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

Abstract

We prove the approximate controllability of the semilinear heat equation in R N, when the nonlinear term is globally Lipschitz and depends both on the state u and its spatial gradient <FONT FACE=Symbol>Ñ</FONT>u. 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.

approximate controllability; optimal control; unbounded domains; weighted Sobolev spaces

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: [e-mail]

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) 1_w 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 u⁰Î L²(W), system (1.1) is said to be approximately controllable at time T in L²(W) if the reachable set

R_NL(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 W_R = W Ç B_R, where B_R 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 W_R 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 u⁰Î L² (W), u¹Î H(W) with 0 < s < 1 and k > 0, let us consider the functional

where u_h denotes the solution of (1.1) with control h. The functional J_k 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 h_k of J_k in L² (W × (0, T)). Let us denote by u_k the solution of (1.1) associated to this minimizer. It is shown in [FZ] that

as k +¥, and therefore u_k(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 K₀is a positive constant. Then, for any u⁰Î L²(^N) and T > 0, the set of reachable states (at time T > 0) given by

R_NL(T) = {u_h,u º (x, T): u_h,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) = e^sN/2u(e^s/2y, e^s – 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 L^p-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

H^s(K) {f Î L² (K) : D^af 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) : u_t Î L²(0, T, H)}, equipped with the norm

u

_W(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 v⁰Î L²(K), there exists a unique solution v in the space W(0, S, H¹(K), L²(K)) of problem (2.12). Moreover, if v⁰Î 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 (f_m)_mÎ with f_m Î L²(K) for all m, such that

Let (p_m)_mÎ be the sequence of solutions corresponding to f_m for each m. Obviously, the function p_n – p_m is the solution by transposition corresponding to f_n – f_m.

From (3.6) and (3.7) we have that (p_m)_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

p_mp strongly in W(0, S, L²(K), (H²(K))^*)

and this space is continuously embedded in C([0, S],(H¹(K))^*). We obtain p_m 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 w¢(s) = e^–s/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, w¢(s) = e^–s/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 v⁰Î 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 v_j,0 by v_j. Given a fixed element v_d Î H¹ (K) and k Î , we introduce the following optimal control problem

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

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

for all x Î L²(0, S, L²(K)), with v_k(S) = v_jk(S).

Taking into account that J_k (j_k) < J_k(0) = for every k Î , we deduce that {v_k(S)-v_d}_kÎ is a bounded sequence in H¹(K) and is bounded in L²(0, S, L² (K)). Then we can extract a subsequence {v_k(S)-v_d}_{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 v_x. 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

v_k(S) v_d weakly in H¹(K).

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

we obtain that v_k(S) v_d strongly in H¹(K).

Corollary 3.1. For every v⁰Î H¹(K) and m Î [0,1), the set (S) is dense in H^m(K).

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

Corollary 3.2. For every v⁰Î L²(K), the set (S) is dense in L²(K).

Proof. Let us consider v¹Î 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 j_n such that the solution of (2.12) with j = j_n satisfies

Let Ñ > 0 be such that where c > 0 is a constant that will appear bellow. Consider v_{j, v0} the solution of (2.12) corresponding to v⁰ and j = j_Ñ. 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 u⁰Î L²(^N) the set of reachable states at time T > 0, R_L(T) = {u_h,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 u¹Î L²(^N) and e > 0. We divide the proof into three steps.

Step 1. The case u⁰ = 0, u¹Î L²(K).

Let us consider v¹(y) = (T + 1)^N/2u¹ ((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, u¹Î 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 h_n such that u_n the solution of (2.1) with h = h_n satisfies

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

Step 3. The general case, i.e. u⁰, u¹Î 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) = e^s(N+2)/2f(e^s/2y, e^s - 1, e^{-s N/2}v, e^-s(N+2)/2Ñ v) and j(y, s) = e^s(N+2)/2h(e^s/2y, 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 v⁰Î L²(K), there exists a unique solution v in the space W(0, S, H¹(K), (H¹(K))^*) of problem (4.1). Moreover, if v⁰Î 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 = v⁰Î 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 j_mj in L²(0, S, L²(K)). Then,

in W(0, S, H¹(K), (H¹(K))^*).

