Periodic Controls in an Oscillating Domain: Controls via Unfolding and Homogenization

Abstract

An optimal control problem in a two-dimensional domain with a rapidly oscillating boundary is considered. The main features of this article are on two points, namely, we consider periodic controls in the thin periodic slabs of period $\epsilon >0$, a small parameter, and height $O(1)$ in the oscillatory part, and the controls are characterized using unfolding operators. We then do a homogenization analysis of the optimal control problems as $\epsilon \rightarrow 0$ with $L^2$ as well as Dirichlet (gradient-type) cost functionals.