Uniform stabilization of the Euler-Bernoulli equation with active Dirichlet and non-active Neumann boundary feedback controls


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Uniform stabilization of the Euler-Bernoulli equation with active Dirichlet and non-active Neumann boundary feedback controls
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vi, 47 leaves : ; 28 cm.
Bartolomeo, Jerry, 1960-
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Boundary layer control   ( lcsh )
Bernoulli polynomials   ( lcsh )
Hilbert space   ( lcsh )
Differential equations, Partial   ( lcsh )
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Thesis (Ph. D.)--University of Florida, 1988.
Includes bibliographical references.
Statement of Responsibility:
by Jerry Bartolomeo.
General Note:
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University of Florida
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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.



ACKNOW LEDGEMENTS ......................................................................................................... ii

ABSTRACT .................................................................................................................. .................... v



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 WELL-POSEDNESS AND STRONG STABILIZATION .......................................... 10

2.1 Preliminaries, Choice of Stabilizing Feedback ............................................ ....... 10
2.2 W ell-Posedness 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


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



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 Euler-Beroulli 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 Co-semigroup which decays uniformly exponentially

to zero as t--o 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

(x-xo) 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.


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 Euler-Bernoulli 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)
-(t, a) = g2(t, a) on (0,T] x r (d)

with non-homogeneous 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. [L-L.1], [R.3] and references cited therein) in the theory of plate equations, of

which the Euler-Bernoulli equation (1.la) is a canonical model, presumably stimulated by two main


i) renewed studies in the dynamics, feasibility, and implementation of so-called 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] [L-T.8] [L-T.9] [L-L-T.1] and of Neu-

mann type [L-T.8], [L-T.9], [L-T.10], [L-T.13], [L-T.14], [L-T.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], [L-T.1],

[L-T.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 [L-M]. Our interest in the present work is on the problem of boundary feedback uniform stabilization

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 [L-T.l], [L-T.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 [F-L-T.1]. Mathematically the present work is

guided by and partially rests upon techniques developed in two main sources:

(i) the studies of exact controllability [L-T.1], [L-T.2] for problem (1.1);

(ii) the study of uniform stabilization of the wave equation with boundary feedback in the Dirichlet

boundary conditions [L-T.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], [L-T.4. sect 4];

b) regularity theory for hyperbolic equations in [L.8], [L-T.8], [L-T.9], [L-L-T.1], as well as

corresponding exact controllability theory [L.6], [L-T.12], [H.1], [T.2], [T.3];

c) exact controllability results for Euler-Bernoulli equations with different boundary conditions

[L.6], [L.7], [L-T.6], [L-L.1]; and, finally,

d) corresponding optimal quadratic cost problems [D-L-T.l], [L-T.11 [F-L-T.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], [L-T.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], [L-T.1] [L-T.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 [B-T.1].

We finally point out that other uniform stabilization problems for plate equations have been/are being

considered following Lagnese-Lions recent monograph [L-L.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-

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 [L-T.1], [L-T.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 H-1 (Q) [L-T.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, self-adjoint operator defined by

Af-A2f; I(A)=H4 (Q)fH(Q) (1.2)
With the operator A defined as such, it then follows that [ L-T.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")
Ilfllo(A,)= IIA'flla = IlfllH.(O= { IVfl2dQ } (1.5)
where the last equivalence follows by Poincare inequality.
Similarly forf E ID(A)
llfll ( =I A3'f { = nV(Af/I2d } (1.6)

As suggested by [L-T.1], [L-T.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 norm-preserving 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 well-known 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], [L-T.1],

[L-T.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 c-d) produce a (feedback) Co-semigroup 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 c-d) produce a (feedback) Co-semigroup 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 [B-T.1] in the general case that the following

choices of Fi and F2

g = F (w,)= -k (x)G*A-law, = -k (x)GAA-mw, =-kI(x) A() ~ (1.12)

