REMARKS ON NULL CONTROLLABILITY FOR SEMILINEAR HEAT EQUATION IN MOVING DOMAINS

We investigate in this article the null controllability for the semilinear heat operator u 0 u + f(u) in a domain which boundary is moving with the time t.


Introduction and Main Result
In this article we consider semilinear parabolic problems in domains which are moving with the time t.Given a time T > 0, the state equation is posed in an open The open set Q is the union of open sets Ω t of R n , for 0 < t < T , which are images of a reference domain Ω 0 by a diffeomorphism τ t : Ω 0 → Ω t .
We identify Ω 0 to a bounded open set Ω of R n and its points are represented by y = (y 1 , y 2 , . . ., y n ) and those of Ω t by x = (x 1 , x 2 , . . ., x n ) are such that x = τ t (y).
We also employ the notation τ (y, t) instead of τ t (y).
Thus, the noncylindrical domain Q of R n+1 is defined by The boundary of Ω t is represented by Γ t and the lateral boundary of Q, denoted by Σ, is given by Σ = 0<t<T {Γ t × {t}}.
We will develope the article under the following assumptions.
(A1) For all 0 ≤ t ≤ T , τ t is a C 2 -diffeomorphism from Ω to Ω t .In this article we will work with the following state equation (1.1) u − ∆u + f (u) = h(x, t)χ q in Q y = 0 on Σ u(x, 0) = u 0 (x) in Ω.
In (1.1) we have u = u(x, t), u = ∂u ∂t ; ∆ is the Laplace's operator in R n ; q is an open, non-empty, subset of Q.We also denote by w t the cross section of q at any 0 < t < T ; χ q the characteristic function of q.The function h(x, t) is the control that acts on the state u(x, t) localized in q.The nonlinear function f is real and globally Lipschitz such that f (0) = 0.This means that there exists a constant K 0 , called Lipschitz constant, such that As we will see later, if u 0 ∈ L 2 (Ω), h ∈ L 2 ( Q) the system (1.1) has a unique solution u ∈ C 0 ([0, T ]; L 2 (Ω t )) ∩ L 2 (0, T ; H 1 0 (Ω t )).
The main result of the present paper is the following: Theorem 1.1.Assume f is C 1 and satisfies (1.2) with f (0) = 0.Then, for all T > 0 and for every u 0 ∈ L 2 (Ω), there exists h ∈ L 2 ( Q) such that the solution u = u(x, t) of (1.1) satisfies (1.3).Moreover, (1.4) holds for a suitable constant C > 0 independent of u 0 .In other words, system (1.1) is null controllable for T > 0.
The methodology of the proof of the Theorem 1.1 is based in the fixed point method, see Zuazua [39,40].There is however a new difficulty related to the fact that Q is noncylindrical.To set up this point we employ the idea contained in [27].
The first step on the fixed point method is to study the null controll for the linearized system.This problem is reduced, by duality, to obtain a observability inequality for the adjoint system.This is get as an application of Carlemann inequalities as in Imanuvilov-Yamamoto [17].This work is organized as follows: Section 2 is devoted to prove the null controllability for the linearized system.In Section 3 we prove Theorem 1.1 by a fixed point method.

Analysis of the Linear Problem
The main result of this article will be proved in Section 3 by means of a fixed point argument.As an step preliminary we need to analyse the null controllability of the following linearized system: (2.1) where the potential is assumed to be in First of all we study the existence and uniqueness of solution of the system (2.1).

