Repositório Institucional da UFPA: Approximate controllability for the semilinear heat equation in RN involving gradient terms

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  Computational and Applied Mathematics Vol. 22, N. 1, pp. 123–148, 2003 Copyright © 2003 SBMAC

  

Approximate controllability for the semilinear heat

N

equation in R involving gradient terms

SILVANO BEZERRA DE MENEZES*

  

Departamento de Matemática, Universidade Federal do Pará

66075-110 Belém, PA, Brazil

E-mail: silvano@ufpa.br

  N

  ,

Abstract. We prove the approximate controllability of the semilinear heat equation in R

when the nonlinear term is globally Lipschitz and depends both on the state u and its spatial

gradient

  ▽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.

  93B05, 73C05, 35B37.

  Mathematical subject classification: approximate controllability, optimal control, unbounded domains, weighted

  Key words: Sobolev spaces.

1 Introduction

  N Let be a domain of R with N

  ⊆ 1. Given T > 0 and an open nonempty subset ω of we consider the following semilinear heat equation u in Q t − u + f (x, t, u, u) = h1 ω = × (0, T )

  (1.1) u = 0 on ∂ × (0, T ) u(x, 0) (x) in .

  = u In (1.1) 1 denotes the characteristic function of ω. Roughly speaking, f is

  ω assumed to be mensurable in (x, t) and globally Lipschitz in the variables u and #543/02. Received: 2/II/02.

  • Partially supported by the Alfa Project of the EU ‘‘Modelisation et Ingenierie Mathéma- tique’’ during the preparation of this work at Universidad Complutense de Madrid.

  N

  124 HEAT EQUATION IN R

  INVOLVING GRADIENT TERMS ▽u. 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 ω. The approximate controllability problem can be formulated as follows: Given a finite time horizon T > 0 and an arbitrary

  2 element u ( ) , system (1.1) is said to be approximately controllable at

  ∈ L

  2 time T in L ( ) if the reachable set

  2 R (T ) (Q)

  N L = {u(x, T ): u is solution of (1.1) with h ∈ L }

  2 is dense in L ( ) .

  There are numerous works treating the approximate controllability in the linear parabolic framework, see [L2] and its bibliography, when 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 is an unbounded set since the com- pacteness 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 controllabil- ity 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 by a sequence of bounded domains , where B denotes the ball centered in zero of radius R. It is then

  R = ∩ B R R proved that, in the limit as R tends to infinity, the approximate controls in

  R provide an approximate control in the unbounded domain . N

  Later on, L. de Teresa [T] proved the approximate controllability when = R 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 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,

  SILVANO BEZERRA DE MENEZES 125 inspired in the work by J.L. Lions [L2] in which, roughly speaking, the ap- proximate controllability is viewed as the limit of a sequence of optimal control s

  2

  1 problems. More precisely, given u ( ) , u ( ) with 0 < s < 1 and

  ∈ L ∈ H k > 0, let us consider the functional T

  1 k

  2

  1

  2

  J (h) h (x, t )dxdt (T ) k = u h − u

  • s

  H ( )

  2

  2 ω where u denotes the solution of (1.1) with control h. The functional J is well h k

  2 defined for h (

  ∈ L × (0, T )). On the other hand, it is shown that, for each

  2 k > 0, there exists a minimizer h of J in L ( k k × (0, T )). Let us denote by u k the solution of (1.1) associated to this minimizer. It is shown in [FZ] that

  1 s u (T ) ⇀ u weakly in H ( ) k

  1

  2 as k (T ) in L ( ) as k

  → +∞, and therefore u k → u → +∞. This fact, s

  2 combined with the fact that H ( ) is dense in L ( ) , guarantees the approximate controllability of system (1.1).

  This paper is devoted to analyze the approximate controllability of system (1.1) N in case when . We adopt the approach in [FZ] but in the more general case

  = R N

  . However, in order to avoid the difficulties associated with the lack of = R 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

   −p t − p + α(x, t)p + div (b(x, t)p) = 0

    in Q

  = × (0, T ) ⇒ p(x, t) = 0 in Q   p(x, t )

  = 0 in q = ω × (0, T ) N

  ∞ ∞ provided a (Q) and b (Q)) .

  ∈ L ∈ (L N N

  Thus, we consider the domain to be R , ω an open nonempty subset of R

  N

  126 HEAT EQUATION IN R

  INVOLVING GRADIENT TERMS and analyze the approximate controllability of the system:

  N R u in t ω

  − u + f (x, t, u, ▽u) = h1 × (0, T ) (1.2)

  N R u(x, 0) (x) in

  = u The main result of this paper is the following:

  Theorem 1.1. Let us assume the following hypotheses:

  f (x, t, θ, η) is a measurable function with respect to N

  1 (1.3)

  (x, t ) function with respect to ∈ R × (0, T ) and a C

  N (θ, η)

  ∈ R × R

  2 N 0, 0) (1.4) f (x, t, (R

  ∈ L × (0, T )) ∂f ∂f

  (x, t, θ, η) (x, t, θ, η) , ≤ K +

  ∂θ ∂η (1.5)

  N N ∀ (x, t, θ, η) ∈ R × (0, T ) × R × R

  2 N

  

where K is a positive constant. Then, for any u (R ) and T > 0, the set

  ∈ L

  of reachable states (at time T > 0) given by

  R (T ) N L = {u h,u ◦ (x, T ): u h,u ◦ is the solution of (1.2) with

  2 N h (R

  ∈ L × (0, T ))}

2 N is dense in L (R ).

  Remark 1.1.

a) As we mentioned above, the method of proof applies in the case where

  N is a cone of R (i.e. λ

  = , ∀ λ > 0). Extending Theorem 1.1 to the case of general open unbounded sets is an open problem. At this respect, the approximation method developed in [TZ] may be the most suitable tool. SILVANO BEZERRA DE MENEZES 127

  1

b) By an approximation argument, the assumption that f is C in the variables

  (u, ▽u) can be easily relaxed. It is sufficient f to be Lipschitz in those variables.

c) The assumption that f is globally Lipschitz in (u,

  ▽u) 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:

  N R u in t ω

  − u + a(x, t)u + b(x, t) · ▽u = h1 × (0, T ) (2.1)

  N R u(x, 0) in

  = u N N N

  ∞ ∞ with a(x, t) (R (R . The

  ∈ L × (0, T )) and b(x, t) ∈ (L × (0, T ))) 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

  N the Sobolev embedding in R 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.

