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Computational & Applied Mathematics

versión On-line ISSN 1807-0302

Comput. Appl. Math. v.22 n.1 São Carlos  2003

 

Approximate controllability for the semilinear heat equation in N involving gradient terms

 

 

Silvano Bezerra de Menezes*

Departamento de Matemática, Universidade Federal do Pará 66075-110 Belém, PA, Brazil E-mail: silvano@ufpa.br

 

 


ABSTRACT

We prove the approximate controllability of the semilinear heat equation in N, when the nonlinear term is globally Lipschitz and depends both on the state u and its spatial gradient Ñu. The approximate controllability is viewed as the limit of a sequence of optimal control problems. In order to avoid the difficulties related to the lack of compactness of the Sobolev embeddings, we work with the similarity variables and use weighted Sobolev spaces.

Mathematical subject classification: 93B05, 73C05, 35B37.

Key words: approximate controllability, optimal control, unbounded domains, weighted Sobolev spaces.


 

 

1 Introduction

Let W be a domain of N with N Í 1. Given T > 0 and an open nonempty subset w of W we consider the following semilinear heat equation

In (1.1) 1w denotes the characteristic function of w. Roughly speaking, f is assumed to be mensurable in (x, t) and globally Lipschitz in the variables u and Ñu. In (1.1) u = u(x, t) is the state and h = h(x, t) is the control function which acts on the system through the subset w. The approximate controllability problem can be formulated as follows: Given a finite time horizon T > 0 and an arbitrary element u0 Î L2(W), system (1.1) is said to be approximately controllable at time T in L2(W) if the reachable set

RNL(T) = {u(x, T) u is solution of (1.1) with h Î L2(Q)}

is dense in L2(W).

There are numerous works treating the approximate controllability in the linear parabolic framework, see [L2] and its bibliography, when W is a bounded set. The first results for nonlinear systems were obtained in [H]. More recently, several situations have been considered by Fabre, Puel and Zuazua [FPZ] for the particular case in which f is a globally Lipschitz function depending only on u., i.e., f = f(u). Their proof is divided in two parts:

a) approximate controllability of the linearized systems;

b) fixed point technique.

This technique cannot be applied when W is an unbounded set since the compacteness of Sobolev's embeddings is one of the main ingredients used in b). In L. de Teresa and E. Zuazua [TZ], they proved the approximate controllability of the semilinear heat equation in unbounded domains by an approximation method, for the case f = f(u), f being globally Lipschitz. The method in [TZ] consists in approximating the domain W by a sequence of bounded domains WR = W Ç BR, where BR denotes the ball centered in zero of radius R. It is then proved that, in the limit as R tends to infinity, the approximate controls in WR provide an approximate control in the unbounded domain W.

Later on, L. de Teresa [T] proved the approximate controllability when W = N by an alternative method that consists in writing the heat equation in the similarity variables and using the weighted Sobolev spaces introduced in [EK] to guarantee the compacteness of the Sobolev embeddings. Then, essentially, the methods in [FPZ] apply.

When W is a bounded set, L. Fernandez and E. Zuazua [FZ] gave a proof of the approximate controllability for the system (1.1), f being globally Lipschitz, inspired in the work by J.L. Lions [L2] in which, roughly speaking, the approximate controllability is viewed as the limit of a sequence of optimal control problems. More precisely, given u0 Î L2 (W), u1 Î H(W) with 0 < s < 1 and k > 0, let us consider the functional

where uh denotes the solution of (1.1) with control h. The functional Jk is well defined for h Î L2(W × (0, T)). On the other hand, it is shown that, for each k > 0, there exists a minimizer hk of Jk in L2 (W × (0, T)). Let us denote by uk the solution of (1.1) associated to this minimizer. It is shown in [FZ] that

as k +¥, and therefore uk(T) u1 in L2(W) as k +¥. This fact, combined with the fact that H(W) is dense in L2(W), guarantees the approximate controllability of system (1.1).

