The Controllability of Systems Governed by Parabolic Differential Equations

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1 JOURNAL OF MAEMAICAL ANALYSIS AND APPLICAIONS 5, ARICLE NO. AY he Controllability of Systems Governed by Parabolic Differential Equations Yanzhao Cao Department of Mathematics, Florida A & M Uniersity, allahassee, Florida 337 Max Gunzburger Department of Mathematics, Iowa State Uniersity, 4 Carer all, Ames, Iowa 5-64 and James urner Department of Mathematics, Florida A & M Uniersity, allahassee, Florida 337 Submitted by Cornelius. Leondes Received February, 997 he main result of this paper is a new, weaker condition for the exact controllability of linear parabolic partial differential equations. he result is derived through an examination of a related optimal control problem. It is shown that if the terminal state belongs to the Sobolev space of functions having two square integrable derivatives, then the parabolic equation is exactly controllable. 997 Academic Press. INRODUCION Let be a bounded domain in d with smooth boundary. For a fixed, let Q Ž,. and Ž,.. On the domain, let A be the second-order elliptic differential operator d y Ay Ý ai, jž x. cž x. y. x x i, j -47X97 $5. Copyright 997 by Academic Press All rights of reproduction in any form reserved. i ž / j 74

2 CONROLLABILIY OF PARABOLIC EQUAIONS 75 We will assume that c onand that the matrix Ža Ž x.. i, j is symmetric and positive definite. Consider the parabolic initial boundary value problem y Ay u t in Q, y on, yž. y on. Ž.. It is well known that if the data y and u are given such that y Ž. and u L Ž Q., then the problem Ž.. has a unique weak solution, y Ž Q.. ere, for real numbers s, s Ž. denotes the standard Sobolev spaces Žsee, e.g.,., Ž. denotes the subspace of Ž. consisting of functions having vanishing traces with respect to, and, Ž Q. is a ilbert space that will be defined at the end of this section. he exact controllability problem we study here is to seek, for given functions y and ˆy L Ž., a function y yt,x and a control ut,x, both defined for Ž t, x. Q, such that y, u satisfy Ž.. and also y,x ˆyŽ x. for x. his problem has been studied extensively in the last thirty years. Some early contributions are due to Egorov, 3 and Gal chuk 9. Many significant developments in the controllability theory for linear parabolic and hyperbolic equations are due to Fattorini and Russell 4, 5, 46. Using a harmonic analysis method, they obtained sufficient conditions for the exact controllability of systems governed by hyperbolic and parabolic equations. More recently, Lions developed the ilbert Uniqueness Method for the exact controllability problem for hyperbolic equations. Other recent developments in exact controllability problems for distributed parameter systems are due to Fursikov and Imanuvilov 7, 8. In this paper, we attempt to study the exact controllability problem for the system Ž.. by examining the limit behavior of a corresponding optimal control problem. his method is different from the traditional harmonic analysis method. It has been used by Fursikov and Imanuvilov 8 in the study of the approximate controllability of systems governed by the NavierStokes equations. Using this method, we obtain a new sufficient condition for the exact controllability of Ž... Specifically, we prove that if ˆy Ž., then the system Ž.. is exactly controllable. he plan of this paper is as follows. In the remainder of this section, we introduce some notation that will be used throughout the paper. hen, in Section, we discuss the exact controllability problem for systems governed by ordinary differential equations using the optimal control method. his section can be considered as motivational for the study of the

3 76 CAO, GUNZBURGER, AND URNER controllability problem for parabolic differential equations which is given in Section 3. For a ilbert space X, define L Ž,; X. ½ f: Ž,. X; fž t. X dt5 which is also a ilbert space with the norm We shall use the notation ž / L Ž,; X. X f f t dt. Ž,. dx for the L Ž. -inner product and the associated norm will be denoted by. For p, q, let p, q Ž Q. L Ž,; q Ž.. p Ž,; L Ž... hese spaces are discussed in, and their norms can be defined by p, q ŽQ. q L Ž,; Ž.. p Ž,; L Ž.. f f f. Define the bilinear form a, : by ž / i, j d až,. Ý aijž x. cž x. dx. x x i j Note that, from the properties of the operator A, we have that there exist constants c, c such that a, c,, až,. c Ž.. Next, we introduce the eigenvalue problem A in, on. It is well known 3 that this problem has a system of eigenfunctions 4 that form a complete orthonormal set in L Ž. j j, with corresponding eigenvalues j as j.

