4 VDWLVILHV WKH UDGLDO YHFWRU ILHOG DVVXPSWLRQ f 7KHQ WKH SUREOHP ,, H LQ 4 Df W!_ RQ U Ef Â‘U R ,, X RQ U Ff $rf_ R9 S RQ U LGf f KDV RQO\ WKH WULYLDO VROXWLRQ M! 1RWHV 6LQFH $AP[ $anHf UaHQ Gf IROORZV IURP f 7KH DERYH /HPPD LV QRW FRYHUHG E\ VWDQGDUG HOOLSWLF WKHRU\ VLQFH RQO\ WKUHH ERXQGDU\ FRQGLWLRQV LQVWHDG RI IRXU DUH LQYROYHG IRU WKH IRXUWKRUGHU HOOLSWLF RSHUDWRU LQ f 5HFDOO WKDW DV JLYHQ LQ $SSHQGL[ & LI K[f [[f WKHQ ,, [f LGHQWLW\ PDWUL[f GLY K Q GLP Â Df Ef ,I S LV D VROXWLRQ RI f WKHQ LW LV LPPHGLDWH WKDW YLD PXOWLSOLFDWLRQ E\ $S DQG G4 ZH KDYH 9$!f G' 9_! G4 f
PAGE 25
3URRI RI /HPPD 0XOWLSO\ ERWK VLGHV RI Df E\ K f 9$`!f DQG LQWHJUDWH /HIW KDQG VLGH :H KDYH E\ $f Gf DEf f $W\fK Â‘ 9$_!fG4 L> 9$!f K f ?G7 A Of;M ,9W!OG4 f 5LJKW KDQG VLGH [M K f 9$ffG4 E\ $f [M $:KYGU;M $fGLYWfKfG4 E\ $fEfEf ;M $!; f 9FSIÂ Q;MA$!_fLLL E\ $fDEf$Of fÂ§ [I _AL90U ,9,$ 9U [I 9SeeO AI ,9W!OG4 UGY U MT T m[I A0U m[I L9MfLGL UY T E\EFf U 9FffK Â‘ YAU f [M 9!G2 %XW VLQFH ff ZH KDYH 9c! , a 7KHUHIRUH S RY [MA f 9$_ffG4 M f [MA 9W! G f 6HWWLQJ f f DQG VLPSOLI\LQJ ZH JHW LI 9$f!fK Â‘ YGU [M ,9ffOGU f U 4L 1RZ LI ZH DQDO\]H f ZH VHH WKDW E\ f WKH OHIW KDQG VLGH LV QRQSRVLWLYH DQG WKH ULJKW KDQG VLGH LV QRQQHJDWLYH WKHUHIRUH ERWK VLGHV PXVW HTXDO ]HUR 7KLV WKHQ LPSOLHV 9E DH DQG KHQFH >! V F DH %XW ZH KDYH WKDW M! HLJHQIXQFWLRQ LPSOLHV f! VPRRWK DQG WRJHWKHU ZLWK f IRUH /HPPD DQG KHQFH 7KHRUHP DUH SURYHG Â’ U ZH KDYH f! r 7KHUH 1RZ WKDW ZH KDYH SURYHQ WKDW $ JHQHUDWHV D & VHPLJURXS RI FRQWUDFWLRQV H RQ = LW IROORZV WKDW
PAGE 26
(Wf(f IRU I! f 7KLV IDFW ZLOO EH XVHG FUXFLDOO\ EHORZ 7KH QH[W FRUROODU\ LV D FRQVHTXHQFH RI WKH GLVVLSDWLYH IHHGEDFN SHUn WXUEDWLRQ RQ WKH ERXQGDU\ &RUROODU\ %\ FKRRVLQJ Z J? *?$ PZf LW IROORZV WKDW *?$ PZ H /rrf /7ff DQG LQ IDFW Â‘[ ,, *?$PZ ,, /!Y f ,, *?$AZW ,,?GW \ef f IRU DOO LQLWLDO FRQGLWLRQV >ZZL@ H = 3URRI RI &RUROODU\ /HW > ZRZM @ H '$f DQG UHFDOO IRU FRQYHQLHQFH (^Wf f 1RZ Â‘aaeLf af GW GW IrO H$L _:_ ,}9 _Y9@ +f B $W _:r ,ZA H :L ?!= !YLf _:Lf_ OrfIfO f YLD WKH SURRI RI /HPPD f 5HPDUN :H VHH WKDW f VKRZV WKDW VXFK D FKRLFH RI J W GRHV OHDG WR DQ HQHUJ\ GHFUHDVH DV ZDV GHPRQn VWUDWHG LQ DQRWKHU ZD\ XVLQJ *UHHQfV IRUPXODf LQ 5HPDUN Â’ &RQWLQXLQJ WKH SURRI QRZ ZH LQWHJUDWH GW ERWK VLGHV WR REWDLQ
PAGE 27
?Hf? OLUD (7f(f / / }a ZKHUH LQ WKH ODVW LQHTXDOLW\ ZH XVHG WKH FRQWUDFWLRQ RI WKH VHPLJURXS LH f ([WHQVLRQ E\ FRQWLQXLW\ \LHOGV f IRU DOO >ZZL@ H = Â’ 7KHRUHP /HW WKH UDGLDO YHFWRU ILHOG DVVXPSWLRQ f RQ 4 KROG 7KHQ IRU DQ\ >ZRZM H = ZH KDYH WKDW (Wf ZW Z Z2 :I :R Z2 ] } DV L RR f 3URRI RI 7KHRUHP 7KH DERYH UHVXOW IROORZV E\ DSSHDOLQJ WR WKH 1DJ\)RLDV)RJHO GHFRPSRVLWLRQ WKHRU\ >/@ 6LQFH HAf LV D &RFRQWUDFWLRQ VHPLJURXS E\ 7KHRUHP WKH +LOEHUW VSDFH = FDQ EH GHFRPSRVHG LQ D XQLTXH ZD\ LQWR WKH RUWKRJRQDO VXP = =FQXp=X f ZKHUH ERWK =FQX DQG =X DUH UHGXFLQJ VXEVSDFHV IRU H DQG LWV DGMRLQW ,W LV DOVR WUXH WKDW Lf RQ =FQX HAf LV FRPSOHWHO\ QRQXQLWDU\ DQG ZHDNO\ VWDEOH LLf RQ =X H A LV D &XQLWDU\ JURXS ,Q RXU FDVH =X ^ ` WKH WULYLDO VXEVSDFH EHFDXVH RWKHUZLVH DQ DSSOLFDWLRQ RI 6WRQHfV WKHRUHP >3@ ZRXOG JXDUDQWHH DW OHDVW RQH HLJHQYDOXH RI $ RQ WKH LPDJLQDU\ D[LV EXW WKLV LV FOHDUO\ IDOVH GXH WR 7KHRUHP +HQFH = =FQX DQG WKHUHIRUH H LV ZHDNO\ VWDEOH RQ = +RZHYHU VLQFH $ KDV FRPSDFW UHVROYHQW LW IROORZV WKDW HAf LV VWDEOH LQ WKH VWURQJ WRSRORJ\ RI =>%@ 7KHUHIRUH HAf] DV W rr IRU DOO ] H = DQG VWURQJ VWDELOLW\ LV YHULILHG Â’
PAGE 28
&+$37(5 81,)250 67$%,/,=$7,21 3UHOLPLQDULHV &KDQJH RI 9DULDEOHV 5HFDOO RXU IHHGEDFN V\VWHP Zf $Z LQ 4 }f[IL Df Zf Z rf r LQ 4 Z? 6L f A>$$0Zf@ Â‘O 29 RQ ( }f [ U Ff GZ Q RQ ( Gf DQG WKH FRUUHVSRQGLQJ HQHUJ\ ( Wf GHILQHG E\ WKH VTXDUHG QRUP RI WKH VHPLJURXS emf emÂ‘f OOm$n_f_OO_OO_Af_OOL ,0n0,Q ,8fÂ§: __Â f :H ZDQW WR VKRZ WKDW XQGHU VXLWDEOH DVVXPSWLRQV RQ e WKH HQHUJ\ ( Wf GHFD\V XQLIRUPO\ H[SRQHQn WLDOO\ WR ]HUR 0RUH SUHFLVHO\ WKHUH H[LVWV FRQVWDQWV & VXFK WKDW IRU DQ\ > YYT:@ @ H = >,'$rf@n [ >c'$\rf< WKH FRUUHVSRQGLQJ VROXWLRQ RI f VDWLVILHV ( &HA(Lf IRU DOO W! f 7KH SURRI RI f ZLOO UHTXLUH D GLIIHUHQW DSSURDFK WKDQ ZDV XVHG LQ &KDSWHU WR DFKLHYH VWURQJ VWDn ELOLW\ 7KH SURRI ZLOO LQYROYH PXOWLSOLHUV EXW EHIRUH ZH FDQ SURFHHG D FKDQJH RI YDULDEOHV PXVW EH LQLn WLDWHG /HW SWf $rZWf IRU >Z: @ H = f 7KHQ VLQFH ZWf H >,'$rf@n LH $ WLZWf H /4f LW IROORZV WKDW $rSWf $Yr$aDZWf $arZWf J /ef 7KXV
PAGE 29
6R LQ SDUWLFXODU SLWf H ='$0f ^Hm}}_U I_UR` Df Ef f 1RZ UHFDOOLQJ f ZH KDYH XVLQJ f SW $PZf $nf>$Z $* ;*?$a[DZ@ $PZ$P**nO$aPZO 'LIIHUHQWLDWLQJ RQFH PRUH LQ WLPH DQG XVLQJ $aPZ $S DQG $aPZf $S ZH JHW SWW $S$ LQ* n$S DQG KHQFH Sf$S $ aP* L r$S 3f 3R $aQZX Sf SL $aPZ $aP*?*?$aP:? f S? Â‘[ LQ 4 LQ 4 RQ e RQ e f f f Df Ef Ff f Gf 8QLIRUP 6WDELOL]DWLRQ 2XU PDLQ UHVXOW UHIHUUHG WR LQ &KDSWHU DV 7KHRUHP LV DV IROORZV 7KHRUHP $VVXPH WKDW WKHUH H[LVWV D SRLQW [ H VR WKDW WKH UDGLDO YHFWRU ILHOG GHILQHG E\ K [f [[f VDWLVILHV L Wf f Y \ RQ7 f ZKHUH Y LV WKH XQLW RXWZDUG QRUPDO WKHQ WKHUH H[LVWV SRVLWLYH FRQVWDQWV & VXFK WKDW (f &HaA(f IRU DOO f 7KH SURRI RI 7KHRUHP ZLOO IROORZ GLUHFWO\ IURP WKH QH[W WKHRUHP E\ WDNLQJ OLP LQ f DQG LQYRNLQJ 3}R 'DWNRfV WKHRUHP >'O@
PAGE 30
7KHRUHP 8QGHU WKH VDPH DVVXPSWLRQ f DV LQ WKH DERYH WKHRUHP WKHUH H[LVWV D FRQVWDQW VXFK WKDW IRU DOO LQLWLDO GDWD >ZTZ @ H = WKHUHIRUH \LHOGLQJ ef rr f DQG DOO 3 LW IROORZV WKDW ?aHarW(WfGW 9A,, _R_ OO]A9A1AS =GW .( f f 3URRI RI 7KHRUHP $ PXOWLSOLHU DSSURDFKf )LUVW ZH WDNH LQLWLDO GDWD VPRRWK >ZT}@ @H e! $f VHH Ef ZKLFK WKHQ JXDUDQWHHV >ZL2Z;Lf@ H &>7@'$ff IRU DOO 7 rr DQG ILQG WKH GHVLUHG HVWLPDWH ZLWK LQGHSHQGHQW RI > :T:M @ DQG RI S 7KHQ H[WHQVLRQ E\ FRQWLQXLW\ \LHOGV f IRU DOO LQLWLDO GDWD >ZZc@ H = 1RZ ZH UHFDOO RXU VWDUWLQJ LGHQWLW\ ZKLFK ZH GHULYHG LQ WKH DSSHQGLFHV YLD WKH PXOWLSOLHUV Ha3fK Â‘ 9$Sf DQG HaA$SGL?K UHFDOO &ff HZ.t3sK 9$!fG, 9$!fK Â‘ ?G/ \H:AMA$SGO 93!L4 9$Sf SAHIn0 f 9$SfG AHnArSW$SG4 Â‘ 9$SfG4 I H _IA *L*O$$SGJ OLP H S U9$Sf !D T U!a/ OLP >HSL SW $S!Q W fÂ§! rr/ R Df 6S 9Sf ,Â6*L a 4]a 4]a 4ra a L Ef ZKHUH WKH } DUH WKH FRUUHVSRQGLQJ LQWHJUDOV RYHU 4 DQG WKH e L DUH WKH FRUUHVSRQGLQJ OLPLW WHUPV 7KH ILUVW WZR LQWHJUDOV DUH PRVW LPSRUWDQW DV ZH VKDOO VHH EHORZ 1H[W ZH SURYH /HPPD 7KHUH H[LVWV D FRQVWDQW & L VXFK WKDW ,, t3 OLS 6&L $\rS __A f
PAGE 31
3URRI )LUVW E\ WUDFH WKHRU\ >/0@ __$S__ &__$"__ +PDf & $S nf VLQFH + 4f LV D VWURQJHU WRSRORJ\ &__$S,,T&__ _YSf_ E\GHILQLWLRQ & $ LQS &&S $ ZS __A E\ QRUP HTXLYDODHQFH A & __ $ PS __ /2f ,, $9 _e &&S __ $rrS ??D & __ $! _e Â’ :H FRQWLQXH WKH SURRI RI 7KHRUHP ZKLFK ZLOO EH GLYLGH LQWR VHYHQ VWHSV 6WHS $EVRUSWLRQ RI WKH ERXQGDU\ WHUPV LQ f /HW 0KE PA[ K DQG FKRRVH eL H VXFK WKDW W[0KE Df Q&? HfÂ§fÂ§fÂ§ D Ef ZKHUH &[ DV LQ /HPPD DQG RT WR EH JLYHQ EHORZ %HIRUH SURFHHGLQJ QRWH WKDW WKH IROORZLQJ LQHTXDOLW\ ZLOO EH XVHG H[WHQVLYHO\ EHORZ )RU DQ\ H DÂ" WD AE rf 1H[W ZH RSHUDWH RQ WKH OHIW KDQG VLGH /+6f RI HTXDWLRQ f E\ XVLQJ rf ZLWK HW DQG H DV ZHOO DV WKH DVVXPSWLRQ RQ K f Y WR REWDLQ I 9$ I HIf_9$!fOL?UGO Â H:AA$SG= O Y H Â‘!e DY fÂ§ > HS_L0_ rG]=0KE? HaA 9$Sf G( I HaA ,9$SfOG/ @ G9 6fÂ§ HI f(f K HA ,9?Sf?GOr AÂ9rf ,, $S ??cGc eM = = = M f ZKHUH WKH (f WHUP IROORZV E\ XVLQJ f DQG ZH KDYH DOVR XVHG /HPPD IRU WKH $S WHUP 1RZ ZH FDQ GURS WKH PLGGOH WHUP EHFDXVH E\ Df DQG XVH WKH IDFW WKDW O_$!__A __ $Z __A eLf WRREWDLQ
PAGE 32
/+6f RI f .[(^M6fAAAHA(^WfÂ£ f ZKHUH .[ fÂ§ SfÂ§ (L H )LQDOO\ XWLOL]LQJ f LQ f DQG LVRODWLQJ WKH LPSRUWDQW WHUPV ZH REWDLQ ? HA:SWAG4 9$Sf?G4 .[(^f aA?aH:(WfGW 4? 4L /? O f 6WHS ,VRODWLQJ WKH HQHUJ\ LQWHJUDO HaA( WfGW )LUVW VLQFH S H '$0f WKHUH H[LVWV D FRQVWDQW &SL VXFK WKDW ,, 0 ??O!&S[ __$!__r>/7@ 6LPLODUO\ S H '^$Zf LPSOLHV WKH H[LVWHQFH RI &S VXFK WKDW __ 9$Sf_ __Â &S __ __Â>/7@ f f 1H[W XVLQJ f DQG f ZH KDYH ,, 0 _e!&S ??$ZS __r &S[ __ $aZZ _e &S5H $nYrZ $A*A$ AZ !D &S[ ??$A*[*?$LDZ?? Q 1RZ WR ERXQG EHORZ ZH XVH ODE ]D E RQ WKH PLGGOH WHUP DQG WDNLQJ H fÂ§ ZH REWDLQ ,, 0 _Â!A_8Z__Â&S __ $a:*[*?$PZ __ $OVR LW LV LPPHGLDWH WKDW ,, 9$Sf_ ,,T &S __ $OrS __A &S __ $0Z __ Q 3XWWLQJ WKH SLHFHV WRJHWKHU LQ D /HPPD /HPPD f f /HW D PLQ ^ &S ` DQG &S[ __ $ P*[ __ /UQ! 7KHQ HaA 9S G4 HaAf 9$Sf?G4 D[MAHA(WfGW .(f f f f
