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 partial differential omega,Gamma 0 of the boundary of a uniformly Lipschitzian domain omega included in Double-struck capital RN from the measurement of the normal derivative partial differential nv on suitable part Gamma 0 of its boundary, where v is the solution of the wave equation partial differential ttv<mml:mfenced close=")" open="(" separators="|">x,t-Delta v<mml:mfenced close=")" open="(" separators="|">x,t+p<mml:mfenced close=")" open="(" separators="|">xv<mml:mfenced close=")" open="(" separators="|">x=0 in omega x<mml:mfenced close=")" open="(" separators="|">0,T and given Dirichlet boundary data. We use shape optimization tools to retrieve the boundary part Gamma of partial differential omega. From necessary conditions, we estimate a Lagrange multiplier k<mml:mfenced close=")" open="(" separators="|">omega 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.</p>