Nonlinear and memory boundary feedback stabilization for Schrödinger equations with variable coefficients

In this paper, the boundary stabilization of Schrödinger equations with variable coefficients by nonlinear and memory feedback is considered. The approch adopted uses Riemannian geometry methods and multipliers techniques.©2020 All rights reserved.

Throughout the paper we assume where k : Γ 1 × R + → R + ∈ C 2 (R + , L ∞ (Ω)) , and : R + → R + is a continuous. we define the corresponding energy functional by The goal of this work is to stabilize the system (1.1a − 1.1d) and (1.3); to find a suitable feedback u = F (x, y t ) such that the energy (1.3) decays to zero exponentially as t → +∞ for every solution y of which E (0) < +∞. The approch adopted uses Riemannian geometry, this method was first introduced into the boundary-control problem by Yao [10] for the exactly controllability of wave equations. The stabilization of partial differential equations has been considered by many authors. The asymptotic behaviour of the wave equation with memory and linear feedbacks with constant coefficients has been studied by Guesmia [4], and by Aassila et al [1] in the case nonlinear .This study has been generalised by Chai Guo [2], with variable coefficients by using a very different method, namely, the Riemannian geometry method.
On the other hand, the stabilization of the Schrödinger equation has been studied by Machtyngier Zuazua [9] in the Neumann boundary conditions, and by Cipolatti et al [3] with nonlinear feedbacks, this study has been considered by Lasiecka Triggani [7] with constant coefficients acting in the Dirichlet boundary conditions. The objective of this work, we consider the boundary stabilization for system (1.1a − 1.1d) and (1.3) with variable coefficients and memory with nonlinear feedbacks by using multipliers techniques. Our paper is organized as follows. In subsection 1.1, we introduce some notations and results on Riemannian geometry. Our main results are studied in section 2. Section 3 is devoted to the proof of the main results.

Euclidean metric on R n
Let (x 1 , . . . , x n ) be the natural coordinate system in R n . For each x ∈ R n , denote the gradient of f and the divergence of X in the Euclidean metric.

Riemannian metric
For each x ∈ R n , define the inner product and the corresponding norm on the tangent space T x R n by Then (R n , g) is a Riemannian manifold with a Riemannian metric g. Denote the Levi-Cevita connection in metric g by D. Let H be a vector field on (R n , g). The covariant differential DH of H determines a bilinear form on T x R n × T x R n . For each x ∈ R n , by where D X H is the covariante derivative of H with respect to X. The following lemma provides some useful equalities.
Let f, h ∈ C 1 Ω and let H, X be vector field on R n .Then with reference to the above notation, we have The gradient ∇ g f of f in the Riemannian metric g is given by

Statement of main result
To obtain the boundary stabilization of problem (1.1a − 1.1d) and (1.2) , the following hyptheses are assumed.
There exists a vector field H on the Riemannian manifold (R n , g) such that and We also assumed the feedback function satisfies and k 0 and k / 0, on Γ 0 × R + . Moreover, We have the following result of existence and uniqueness of strong and weak solution to (1.1a − 1.1d) and (1.2) . 1a − 1.1d) and (1.3) admits a unique global weak solution y ∈ C(R + , V). Furthermore, if y 0 ∈ H 3 (Ω) ∩ H 1 Γ 0 (Ω) and ∂y 0 ∂ν A = − 1 2 ky(0) on Γ 1 , then the solution has the regularity y ∈ C 1 (R + , V).
and y = 0 is the unique solution of the problem. Then for all given initial data y 0 ∈ H 1 Γ 0 (Ω) , there exist two positive constants M > 0 and ω > 0 such that

PROOF OF MAIN RESULT
For simplicity, we assume that y is a strong solution. By a classical density argument, Theorem 2.2 still holds for a weak solution.
|y| ds, and whenever 0 < T < ∞. Proof. Differentiating the energy E (.) defined by (1.4) and using Green's second theorem, we have By the boundary condition, and by the assumption (2.5) and several technics we obtain that This completes the proof of lemma 3.1.

Lemma 3.2.
For all 0 S < T < ∞ we have Thus using (1.1c) and (3.2), we find that this simplifies to the sought-after identity .

4.
Completion of the proof of theorem 2.1.
We also have the estimates, in (3.3c) and (3.3d) ε is an arbitary positive constant.
Now using a compactness-uniqueness argument in the style of Lasiecka and Triggiani [7] , we deduce