  N

  128 HEAT EQUATION IN R

  INVOLVING GRADIENT TERMS We introduce the new space-time variables x y s (2.2) = √ ; = log(t + 1). t

  • 1 Then, given u

  = u(x, t) solution of (2.1) we introduce v(y, s) = sN/ 2 s/ 2 s e u(e y, e

  − 1). It follows that u solves (2.1) if and only if v satisfies y N · ▽v

  ′

  v v (s) s − v − + A(y, s)v + B(y, s) · ▽v − = ϕ(y, s)1 ω

  2

  2 N R in × (0, S)

  N R v(y, 0) (y) (y) in

  = v = u (2.3) where s s/ 2 s

  A(y, s) a(e y, e = e − 1) s/ 2 s/ 2 s

  B(y, s) b(e y, e = e − 1) s(N s/

  2

  2

  • 2)/2

  ϕ(y, s) h(e y, e (2.4) = e − 1)

  S = log(T + 1)

  ′ −s/2 w (s) ω

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

  1 · ▽v

  Lv div (K(y) (2.5) = − v − = − ▽ v)

2 K(y)

  2 where K

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

  1/p p p

  L (K) = : |f | = |f (y)| ∞

  p L (K) f K(y)dy < .

  

N

  R

  2

  = 2, L | · | L (K) inner product (f, g) N f (y)g(y)K(y)dy . We then define the unbounded = R y

  2 For p (K) is a Hilbert space and the norm is induced by the

  2 ·▽f operator L on L (K) by setting Lf as above, and D(L)

  = − f − = {f ∈

  2

  2

2 L (K) (K)

  : Lf ∈ L }. Integrating by parts it is easy to see that

  2 (Lf )f K dy K dy. (2.6)

  = | ▽ f |

  N N

  R R

  SILVANO BEZERRA DE MENEZES 129

  1 Therefore it is natural to introduce the weighted H -space: ∂f

  1

  2

  2 H (K) f (K) (K), i = ∈ L : ∈ L = 1, . . . , n

  ∂y i

  1/2

  2

  2 endowed with the norm N ( )K dy . In a similar f H

  1 (K) = R |f | + | ▽ f |

  way, for any s ∈ N and multi-index α we may introduce the space s 2 α

  2 H (K) f (K) f (K), .

  ∈ L : D ∈ L ∀ α, |α| ≤ s The following properties were proved in [EK] and [K]:

  2

  2

  2 K(y) dy K(y) dy (2.7) |f (y)| |y| ≤ c | ▽ f |

  N N

  R R

  2

  2 The embedding H (K) ֒ (K) is compact. (2.8) → L

  1

  1 ∗

  L (K) (K)) (2.9) : H → (H

  1

  1 ∗ is an isomorphism where (H (K)) denotes the dual space of H (K) .

  2 D(L) (K) (2.10) = H

  2

  2 −1

  L (K) (K) is self-adjoint and compact. (2.11) : L → L

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

  ′

  v v s + Lv + A(y, s)v + B(y, s) · ▽v − = ϕ(y, s)1 ω (s)

  2 N (2.12)

  R in × (0, S)

  N R v(y, 0) (y) in

  = v can be studied in the same manner as the heat equation in a bounded region of N R

  . 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

  ⊂ H with continuous embedding, V being dense in

  2 H , let us consider the Hilbert space W (0, T , V , H ) ( 0, T , V ) t = {u ∈ L : u ∈

2 L ( 0, T , H )

  }, equipped with the norm 1/2

  2

  2

  2 2 see, for instance [DL].

  W ( 0,T ,V ,H ) t u = |u| + |u |

  L ( 0,T ,V ) L ( 0,T ,H )

  N

  130 HEAT EQUATION IN R

  INVOLVING GRADIENT TERMS We have

  1

  1

  2

  2 ∗

  W ( 0, T , H (K), (H (K)) ) ( 0, T , L (K)) ⊂ L with compact embedding (2.13)

  1

  1

  2 ∗

  W ( 0, T , H (K), (H (K)) ) (K)) ⊂ C([0, T ], L with continuous embedding (2.14)

  2

  2

  2

  1 W ( 0, T , H (K), L (K)) ( 0, T , H (K)) ⊂ L with compact embedding (2.15)

  2

  2

  1 W ( 0, T , H (K), L (K)) (K)) ⊂ C([0, T ], H with continuous embedding (2.16)

  We represent by ( , ), | · |, (( , )) and · the inner product and norm, respectively,

  2

  1 of L (K) and H (K) . We observe that the operator L is defined by the triplet

  1

2 H (K), L (K), (( , )) , with ((u, v)) = ▽u · ▽vK(y)dy.

  N

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

  2

  2

  2 Proposition 2.1. Given ϕ ( 0, S, L (K)) and v (K), there exists ∈ L ∈ L

  1

  2

a unique solution v in the space W ( 0, S, H (K), L (K)) of problem ( 2.12).

  1

  2

  2 Moreover, if v (K), then v (K), L (K)).

  ∈ H ∈ W (0, S, H

3 Solutions by transposition and the linear case

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

  N

  1 s p y

  −p + Lp + Ap − div(Bp) − − · Bp = 0

  2

  2 N (3.1)

  R in × (0, S)

  N R p(y, S) in ,

  = f

  SILVANO BEZERRA DE MENEZES 131

  1

  1

  1 ∗ ∗ when f (K)) . Here (H (K)) denotes the dual of H (K) . A function

  ∈ (H

  2

  2 p ( 0, S, L (K)) is called an solution of (3.1) obtained by transposition if

  ∈ L S p(y, s)ϕ(y, s)K(y)dy

  = f, v(S)

  N

  R

  2

  2 for any solution v of problem (2.12) with ϕ ( 0, S, L (K)) and v (y)

  ∈ L = 0 N

  1

  1 ∗ in R . We represent by (K)) and H (K) .

