Recovering a potential in damped wave equation from Dirichlet-to-Neumann operator

The inverse problem of recovering the potential q ( x ) in the damped wave equation m ( x ) u t t + μ ( x ) u t = r ( x ) u x x + q ( x ) u , ( x , t ) ∈ Ω T ≔ (0, ℓ ) × (0, T ) subject to the boundary conditions u (0, t ) = ν ( t ), u ( ℓ , t ) = 0, from the Neumann boundary measured output f ( t ) ≔ r (0) u x (0, t ), t ∈ (0, T ] is studied. The approach proposed in this paper allows us to derive behavior of the direct problem solution in the subdomains defined by characteristics of the wave equation and along the characteristic lines, as well. Based on these results, a local existence theorem and the stability estimate are proved. The compactness and Lipschitz continuity of the Dirichlet-to-Neumann operator are derived. Fréchet differentiability of the Tikhonov functional is proved and an explicit gradient formula is derived by means of an appropriate adjoint problem. It is proved that this gradient is Lipschitz continuous.