PAGE 33
3URRI I Ha:? 9$Sf?G4 ? HA:SAG4 M Q Q E\ f f &S?aH: __ $AZ _e GU A9nn ,, $ nZ _?DGW &S[?aH: __ $A**?$AZW ? eG! Am9A __ $afZ _e __ $ Z_e@ GW&S __ $* __ /Uf ,, *M$AZ __?GW E\ f RTM9A(WfGW &S[ O8A* __ OW Q!(f Â’ &ROOHFWLQJ RXU UHVXOWV VR IDU ZH KDYH YLD Ef f f WKDW DA H Af(WfGL .(f /O / 4L 4 &" ZKHUH DQG .c N Q&? D DL eaa f Df Ef 6WHS +DQGOLQJ WKH OLPLW WHUPV DULVLQJ IURP LQWHJUDWLRQ E\ SDUWV LQ WLPH 5HPDUN 7KH IROORZLQJ ZLOO EH XVHG LQ HYDOXDWLQJ WKH OLPLW WHUPV LQ f 6LQFH ZH DUH WDNLQJ LQLWLDO FRQGLn WLRQV > ZZM @ J '$f ZH KDYH E\ WKH UHJXODULW\ WKHRU\ >/7@ >Zf Zf@ H *>f@ '$ff LH ZWf H &>U@ +nP DQG Zf &>7@ >'$f@ ,ZL $ A p Z ? $f $f :T : :T ,, $H$n_f,, ] OOm$n$_:__ ,, ] V ,, $_Z_ +] f ZKHUH ZH KDYH XVHG J S$f TXRWHG LQ 7KHRUHP +HQFH ZH KDYH WKDW ,Z_ ,, Z __ :nWLf __ $ PZ __Q ,, $MZfL M ,, ] IRU DOO f
PAGE 34
1RZ OHWWLQJ 0N K ZH KDYH _UOLP 3W7f K f 9$S7ff !D P Har7> __ S7f _Â$Âf __ $\rS7f __ E\ f f OLP H A7 7 fÂ§!m} __ $n __ /Qf __ $aYrZ7f __ D __ $ A*;*?$A __ 8Qf __ $0Z7f __Q 0N(f E\ f f U}HBA7 *(f ,, $aP*?*?$nr ,, /Qf O_$_Zr_ +]` r f $ VLPLODU FRPSXWDWLRQ VKRZV WKDW MAOLP H 3U S7f $S7f !4 M f +HQFH ZH KDYH WKDW / S? K f 9$Sf !Q \ S? $S !Q $nZ $X**M$BLD:L K f 9$Sf !Q \ $Z $n**$Z $S !Q f 6WHS +DQGOLQJ WKH WHUPV SUHPXOWLSOLHG E\ S LH WKH WHUPV 4 ? DQG 4 )LUVW 0: H :ScK Â‘ 9$SfG4 639r __S _e 3$fHA __ _9$Sf_ _?DGW E\ f 9??$n}??XDfcaH9??$nm:??4GW 3 __ $ P*@ __ /URfAf3Â ,, *W$n9 _e GW S$&S9S __ _eGL S __ $f0 __ /27:r&}f,?$ P*[ __ /URf (f ZKHUH ZH KDYH XVHG WKH IDFWV WKDW __ $fZ __I e Wf DQG VLPLODUO\
PAGE 35
$ Z _Q ( Wf DV ZHOO DV ff V> ,, Ân ,, /} 0}&3L ,, $aP* ,, /U!Qf @ (f r(f ZKHUH ZH DOVR KDYH XVHG (Wf( f IRU DOO U FRQWUDFWLRQ RI WKH VHPLJURXSf DQG 3 $ VLPLODU FRPSXWDWLRQ VKRZV WKDW WKHUH H[LVWV D FRQVWDQW .V LQGHSHQGHQW RIIf f VXFK WKDW 4 .(f 7KXV ZH DUULYH DW f f f /HPPD /HW .L . f 7KHQ D?rrH:(WfGW.(f 4 4 $PZ $aP*[*?$ [DZA K f 9$Sf !Q R M $PZ $nP*[*?$PZ[ $S!Q f 3URRI )ROORZV LPPHGLDWHO\ E\ XVLQJ f f f LQ f Â’ 1RZ KDYLQJ GLVSHUVHG ZLWK WKH ORZHU RUGHU WHUPV RXU WDVN LV WR DEVRUE WKH PRUH GLIILFXOW LQWHULRU LQWHJUDOV 6WHS 7KH WHUP c :H DSSO\ LQWHJUDWLRQ E\ SDUWV LQ WLPH ZLWK GY $aP*?*?$nPZf X HaAfK Â‘ 9$Sf WR REWDLQ > $aP*? *?$aPZWWK Â‘ 9$SfG4 OLP ÂH_ $aP*?*?$aPZ K Â‘ 9$Sf !Q 4 W fÂ§!m!/ R SAUL$**$Z] f 9$SfG4 AHA$A*A$nAZK Â‘ 9$SfG4 f
PAGE 36
3DUW $ 6LQFH __ $aP*L*?$APZ7f __A __ $A*;*?$ A __/Âf __ __ $ A7f _e $ ,Z_ E\f ,8*M*0,,A __$___] DQ DUJXPHQW H[DFWO\ OLNH WKH RQH XVHG WR REWDLQ f \LHOGV ?LPHaA7 $P*[*?$ PZWf K f 9$S7ff !D f 7KXV ZH RQO\ KDYH D FRQWULEXWLRQ IURP W LH 3DUW % SRrrH3n $ P*L*c$ZW K Â‘ 9$Sf !DGW SM 9A __ $nr*[*?$ PZ _ÂGL S$Â9r __ 9$Sf __GW __ $*L ??/U LOf0K&Sf(f f ZKHUH ZH KDYH XVHG f f DQG f 3DUW& 7KH IROORZLQJ /HPPD ZLOO HQDEOH XV WR KDQGOH WKH UHPDLQLQJ WHUP DQG KHQFH FRPSOHWH 6WHS /HPPD )RU K f >&4f@E ZH KDYH WKDW WKHUH H[LVWV D FRQVWDQW &K VXFK WKDW ??$aPK f 9$Sff __ &K __ $ 9 __ Df 5HPDUN 6LQFH c'$nDf +T4f VHH >/7 $SSHQGL[ &@f VR WKDW >,'$Pf< a LU4f ZLWK HTXLYDOHQW QRUPVf ZH KDYH __ $aP^K f 9$Sff __ Q __ K f 9$Sf __ >'$,f@n ,8 n 9$SWf __ +ADf Ef &ODLP /HW ] H 74f K H &4f 7KHQ K] J +a4f 3URRI RI FODLP 6HH DOVR >/0 9RO 7KHRUHP S@f %\ DVVXPSWLRQ ] J !4 ]JGeOrr ?J +T4f DQG ZH ZDQW WR VKRZ WKDW
PAGE 37
K] J! ?4]KJGQ rr JH +LOf 7KXV LW VXIILFHV WR VKRZ KJ H OO? ILf LI J H Â4f &OHDUO\ KJ H +4f 6R ZH PXVW VKRZ KJ B ÂKJf U Y U EXWb_ LO J_ VLQFHJ H +T4f Â‘U rU fU $OVR A? 9bf f 9 ^ KJ? f f f : f >9LY f f f 9r@ ` ^ ,.6 KJ[@9L >K;WJ KJ;L@Y f f f >KAJ KJA@Yf ` VLQFH J J i4f LPSOLHV J? J[c J] ? f t Â’ fU nU nU Â‘U 3URRI RI /HPPD 8VLQJ WKH DERYH FODLP LQ Ef ZH KDYH ,, $aPK f 9W\ff __ Q __ K f 9$Sff __ f4f fÂ§ $f __ :r4f A&0K __ __ +?Df E\ >/0 9RO ,@ &0K&S[ __ 9S_ __4 E\ 3RLQFDUH ,QHTXDOLW\ &K $ PS __ D YLD QRUP HTXLYDOHQFH Â’ &RQWLQXLQJ 3DUW & FKRRVH H VXFK WKDW &$H D 1RZ ZH FRPSXWH M H Af$ P*[*?$ PZK Â‘ 9$SfG4 *[*?$aP:O $ P^K Â‘ 9$Sff !R_FLU E\ Df f
PAGE 38
E\ frf 6 ,, & __ /U Rf(f 9}n ,, 9f*_J &HfH}n _8Z __ G r OO/UDfY,0 f*n XUp P ;WRIU9(P $7ef &rH9SefG f $JDLQ RUJDQL]LQJ RXU UHVXOWV ZH KDYH /HPPD /HW. . .IL f DQG D D &rH f WKHQ D I 9Ser re f J $Z K f 9$!f !Q R $PZ $P*[*?$PZ[ $S !D f 3URRI 7KH SURRI IROORZV GLUHFWO\ E\ XWLOL]LQJ f f DQG f LQ f DQG WDNLQJ DGYDQWDJH RI WKH FDQFHOODWLRQ RI WHUPV $PZ $aP**?$ aPZ[ K f9$Sf!Q $P*[*?$aPZ[ K f 9$Sf !Q $fZ K f 9$SRf !Q Â’ %HIRUH EHJLQQLQJ 6WHS WKH IROORZLQJ /HPPD ZLOO EH QHHGHG /HPPD 7KH /QRUP RI $S LV ERXQGHG E\ WKH LQLWLDO HQHUJ\ (f LH ,, OLR ,, fR ,Â‹ __frr!OOÂ6 ,8f+/Tf ,_R,,T ,,f,,/4f ,8f},_ ,_Df__/Tf(V ,8nf ,_/4fHf f
PAGE 39
3URRI ,, $PS __ $S S !Q $$Sf S !D E\ $2f $S $S !D cUASGUcU$SAG9 E\ DEf $S $S !D ,, $S __ Â’ 6WHS 7KH WHUP 4 $V LQ 6WHS ZH EHJLQ ZLWK LQWHJUDWLRQ E\ SDUWV LQ WLPH ZLWK GY $aP*[*?$aPZf X HaAW$S WR REWDLQ m7 HA$P*[*?$LDZWW$SG4 A 8P >HBSL $A*A$AZ $S !4 r 4 U}rr/ R IF T If f 1RZ ZH EHJLQ D WKUHH SDUW SURFHVV DV LQ VWHS 3DUW $ 8VLQJ f DQG f LW IROORZV GLUHFWO\ WKDW e OLP 7SU $P*[*?$ PZWf $S7f !D e& OLP HA f Wr! W 6R DJDLQ ZH KDYH RQO\ D FRQWULEXWLRQ IURP U LH rr $P*L*r$r1:L $S !4 U A $A*M*W$AZ $Sf !Q f R fÂ§ OLP U 3DUW % QSM9A $A*M*M$} $S !QGI QS __ $aP*? __/U Qf 9 __ *O$aPZW __r QS4 HfSL __ $S __rGW E\ f f f Q __ $aP*@ ,, OW Lf a m, A ,, /ef (f :( f 3DUW & )RU & DV JLYHQ EHORZ FKRRVH H! VXFK WKDW HQ& __$0 __ /Qf D _IM HB3n $P**?$aPZW $S !QG_ f f n4 HB3Â_ *L*r$fnKfnf $$3f!m_G e9r __ **0fZ _Âm OO$ f9f _Âm f
PAGE 40
([DPLQH WKH LQWHJUDO RQ WKH ULJKW LQ f 8VLQJ WKH IDFWV WKDW ,'$ Pf ccO 4f >/7 $SSHQGL[ &@ Df DQG $ >^^LOf & __ e >/0 9RO @ Ef :H KDYH __$ $S__Â,, ,, >'WOf@n ,, $3} ,, >J4f@n __$IW,,ÂAA__$__A&__$__D/Zf ??$nrSW??D f VR FRQWLQXLQJ f __ *[ ,, /7IOferf rr r rr /Lf H A + $r13f +DGW A + *! +/UQferf (A ,, $ ??OH9(0 ]Q&??$nr*[ +/U Qf(f $7fef HQ& __ $n __ /4fcfHA(WfGW f /HPPD /HW . .W .Q f DQG D D HQ& __ $ __ f 7KHQ D ?rrHA(^WfGW .Q(f $PZ K f 9$Sf !Q R fÂ§ $ Z $SR f 3URRI 7KH SURRI IROORZV DJDLQ GLUHFWO\ XVLQJ f f f DQG DJDLQ XWLOL]LQJ WKH FDQFHOODWLRQ RI WHUPV $ PZ $ P*[*?$ PZ[ $S !DaA $ **Â$nKn $S !Q \ $aPZr A $3R!Q Â’ 6WHS 7KH LQLWLDO FRQGLWLRQV )LUVW $fZ K Â‘ 9$Sf !T O ,, $fZ __ 0N&S __ $: __ ,, $n __/Qf 0N&SA(f .@(f f
PAGE 41
6HFRQG O\ $ $SD !D M> __ $ Z __A __ $S __AM I> ,, $nf /LLf( f +$f0 +8+ m ,00 +/Qfef ALef ZKHUH ZH KDYH DSSOLHG /HPPD IRU WKH $S WHUP )LQDOO\ XVLQJ f f LQ f ZH KDYH A.O(f R ZKHUH ? .? . A L 'LYLGLQJ f E\ RF DQG OHWWLQJ fÂ§fÂ§ 7KHRUHP LV SURYHG Â’ &Â 1RZ WKH SURRI RI WKH PDLQ UHVXOW 7KHRUHP f ZLOO IROORZ HDVLO\ IURP WKH EHORZ TXRWHG UHVXOW f f f 7KHRUHP 'DWNRf ,% SDJH 6XSSRVH 7Wf LV D VHPLJURXS RQ D %DQDFK VSDFH ; ,I IRU DOO [ H ; f __ ;rAa WKHQ WKH VHPLJURXS LV XQLIRUPO\ H[SRQHQWLDOO\f VWDEOH LH WKHUH H[LVWV FRQVWDQWV & VXFK WKDW ,, A ,, /&;f IRU DOO L! Â’ 3URRI RI 7KHRUHP :H KDYH VHHQ WKDW > HnA(AGW .(f ZKHUH $ LV D FRQVWDQW LQGHSHQGHQW RI 3 H @ DQG RI R > YYR: @ H '$f 7KXV WDNLQJ WKH OLP ZH JHW 3fÂ§! ?rr(WfGW.( f f R %XW (Wf LV VLPSO\ WKH VTXDUHG QRUP RI WKH VHPLJURXS DSSOLHG WR DQ\ LQLWLDO FRQGLWLRQV > Z!YL @ H M'$f DQG KHQFH E\ FRQWLQXLW\ DOO > @ H = LH $ , VXEVWLWXWLQJ (Wf __ __ LQWR f _+!L = ZHJHW R ??Hrf?Z@??]GW.(pf 22
PAGE 42
7KXV E\ 'DWNRfV 7KHRUHP WKHUH H[LVWV & c @ VXFK WKDW ,8$n ,, /=f =&An DQGKHQFH ,, ,, O=f a &nH A WKHUHIRUH E\ WDNLQJ & &M DQG [ ZH KDYH r ,8Dn4 ,8$nOO/=fOO__OOOVFHef ZKLFK LV H[DFWO\ RXU XQLIRUP VWDELOL]DWLRQ UHVXOW Â’
PAGE 43
$33(1',; $ %$6,& ,'(17,7,(6 /HW J EH VFDODU IXQFWLRQV DQG K [f H &4f YHFWRU ILHOG 7KH IROORZLQJ LGHQWLWLHV ZLOO EH QHHGHG *UHHQfV 6HFRQG 7KHRUHP MA$JGe M $AD4 tJGU,IGU UGY }UY $2f *UHHQfV )LUVW 7KHRUHP I $IJG4 I UAJG7 I UGY T 9I9JG4 $Of 'LYHUJHQFH 7KHRUHP AGLY$GLO fYGU $f GLYLf K f 9GLYL $f IK f 9 r f 9f $f $ f 9GIO GLY$ GLYILfGL ILYLU AIGLYKG4 $f 9 f 9$ Â‘ 9f 9 f 9 \ $ 9O 9Of $f GK? $L G[c ZKHUH +[f GKQ $E G[L Dr% 9 f 9GLY$f 9GLYLf f 9 9O GLYIL $f IK Â‘ 9GLO UA$ f 9G7 _U YOIL f YGU 9 f 9G 9O GLY:L $f
PAGE 44
3URRI RI $f 0XOWLSO\ $E\ K Â‘ 9 LQWHJUDWH DQG VXFFHVVLYHO\ DSSO\ $Of $f DQG $f MA$IK Â‘ 6IG4 f 6IG7 9 9L f 9fGLL f YGU 49 Â‘ 9Ge _TK f 9O 9O fG4 eW f 9GU n9 Â‘ 9IG4 \U 9O L f YGU \ 9O GLYLGe $QRWKHU XVHIXO LGHQWLW\ DULVHV E\ DSSO\LQJ $Of $f WR JHW M A$IIGL?KG AAGLYLGU 9 Â‘ 9GLYcfGIO AGLYLGU A9GLY$f f 9G4 9O GLYIFGL $f