This paper is devoted to analyze the approximate controllability of system (1.1) in case when W = N. We adopt the approach in [FZ] but in the more general case W = N. However, in order to avoid the difficulties associated with the lack of compactness of the Sobolev embeddings, we work with similarity variables and the weighted Sobolev spaces as in [T]. We should point out we have essentially combined the techniques from both works [FZ] and [T]. A key point in the proof of our result is aresult of unique continuation by C. Fabre [Fa] in the context of linear heat equations involving gradient terms. The result in [Fa] allow us to conclude that

provided a Î L¥ (Q) and b Î (L¥ (Q))N.

Thus, we consider the domain W to be N, w an open nonempty subset of N and analyze the approximate controllability of the system:

The main result of this paper is the following:

Theorem 1.1. Let us assume the following hypotheses:

where K0 is a positive constant. Then, for any u0 Î L2(N) and T > 0, the set of reachable states (at time T > 0) given by

RNL(T) = {uh,u º (x, T): uh,u º is the solution of (1.2) with
h Î L2 (N x (0, T))}

is dense in L2(N).

Remark 1.1.

a) As we mentioned above, the method of proof applies in the case where W is a cone of N (i.e. lW = W, "l > 0). Extending Theorem 1.1 to the case of general open unbounded sets W is an open problem. At this respect, the approximation method developed in [TZ] may be the most suitable tool.

b) By an approximation argument, the assumption that f is C1 in the variables (u, Ñu) can be easily relaxed. It is sufficient f to be Lipschitz in those variables.

c) The assumption that f is globally Lipschitz in (u, Ñu) might seem to be artificial. But it is not. As it is shown in E. Fernández-Cara [F], there are nonlinearities growing at infinity in a slightly superlinear way and for which the approximate controllability fails.

The rest of the paper is organized as follows: in section 2, we introduce the similarity variables and use weighted Sobolev spaces. We prove basic results on existence, uniqueness and regularity of solutions. In section 3 we state some preliminary results concerning the solutions by transposition and we prove the approximate controllability of the linear equation. Section 4 is devoted to prove the main result. Finally, in section 5, we briefly comment the case of general conical domains.

 

2 Similarity variables and weighted Sobolev spaces

In this section we recall some basic facts about the similarity variables and weighted Sobolev spaces for the heat equation. We refer to [EK] and [K] for further developments and details. As we said above, to prove Theorem 1.1 we follow the approach in [FZ]. Thus, first we consider the problem of approximate controllability for the linearized system with potentials:

with a(x, t) Î L¥ (N × (0, T)) and b(x, t) Î (L¥ (N × (0, T)))N. The approximate controllability of (2.1) is then obtained as a singular limit of a sequence of optimal control problems. But, to do that, the lack of compactness of the Sobolev embedding in N is an obstacle. To avoid it we work on the similarity variables of the heat equation and the weighted Sobolev spaces introduced in [EK] that we recall now.

We introduce the new space-time variables

Then, given u = u(x, t) solution of (2.1) we introduce v(y, s) = esN/2u(es/2y, es – 1). It follows that u solves (2.1) if and only if v satisfies

where

The elliptic operator appearing in (2.3) may also be written as

where K = K(y) is the Gaussian weight K(y) = exp(|y|2/4). We introduce the weighted Lp-spaces:

For p = 2, L2(K) is a Hilbert space and the norm |·|L2(K) is induced by the inner product (f, g) = f(y)g(y)K(y)dy. We then define the unbounded operator L on L2(K) by setting Lf = –Df as above, and D(L) = {f Î L2(K) : Lf Î L2(K)}. Integrating by parts it is easy to see that

Therefore it is natural to introduce the weighted H1-space:

endowed with the norm . In a similar way, for any s Î and multi-index a we may introduce the space

Hs(K) {f Î L2 (K) : Da f Î L2 (K), "a, |a| < s}

The following properties were proved in [EK] and [K]:

is an isomorphism where (H1(K))* denotes the dual space of H1(K).