2 = F2(w,) = -k2(x)A2GAA-Mw, = k2(x)A2A(A-w,) (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 well-posed in

the semigroup sense in Z and the Z-norm of all of its solutions originating in Z decreases as t -+ +0 ( this,

however, does not say that such Z-norms 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'-'(), self-adjoint on L2() (1.15)
so that
l|AgL'( 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)
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 [L-M, 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 [L-T.2, Lemma 2.0 and Lemma 4.0, respectively]

G*IAf = l fE D(A)
0-V Ir



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(A-3aw)I on E (d)

Using the techniques of [T.4], problem (1.24) can be re-written more conveniently in abstract form as

dt ;II=AI

A= 0 I
A= I-A -A[kxG1GIA-1 +k2G2A2G2A-1']

o)(A) = yeZ: AyEz

A more explicit description of D(A) will be given below. Our main results are as follows.




Theorem 1.1

(i) Well-posedness 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 (Q-A) = Z for X > 0. Thus, by Lumer-Phillips 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

k1G*A-'aw,=k E L2((0,o) ; L2(P)) (1.27)

k2A2GA-2w, = k A2A(A-mw,) e L2( (0,*) ; H-1 (T) (1.28)
more precisely

o IIG k A-w, 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 [B-T.1] in the general


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 Me-u 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 (x-xo) -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 (x-xo). This will be done in [B-T.1]. Also Theorem 1.2--the general result

with two feedback operators and no geometrical condition--will likewise appear in [B-T.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 techniques--based on the operator model (1.25), (1.26) and the Nagy-Foias-Foguel

decomposition for contraction semigroups--can be carried out along the lines of arguments first used for

boundary control problems for second order hyperbolic equations in [L-T. 16], [L-T.17], [L-T.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 eigen-values along the imaginary axis. This is Lemma 2.7, where only three

homogeneous boundary conditions (2.31b-c-d) are in place for the 'eigen-problem' (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 [L-T.16], [L-T.17], [L-T.4], [T.1], the

corresponding 'eigen-problem' 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 [L-T.1], [L-T.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 3--which 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



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 closed-loop 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, self-adjoint 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>r-r

=< A, Af>n= II Afll


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 [L-T.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

Ilxll(Ai)= IIAx lln; IlxltII r((P = IIA-xlln a,p >0 (2.5)
Below we state the regularity result as well as the exact controllability result.

Theorem 2.0 (Regularity) [L.71, rL-T.21

Consider the problem (2.1) subject to [w0,w1] EZ, ge L2((0,T); L2(r)),

g2 E L2((0,T); H-t'()). Then the map {wo,w1,gl,g2) [w(t),w,(t)] e C([0,T] ;Z) is continuous


Theorem 2.1 (Exact Controllability) fL-T.21

(i) Assume there exists a point xo E R?" such that (x-xo) 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 + IA-34w,(t) I (2.6)
Next we seek a candidate g which at least produces 0, i.e., energy "decrease." This does not,

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 well-posedness in Z, with g2 = 0. Then since w, e [ID(A )]', it follows that

A-3w, = A -3A- 4w, e D(A 3) a V. Therefore, A -w, satisfies the required boundary conditions.

A-3a2w =a(A-sw,) = 0 (2.7)

By writing E(t)= < A-"w A -w >Q + < A-4w,, A 4w, >0 and differentiating with respect to t

we have that

dE= < A-'w, < -1/4A- w, A-4w, >a
by (2.1a)

= < w, A-U2w > < Aw, A-mw, >a = < w, A-1aw, >Q < A(Aw), A-3w, >0

by (A.O)

= < w A-2w, >a < (Aw) A-'w, >r < Aw, (A- w,) >r + < Aw A(A-32w) >0

by (2.7)

= < w A-'w, >n < Aw, A(A-mw,) >n


by (A.0)

= < w, A-nw, >0 < A(A- w,) >r < w, [A(A-awt)] >r + < w, A(A-mw) >n

= < w, A- > + < w, [A(A-w,)] >r < w, A-w, >a = < w -[A(A-w)] >r
ov dv

Therefore, by selecting the simplest choice

wI =g =- A-[A(A-2w,)] (2.8)
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 [L-T.1],[L-T.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 [L-T.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(A-w,)] = -GIA(A-ww,)= -G*A-a2w, (2.11)

Using elliptic theory [ L-M, 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)


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 Well-Posedness 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*,A-aI (2.16)

More explicitly if ye D(A) then we can write

.A Io 1 + GIG'A-1 y2
Ay -AOI il (2.17a)

Thus,)(A)= { [y y21E Z:y2 [)(AI )]and-A [y, +G G*IA-Lay2] [D(A4)]' ,i.e.,

yi + GGIA-my2 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)]'.