Strong and Weak Solutions
We distinguish three classes of solutions for the system (2.1), as follows: strong, weak and ultra weak solutions defined by transposition.Definition 2.1.a) A real function u = u(x, t) defined in Q is said to be a strong solution for the boundary value problem (2.1) if and the three conditions in (2.1) are satisfied almost everywhere in their corresponding domains.
EJQTDE, 2003 No. 16, p. 5 b) We say that the function u is a weak solution of (2.1) if and (2.4) Theorem 2.1.Assume that the noncylindrical domain Q satisfies the conditions of the Section 1.Then, if ), the problem (2.1) has a unique strong solution.
Moreover, there exists a positive constant C (depending on Q but independent of u 0 and h) such that and for f ∈ L 2 (0, T ; Proof: As in [27] we employ the argument consisting in transforming the heat equa- Here and in the following τ −1 t denotes the inverse of τ t , which, according to assumption (A1) is a C 2 map from Ω t to Ω, for all 0 ≤ t ≤ T .This map will be denoted by ρ t .We shall also employ the notation ρ(x, t) = ρ t (x), y j = ρ j (x, t), We obtain, where • is the scalar product in R n .In other words, where b(y, t) denotes a vector field (2.9) b(y, t) = ∂ ∂t ρ(x, t).
On the other hand, Thus by the mapping x = τ −1 (y) that takes Q into the cylinder Q we transform (2.1) in an equivalent problem (2.11) given by (2.11) and Gauss' Lemma we obtain the bilinear form a(t, v, ϕ) defined by This bilinear form is bounded because ρ ∈ C 2 (−Ω) by assumption (A2).Let us prove that it is H 1 0 (Ω)-coercive.In fact, set ϕ = v ∈ H 1 0 (Ω).We have given by But, by assumption, M is bounded and invertible what comes from assumptions on x = τ t (y).Then, Thus, returning to the quadratic form, we obtain Thus, from (2.11) we obtain for (2.1) in Q the following system (2.12) Note that (2.12) is a linear parabolic system with variable coefficients in a cylin- ercive the boundary value problem (2.12) is a classical problem studies in Lions-Magenes [24].If we take u 0 ∈ H 1 0 (Ω) and h ∈ L 2 (0, T ; L 2 (Ω)) then (2.12) has EJQTDE, 2003 No. 16, p. 9 In both cases we have uniqueness.From the assumption (A1) and (A2) the transformation To prove the estimate (2.5) we first establish the classical energy estimate.In fact, multiplying (2.1) by u integrating for x ∈ Ω t and 0 < t < T , we get (2.13) Note that since u vanishes on the lateral boundary Σ of Q we have (cf.Duvaut [11], p.26): (2.14) As a consequence of the assumptions (A1) and (A2) it follows that the Poincaré inequality is satisfied, uniformly, in the domain Ω t for all 0 ≤ t ≤ T .Thus, in view of (2.13) and (2.14) we have (2.15) for a constant C > 0.
In particular, strong solutions satisfies the energy estimate with C > 0 constant independent of the solution.
Solving this differential inequality we deduce the existence of a constant C such that A variation of this argument alows also to get In fact, to obtain (2.24) instead of (2.23) it is sufficient to estimate the term Ωt h∆u dx as follows This complete the proof of the Theorem 2.1.
Remark 2.1 Note that we could also have obtained the above estimates using existence results for the variable coefficients parabolic equation satisfied by v and then doing the change of variables , there exists a unique weak solution of (2.1).Moreover, there exists a constant C > 0 (depending on Q but independent of u 0 and h) such that EJQTDE, 2003 No. 16, p. 12 A similar argument allows to replace h ∈ L 2 (0, T ; H −1 (Ω t )) by the assumption h ∈ L 1 (0, T L 2 (Ω t )) and to obtain the estimate Proof: We follow the argument of reference [27].We proceed by steps.
Step 1 (Existence).Let u 0 m ∈ H 1 0 (Ω) and h m ∈ L 2 ( Q) be a sequence of regularized initial data and right hand side terms, respectively, such that u 0 m → u 0 strongly in Then, for each m ∈ N, let us consider the unique strong solution u m of (2.1) with initial data u 0 m and right side h m .Thus, for any n, k ∈ N we have By the energy estimate (2.18) we obtain that (u m ) is a Cauchy sequence in the space Thus it converges, as m → ∞, to a limit u ∈ C([0, T ]; L 2 (Ω t )) ∩ L 2 (0, T ; H 1 0 (Ω t )).The limit u is a weak solution of (2.1) satisfying (2.4) and the estimate (2.25).In fact, u m is strong solution for every m ∈ N.Then, multiplying the equation with u m by a test function ϕ and integrating by parts we deduce that u m satisfies the weak formulation (2.4).
The convergence of u m to u in the space EJQTDE, 2003 No. 16, p. 13 allows to pass to the limit in the weak formulation to conclude that u satisfy (2.4).
Step 2 (Uniqueness).Assume that the system (2.2) admits two weak solution u and û satisfying (2.4).Introduce w = u − û.Then, w belongs to C 0 ([0, T ]; L 2 (Ω t )) ∩ L 2 (0, T ; H 1 0 (Ω t )) and satisfies for all test function ϕ.In order to conclude that w = 0, it is sufficient to consider w = ϕ as a test function.Of course we cannot do it directly since w is not a test function.It is justified by regularization and cut-off argument.
In this way we complete the proof obtaining the energy estimate for w what guarantees that Then w = 0 because w(0) = 0.
Step 3. To prove the estimate (2.26), it suffices to employ in (2.18) the estimate

