Asymptotic analysis and error estimates for an optimal control problem with oscillating boundaries

In this article, we consider a distributed optimal control problem associated with the Laplacian in a domain with rapidly oscillating boundary. For simplicity, we consider a rectangular region in 2d with oscillations on one part of the boundary. We consider two types of functionals, namely a functional involving the L ²-norm of the state variable and another one involving its H ¹-norm. The homogenization of the optimality system is obtained and then we derive appropriate error estimates in both cases.