Exact Internal Controllability for the Semilinear Heat Equation

Transcription

1 JOURNAL OF MAEMAICAL ANALYSIS AND APPLICAIONS, ARICLE NO AY Exact Internal Controllability for the Semilinear eat Equation Weijiu Liu* and Graham Williams Department of Mathematics, Uniersity of Wollongong, Northfields Aenue, Wollongong, NSW 5, Australia Submitted by George Leitmann Received September 6, 996 Using the BrowderMinty surjective theorem from the theory of monotone operators, we consider the exact internal controllability for the semilinear heat equation We show that the system is exactly controllable in L if the nonlinearities are globally Lipschitz continuous Furthermore, we prove that the controls depend Lipschitz continuously on the terminal states, and discuss the behaviour of the controls as the nonlinear terms tend to zero in some sense A variant of the ilbert Uniqueness Method is presented to cope with the nonlinear nature of the problem 997 Academic Press INRODUCION Of recent years, there has been some study on the problem of approximate controllability for the semilinear heat equation Combining a variational approach and Kakutani s fixed point theorem, Fabre, Puel, and Zuazua 5 studied the approximate controllability for the semilinear heat equation y y y f y h in, ž y t/ y y in, y on * address: wliu@uoweduau address: ghw@uoweduau -47X97 $5 Copyright 997 by Academic Press All rights of reproduction in any form reserved 58

2 EXAC CONROLLABILIY 59 n In, is a bounded domain nonempty, open, and connected in with suitably smooth boundary say C, and, for y y x, hh x,t represents the control function, is an open and nonempty subset of, and is the characteristic function of hey proved that this system is approximately controllabel if f is globally Lipschitz Naturally, we would ask: Is the semilinear heat equation exactly controllable? hat is to say, for suitable, is it possible, for eery initial and final data y and z gien in suitable ilbert spaces, to find a corresponding control h driing the system y y f t, y h in, y y in, y on, to the state z at time, ie, such that the solution y y x,t; h of satisfies y x, z in? 3 In, we have allowed f to depend on t as well as y It is the purpose of this paper to positively answer this question o achieve this goal, we will introduce a Monotone Operator Method ab- breviated to MOM he idea of this method is to first construct a nonlinear, monotone, and continuous operator by coupling a linear heat equation with a semilinear heat equation, and they apply the famous BrowderMinty surjective theorem see, p 557 from the theory of monotone operators hroughout this paper let be a bounded domain nonempty, open, n and connected in with suitably smooth boundary say C Let and set, and, In the sequel, s always denotes the usual Sobolev space and s denotes its norm for any s Let X be a Banach space We denote by C k,, X the space of all k times continuously differentiable functions defined on, with values in X, and write C,, X for C,, X We make the following hypothesis on f: Assume the function f t, y is continuous in t on, and globally Lipschitz continuous in y on, that is, there exists a positive constant l such that f t, y f t,t lyy, for all y, y 4

3 6 LIU AND WILLIAMS he main result of this paper is as follows EOREM Assume holds hen there exists a such that for system is exactly controllable in L at time, that is, for any initial state y and any terminal state z L, there exists an internal control function h x, t h x,t; y, z L, ; such that the solution of with satisfies 3 Furthermore, for any fixed y L, the control function is Lipschitz continuous h x,t; y, z : L L,; h As remarked in 6, if is a proper subset of, the exact internal controllability for the linear heat equation is going to be impossible hus we cannot expect the exact internal controllability for the semilinear heat equation if is a proper subset of Compared with existing results, the result obtained here is essentially different from Fabre, Puel, and Zuazua s results 5 since we here consider the exact controllability In addition, they generalize the relevant theorems of, 6, 8, from the linear to the nonlinear case he rest of this paper is organised as follows For completeness, in Section we present some notions and a main theorem about monotone operators hen we construct a nonlinear operator F in Section 3 and prove the monotonicity of the operator F in Section 4 heorem is proved in Section 5 Finally, we discuss the behaviour of controls as the nonlinear terms tend to zero in some sense in Section 6 PRELIMINARIES For convenience, we recall some basic notions and a main result about monotone operators For details, we refer to 4, Chap 3;, Chaps 56 DEFINIION see, pp 47 and 5 Let X be a real Banach space, X* its dual space, and ², : the duality pairing between X and X*, and let F : X X* be an operator hen i F is called monotone iff ² : Fu F, u, for all u, X ii F is called strictly monotone iff ² : Fu F, u, for all u, X with u

