Abstracts
We investigate finite approximate controllability for semilinear heat equation in noncylindrical domains. First we study the linearized problem and then by an application of the fixed point result of Leray-Schauder we obtain the finite approximate controllability for the semilinear state equation.
heat operator; finite approximate controllability; Leray-Schauder fixed point; non-cylindrical
Este artigo é dedicado ao estudo da controlabilidade finito-aproximada para a equação não linear de transferência de calor em domínios com fronteira móvel. A demonstração do resultado principal baseia-se no princípio de continuação única de Carolina Fabre 1996 e em argumentos de ponto fixo do tipo Leray-Schauder.
transferência de calor; controlabilidade aproximada finita; ponto fixo de Leray-Schauder; não cilíndrico
MATHEMATICAL SCIENCES
Finite approximate controllability for semilinear heat equations in noncylindrical domains
Silvano B. de MenezesI, Juan LimacoII; Luis A. MedeirosIII
IDepartamento de Matemática, Universidade Federal do Pará, 60075-110 Belém, PA, Brasil
IIInstituto de Matemática, Universidade Federal Fluminense, 24210-110 Niterói, Rio de Janeiro, Brasil
IIIInstituto de Matemática, Universidade Federal do Rio de Janeiro, Caixa Postal 68530, 21945-970 Rio de Janeiro, RJ, Brasil
Correspondence Correspondence to Luis A. Medeiros E-mail: lmedeiros@abc.org.br
ABSTRACT
We investigate finite approximate controllability for semilinear heat equation in noncylindrical domains. First we study the linearized problem and then by an application of the fixed point result of Leray-Schauder we obtain the finite approximate controllability for the semilinear state equation.
Key words: heat operator, finite approximate controllability, Leray-Schauder fixed point, non-cylindrical.
RESUMO
Este artigo é dedicado ao estudo da controlabilidade finito-aproximada para a equação não linear de transferência de calor em domínios com fronteira móvel. A demonstração do resultado principal baseia-se no princípio de continuação única de Carolina Fabre 1996 e em argumentos de ponto fixo do tipo Leray-Schauder.
Palavras-chave: transferência de calor, controlabilidade aproximada finita, ponto fixo de Leray-Schauder, não cilíndrico.
1 INTRODUCTION AND MAIN RESULT
We consider a semilinear parabolic problem in a domain of n+1 = n × t which is not a cylinder, but it is a union of open domains Wt of n, 0 < t < T, images of a reference domain W0 by a diffeomorphism tt : W0® Wt.
To make clear the notation, we identify W0 to a non empty bounded open et W of n. The points of W are represented by y = (y1, y2, ..., yn), yi Î , for i = 1, 2, ..., n and those of Wt are represented by x = (x1, x2, ...,xn), xi Î , i = 1, 2, ..., n. Thus, we have x = tt(y) or x = t(y, t), 0 < t < T.
We define the noncylindrical domain contained in n+1 by
The boundary of Wt is represented by Gt and the lateral boundary of is defined by
By Q we represent the cylinder
with W the reference domain, with lateral boundary S given by
where G is the boundary of W.
Thus we have the diffeomorphism tt : Q ® given by
We need the following assumptions:
(A1) tt is a C2 diffeomorphism from W to Wt, for all 0 < t < T.
(A2) t(y, t) Î Cº(0, T; C2()).
Note that W is a non empty bounded open set of
n which boundary G we suppose C2.In this article we investigate finite approximate controllability for the following semilinear parabolic system
In (1.1) we denote u = u(x, t), , D is the Laplace operator in n; is an open subset contained in . Note that we consider the image by tt of a cylinder q contained in Q, of the type q = w × (0, T), w Ì W. We denote by wt the sections of at level 0 < t < T, wt Ì Wt. By we represent the characteristic function of . The function h = h(x, t) is the control function such that h(x, t) acts on the state u = u(x, t) localized in .
The non linear function f = f(u) is real, globally Lipschitz and we suppose that f(0) = 0, that is, there exists a positive constant K0 , such that
for all x, h Î .
To formulate our problem, let E be a finite dimensional subspace of L2(WT) and denote by pE the orthogonal projection from L2(WT) onto E.
