On $S$-Homogenization of an Optimal Control Problem with Control and State Constraints

Abstract

We study the limiting behavior of an optimal control problem for a linear elliptic equation subject to control and state constraints. Each constituent of the mathematical description of such an optimal control problem may depend on a small parameter $ϵ$. We study the limit of this problem when $ϵ\to 0$ in the framework of variational $S$-convergence which generalizes the concept of $\Gamma$-convergence. We also introduce the notion of $G\ast$-convergence generalizing the concept of $G$-convergence to operators with constraints. We show convergence of the sequence of optimal control problems and identify its limit. We then apply the theory to an elliptic problem on a perforated domain.