Approximate controllability of a semilinear elliptic problem in periodically perforated domains

Abstract

In this paper, we study the controllability of a semilinear second order elliptic boundary value problem in a periodically perforated domain, where the control is distributed in an open subset of the domain. Compared to existing literature, the holes in this control problem can intersect the control zone. We study L2 and H1-approximate controllability for the problem and we prove the homogenization results for the control problem in an ε-periodically perforated domain. The holes are of same size as ε with the coefficients of state equation, and the cost functional being rapidly oscillating.