PAGE 45
$33(1',; % 72 +$1'/( ',))(5(1&( 2) (1(5*< 7(50 :H PXOWLSO\ ERWK VLGHV RI RXU HTXDWLRQ Sf $S $ P*[*?$S E\ H A$SGLYK ZLWK 3 DQG LQWHJUDWH ERWK VLGHV RYHU 4 MA4 TSf WHUP )LUVW LQWHJUDWH E\ SDUWV LQ WLPH DQG WKHQ DSSO\ $f WR JHW I SfHaA$SGLYKGWG4 OLP>HBA S $SGLYL !Q U!a/ R 3HBILL $SSGLYKG4 M AHA $SSGL?KG4 OLP >HBS S $SGLYL !QO SI HaAf$SSGL?KG4 U!a/ R 4 MAH 3WASGcYKGn/ MAH AS9GLYLf f 9SG MAH A 9S GLY]G n4 %Of $G WHUP )LUVW DSSO\ $Of DQG WKHQ $f WR JHW M H AW$$Sf$SG?YKG4 AH $SGL[KG/ MAH A9$Sf f 9$SGLYKfG I HaA $SGLYKGO I HA$S9IGLYLf f 9$SfG4 HA_ 9$Sf?GL?KG4 %f GY T T 1RZ LVRODWLQJ GLIIHUHQFH RI HQHUJ\ WHUP DQG PXOWLSO\LQJ E\ a ZH JHW I H S 9S 9$Sf GL?KG4 fÂ§OLP ÂH Af S $SGLYL !Q MT / R SMAHA$SSALYLG4 \MAHSLASGLYLG/ \MAHa_L3LAGnYAf f 9SG4 \MAHA $SGLYLM( \_AHfAL$S9GLYLf f 9SG4 AAHnA $aP*[*?$S$SG??KG4 %f
PAGE 46
$33(1',; & 72 2%7$,1 *(1(5$/ ,'(17,7< :H PXOWLSO\ RXU HTXDWLRQ Sf $S $ P*[*?$S E\ H f 9$Sf ZLWK 3 DQG LQWHJUDWH ERWK VLGHV RYHU 4 DV LQ $SSHQGL[ % Sf WHUP )LUVW LQWHJUDWH E\ SDUWV LQ WLPH WKHQ VXFFHVVLYHO\ DSSO\ $f DQG $f WR JHW I I SfH SnL Â‘ 9$SfGWG4 OLP H SL S K9$Sf !4 TUA/ R 3 I H:SK f 9$SfG4 ?H:3OK Â‘ 9$SfG4 4 4 OLP>USL S K:$Sf !T_ S_ HaASK Â‘ 9$SfG4 Ura/ R 4 HaAf$SSK f YGe HaA $SWG??SKfG4 A 4 OLP 7fÂ§!rr/ >FS 3W L9$Sf!QOU S> H93OK9$SfG4? HA$SSKYGO L R MR \ I#H SK f 9SG4 MQH Af$SSGL?KG4 4 &Lf 1H[W ZH ZLOO DSSO\ $f WR M H A$SK f 9SG4 DQG $f WR MAH Af$SSÂ£L?KG4 VLPXOWDQHRXVO\ WR REWDLQ 8 SH OLP >HSL S K9$Sf !4 7 SI HaPSK Â‘ 9$SfG4 I HA$SSWK f YGO Ura/ R 4 P G3W * A K Â‘ 9SGO S 9S K Â‘ ?G= m A*9S f 9SG4 S 9S GL?KG4 GS RDL?QDÂM L H SASGLY]G6H SLS9GLYLf f 9SG4 H A 9S GLYLG &f
PAGE 47
$S WHUP $SSO\LQJ $f ZH JHW MAHaA $$SfK Â‘ 9$SfG4 n 9$SfG= HBIn9$3fn f Â4H Af+ 9$Sfn 9$SfG& LJ HB3Â 9$Sf GLY: &f 1RZ ZH SXW DOO ERXQGDU\ WHUPV RQ WKH OHIW VLGH DOO LQWHULRU WHUPV RQ WKH ULJKW VLGH DQG VLPSOLI\ WR REWDLQ I HA$SSW f YGe H f 9SG=H SL ,9SO]YG( f ; &U9 m e HaAASGLYKG= L f 9$SfG/ AHBS 9$"f L f YG= OLP >HBA S U9$Sf !QO I HBSLSW f 9$SfG4 7r}O R H 9$Sf f 9$SfG4 I#H pO+9S f 9SG4 m SLS9GL:Lf f 9SG 3Â * ,9SO ,9$SfO GLYKG4 a IAHaA$ P*L*?$SK Â‘ 9$SfG4 %HIRUH SURFHHGLQJ ZH PDNH VRPH VLPSOLILFDWLRQV ZKLFK DULVH GXH WR WKH ERXQGDU\ FRQGLWLRQV &f 6LQFH S LW LV LPPHGLDWH WKDW ,\ G9 O\ GS S?a DY R $OVR 9SAUDQG 9SAU LPSO\ 9S , RQ 7 R9 DQG 9S , RQ 7 GY $QG E\ XWLOL]LQJ K rf [[f WKH UDGLDO YHFWRU ILHOG ZH KDYH WKDW rf LGHQWLW\ PDWUL[ GLY$ Q GLP e 9GLY$f &f &f &f
PAGE 48
1RZ LQVHUWLQJ %f IRU WKH GLIIHUHQFH RI HQHUJ\ WHUP LQ &f DQG XWLOL]LQJ WKH VLPSOLILFDWLRQV &f &f DQG &f ZH ILQDOO\ DUULYH DW WKH GHVLUHG LGHQWLW\ Hrf 9$f n EH OLP>HSL S K9$Sf !QO 7 A ?LP?HaA S $S !Ura/ R W[, R SH:SK Â‘ 9$SfG4 QAHAS$SG4 H: 93O ?G4> Ha: 9$"f G4 4 4 > HA$aP**?$3OK Â‘ 9$SfG4 eI HaA$aP**?$SW$SG4 &f 4 4
PAGE 49
5()(5(1&(6 >%@ $9 %DODNULVKQDQ $SSOLHG )XQWLRQDO $QDO\VLV QG HGLWLRQ 6SULQJHU9HUODJ 1HZ %7O@ %DUWRORPHR DQG 5 7ULJJLDQL 8QLIRUP VWDELOL]DWLRQ RI WKH (XOHU%HPRXOOL (TXDWLRQ :LWK 'LULFKOHW DQG 1HXPDQQ %RXQGDU\ )HHGEDFN 5HSRUW 'HSW RI $SSOLHG 0DWKHPDWLFV 8QLYHUn VLW\ RI 9LUJLQLD >&O@ &KHQ (QHUJ\ 'HFD\ (VWLPDWHV DQG ([DFW &RQWUROODELOLW\ RI WKH :DYH (TXDWLRQ LQ D %RXQGHG 'RPDLQ RXUQDO GH 0DWKHPDWLTXHV 3XUV HW $SSOLTXHHV f f >&@ &KHQ $ 1RWH RQ WKH %RXQGDU\ 6WDELOL]DWLRQ RI WKH :DYH (TXDWLRQ 6,$0 &RQWURO f >'O@ 'DWNR ([WHQGLQJ D 7KHRUHP RI /LDSXQRY WR +LOEHUW 6SDFHV 0DWKHP $QDO DQG $SSOLF f >'/7@ 'D3UDWR /DVLHFND DQG 5 7ULJJLDQL $ 'LUHFW 6WXG\ RI WKH 5LFFDWL (TXDWLRQ $ULVLQJ LQ +\SHUEROLF %RXQGDU\ &RQWURO 3UREOHPV 'LIIHUHQWLDO (TXDWLRQV f >)/7@ ) )ODQGROL /DVLHFND DQG 5 7ULJJLDQL $OJHEUDLF 5LFFDWL (TXDWLRQV :LWK 1RQVPRRWKLQJ 2EVHUYDWLRQ $ULVLQJ LQ +\SHUEROLF DQG (XOHU%HPRXOOL (TXDWLRQV $QQDOL GL 0DWHPÂ£WLFD 3XUD H $SSOLFDWD WR DSSHDU >+O@ / ) +R 2EVFUYDELOLWHn )URQWLHUH GH /fHTXDWLRQ GHV 2QGHV &5$6 f >/@ 1 /HYDQ 7KH 6WDELOL]DWLRQ 3UREOHP $ +LOEHUW 6SDFH 2SHUDWRU 'HFRPSRVLWLRQ $SSURDFK ,((( 7UDQV &LUFXLWV DQG 6\VWHPV $6 f >/O@ /DJQHVH D SDSHU SUHVHQWHG DW WKH ,QWHUQDWLRQDO :RUNVKRS KHOG LQ 9RUDX $XVWULD XO\ >/@ /DJQHVH %RXQGDU\ 6WDELOL]DWLRQ RI 7KLQ (ODVWLF 3ODWHV WR DSSHDU >/@ /DJQHVH 8QLIRUP %RXQGDU\ 6WDELOL]DWLRQ RI +RPRJHQHRXV ,VRWURSLF 3ODWHV LQ /HFWXUH 1RWHV LQ &RQWURO 6FLHQFH f 6SULQJHU9HUODJ 1HZ
PAGE 50
>/@ >fÂ§@ >/@ >/@ >/@ >/0@ >//O@ >//7O@ >/7O@ >/7@ >/7@ >/7@ >/7@ >/7@ >/7@ /DJQHVF 'HFD\ RI 6ROXWLRQV RI :DYH (TXDWLRQV LQ D %RXQGHG 5HJLRQ ZLWK %RXQGDU\ 'LVVLn SDWLRQ 'LIIHUHQWLDO (TXDWLRQV f /DJQHVH $ 1RWH RQ %RXQGDU\ 6WDELOL]DWLRQ RI :DYH (TXDWLRQV 6,$0 &RQWURO WR DSSHDU / /LRQV ([DFW &RQWUROODELOLW\ 6WDELOL]DWLRQ DQG 3HUWXUEDWLRQV 6,$0 5HYLHZ 0DUFK / /LRQV $ 5HVXOWDW GH 5HJXODQWH SDSHU GHGLFDWHG WR 6 0L]RKDWDf &XUUHQW 7RSLFV RQ 3DUn WLDO 'LIIHUHQWLDO (TXDWLRQV .LQLNXQL\D &RPSDQ\ 7RN\R / /LRQV &RQWUROH GHV 6\VWHPHV 'LVWULEXHV 6LQJXOLFUV *DQWKLHU 9LOODUV 3DULV / /LRQV DQG ( 0DJHQHV 1RQKRPRJHQHRXV %RXQGDU\ 9DOXH 3UREOHPV DQG $SSOLFDWLRQV 9ROV ,, 6SULQJHU9HUODJ %FUOLQ+HLGFOEHUJ 1HZ
PAGE 51
>/7@ /DVLHFND DQG 5 7ULJJLDQL $ &RVLQH 2SHUDWRU $SSURDFK WR 0RGHOLQJ /7 /Uff ERXQGDU\ ,QSXW +\EHUEROLF (TXDWLRQV $SSOLHG 0DWKHP DQG 2SWLPL] f >/7@ /DVLHFND DQG 5 7ULJJLDQL 5HJXODULW\ RI K\SHUEROLF HTXDWLRQV XQGHU /7 /7ff 'LULFKHOW %RXQGDU\ WHUPV $SSOLHG 0DWKHP DQG 2SWLPL] f >/7@ /DVLHFND DQG 5 7ULJJLDQL $ /LIWLQJ 7KHRUHP IRU WKH 7LPH 5HJXODULW\ RI 6ROXWLRQV WR $EVWUDFW (TXDWLRQV ZLWK 8QERXQGHG 2SHUDWRUV DQG $SOOLFDWLRQV 7KURXJK +\SHUEROLF (TXDn WLRQV 3URFHHGLQJV $PHULFDQ 0DWKHPDWLFDO 6RFLHW\ WR DSSHDU >/7OO@ /DVLHFND DQG 5 7ULJJLDQL 5LFFDWL (TXDWLRQV IRU +\SHUEROLF 3DUWLDO 'LIIHUHQWLDO (TXDWLRQV ZLWK/U /Uff'LULFKOHW %RXQGDU\ 7HUPV 6,$0 &RQWURO DQG 2SWLPL] f f >/7@ /DVLHFND DQG 5 7ULJJLDQL ([DFW &RQWUROODELOW\ IRU WKH :DYH (TXDWLRQ ZLWK 1HXPDQQ %RXQn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f >/7@ /DVLHFND DQG 5 7ULJJLDQL 'LULFKOHW %RXQGDU\ 6WDELOL]DWLRQ RI WKH :DYH (TXDWLRQ YLD %RXQn GDU\ 2EVHUYDWLRQ 0DWKHP $QDO DQG $SSOLF f >3@ $ 3D]\ 6HPLJURXSV RI /LQHDU 2SHUDWRUV DQG $SSOLFDWLRQV WR 3DUWLDO 'LIIHUHQWLDO (TXDWLRQV 6SULQJHU9HUODJ 1HZ 5O@ '/ 5XVVHOO ([DFW %RXQGDU\ 9DOXH &RQWUROODELOLW\ 7KHRUHPV IRU :DYH DQG +HDW 3URFHVVHV LQ 6WDU &RPSOHPHQWHG 5HJLRQV LQ 'LIIHUHQWLDO *DPHV LQ &RQWURO 7KHRU\ 'HNNHU 1HZ 5@ '/ 5XVVHOO $ 8QLILHG %RXQGDU\ &RQWUROODELOLW\ 7KHRU\ IRU +\SHUEROLF 3DUWLDO 'LIIHUHQWLDO (TXDWLRQV 6WXG $SSO 0DWK f >5@ '/ 5XVVHOO 0DWKHPDWLFDO 0RGHOV IRU WKH (ODVWLF %HDP DQG 7KHLU &RQWUROWKHRUHWLF ,PSOLFDn WLRQV LQ 6HPLJURXSV 7KHRU\ DQG $SSOLFDWLRQV 3LWPDQ 5HVHDUFK 1RWHV LQ 0DWKHPDWLFV >6O@ : 6\PHV $ 7UDFH 7KHRUHP IRU 6ROXWLRQV RI :DYH (TXDWLRQV 0DWKHPDWLFDO 0HWKRGV LQ WKH $SSOLHG 6FLHQFHV f
PAGE 52
>7O@ 5 7ULJJLDQL :DYH (TXDWLRQ RQ D %RXQGHG 'RPDLQ ZLWK %RXQGDU\ 'LVVLSDWLRQ DQ 2SHUDWRU $SSURDFK 0DWKHP $QDO DQG $SSOLF WR DSSHDU >7@ 5 7ULJJLDQL ([DFW &RQWUROODELOLW\ RI :DYH DQG (XOHU%HPRXOOL (TXDWLRQV LQ WKH 3UHVHQFH RI 'DPSLQJ 3URFHHGLQJV RI ,QWHUQDWLRQDO &RQIHUHQFH RQ 'LIIHUHQWLDO (TXDWLRQV KHOG LQ &ROXPEXV 2KLR 0DUFK >7@ 5 7ULJJLDQL ([DFW &RQWUROODELOLW\ RQ /ef [ B Âf IRU WKH :DYH (TXDWLRQ ZLWK 'LULFKOHW &RQWURO $FWLQJ RQ D 3RUWLRQ RI WKH %RXQGDU\ DQG 5HODWHG 3UREOHPV $SSOLHG 0DWK DQG 2SW f >7@ 5 7ULJJLDQL $ &RVLQH 2SHUDWRU $SSURDFK WR 0RGHOLQJ /7 /'fERXQGDU\ ,QSXW 3UREn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f
PAGE 53
%,2*5$3+,&$/ 6.(7&+ HUU\ %DUWRORPHR ZDV ERP XQH LQ %URRNO\Q 1HZ
PAGE 54
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` RI LQ WKH &ROOHJH RI DQG WR WKH *UDGXDWH 6FKRRO DQG ZDV DFFHSWHG DV SDUWLDO IXOILOOPHQW RI WKH UHTXLUHPHQWV IUR WKH GHJUHH RI 'RFWRU RI 3KLORVRSK\ 'HFHPEHU 'HDQ &ROOHJH RI 0DWKHPDWLFV 'HDQ *UDGXDWH 6FKRRO
PAGE 55
81,9(56,7< 2) )/25,'$
UNIFORM STABILIZATION OF THE EULERBERNOULLI EQUATION
WITH ACTIVE DIRICHLET AND NONACTIVE NEUMANN
BOUNDARY FEEDBACK CONTROLS
By
JERRY BARTOLOMEO
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1988
U 0E F LIBRARIES
ACKNOWLEDGEMENTS
I express my sincere appreciation to my advisor Dr. Roberto Triggiani for his leadership and support
over the past four years and for directing this research project. I would also like to thank the other
members of my committee, especially Dr. Irena Lasiecka who also played a major role in my education.
