Alberto Mercado

Inverse problems for the Schrödinger equation via Carleman inequalities with degenerate weights

Baudouin and Puel (2002 Inverse Problems 18 1537-54), investigated some inverse problems for the evolution Schr¨odinger equation bymeans of Carleman inequalities proved under a strict pseudoconvexity condition. We showhere that new Carleman inequalities for the Schr¨odinger equationmay be derived under a relaxed pseudoconvexity condition, which allows us to use degenerate weights with a spatial dependence of the type ψ(x) = x * e, where e is some fixed direction in RN. As a result, less restrictive boundary or internal observations are allowed to obtain the stability for the inverse problem consisting in retrieving a stationary potential in the Schr¨odinger equation from a single boundary or internal measurement.

A global Carleman estimate in a transmission wave equation and application to a one-measurement inverse problem

We consider a transmission wave equation in two embedded domains in , where the speed is a1 > 0 in the inner domain and a2 > 0 in the outer domain. We prove a global Carleman inequality for this problem under the hypothesis that the inner domain is strongly convex and a1 > a2. As a consequence of this inequality, uniqueness and Lipschitz stability are obtained for the inverse problem of retrieving a stationary potential for the wave equation with Dirichlet data and discontinuous principal coefficient from a single time-dependent Neumann boundary measurement.

Lipschitz stability in an inverse problem for the Kuramoto-Sivashinsky equation

In this article, we present an inverse problem for the nonlinear 1D Kuramoto–Sivashinsky (KS) equation. More precisely, we study the nonlinear inverse problem of retrieving the anti-diffusion coefficient from the measurements of the solution on a part of the boundary and also at some positive time in the whole space domain. The Lipschitz stability for this inverse problem is our main result and it relies on the Bukhgeĭm–Klibanov method. The proof is indeed based on a global Carleman estimate for the linearized KS equation.

An inverse problem for Schrödinger equations with discontinuous main coefficient

This article concerns the inverse problem of retrieving a stationary potential for the Schrödinger evolution equation in a bounded domain of ℝ N with Dirichlet data and discontinuous principal coefficient a(x) from a single time-dependent Neumann boundary measurement. We consider that the discontinuity of a is located on a simple closed hyper-surface called the interface, and a is constant in each one of the interior and exterior domains with respect to this interface. We prove uniqueness and Lipschitz stability for this inverse problem under certain convexity hypothesis on the geometry of the interior domain and on the sign of the jump of a at the interface. The proof is based on a global Carleman inequality for the Schrödinger equation with discontinuous coefficients, result also interesting by itself.

On the determination of the principal coefficient from boundary measurements in a KdV equation

Abstract This paper concerns the inverse problem of retrieving the principal coefficient in a Korteweg–de Vries (KdV) equation from boundary measurements of a single solution. The Lipschitz stability of this inverse problem is obtained using a new global Carleman estimate for the linearized KdV equation. The proof is based on the Bukhgeĭm–Klibanov method.

Null Controllability of the Stabilized Kuramoto--Sivashinsky System with One Distributed Control

This paper presents a control problem for a one-dimensional nonlinear parabolic system, which consists of a Kuramoto--Sivashinsky--Korteweg de Vries equation coupled to a heat equation. We address the problem of controllability by means of a control supported in an interior open subset of the domain and acting on one equation only. The local null-controllability of the system is proved. The proof is based on a Carleman estimate for the linearized system around the origin. A local inversion theorem is applied to get the result for the nonlinear system.