Let z E Z, then using below the skew-adjointness 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")]' = < A-4AG1G*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


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 Lumer-Phillips Theorem [P], A generates a Co-semigroup 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(^ A-11 (2.19a)
where V(X) = [I + XG1G IA-1 + 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.


(iii) If the domain l is such that there exists a point x0 E R" such that
(x-xo)-v>y>0 onr (2.20)

then R(1; A) is well-defined 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)


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 A-yz2) + y2 = z2 [D)(A)]' (2.22b)
for y e D(A). We apply A-1 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

A-4V(X)A = = I + A-G ,1GA-~' + 2A-' is boundedly invertible on L2(Q) (being self-adjoint, strictly

positive on L2() ) with inverse

A -' V-1 (X)A/4 e L(Q) (2.24)
Thus, from (2.23)
Y2 = V-1'() (A-'z2 1) e [(D(A 14)]' (2.25)
which then inserted in (2.22a) yields

Y = zi- L 1 + V-' 2 (2.26)


Then (2.19a) follows from (2.25) and (2.26). Note that from (2.22b) and (2.25) that
Yi + G1GAA-l2y2 =A-zz 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


A-"4(I V- (X) )A 1 (2.28a)
A -"4 V-1 ()A-14 = A-" V-1 (X)A '1A-/2 (2.28b)
A -34V-1 ()A 1' = A -1A-1 V-'1 (XA 1 (2.28c)
A-4 V-1 (X)A -14 = A -UA-l'V-1 (X)A A -l/2 (2.28d)

First, compactness of the operators (2.28b-c-d) on L2(0) is plain from (2.24) and A -a a> 0 being

compact on L2(9). For (2.28a) apply V-1 (X) on (2.19b) so that

I = V-'1() + XV-'(X)G G*A-2 + 2V-1 ()A-1
and then
A-"4 [I V-1 () ] A 4 = A -4V-1 (X)G iGIA-"4 + X2A- -1 ()A-3/4
= a -'"V- (X)A A-'G G*IA -'4 + 2A-4V- (X)A 14A-1
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< G1GAA-lx, x >[D(A")]' -r2< A-1x, x >[ID(A14)]'
= < A-ax x >O +ir II G;A-mx 1| r2< A-x 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] = [AA-A3x)lr=0 (2.31)
Ir 0-IV


Also, we have that by (2.30), A-'2x = r2A-Mx, 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)

=0 on r (c)

| 0 on r (d)

has only the trivial solution a 0.


1. Since A-3x = A-3e, = r-Me, ; (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 fourth-order elliptic operator in (2.33).

3. Recall that as given in Appendix C, if h(x) = (x-xo) 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)


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), (2.34a-b), (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
=J IVI2h-vdr+(- +1) I VI2d2

Butsinceol = 0wehave I V -I = I =0. Therefore,

1 Oh V(AO)df = (- + 1) X IV,122d (2.37)
Sfa 2 k-Z
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 Co-semigroup of contractions eAt on Z, it follows that


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*A-2w, E L2((O,oo); L2(r)) and in fact

I GIA-m = 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< A|Atwj, e AjI:wI >Z=R>< A W,(1)1 ,,,,(t)W >Z

via the proof of Lemma 2.5

=- II G*A-aw, 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:


1iM T d 1 'd (t)dt
f i 11 A-'2w, II dt E()dt lim E(t)d
2T-f- o d t 2T, dt
1 1
-E (0)- lim E(T) < E (0)
2 2T-o-
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 Nagy-Foias-Fogel decomposition theory [L]. Since

eAt is a Co-contraction 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 Co-unitary 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


