Inverse Problem Related to Boundary Shape Identification for a Hyperbolic Differential Equation

In this paper, we are interested in the inverse problem of the determination of the unknown part ∂Ω, Γ0 of the boundary of a uniformly Lipschitzian domain Ω included in ℝN from the measurement of the normal derivative ∂nv on suitable part Γ0 of its boundary, where v is the solution of the wave equation ∂ttv(x, t) − Δv(x, t) + p(x)v(x) = 0 in Ω × (0, T) and given Dirichlet boundary data. We use shape optimization tools to retrieve the boundary part Γ of ∂Ω. From necessary conditions, we estimate a Lagrange multiplier k(Ω) which appears by derivation with respect to the domain. By maximum principle theory for hyperbolic equations and under geometrical assumptions, we prove a uniqueness result of our inverse problem. The Lipschitz stability is established by increasing of the energy of the system. Some numerical simulations are made to illustrate the optimal shape.