The problem of finite approximate controllability for (1.1) can be formulate as follows: given T > 0, uº Î L2(W), u1Î L2(WT) and e > 0, to find a control h Î L2() such that the corresponding solution u = u(x, t) of (1.1), satisfies the conditions
This means that the control h = h(x, t) can be chosen such that u(T) satisfies (1.3) and simultaneously a finite number of constrains, that is the condition (1.4).
There is an extensive literature about finite-approximate controllability for linear and semilinear heat equation in cylindrical domains. Among these works, it is worth mentioning the articles of Fernandez and Zuazua 1999, Lions 1991 and Zuazua 1997, 1999. For basic results on Sobolev spaces see Lions and Magenes 1968.
In the context of linear heat equation in noncylindrical domains, Limaco et al. 2002 proved the finite-approximate controllability. In Menezes et al. 2002 is given a proof of null controllability of the semilinear case (1.1). In Limaco et al. 2002 is mentioned some basic references on the mathematical analysis of partial differential equations in noncylindrical domains.
The main result in the present article is:
THEOREM 1.1. Assume f is C1and satisfies (1.2) with f(0) = 0. Then, for all T > 0, the system (1.1) is finite-approximately controllable.
This means, for any finite-dimensional subspace E of L2(WT), uº Î L2(W), u1Î L2(WT) and e > 0, there exists a control h Î L2() such that the solution u of (1.1) satisfies (1.3) and (1.4), for T > 0 given.
The methodology of the proof of the Theorem 1.1 is based in the fixed point argument, see Zuazua 1999.
There is however a new and no trivial difficulty related to the fact that is noncylindrical. To set up this point we employ the idea contained in Limaco et al. 2002.
The first step in the fixed point method is to study the finite-approximate controllability for a linearized system. The application of this method is based on the unique continuation property due to Fabre 1996.
The present paper is organized as follows: Section 2 is devoted to prove the finite-approximate controllability for the linearized system. In Section 3 we prove Theorem 1.1 by a fixed point method.
2 ANALYSIS OF THE LINEARIZED SYSTEM
The main result of this paper is proved in Section 3 by argument of fixed point. As an step preliminary we need to analyse the finite-approximate controllability of the following linearized system:
where the potential a = a(x, t) is assumed to be in L¥().
As in Limaco et al. 2002 a function u = u(x, t) defined in is said to be strong solution for the boundary value problem (2.1), if
and the conditions in (2.1) are satisfied almost everywhere in their corresponding domains.
We say that a function u = u(x, t) is a weak solution of (2.1) if
and
for all j Î L2(0, T; H1(Wt)) Ç C1([0, T]; L2(Wt)) such that j(T) = 0.
THEOREM (Existence of Strong and Weak Solution).
a) If uº Î (W), a(x, t) Î L¥() and h Î L2(0, T; L2(Wt)), the problem (2.1) has a unique strong solution.
b) Given uº Î L2(W), a(x, t) Î L¥() and h Î L2(0, T; H-1(Wt )), there exists a unique weak solution of (2.1).
See Menezes et al. 2002 where the authors employ the argument consisting in transforming the heat equation in the noncylindrical domain into a variable coefficients parabolic equation in a reference cylinder Q, by means of the diffeomorphism
and 0 < t < T, that is, for (x, t) Î and (y, t) Î Q.
In fact, set
Equivalently
where tt-1 denotes the inverse of tt, which, according to (A1) is a C2 application from Wt to W for 0 < t < T which we denote by rt. We shall also employ the notation r(x, t) = rt(x), yj = rj(x, t), 1 < j < n.
It follows that u = u(x, t) solves (2.1) if and only if v = v(y, t) satisfies:
In (2.7) we have . Note that according to (A1) we have
ANALYSIS OF THE OPERATOR. A(t)v =
For v, j Î L2(0, T; (W)) and Gauss' Lemma we obtain the bilinear form a(t, v, j) defined by
This bilinear form is bounded and (W)-coercive.
In (2.7) set
Thus, from (2.7) the transformed of (2.1) in Q is the following system
Note that (2.10) is a linear parabolic system with variable coefficients in a cylinder Q = W × (0, T), W a regular bounded non empty open set of n. Since A(t) is coercive, the initial boundary value problem (2.10) is a classical problem investigated in Lions and Magenes 1968.
