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

### Peter I. Kogut

Dnipropetrovsk National University, Ukraine### Günter Leugering

Universität Erlangen-Nürnberg, Germany

## 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 $\epsilon$. 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.

## Cite this article

Peter I. Kogut, Günter Leugering, On $S$-Homogenization of an Optimal Control Problem with Control and State Constraints. Z. Anal. Anwend. 20 (2001), no. 2, pp. 395–429

DOI 10.4171/ZAA/1023