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(A-mw,)] 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 A-W 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. A-wt(t) e Lf 2), it follows that

A p (t)= AwA-mw,(t)= A-'3w,(t)e L2(Q). Thus



p(t) ID(A")= {fE H3(Q):f'r=- =01

So in particular

p\ =0 (a)
=1 0 (b)

Now recalling (2.15) we have using (3.3)
p, =A-saw, =A-a[-Aw -AG 1iGA-2w,] (3.5)
= -A-raw -A-aG1GGA-1aw, (3.6)
Differentiating once more in time and using A-'aw, = Ap and A-2wa = Ap, we get
P. = -Ap-A-aG 1G 'Ap, (3.6)
and hence
p,+A2p = -A-aG IG'Ap, in Q (a)
p(0,.)=po =A-MW, p,(0,)=pl =-A-lawo-A-1naG 1GA-l'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) = (x-xo)


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) < Ce-EE (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

Datko's theorem [D.1].


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

e-0th V(Ap) and e-2tApdivh (recall (C.8)):

Je-2pt (L)h- V(Ap)d -' e-2t I V(Ap) I2h vdE+ n e-2t- ApdL

= J e-P I Vp, 12dQ + J e-2 I V(Ap) 12dQ 2p e-2lptp,h V(Ap)dQ

-n BQe-l2p,ApdQ Qe- A-aGiGAl p,h -V(Ap)dQ

n 2Qe-*A-mGiGIApApdQ lim e-2< p, hV(Ap) >Q
2 Q T-L.- 0i T
n lim e-t< p,, Ap > T (3.11a)
2 rT--L To

=- e-2tIVp 12dQ+ Qe-2tIV(Ap)12dQ-Q1 -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)


2 2
First by trace theory [L-M] II Ap II, -< C II Ap II Hm(a)
< 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)
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 e-2 I V(iAp) 12h vd + -fe-2 Apd

f _2~ 2 a() 2d+eMbf e-2 lV(Ap)12dZ- e-2 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:


(L.H.S.) of (3.11) < K1E(0) + -' e-WE(t)dt (3.14)
2 n
where K = + -
el 2e2
Finally, utilizing (3.14) in (3.11) and isolating the important terms we obtain:
Qe-2t IVp1dQ + Qe-2t IV(Ap)12dQ K E (O) + -2nC -2eE (t)dt
+Q1+Q2+Q3+Q4+L1 +L2 (3.15)

Ste 2: Isolating the energy integral oe-tE (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 [L-T.2] (3.16)
Similarly p E D(A 4) implies the existence of Cp3 > 0 such that

|| IV(Ap)| 11 >Cp3 JJAp |[L-T.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-'GIGlA-uZw, > a+Cpl I A- LGIG A-2w, 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)
and K2=Cpj II A-'G, II L(r.a) (3.21)
Then e-2P I Vp, 12dQ + Qe-2 IV(Ap)I2dQ >a e-0E(t)dt -K2E(O) (3.22)



Q e-I V(Ap)12dQ + Qe-IVp,12dQ
by (3.18), (3.19)

SCps Je d + e-1 A-, w I2 dt Cp1 oe- II A-1 GG*iA-U2w, II2dt

>,afe -20t[ IlA A4wI 2+ hiA-"w, 2II dt-Cpl JiA-UGi [I r.)l II GA-'2w, Idt
>by(2.40) aiJoe-P 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

ac2Je-2PtE(t)dt 5K3E(0)+i+LZ2+Ql+Q2+Q3+Q4 (3.23)
K3 =K + K2 (3.24a)
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 [L-T.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)


Now letting M = I h 12 we have

lim e-21 < p,(T), h V(Ap(T)) >a lim e-2 I p,(T) 2 + M 11 A31'p(T 112
by (3.5), (3.9)
m e-l T 2 II A L(4 IIA-w(T) IL
2 12
+2I A-'GiG*A-'" II L(o) II A-wr(T) II + ME(0)

by (3.26)