Since the operator L defined above has compact inverse in L2 (K), the equation

can be studied in the same manner as the heat equation in a bounded region W of N. Let us recall some important and useful facts about the spaces appearing this paper. First, we introduce some notation. In fact, given two separable Hilbert spaces V and H such that V Ì H with continuous embedding, V being dense in H, let us consider the Hilbert space W(0, T, V, H) = {u Î L2(0, T, V) : ut Î L2(0, T, H)}, equipped with the norm

uW(0,T,V,H) = see, for instance [DL].

We have

We represent by (, ), |·|, ((, )) and · the inner product and norm, respectively, of L2(K) and H1(K). We observe that the operator L is defined by the triplet

Therefore we have the following result about existence and uniqueness of the linear system (2.12). (See, for instance, [L1]).

Proposition 2.1. Given j Î L2 (0, S, L2(K)) and v0 Î L2(K), there exists a unique solution v in the space W(0, S, H1(K), L2(K)) of problem (2.12). Moreover, if v0 Î H1(K), then v Î W(0, S, H2(K), L2(K)).

 

3 Solutions by transposition and the linear case

In this section we give a precise definition of (2.12) in the sense of transposition and prove an existence and uniqueness result. The main question we are concerned here consists in finding a solution p of the parabolic problem:

when f Î (H1(K))*. Here (H1(K))* denotes the dual of H1(K). A function p Î L2(0, S, L2(K)) is called an solution of (3.1) obtained by transposition if

for any solution v of problem (2.12) with j Î L2(0, S, L2(K)) and v0(y) = 0 in N. We represent by á , ñ the duality pairing between (H1(K))* and H1(K). We have

Proposition 3.1. If f Î (H1(K))* then there exists an unique solution p Î L2(0, S, L2(K)) Ç C([0, S], (H1(K))*) of problem (3.1).

Proof. Let us consider the linear form F : L2 (0, S, L2(K)) defined by

where v is the solution of (2.12), with v0 = 0, corresponding to j.

Since v0 = 0, it is easy see that

Thus, F is a continuous linear form in L2(0, S, L2(K)). By Riesz's representation theorem, there exists a unique p Î L2(0, S, L2(K)) such that

The uniqueness is consequence of the Du Bois Raymond Lemma. Moreover,

Thus, by (3.2), (3.3) and (3.5), we have

Since f Î (H1(K))*, there exists (fm)mÎ with fm Î L2(K) for all m, such that

Let (pm)mÎ be the sequence of solutions corresponding to fm for each m. Obviously, the function pn – pm is the solution by transposition corresponding to fn – fm.

From (3.6) and (3.7) we have that (pm)mÎ is a Cauchy sequence in L2(0, S, L2(K)) and

On the other hand,

Then, for each y Î L2 (0, S, H2(K)) we have

and this implies that ()mÎ is a Cauchy sequence in L2(0, S, (H2(K))*).

Therefore, we have

pm p strongly in W(0, S, L2(K), (H2(K))*)

and this space is continuously embedded in C([0, S],(H1(K))*). We obtain pm p strongly in C([0, S], (H1(K))*). In particular, p Î C([0, S], (H1(K))*).

Through this work we set the notation q¢ = {(y, s), s Î (0, S), y Î w¢(s)}, where (s) = es/2 and w is an open nonempty subset of N.

To prove the approximate controllability of system (2.12) we employ the theorem of unique continuation due to C. Fabre [Fa], Theorem 1.4 of [Fa]. We have

Proposition 3.2. Let w be an open and nonempty set of N, (s) = es/2w and q¢ defined above. Assume that A(y, s) Î L¥(N × (0, S)), B(y, s) Î (L¥(N × (0,S)))N. Let p Î L2 (0, S, H1(K)) be such that

Then p º 0.

