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 1-D wave equation. The problem consists in seeking simultaneously the spatiotemporal 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 2-D divergence free-vector as rotated gradient. We transform the original optimal design problem into a non-convex 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 sub-domains and our contribution consists in understanding their mutual interaction.