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

Abstract

In this paper, we are interested in the inverse problem of the determination of the unknown part $\partial \mathrm{\Omega },{\mathrm{\Gamma }}_0$ of the boundary of a uniformly Lipschitzian domain $\mathrm{\Omega }$ included in ${ℝ}^N$ from the measurement of the normal derivative ${\partial }_{n}v$ on suitable part ${\mathrm{\Gamma }}_0$ of its boundary, where $v$ is the solution of the wave equation ${\partial }_{tt}v\left(x,t\right)-\mathrm{\Delta }v\left(x,t\right)+p\left(x\right)v\left(x\right)=0$ in $\mathrm{\Omega }\times \left(0,T\right)$ and given Dirichlet boundary data. We use shape optimization tools to retrieve the boundary part $\mathrm{\Gamma }$ of $\partial \mathrm{\Omega }$ . From necessary conditions, we estimate a Lagrange multiplier $k\left(\mathrm{\Omega }\right)$ 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.