Suboptimal boundary controls for elliptic equation in critically perforated domain

Abstract

In this paper we study the asymptotic behaviour, as ɛ tends to zero, of a class of boundary optimal control problems ${\mathbb{P}}_{\varepsilon }$, set in ɛ-periodically perforated domain. The holes have a critical size with respect to ɛ-sized mesh of periodicity. The support of controls is contained in the set of boundaries of the holes. This set is divided into two parts, on one part the controls are of Dirichlet type; on the other one the controls are of Neumann type. We show that the optimal controls of the homogenized problem can be used as suboptimal ones for the problems ${\mathbb{P}}_{\varepsilon }$.