  , the duality pairing between (H We have

  1 ∗

  Proposition 3.1. If f (K)) then there exists an unique solution p

  ∈ (H ∈

  2

  2

  1 ∗ L ( 0, S, L (K)) (K)) ) of problem ( 3.1).

  ∩ C([0, S], (H

  2

  2 Proof. Let us consider the linear form F ( 0, S, L (K)) : L → R defined by

  2

  2 F (ϕ) ( 0, S, L (K)) (3.2) = f, v(S) for all ϕ ∈ L where v is the solution of (2.12), with v

  = 0, corresponding to ϕ. Since v

  = 0, it is easy see that

  1

  2

  2

  . (3.3) v(s) H (K) ≤ C|ϕ| L ( 0,S,L (K))

  2

  2 Thus, F is a continuous linear form in L ( 0, S, L (K)) . By Riesz’s representa-

  2

  2 tion theorem, there exists a unique p ( 0, S, L (K)) such that

  ∈ L S

  2

  2 F (ϕ) pϕK(y) dyds, for all ϕ ( 0, S, L (K)). (3.4) = ∈ L

  N

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

  2

  2

  2

  2

  . (3.5) |F | L ( 0,S,L (K)) = |p| L ( 0,S,L (K))

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

  2

  2

  1 ∗ . (3.6)

  |p| L ( 0,S,L (K)) ≤ C|f | (H (K))

  1

  2 ∗

  Since f (K)) , there exists (f ) with f (K) for all m, such that m m m

  ∈ (H ∈ L ∈N

  1 ∗ f (K)) . (3.7) m → f strongly in (H N

  132 HEAT EQUATION IN R

  INVOLVING GRADIENT TERMS Let (p ) be the sequence of solutions corresponding to f for each m. Ob- m m m

  ∈N viously, the function p is the solution by transposition corresponding to n m

  − p f . n m

  − f From (3.6) and (3.7) we have that (p ) is a Cauchy sequence in m m

  ∈N

  2

2 L ( 0, S, L (K)) and

  2

  2 p ( 0, S, L (K)). (3.8) m

  → p strongly in L On the other hand,

  ′ ′ p ) ) )

  − p = L(p m − p n + A(p m − p n − div (B(p m − p n m n

  N

  1 (3.9)

  (p m n ) y m n ) − − p − · B(p − p

  2

  2

  2

  2 ∗ in L ( 0, S, (H (K)) ).

  2

2 Then, for each ψ ( 0, S, H (K)) we have

  ∈ L ′ ′

  , ψ , Lψ ), ψ m n m n

  | p m − p n | ≤ | p − p | + | A(p − p | N

  , B , ψ m n m n

  • | p − p · ▽ψ | + | p − p |

  2

  2

  2

  2

  2

  m n 0,S,L 0,S,H ≤ c|p − p | L ( (K)) |ψ| L ( (K))

  2

  2 ′

  ∗ and this implies that (p ) is a Cauchy sequence in L ( 0, S, (H (K)) ) . m m

  ∈N Therefore, we have

  2

  2 ∗ p (K), (H (K)) ) m → p strongly in W (0, S, L

  1 ∗ and this space is continuously embedded in C( (K)) ) . We

  [0, S], (H

  1 ∗ obtain p (K)) ) . In particular, p m → p strongly in C([0, S], (H ∈

  1 ∗ C( (K)) ) .

  [0, S], (H ′ ′

  Through this work we set the notation q (s) = {(y, s), s ∈ (0, S), y ∈ w },

  N ′ −s/2 where ω (s) and ω is an open nonempty subset of R .

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

  SILVANO BEZERRA DE MENEZES 133 N ′ −s/2

  

Proposition 3.2. Let w be an open and nonempty set of R , ω (s) ω

  = e N

  ′ ∞

  and q defined above. Assume that A(y, s) (R

  ∈ L × (0, S)), B(y, s) ∈ N N

  2

  1 ∞

  (L (R . Let p ( 0, S, H (K)) be such that × (0, S))) ∈ L

  N

  1 p y s

  −p + Lp + Ap − div (Bp) − − · Bp = 0

  2

  2 N (3.10)

  in R

  × (0, S) ′ p

  = 0 in q

  Then p ≡ 0.

  

Proof. L , A and B do satisfy the conditions of Theorem 1.4 due Fabre [Fa]

  2

  1 N and the assumption of p implies that p ( 0, S, H (R )) . Consequently

  ∈ L loc loc p ≡ 0.

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

  1 Theorem 3.1 (Approximate controllability in H (K)). Given bounded, mea-

  1

  

surable potentials A and B with S > 0, for every v (K), the set of reach-

  ∈ H

  

able states at time S > 0, ˆ R (S) (S) is the solution of ( 2.12) with

  L = {v ϕ,v : v ϕ,v

  2

  2

  1 ϕ ( 0, S, L (K)) (K).

  ∈ L } is dense in H Proof. The proof follows the arguments in [FZ]. Because of the linearity of (2.12), we can assume without loss of generality that

  1 v by v . Given a fixed element v (K)

  ϕ, ϕ d = 0 and we denote the solution v ∈ H and k

  ∈ N, we introduce the following optimal control problem 

  ′ 1 k

  2

  2  minimize J (ϕ)

  ϕ K(y) dyds (S)

  k ϕ d = v − v

  • 1

  H (K) P (k)

  2

  2 q

  

  2

  2 over ϕ ( 0, S, L (K))

  ∈ L For all k k is lower semicontinuous and strictly convex. ∈ N the functional J

  2

  2 Observe that the functional J k is coercive in the Hilbert space of L ( 0, S, L (K)) ′ of functions with support in q . Then, there exists a solution ϕ k of (P k ) for each is given by k k (ϕ)

  ∈ N. The derivative of J ′

  ϕ d ξ = + k((v − v H (K) k

  1 J (ϕ)ξ ϕξ K(y) dyds (S) , v (S))) (3.11)

  ′

  q

  N

  134 HEAT EQUATION IN R

  INVOLVING GRADIENT TERMS

  2

  2 ′ for all ξ ( 0, S, L (K)) . If we evaluate J (ϕ) of (P ) k k

  ∈ L · ξ in the solution ϕ k we must have

  ϕ ξ K(y) dyds (S) , v (S))) (3.12) k + k((v k − v d ξ = 0

  ′

  q

  2

  2 for all ξ ( 0, S, L (K)) , with v (S) (S) .

