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Stability of the determination of a coefficient for the wave equation in an infinite wave guide

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 $\partial_t^2u-Δu+q(x)u=0$ in an unbounded wave guide $Ω=ω\times\mathbb R$ with $ω$ a bounded smooth domain of $\mathbb R^2$, 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 $\barΩ$, we extend this result to the same inverse problem with measurements on a bounded subset of the lateral boundary $(0,T)\times\partialΩ$.