Also, I would like to thank my parents, James and Elisa Bartolomeo, and my wife Michelle, for their conÂ¬
tinued encouragement. Finally, special thanks are due to John Holland who did an excellent job in preparÂ¬
ing this document.
 u 
TABLE OF CONTENTS
page
ACKNOWLEDGEMENTS ii
ABSTRACT v
CHAPTERS
1 INTRODUCTION, PRELIMINARIES, STATEMENT OF MAIN RESULTS 1
1.1 Introduction And Literature 1
1.2 Formulation of the Uniform Stabilization Problem and Main Statements 3
Choice of Operators F i and F2 5
Theorem 1.1 7
Theorem 1.2 8
Theorem 1.3 8
2 WELLPOSEDNESS AND STRONG STABILIZATION 10
2.1 Preliminaries, Choice of Stabilizing Feedback 10
2.2 WellPosedness and Semigroup Generation 14
Lemma 2.5 14
Theorem 2.6 15
Proof of Theorem 2.6 16
Lemma 2.7 18
Proof of Lemma 2.7 19
Corollary 2.8 20
Theorem 2.10 21
Proof of Theorem 2.10 21
3 UNIFORM STABILIZATION 22
3.1 Preliminaries, Change of Variables 22
3.2 Uniform Stabilization 23
Theorem 3.1 23
Theorem 3.2 24
Proof of Theorem 3.2 (A multiplier approach) 24
Lemma 3.3 24
Lemma 3.4 26
Lemma 3.7 29
Lemma 3.8 30
Proof of Lemma 3.8 31
 iii 
Lemma 3.9 32
Lemma 3.10 32
Lemma 3.11 34
Proof of Theorem 3.1 35
APPENDICES
A BASIC IDENTITIES 37
B TO HANDLE DIFFERENCE OF ENERGY TERM 39
C TO OBTAIN GENERAL IDENTITY 40
REFERENCES 43
BIOGRAPHICAL SKETCH 47
 iv 
Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy
UNIFORM STABILIZATION OF THE EULERBERNOULLI EQUATION
WITH ACTIVE DIRICHLET AND NONACTIVE NEUMANN
BOUNDARY FEEDBACK CONTROLS
By
Jerry Bartolomeo
December 1988
Chairman: Dr. Roberto Triggiani
Major Department: Mathematics
Given Q an open bounded domain in lRn with sufficiently smooth boundary T, we consider the
nonhomogeneous EulerBemoulli equation in the solution w (t,x):
wâ€ž + A2w = 0 in Q = (0,Â°Â°) x Ã2 (a)
w(0,) = wo ; w,(0,) = w1 in Ã2 (b)
' w I =gi e L2(I) = L2((0,oo);L2(r)) on E = (0,Â°Â°) x T (c) Ã1)
'l
trls=^=o Â°nz w
We seek to express the nonzero control function gÂ¡ as as suitable linear feedback applied to the veloÂ¬
city wâ€ž i.e., w I = Fwâ€ž such that Fw, e L2( (0,Â°Â°); L2(T)), and the corresponding closed loop system
obtained by using such a feedback in (1) generates a C0semigroup which decays uniformly exponentially
to zero as in the uniform topology of Z = [ID{AVi)Y x /D(A3'4)]':
 w(i),w,(0 II z ^ Ce [ hâ€™o.hâ€™! ]  z for all /> 0 and some C, 8 > 0 (2)
 v 
Having identified the candidate Fw, = â€”[A(A
3/2w,)], where A is the operator defined by
Af = A2/; ID(A)= { fe L2(Q): A2/e L2(Q), / =JÂ£ = 0 }, we prove two stabilization results,
the second of which (2) is the goal of this thesis and implies the first However, we include both results to
illustrate a contrast in modem day control methods.
Specifically if the domain Â£2 satisfies a radial vector field assumption
(jcâ€”JC0) â€¢ v > y > 0 on r (3)
where v is the outward unit normal vector, we prove strong stabilization, i.e., solutions go to zero in the
strong topology of Z : lim  [w(f),w,(0] II X = 0, by the use of a Hilbert space decomposition for contrac
tâ€”>oo
tive semigroups. Finally, if Q. satisfies (3), we obtain the desired uniform stabilization (2) via a change of
variables followed by the use of multipliers.
 vi 
CHAPTER 1
INTRODUCTION, PRELIMINARIES, STATEMENT OF MAIN RESULTS
LI Introduction And Literature
Let Q be an open bounded domain in IRn, n typically > 2, with sufficiently smooth boundary T. In
Q we consider the EulerBemoulli mixed problem in w(t,x) on an arbitrary time interval (0,7â€™ ]:
wâ€ž + A2w = 0
in
(0,T] x fi
(a)
w(0, x) = w0(x); w,(0, x) = Wi (x)
in
Q
0b)
w(t,a) = gx(t,o)
on
(0,7] x r
(c)
^(/,a) = g2(i, ct)
on
(0,7] X r
(d)
with nonhomogeneous forcing terms (control functions g\ and g2 in the Dirichlet and Neumann boundary
conditions). In (l.ld), v denotes the unit normal to T pointed outward. Recently there has been a keen
resurgence of interest (e.g. [LL.l], [R.3] and references cited therein) in the theory of plate equations, of
which the EulerBemoulli equation (1.1a) is a canonical model, presumably stimulated by two main
sources:
i) renewed studies in the dynamics, feasibility, and implementation of socalled large scale flexible
structures envisioned to be employed in space;
ii) recent mathematical advances in regularity theory of second order mixed hyperbolic problems
(canonically, the wave equation of both Dirichlet Type [L.8] [LT.8] [LT.9] [LLT.l] and of NeuÂ¬
mann type [LT.8], [LT.9], [LT.10], [LT.13], [LT.14], [LT.15], [S.l]) withL2 boundary data.
In either case, a prime thrust of motivation has come from dynamical control studies, at either an
engineering or a theoretical level. With reference to the specific problem (1.1), we cite, [L.7], [LT.l],
[LT.2] for optimal regularity theory and exact controllability theory with respect to classes of interest for
the initial data [ w'o.w'j ] and for the boundary data [gi,g2], which markedly improved upon prior literaÂ¬
ture [LM], Our interest in the present work is on the problem of boundary feedback uniform stabilization
 1 
2
for the dynamics (1.1) by explicit feedback operators, to be more properly defined below. Our results are
fully consistent with the corresponding exact controllability results [LT.l], [LT.2], not only with respect
to the function spaces for [g\,g2] and [w,w(] as mentioned above, but also with respect to the lack of
geometrical conditions on Â£2 when both g j and g 2 are active, or else with respect to the presence of similar
geometrical conditions on Â£2 when only g t is active and g2 is taken g2 = 0. This consistency is most desirÂ¬
able, and indeed has to be expected, in view of the known relationship between exact controllability and
uniform stabilization for time reversible dynamics such as (1.1) [R.l] [R.2]. We note, in passing, that uniÂ¬
form stabilization of problem (1.1) by means of a feedback operator (acting on [w.w,]) which is defined in
terms of the algebraic Riccati operator (which arises in the study of the optimal quadratic cost problem on
(0,Â°Â°)) was already achieved in the abstract treatment of [FLT.l], Mathematically the present work is
guided by and partially rests upon techniques developed in two main sources:
(i) the studies of exact controllability [LT.l], [LT.2] for problem (1.1);
(ii) the study of uniform stabilization of the wave equation with boundary feedback in the Dirichlet
boundary conditions [LT.4] and in the Neumann boundary conditions [T.l],
Of course these studies have to be seen in the context of recent investigations including:
a) uniform stabilization of the wave equation with feedback in the Neumann boundary conditions
[C.l], [C.2], [L.4], [L.5], [LT.4. sect 4];
b) regularity theory for hyperbolic equations in [L.8], [LT.8], [LT.9], [LLT.l], as well as
corresponding exact controllability theory [L.6], [LT.l2], [H.l], [T.2], [T.3];
c) exact controllability results for EulerBemoulli equations with different boundary conditions
[L.6], [L.7], [LT.6], [LL.l]; and, finally,
d) corresponding optimal quadratic cost problems [DLT.l], [LT.l 1], [FLT.l],
A point of view which we stress is the following: we choose g 1 in â€˜open loopâ€™ form to be in
L2((0,7);L2(O), T < OO. This determines that the corresponding solution of problem (1.1) with
w0 = H'1=g2 = 0 satisfies [w.w,] e C([0, T]; Z), where Z is the space identified in (1.7) below. This is
an optimal regularity result [L.7], [LT.2], Next, we choose g2 such that the corresponding solution of
3
(1.1) with gi = w0 = Wj = 0 also produces [w,w,] g C([0, 7]; Z), again as an optimal regularity result
This leads to [w,w,] e Z and g2 e L2( (0,7); H~\T)) [L.7], [LT.l] [LT.2],
In other words, only one choice is made, that gx e L2( (0,7) ; L2(V)); then, we work with data and
solutions in the corresponding optimal spaces. Our solution of the uniform stabilization problem below is
fully consistent with these â€˜open loopâ€™ considerations: uniform stabilization will be achieved in the space
Z with controls in feedback form gx e L2( (0,Â°Â°); L2(F)) and g2 e L2( (0,Â°Â°); //â€'(r)), (see Theorem
1.2 below). However, the present thesis will treat only the case gi e L2( (0,Â°Â°); L2(T)) in feedback form
and g2 = 0, leaving the more general case to our successive effort [BT.l],
We finally point out that other uniform stabilization problems for plate equations have been/are being
considered following LagneseLions recent monograph [LL.l] on plates [L.l], [L.3]. In these works,
however, different boundary conditions occur, typically of higher order, e.g., Aw and As a conse
ov
quence, uniform stabilization results are sought and obtained in higher topologies on Â£2 (so called
corresponding â€˜energy spacesâ€™). The problem considered in this thesis with boundary conditions as in
(1.1c), (l.ld) of lower order have a natural (and optimal, in fact) setting in very low topologies on Â£2; see
the optimal space below. This produces additional mathematical difficulties. To overcome these obstacles,
it will be necessary to introduce a new variable, the variable p below in (3.3), which lifts the topologies on
Â£2 to the level where the multiplier techniques which were successfully used in [LT.l], [LT.2] for the
corresponding exact controllability problem are applicable. A similar lifting was employed in the case of
the uniform stabilization problem for wave equations with Dirichlet feedback in the (low, but optimal)
topology L2(Â£2) x /Y"1 (Â£2) [LT.4],
1.2 Formulation of the Uniform Stabilization Problem and Main Statements
Throughout the paper we let < , >n denote the L2(Â£2)inner product with associated norm
  q, and < , >r denote the L2(I>inner product with associated norm   r. In addition, L(Â£2)
denotes the Hilbert space of all bounded linear operators on L2(Q) and L(T,Â£2) denotes the Hilbert space
of all bounded linear operators from L2(T) into L2(Q). Finally, L(X) will denote the Banach space of all
bounded linear operators on X.
4
We begin by letting A: ID(A) c Â¿2(Q) > Â¿2(Ãl) be the positive, selfadjoint operator defined by
Af= A2/; D(.A) = H\Q.)nHl(.Q) (1.2)
With the operator A defined as such, it then follows that [ LT.2, Appendix C ]
ZD(A1M) = Hq(Q) = {feH'(Q) : / =0 } (1.3)
/D(A3'4) = V= {/6//3(Q) :/r=^r = 0 } (1.4)
where we use = to denote norm equivalence.
Thus, for fe 1D(A1H)
1/2
ll/lllXA*)= II^'VIL = /W;(Q)= { IV/2rfi2 } (1.5)
where the last equivalence follows by Poincare inequality.
Similarly for/e Â©(A3"1)
1/2
ll/llf**) = II A3'4/1 q = { Jq I V(A/I2dÂ£2 } (1.6)
As suggested by [LT.l], [LT.2], our optimal space in which to study stabilization will be,
Z = H~l (fi) xV'= [/D(A "*)]' x [ZD(A3/4)]' (1.7)
where ' denotes duality with respect to the L2{Q.) topology. Next, let g\=g2 = 0 in (1.1). Then, the
corresponding evolution of (1.1) is governed by the operator
which generates a strongly continuous unitary group ( on the space ID(Am)xL2(Q) with domain
ZD(A0) = /D(A)x/D(A1'2) and hence ) on the space Z of our interest with domain
ZD(Ao) = ID (A w) x [ZD(A1/4)]' = Hl0 (Q) x H~l (Q). We denote this unitary group by e . Thus, the free
solutions of (1.1) with gj = g2  0 are normpreserving in Z:
II [w(0,w,(0] II z= II e^â€™\ wq.Wj ]  z = II [w0,wi ] z. for all fe R
With this wellknown result at hand, we can state the aim of the paper. Motivated by and consistent
with the function spaces in the optimal regularity and exact controllability theory of (1.1) [L.7], [LT.l],
[LT.2], we shall study the question of existence and construction of explicit boundary feedback operators
F x and F2 based on the â€˜velocityâ€™ w,
5
(1.9)
(1.10)
^i(w,)eL2((O,) ,L\D)
F2(w,)eL2((0,~) Â¡//''(O)
such that the boundary feedback functions
*i=^i(wÂ»), gi = F2(yv,) (1.11)
once inserted in (1.1 cd) produce a (feedback) C0semigroup e^! which is exponentially stable in the uniÂ¬
form operator norm L(Z) of the space Z in (1.7): namely, there exist constants M > 1,8 > 0 such that for
all t>0
IUa'IIl(z,sÂ«Ãâ€œ (u.s.)
We then say that such operators F x and F2 uniformly (exponentially) stabilize the original normÂ¬
preserving (conservative) dynamics (1.1) with gj = g2 = 0.
A weaker concept is that of â€˜strong stabilizationâ€™, by which we seek operators Ft and F2 as in
(1.11) such that once inserted in (1.1 cd) produce a (feedback) C0semigroup e^â€˜ on Z which decays
strongly to zero.
\\e^1  z â€”^ 0 ast â€”for all z g Z (S.S)
We note that for compact semigroups such as those arising in parabolic equations on a bounded
domain Â£1, the concept of stability in the strong topology (even weak topology) is equivalent to the concept
of stability in the uniform topology. However, for second order hyperbolic problems and plate problems
such as (1.1), the two concepts of strong and uniform stability are distinct. For recent optimal results on
the lack of uniform stabilization, see [T.5], [T.6].
Choice of Operators F, and F
It is justified in Chapter 2 in the case of F x and in [BT.l] in the general case that the following
choices of Fl and F 2
6
gx =Fl(wl) = kl(x)G\Ainwt = kl(x)G\AA3nwl = kl(x)â€”:L^!L\ (U2)
g 2 = P^Â¡) = k2(x)h2G'2AA'!'rlwt = Â¿2(^)A2A(/4^w,) (1.13)
â– l
provide reasonable candidates for the uniform stabilization problem of (1.1), in the sense that the closed
loop feedback dynamics with (1.12) and (1.13) inserted in (1.1c) and (l.ld), respectively, is wellposed in
the semigroup sense in Z and the Znorm of all of its solutions originating in Z decreases as t +Â°Â° (this,
however, does not say that such Znorms decreases to zero as t â€”Â» kÂ», (strong stabilization), let alone in
the uniform norm of L(Z) ). To show this conclusion will be our major task in Chapter 3, at least in the
case where g2 = 0 at the outset. In (1.12), (1.13) we have that:
a) ki(x) = smooth functions on T, kÂ¡(x) > 0 ;
(1.14)
b) A : (onto) isomorphism //*(!") Â»IIs 1 (T)> selfadjoint on L2(D
(U5)
so that
1/2
\
llA*llrcn= IUII
n\r>= {Jrlv
(1.16)
where V6 denotes the tangential gradient on T;
c) The operators G] are the adjoints, in the sense that
< GÂ¡g, z >a = < g ,
G\z >r, g e L2(D, z e L2(Q)
(1.17)
of the operators Gt defined by:
A2x = 0 in Q
(a)
Gig=x if and only if
xI = g] on T
â€¢r
(b)
(1.18)
^1 =0 onT
L 3v lr
(c)
A2y = 0 in Q
(a)
G2g=y if and only if
y I =0 on T
â€¢r
(b)
(1.19)
l^lr*S!0,,r
(c)
7
Elliptic theory [LM, Vol. 1] gives for any s e R
G\ : continuous H*(T) 1/2(Q)
G2 : continuous HS(T) >//, + 3/2(fi)
Moreover, by Greenâ€™s theorem it is proved that [LT.2, Lemma 2.0 and Lemma 4.0, respectively]
(1.20)
(1.21)
(1.22)
G'2Af=~(Af) I , / g ID(A) (1.23)
â€¢r
Identities (1.22), (1.23) are used in the last step of (1.12), (1.13), respectively. Thus, the resulting candiÂ¬
date feedback system, whose stability properties in Z we shall investigate is,
wâ€ž + A2w = 0
in (0,oo) x Q = Q
(a)
w(0,x) = w0(x); w,(0,x) = WjCx)
in Q
(b)
1 , , s 3A(A"ww,) 
Sv (
on (0,Â°o) x T = Z
(c)
(1.24)
on E
(d)
Using the techniques of [T.4], problem (1.24) can be rewritten more conveniently in abstract form as
A1 0 '
I A A[kxG,G\Am + k2G2\2G'2Am]
D(A) = { y e Z : Ay eZ }
A more explicit description of D(A) will be given below. Our main results are as follows.