4 EXAC CONROLLABILIY 6 iii F is called strongly monotone iff there is a such that ² : Fu F, u u, for all u, X iv F is called hemicontinuous iff ² Fut,w : is continuous in t on, for all u,, w X v Fis called coercive iff lim u ² : Fu, u u Obviously, strong monotonicity implies coercivity EOREM Browder-Minty, see, p 557 Let X be a real reflexie Banach space, and F : X X* be a monotone, coercie, and hemicontinuous operator hen F is onto X* Furthermore, if F is strongly monotone, then the inerse operator F : X* X exists and is Lipschitz continuous 3 CONSRUCION OF A NONLINEAR OPERAOR We construct a nonlinear operator F o do this, we first consider the following problem with a given terminal state u x : uu in, u x, u x in, 3 u on Concerning problem 3, the following results are classical LEMMA 3, p is the infinitesimal generator of both a C semigroup of contractions and an analytic semigroup of operators on L LEMMA 3 3, pp 553; p i Let be a bounded domain in n with a Lipschitz boundary hen for all u L there exists a unique solution u u x,t of 3 with Moreoer, uc, ; L L,; 3 u t u, t, 33 n ii Let be a bounded domain in with a boundary of class C hen for all u D with D 34

5 6 LIU AND WILLIAMS there exists a unique solution u u x,t of 3 with uc, ; 35 Moreoer, there exists a constant c independent of such that u t cu, t, 36 Using the solution u of 3 we then consider the problem with any fixed initial state y x L : y y f t, y uu in, y x, y in, 37 y on It follows from the assumption and the classical semigroup theory, heorem, p 84 that problem 37 admits a unique weak solution yc, ; L since u u C, ; L for u D Now we define for any fixed initial state y x L the nonlinear operator F y,: D L L by F y,u y x, 38 4 MONOONICIY OF E NONLINEAR OPERAOR F We will prove that the operator F defined by 38 is Lipschitz continuous and strongly monotone LEMMA 4 Assume holds hen the operator F defined by 38 is Lipschitz continuous, that is, there exists a positie constant c c such that F y,u F y,u c y y u u, for any u, u D and any y, y L Proof Let u and u be solutions of 3 with terminal states u and u D, and y and y be the solutions of 37 corresponding to u, y and u, y hen we have y y y y f t, y f t, y u u u u in, 4 y x, y x, y y in, y y on

6 EXAC CONROLLABILIY 63 Multiplying 4 by y y and integrating over, t t, we obtain y t y t dx y y dx y y dxdt t y y u u dxdt u u y y dxdt t t f t, y f t, y y y dxdt t u u u u dxdt y y y y dxdt l y y dxdt t t 43 hus, y t y t dx y t y t dx y y dxdt u u u u dxdt y y dx t t l y t y t dxdt 44 Gronwall s inequality see 7, p 36 gives y t y t dx ž / lt e u u u u dxdt y y dx 45

7 64 LIU AND WILLIAMS But there exists a positive constant c such that u u u u dxdt c u u 46 herefore, 4 follows from 45 and 46 Remark 4 By Lemma 4, for any fixed y L, the operator F y, can be extended to L o do so, let u L and u 4 n be a sequence in D such that u u in L n hen it follows from Lemma 4 that F y,u 4 is a Cauchy sequence in L We set n n F y,u lim F y,u n It is clear that F y,u is independent of the choice of the sequence u 4 n Furthermore, it is easy to show that the extension of F y, is still Lipschitz From now on, F is thought of as an operator defined on L L o prove the strong monotonicity of F y, for any fixed y L, we need the exponential decay rate for solutions of the heat equation LEMMA 43 Let u be the solution of 3 hen there is a constant such that Proof t u t dxe u dx, for t, 47 Since d t t t dt e u t dx e u t u t dxe u t dx if, we obtain t t e u t dxe u t dx t t e u t dxe u t dx t e u t dx, t u t dxe u dx, for t,

