On $S$-Homogenization of an Optimal Control Problem with Control and State Constraints
Peter I. Kogut, Günter R. Leugering · Zeitschrift für Analysis und ihre Anwendungen · 2001
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 \epsilon . We study the limit of this problem when \epsilon \rightarrow 0 in the framework of variational S -convergence which generalizes the concept of \Gamma -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.