  ∈ L k = v ϕ k k

  2 Taking into account that J (ϕ ) ( 0)

  1 for every k

  k k ≤ J k = v d ∈ N, we H (K)

  2

  1

  1 deduce that (S) is a bounded sequence in H (K) and ϕ 1 ′ k d k k ω

  {v −v } √ ∈N k k

  ∈N

  2

  2 is bounded in L ( 0, S, L (K)) . Then we can extract a subsequence (S) k

  {v − v such that d k

  } ∈N

  1 v (S) ⇀ ψ weakly in H (K). (3.13) k − v d

  From (3.12) we obtain:

  2

  2 ((ψ, v (S))) ( 0, S, L (K)). (3.14)

  ξ = 0 for all ξ ∈ L By transposition we define the adjoint state p as the unique solution (cf. section 3) of the problem:

  N

  1 s p y

  −p + Lp + Ap − div (Bp) − − · Bp = 0

  2

  2 N (3.15)

  R in × (0, S)

  N p(y, S)

  = Lψ in R We obtain

  S N p(y, s) z z K(y) dyds s + Lz + Az + B · ▽z −

  N

  R

  2 (3.16)

  = Lψ, z(S)

  2

  2 for all z (K), L (K)) , such that z(0) ∈ W (0, S, H = 0.

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

  2

  2

  1

  1 ∗ ∗ p ( 0, S, L (K)) (K)) ) since Lψ (K)) ),

  ∈ L ∩ C([0, S], (H ∈ (H cf. Proposition 3.1.

  SILVANO BEZERRA DE MENEZES 135 Let us consider z the solution of (2.12) when ϕ

  = ξ; this solution we denote by v . From relation (3.16) it follows that ξ

  S p(y) ξ K(y) 1 ′ dyds (S)))

  N

  ω ξ = ((ψ, v = 0, ∀ ξ.

  R ′

  It follows that p(x, t) = 0, a.e. (y, s) ∈ w × (0, S) and by Proposition 3.2 we

  N

  1 ∗ have p(x, t)

  (K)) ) = 0, a.e. (y, s) ∈ R × (0, S). Since p ∈ C([0, S], (H

  1 and p(S) (K) we have ψ

  = Lψ with ψ ∈ H = 0. From (3.13) we have

  1 v (S) ⇀ v weakly in H (K). k d

  The strong convergence follows from (3.12) with ξ . Indeed, letting k = ϕ k → ∞ in

  1

  2

  2

  1

  ϕ K(y) dyds (S)

  1 (S) , v ))

  • v k − v d = −((v k − v d d H (K)

  k H (K)

  ′

  k q

  1 we obtain that v (S) strongly in H (K) . k → v d

1 Corollary 3.1. For every v (K) and m R (S) is dense

  ∈ H ∈ [0, 1), the set ˆ L m in H (K).

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

  2

  2 Corollary 3.2. For every v (K), the set ˆ R (S) is dense in L (K).

  ∈ L L

  1

  2

  1

  2 Proof. Let us consider v (K) and ε > 0. Since H (K) (K) with ∈ L ⊂ L

  1 dense inclusion, there exists a sequence (K) such that v

  {v } ⊂ H → v n n

  2 strongly in L (K) .

  From Corollary 3.1, we know the existence of controls ϕ such that the solution n n

  n

  v of (2.12) with ϕ satisfies = ϕ n

  ϕ n ,v ε n (S) < .

  1 − v v ϕ ,v

  n

  2 n

  L (K)

  2 N

  136 HEAT EQUATION IN R

  INVOLVING GRADIENT TERMS

  S

  ε −c

  2

  |v − v | L (K) ≤ e

  2 N will appear bellow. Consider v the solution of (2.12) corresponding to v and

2 Let N > 0 be such that where c > 0 is a constant that

  ϕ,v ϕ . Let z . Then z satisfies

  = ϕ N = v ϕ,v − v ϕ,v

  N

  N N z z s

  • Lz + Az + B · ▽z − = 0 in R × (0, S)

  (3.17)

  2 N R z(y, 0) (y) in = z = v − v

  N N We multiply (3.17) by zK(y) and integer over R . Then, there exists a constant

  S

  c

  2

  2

  

2

  c > 0 such that and, therefore, |z(S)| L (K) ≤ e |z | L (K)

  1

  1

  

2

  v , v (S) (S)

  • 2

  L (K)

  2 N

  ϕ − v ≤ |z(S)| L (K) ϕ,v − v ≤ ε . v

  L (K) We conclude with the

  2 N

  

Theorem 3.2. Given u (R ) the set of reachable states at time T > 0,

  ∈ L

  2 N R (T ) (T ) is the solution of (2.1) with h (R

  L = {u h,u : uh, u ∈ L × (0, T ))}

2 N is dense in L (R ).

  1

  2 N

  

Proof. Let us consider u (R ) and ε > 0. We divide the proof into three

  ∈ L steps.