8 EXAC CONROLLABILIY 65 ere we have used Poincare s inequality see, p 7 and is Poincare s constant his is valid for the solution u of 3 by the definition of D LEMMA 44 Assume holds Suppose l or is so small that l l M l, e 48 l hen for any fixed y L the operator F y, : L L is strongly monotone Proof We first assume u, u D Multiplying 4 by u u, integrating over, and noting y y y, we obtain F y,u F y,u u u dx y y u u dx u u u u dxdt f t, y f t, y u u dxdt u u u u dxdt u u dxdt Moreover, l y y dxdt u u u u dxdt use 45 l lt u u u u dxdt e dt M l, u u u u dxdt u u dx u u dx u u dxdt

9 66 LIU AND WILLIAMS hus, it follows from Lemma 43 that F y,u F y,u u u dx ž / ž / M l, u u dx u u dx M l, u u dxe u u dx 4 4 M l, e u u dx 49 By taking a limit, we can show 49 holds for any u, u L hus we have proved A y, is strongly monotone Remark 45 here are many function f t, y that satisfy the conditions of Lemmas 4 and 44 For example, f t, y t sin y, provided is so small that e 5 PROOF OF EOREM Lemma 44 shows that the constant l is required to be small enough so that F is strongly monotone o overcome this drawback, we introduce a Domain Expansion Method to prove heorem his method is general and can be applied to other situations Proof of heorem For, we introduce a function f by and a domain Set ž / t f t,u f,u, 5 x: x 4,,, x, t : x, t, 4

10 EXAC CONROLLABILIY 67 Instead of 3 and 37, we consider and where u u in, u x, u x in, 5 u on, w w f t,w uu in, w x, w x in, 53 w on, ž / x w x y, x 54 hen the operator F defined by 38 becomes F w,u w x, 55 he constant l in for f now is l, and the constant M defined by 48 now is ž / ž / ž / l l l M, exp l Let be such that Ml, hen we have Ml, for It therefore follows from Lemmas 4 and 44 that F is continuous and strongly monotone on L It then follows from heorem that for any z L there exists u L such that where F w,u, ž / x x z, x 56 Moreover, F w, is Lipschitz continuous hen we solve problem 5 with the terminal state u hus we have found an internal control function u u L,; such that the solution of 53 satisfies w x,, x

11 68 LIU AND WILLIAMS Setting y x,t w x, t, x, t, h x,t; y, z u x, t u x, t, 57 then y satisfies and Set ž / y x, z, x hen we have proved that for system is exactly controllable in L at time Furthermore, it follows from 57 and the Lipschitz continuity of F that the following c s denoting various constants L,; h x,t; y, z h x,t; y, z cu u u u L,; cu u L,; c u u,, c F w, F w, c u, cz z, We call the method of the above proof the Domain Expansion Method Remark 5 It should be understood that the solution of 53 for Fw,u Fw,u :u D 4 is the limit of w 4 n in C, ; L, where the w are the weak solutions of 53 n with u replaced by the solutions u of 5 with terminal states u D n n such that u u in L n 6 BEAVIOUR OF CONROLS AS f Let c, c with c c be two fixed constants It is known from heorem that if c, c, then we can find independent of such that for any terminal state z L there exists an internal control

12 EXAC CONROLLABILIY 69 h x,t such that the solution y, which depends on, of satisfies y y f t, y h in, y 6 x, y in, y on, y x, z iin 6 We now study the behaviour of h x,t as ifc,c his is a kind of nonlinear perturbation, which is motivated by Lions work 9, where the problems of linear perturbation have been studied in detail Let f be replaced by f in the proof of heorem Obviously, the operator F defined by 55 now depends on So we write F for F For a fixed z L, let u be the solution of the operator equation F w,u, 63 and u x,t be the solution of 5 with u u, where w, are given by 54 and 56, respectively aking u u and u and in 49, we obtain F w, u dx F w,u F w, u dx 4 M l, e u dx 8 e u dx, 64 for c, c if is sufficiently large so that Ml, for c,c On the other hand, as in the proof of Lemma 4, we can prove for the solution w x,t F w, of 53 with u u w x,t dx l exp t m w x dx f t, dt, 4 for t, 65

