Carleman Estimate for a Singulary Perturbed Degenerated Parabolic Equation

: In this paper, we are concerned with the the internal control of an elliptic singularly perturbed degenerated parabolic equation. This parabolic equation models sand transport problem near the coast in areas subjected to the tide. We study ﬁrst the null controllability result of the parabolic equation modeling sand transport equation.The limit problem obtained by homogenization problem is also considered. We use distributed and bounded controls supported on a small open set of the initial domain. We prove the null controllability of the system at any time by using observability inequality for both problem. For this purpose, a speciﬁc carleman estimate for the solutions of degenerate adjoint limit problem is also proved.


Introduction and Results
This paper, we focus on the study of null controllability of a singularly perturbed and degenerated parabolic partial differential equation. This kind of PDE arises from the modeling sand transport on the seabed in short time. The model is described in I. Faye and al. [9]. In the case of linear systems, exact controllability or null controllability is a widely study by many authors and for various methods see for example Russell [15], Lebeau-Robbiano [12], J. L. Lions [13]. Many other results of the null controlability of heat equation can also be found in G. Lebeau and L. Robbiano [12] and A. V. Fursikov and O. Yu. Imanuvilov [11] and Dubova et al. [6,7,8].
We are particularly interested on models on the general form of: Where A and C are regular coefficients and z 0 ∈ L 2 (T 2 ). The function z = z (t, x), is the dimensionless seabed altitude at t and in x. For a given constant T, t ∈ [0, T ), stands for the dimensionless time and x = (x 1 , x 2 ) ∈ T 2 , T 2 being the two dimensional torus R 2 / Z 2 , is the dimensionless position variable.
Existence and uniqueness of solutions to (1) has been studied in [9] in the framework of periodic solutions if A and C and regular and bounded functions and can be set in the form A (t, x) = A (t, t , x) and C (t, x) = C (t, t , x), where θ → A (t, θ, x), C (t, θ , x) are 1-periodic in θ. We have also to recall that the coefficients A and C can vanish and if is too small, the diffusion coefficient tends to infinity. The solution z to (1) is bounded in this space and belongs to L ∞ ([0, T ), L 2 (T 2 )).
In this paper, we consider the following control problem: let ω be an open subset of T 2 and z the solution to the problem where f ∈ L ∞ ((0, T ), L 2 (Ω)) and h is the control function and z 0 ∈ L 2 (Ω), χ ω represent the charasteristic function of ω ⊂ T 2 .
We consider also the following controlled problem, Before going further, we will recall the following notions of controllability.

On the Control Problems
In this paper, we will consider a null controllability result for the systems ∂U ∂θ − ∇ · A(t, ., .)∇U = ∇ · C(t, ., .) and where T > 0 is given, z 0 , z 1 ∈ L 2 (Ω) are the initial and final data. Moreover, h and h are locally distributed control acting on the control region ω ⊂ Ω.
The aim of this paper is to analyze the null controllability of equations (5) and (6) . In perspective, we will look for the behavior of h (t, x, U ) in system (6) when goes to 0.

Presentation of the Results
In this section, we analyze the null controllability, using a locally distributed control acting on the control region ω.
We assume that ω is bounded open set included in the two dimensional torus T 2 . In the following, we consider that A ∈ L ∞ ([0, T ], L ∞ # (R, L 2 (T 2 ))) and C ∈ are respectively the two- The coefficients A and C satisfy the following hypotheses and where γ and G thr are constant positifs, not depending on . For the notion of two scales limits, we refer to [1,9,16] We have the following theorem. Theorem 1.1. Let > 0, for any T > 0, under hypotheses (7)-(9), there exists a unique solution z ∈ L 2 ([0, T ), L 2 (T 2 )) solution to (1). This solution satisfies where γ is a constant not depending on . Moreover, the solution z to (1) two scales converges to U ∈ L ∞ ([0, T ), L 2 # (R, L 2 (T 2 ))) unique solution to (2). Proof The proof of this theorem is done [9]. We have also the following theorem Theorem 1.2. Let ω ⊂ T 2 and > 0, under assumptions (7)-(9), the system (5) is null controllable at any time T > 0. In We have our second null controllability result via the following theorem Theorem 1.3. Assume that A and C are the two scales limits of A and C satisfying (7)-(9), there exists h ∈ L ∞ # (R, L 2 (R, T 2 )) such that the solution U to (5) satisfies The proof of theorem 1.2 and theorem 1.3 are done in section 3.
In the following, we consider the solution w to the adjoint problem to (6) given as follows and the solution y to the adjoint state (5): The corresponding observability inequality is given by the following result.
In the same way, assuming that hypotheses (7)-(9) holds, and considering y the solution to the adjoint problem (12) we have 2. Inequality of Observability