It follows that if we take uº Î (W) (resp. uº Î L2(W)) and h Î L2(0, T; L2(W)) (resp. h Î L2(0, T; H-1(W)), then (2.10) has a unique strong (resp. weak) solution v Î Cº([0, T]; (W)) Ç L2(0, T; H2(W)) Ç H1(0, T; L2(W)) (resp. v Î Cº([0, T]; L2(W)) Ç L2(0, T; (W)).
By means of the change of variable y ® x we deduce the existence of a unique strong (resp. weak) solution u for the system (2.1).
At this point we underline that, under assumptions (A1) and (A2), the transformation y ® x does indeed, map the space of functions given in (2.2) (resp. (2.3)) into the space
(resp. Cº([0, T]; L2(Wt)) Ç L2(0, T; (Wt))).
We have the following result
THEOREM 2.2. The system (2.1) is finite-approximate controllable. More precisely, for any T > 0, uº Î L2(W), u1Î L2(WT), e > 0 and E, finite dimensional subspace of L2(WT), there exists a control h Î L2() such that the corresponding solution of (2.1) satisfies
Moreover, for any R > 0 there exists a constant C(R) > 0 such that
for any a Î L¥() satisfying
REMARK 2.1. Theorem 2.2 does not provide any estimation how the norm of the control h depends S, uº, u1 and e > 0. However, (2.13) guarantees that h remains uniformly bounded when the potential a remains bounded in L¥.
The control h is not unique. The method we develop below provides the control of minimal L2-norm. It is this control of minimal norm which satisfies the uniform boundness condition (2.13).
PROOF OF THEOREM 2.2. To prove it we employ exactly the same argument as in Limaco et al. 2002. Without loss of generality we may assume uº = 0. Indeed, otherwise, we consider the solution z of
Then, the solution u of (2.1) may be decomposed as u = w + z where w solves
Then (2.11)-(2.12) are equivalent to
Therefore we will consider uº = 0. The regularizing effect of the heat equation allows to show that
Let us consider the adjoint system
Taking in account that the potential a is bounded it follows that for any jº Î L2(WT), the system (2.15) has a unique solution j Î Cº([0, T]; L2(Wt)) Ç L2(0, T; (Wt)).
We consider the functional J defined in L2(WT) by
where j is the unique solution of (2.15) with initial data jº.
The functional J is continuous and strictly convex in L2(WT). More precisely, by the argument of Limaco et al. 2002 we can prove that
One of the key point of the proof of (2.17) is the unique continuation result of Fabre 1996.
Thus, it follows that J has a unique minimizes in L2(WT). If is the solution of (2.15) associated to the minimizer , then the control h = is such that the solution of (2.1) satisfies (2.11) and (2.12).
Thus it is proved the finite-approximate controllability.
In order to prove the uniform boundedness (2.13) for h = , the problem may be reduced to the case uº = 0 as indicated above, provided u1 belongs to a relatively compact set of L2(WT).
PROPOSITION 2.3. Let R > 0 and K be a relatively compact set of L2(WT). Then, the coercivity property (2.7) holds uniformly for u1Î K and potential a satisfying (2.14).
REMARK 2.2. Note that the functional J depends on the potential a and the target u1. Thus, the Proposition 2.3 guarantees the uniform coercivity of the functional when u1Î K, K compact set in L2(WT) and the potential a is uniformly bounded.
As a consequence of the Proposition 2.3 we deduce that the minimizers of the functional J are uniformly bounded, u1Î K and the potential a is uniformly bounded. Consequently, the controls h = are uniformly bounded.
Therefore, the proof of Theorem 2.2 is a consequence of the proof of Proposition 2.3.
PROOF OF PROPOSITION 2.3. We argue by contradiction. If the coercivity property (2.17) does not hold uniformly, we deduce the existence of sequences
and
such that
for some 0 < d < e.
Note that Jj is the functional (2.16) corresponding to and aj.
Set
We have
By (2.19) and (2.21) we deduce
Extracting a sequence we also have
By (2.24) we have
On the other hand, solves
Note that a is given by (2.28) and by (2.25).
In order to obtain (2.30) we need to show that
In fact, with the change of variables x ® y form into Q, the system
is transformed in a new system in Q with yj(x, t) = jj(y, t) for y = tt(x), given by
For the parabolic problem with unknow yj, in cylindrical domain Q, we obtain estimates which permites to apply the Aubin-Lions compactness argument for yj. Then when we change variables y ® x we obtain a subsequence () in L2() such that
This implies
This justifies the conclusion that is solution of (2.30).