13 7 LIU AND WILLIAMS In fact, multiplying 53 by w, integrating over, t t,we obtain w x,t dx w x dx w dxdt t f t,w w dxdt f t,w f t, w dxdt f t, w dxdt t t m l f t, dt w dxdt, 4 t t where m denotes the Lebesgue measure of hus, w x,t dx m w x dx f t, dt 4 l t w x,t dxdt, which, by Gronwall s inequality, gives 65 Relations 64 and 65 show that u 4 is bounded in L for c c Furthermore, since u u and is an isomorphism from onto, it follows from 57 that there exists a constant c such that h cu u L,; L,;, c u dt 66 Moreover, by heorem 3 of 3, p 5, there is a constant c such that u u dxdt cu 67, It therefore follows from 66, 67, and the boundedness of u : c 4 4 c that the set h : c c is bounded in L, ; Let c, c Let the subsequence h 4 of h : c c 4 be such i that h h weakly in L, ; as i i

14 EXAC CONROLLABILIY 7 then Let y be the solution of y y h in, y y in, 68 y on, y z in 69 Indeed, subtracting 68 from 6, we obtain y y y y f t, y hh in, y 6 y in, y y on For any L, let be the solution of in, in, 6 on Multiplying 6 by and integrating by parts, we obtain z y dx f t, y dxdt h h dx 6 In addition, it follows from the assumption that f t, y ly f t, m 63 herefore, letting in 6, we obtain i z y dx 64 his implies 69 because is arbitrary in L In summary, we have proved EOREM 6 Let h be internal control functions obtained as in heo- rem, driing system 6 from an initinal state y L to a terminal state z L hen for any fixed constants c, c with c c, the set 4 h : c c is relatiely weakly compact in L, ; Furthermore, if c, c, then any weak limit h of a subsequence h 4 i of 4 h : c c in L, ; as i is an internal control function driing system 68 from the initinal state y to the terminal state z

15 7 LIU AND WILLIAMS ACKNOWLEDGMENS he first author is grateful for the Overseas Postgraduate Research Scholarship from Australian Government, the University of Wollongong Postgraduate Research Award, and the Department of Mathematics Analysis Research Group Scholarship from the Faculty of Informatics at the University of Wollongong REFERENCES C Carthel, R Glowinski, and J L Lions, On exact and approximate boundary controllabilities for the heat equation: A numerical approach, J Optim heory Appl 8, No 3, 994, R Dautray and J L Lions, Functional and variational methods, in Mathematical Analysis and Numerical Methods for Science and echnology, Vol, Springer-Verlag, Berlin, 99 3 R Dautray and J L Lions, Evolution problems, I, in Mathematical Analysis and Numerical Methods for Science and echnology, Vol 5, Springer-Verlag, Berlin, 99 4 K Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, C Fabre, J P Puel, and E Zuazua, Approximate controllability of the semilinear heat equation, Proc Roy Soc Edinburgh Sect A 5 995, 36 6 R Glowinski and J L Lions, Exact and approximate controllability for distributed parameter systems, Acta Numerica 994, J K ale, Ordinary Differential Equations, WileyInterscience, New York, G Lebeau and L Robbiano, Controle ˆ exact de l equation de la chaleur, Comm Partial Differential Equations, Nos 995, J L Lions, Perturbations, in Controlabilite ˆ exacte perturbations et stabilisation de systemes ` distribues, Vol, Masson, Paris, 988 J L Lions, Exact controllability for distributed systems: Some trends and some problems, in Applied and Industrial Mathematics R Spigler, Ed, pp 5984, Kluwer Academic, DordrechtNorwell, MA, 99 A Pazy, Semigroup of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 983 E Zeidler, Nonlinear monotone operators, in Nonlinear Functional Analysis and its Applications, IIB, Springer-Verlag, New York, 99