< lim e-A CE(0) + 2 11 A-aGIGA- 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
< A-"lwo +A-l'GiGAA-l'aw, h V(Apo) >a+ -< A- Awo +A-GG w, Apo >n (3.29)

Ste4 : Handling the terms premultiplied by 0, i.e. the terms Q and Q2. First

Q I =2Pj e-p,h -V(Ap)dQl

< PjIe-* I P, 12 + 1 Mhf e-t II V(AP) I dt
by (3.5)

<23 II Al' II L(n)Je I-2I AII A-w 12dt

+ 2 II A-G, II L(r. o)Jfe- II G A-r"w, 2d + MACP3 e-2 II A-4w, I\dt
2 2
< 2( 11 A-"4 II L()+MhCp3)0 e-2 E(t)dt + II A-mGI II (r,a) E(0)
(where we have used the facts that I A -"w II 5E(t) and similarly


II A-3w, 112 E(t) as well as (2.40))
2 2
<[ 1 A-1II 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
fe-dt =- (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 +A-IRGIGGA-12wl Apo >n (3.34)

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


Step : The term Q3. We apply integration by parts in time with dv=A- 'G1G*A-'w,,

u = e-2th V(Ap) to obtain

Q T-+-[L 0
+ e-2 e-aGIGA-Gaw,,h V(Ap)dQ = lim [e-p < A-2G I GA-Law, h V(Ap) >n (3.
+2a2J e A A-aG1G*IA- 'awh V(Ap)dQ Je-A-aG1Ga A-I wh V(Ap,)dQ (3.35)


Part A
Since A-'G,G*A-aw,(T) I < II A-'GG*IA-^ 1 L() | IIA-'w,(T)

by(3.26) < IIA-lGiG*IA-"M IL(n)2 IIAJw 0

an argument exactly like the one used to obtain (3.27) yields:
lime-2PT< A-i`G GI*GA-"aw,(T), h V(Ap(T)) >Q = 0 (3.36)
Thus we only have a contribution from t = 0, i.e.,

lim [e-2< A-mGIGIA- w,, h V(Ap) >] =-< A-2GGIA-'2aw, h -V(Apo) >0 (3.37)
T-+-L 00

Part B
2PJ e-2 < A-IGIG A-"2w h -V(Ap) >ndt

< 2I e-2Mt A-1G1G*GA-aw, II Idt + 2P3M o e-t II V(Ap) I|dt

<(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 A-2(h V(Ap,)) |I1 CI, || Ap, |1| (3.39a)


Since D(A 1')= H2() (see [L-T.2, Appendix C]) so that [ID(A1')] '=H-2() (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 H-2(0), h e C2(0). Then hz e H-2(4).

Proof of claim (See also [L-M, Vol I, Theorem 7.3, p31])

By assumption o= zgd< *a ;gE HI () and we want to show that
By~f as0plo


< 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 H-a)
< 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 e-2AA-'G GA-2w, h V(Ap,)dQ

< ,e-2jt I< GIGA-a2w,, A-a(h V(Ap,)) >n dt
by (3.39a)

5< JGIIGL(, 11G1 GIIG1A -2w, Ildt +C 0 e2 11 A"4p, IL dt


by (2.40),(*)

< II G L f A)E(0)+- l A-GGA-"2w, 112dt +2CAJ e-2X 11A-I'w ||dt

<- II G lLrn) + 3 IA- "G1 II E (0)+2Che3J e-tE(t)dt

= KgE(O)+ 2ChSe-2 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 e-E(t)dt <5KE(0) +Q4 + < A-awo h V(Apo) >
+ n < A-wo +A-2G1GiA-lzwl Apo > (3.44)


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-a2wo +A-aGjG*A-`awl h V(Apo) >a- < A-flGiGIA-awi h V(Apo) >a
= < A-wWo h V(Apo) >n 1

Before beginning Step 6, the following Lemma will be needed.