Ultra Weak Solutions by Transposition Method
In this section we address the question of finding solutions u of the boundary value problem (2.28) We employ the transposition method as in Lions-Magenes [23].First of all we define what we understand by ultra weak solution by this method.
A function u = u(x, t) is said to be ultra solution of (2.28) or solution by transposition if and (2.30) where ϕ is the unique strong solution of the adjoint system (2.31) Here, , denotes the duality passing between H −1 (Ω) and H 1 0 (Ω).According to Theorem 2.1, the system (2.31) admits a unique strong solution ϕ.
Thus the definition of ultra weak solution makes sense.
Note that the strong solution ϕ satisfies the following estimates: These estimates were proved in Theorem 2.1.Indeed, it is sufficient to make the change of variables t → T − t to reduce the system (2.31) to (2.1).

EJQTDE, 2003 No. 16, p. 15
By Riesz-Fréchet theorem we deduce that there exists a unique ultra weak solution in the class (2.29).More precisely, in view of (2.32) we deduce the existence of unique solution u ∈ L 2 ( Q) and the second estimate (2.33) provides the aditional regularity u ∈ L ∞ (0, T ; H −1 (Ω t )).Moreover, one deduces the existence of a constant, independent of u 0 , such that In order to show that u ∈ C 0 ([0, T ]; H −1 (Ω t )) we use a classical density argument.
When u 0 is smooth enough, u is a weak or strong solution, therefore u is continuous with respect to time with values in H −1 (Ω t ).According to (2.34), by density, we To complete this section, we observe that when u 0 is smooth so that exist weak or strong solution then they are also ultra weak solutions.It is sufficient to integrate by parts in the strong formulation of (2.1) or consider the weak formulation.

Observability of the Linearized Adjoint System
As we said before we employ a fixed point argument in order to prove our results in the semilinear case.However, first we analyse the null controllability for the following linearized system: (2.35) where the potential a = a(x, t) is assumed to be in L ∞ ( Q). Remember we denote by ût the cross section of q at any 0 < t < T .
As we know, the null controllability of (2.35) is equivalent to a suitable observability property for the adjoint system of ( for which we have the following observability property.
Proposition 2.1.For all T > 0 and R > 0 there exists a positive constant C > 0 such that for every solution of (2.36) and for any a Remark 2.2.The constant C in (2.37) will be referred to as the observability constant.It depends on Q, q the time T and the size R of the potential but does not depend of the solution ϕ of (2.36).
Proof of the Proposition 2.1: The inequality (2.37) is a consequence of the results in [17].In fact, by the change of variables x → y, from Q into Q, the adjoint system (2.36) is transformed into a variable coefficient parabolic equation of the form (2.38) as in (2.12) with h = 0. Thus the coefficient of the principal part A(t), according to the assumption (A1) and (A2) are of class C 1 and a and b are bounded.Then, EJQTDE, 2003 No. 16, p. 17 the observability inequalities in [17] guarantee that for every T > 0 and every open subset q of Q, there exists a constant C > 0 such that In particular it is true for q ⊂ Q image of q by x → y.Thus estimate (2.37) for ϕ is obtained from (2.39) for ψ by the change of variables y → x.

Approximate Controllability for the Linearized System
From the observability inequality (2.37) the null controllability result for the linearized system can be proved as the limit of an approximate controllability property.
In fact, given u 0 ∈ L 2 (Ω) and δ > 0 we introduce the quadratic functional where ϕ is the solution of (2.36) with initial data ϕ 0 .The functional J δ is continuous and strictly convex in L 2 (Ω t ).Moreover, J δ is coercive.More precisely, in view of (2.37) we have (2.41) lim inf To prove (2.41) we follows the argument used in [27] which we will not repeat here.
Thus J δ has a unique minimizer in L 2 (Ω T ).Let us denote it by φ0,δ .It is not difficult to prove that the control h δ = φδ , where φδ is the solution of (2.36) associated to the minimizer φ0,δ is such that the solution u δ of (2.1) satisfies We refer to [12] for the details of the proof.