b) Let us suppose

v⁰ in L²(K) and j_mj 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 v⁰Î H¹ (K), the functional F : L²(0, S, L²(K)) H¹(K) defined by F(j) = v_{j, v0} (S) is differentiable. Moreover, DF (j)y = z_y (S), where z_y 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 v_l = v_j+ly,v⁰, v = v_j,v⁰ and z_l = (v_l - v). To show that F is Gâteaux differentiable we have to prove that

It is clear that for each l we have

where z_l,s denotes .

By the Mean Value Theorem, we have

where w_l = v + x(v_l - v), 0 < x < 1 depends on y, s and l. Applying (4.5) to z_lK(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

|z_l(s)|_L²(K)< e^c1^S|y|_{L²(0,S,L²(K))}, " s Î [0, S], " l Î (0, 1).

Thus, the sequence {z_l} is bounded in ([0, S], L²(K)) and by (4.7), {z_l} is bounded in L²(0, S, H¹(K)). In fact, |z_l|_{L²(0,S,H¹(K))}< e^c1^S |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

|z_l|_W(0,S,H¹_(K),H¹_(K))*)< c₂|y|_L²_(0,S,L²_(K)), " l Î (0, 1).

In view of the expression of g_l 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 = z_y 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 v⁰Î H¹(K) the set of admissible states at S > 0,

(S) = {v_j,v⁰ (S): v_j,v⁰is solution of (4.1) with j Î L²(0, S, L²(K))}

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

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

In fact, let us take v_d 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 (P_k) given by:

has at least one solution j_k.

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

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

By the minimum condition we have J_k(j_k) < J_k(0) for every k Î . It follows that {v_k(S) - v_d}_kÎ is a bounded sequence in H^m(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 z_y 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 {z_y(S)} is dense in H¹(K) and, therefore, in H^m(K), when y varies in L²(0, S; L²(K)). Consequently, (4.16) implies that q º 0. Therefore, according to (4.12), v_k(S) v_d weakly in H^m(K). Since the embedding H^m(K) Ì H^m¢(K) is compact for m¢ < m, we conclude the strong convergence in H^m¢(K), for any m¢ < m. Taking into account that this holds for all m < 1, we conclude the density in H^m(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 {}_kÎ 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 = z_y. On the other hand, from (4.20) and the continuous embedding I4) we have:

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

Corollary 4.1. Given v⁰Î L² (K), the set (S) is dense in L²(K).

Proof. As in the proof of Corollary 3.2, let us consider v¹Î L²(K) and e > 0. Then, there exists a sequence {}_nÎÌ H¹(K) such that

v⁰ strongly in L²(K). Moreover, as we saw previously, there exists a sequence of controls j_n such that , the solution of (4.1) corresponding to and j_n, satisfies

Let Ñ > 0 be such that

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

Consider v_j,v⁰ the solution of (4.1) corresponding to v⁰ and j = j_Ñ. 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 u¹Î 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 v¹Î 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), u¹Î L²(^N).

Since L²(K) Ì L²(^N) with dense inclusion, there exists a sequence {}_nÎÌ L²(K) such that

u¹ strongly in L²(^N). From the First Step, we know that existence of controls h_n such that u_n the solution of (1.2) satisfies . Let Ñ > 0 such that for every n > Ñ, .

Then, u the solution of (1.2) with h = h_Ñ satisfies |u(T) - u¹|_L²(

₎< a.

Third step. The case u⁰Î L²(^N).

Then there exists a sequence {} with Î L²(K) such that

u⁰ strongly in L²(^N). Given u¹Î L²(^N) and a > 0, as we saw previously, there exists a sequence of controls h_n such that u_n, the solution of (1.2) corresponding to and h_n, 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 = h_Ñ. Let z = u-u_Ñ. 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.

Received: 2/II/02.

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*

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

Publication Dates

• Publication in this collection
  19 July 2004
• Date of issue
  2003

History

• Received
  02 Feb 2002