(1.25)
(1.26a)
(1.26b)
Theorem LI
(i) Wellposedness on Z
The operator A in (1.26) is dissipative on Z = [7D(A1/4)]' x [ID{A3/4)]', see (1.7), and satisfies here:
range (XI A )=z for X > 0. Thus, by LumerPhillips theorem [P], A generates a strongly continuous
contraction semigroup e^â€˜ on Z, and the resolvent operator /?(X; A) is compact on Z, for ReX> 0.
8
Moreover 0 e p(A), the resolvent set of A.
(ii) Boundedness of feedback operators
For [ vvo.Hâ€™j ] e Z, we have
more precisely
, 3A(A 3/2 w,)
kxGxAiaw, = kxâ€”^â€” e L2( (0,Â°Â°) ;L2(T))
3v
k2A2G*2A~mw, = k2A2A(Aiawt) e L\ (0,Â°Â°); H~\T))
2
z
(1.27)
(1.28)
(1.29)
(1.30)
J0 \\kxG\Amw,\?Tdt<  [h'o.h'i ] 
JQ II k2A2G2A'mw, 2rdt <  [ w0,w, ]  z
The proof of Theorem 1.1 will be given in Chapter 2, in the case of g2 = 0, and in [BT.l] in the general
case.
Theorem 1,2 ( Uniform stabilization on Z with both feedback operators in the absence of geometrical conÂ¬
ditions on Â£2)
The following property holds for the feedback problem (1.24), or (1.25), (1.26): there are constants
M > 1 and 8 > 0 such that for all t>0
II
W(0
U(ol
II z =
(1.31)
Theorem 1.3 ( Uniform stabilization on Z with only the first feedback operator gx and g2 = 0, in the presÂ¬
ence of geometrical conditions on Q).
Consider the feedback problem (1.1) with gx given by (1.12) while g2 = 0. Then there is a constant
8 > 0 such that the uniform decay (1.31) holds true, provided Q satisfies the following geometrical condiÂ¬
tion (radial vector field assumption):
there exists a pointx0 e Rn such that (xx0) â€™V >y>0on T (1.32)
Theorem 1.3 is the main result proved in this thesis. It may be extended to more general domains Â£2
which satisfy a weaker geometrical condition than (1.32), expressed in terms of a more general vector field
than the class of radial fields (jcx0). This will be done in [BT.l], Also Theorem 1.2the general result
9
with two feedback operators and no geometrical conditionwill likewise appear in [BT.l], Instead, in
Chapter 2 of this thesis we shall also prove a strong stabilization result (Theorem 2.10) with gj as in (1.12),
( and Â¿!=1 ) and g2 = 0, under the same geometrical assumption (1.32). Though the strong stabilization
result of Theorem 2.10 in Chapter 2 is implied by the uniform stabilization result of Theorem 1.3 in
Chapter 3, we feel that its inclusion in this thesis is justified by the following considerations. It shows â€˜how
farâ€™ the purely operator techniquesbased on the operator model (1.25), (1.26) and the NagyFoiasFoguel
decomposition for contraction semigroupscan be carried out along the lines of arguments first used for
boundary control problems for second order hyperbolic equations in [LT.16], [LT.17], [LT.4] and also in
[T.l] for different feedback operators. A new obstacle arises, however, in the case of plate problems
(fourth order in the space variable rather than second order in space as in hyperbolic problems) at the level
of excluding the presence of eigenvalues along the imaginary axis. This is Lemma 2.7, where only three
homogeneous boundary conditions (2.31bcd) are in place for the â€˜eigenproblemâ€™ (2.31a), as opposed to
the four homogeneous boundary conditions for a fourth order operator covered by standard theory. (In preÂ¬
vious arguments for second order hyperbolic problems as in [LT.16], [LT.17], [LT.4], [T.l], the
corresponding â€˜eigenproblemâ€™ has two homogeneous boundary conditions for a second order operator and
hence is covered by standard theory). A novelty is then that Lemma 2.7 uses a multiplier technique proof,
which is supported by the exact controllability problem [LT.l], [LT.2], as applied however to the (staÂ¬
tionary) elliptic problem this time. It is the multiplier technique that requires the geometrical condition
(1.32) in the proof of Lemma 2.7. Lemma 2.7 appears to be new in elliptic theory. This fact alone would
justify its inclusion here, even though Lemma 2.7 plays only a secondary role in the problem of stabilizaÂ¬
tion considered here, as it leads only to strong stabilization (Theorem 2.10). It is through the more elaÂ¬
borate and lengthier arguments of Chapter 3which are necessitated by use of the same multiplier applied
to the feedback dynamics (1.24)that we will eventually obtain the uniform stabilization result of Theorem
1.3.
CHAPTER 2
WELLPOSEDNESS AND STRONG STABILIZATION
2,1 Preliminaries. Choice of Stabilizing Feedback
Let Q be an open bounded domain in //?", n>2 with sufficiently smooth boundary T. Consider the
nonhomogeneous problem in the solution w (t,x):
wâ€ž + A2w  0
in Q = (0,Â°o) x Q
(a)
w(0,) = w0 w,(0,) = w1
in ii
(b)
w =g1eL2(Â£) = L2((0,oo);Â¿2(r))
â– l
on E = (0,oo) x T
(c)
(2.1)
dw I â€ž
on E
(d)
The goal of this chapter is to obtain strong stabilization of the system (2.1) via a closedloop feedback g j
based on the velocity w,. However, the optimal function space in which to work (a cross product space for
position and velocity) is obtained from the exact controllability result to be summarized below.
First, we define the positive, selfadjoint operator A: ID(A) c L2(il) â€”> L2(Q) by
Af = A2/ (2.2a)
2D(A)= {/eL2(Â£2):A2/eL2(Q), /^ = JÂ£ =o} s O//Â§(Q) (2.2b)
Since Q is bounded in Rn, then A has compact resolvent R( X ; A). Also, if Af= 0 for / 6 ID {A) then by
(A.O) and (2.1c,d) we have
0 =n = < A(A/) , / >n
= < A/, Af >q + < f>r< Af, >r
= < Af, A/>q = A/Â¿
 10
11 
This implies/= 0, so therefore,
A~x e L(Q)
(2.3)
Next, we let
and consider the space Z = H '(Ã2) x V. As shown in [LT.2], Z can be characterized by using equivalent
norms as
Z = [ID(Am)]' x [2D(A3'4)]'
where ' denotes duality with respect to the Â¿2(Ã2)topology.
(2.4b)
The norms on these spaces are given by
(2.5)
IMI Â£>(*â€œ) = IUâ€œ*lln; II X II (zdcaP)]' = IU Min a,p>0
(2.5)
Below we state the regularity result as well as the exact controllability result.
Theorem 2.0 (Regularity) ÃL.71. fLT.21
Consider the problem (2.1) subject to [w0.wJeZ, gx e L2( (0,7); L2(T)),
g2 e L2((0,7) ;//_1(r)). Then the map {w0,w^gi,g2} > [w(i),w,(i)] 6 C([0,7] ;Z) is continuous
for any 0
Theorem 2.1 (Exact Controllability) fLT.21
(i) Assume there exists a point x0 e Rn such that (xx0) â€¢ v > y > 0 on T where v is the unit outÂ¬
ward normal vector. Let 0 < T < Â°Â° be arbitrary. If [wo,^ ] e Z arbitrary, then there exists a suitable conÂ¬
trol function gj e L2( (0,T); L2(F)), such that the corresponding solution of (2.1) with g2 = 0 satisfies
w(T,) = w,(T,) = 0 and in addition [w.w,] e C( [0,7]; Z )
(ii) The same conclusion holds true without geometrical conditions if g 2 is taken within the class of
L2( (0,7) ;//â€™(r)) controls. â–¡
By time reversibility, we see that at any finite 7 the totality of all solution points [w(T),w,(T)} of
problem (1.1) with w0 = W] = 0 fills all of the space Z when either g[ runs over all of L2( (0,7); L2(T))
12
and g2=0 under geometrical conditions on Q, or else when lg\,gj} nins over all of
L2((P,T);L2(r))xL2((P,T)]H~l(T)) without geometrical conditions. Therefore, since the space of
exact controllability is the space of maximal regularity, we seek stabilization in exactly this space Z.
We define the "energy" E(t) for the dynamics (2.1) over the space Z = [K){A1/4)]' x [lD(Ayi)]' by
IW (t) I o 2 2
w,(i)l HZ = II II [D(A'*)Y X II wâ€˜Â® II [Â©(A*4)]'
= IU1/4w(0lÂ£+ IU'â€œw,(0q (2.6)
dE
Next we seek a candidate gi which at least produces â€”â€” < 0, i.e., energy "decrease." This does not,
at
however, guarantee lim E(t) = 0 (which is precisely strong stability of (2.1)), let alone uniform stability.
Remark 2.2
Below we shall show wellposedness in Z, with g2 = 0. Then since w, e [/D(A3/4)]', it follows that
A ~yiw, = A 3/4 A 3/4 w, e ID (A3/4) = V. Therefore, A 3,2 w, satisfies the required boundary conditions.
A3'2
w,
(2.7)
By writing E(t)= < A 1,4w , A I,4w >a + < A 3/4 w,, A 3Mw, >n and differentiating with respect to t
we have that
 < A~Mw , A1,4w( >n + < A3/4wâ€ž , A^w, >n
2 ai
by (2.1a)
= < w , A_1/2w, >a~< Aw , A'^w, >n = < w , A'1,2w, >n  < A(Aw), A'3/2w, >n
by (A.O)
= < w , A~ll2wÂ¡ >n  < ^f(Aw), A_3/2w, >r  < Aw , ^(A_3/2w,) >r + < Aw , A(A~iawt) >n
dv ov
by (2.7)
= < w , A iaw, >q  < Aw , A(A mw,) >n
13
by (A.O)
= < w, A'mwt >a 
< . A(A'3,2h',) >r ~ < w , ^j[A(A 3nw,)] >r + < w, A2(A~y2wt) >a
= < w , A inw, >q + < w , ^[Â¿(A3'2^)] >r  < w , A mwt >n = < w , ^[A(A~3/2w,)] >r
dv dv
Therefore, by selecting the simplest choice
w\ =Si =  ^[ACA30^,)]
11 ov
(2.8)
We obtain â€” = 2 11 g j Ã. <0, our desired energy decrease. â–¡
at 1
Next we will show how our feedback can be expressed in terms of an operator (Green map) which
acts from boundary r to interior Q. Following [LT.1],[LT.2] we define G j : L2(V) Â» L2(Q) by
Â£
il
o
in Q
(a)
Gig =y if and only if â€¢
y =g
â– r
on r
(b)
(2.9)
r
il
o
on r
(c)
We quote the following Lemma which will be used below.
Lemma 2,3 ÃLT.21
Let Gj :L2(Q) Â»L2(r) denote the continuous operator defined by
Q = r, g e L2(T), v e L2(Q), i.e., G\ is the adjoint of G\. Then
Gji4/=j(A/) for/e ZD(A) (2.10)
Now using (2.8) and (2.10) we see that
W\ =^[MAmw,)] = G\A(Ay2w,) = G\Al,2Wl (2.11)
Using elliptic theory [ LM, Vol I, p. 188 ] we have that for any s real
Gi : continuous HS(T) Â»H1 + 1/2(Q) (2.12a)
and in particular for s  0
G] : continuous L2(f) â€”Â» //1/2(Q) (2.12b)
 14
We also have that by duality on (2.12a) with s =  3/2 that
G\ : continuous //Â¿(Ã2)>//3,2(r) (2.13)
so that (2.12a), (2.13) imply
G,Gl : continuous Â©(A **) =//Â¿ (Q) >//2(Q) (2.14)
to be used below (2.17b) in the description of the domain of the feedback generator.
2,2 WellPosedness and Semigroup Generation
First we want to introduce an abstract operator model for problem (2.1). According to [T.l], [T.4],
problem (2.1) with g 2 = 0 admits the following abstract versions:
as a second order equation
w = A [wGjg^A [w + GxG\Amw] (2.15a)
or else as a first order system
[w,
w] eZ = [/D(A1M)]'x [Â©(A3'4)]'
(2.15b)
Â° I .
where
A =
lA AG\G\A~xa\
(2.16)
More explicitly if y â‚¬ Â©(A) then we can
write
1 0 7I
1 i>i + G\Amy2.
Ay =
1 A o
II 1
(2.17a)
Thus, /0(A) = { [y, , y2]eZ:y2e [Â©(A1/4)]' andA [yx + G1G'lA~iay2] e [Â©(A*4)]' ,i.eâ€ž
y, +GXG\Amy 2 e /D(A1/4) = //Â¿ (Q) which implies yx e H\SH)} (2.17b)
The operator A defined above is our candidate to be the generator of a feedback semigroup. The first step
in this direction is the following Lemma.
Lemma 2,5
The operator A is dissipative on Z = [!D(A1/4)]' x [Â£)(A3'4)]'.
15
Proof
Let z e Z, then using below the skewadjointness of  q  we *iave fÂ°r 2 6 Â® (A)
Re< Ai, i>z=Re< â€œ 'n;;i.i;;i>z
iÂ° 0 nZii iZii
+ Re< lO AGjGjAI/2I U2I â€™ Ul >Z
= 0< AGjGtA^zz, z2 >[/D(A3,4)]' = < A3'4AG,GÃA1/2z2, A"3/4z2 >a
=   G\A~iaz2 p <0 and dissipativity holds. â–¡
The above proof is a reformulation of our argument below (2.7). Now we come to our result on semigroup
generation.
Theorem 2,6
(i) The dissipative operator A in (2.16) also satisfies range (XI  A) = Z on Z for X > 0.
Thus, by the LumerPhillips Theorem [P], A generates a C0semigroup of contractions e^â€˜ on Z, t >0
and the solution of (2.1),(2.11) is given by
and in fact
,w(r, w0 , wO, A .Wo,
,, \l=e M I all / >0, [w0 , wj]eZ
w,(f. Wo. w,) Wi I 01
W0, 2
(ii) The resolvent operator R( X; A) of A is given by
i  w1
R(X;A)=
P(X)]A
Ui
 v(X)1 xv^r'A1
where V(X) = (I + XG ]G*A'm + X2A~
(2.18a)
IUAÃ Â° II Z = E(0 = f f IA1Mwl2+ IA^vv,!2} dil (2.18b)
I W] 
(2.19a)
(2.19b)
at least for all X satisfying ReX > 0 . Moreover X = 0 belongs to the resolvent set of A and R(X; A) is
compact on Z.
16
(iii) If the domain Â£2 is such that there exists a point x0 e Â¡Rn such that
(jcjco) â€¢ v > y > 0 on T (2.20)
then R( X; A) is welldefined and compact on Z also on the imaginary axis and hence, for all X satisfying
ReX> 0. Thus, the spectrum (point) of A satisfies
a(A) c { X: ReX < 0 } (2.21)
Remark
A stronger result will follow below once we prove our uniform stabilization Theorem 1.3, that in fact
o(A)c { X:ReX<  S < 0 }. â–¡
Proof of Theorem 2.6
Dissipativity of A on Z was already shown in Lemma 2.5. Next, fix X > 0 and let z e Z and we
want to solve (XI  A)y = z, i.e.,
Xy1y2 = z1e [Â©(A1*)]' (2.22a)
A(y, +GlG*All2y2) + Xy2 = z2 e [Â©(A3*)]' (2.22b)
for y e ID{A). We apply A1 to (2.22b), multiply (2.22a) by X and subtract to obtain:
V(X)y2  XAâ€™z2  z, e [Â£>(A1M)]' (2.23)
with V(X) defined in (2.19b).