4 CONROLLABILIY OF PARABOLIC EQUAIONS 77 We now consider the weak formulation of problem. with u : ž / y, až y,. Ž., t Ž.. yž,. y on. It is also well known that if y L Ž., then Ž.. has a unique solution yl Ž, ; Ž... he exact solution of Ž.. can be represented as j t Ý j j j yž t, x. Ž y,. e Ž x.. Ž.3.. EXAC CONROLLABILIY FOR FINIE-DIMENSIONAL SYSEMS Consider the following finite-dimensional system, yž t. AyŽ t. BuŽ t. for i ti, yž., Ž.. nn nk Ž. n Ž. k where A, B, y yt, and u ut. ere u represents the control ariable and y the state ariable. Define a cost function as J Ž u. yž ; u. y ˆ u Ž t. dt, where yt;u is the solution of Ž.. for a given u and denotes the Euclidian norm in n. In this setting, the optimal control problem for the system Ž.. is to find u Ž t. such that J Ž u. min J Ž u.. Ž.. ul Ž,. he optimality system of the problem Ž.. can be calculated as follows. For L Ž,., d JŽ u. Ž yž ; u. y ˆ. yž,. u dt, Ž.3. d where denotes the Euclidean inner product. Let pt Ž. be the solution of the following adjoint problem, pž t. A*pŽ t. for t, pž. yž. y, ˆ Ž.4.

5 78 CAO, GUNZBURGER, AND URNER where * denotes the matrix transpose. hen, combining.3 and.4, d JŽ u. pž. yž ;. u dt d hus, d Ž py. dt u dt dt ž / d d py yp dt u dt dt dt Ž. A*p y Ay B p dt u dt Ž B*pu. dt. ub*p. Ž.5. Equations Ž.., Ž.4., and Ž.5. together form the optimality system for the problem Ž... From the optimality system we have that where hus, or t AŽ. AŽ. yž ; u. e Bu d e BB*pd AŽ. A*Ž. e BB*e yž ; u. y ˆ d UŽ. Ž yž ; u. y ˆ., UŽ. eaž. BB*eA*Ž. d. yž ; u. UŽ. yž ; u. y ˆ Ž. ˆ ˆ y ; u y IU y,.6 where I denotes the identity matrix. he following result can be found in 5.

6 CONROLLABILIY OF PARABOLIC EQUAIONS 79 n EOREM.. he system. is exactly controllable for any ˆy if and only if UŽ. is positie definite. Using.6, we have the following result. EOREM.. he system Ž.. is exactly controllable if and only if for any ˆy n and any, there exists a constant C, independent of, such that yž ; u. y ˆ C. Ž.7. n Proof. Assume that. is exactly controllable for any ˆy. hen, by heorem., U is positive definite. hus, by Ž.6., there exists a constant C, independent of, such that y;u yc y. ˆ ˆ Setting CC y, we obtain Ž.7. ˆ. Now, assuming that Ž.7. is valid, we shall establish that U is positive definite. Suppose that U is not positive definite. hen, there exists ˆy such that U ˆy and ˆy. hen, from Ž.6., we have that y;u which implies that Ž.7. is not valid. his contradiction proves that U is positive definite. In the next section, we show that a condition analogous to.7 is sufficient for the exact controllability of systems governed by linear parabolic differential equations. 3. EXAC CONROLLABILIY OF LINEAR PARABOLIC DIFFERENIAL EQUAIONS 3.. race heorems and Regularity Properties of Parabolic Equations We first quote some trace theorems and regularity properties for parabolic equations; see for details. p, q EOREM 3., Vol., p. 9. For u Q with q, p, we may define, for a fixed t, and k where p q pk k u x, t on if k p t k k u p Ž x, t. k Ž., t k p.

7 8 CAO, GUNZBURGER, AND URNER Now, consider the initial boundary value problem for linear parabolic differential equations, A g in Q, t on, h in for t, where A,, Q, and are defined in Section. Ž.Ž. EOREM 3.. See, Vol., p. 83. If g L Q and h Ž.,, then Ž Q. and, C g h. ŽQ. L ŽQ. ŽQ. Ž.Ž. See, Vol., p. 84. If g L Q and h Ž., then 4, Ž Q. and 4, C g h. ŽQ. L ŽQ. ŽQ. Ž., 3 If g L Q and h L, then Q and, C g h. ŽQ. L ŽQ. L ŽQ. Ž. 34, 3 4 If g L Q and h, then Q and 34,3 C g h. ŽQ. L ŽQ. ŽQ. Ž. Ž. Proof of 3 and 4. Let 3. hen so that by interpolation theorem we have that if L Ž., then and 4,, Q Q, C g h. ŽQ. L ŽQ. L ŽQ. Ž. Ž. his proves 3. he proof of 4 is similar with Optimal Control and the Representation of the erminal State Recall the control system governed by the linear parabolic differential equation: y Ay u in Q, t y on, yy on for t. Ž 3..