Proof. L, A and B do satisfy the conditions of Theorem 1.4 due Fabre [Fa] and the assumption of p implies that p Î (0, S, (N)). Consequently p º 0.

Now, we have all the ingredients to prove our result:

Theorem 3.1 (Approximate controllability in H1(K)). Given bounded, measurable potentials A and B with S > 0, for every v0 Î H1(K), the set of reachable states at time S > 0, is the solution of (2.12) with j Î L2 (0, S, L2(K))} is dense in H1 (K).

Proof. The proof follows the arguments in [FZ].

Because of the linearity of (2.12), we can assume without loss of generality that v0 = 0 and we denote the solution vj,0 by vj. Given a fixed element vd Î H1 (K) and k Î , we introduce the following optimal control problem

For all k Î the functional Jk is lower semicontinuous and strictly convex. Observe that the functional Jk is coercive in the Hilbert space of L2(0, S, L2(K)) of functions with support in q¢. Then, there exists a solution jk of (Pk) for each k Î . The derivative of Jk(j) is given by

for all x Î L2(0, S, L2(K)). If we evaluate · x in the solution jk of (Pk) we must have

for all x Î L2(0, S, L2(K)), with vk(S) = vjk(S).

Taking into account that Jk (jk) < Jk(0) = for every k Î , we deduce that {vk(S)-vd}k Î is a bounded sequence in H1(K) and is bounded in L2(0, S, L2 (K)). Then we can extract a subsequence {vk(S)-vd}k Î such that

From (3.12) we obtain:

By transposition we define the adjoint state p as the unique solution (cf. section 3) of the problem:

We obtain

for all z Î W(0, S, H2(K), L2(K)), such that z(0) = 0.

The solution p of (3.15) defined by (3.16) has the regularity:

cf. Proposition 3.1.

Let us consider z the solution of (2.12) when j = x; this solution we denote by vx. From relation (3.16) it follows that

It follows that p(x, t) = 0, a.e. (y, s) Î w¢ × (0, S) and by Proposition 3.2 we have p(x, t) = 0, a.e. (y, s) Î N × (0, S). Since p Î ([0, S], (H1(K))*) and p(S) = Ly with y Î H1(K) we have y = 0. From (3.13) we have

vk(S) vd weakly in H1(K).

The strong convergence follows from (3.12) with x = jk. Indeed, letting k ¥ in

we obtain that vk(S) vd strongly in H1(K).

Corollary 3.1. For every v0 Î H1(K) and m Î [0,1), the set (S) is dense in Hm(K).

Proof. This is a direct consequence of Theorem 3.1 and the density of H1(K) in Hm(K).

Corollary 3.2. For every v0 Î L2(K), the set (S) is dense in L2(K).

Proof. Let us consider v1 Î L2 (K) and e > 0. Since H1 (K) Ì L2(K) with dense inclusion, there exists a sequence {} Ì H1 (K) such that strongly in L2(K).

From Corollary 3.1, we know the existence of controls jn such that the solution of (2.12) with j = jn satisfies

Let Ñ > 0 be such that where c > 0 is a constant that will appear bellow. Consider vj, v0 the solution of (2.12) corresponding to v0 and j = jÑ. Let z = . Then z satisfies

We multiply (3.17) by zK(y) and integer over N. Then, there exists a constant c > 0 such that |z(S)|L2(K) < and, therefore,

We conclude with the

Theorem 3.2. Given u0 Î L2(N) the set of reachable states at time T > 0, RL(T) = {uh,u0(T) : uh, u0 is the solution of (2.1) with h Î L2(N × (0, T))} is dense in L2(N ).

Proof. Let us consider u1 Î L2(N) and e > 0. We divide the proof into three steps.

Step 1. The case u0 = 0, u1 Î L2(K).

Let us consider v1(y) = (T + 1)N/2 u1 ((T + 1)y) Î L2(K). From Corollary 3.2 we know the existence of j Î L2(0, S, L2(K)) such that |v(S) - v1|L2(K) < e, with S = log(T + 1). Then

is the solution of (2.1) with

and

Step 2. The case u0 = 0, u1 Î L2(N).

