Global Carleman estimates for waves and applications

TL;DR

This paper develops Carleman estimates for the wave equation, enabling new methods for controllability and inverse problems, including constructive algorithms for potential recovery from boundary flux measurements.

Contribution

It introduces a comprehensive framework of Carleman estimates for waves and applies it to controllability and inverse problems with constructive solution methods.

Findings

Constructive control methods weakly dependent on potentials.

Algorithm for recovering unknown potentials from boundary flux.

Enhanced Carleman estimates for wave equations.

Abstract

In this article, we extensively develop Carleman estimates for the wave equation and give some applications. We focus on the case of an observation of the flux on a part of the boundary satisfying the Gamma conditions of Lions. We will then consider two applications. The first one deals with the exact controllability problem for the wave equation with potential. Following the duality method proposed by Fursikov and Imanuvilov in the context of parabolic equations, we propose a constructive method to derive controls that weakly depend on the potentials. The second application concerns an inverse problem for the waves that consists in recovering an unknown time-independent potential from a single measurement of the flux. In that context, our approach does not yield any new stability result, but proposes a constructive algorithm to rebuild the potential. In both cases, the main idea is to introduce weighted functionals that contain the Carleman weights and then to take advantage of the freedom on the Carleman parameters to limit the influences of the potentials.