We next note that L(A.) is boundedly invertible on [ID(A1/4)]' since equivalently
A1I4V(X)A1,4 =1 + XA~WG\G\A~xi* + X2A~l is boundedly invertible on L2(i2) (being selfadjoint, strictly
positive on Lz(Â£2)) with inverse
A"1'4r1WAw e L(Q)
Thus, from (2.23)
y2 = K'(X)(XA1z2z1)e [/D(Am)]'
which then inserted in (2.22a) yields
r IV~\X)
(2.24)
(2.25)
y\ =
zl+V\X)A'z2
(2.26)
Then (2.19a) follows from (2.25) and (2.26). Note that from (2.22b) and (2.25) that
yl+G1G\AMy2=A1z2U1y2e D(A'â€œ) (2.27)
So that recalling (2.17b) we see that from (2.25) and (2.27) it is verified that y e ZD( A). The compactness
of R(X; A) on Z is readily seen from (2.19a) to be equivalent to compactness on L2(Q) of the following
operators:
AUA(I V~l(K))A114 (2.28a)
A1My_1(X)A1/4 = A wV\X)AmA~m (2.28b)
A 3MV~' (X)A1/4 = A mA 1M V1 (k)A1/4 (2.28c)
A V~l (X)A m = A ~mA "1/4 V1 (X)A1,4 A ~m (2.28d)
First, compactness of the operators (2.28bcd) on L2(Q) is plain from (2.24) and A â€œ , a > 0 being
compact on L2(il). For (2.28a) apply V~l (X) on (2.19b) so that
I = V~1 (X) + XV1 (X)G, G\Am + X2V~x (X)A
and then
Am[I V1 (X) ] A1,4 = XA1/4F_1 (X)G ,GlA 1/4 + X2A"1MF_1 (X)A3M
= XA^V\X)AmA^GxG\A~l'A + X2A1/4F"â€˜(X)A 1,4A_1
which is compact on L2(Q) by (2.34) since A_1/4G j G*A 1,4 e L(il) â–¡
To complete the proof of Theorem 2.6, we must show that ct(A) does not contain any points on the
imaginary axis (we already know that a(A) does not contain points in { ReX > 0 } since A is the generaÂ¬
tor of a contraction semigroup).
Thus, we need to show that
V(X)1 e L( [ZD(A1/4)]') for X = ir , r e IR , r *0 (2.29)
To this end let x g [ID(A1/4)]' and suppose V(X)x = 0 for X = ir. Then from (2.19b),
0 = < F(X)x , x >[ID(AW)]â€™
= < x , x >[2D(Aâ€˜'4)]' +ir< GxG\A~mx , x >[Â¡D(A"*)Y ~ *2< , x >[JD(AW)Y
= < A~mx , x >n +ir  G*Aâ€œ1/2x ^. r2< A~mx , x >a
Since the middle term in (2.30) is purely imaginary we must have that via (2.10)
(2.30)
18
Also, we have that by (2.30), A mx = r2A mx, i.e.,
Ax = r2x (2.32)
which means that x must be an eigenvector of A say x = en with eigenvalue r2. Therefore, since
en e ID (A) we have that it satisfies the two zero boundary conditions associated with ID{A) (see (2.2b)), as
well as (2.31). Therefore, the following Lemma will complete the proof of Theorem 2.6.
Lemma 2.7
Let X = r2 > 0 and suppose Q satisfies the radial vector field assumption (2.20). Then the problem
3
II
e
in Q
(a)
 =o
on r
(b)
â– r
o
II
u
on r
(c)
3(A*) =0
oV 1 p
on r
(d)
(2.33)
has only the trivial solution = 0.
Notes
1. Since A ~y2x = A ~3l2eH = ry2eâ€ž ; (2.33d) follows from (2.31).
2. The above Lemma is not covered by standard elliptic theory since only three boundary conditions,
instead of four, are involved for the fourthorder elliptic operator in (2.33).
3. Recall that as given in Appendix C, if h(x) = (xx0) then
II (x) = 1 (identity matrix)
div h  n  dim Ã2
(2.34a)
(2.34b)
4. If 4> is a solution of (2.33) then it is immediate that via
multiplication by A(p and
I dQ we have
I V(A) 12dD. = I Vtp 12dQ
(2.35)
19
Proof of Lemma 2.7
Multiply both sides of (2.33a) by h â€¢ V(A<}>) and integrate J dÂ£2.
Left hand side: We have by (A.8), (2.33d), (2.34ab), (2.35)
J A(A<)))h â€¢ V(A<())dQ = yJ I V(Al2dQ (2.36)
Right hand side:
xj Â§h â€¢ V(A<())dQ
by (A.5)
= xj A* vdTXj Adiv(/i)dQ
by (A.3),(2.33b),(2.34b)
= Xj A4>X â€¢ V<)(iii
by (A.8),(2.34a,b),(A.l)
= â€” x[ ^h â– V({)Ãfr+4Ã IV0IA VÃ/r + xf IVl2dÂ£2^rf IVl2dQ
Jrdv 2Jr jq 2 Jq
nXÃ ^dr + /iXf IV(J)l2di2
Jr3v Jq
by(2.33bc)
= JJr 1 V I2h â– \dT + (y + 1) xj I V<> 12dO.
But since <)) I = 0 we have I V I = I ~ I = 0. Therefore,
I p ov
xj # â€¢ V(A<>)dQ = ( +1) xj^ I V 12dn (2.37)
Setting (2.36) = (2.37) and simplifying we get
â€¢if I V(A<)>)12h â– vdT = 2xj IV())l2dr (2.38)
2 r Qi
Now if we analyze (2.38) we see that by (2.32) the left hand side is nonpositive and the right hand
side is nonnegative, therefore both sides must equal zero. This then implies V
a.e. But we have that eigenfunction implies <)> smooth, and together with (J)
fore, Lemma 2.7 and hence Theorem 2.6 are proved. â–¡
r
= 0 we have <)> = 0.
There
Now that we have proven that A generates a C 0semigroup of contractions e on Z, it follows that
20
E(t)0 (2.39)
This fact will be used crucially below. The next corollary is a consequence of the dissipative feedback perÂ¬
turbation on the boundary.
Corollary 2.8
By choosing w = g\ = G\A mwâ€ž it follows that G\A mw, e L2((0,Â°Â°); L2(T)) and in fact
lz
II G\Amw, II 2L.m = Jâ€œ II G\A^w, II\dt < yÂ£(0) (2.40)
for all initial conditions [m'0,w1] e Z.
Proof of Corollary 2,8
Let [ wo.wj ] e /D(A) and recall for convenience
E(t) =
(2.41)
Now â– ~~Â£(i) = ~â€œ<
2 dt 2 dt
fÂ°!
. *A'
W0
I*T
vV] 1
.Hâ€™0!
_ At I
WÂ°
In'! 
, e
Wi 1
\>Z
lw(0l
iw(0i
l*â€™,(f)l â€™
U(0l
via the proof of Lemma 2.5
(2.42)
Remark 2,9
We see that (2.42) shows that such a choice of g t does lead to an energy decrease as was demonÂ¬
strated in another way (using Greenâ€™s formula) in Remark 2.2. â–¡
Continuing the proof now we integrate dt both sides to obtain:
21 
Ã~Â»G^m^ld'bTXÃE^â€˜b^JrÃEm
= ^E(0)\ lim E(T)
L L 7Â»~
where in the last inequality we used the contraction of the semigroup, i.e., (2.39). Extension by continuity
yields (2.38) for all [w0,wi] e Z. â–¡
Theorem 2.10
Let the radial vector field assumption (2.20) on Q hold. Then for any [wq.W]] e Z we have that
E(t) =
w(i , W0 , Wi)
W,(f , Wo , Wj)
2
11 z Â» 0 as i > +oo
(2.43)
Proof of Theorem 2,10
The above result follows by appealing to the NagyFoiasFogel decomposition theory [L], Since
e^â€˜ is a Cocontraction semigroup by Theorem 2.6, the Hilbert space Z can be decomposed in a unique
way into the orthogonal sum;
Z = ZcnuÂ®Zu (2.44)
where both Zcnu and Zu are reducing subspaces for e and its adjoint.
It is also true that
(i) on Zcnu, e^â€˜ is completely nonunitary and weakly stable
(ii) on Zâ€ž, e^â€˜ is a C0unitary group.
In our case, Zu= { 0 } , the trivial subspace, because otherwise an application of Stoneâ€™s theorem
[P] would guarantee at least one eigenvalue of A on the imaginary axis, but this is clearly false due to
Theorem 2.6. Hence Z = Zcnu and therefore e is weakly stable on Z. However, since A has compact
resolvent, it follows that e^â€˜ is stable in the strong topology of Z[B], Therefore, e^â€˜z > 0 as t > +Â°Â° for
all z e Z and strong stability is verified. â–¡
CHAPTER 3
UNIFORM STABILIZATION
3.1 Preliminaries. Change of Variables
Recall our feedback system
wâ€ž + A2w = 0
in Q = (0,<
Â»)xfi
(a)
w(0,) = w0 w,(0,) = w,
in Q
00
w\ =Si = ~ t[4(i4_ww,)]
â– l OV
on Z = (0,Â«
Â») x r
(c)
dw I ~
WS!=0
on Z
(d)
and the corresponding "energy" E (t) defined by the squared norm of the semigroup
iwiw iuA,*Â° lt = II Z,col III  IU*v,lÂ£ (3.D
We want to show that under suitable assumptions on Â£2 the energy E (t) decays uniformly, exponenÂ¬
tially to zero. More precisely there exists constants C, 5 > 0 such that for any
[ Wo,W] ] e Z = [ID(A11*)]' x [Â¡D(Ay*)Y the corresponding solution of (3.0) satisfies
E (0 < Ce^Ei0) for all t> 0 (3.2)
The proof of (3.2) will require a different approach than was used in Chapter 2 to achieve strong staÂ¬
bility. The proof will involve multipliers, but before we can proceed a change of variables must be iniÂ¬
tiated. Let
p(t) = A~3nw,(t) for [ wq.wj ] e Z (3.3)
Then since w,(0 e [ID(A31*)]' i.e. Aâ€œ3Mw,(f) e L2(Â£2), it follows that
A3*p (t) = Av*A~3aw,(t) = A~3*w,(t) g L2(Â£2). Thus
22
23
So in particular
pit) e ID(A31*) =
{/e//>(a):/r=fr=o}
(a)
(b)
(3.4)
Now recalling (2.15) we have using (3.3)
p, = Amwâ€ž = A^[Aw AG\G\A1/2w,]
= AmwAinGiGÂ¡A~mwl
Differentiating once more in time and using A~mw, = Ap and A~mwâ€ž = Ap, we get
ptt = ApA mG\G*Ap,
and hence
pâ€ž+A2p = A ~mG i G *Ap,
P(0,)=Po=A~3nwl, p,(0,) = pi =A~mw0  A~mG\G\A~mW\
â€¢ p\ =0
â– l
in Q
in Q
on Z
on Z
(3.5)
(3.6)
(3.6)
(.a)
0b)
(c) (3.7)
Cd)
3.2 Uniform Stabilization
Our main result referred to in Chapter 1 as Theorem 1.3 is as follows:
Theorem 3.1
Assume that there exists a point x0 e IR" so that the radial vector field defined by h (x) = (xx0)
satisfies
h (x) â€¢ v > y > 0 on T (3.8)
where v is the unit outward normal, then there exists positive constants C, 8 such that
E(/) < Ce~^E(0) for all />0 (3.9)
The proof of Theorem 3.1 will follow directly from the next theorem, by taking lim in (3.10) and invoking
pÂ»o
Datkoâ€™s theorem [D.l],
24
Theorem 3.2
Under the same assumption (3.8) as in the above theorem, there exists a constant K such that for all
initial data [w'0,w'1] e Z (therefore yielding Â£(0) < +Â°Â° ) and all 0 < P <1 it follows that
1*^1 llz* = jV2prllÂ«AiM,â€œ \\zdt
(3.10)
Proof of Theorem 3.2 (A multiplier approach)
First, we take initial data smooth [w0,W] ]e Â£>( A) (see 2.17b), which then guarantees
[w(i),w,(f)] e C([0,T];/D(A)) for all T < Â°Â° and find the desired estimate with K independent of
[ wq.Wj ] and of p. Then extension by continuity yields (3.10) for all initial data [wd.wJ e Z.
Now we recall our starting identity which we derived in the appendices via the multipliers
e~2â– V(Ap) and e~2^Apdi\h (recall (C.8)):
J eW(KÂ±P)_h . s/{Ap)d'L 1J eW\V(Ap)\2h vdÂ£ + e^^^ApdZ
= jQe~m 1 VP>11(iQ + iQe~^â€˜1 V0V) 12dQ ~ 2fij e'^p,/! â– V(Ap)dQ
 npj^e'2^1 ptApdQ  j^e~2^â€˜A~mGiG\A~inp,h â– V(Ap)dQ
 4f e 2IM4 1/2GiG\AptApdQ  lim e p,, /tV(Ap) >n
2 Jq r>~L
 4 lim [e_2pi< pt, Ap> a
2 Ãt â€”> Â°Â°L I 0
(3.11a)
= I Vp, 12dQ + f eV I V(Ap)\2dQ  Qx  Q2Q2Q<L,  L2
2 (3.Hb)
where the QÂ¿, i = 1,2,3,4 are the corresponding integrals over Q and the Â£,, i = 1,2 are the corresponding
limit terms. The first two integrals are most important as we shall see below. Next we prove:
Lemma 3.3
There exists a constant C 1 such that
II &P lip SCi 11 A y*p ^
(3.12)
25
Proof
2 2
First by trace theory [LM] Ap
2
< C 11 Ap 11 since II1 (Q) is a stronger topology
= C  Ap +C  v(Ap) by definition
< C 11 A mp 1+ CCp3 11 A wp ^ by norm equivalaence
^ C  AWp  L(Q) II A3/V Â£ + CCp3 II A**p Â£ = C,  A*p \\2a â–¡
We continue the proof of Theorem 3.2 which will be divide into seven steps.
Step 1 : Absorption of the boundary terms in (3.11). Let Mhb = m^x\h\2 and choose Ej > 0, e2 > 0
such that
z,Mhb < 1 (3.13a)
nC\
e2â€”â€”â€” < cq (3.13b)
where as in Lemma 3.3 and oq to be given below.
Before proceeding, note that the following inequality will be used extensively below:
For any e > 0 ; 2aÂ¿? < ea2 + ^b2 (*)
Next we operate on the left hand side (L.H.S) of equation (3.11) by using (*) with et and e2 as well
as the assumption on h â– v to obtain:
f W
Jl 3v 2Jl 2 av
< â€”f Â£20*  1+ oMhb f e2p/IV(Ap)l2dI^[ e~2^ IV(Ap)l2dZ,
C] av 2 â– >Â£
+ Â¿4e_2Pâ€˜l^2Â¿Â£+E4l/_ai'lipl2râ€˜'1
s(â€” + e2f )E(0) + (e,m* h\ e',?Jm,\p)\2dl+ IIAâ€œp ll>
El L L Zj0 â€œ
where the E(0) term follows by using (2.40) and we have also used Lemma 3.3 for the Ap term.
Now we can drop the middle term because by (3.13a) e1MWj^<0 and use the fact that
IU3aVIIq= A3/4w, ^ <Â£(i) toobtain:
26
(L.H.S.) of (3.11) <ÃT1Â£;(0) + ^^Jâ€œe"2pÃÂ£:(Ã)dÃ (3.14)
where Ki = â€” + â€”.
Ei 2e2
Finally, utilizing (3.14) in (3.11) and isolating the important terms we obtain:
\ \Vp,\2dQ + Â¡ e^lV(Ap)\2dQ
+ Q\ + Qi + 03 + <2 4 +L\ +l2 (3.15)
Step 2 : Isolating the energy integral e~2^E (t)dt. First since p, e D(A1M) there exists a constant
Cp i > 0 such that
II M \\l>CPl U>,Â¿[LT.2]
Similarly p e 1D{Aw) implies the existence of Cp2 > 0 such that
  V(Ap) ^  CPi II AMp ^[LT.2]
(3.16)
(3.17)
Next using (3.15) and (3.16) we have
II M Â£ >Q>, \\AlÂ«p,\\l
= Cpx  A~ww 2 + 2Cp,Re< , A'â„¢GxG\A~mwt >a + Cpx  A^G^A^w, 2
a
1
Now to bound below we use lab > za2 b2 on the middle term and taking e = â€” we obtain
II M Â£ >^~ A1/4w \\2aCpx \\AwG,G\Amwt 
Also it is immediate that
 V(Ap) Â¿>Cp3A>Â¿=Cp3lMww(2
n
Putting the pieces together in a Lemma:
Lemma 3.4
(3.18)
(3.19)
Let a! = min { Cp3 }
and K2Cp\  A mG\  L(r,Q)
Then J e~2^ I Vp, 12dQ + J e~^â€˜ I V(Ap)\2dQ > axj^e^E(t)dt  K2E(0)
(3.20)
(3.21)
(3.22)
27
Proof
f eâ€œ2pÃ I V(Ap) 12dQ + \ eW I Vp, 12d<2
by (3.18), (3.19)
> Cp3\~eW  \2adi + ^JV2*  A'1'4* *Ã¡rC/Â»1jV2pÃ  A^G^A"Â»*, \dt
^Â«.JV2^  A w \2a +  A w, Â£] *CPl II Aâ€”G,  L
>by (2.40) ajJV^EÃOdÃCpj A1MG, Iko.Q^W. â–¡
Collecting our results so far we have via (3.13b), (3.15), (3.22) that
a2Jo e ^â€˜E(t)di
where
K3=Kx+K2
and
/1C1
â€œ2 = ai _ Â£2~2~ > 0
Step 3 : Handling the limit terms arising from integration by parts in time.
