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Zeitschrift für Analysis und ihre Anwendungen

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Volume 20, Issue 2, 2001, pp. 395–429
DOI: 10.4171/ZAA/1023

Published online: 2001-06-30

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

Peter I. Kogut[1] and Günter Leugering[2]

(1) Dnipropetrovsk National University, Ukraine
(2) Universität Erlangen-Nürnberg, Germany

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.

Keywords: Homogenization, S-convergence, optimal control

Kogut Peter, Leugering Günter: On $S$-Homogenization of an Optimal Control Problem with Control and State Constraints. Z. Anal. Anwend. 20 (2001), 395-429. doi: 10.4171/ZAA/1023