  1

2 Step 1. The case u (K) .

  = 0, u ∈ L

  1

  1 N/

  2

  1

  2

  = (T + 1) + 1) ∈ L

  2 Let us consider v (y) u ((T y) (K) . From Corollary

  2

  2

  1 3.2 we know the existence of ϕ ( 0, S, L (K)) such that

  2

  ∈ L v(S) − v ≤ L (K)

  ε , with S = log(T + 1). Then

  N

  x −

  2

  u(x, t ) v , log(1 = (1 + t) √ + t)

  1

  • t is the solution of (2.1) with

  N

  x −

  2 −1

  h(x, t ) ϕ , log(1 = (1 + t) √ + t)

  1

  • t
SILVANO BEZERRA DE MENEZES 137 and

  2

  2

  1 −N |x|

2 N (

  1 e 4(1 u(T ) − u ≤ + t) + T ) L (R )

  N

  R

  2 x x

  1 , S dx

  √ − v √ v

  1

  1

  • t + t

  N

  2

  1 −

  1 (y) dy ≤ + T ) v(y, S) − v

  2 K(

  N

  R

  N

  2

  1

  2 −

  2

  ≤ (1 + T ) v(S) − v ≤ ε L (K)

  2 .

  1

2 N Step 2. The case u (R ) .

  = 0, u ∈ L

  2

  2 N

  1 Since L (K) (R ) with dense inclusion, there exists a sequence ⊂ L

  {u } and n

  ε

  1

  1 u

  N > 0 such that 2 N for every n > N .

  − u ≤ n

  L (R )

2 From the first step, we know the existence of controls h such that u the

  n n solution of (2.1) with h satisfies

  = h n ε

  1 u (T ) . n

  2 N

  − u n ≤ L (R )

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

  N

  1

  L (R )

  2 N u(T ) − u ≤ ε .

  1

  2 N Step 3. The general case, i.e. u , u (R ) arbitrary.

  ∈ L We write u

  = z + Y where z is the solution of N z t

  − z + a(x, t)z + b(x, t) · ▽z = 0 in R × (0, T ) (3.18)

  N R z(x, 0) (x) in .

  = u

  2 N

  1 Then z(T ) (R ) . We construct h ∈ L = h(u − z(T )) such that the solution of

  N R Y in t ω

  − Y + a(x, t)Y + b(x, t) · ▽Y = h1 × (0, T ) (3.19)

  N Y (x, 0)

  = 0 in R satisfies

  1

  L (R )

  2 N (3.20) Y (T ) − (u − z(T )) ≤ ε.

  1 Therefore in (2.1) it is enough to chose h = h(u − z(T )) since the unique solution is u

  = z + Y and in view of (3.20) we conclude the proof.

  N

  138 HEAT EQUATION IN R

  INVOLVING GRADIENT TERMS

4 The non linear case

  We will now study the same problem for the semilinear heat equation (1.2). As

  N

  s s/ 2 s

  2

  in the linear case, a change of variables v(y, s) u(e y, e = e −1) transforms the system (1.2) in

  N N R v v ′ in s ω (s)

  • Lv + g(y, s, v, ▽v) − = ϕ(y, s)1 × (0, S)

  (4.1)

  2 N R v(y, 0) (y) in = v s(N s/ 2 s

  • 2)/2 −sN/2 −s(N+2)/2

  with g(y, s, v, f (e y, e v, e ▽v) = e − 1, e ▽ v) and s(N s/

  2

  2

  • 2)/2

  ϕ(y, s) h(e y, e = e − 1).

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

  2

  2 N Observe that by (1.4) and because L (K) is dense in L (R ) we can con-

  2

  2 sider f (x, t, 0, 0) in L ( 0, T , L (K)) . This guarantees that g(y, s, 0, 0)

  ∈

  2

2 L ( 0, S, L (K)) . Thus, we suppose that f also satisfies

  2

  2 f (x, t, 0, 0) ( 0, T , L (K)), (4.2)

  ∈ L 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)

  2

  2

  2

  

above. Then, given ϕ ( 0, S, L (K)) and v (K), there exists a unique

  ∈ L ∈ L

  1

  1 ∗

  

solution v in the space W ( 0, S, H (K), (H (K)) ) of problem ( 4.1). Moreover,

  1

  2

  2 if v (K), hence v (K), L (K)).

  ∈ H ∈ W (0, S, H 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 u =

  2

  2

  2 v (K) and f ( 0, T , L (K)) then there exists a unique solution u

  ∈ L ∈ L ∈

  1

  1 ∗ W ( 0, T , H (K), (H (K)) ) of problem (1.2).

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

  SILVANO BEZERRA DE MENEZES 139

  

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

the conditions above.

  2

  2

  2

a) Let us suppose v in L (K) and ϕ ( 0, S, L (K)). Then,

  m m → v → ϕ in L

  1

  1 ∗ v in W ( 0, S, H (K), (H (K)) ).

  ϕ m ,v → v ϕ,v

  m

  2

  1

  2 b) Let us suppose v in L (K) and ϕ ( 0, S, L (K)). m m → v → ϕ in H

  2

  2 ∗ Then, v in W ( 0, S, H (K), (L (K)) ).

  ϕ m ,v → v ϕ,v

  m

  To show the approximate controllability property of the semilinear heat equa- tion, 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

  1

  

that g satisfies the conditions above. Given v (K), the functional

  ∈ H

  2

  2

1 F ( 0, S, L (K)) (K) defined by F (ϕ) (S) is differen-

  : L → H = v ϕ,v

  

tiable. Moreover, DF (ϕ)ψ (S), where z is the unique solution in

  = z ψ ψ

  2

2 W ( 0, S, H (K), L (K)) of the following linearized problem

  ∂g ∂g z (y, s, v , )z (y, s, v , ) s

  • Lz + + ϕ,v ▽v ϕ,v ϕ,v ▽v ϕ,v

  ∂θ ∂η N

  N (4.3) z

  · ▽ z − = ψ in R × (0, S)

  2 N z(y, 0) = 0 in R Proof.

  The functional F is well defined by Proposition 4.1 and the embedding

  2

  2 (2.16). Given ϕ, ψ 0, S, L and λ

  ( (K)) λ ∈ L ∈ (0, 1), let us denote v =

  1 v , v and z (v ϕ = v ϕ,v λ = λ −v). To show that F is Gâteaux differentiable

  • λψ,v

  λ we have to prove that

  1 in as (4.4) z λ (S) ψ (S) H (K) λ

  → z → 0.

