Stability of the determination of a coefficient for wave equations in an infinite waveguide

We consider the stability in the inverse problem consisting in the determination of an electric potential q, appearing in a Dirichlet initial-boundary value problem for the wave equation ∂2 t u − ∆u + q(x)u = 0 in an unbounded wave guide Ω = ω × R with ω a bounded smooth domain of R2, from boundary observations. The observation is given by the Dirichlet to Neumann map associated to a wave equation. We prove a Hölder stability estimate in the determination of q from the Dirichlet to Neumann map. Moreover, provided that the gap between two electric potentials rich its maximum in a fixed bounded subset of Ω, we extend this result to the same inverse problem with measurements on a bounded subset of the lateral boundary (0, T ) × ∂Ω. Stability of the determination of a coefficient for the wave equation 2