(3.23)
(3.24a)
(3.24b)
Remark 3.5
The following will be used in evaluating the limit terms in (3.23). Since we are taking initial condiÂ¬
tions [ w0,wj ] e Â£>(A) we have by the regularity theory [LT.5], [w(/), w,(/)] e C([0,7â€™]; 10(A)), i.e.,
w(t) e C([0,r]; H'm and w,(t) e C([0,T] ; [/D(A1,4)]0.
I VV I A I ^0 I
Computing  I^J  /D(A) = IUA'W] H Â®(A)
. .Wn, . , Wn 1 1 Wq ,
= i AeA'(V I  z = IIÂ«A'Ab,J IIzS IAwJz (3.25)
where we have used 0 e p(A) quoted in Theorem 2.6. Hence we have that
2 1 . w01 2
II w  //'(Q)+  A 1/4w( Q <  AWi j  z for all / >0
(3.26)
28
Now letting Mh = I h 12 we have
rlim Pt(T) , h â– V(Ap(T)) >a  < II P'
by (3.5), (3.9)
< lim eW
2  A1'4  L(n)  A~y*w(T)  n
2  A~mG\ G\A~xi*  L(n) II AMw,(T) n +A/*Â£(0)
by (3.26)
â€œ rÃÃÂ»e_2^T ( CE(0) + 2 II A~mG\G\A'*  l
(3.27)
A similar computation shows that j^lim e 2PT< pt(T), Ap(T) >Q j = 0 (3.28)
Hence we have that/,! +L2 = < p\ , h â€¢ V(Ap0) >n  y< p\ , Ap0 >a
= < Aiaw0+AmGlG\Amwl,hV(Ap0)>il + ^< A'mw0 + A^G xG\A~mw x, Ap0 >n (3.29)
Step 4 : Handling the terms premultiplied by p, i.e. the terms Q \ and Q2. First
MW0 e WpÂ¡h â– V(Ap)dQ 
II p, Â£ + PtfJV* II V(Ap) \2adt
by (3.5)
Ã2V\\A'Â»\\ua)Â¡~e2V\\A'Â«W\\2Qdt
+ 2P  A~mG\  L(r.a)\^e~2^ II G^w, \dt + pA1hCp,\~e~W  A^w, Â¿di
< 2p(  A_1M  2Un)+MhCp,)\â„¢eWE(t)dt + \ AWG,  L(r,o) E(0)
2
(where we have used the facts that  Aâ€œ1/4w 
29
A 3,4 w,  n < E (t) as well as (2.40))
<[ II A1*  L(n)+A/*C/>3+  A~mG,  L(r>n)] E(0) = K4E(0)
where we also have used E(t) 0 (contraction of the semigroup) and
2P
A similar computation shows that there exists a constant K5 (independent off) ) such that
Q21
Thus we arrive at
(3.30)
(3.31)
(3.32)
Lemma 3.7
Let AT6 = AT3 t t Ars (3.33)
Then a2\Â°Â°eWE(t)dtn
Jo
+ j< A1/2w0 + AmGxG\A'mwx , Ap0>n (3.34)
Proof
Follows immediately by using (3.29), (3.30), (3.32) in (3.23). â–¡
Now having dispersed with the "lower order" terms, our task is to absorb the more difficult interior
integrals.
Step 5 : The term Â¡23. We apply integration by parts in time with dv = A"mG\G\A~mwt,
u = e~2^â€˜h â– V(Ap) to obtain
f e~2^A~mGxG\A~mwtth â– V(Ap)dQ = lim Ãe23Ã< A~mG\G\A~mwt, h â– V(Ap) >n
JQ t â€”>Â«>L J o
+ itf^eWA ^G^A ^w.h â– V(Ap)dQ  f e^A^GiGÂ¡A'*1wlh â€¢ V(Ap,)dQ (3.35)
30
Part A
Since \\A ^GxG\A^Wl(T)\\2a< \\Aâ„¢GXG\A'* 2L(Q)  A ww,(r)Â¿
, A Iw0 ,
by(3.26) < A1'2G1GlA1*L(il) AWJ \\z
an argument exactly like the one used to obtain (3.27) yields:
lme~^T< AmGxG\A mwt(J), h â€¢ V(Ap(T)) >a = 0
(3.36)
Thus we only have a contribution from t = 0, i.e.,
Part B
2pJoÂ°Â°e2P'< AmGxG\Amw,, h â– V(Ap) >adt
< 2pj V2^  A mGlG*1A1/2wl \+ 2pM*J V2*  V(Ap) \dt
<(2  A1/2Gi \\2L(r il)+MhCp3)E(0)
(3.38)
where we have used (2.39), (2.40) and (3.31)
PartC The following Lemma will enable us to handle the remaining term and hence complete Step 5.
Lemma 3.8
For h â‚¬ [C2(Q)]b we have that there exists a constant Ch such that
 A~m(h â€¢ V(Ap,)) *
(3.39a)
Remark
Since Â¡D(A'a) = Hq(Q.) (see [LT.2, Appendix C]) so that [Â¡D(Am)Y = 2) (with equivalent
norms) we have
 A~iri(h â€¢ V(Ap,))  n =  h â€¢ V(Ap,)  [JD(A10)]' = IU â€¢ V(Ap,)  H^(a) (3.39b)
Claim Let z e /T2(i2), h e C2(Q). Then hz e H~2(Q).
Proof of claim (See also [LM, Vol I, Theorem 7.3, p31])
By assumption a = I zgdQ.<Â°Â° \g e Hq(Ã¼) and we want to show that
31 
< hz, g> = \Qzhgdn <Â°Â° ; ge H20(il).
Thus it suffices to show hg e ll\ (Q), if g e //o(Q). Clearly hg e H2(Q). So we must show
= 0
r
but% =/il g = 0 since# e //o(Â£2).
â€¢r *r â€˜r
Also ^\= V(/tf) â€¢ v = { , 0hg\ , â€¢ â€¢ â€¢ ,(hg\] â– [v,,v2, â€¢ â€¢ â€¢ ,vâ€ž] }
= { lhx,g + hgx>]vj + [hXtg + hgXi]v2 + â€¢ â€¢ â€¢ + [h^g + /z#*Jvâ€ž }  = 0
sinceg Â£ //Â¡Â¡(Q) implies g\ =gx< =gx I = â€¢=&. =0 â–¡
â€¢r 'r 'r â– r
Proof of Lemma 3.8
Using the above claim in (3.39b) we have
II A~m(h â– V(/ty,))  o =  h â€¢ V(Ap,))  w(n)
â€” 11 ^(ApÂ¡)  W*(Q)
iCMh\\p,\\H\a) by [LM, Vol I]
< CMhCpx   Vp,  Ã¼ by Poincare Inequality
Continuing Part C, choose e3 > 0 such that 2CAe3 < a2.
Now we compute
mw,h â– V(Ap,)dQ 
,JV< GxG\Ainw,, A~ia(h â– V(Ap,)) >a\dt
(3.40)
by (3.39a)
<  G, 2L(rjfÃ)jo00GlA1'2w,^Ã + C*Jo00e2â€˜lÃA1'V,Â¿dÃ
32
by (2.40),(*)
<  C, ll2L(r.a)Â£<Â°)+ T1!Â»"'"Â®'11 lÂ£<* +2Crf,JVÂ»' A"V *i
II G, 
E(0) + 2Che3\~eWE(t)dt
= AT8Â£(0) + 2C*e3jV2pÃÂ£(f)Ã¡r (3.41)
Again organizing our results we have
Lemma 3.9
LetK9 = K6 + K1 + Kii (3.42)
and a3 = a2  2C*e3 > 0 (3.43)
then a3 f V2p,Â£(f)<* <^9Â£(0) + g4 + < A~mw0 , h â€¢ V(Ap0) >n
Jo
+ < A^Wo+A^GiGlA^W! , Ap0 >a (3.44)
Proof
The proof follows directly by utilizing (3.36), (3.38) and (3.41) in (3.34) and taking advantage of the
cancellation of terms
< A^wo + A^CjGlAâ€™^! , h â– V(Ap0)>n< A'aGxG\A~mwx , h â€¢ V(Ap0) >n
= < AWw0 , A â€¢ V(Ap0) >n â–¡
Before beginning Step 6, the following Lemma will be needed.
Lemma 3.10
The L2norm of Ap is bounded by the initial energy Â£(0), i.e.,
llalla IM>Hq= lAâ€œit*â€˜/>Â¿s IUâ€œ2L(Q) a*v>Â£
 HA 'lli^, IUâ€œÂ».,2 s IUII^ews IU ll2L(O)Â£<0)
(3.45)
33
Proof
II Amp \\^ = < Ap , p >a = < A(Ap), p >a
by (A.O) = < Ap , Ap >a + Â¡r^pdrÂ¡rAp^dV
by (3.4a,b) = < Ap , Ap >a =  Ap 2 â–¡
Step 6 : The term g4. As in Step 5, we begin with integration by parts in time with
dv = A~iaGiG*A~mwa, u = e~2^tAp to obtain:
Â«T eWAv2G G]Av2WttApdQ = 1 Um \eVâ€˜< AmG,G\Amwt, Ap >J T
2 JQ 2rÂ»ooL J o
+ npf eWA ^G^A^W'ApdQ^i e'^A^G xG\Amw,AptdQ
JQ 2Jf)
(3.46)
Now we begin a three part process as in step 5.
Part A Using (3.26) and (3.45) it follows directly that
Â£ lim a<^C lim e~^T = 0 (3.47)
2 7>~ 2 T
So again we have only a contribution from r = 0 i.e.
2pi< A^GiG*AinWi t A/, >1 r = _ A< , Apâ€ž >n (348)
Jo 2
â€” lim
2 T>~
Part B
npjV2^ AlflG1GlA1/2wl, Ap >nd/
by (2.40), (3.32), (3.45) <
Â»IUâ€œG12L(r>Q)+Au*
2
L(Â£2)
E(0) = KwE(0)
Part C For C2 as given below, choose e4>0 such that
e4nC2 A1M  L(n)
fj e_2P'< A1/2G,G;A1/2w( , Ap, >ndQ 
(3.49)
(3.50)
â€œ f CiGjA^w,, A w(Ap,)>iJde
s Â£JV*  G,GM"â€œw, Â¿Â« + ^JVÂ® Uâ€œ(/V,) lÂ£Â«
(3.51)
34
Examine the integral on the right in (3.51)
Using the facts that ID(A m) = Â¡Â¡l (Q) [LT.2 , Appendix C] (3.52a)
and 11 A/ 112[{.2{il) < C2  /1Â£ [LM, Vol. 1] (3.52b)
2 2 2
We have  A 12Ap, ^ =  Ap,  = II 4Pt II [//g(ii)]'
= AftIIÃ^^2A^C2AaLw) \\A'Â»Pl\\2a (3.53)
so continuing (3.51) <  Gx 11 L(r, O)g (Â°)+ *2 * ^ A^L(il)i0 e^ W^P* Hadt
* Â£ II G> ll2L(r.fl)Â£(Â°) + E^2 II A W2Ua)\~e~mEWt
+ Ã4Â«c2awg1 ll2L(rÃl)Â£(0)
= KnE(0) + e4nC2  A1'4 j~e*E(f)dt (3.54)
Lemma 3.11
Let Ki2=K9 + Kl0 + KU (3.55)
and 0(4 = a3  e4nC2  A 1/4  > 0 (3.56)
Then 0(4 f e'^E^dt n
Jo
+ â€” < A 1,2w0 , Apo (3.57)
Proof
The proof follows again directly using (3.48), (3.49), (3.54) and again utilizing the cancellation of
terms:
Amw0 + A mGxG\A I/2w, , Ap0 >qj< A 1/2G1GÃA"1/2h'1 , Ap0 >n
= Y< A~mwo , Ap0 >n â–¡
Step 7 : The initial conditions.
First, I < Aâ€œ1/2w0 , h â– V(Ap0) >q l<
<
II Aâ€1/2w0 ^ + MhCpi  A3/4W! 2
11 A 1/4 2L(n) + A/*Cp3J Â£(0) = AT 13Â£ (0)
(3.58)
35
Second, ly< A , Ap0 >QI <  A 1/2w0 ^ +  Ap0 ^j
Jo
[ vvo.W! ] e ID(A). Thus taking the lim we get
Pâ€”>0
\Â°Â°E(t)dt
Jo
But E(t) is simply the squared norm of the semigroup applied to any initial conditions [ w0,>vi ] e jD(A)
and hence by continuity all [ Wo.wq ] e Z, i.e,
A I I 2
substituting E(t)=  I 7 into (3.62),
Wi I z,
weget Jo eA/JWiJ \\^dt
OO
36
Thus by Datkoâ€™s Theorem there exists C Â¡ ,8] >0 such that
IUA' II L(Z) ^Cle~s't andhence II II l(Z) ~ C'e ^
therefore by taking C Cj and 8 = 25x we have
Â£(0=IUa'Q ills IUA,llÃ.(Z)ll";lllsceÂ£(0)
which is exactly our uniform stabilization result. â–¡
APPENDIX A
BASIC IDENTITIES
Let/, g be scalar functions, and h (x) e C2(Q) vector field. The following identities will be needed:
Greenâ€™s Second Theorem j^A/gdÂ£2  j
A^/aQ=J
%gdr f f/dr
rdv Jr^v
(A.O)
Greenâ€™s First Theorem f AfgdQ = f ^gdT f
>r3v6 JfÃ
VfVgdQ.
(A.l)
Divergence Theorem J^div/idÃl = J
â€¢vdr
(A.2)
div(/7i) = h â€¢ V/+/div/i
(A.3)
fh â€¢ V/= 1* â€¢ V(/2)
(A.4)
 A â€¢ VfdSl = j (divfh  fdi\h)dÂ£l =
J^vdr
j^/div/idQ
(A.5)
V/V(/2 â– V/) = //V/V/+/I
V(l V/l2)
(A.6)
dh\
3/ii
3x]
3xb
where //(x) 
dhn
a/zB
dxi
3xb
V/ â€¢ V(/div/i) = /V(div/i) â€¢ V/+ I V/l 2divfi (A.7)
fh â– V/dÂ£l = â€¢ V/dr Jr I v/l2/, â€¢ vdr
 J //'V/ â€¢ Vfdi2 + I V/l 2d\\hdQ. (A.8)
37
38
Proof of (A.8):
Multiply A/by h â€¢ V/, integrate j^dQ, and successively apply (A.l), (A.6), and (A.5).
j^Afh â€¢ V/dQ = JÂ£/i â€¢ v/dr  Jq V/ â€¢ v(* â€¢ YfldQ
= J JÂ£
= J JÂ£/t â€¢ V/dr  J^//V/ â€¢ Y/dÂ£2  yJr i V/l 2/t â€¢ Vdr + yI V/l 2diWidQ
Another useful identity arises by applying (A.l), (A.7) to get:
J^Aj/div/idQ = j^4y/div/idr  V/ â– V(/div/i)dQ
= J^/div/idr  J^/V(divA) â€¢ V/dQ  I V/l 2divfcdQ
(A.9)
APPENDIX B
TO HANDLE DIFFERENCE OF ENERGY TERM
We multiply both sides of our equation pâ€ž + A2p = A mGÂ¡G\Ap, by e ^Apdivh with 0
integrate both sides over Q: j^Q = JqJ0 <^t^
pâ€ž term: First integrate by parts in time, and then apply (A.9) to get:
f plte~2^Apdi\hdtdil = limie2^^ p,, Apdiv/i >n
r^oL J o
+ 2$^ ^e'2^1 App,di\hdQ  jAp,p,di\hdQ
= lim [e_2p,< p,, Apdiv/i >nl + 2pf e~2^â€˜App,divhdQ
r>~L Jo JQ
 j^e 2^t^ptd\\hd'L + j^e 2^p,V(div/i) â€¢ VpidQ + j^e Wp, \2divhdQ
'Q
(B.l)
A2p term : First apply (A.l) and then (A.7) to get:
j e 2^tA(Ap)ApdivhdQ = j^e ApdivhdI.  2^â€˜V(Ap) â– V(Apdi\h)dQ
= f e~2^ ApdivhdI,  f e^ApVfdiv/i) â€¢ V(Ap)dQ  f e~2^ I V(Ap)\2di\hdQ (B.2)
dv JQ JQ
Now isolating difference of energy term and multiplying by ~ we get:
4f e 2p/ I Wp, 121 V(Ap) 121 di\hdQ = â€”lim Ãc 2^,< />, , Apdiv/i >n
2 L J 2 J o
 pj^e^Ap/^div/idQ + y _2piJ^p,divhd'L yj^e2(5Ip,V(div/i) â– Vp,dQ
 yj^e2^ Apdiv/idE + yj^e^ApVidiv/i) â€¢ Vp,dQ
 ^ ^e'2^1 A~mG xG\Ap,Apd\\hdQ
(B.3)
39
APPENDIX C
TO OBTAIN GENERAL IDENTITY
We multiply our equation pâ€ž + A2p = A mGlG\Ap, by e â€¢ V(Ap) with 0 < P < 1, and
integrate both sides over Q as in Appendix B.