Lemma 3.10

The L2-norm of Ap is bounded by the initial energy E (0), i.e.,

11 AP 112= I I A 1II2 = 1 AI A-1^4A3p I2 < IA-' I12 1 IAL p I(2
= IIA-'4I2 IA-w, ll < 1IA-' 112\ )E(t),5 A-I4 11 E(0) (3.45)
L(Q) a- L(QL



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 =A-flG1G A-lawu, u = e-2Ap to obtain:

n-f e- A-2GiGGA-wuApdQ = n lim e-2P < A-"2GG A- a1, Ap > T
2 Q 2T-,-[ Jo
+n pfe-A-2GIGA-2wApdQ e-2PA-t2GiG*A-2wsApdQ (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 e-2T< A-2G I GA-'aw,(T), Ap(T) >a < n-C lim e-2 = 0 (3.47)
2rT-4 2 Tr-
So again we have only a contribution from t = 0 i.e.
2- lim e-2P< A-'aG,GiA-1w, Ap >] T=-n< A-2Gr1GA-awl Apo >n (3.48)
2-4- o 2
Part B
nPf e-2t< A-aG 1G;A-aw,, Ap >adt

nI n A-IGI A12G1 IIL 11e GIA-"w, 112dt +nl 1 e- II A 11

by (2.40), (3.32), (3.45) n IIA-G I L2 ) + 11lA-1'4 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

Qe-2P< A-'2G1lGiA-2w,, Ap,>QdQ

5 nJe- < GiG'1A-'2w, A-'2(AP) >.IdQ

< e--2- II GIGIA-"w, 11 dQ + -E4n T II A-'aAp,) 12dQ (3.51)


Examine the integral on the right in (3.51)
Using the facts that Da(A la)= H2 () [L-T.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) oe-2 IIA"p, la dt
F4 L(r, 1) 22

Si IG, II E(0)+ e4nC2 I A-" 1 L( e-E(t)dt
2F4 L(rQL)

+ E4nC2 IIA-'G1 I L(r, )E (0)

= KIE(0)+ E4nC2 IIA-"4 II e-E (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 e-E(t)dt < K2E(0) + < A-'wo, h *V(Apo) >0
+ -< A-l"wo, APo >n (3.57)


The proof follows again directly using (3.48), (3.49), (3.54) and again utilizing the cancellation of


n < A-"2Wo +A-_2GiGA-"aw, Apo >o n< A-'G,1GA-"'wl Apo >n
2 2
n < A-2wo, Apo >0a
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)


Second, I < A-2wo, Apo >nl < [ HA-12wo 11 + IIApo0 11]

<-5 A 1IM E2(0)+ IIA- 1 ) E(0)] =n A-A 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)
Dividing (3.60) by o4 > 0 and letting K = Theorem 3.2 is proved. O
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 Je-E (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)= Ile-Aw l w IZ into (3.62),

weget |oII eAl l Ilzd1 KE(0)


Thus by Datko's Theorem there exists C 1,81 > 0 such that

II eAt IIL(Z)> Cle-kt and hence II eAt IL(Z) < Ce-2
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


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. Vfdr--1 IVfl2h vdr

HVf VfdM + -1 Vf I2divhd (A.8)



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(= F-fdivhdFr- Vf V(fdivh)dQ

= fr-aydivhdr- fV(divh) Vfd f I Vf lI divhd
Nrov l'Q f' Q



We multiply both sides of our equation pa + A2p = -A-'2G 1GAp, by e-2tApdivh 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 Pe-2ApdivhdtdQ = lim e-2 < p,, Apdivh >
+ 2p e-2ftAppdivhdQ e-2stAppdivhdQ

= lim e-2< p,. Apdivh >] + 2 e-WAppdivhdQ
rT--L 0o Q
I f, -2 p,divhdE+ e-2tp,V(divh) Vp,dQ + e-2Vp, 2divhdQ (B.1)
A2p term: First apply (A.1) and then (A.7) to get:

JIe-2A(Ap)ApdivhdQ = e-2t~) ApdivhdE- Q e-2V(Ap) *V(Apdivh)dQ
= fe-2pr '(MAvApdivhdE- e-ApV(divh) V(Ap)dQ f e- I V(Ap) 12divhdQ (B.2)
Now isolating difference of energy term and multiplying by we get