  N

  140 HEAT EQUATION IN R

  INVOLVING GRADIENT TERMS It is clear that for each λ we have

  1 N z g(y, s, v , ) z λ,s λ λ ▽v λ − g(y, s, v, ▽v) − λ = ψ + Lz +

  λ

  2 N (4.5)

  R in × (0, S)

  N z (y, 0)

  λ = 0 in R dz

  λ where z denotes . λ,s ds

  By the Mean Value Theorem, we have

  1 g (y, s) (g(y, s, v (y, s), (y, s))

  λ = λ ▽v λ − g(y, s, v(y, s), ▽v(y, s))

  λ

  (4.6)

  ∂g ∂g

  = λ λ λ ▽w λ · ▽z λ ∂θ ∂η

  • (y, s, w (y, s))z (y, s, w (y, s), (y, s)) .

  where w λ λ

  = v + ξ(v − v), 0 ≤ ξ ≤ 1 depends on y, s and λ. Applying (4.5) to z K(y) , integrating by parts, using the fact that g is globally Lipschitz in the λ variable (v,

  ▽v) and Young’s inequality, we deduce the existence of a constant c > 0 such that

  1 ˜s ˜s

  2

  2

  2

  2 ds

  |z λ ˜s)| λ | ≤ K 1 |z λ | |▽z

  • ( K dyds (s)

  L (K)

  N

  R (4.7)

  ˜s

  2

  • 2 ds, |ψ(s)| ∀ ˜s ∈ [0, S] and ∀ λ ∈ (0, 1).

  L (K) Using Gronwall inequality, we have c s

  1

  2

  2

  

2

  λ (s) , |z | L (K) ≤ e |ψ| L ( 0,S,L (K)) ∀ s ∈ [0, S], ∀ λ ∈ (0, 1).

  2 Thus, the sequence (K)) and by (4.7), λ

  λ {z } is bounded in C([0, S], L {z } is

  2 1 c

  1 S

  2

  1

  2

  2

  bounded in L ( 0, S, H (K)) . In fact, , λ

  |z | L ( 0,S,H (K)) ≤ e |ψ| L ( 0,S,L (K)) ∀ λ ∈ (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 c

  2 (independent on λ and ψ) such that

  1

  1

  2

  |z λ | ≤ c 2 |ψ| ∀ λ ∈ (0, 1).

  2 ∗ ,

  W ( 0,S,H (K),(H (K)) ) L ( 0,S,L (K)) In view of the expression of g and using classical estimates, we deduce the

  λ existence of a positive constant c (still independent of λ and ψ) such that

  3

  2

  2

  2

  2

  (4.8) λ 3 , |z | W ( 0,S,H (K),(L (K)) ≤ c |ψ| L ( 0,S,L (K)) ∀ λ ∈ (0, 1). SILVANO BEZERRA DE MENEZES 141 Thus, by extracting subsequences,

  2

  2 z ⇀ (K), L (K)), as λ (4.9) λ ˆz weakly in W (0, S, H → 0.

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

  ∂g ∂g

  2

  2 g (y, s, v, (y, s, v, ( 0, S, L (K)). λ

  → ▽v)ˆz + ▽v) · ▽ˆz in L ∂θ ∂η

  It is then easy to see, by the well-posedness properties of (4.5), that and ψ

  ˆz = z 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):

1 Theorem 4.4. For each v

  (K) the set of admissible states at S > 0, ∈ H

  2

  2 ˆ

  R (S) v (S) is solution of (4.1) with ϕ ( 0, S, L (K)) N L

  = ϕ,v : v ϕ,v ∈ L m

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

  1 m

  Proof. We know that H (K) is dense in H (K) for 0

  ≤ m < 1. Then, it is 1 m sufficient to prove that ˆ R (S) is dense in H (K) with the norm of H (K) for

  N L ≤ m < 1.

  1 In fact, let us take v (K), d ∈ H ≤ m < 1 and consider the functional defined

  2

  2 in L ( 0, S, L (K)) by: 1 k

  2

  2 v

  k = ϕ,v − v d H (K)

  • J (ϕ) K(y)ϕ dyds (S) m .

  2 ′

  2 q

  This functional is lower semicontinuous and coercive in the Hilbert space of

  2

  2 ′

  L ( 0, S, L (K)) of functions with support in q . It follows that the minimization problem (P ) given by: k

  2

  2 Min J (ϕ) for all ϕ ( 0, S, L (K)) (P ) k ∈ L k has at least one solution ϕ . k N

  142 HEAT EQUATION IN R

  INVOLVING GRADIENT TERMS Thanks to Proposition 4.3, the first order optimality condition associated with the minimization problem (P ) at this minimum gives k k

  ′

  m

  J (ϕ )ψ K(y)ϕ ψ dyds (S) , z (S)) (4.10) k = k + k(v k − v d H (K) = 0 k

  ψ

  ′

  q k

  2

  2 for all ψ ( 0, S, L (K)) , where v and z is the unique function in k

  ∈ L = v ϕ k ,v ψ

  2

2 W ( 0, S, H (K), L (K)) solution of the problem

  ∂g ∂g

  s + Lz + k ▽v k k ▽v k · ▽z ∂θ ∂η

  • z (y, s, v , )z (y, s, v , )

  N (4.11)

  N R

  ′

  z ω in − = ψ1 × (0, S)

2 N

  0) z(y, = 0 inR

  By the minimum condition we have J (ϕ ) ( 0) for every k k k ≤ J k ∈ N. It follows

  1 m that (S) is a bounded sequence in H (K) and ϕ 1 ′ is a k d k k ω

  {v − v } √ ∈N k k

  ∈N

  2

  2 bounded sequence in L ( 0, S, L (K)) . Thus, by extracting subsequences, m v (S) ⇀ θ weakly in H (K). (4.12) k d

  − v N N N

  ∞ ∞ Moreover, there exist c (R (R such

  ∈ L × (0, S)) and d ∈ (L × (0, S))) that ∂g

  N ∞

  (y, s, v , ) ⇀ c weak* L (R k k

  ▽v × (0, S)) ∂θ

  (4.13) ∂g

  N N ∞

  (y, s, v , ) ⇀ d weak* (L (R k ▽v k × (0, S)))

  ∂η Let z be the unique solution of the problem

  ψ N N

  ′

  z z in R s + Lz + c(y, s)z + d(y, s) · ▽z − = ψ1 ω × (0, S)

  2 (4.14) N z(y, 0) .

