Homogenization of Semi-linear Optimal Control Problems on Oscillating Domains with Matrix Coefficients

Abstract

In this article, we study the homogenization of optimal control problems subject to second-order semi-linear elliptic PDEs with matrix coefficients in two different types of oscillating domains: a circular domain and a domain with general low-dimensional oscillations. The cost functionals considered are of general energy type with oscillating matrix coefficients, and the coefficient matrix in the cost functional is allowed to differ from the coefficient matrix in the constrained PDE. We prove well-defined limit problems for both domains and obtain explicit forms for the limiting coefficient matrices of the cost functionals and constrained PDEs. As expected, the coefficient matrix of the limit cost functional is a combination of the original cost functional’s and constrained PDE’s coefficient matrices.