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

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 → 0 in the framework of variational S-convergence which generalizes the concept of Γ-convergence. We also introduce the notion of G∗-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.