8 CONROLLABILIY OF PARABOLIC EQUAIONS 8 Similar to the finite dimensional case, we define an optimal control problem for the system 3. as follows. First, we define a cost functional JŽ u. Ž yž, x; u. y ˆŽ x.. dx u Ž t, x. dxdt, where y yt,x;u satisfies Ž 3... u Ž t, x. is said to be an optimal control if it satisfies J Ž u. inf J Ž u.. ul Ž Q. We let y yt,x;u denote the solution of Ž 3.. with u u. he optimality system is given by Žsee.Ž 3.. and p Ap in Q, t p on, Ž 3.. p y y ˆ in for t, p u in Q. From now on, we suppress the explicit dependence on x, e.g., we will use ut Ž. to mean ut,x. Let Et Ž. denote the solution operator for the problem i.e., w Aw on Q, t hen, from 3., we have that and w wz on, wž t. EŽ t. z. Q in for t, p Ž t. EŽ t. y Ž. y ˆ už t. EŽ t. Ž yž. y ˆ.. Ž 3.3. Combining 3. with 3.3, we obtain y Ž. EŽ. y EŽ t. EŽ t. y Ž. y ˆ dt.

9 8 CAO, GUNZBURGER, AND URNER Now, define a linear operator R from L Ž. to itself as follows. For zl Ž., we have that Rz EŽ t. EŽ t. zdt. R will play a central role in obtaining the main result of this paper. Using the operator R, we have that or yž. EŽ. y R yž. y ˆ ˆ ˆ y y ei R E y y, 3.4 where I denotes the identity operator on L Ž Q.. he next two lemmas describe some properties of the operator R. LEMMA 3.3. R is a symmetric and nonnegatie operator from L to itself. Also, Ker R. Proof. For z, z L we have that Ž Rz, z. EŽ t. EŽ t. z, z dt E t z, E t z dt Ž. z, EŽ t. EŽ t. z dt Ž z, Rz.. hus, R is symmetric. Using the same argument, we have, for z L, that L Ž Rz, z. EŽ t. z dt. hus, R is also nonnegative. If, for some z L Ž., Rz, then from Ž 3.7. we have that hus z. hus, Ker R. EŽ t. z for t.

10 CONROLLABILIY OF PARABOLIC EQUAIONS 83 LEMMA 3.4. R is a compact operator from L to itself. Proof. Let z L. By the semigroup property of Et and 3 of heorem 3., we have that Rz Ž. EŽ t. EŽ t. z dt EŽ t. z Ž. dt ž /, Ž,;. C E t z dt C E z C EŽ. z, C z Ž,;. L Ž.. hus, R is a compact operator from L to itself Conergence of the Final Optimal State and Approximate Controllability We now show that the final state y Ž. of the optimal control problem converges, as, to ˆy. EOREM 3.5. Let ˆy L Ž.. hen, lim y Ž. y. ˆ Proof. By Lemmas 3.3 and 3.4 and the ilbertschmidt heorem 3, the operator R has a system of eigenfunctions e 4 j jwith corresponding eigenvalues 4 j as j. Moreover, ej j forms an orthonormal basis in L Ž.. For y, y L Ž., let hen, Using 3.4, we have that hus, ˆ ˆy EŽ. y ye. Ý j j j R ˆyEŽ. y ye. Ý j j j j y j yž. y ˆ Ý e j. j j y y y y y. e N j j L Ž. j j Ž j. j Ž j. jn ˆ Ý Ý Ý For any given, there exists N such that Ý y j. jn