Since L2(K) Ì L2(N) with dense inclusion, there exists a sequence {} and Ñ > 0 such that | - u1|L2() < for every n > Ñ.

From the first step, we know the existence of controls hn such that un the solution of (2.1) with h = hn satisfies

Then, the solution u of (2.1) with h = hÑ satisfies

Step 3. The general case, i.e. u0, u1 Î L2(N) arbitrary.

We write u = z + Y where z is the solution of

Then z(T) Î L2(N). We construct h = h(u1 - z(T)) such that the solution of

satisfies

Therefore in (2.1) it is enough to chose h = h(u1 - z(T)) since the unique solution is u = z + Y and in view of (3.20) we conclude the proof.

 

4 The non linear case

We will now study the same problem for the semilinear heat equation (1.2). As in the linear case, a change of variables v(y, s) = transforms the system (1.2) in

with g(y, s, v, Ñ v) = es(N+2)/2 f(es/2 y, es - 1, e-s N/2 v, e-s(N+2)/2 Ñ v) and j(y, s) = es(N+2)/2 h(es/2 y, e2 - 1).

We must remark the function g = g(y, s, q, h) possesses the same properties as f. In particular, g is globally Lipschitz in the variables (q, h).

Observe that by (1.4) and because L2(K) is dense in L2(N) we can consider f(x, t, 0, 0) in L2 (0, T, L2(K)). This guarantees that g(y, s, 0, 0) Î L2(0, S, L2(K)). Thus, we suppose that f also satisfies

besides (1.3) and (1.5).

From the embedding (I1)-(I4) and g globally Lipschitz, it follows the following result on existence and uniueness for system (4.1).

Proposition 4.1. Suppose f satisfying the conditions (1.3), (1.5) and (4.2) above. Then, given j Î L2 (0, S, L2(K)) and v0 Î L2(K), there exists a unique solution v in the space W(0, S, H1(K), (H1(K))*) of problem (4.1). Moreover, if v0 Î H1(K), hence v Î W(0, S, H2(K), L2(K)).

Proof. The proof is the same given for Theorem 1 in [FZ].

We observe that, in view of Proposition 4.1 and change of variables, if u0 = v0 Î L2(K) and f Î L2(0, T, L2(K)) then there exists a unique solution u Î W(0, T, H1(K), (H1(K))*) of problem (1.2).

As usual, it is possible to derive the continuous dependence of the solution with respect to the initial data:

Proposition 4.2 (Continuity with respect to the data). Suppose g satisfying the conditions above.

a) Let us suppose v0 in L2(K) and jm j in L2(0, S, L2(K)). Then, in W(0, S, H1(K), (H1(K))*).

b) Let us suppose v0 in L2(K) and jm j in H1(0, S, L2(K)). Then, in W(0, S, H2(K), (L2(K))*).

To show the approximate controllability property of the semilinear heat equation, we are going to introduce a family of optimal control problems. A previous step for establishing the optimality conditions corresponding to their solutions in the study of the differentiability of the functional involved.

Proposition 4.3 (Differentiability with respect to the data). Suppose that g satisfies the conditions above. Given v0 Î H1 (K), the functional F : L2(0, S, L2(K)) H1(K) defined by F(j) = vj, v0 (S) is differentiable. Moreover, DF (j)y = zy (S), where zy is the unique solution in W(0, S, H2(K), L2(K)) of the following linearized problem

Proof. The functional F is well defined by Proposition 4.1 and the embedding (2.16). Given j, y Î L2(0, S, L2(K)) and l Î (0,1), let us denote vl = vj+ly,v0, v = vj,v0 and zl = (vl - v). To show that F is Gâteaux differentiable we have to prove that

It is clear that for each l we have

where zl,s denotes .

