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.Theorem 1.1. Assume the multiplier condition (1.2) and the time condition (1.3). Let β ∈ (0, 1) be such that sup x∈Ω |x − x0| < βT.(1.6)Then for any m > 0, there exist λ > 0 independent of m, s0 = s0(m) > 0 and a positive constant M = M (m) such that for ϕ being defined as in (1.5), for all p ∈ L ∞ ≤m (Ω × (−T, T )) and for all s ≥ s0:for all z ∈ L 2 (−T, T ; H 1 0 (Ω)) satisfying ∂ 2 t z − ∆z + pz ∈ L 2 (Ω × (−T, T )) and ∂νz ∈ L 2 (Γ0 × (−T, T )).Theorem 1.2. Under the assumptions of Theorem 1.1, if z furthermore satisfies z(·, 0) = 0 in Ω, one also hasIn particular, if z(·, 0) = 0 in Ω and q ∈ L ∞ ≤m (Ω), then for all z ∈ L 2 (0, T ; H 1 0 (Ω)) satisfying ∂ 2 t z − ∆z + qz ∈ L 2 (Ω × (0, T )) and ∂νz ∈ L 2 (Γ0 × (0, T )) and for all s ≥ s0(m), s 1/2 Ω e 2sϕ(0) |∂tz(0)| 2 dx + s T 0 Ω e 2sϕ |∂tz| 2 + |∇z| 2 dxdt + s 3 T 0 Ω e 2sϕ |z| 2 dxdt ≤ M T 0 Ω Js,q[µ, g](z) =

This article develops the numerical and theoretical study of a reconstruction algorithm of a potential in a wave equation from boundary measurements, using a cost functional built on weighted energy terms coming from a Carleman estimate. More precisely, this inverse problem for the wave equation consists in the determination of an unknown time-independent potential from a single measurement of the Neumann derivative of the solution on a part of the boundary. While its uniqueness and stability properties are already well known and studied, a constructive and globally convergent algorithm based on Carleman estimates for the wave operator was recently proposed in [BdBE13]. However, the numerical implementation of this strategy still presents several challenges, that we propose to address here.

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The aim of this paper is to propose a new method to solve the inverse scattering problem. This method works directly in the time-dependent domain, using the wave equation and proceeds in two steps. The first step is the Time-Reversed Absorbing Condition (TRAC) method to reconstruct and regularize the signal and to reduce the computational domain. The second step is the Adaptive Inversion method to solve the inverse problem from the TRAC data, by using basis and mesh adaptation. This strategy allows to recover the position, the shape and the properties of the scatterer in a precise and robust manner.