11 84 CAO, GUNZBURGER, AND URNER For this fixed N, there exists such that hus, N yj Ý j j L for. y Ž. y ˆ for. he proof is complete. DEFINIION 3.. he system Ž 3.. is said to be approximately controllable if, for any given ˆy and, there exists a control ut Ž. and a function yt Ž. such that Žyt,ut Ž. Ž.. is a solution of Ž 3.. and yž. y ˆ. he approximate controllability of the system Ž 3.. is an immediate consequence of heorem 3.5. COROLLARY. Let ˆy L Ž.. hen, the system Ž 3.. is approximately controllable. Approximate controllability has been proved in. Our proof is different in that it is constructive Exact Controllability We first prove the following theorem. We shall see that it is equivalent to our main result. EOREM 3.6. Assume that y. If y L Ž. ˆ satisfies the condition Ý j j y j, Ž 3.5. where y, are defined in heorem 3.5, then Ž 3.. j j is exactly controllable, that is, there exists u L Ž Q. such that the solution y of Ž 3.. also satisfies yž. y. ˆ Proof. By the proof of the heorem 3.5 we have that y y ˆ L Ž. C. 3.6

12 CONROLLABILIY OF PARABOLIC EQUAIONS 85 hen, using the optimality system Ž 3.. and the regularity properties of parabolic equations Ž see Subsection 3.., we have that p, C p Ž Q. L Ž. C. hen, by the optimality system 3.4 again we obtain u, p, Ž Q. Ž Q. C. hus, there exists a sequence,, n, and u L Ž Q. such that n n u u strongly in L n Ž Q.. Let y be the solution of the problem y Ay u in Q, t y yž. y on, on. By the regularity property of the solutions of linear parabolic differential equations see Subsection 3. we have that y, Ž Q.. By the trace theorem, we have that y,y Ž. L Ž.. Let y y. n n n We have that Multiplying Ž 3.7. by ž / n A u u in Q, n t n n n on, Ž 3.7. n Ž. in. and integrating the result over Q, we obtain n, n dt Ž A n, n. dt Ž u u n, n. dt. t

13 86 CAO, GUNZBURGER, AND URNER Integrating by parts and using the coercivity property of a,, we have that herefore, Ž. n n n y y a, dt n n uu, dt But, from 3.6, we have that so that ž / c uu dt n L Ž. n L Ž. c n L Ž Q. n n C uu a, dt. y y in L n Ž.. y Ž. y ˆ in L Ž. n yž. y. ˆ Remark 3.. Let R be the inverse of operator R. hen, the condition Ž 3.5. is equivalent to Remark 3.. then ˆy EŽ. ydž R.. From the proof of heorem 3.7, we see that if Ž 3.5. holds, ˆ y, y L C. It is easy to show that 3.5 is actually a necessary condition for the above inequality to hold. In fact, assume that the above inequality holds. hen, by the proof of heorem 3.5, we have that hus, Letting, we obtain 3.5. y j yž,. y ˆ Ý C. L j Ž j. y j C. Ý j Ž j.

14 CONROLLABILIY OF PARABOLIC EQUAIONS 87 We are now ready to prove the main result of this paper. EOREM 3.7. Assume that y. If y Ž. ˆ, then the system Ž.. is exactly controllable. Proof. Recall that for z L, Rz EŽ t. EŽ t. zdt. Let, 4 j j be the eigensystem of A. hen by the analyticity of the semigroup E Žsee. 6, we have that Žt.j Žt.j Žt.j j j j R e e dt e dt Žt. e j e j j j. j j hus, the eigensystem of R is given by ence, the eigensystem of R is ½ 5 e,, e j j j j. j 4 j e j, j j,. e j 4 ½ 5 Since, j, we can find two positive constants C and C such j that C C, j,,.... j j j hus, R and A have equivalent eigensystems and therefore DŽ R. DŽ A. Ž.. It is easy to see from Ž.3. that E y Ž.. he result of the theorem is then the direct consequence of Remark 3.. Remark 3.3. In 4, Russell gave a sufficient condition on exact controllability of the system Ž.. using the harmonic analysis method. he condition is C' A ˆye, Ž 3.8.

15 88 CAO, GUNZBURGER, AND URNER where C is a constant and A is the elliptic operator that appears in Ž 3... We now construct an example in which our condition Ž 3.5. is satisfied but Ž 3.8. is not satisfied. Let Ž,., a, and c. It is well known that the eigensystem of the operator A is given by Let k,k k, sinž kx. k. 4 4 Ý k u. k k he final state corresponding to the control u is given by where yž. Ý y k k, k k Žt. k yk e dt Ž e.. 3 k k By the construction of the operator R, the eigenvalues of R are given by It is easy to see that k Žt. k k e dt Ž e.. k y k Ý k k, i.e., y satisfies the condition Ž But, as pointed out in 4, the condition Ž 3.8. is not even nearly satisfied. ACKNOWLEDGMEN he first author thanks David Russell for many helpful discussions related to this paper.

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