By the Mean Value Theorem, we have

where wl = v + x(vl - v), 0 < x < 1 depends on y, s and l. Applying (4.5) to zlK(y), integrating by parts, using the fact that g is globally Lipschitz in the variable (v, Ñv) and Young's inequality, we deduce the existence of a constant c1 > 0 such that

Using Gronwall inequality, we have

|zl(s)|L2(K) < ec1S|y|L2(0,S,L2(K)), " s Î [0, S], " l Î (0, 1).

Thus, the sequence {zl} is bounded in ([0, S], L2(K)) and by (4.7), {zl} is bounded in L2(0, S, H1(K)). In fact, |zl|L2(0,S,H1(K)) < ec1S |y|L2(0,S,L2(K)), " l Î (0,1). Taking into account this estimate together with the equation (4.5), (4.6) and hypothesis on g, we conclude that there exists a positive constant c2 (independent on l and y) such that

|zl|W(0,S,H1(K),H1(K))*) < c2|y|L2(0,S,L2(K)), " l Î (0, 1).

In view of the expression of gl and using classical estimates, we deduce the existence of a positive constant c3 (still independent of l and y) such that

Thus, by extracting subsequences,

Combining this convergence with the compact imbedding (2.15) and Proposition (4.2b), we deduce that

It is then easy to see, by the well-posedness properties of (4.5), that = zy and that the convergence in (4.9) holds in the strong topology. Using the embedding (2.16), we obtain (4.4).

We have the following result on approximate controllability for the system (4.1):

Theorem 4.4. For each v0 Î H1(K) the set of admissible states at S > 0,

(S) = {vj,v0 (S): vj,v0 is solution of (4.1) with j Î L2(0, S, L2(K))}

is dense in Hm(K), for each 0 < m < 1.

Proof. We know that H1(K) is dense in Hm(K) for 0 < m < 1. Then, it is sufficient to prove that (S) is dense in H1(K) with the norm of Hm(K) for 0 < m < 1.

In fact, let us take vd Î H1(K), 0 < m < 1 and consider the functional defined in L2(0, S, L2(K)) by:

This functional is lower semicontinuous and coercive in the Hilbert space of L2(0, S, L2(K)) of functions with support in q¢. It follows that the minimization problem (Pk) given by:

has at least one solution jk.

Thanks to Proposition 4.3, the first order optimality condition associated with the minimization problem (Pk) at this minimum gives

for all y Î L2 (0, S, L2(K)), where vk = vjk,v0 and is the unique function in W(0, S, H2(K), L2(K)) solution of the problem

By the minimum condition we have Jk(jk) < Jk(0) for every k Î . It follows that {vk(S) - vd}k Î is a bounded sequence in Hm(K) and is a bounded sequence in L2(0, S, L2(K)). Thus, by extracting subsequences,

Moreover, there exist c Î L¥(N × (0, S)) and d Î (L¥(N × (0, S)))N such that

Let zy be the unique solution of the problem

Suppose that

By (4.10), (4.12) and (4.15) we obtain:

and when k ¥, we have

By Theorem 3.1 the set {zy(S)} is dense in H1(K) and, therefore, in Hm(K), when y varies in L2(0, S; L2(K)). Consequently, (4.16) implies that q º 0. Therefore, according to (4.12), vk(S) vd weakly in Hm(K). Since the embedding Hm(K) Ì Hm¢(K) is compact for m¢ < m, we conclude the strong convergence in Hm¢(K), for any m¢ < m. Taking into account that this holds for all m < 1, we conclude the density in Hm(K), as stated in Theorem 4.4.

To complete the proof we need to proof the convergence (4.15). In fact, multiplying (4.11) by K(y), integrating in N and observing that

we have

Multiplying (4.11) by K(y) ()¢ and integrating in N, we obtain:

From (4.18) and (4.19) it follows that {}k Î is bounded in W(0, S, H2(K), L2(K)). Since W(0, S, H2(K), L2(K)) is compactly embedded in L2(0, S, H1(K)), there exists subsequences such that

From (4.15) and (4.20) we can pass to the limit in (4.11) obtaining that c is the solution of (4.14) and by uniqueness c = zy. On the other hand, from (4.20) and the continuous embedding I4) we have:

and k tends to +¥. The embedding of H1(K) into Hm(K) into Hm(K) being compact, for m < 1. This proves (4.15).

