NULL CONTROLLABILITY FOR A HEAT EQUATION WITH DYNAMIC BOUNDARY CONDITIONS AND DRIFT TERMS

. We consider the heat equation in a bounded domain of R N with distributed control (supported on a small open subset) subject to dynamic boundary conditions of surface diﬀusion type and involving drift terms on the bulk and on the boundary. We prove that the system is null controllable at any time. The result is based on new Carleman estimates for this type of boundary conditions.

As in the literature, by duality, to obtain this aim, we show an observability inequality to its associated backward adjoint problem that is an estimate of type where C T is a positive constant, which will be explicitely defined later. Note here that, as B and b are only in L ∞ (Ω) N and L ∞ (Γ) N , the terms div(ψB(x)) and div Γ (ψ Γ b(x)) are understood in the sense of distributions.
The observability estimate (3) will be derived thanks to a suitable Carleman estimate fulfilled by the solution to (2). The main result of this paper will be the establishment of a Carleman estimate for the following intermediate equation in Ω T , in Ω × Γ for F 0 ∈ L 2 (0, T ; L 2 (Ω)), F ∈ L 2 (0, T ; L 2 (Ω) N ), F 0,Γ ∈ L 2 (0, T ; L 2 (Γ)), F Γ ∈ L 2 (0, T ; L 2 (Γ) N ), which will be obtained by showing another Carleman estimate fulfilled by the solution of an optimal control problem thanks to the Carleman estimate of [32]. The wellposedness and regularity of these two equations were extensively studied in the literature by several authors, in [17,20,22,29,34,35,37], without drift terms, i.e., B = b = 0, and recently with drift terms in [28]. Null controllability of parabolic equations, using Carleman estimates, is extensively studied in the case of static boundary conditions (Dirichlet, Neumann, or Fourier boundary conditions), see e.g. [4,6,11,12,13,15,25,31,33]. Recently, in the case of dynamic boundary conditions as above, Maniar et al. [32] achieved this aim in the absence of drift terms. Dynamic surface and interface processes have attracted a lot of attention in recent years in the mathematical and applied literature in various contexts (such as, diffusion phenomena in thermodynamics, phase-transition phenomena in a material science, climate science, control theory, in chemical reactor theory, and in colloid chemistry and special flows in hydrodynamics and in a class of problems which arise when the diffusion or flow takes place between a solid and a fluid, see [30,3,5,7,10,18,19,27,20,21,22,24,34,38,41]. For some applications of dynamic boundary conditions for physiologically structured populations with diffusion, we refer the reader, for instance, to [9]. In the context of reaction-diffusion equations, dynamic boundary conditions have been rigorously derived in [18] based on first and second thermodynamical principles and their physical interpretation was also given in [22]. This work have focused, as a first step, on the heat equation with constant diffusion coefficients d, but as in the case of static boundary conditions presumably our results also hold for general elliptic second order operators, with diffusion coefficients d(x), or even depending on time d(x, t). Recently, many results are obtained even in the case where diffusion coefficients degenerate in the case of static boundary conditions, see [1] and the references therein. We can also consider possible extensions of the null controllability results above to nonlinear problems of the form where F : R × R N → R and G : R × R N → R are locally Lipschitz-continuous functions with F (0) = G(0) = 0 as in the case of static boundary conditions (Dirichlet, Neumann boundary, or Fourier boundary conditions) see [6,14]. The case F = F (s), G = G(s), with F , G ∈ C 1 (R) was already studied in [32]. The general case of nonlinearities leading to a blowup will be treated in a forthcoming paper. 2. The wellposedness. We refer to paper [28], where a complete study of the wellposedness and regularity properties of solutions to the inhomogeneous linear system and the inhomogeneous backward adjoint problem (in the transposition sense) with f ∈ L 2 (0, T ; (H 1 (Ω)) ) and g ∈ L 2 (0, T ; H −1 (Γ)), is established. Recall briefly the functional spaces introduced and the main wellposedness results of [28]. Define the Hilbert space L 2 := L 2 (Ω) × L 2 (Γ) with the scalar product f , g L 2 dt, and its associated norm will be denoted by · E1 . For the wellposedness of the adjoint problem, we need also the Banach space with the norm The space W 1 is continuously embedded in C([0, T ]; L 2 ), see [28]. System (4) can be written equivalently as a Cauchy initial value problem where and g(t,·) . We have shown, in [28], that the operator A generates an analytic semigroup on L 2 . We adopt the following notion of solutions to system (4). Definition 2.1. A function U := (u, u Γ ) is said to be a strong solution of (4) if U ∈ E 1 and fulfills (4) a.e., t ∈ [0, T ]. A function U := (u, u Γ ) is called a mild solution of (4) if U ∈ C([0, T ]; L 2 ) and satisfies We have established the following existence, uniqueness and regularity results. Proposition 1. Let Y 0 := (y 0 , y Γ,0 ) ∈ L 2 and F := (f, g) ∈ L 2 (Ω T ) × L 2 (Γ T ), then the following assertions are true. 1) The Cauchy problem (6), and hence system (4), has a unique mild solution U given by Moreover, there exists a constant C > 0 such that

NULL CONTROLLABILITY FOR A HEAT EQUATION 539
2) If Y 0 ∈ H 1 , the Cauchy problem (6) and hence system (4) has a unique strong solution U ∈ E 1 , which is also a mild solution. Moreover, there exists a constant C > 0 such that We also gave in [28] the definition of solutions to the backward adjoint problem (5) and proved results of existence and uniqueness of such solutions.
We proved two useful characterizations of the weak solution to (5).
Proposition 2. The following conditions are equivalent.

