Approximate controllability of a semilinear elliptic problem with robin condition in a periodically perforated domain

Abstract

In this article, we study the approximate controllability and home-genization results of a semi-linear elliptic problem with Robin boundary condition in a periodically perforated domain. We prove the existence of minimal norm control using Lions constructive approach, which is based on Fenchel-Rockafeller duality theory, and by means of Zuazua's fixed point arguments. Then, as the homogenization parameter goes to zero, we link the limit of the optimal controls ( the limit of fixed point of the controllability problems) with the optimal control of the corresponding homogenized problem.