Global Carleman Estimates for Waves and Applications

By Lucie Baudouin, Maya de Buhan, Sylvain Ervedoza. 2011

90 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/06146f4030ab786756ac95f849f30a5f10d46ff2

Citations in the corpus

• Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs (2020, unknown): “Alternative proofs use Carleman estimates [4].”