For any
has a unique weak solution.
3. For every Φ T := (ϕ T , ϕ T,Γ ) ∈ L 2 (resp. Φ T ∈ D(A * )), the homogeneous backward problem (2) has a unique mild solution (resp. classical solution Moreover, there exists a constant C = C(Ω, ω) > 0 such that for all 3. Carleman estimates. To establish null controllability of the linear equation (1), we will first prove an observability inequality for the associated backward adjoint problem (2). This will be done by establishing first a Carleman estimate for the following intermediate system We begin by recalling a lemma due to Fursikov-Imanuvilov in [25].
For any nonempty open set ω Ω, there is a function η 0 ∈ C 2 (Ω) such that Given ω ω an nonempty open subset, we take λ, m > 1 and η 0 with respect to ω as in Lemma 3.1. Following [11] and [32], we define the weight functions α and ξ by for x ∈ Ω and t ∈ [0, T ]. Note that α and ξ are C 2 and positive on (0, T ) × Ω and blow up as t → 0 and t → T . Observing that We collect in the following lemma some useful properties and estimates fulfilled by the above weight functions, see [2].
(c) For any σ > 0 and µ ∈ R there exists C > 0 (only depending on Ω, ω, σ and µ) such that for any s ≥ σ(T + T 2 ) (d) For any σ > 0, there exists C > 0 (only depending on Ω, ω and σ) such that for any s ≥ σ(T + T 2 ) We recall also the following Carleman estimate from [32], needed to show our main results.
The main result of this section is the next Carleman estimate which will be the key to prove the null controllability of equation (1).
1. An intermediate Carleman estimate. Following the approach of [20], we introduce the solution of a variational problem on the space E 1 involving the solution ϕ of (9) which will be used to derive the Carleman estimate (15). For Y = (y, y Γ ) ∈ E 1 , we set Let (ϕ, ϕ Γ ) be the unique weak solution to (9), and consider the variational problem To establish the existence and uniqueness of the solution to this variational problem, we use Lax-Milgram Theorem. For this, we need to prove that E 1 is a Hilbert space for a new suitable scalar product. Note that this idea was firstly used by Imanuvilov in [25] and Imanuvilov-Yamamoto in [26], by considering the unique solution of an auxiliary extremal problem and solvability of their optimal systems. Lemma 3.5. For a real positive s, let Then, Proof. For assertion (i), it is obvious that ·, · s,λ is a scalar product on X. Set · λ,s the associated norm to this scalar product and let ((f n , g n )) n∈N be a Cauchy sequence of (X, · λ,s ). Hence, ((e αs f n , e αs g n )) n∈N is a Cauchy sequence in the space (L 2 , · L 2 ), and thus it converges to an element (f, g) ∈ L 2 . Therefore, ((e αs f n , e αs g n )) n∈N converges to (e −sα f, e −sα g) in X.

ABDELAZIZ KHOUTAIBI AND LAHCEN MANIAR
Thus, On the other hand, (U n − U ) is the strong solution to the system   Proof. Since Φ is a scalar product on E 1 , it is continuous and coercive. On the other hand, define the linear form L on E 1 by By Young and Carleman estimate (14), we have for all U ∈ E 1 , and this yields that L is continuous on E 1 . Finally, by Lax-Milgram Theorem, we deduce that the variational problem Φ(U, V ) = L(V ) for all V ∈ E 1 has a unique solution U ∈ E 1 .

Now, we show an intermediate Carleman estimate.
Lemma 3.6. Let λ ≥ λ 0 , s ≥ s 0 and U = (u, u Γ ) be the unique solution to the variational problem (16), and set Then, the following assertions hold.
Step 1. Estimate of the three first terms. As in [11], multiplying the first equation of (18) by u and integrating on Ω T , we obtain Similarly, multiplying the second equation in (18) by u Γ and integrating by parts, we obtain Adding (20) and (21), we obtain Using Young inequality, for all > 0, we obtain

3.2.
Proof of Theorem 3.4. By the preparations of the above section, we are now able to show the main Carleman estimate (15). The proof will be done in three steps. Proof.
4. Null controllability. In this section, we apply Carleman estimate (31) to show null controllability for (1). In the following proposition, we show first the observability inequality for the backward adjoint problem (2).
Step 1. First, assume that Φ T ∈ D(A * ) and let Φ be the unique strict solution to the backward adjoint problem (2)  where C = max(C 0 , C 2 , 2) and the constant K is given by (35).
Finally, we can state and show the aim result of this work.
Proof. The null controllability of the system (1) will follow by the observability inequality (34) and a classical duality argument as in [32]. Define the bounded linear operator T : L 2 (ω T ) → L 2 by Using the continuous embedding of Z Ω T ×Z Γ T in L 2 (Ω T )×L 2 (Γ T ), we also introduce the bounded linear operator R : L 2 × Z Ω T × Z Γ T → L 2 given by R(Y 0 , f, g) = S(T )Y 0 + T 0 S(T − s)(f (s), g(s))ds.