  = 0 in R Suppose that k m z (S) ⇀ z (S) strongly in H (K). (4.15)

  ψ ψ

  By (4.10), (4.12) and (4.15) we obtain: 1 ϕk k

  m

  ψ K dyds (S) , z (S)) √ √ + (v k − v d H (K) = 0

  ψ

  ′

  k q k

  SILVANO BEZERRA DE MENEZES 143 and when k

  → ∞, we have

  2

  2

  m

  (θ, z (S)) ( 0, S, L (K)). (4.16) ψ H (K) = 0, for each ψ ∈ L 1 m

  By Theorem 3.1 the set (S) (K) and, therefore, in H (K) , ψ

  {z } is dense in H

  2

  2 when ψ varies in L ( 0, S (K)) . Consequently, (4.16) implies that θ ; L ≡ 0. m

  Therefore, according to (4.12), v (S) ⇀ v weakly in H (K) . Since the em- k d

  ′

  m m ′ bedding H (K) (K) is compact for m < m , we conclude the strong

  ⊂ H

  ′

  m ′ convergence in H (K) , for any m < m . Taking into account that this holds for m all m < 1, we conclude the density in H (K) , as stated in Theorem 4.4.

  To complete the proof we need to proof the convergence (4.15). In fact, mul- k N tiplying (4.11) by K(y)z , integrating in R and observing that

  ψ ∂g ∂g

  (y, s, v , ) (y, s, v , ) (4.17)

  k k k ▽v ▽v

  • k

  < K ∂θ ∂η we have

  S

  2 k k

  2 z (s)

  1 ψ z ψ k N

  • (s) ds < c . (4.18)

  ′ Multiplying (4.11) by K(y)(z ) and integrating in R , we obtain:

  ψ S

  2 k k

  2

  2 ′

  (z ) ds (s) < c . (4.19)

  • z

  ψ ψ k

  2 From (4.18) and (4.19) it follows that is bounded in W (0, S, H (K) , k {z }

  ψ ∈N

  2

  2

  2

  2 L (K)) . Since W (0, S, H (K), L (K)) is compactly embedded in L ( 0, S,

  1 H (K)) , there exists subsequences such that k

  2

  1 z ( 0, S, H (K))

  → χ strongly in L ψ

  (4.20) k

  2

  2 z (K), L (K))

  → χ weakly in W (0, S, H ψ

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

  1 z ⇀ z weakly in H (K)

  ψ ψ 1 m m and k tends to (K) into H (K) into H (K) being

  • ∞. The embedding of H compact, for m < 1. This proves (4.15).

  N

  144 HEAT EQUATION IN R

  INVOLVING GRADIENT TERMS

  2

  2 Corollary 4.1. Given v (K), the set ˆ R (S) is dense in L (K). N L

  ∈ L

  1

  2 Proof. As in the proof of Corollary 3.2, let us consider v (K) and ε > 0.

  ∈ L

  1 Then, there exists a sequence (K) such that v strongly in n {v n } ⊂ H n → v

  ∈N

  2 L (K) . Moreover, as we saw previously, there exists a sequence of controls ϕ n such that v , the solution of (4.1) corresponding to v and ϕ , satisfies n

  ϕ n ,v n

  n

  ε

  1 v (S) .

  ϕ n ,v − v

  2 ≤ n

  L (K)

  2 Let N > 0 be such that

  cS

  ε −

  2

  v ,

  − v

  2 ≤ e

  N L (K)

  2 where c > 0 is a constant that will be timely introduced. Consider v the solution of (4.1) corresponding to v and ϕ . Let

  ϕ,v = ϕ

  N z . Then, z satisfies

  = v ϕ,v − v ϕ,v

  N

  z , ) , ) s + Lz + g(y, s, v ϕ,v ▽v ϕ,v − g(y, s, v ▽v

  ϕ,v ϕ,v

  N N

  N N (4.21) z

  − = 0 in R × (0, S)

2 N R z(y, 0) (y) in .

  = z = v − v N N

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

  cS

  K N

  2

  2

  z

  2 , where c , and therefore

  |z(S)| L (K) ≤ e = K + + L (K)

  2

  2

  1

  1

  

2

  v (S) (S)

  − v + v L (K)

  2 ϕ,v ≤ |z(S)| L (K) ϕ,v − v ≤ ε.

  2 N

  L (K) We conclude with the proof of Theorem 1.1.

  1

  2 N

  

Proof of Theorem 1.1. Let us consider α > 0 and u (R ) . As in the

  ∈ L proof of Theorem 3.2 we divide into several steps. SILVANO BEZERRA DE MENEZES 145

  1

  2 First step. The case u , u in L (K) .

  1 N/

  2 1 1/2 We make the change of variables v (y) u (( 1 y) and

  = (1 + T ) + T )

  1

  2 S (K) and by Corollary 4.1, there exists ϕ = log(1 + T ). Then v ∈ L ∈

  2

  2 L ( 0, S, L (K)) such that the solution v of (4.1) corresponding to v and ϕ, satisfies

  1

  L (K) We define x

  2 v(S) − v ≤ α.

  −N/2 u(x, t ) v , log(1 .

  = (1 + t) √ + t)

  1

  • t Then u is solution of (1.2) with

  N

  x −

  2 −1

  log(1 h(x, t ) ϕ , = (1 + t) √ + t)

  1

  • t

  2

  1

  2 and 2 N (cf. Theorem 3.2, Step 1). u(T ) − u ≤ α

  L (R )

  2

  1

  2 N Second step. The case u (K), u (R ) .

  ∈ L ∈ L

  2

  2 N Since L (K) (R ) with dense inclusion, there exists a sequence

  ⊂ L

  1

  2

  1

  1

  2 N (K) such that u strongly in L (R ) . From the First n

  {u } ⊂ L → u n ∈N n

  Step, we know that existence of controls h such that u the solution of (1.2) n n

  α

  1 satisfies u (T ) . Let N > 0 such that for every n > N , n

  2 N

  − u n ≤ L (R )

  2 α

  1

  2 N

  − u ≤ n

  1 u .

