Homogenization of boundary optimal control problems with oscillating boundaries

Abstract

This article is devoted to the study of optimal boundary control problems associated with Laplacian posed on a domain having rapidly oscillating boundary. A rectangular region with oscillations on the top boundary is considered as a domain for simplicity. A control is applied on the regular bottom boundary part, away from the oscillatory one. We discuss both, Dirichlet as well as Neumann boundary control problem. In both of the cases the $L^2$- cost functional is taken into account. A complete asymptotic analysis of the optimality system is obtained and then we derive appropriate error estimates. Homogenization is quite similar in both and is not very difficulty. But the major contribution of the work in this paper is the error analysis and we need to construct different test functions for Dirichlet and Neumann boundary conditions.