Null Controllability of the Linearized System
The null controllability property may be obtained as the limit when δ tends to EJQTDE, 2003 No. 16, p. 18 zero of the approximate controllability property above obtained.However, to pass to the limit we need a uniform bound of the control.To obtain this bound we observe that, by (2.37), (2.43) when C > 0 is independent of δ.On the order hand, (2.44) Writing (2.43) for φ0,δ instead of ϕ, with ϕ 0,δ the minimizer of J δ in L 2 (Ω T ) and combining it with (2.44), we deduce that (2.45) for all δ > 0.
In other words, We can prove that the limit h is such that the solution u of (2.1) satisfies (1.3).
Moreover, by the lower semicontinuity of the norm with respect to the weak topology and by (2.46) we deduce that By the process we complete the proof of the following result.
Theorem 2.3.Assume that the noncylindrical domain Q satisfies the conditions fixed in Section 1 and that a(x, t) ∈ L ∞ ( Q).Then, for every T > 0 and u 0 ∈ L 2 (Ω).
EJQTDE, 2003 No. 16, p. 19 there exists h ∈ L 2 ( Q) such that the solution of (2.1) satisfies (1.3).Moreover, there exists a constant C > 0, depending on R > 0, but independent of u 0 , such that (1.4) holds for every potential a = a(x, t) in

Proof of the Main Result
This section is devoted to prove Theorem 1.1.As we said, in the Introduction, it will be a consequence of Theorem 2.3 above and a fixed point argument.
In order to be self contained we will prove existence result.
Theorem 3.1.Assume that f : R → R is C 1 and globally Lipschitz function, such that f (0) = 0. Let u 0 ∈ L 2 (Ω) and h ∈ L 2 ( Q).Then, there exists a unique solution Then we know that (3.1) admits a unique solution By the change of variable y → x we deduce the existence of a unique solution u of (1.1) in the class EJQTDE, 2003 No. 16, p. 20 As in [12] we introduce the nonlinearity Given any z ∈ L 2 ( Q) we consider the linearized system Observe that (3.3) is a linear systsem in the state u = u(x, t) with potential With this notation, the system (3.3) can be written as By the subsection 2.4, if δ > 0 is fixed, for each z ∈ L 2 ( Q) we can define a control Moreover, for every R > 0 and all potential a = a(x, t) EJQTDE, 2003 No. 16, p. 21 for all δ > 0. Therefore, the controls h δ are uniformly bounded (with respect to z and δ) in L 2 ( Q).
This result allows to define a nonlinear mapping where u satisfies (3.5) and (3.6).
In this way, the approximate controllability problem for (1.1) is reduced to find a fixed point for the map 3) is solution of (1.1) Then the control h δ = h δ (z) is the one we were looking for, since, by construction, u δ = u δ (z) satisfies (3.6).
As we shall see, the nonlinear map N δ satisfies the following properties: (3.9) N δ is continuous and compact, (3.10)The range of N δ is bounded, i.e., exists Therefore, by (3.9), (3.10) and Schauder fixed point theorem, it follows that N δ is a fixed point.
By the moment, assume that (3.9) and (3.10) are true which proof comes after.
Then if (3.9) and (3.10) are true it follows, by Schauder's fixed point theorem, that we have a control h δ in L 2 ( Q) such that the solution u δ of (3.11) with C independent of δ.
Passing to the limit as δ → 0, as in Section 2, we deduce the existence of a limit control h ∈ L 2 ( Q) such that the solution u of (1.1) satisfies (1.3) and (1.4).
To complete the argument we need to prove (3.9) and (3.10).

Continuity of N
In fact, we have by hypothesis, K 0 the Lipschitz constant of f , then |g(z j )| p ≤ K p 0 , 1 ≤ p < ∞.We also have z j → z in L 2 ( Q) and consequently a subsequence z j → z a.e. in It follows by Lebesgue's bounded convergence theorem that with the initial data φ0 j that minimizes the correspondent functional J δ in L 2 ( Q).We also have By extracting a subsequence ( φ0 j ) we have We will prove that χ = φ and φ solves It is sufficient to prove that g(z j ) φj g(z) φ weakly in L 2 (Ω × (0, T )).
For the parabolic problem for ψ j we obtain estimates in the cylinder which permits to employ compacteness argument of the type Lions-Aubin for ψ j .When we change the variables y → x we obtain subsequence ( φj ) in L 2 ( Q) such that φj → φ strong L 2 ( Q).
Note that u j and u solve (3.11), what implies, by the estimates, that where u solves To complete the proof of the continuity of N δ it is sufficient to check that the limit φ0 obtained in (3.20) is the minimizer of the functional J δ associated to the limit control problem (3.21) and (3.22).

Compactness of N
) and v varies in a bounded set of L 2 (0, T ; L 2 (Ω t )).Thus by Aubin-Lions compactness result, v varies in a relatively compact set of L 2 (0, T ; L 2 (Ω t )).It then follows that u = w + v, with w in L 2 (0, T ; L 2 (Ω t )) varies in a relatively compact set of L 2 (0, T ; L 2 (Ω t )).