Corollary 4.1. Given v0 Î L2 (K), the set (S) is dense in L2(K).

Proof. As in the proof of Corollary 3.2, let us consider v1 Î L2(K) and e > 0. Then, there exists a sequence {}n Î Ì H1(K) such that v0 strongly in L2(K). Moreover, as we saw previously, there exists a sequence of controls jn such that , the solution of (4.1) corresponding to and jn, satisfies

Let Ñ > 0 be such that

where c > 0 is a constant that will be timely introduced.

Consider vj,v0 the solution of (4.1) corresponding to v0 and j = jÑ. Let . Then, z satisfies

We multiply (4.21) by zK(y) and integer over N. Since g satisfies (4.17), we obtain

We conclude with the proof of Theorem 1.1.

Proof of Theorem 1.1. Let us consider a > 0 and u1 Î L2(N). As in the proof of Theorem 3.2 we divide into several steps.

First step. The case u0, u1 in L2(K).

We make the change of variables v1(y) = (1 + T)N/2u1((1 + T)1/2y) and S = log(1 + T). Then v1 Î L2(K) and by Corollary 4.1, there exists j Î L2(0, S, L2(K)) such that the solution v of (4.1) corresponding to v0 and j, satisfies

We define

Then u is solution of (1.2) with

and |u(T) – < a2 (cf. Theorem 3.2, Step 1).

Second step. The case u Î L2(K), u1 Î L2(N).

Since L2(K) Ì L2(N) with dense inclusion, there exists a sequence {}n Î Ì L2(K) such that u1 strongly in L2(N). From the First Step, we know that existence of controls hn such that un the solution of (1.2) satisfies . Let Ñ > 0 such that for every n > Ñ, .

Then, u the solution of (1.2) with h = hÑ satisfies |u(T) - u1|L2() < a.

Third step. The case u0 Î L2(N).

Then there exists a sequence {} with Î L2(K) such that u0 strongly in L2(N). Given u1 Î L2(N) and a > 0, as we saw previously, there exists a sequence of controls hn such that un, the solution of (1.2) corresponding to and hn, satisfies

Let Ñ > 0 be such that

K0 the constant given in (1.5).

Consider u the solution of (1.2) corresponding to u0 and h = hÑ. Let z = u-uÑ. Then, z satisfies

We multiply (4.22) by z and integer over N. Since f satisfies (1.5), we obtain

Then, and therefore

 

5 Conical domains

Consider a cone-like domain W satisfying:

All the results of these paper hold for the semilinear heat equation (1.1) when W is a cone-like domain. In fact, consider a domain W satisfying (5.1). Then if one consider the evolution equation

we can study the controllability of the equation (5.2) in the same way as the case of the whole space N. Observe that defining v, g and j like in this section we obtain that v satisfies

The operator Lf = - div(K Ñ f), K(y) = exp (|y|2/4), is self-adjoint in L2(K, W) with compact inverse and

where

Indeed, the methods of the previous sections apply. First of all, the approximate controllability of the linearized equation can be obtained as a consequence of the unique continuation result in [Fa]. Then, the approximate controllability for the semilinear system can be proved by viewing this property as the limit of optimality conditions of penalized optimal control problems.

 

6 Acknowledgement

My appreciation to E. Zuazua who drew my attention to this problem, for reading the manuscript and for his valuable suggestions.

 

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Received: 2/II/02.

 

 

#543/02.
*Partially supported by the Alfa Project of the EU ''Modelisation et Ingenierie Mathématique'' during the preparation of this work at Universidad Complutense de Madrid.