  L (R )

  2

  1 Then, u the solution of (1.2) with h satisfies

  L (R )

  2 N = h N u(T ) − u ≤ α.

2 N Third step. The case u (R ) .

  ∈ L

  2 Then there exists a sequence (K) such that u strongly {u } with u ∈ L → u n n n

  2 N

  1

  2 N in L (R ) . Given u (R ) and α > 0, as we saw previously, there exists a

  ∈ L sequence of controls h such that u , the solution of (1.2) corresponding to u n n n and h , satisfies n

  α

  1 u

  (T ) 2 N . n − u ≤

  L (R )

  2 Let N > 0 be such that

  K2 T

  α )

  −(K u

  • 2

  2

  ,

2 N

  − u ≤ e N L (R )

  2 N

  146 HEAT EQUATION IN R

  INVOLVING GRADIENT TERMS K the constant given in (1.5).

  Consider u the solution of (1.2) corresponding to u and h . Let z = h N = u . Then, z satisfies

  − u N z , ) t

  − z + f (x, t, u, ▽u) − f (x, t, u N ▽u N = 0 N in R

  (4.22) × (0, T )

  N R z(x, 0) (x) in .

  = z = u − u N N

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

  2 1 d K

  1

  2

  2

  2

  2

  2

  2 N

  1 N

  2 N

  2 N 1 N .

  |z| + z ≤ K |z| |z| z + + L (R ) H (R ) L (R ) L (R ) H (R )

  2 dt

  2

  2 K2

  T

  • (K )

  2

  2

  2 N

  z Then,

  2 N , and therefore

  |z(T )| L (R ) ≤ e L (R )

  1

  1

  

2 N

  u

  2 N (T )

  N L (R ) L (R )

  2 N u(T ) − u ≤ |z(T )| + L (R ) − u ≤ α.

5 Conical domains

  Consider a cone-like domain satisfying: λx (5.1) ∈ ¯ , ∀ λ > 0, ∀ x ∈ , ∈ . All the results of these paper hold for the semilinear heat equation (1.1) when is a cone-like domain. In fact, consider a domain satisfying (5.1). Then if one consider the evolution equation u in Q t − u + f (x, t, u, ▽u) = h1 ω = × (0, T )

  (5.2) u = 0 on ∂ × (0, T ) u(x, 0) (x) in

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

  N v v ′ in s + Lv + g(y, s, v, ▽v) − = ϕ1 ω (s) × (0, S)

  2 v = 0 on ∂ × (0, S) v(y, 0) (y) in

  = v

  SILVANO BEZERRA DE MENEZES 147

  1

  2 The operator Lf div(K / 4), is self-adjoint in = − ▽ f ), K(y) = exp(|y|

  K

2 L (K, ) with compact inverse and

  1

  2 D(L) (K, ) (K, ) = {f ∈ H : Lf ∈ L }, where

  2

  2 L (K, ) v K(y)dy < ,

  = : |v| ∞

  1

  2

2 H (K, ) v K(y)dy < v and

  = : |v| + | ▽ v| ∞, ∂ = 0    

  1/2

     

  m 2 α

  2

  m   H (K, ) v (K, ) v

  2 &lt; .

  = ∈ L : v H (K, ) = D ∞

  L (K, )

     

  |α|≤m 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.

  

REFERENCES

[CMZ] V. Cabanillas, S. Menezes and E. Zuazua, Null controllability in unbounded domains for

the semilinear heat equation with nonlinearities involving gradient terms , J. Optm. Theory Appls., 110, 2, (2001), 246-264.

  

[DL] R. Dautray and J.L. Lions, Mathematical Analysis and Numerical Methods for Science and

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[EK] M. Escobedo and O. Kavian, Variational Problems Related to Self-Similar Solutions of Heat

Equation , Nonl. Anal. TMA, 11, 10, (1997), 1103-1133.

  

[EZ] M. Escobedo and E. Zuazua, Large Time Behaviour for Convection-diffusion Equations in

  n J. Funct. Anal., 100 (1991), 119-161.

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[F] E. Fernández-Cara, Null controllability of the semilinear heat equation, ESAIM: COCV

2 (1997), 87-107.

[Fa] C. Fabre, Uniqueness results for Stokes equations and their consequences in linear and

nonlinear control problems , ESAIM: COOCCV, 1 (1996), 267-302.

[FI] A. Fursikov, O. Yu, Imanuvilov, Controllability of evolution equations, Lecture Notes, Vol.

  34 (1996), Seoul National University, Korea.

  

[FPZ] C. Fabre, J.P. Puel and E. Zuazua, Approximate controllability of the semilinear heat

equation , Proc. Royal Soc. Edinburgh, 125A (1995), 31-61.

[FZ] L.A. Fernández and E. Zuazua, Approximate controllability for the semilinear heat equation

involving gradient terms , J. Opt. Theo. and Appl., 101 (1999), 307-328.

  

[FCZ1] E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat

equation: the linear case , Adv. Diff. Equations, 5, 4-6, (2000), 465-514.

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[H] J. Henry, Contrôle d’un réacteur enzymatique à l’aide de modèles à paramètres distribués.

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[IY] O. Yu, Imanuvilov and M. Yamamoto, On Carleman inequalities for parabolic equations in

Sobolev spaces of negative order and exact controllability for semilinear parabolic equations ,

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[K] O. Kavian, Remarks on the large time behaviour of a nonlinear diffusion equation, Ann. Inst.

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20 (1995), 335-356.

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[L2] J.L. Lions, Remarques sur la contrôlabilité approchée. In Jornadas Hispano-Francesas sobre

Control de Sistemas Distribuidos, pages 77-87, University of Málaga (Spain), 1991.

  

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differential equations , Studies in Appl. Math., 52 (1973), 189-221.

  N , SIAM J. Cont.

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