pâ€ž term: First integrate by parts in time, then successively apply (A.5) and (A.3) to get:
f f Pae ,2p/h â– V(Ap)dfdÂ£2 = lim e 2p/< p,, hV(Ap) >Q1
JqJ0 r^L J o
+ 2P f eWp.h â€¢ V(Ap)dQ  \ eWpth â– V(Ap,)dQ
JQ JQ
= lim [Q1 + 2p e~2fapth â– V(Ap)dQ
r>~L Jo JQ
 J e^AptPth â€¢ vdZ +1 e~2^1 Ap,div(p,h)dQ
^ Q
 lim
Tâ€”>Â°Â°L
[e~2tu< p,, hV(Ap) >Q1 T + 2pf e^p'h â– V{Ap)dQ\ e~^1 Aptp,h â– vdl
i Jo jo Jy.
f@e ^â€˜Ap,h â€¢ Vp,dQ + jne 2^â€˜Ap,p,divhdQ
Q
(C.i)
Next, we will apply (A.8) to j e 2^Ap,h â€¢ Vp,dQ and (A.9) to j^e 2^â€˜ Ap,p,di\hdQ simultaneously to
obtain:
U p,,e
= lim Iâ€Q1 T + 2pf e^p.h â– V(Ap)dQ  f e^Ap.p.h â– vdl
r*~L J o JQ
261 dPt
2 JQ
^ h â– Vp,dl.  2pÃ I Vp, 12h â– vdl  J Â« ^H^Pt' Vp,d(2 + ^jne 2PÃ' ^Pâ€˜'2div/zdQ
dp,
d,qi\naÂ¿j i
+ J e 2pÃ^p,div/zdZJ e 2p,p,V(div/i) â€¢ Vp,dQ  J e ^ I Vp, 12di\hdQ (C.2)
40
41 
A2p term: Applying (A.8) we get:
f p)h â€¢ V(Ap)dQ â– V(Ap)dl
2 4e_2f5Ã'V(AP)'2/1 â€œ ÃQe ^â€˜H V(Ap)' V(Ap)dC + iig e~m 1 V(Ap) 12divW2 (C.3)
Now we put all boundary terms on the left side, all interior terms on the right side and simplify to obtain:
f e ^AptPth â€¢ vdÂ£ + J e â€¢ Vp,dSe 2^IVp,l2/zvdl
+  e~2^^p,divhdZ + /i â€¢ V(Ap)dZ ^ W(&P)\2h â€¢ vdZ
=  lim [e_2^< p,, /rV(Ap) >nl  2fif e_2pip,A â€¢ V(Ap)dQ
T>~ L Jo
+ 2(li//V(Ap) â€¢ V(Ap)dQ + f@e ^//Vp, â€¢ Vp,d(2
+ J e ^(VCdiv/i) â€¢ Vp,d(2 +
2PÃ
2JG
IVp,l2 IV(Ap)l:
div/idg
 f^e~2^AmGxG\Ap,h â– V(Ap)dQ
Before proceeding we make some simplifications which arise due to the boundary conditions.
(C.4)
Since p I = 1 = 0 it is immediate that
Iy dV ly
dp,
p\~ av
= o
Also Vp^rand Vp,^r imply I Vp I = I 1 = 0 on T
oV
and I Vp, I = I 1 = 0 on T
dv
And by utilizing h (*) = (xx0) the radial vector field, we have that
// (*) = 1 identity matrix
 divA = n = dim Â£2
V(divA) = 0
(C.5)
(C.6)
(C.7)
42
Now inserting (B.3) for the difference of energy term in (C.4) and utilizing the simplifications (C.5), (C.6)
and (C.7), we finally arrive at the desired identity:
1 V(A/,) 12/1 â– vdS + 2 ~vt^dÂ¡?'ApdT'
= lim[e2pi< p,, hV(Ap) >Q1 ^ lim [e2pi< p,, hp >nl
r*~L Jo 2 tâ€”xÂ»l J o
 2pf e^p,h â– V(Ap)dQ  ntfeWp.ApdQ
+ f e2pÃ I Vp, \2dQ+[ e~W I V(Ap) 12dQ
JQ JQ
[ e^A^G^Ap.h â€¢ V(Ap)dQ  â– Â£[ (C.8)
Je 2 J<2
REFERENCES
[B] A.V. Balakrishnan, "Applied Funtional Analysis," 2nd edition, SpringerVerlag, New
York/Berlin, (1981).
[BT.l] J. Bartolomeo and R. Triggiani, Uniform stabilization of the EulerBemoulli Equation With
Dirichlet and Neumann Boundary Feedback, Report, Dept, of Applied Mathematics, UniverÂ¬
sity of Virginia, 1988.
[C.l] G. Chen, Energy Decay Estimates and Exact Controllability of the Wave Equation in a
Bounded Domain, Journal de Mathematiques PurÃ©s et Appliquees (9) 58 (1979), 249274.
[C.2] G. Chen, A Note on the Boundary Stabilization of the Wave Equation, SIAM J. Control 19
(1981), 106113.
[D.l] I. Datko, Extending a Theorem of Liapunov to Hilbert Spaces, J. Mathem. Anal, and Applic.,
32 (1970), 610613.
[DLT] G. DaPrato, I. Lasiecka and R. Triggiani, A Direct Study of the Riccati Equation Arising in
Hyperbolic Boundary Control Problems, J. Differential Equations 64 (1986), 2647.
[FLT] F. Flandoli, I. Lasiecka and R. Triggiani, Algebraic Riccati Equations With Nonsmoothing
Observation Arising in Hyperbolic and EulerBemoulli Equations, Annali di MatemÃ¡tica Pura
e Applicata, to appear.
[H.l] L. F. Ho, Obscrvabilite' Frontiere de Lâ€™equation des Ondes, CRAS, 302 (1986), 443446.
[L] N. Levan, The Stabilization Problem: A Hilbert Space Operator Decomposition Approach,
IEEE Trans Circuits and Systems AS2519 (1978), 721727.
[L.l] J. Lagnese, a paper presented at the International Workshop held in Vorau, Austria, July 10
16,1988.
[L.2] J. Lagnese, "Boundary Stabilization of Thin Elastic Plates", to appear.
[L.3] J. Lagnese, Uniform Boundary Stabilization of Homogeneous, Isotropic Plates in "Lecture
Notes in Control Science #102â€, SpringerVerlag, New York, pp. 204215, 1987 Proceedings
of the 1986 Vorau Conference on Distributed Parameter Systems.
43
44
[L.4]
[L.5]
[L.6]
[L.7]
[L.8]
[LM]
[LL.l]
[LLT.l]
[LT.l]
[LT.2]
[LT.3]
[LT.4]
[LT.5]
[LT.6]
[LT.7]
J. Lagnesc, Decay of Solutions of Wave Equations in a Bounded Region with Boundary DissiÂ¬
pation, J. Differential Equations 50 (1983), 163182.
J. Lagnese, A Note on Boundary Stabilization of Wave Equations, SIAM J. Control, to appear.
J. L. Lions, Exact Controllability, Stabilization and Perturbations, SIAM Review, March 1988.
J. L. Lions, A Resultat de Regulante (paper dedicated to S. Mizohata), "Current Topics on ParÂ¬
tial Differential Equations," Kinikuniya Company, Tokyo, 1986.
J. L. Lions, "Controle des Systemes Distribues Singulicrs", Ganthier Villars, Paris, 1988.
J. L. Lions and E. Magenes, "Nonhomogeneous Boundary Value Problems and Applications,"
Vols. I, II, SpringerVerlag, BcrlinHeidclberg, New York, 1972.
J. Lagnese and J. L. Lions, "Modeling, Analysis and Control of Thin Plates," Masson, Paris,
1988.
I. Lasiecka, J.L. Lions and R. Triggiani, Nonhomogeneous Boundary Value Problems for
Second Order Hyperbolic Operators, Journal de Malhematiques PurÃ©s et Appliquees 65
(1986), 149192.
I. Lasiecka and R. Triggiani, Exact Controllability of the EulerBemoulli Equation with
L2(I)control Only in the Dirichlet Boundary Conditions, Atti della Accademia Nazionale dei
Lincei, Rendiconli Classe di Scienze fisiche, 81 (August 1987).
I. Lasiecka and R. Triggiani, Exact Controllability of the EulerBemoulli Equation with ConÂ¬
trols in the Dirichlet and Neumann Boundary Conditions: a Nonconservative Case, SIAM J.
Control and Optimization, to appear.
I. Lasiecka and R. Triggiani, A Direct Approach to Exact Controllability for the Wave EquaÂ¬
tion with Neumann Boundary Control and to an EulerBemoulli Equation, Proceedings 26th
IEEE Conference, pp. 529534, Los Angeles, December 1987.
I. Lasiecka and R. Triggiani, Uniform Exponential Energy Decay of the Wave Equation in a
Bounded Region with L2(0,Â°Â° ; L2(D)fccdback Control in the Dirichlet Boundary Conditions,
J. Diff. Eqts. 66 (1987), 340390.
I. Lasiecka and R. Triggiani, Regularity Theory for a Class of Nonhomogeneous Euler
Bemoulli Equations: a Cosine Operator Approach, Bollettino unione matemÃ¡tica Italiana, (7)
2B, December 1988.
I. Lasiecka and R. Triggiani, Exact Controllability of the EulerBemoulli Equation with BounÂ¬
dary Controls for Displacement and Moments, J. Mathem. Analysis and Applic., to appear.
I. Lasiecka and R. Triggiani, Uniform Exponential Energy Decay of the EulerBemoulli EquaÂ¬
tion on a Bounded Region with Boundary Feedback Acting on the Bending Moment, to
appear.
45
[LT.8] I. Lasiecka and R. Triggiani, A Cosine Operator Approach to Modeling L2(0,T ;L2(T))
boundary Input Hyberbolic Equations, Applied Mathem. and Optimiz. 7 (1981), 3593.
[LT.9] I. Lasiecka and R. Triggiani, Regularity of hyperbolic equations under L2(0,T ;L2(T))
Dirichelt Boundary terms, Applied Mathem. and Optimiz. 10 (1983), 275286.
[LT.10] I. Lasiecka and R. Triggiani, A Lifting Theorem for the Time Regularity of Solutions to
Abstract Equations with Unbounded Operators and Apllications Through Hyperbolic EquaÂ¬
tions, Proceedings American Mathematical Society, to appear.
[LT.ll] I. Lasiecka and R. Triggiani, Riccati Equations for Hyperbolic Partial Differential Equations
withL2(0,r ; L2(r))Dirichlet Boundary Terms, SIAM J. Control and Optimiz. (5) 24 (1986),
884926.
[LT.12] I. Lasiecka and R. Triggiani, Exact Controllabilty for the Wave Equation with Neumann BounÂ¬
dary Control, Appl. Math, and Optimiz., to appear.
[LT.13] I. Lasiecka and R. Triggiani^Sharp Regularity Theory for Second Order Hyperbolic Equations
of Neumann Type. Part I: L Nonhomogenous data, to appear.
[LT.14] I. Lasiecka and R. Triggiani, Sharp Regularity Theory for Second Order Hyperbolic Equations
of Neumann Type. Part II: The General Boundary Data, to appear.
[LT.15] I. Lasiecka and R. Triggiani, Trace Regularity of the Solutions of the Wave Equations with
Homogeneous Neumann Boundary Conditions and Compactly Supported Data, J. Mathem.
Anal, and Applic., to appear.
[LT.16] I. Lasiecka and R. Triggiani, Nondissipative Boundary Stabilization of the Wave Equation via
Boundary Observation Journal de Mathemaliques PurÃ©s et Appliquees 63 (1984), 5980.
[LT.17] I. Lasiecka and R. Triggiani, Dirichlet Boundary Stabilization of the Wave Equation via BounÂ¬
dary Observation J. Mathem. Anal, and Applic., 87 (1983), 112130.
[P] A. Pazy, "Semigroups of Linear Operators and Applications to Partial Differential Equations,"
SpringerVerlag, New York, (1983).
[R.l] D.L. Russell, Exact Boundary Value Controllability Theorems for Wave and Heat Processes in
Star Complemented Regions, in "Differential Games in Control Theory," Dekker, New York,
1974.
[R.2] D.L. Russell, A Unified Boundary Controllability Theory for Hyperbolic Partial Differential
Equations, Stud. Appl. Math. 3 (1973), 189211.
[R.3] D.L. Russell, "Mathematical Models for the Elastic Beam and Their Controltheoretic ImplicaÂ¬
tions; in Semigroups Theory and Applications, Pitman Research Notes in Mathematics 152,
1986.
[S.l] W. Symes, A Trace Theorem for Solutions of Wave Equations, Mathematical Methods in the
Applied Sciences 5 (1983), 131152.
46
[T.l] R. Triggiani, Wave Equation on a Bounded Domain with Boundary Dissipation: an Operator
Approach, J. Mathem. Anal, and Applic., to appear.
[T.2] R. Triggiani, "Exact Controllability of Wave and EulerBemoulli Equations in the Presence of
Damping," Proceedings of International Conference on Differential Equations held in
Columbus, Ohio, March 2115, 1988.
[T.3] R. Triggiani, "Exact Controllability on L2(Â£l) x//_1(Q) for the Wave Equation with Dirichlet
Control Acting on a Portion of the Boundary, and Related Problems", Applied Math, and Opt.
18 (1988), 241277.
[T.4] R. Triggiani, A Cosine Operator Approach to Modeling L2(0,T ; L2(D)boundary Input ProbÂ¬
lems for Hyperbolic Systems, pp. 380390, SpringerVerlag, 1978. Proceedings 8th IFIP
Conference, University of Wurzburg, W. Germany, 1977.
[T.5] R. Triggiani, Lack of Uniform Stabilization for Noncontractive Semigroups, Proceedings
Amer. Mathem. Soc., to appear.
[T.6] R. Triggiani, Finite Rank, Relatively Bounded Perturbations of Semigroup Generators. Part III:
A Sharp Result of the Lack of Uniform Stabilization, Proceedings of First Conference on
Communication and Control Theory, Washington, D.C., June 1987, to appear.
[T.7] R. Triggiani, On the Stabilization Problem in Banach Space, J. Mathem. Anal, and Applic., 52
(1975), 383403.
BIOGRAPHICAL SKETCH
Jerry Bartolomeo was bom June 17, 1960 in Brooklyn, New York. In 1972, he moved to HomeÂ¬
stead, Florida, and graduated from South Dade High School in June 1978. After entering the University of
Florida in Fall 1978, he received his B.A. in mathematics in 1981, and then his M.S. in mathematics in
1984. While working on his Ph.D., he spent a year teaching at the University of Virginia, School of
Engineering. He is currently living with his wife and son in Miami, Florida, and working as an instructor
at Nova University in Ft. Lauderdale, Florida. Upon receiving his Ph.D. in mathematics, he plans to conÂ¬
tinue working in the area of Boundary Control for Partial Differential Equations and continue his involveÂ¬
ment in program development at Nova.
47
I certify that I have read this study and that in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor
of Philosophy.
Roberto Triggiani, Chair
Professor of Mathematics
I certify that I have read this study and that in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor
of Philosophy.
fjJb
Irena Lasiecka
Professor of Mathematics
I certify that I have read this study and that in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor
of Philosophy.
I certify that I have read this study and that in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor
of Philosophy.
Vi.wypyk
Murali Rao
Professor of Mathematics
I certify that I have read this study and that in my opinion it conforms to acceptable standards of
scholarly presentation and is fully adequate, in scope and quality, as a dissertation for the degree of Doctor
of Philosophy.
Carmen Lanciani
Professor of Zoology
This dissertation was submitted to the Graduate Faculty Department { OR School } of ... in the
College of... and to the Graduate School and was accepted as partial fulfillment of the requirements fro
the degree of Doctor of Philosophy.
December 1988
Dean, College of Mathematics
Dean, Graduate School
UNIVERSITY OF FLORIDA
3 1262 08554 0259