According to Fabre 1996 and (2.29) and (2.30) we have = 0 in . If we observe (2.25) we conclude that
By (2.36) and (2.27) we conclude
But, since E is finity dimensional, we have
and therefore
because = 1 for all j.
Therefore,
which is a contradiction with (2.21).
This complete the proof of Proposition 2.3 and consequently that of Theorem 2.2.
3 PROOF OF THE MAIN RESULT
Now we will prove Theorem 1.1 as a consequence of Theorem 2.3 and a fixed point argument. We will begin with an existence theorem.
THEOREM 3.1. Let f : ® be a C1function, globally Lipschitz such that f(0) = 0. Consider uº Î L2(W) and h Î L2(). Then, there exists only one function
solution of the problem (1.1).
PROOF. We consider the change of variable (2.5) or (2.6) transforming (1.1) into
We know that (3.1) admits a unique solution
By the change of variable y ® x we deduce the existence of a unique solution u of (1.1) in the class
PROOF OF THEOREM 1.1. As in Menezes et al. 2002 we define the function
Given z Î L2() let us consider the linearized system
The system (3.3) is linear in u = u(x, t) with potential a = f(z) Î L¥() and
Thus, will be rewritten (3.3) as
By Theorem 2.2, for each z Î L2() we find a control h = h(x, t) Î L2() such that the solution u of (3.5) satisfies (1.3) and (1.4). Moreover, for every R > 0 and all potential a = a(x, t) Î L¥(Q) such that < R we have
Now we define the nonlinear mapping N
The problem is reduced to finding a fixed point for N. Indeed, if z Î L2() is such that N(z) = u = z, u solution of (3.3) is actually solution of (1.1). Then the control h = h(x, t) is that one we were looking for, since, by construction, u = u(z) satisfies (1.3) and (1.4).
The nonlinear map N : L2() ® L2() satisfies the properties:
The proof of (3.8) and (3.9) is given in Menezes et al. 2002.
It follows from (3.8), (3.9) and Leray-Schauder's fixed point theorem, the mapping N has a fixed point. This completes the proof of Theorem 1.1.
ACKNOWLEDGMENTS
We take this opportunity to express our appreciation to Enrique Zuazua for stimulating conversation about the subject and for constructive remarks.
Manuscript received on November 3, 2003; accepted for publication on May 19, 2004;
contributed by LUIS A. MEDEIROS* * Member of Academia Brasileira de Ciências AMS Classification: 35B99; 35K05; 93B05.
- FABRE C. 1996. Uniqueness results for Stokes equations and their consequences in linear and nonlinear control problems, ESAIM: COCV: 267-307.
- FERNANDEZ LA AND ZUAZUA E. 1999. Approximate Controllability for Semilinear Heat Equation Involving Gradient Terms. J Opt Theory and Appl 101: 307-328.
- LIONS J-L. 1991. Remarques sur la controlabilité approché, in Jornadas Hispano Francesas sobre controle de Sistemas Distribuidos - Univ. Malaga - Espanha: 77-87.
- LIONS J-L AND MAGENES E. 1968. Problčmes aux limites non homogčnes et applications, Vol. I, Dunod, Paris.
- LIMACO J, MEDEIROS LA AND ZUAZUA E. 2002. Existence, uniqueness and controllability for Parabolic Equations in noncylindrical domains. Matemática Contemporânea, 23 (Part II): 49-70.
- MENEZES SB, LIMACO J AND MEDEIROS LA. 2002. Remarks on null controllability for heat equations in thin domains. Compt and Appl Math 21: 47-65.
- ZUAZUA E. 1997. Finite dimensional controllability for semiliner heat equations. J Math Pure Appl 76: 237-264.
- ZUAZUA E. 1999. Approximate controllability for semilinear heat equation with globally Lipschitz nonlinearity, Control and Cybernetics 28: 665-683.
Correspondence to
Publication Dates
-
Publication in this collection
23 Aug 2004 -
Date of issue
Sept 2004
History
-
Received
03 Nov 2003 -
Accepted
19 May 2004