Optimal internal stabilization of a damped wave equation by a level set approach

Abstract. We consider a damped linear wave equation defined on a bi-dimensional domain Ω. Due to the damping term defined on the sub-domain ω ⊂ Ω, the system is dissipative. We address the problem of the optimal position and shape design of the support ω in order to minimize the energy of the system at a given time T . Introducing an adjoint problem we first obtain explicitly the (shape) derivative of the energy at time T with respect to the variation of ω. Expressed as a boundary integral on ∂ω, this derivative is then used as an advection velocity in an Hamilton-Jacobi equation for changing the shape. We use the levelset methodology on a fixed working Eulerian mesh. We also consider the optimization with respect to the value of the damping. The numerical approximation is presented in detail and leads to several numerical experiments that indicate the efficiency of the method.