On the optimal control for the wave equation in heterogeneous materials

Abstract

This article deals with the wave equation in heterogeneous materials. Under this premise, we obtain a result for the following optimal interior control problem: reach optimally (in finite time T) any point on a given trajectory (called the reference trajectory) determined by the solution of the wave equation in a homogeneous medium. In this article, we explicitly build an optimal control function using the decomposition into eigenmodes (or eigenfunctions) of the associated Laplacian operator, emphasizing that our approach is more advantageous than the classical numerical techniques when we need to control only some eigenmodes of the wave equation while ignoring others. As a main result, we obtain a useful expression for the optimal control solution based on the calculation of effective wave coefficients. This expression is given as a linear combination of control terms, each of them controlling an eigenmode associated to the heterogeneous system.