21 e-2 I Vp.l 2 -V(Ap)12] divhdQ =-lim[e-2pt< pr, Apdivh >n]

PQe- tAppdivhdQ + e-2t -pdivhd- 1 e- 'pV(divh) Vp,dQ
e-2Ap l) ApdivhdZ + Q e-2tApV(divh) Vp,dQ
S2 Qe- -GGA pdivhdQ B.3)2
1l e-VA-v2GG-ApApdivhdQ (B.3)


We multiply our equation p,,+A2p=-A-aG1GiGAp, by e-2th 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,,e-h V(Ap)dtd = lim e- t,< p, hV(Ap) > T

+ 2pQ e-tp,h V(Ap)dQ Qe-~'p,h V(Ap,)dQ

= lim e-2t< p,, hV(Ap) >0 + 20p e- p,h V(Ap)dQ

Je- 2Apph vd + Qe-XApdiv(ph)dQ

= lim e-2< p,, h-V(Ap) >o] +2o e ph V(Ap)dQ-e-2tApph *vdZ
T-L- o 0 Q -
+ fe-2PtAph Vp,dQ + ae-2AppdivhdQ (C.1)

Next, we will apply (A.8) to J e-2Ap,h Vp,dQ and (A.9) to IQe-tApp,divhdQ simultaneously to


2ofPce-20 h V(Ap)dtdG

= lim e-W2< p,. h-V(Ap) > + 2 2pfQe-tp,h V(Ap)dQ Ie-2tApp,h vdE

+ e- P'hVp,dZ J e-t IVp,12h vdE-IQe-2P tHVp 'Vp,dQ + J e-~ lVp 12divhdQ

+ f e-2 p, divhdY- e-2fp,V(divh) VpdQ e-2t I Vp, 12divhdQ (C.2)
W Q 0



A2p term: Applying (A.8) we get
J e- A(Ap)h V(Ap)dQ = e-2t h V(Ap)dE

e-Z I V(Ap) I 2h vdE Je-2tHV(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 e-tIVp, 12h -vdW

fE fv Y. 2y e
+ e-2 pt p,divhd + f e-2a(P) h V(p)d 1Je- I V(Ap) 1 h vdW
N2 If av 2
= lim e-2t< p,, h-V(Ap) >a -2 e-2p,h V(Ap)dQ
r--L o 0 Q
+ IQe-2PHV(Ap) V(Ap)dQ + f e- HVp, VpdQ

+ Je-p, V(divh) Vp,dQ + e-2 [ I Vp, 12 I V(Ap) 12] divhdQ

fe-2W A-l2G 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
and IVp,I= I-I 0 on (C.6)
And by utilizing h (x) = (x-xo) the radial vector field, we have that

H (x)= 1 identity matrix
divh = n = dim t (C.7)
V(divh) 0


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:

,e-2~P h V(Ap)dE f e- I V(Ap) I2h vdC + -,e-2ze pd

= -i e-2Ee < p,, h-V(Ap) >a -lim [e-2 p,, Ap >O]
T,-L o0 2 TJ---L 0

2p| e- p,h V(Ap)dQ n 3Q e -p,4p dQ

+ e- IVp,12dQ +f e-2 t IV p) 2dQ

fe-2A- l'2CG Ap,h V(Ap)dQ n fe-A-AGiG GArApdQ (C.8)


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[T.2] R. Triggiani, "Exact Controllability of Wave and Euler-Beroulli Equations in the Presence of
Damping," Proceedings of International Conference on Differential Equations held in
Columbus, Ohio, March 21-15, 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), 241-277.

[T.4] R. Triggiani, A Cosine Operator Approach to Modeling L2(O,T; L2(C))-boundary Input Prob-
lems for Hyperbolic Systems, pp. 380-390, Springer-Verlag, 1978. Proceedings 8th IFIP
Conference, University of Wurzburg, W. Germany, 1977.

[T.5] R. Triggiani, Lack of Uniform Stabilization for Non-contractive 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), 383-403.


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.


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

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

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