A spatio-temporal design problem for a damped wave equation

We analyze in this work a spatio-temporal optimal design problem governed by a linear damped one-dimensional wave equation. The problem consists of simultaneously seeking the spatio-temporal layout of two isotropic materials and the static position of the damping set in order to minimize a functional depending quadratically on the gradient of the state. The lack of classical solutions for this kind of nonlinear problem is well known. We examine a well-posed relaxation by using the representation of a two-dimensional divergence-free vector as a rotated gradient. We transform the original optimal design problem into a nonconvex vector variational problem. By means of gradient Young measures we compute an explicit form of the "constrained quasi convexification" of the cost density. Moreover, this quasi convexification is recovered by first order laminates which give the optimal distribution of materials and damping set at every point. Finally, we analyze the relaxed problem, and some numerical experiments are performed. The novelty here lies in the optimization with respect to two independent subdomains, and our contribution consists of understanding their mutual interaction.