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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) 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>r =< 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 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 + 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 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 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. 