Proof of Theorem 1.4
As is classical in controllability theory, the result of theorem 1.4 can be given a dual form, introducing the so-called adjoint system of (5) In this equation, t is only a parameter. The function θ → w(t, θ, x) is 1 periodic.
The null controllability of (5) is equivalent to the following observability of the adjoint problem Theorem 2.1. Let A and C be the two scale limits of A and C . Let T > 0, be given and ( 15) satisfies the following observability result Proof Multiplying (15) by w and integrating over the T 2 , we get Because of the fact that, the second terme is positive, we get proving that the application t → T 2 w 2 (t, θ, x)dx is nondecreasing. Then we have, Integrating from from θ1 Let R, σ and ρ given as follows, then, the following equality holds Following the idea developed by Cannarsa et al [2], there exists a constant C depending on T 2 such that for all θ 1 ∈ [0, 1] Hence, we get We consider also, the adjoint state of (2) We have also, the following lemma Theorem 2.2. Let T > 0 and > 0, under assumptions (7)-(9), there exists a constant C 0 (T 2 , ω, γ, G th ) such that the solution y to (24) satisfies the following equality Proof We proceed in the same way, as in the proof of the above theorem. Multiplying equation (24) by y and integrating over T 2 we get 1 2 As the second term is positive, we get from the last equality the following equality Hence, the application t → T 2 (y ) 2 dx is nondecreasing and we have giving where R, σ and ρ are given in the proof of the above thoerem. Using a result Cannarsa et al. [2] recalled in the proof of the above theorem, we where C depend on the domain T 2 , ω and T.

Equivalence Between Null Controllability and Observability
This section is devoted to the proof of theorem 1.2 and theorem 1.3. Thus, in other words, (5) and (6) are null controllable. Suppose (5) is null controllable and the control h is bounded. Let U be the solution of (5) and let h ∈ L 2 ((0, T ), L 2 (Ω) ∈ L 2 (Ω) be a control steering the solution U of (5) with U (0, 0, ·) = z 0 (x) such that Then, multiplying (5) by w and (24) by U, and integrating by parts over T 2 leads to then, we get, by summing the two expressions Integrating from 0 to θ 1 ∈ [0, T ], and taking into account that U (t, θ 1 , 0) = 0 we get giving the result. Conversely, assume that we have an observability equality for the solution w to (15) and let z 0 ∈ L 2 (T 2 ). For any > 0, we consider the functional where, for every h ∈ L 2 (R × T 2 ), u h denotes the corresponding solution of (5). By a straightforward convexity argument, J attains its minimum at a unique point, say h ∈ L 2 (R × T 2 ). Then, writing u for u h , Fermat's rule yields, for all g ∈ L 2 (Ω), where U g is the solution of the problem Now, let v be the solution of (15) with and multiplying (36) by v and (15) by U g we get which, combined with (35), implies that h = −χ ω v . Therefore, by a same argument, we get from (36) and (15), Then, integrating over [0, 1] and recalling that w (1, x) = 1 U (1, x), we get Thus, using observability inequality (25) to bound the L 2 -norm of v (0, .), one obtains the last inequality reads as 1 So, the weak limit, say h 0 , of h i along a suitable sequence → 0 satisfies (5).

Carlman Estimate
In this section, we give Carleman estimate of the solution W solution to (5) . Carleman estimates are weighted Sobolev inequalites satisfied by the solution. For all s > 0, let us define the weight function ϕ(θ, x) such that where U is solution (5). Then we have Replacing U in (5), we get In the following, we recalculate, based on data, all the terms of the equation (39) in order to reconstruct it. Because of this, we get Then, equation (5) becomes Then, the solution W solves the following system In the following, let's define the following operator P by We are now interested in the adjoint operator P * of P defined as follows (P W, V ) = (W, P * V ) .
We have Multiplying (44) by V and integrating, we get Thus, the adjoint operator P * V is identified as follows: and we define the two operators P + W and P − W, as the following giving directly From (44) and (48), we have and from which we have P − W = ∂W ∂θ + 2s A∇ϕ∇W + s∇ · ( A∇ϕ))W We have the following lemma.
The following identity holds Proof We have the following equality Then we have, P W 2 = P + W 2 + 2 < P + W, P − W > + P − W 2 , Replacing each operator by it's expression we have and because of the linear of the scalar product we get directly.
Developing term by term we have, and Combining the formulas giving in (55), (56) and (57), we get for (P + W, P − W ) L 2 (D) the following expression . giving Proposition 3.1 The following identity holds Proof It is enough to develop the integrals I 1 , I 2 and I 3 and by simplifying some expressions to obtain the result. We also consider the fact that the function ϕ(x, θ) belongs to C ∞ ([0, 1] × T 2 ) and is 1-periodic with respect to the variable θ. Following this idea, we have Integrating by parts the integral I 1 with respect to the variable θ, and using periodicity, we get The last term of (58) can be written as follow Integrating J 2 by parts in the first term, we have and then, and Then we have For the second term, we have also .
The first scalar product can be written as follows giving Deriving and regrouping the semblable term, we get The second term of I 2 satisfy the following Developing the expression J 4 , we get giving for I 2 the following equality By combining the formulas (63), (70), (71), we get the desired result. Lemma 3.2 Let W be the solution of (37) and P + and P − defined by (52) and (53). Then we have the following estimates: proof The proof of this lemma follows from minimization of the each term I 1 , I 2 and I 3 .
we deduce that,

Conclusion
In this work, we are interested in the null controlability of a degenerated and singularly perturbed partial differential equation. We therefore show the controlability at any times T by taking inspiration from an observability result. A carlmann estimate is also proven. It would therefore be interesting to look at the numerical aspects of these problems.