Inverse problem related to boundary shape identification for a hyperbolic differential equation
Summary: In this paper, we are interested in the inverse problem of the determination of the unknown part \(\partial\operatorname{\Omega},\operatorname{\Gamma}_0\) of the boundary of a uniformly Lipschitzian domain \(\operatorname{\Omega}\) included in \(\mathbb{R}^N\) from the measurement of the normal derivative \(\partial_nv\) on suitable part \(\operatorname{\Gamma}_0\) of its boundary, where \(v\) is the solution of the wave equation \(\partial_{tt}v \left(x, t\right)-\operatorname{\Delta}v \left(x, t\right)+p \left(x\right)v \left(x \right)=0\) in \(\operatorname{\Omega}\times \left(0, T\right)\) and given Dirichlet boundary data. We use shape optimization tools to retrieve the boundary part \(\operatorname{\Gamma}\) of \(\partial\operatorname{\Omega}\). From necessary conditions, we estimate a Lagrange multiplier \(k\left(\operatorname{\Omega}\right)\) 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.
scientific article; zbMATH DE number 7459468
scientific article