Executive Summary : | This project aims to study the homogenization of optimal control problems (OCPs) governed by partial differential equations, considering either a fixed domain or a highly oscillating domain. The problems involve applications such as understanding fluid flow in channels with rough boundaries, heat transmission in winglets, and jet engines. The project considers five model problems: distributive OCP subject to the more generalized stationary Stokes equation involving highly oscillating coefficients posed in an n-dimensional domain, the wave equation governed by the wave equation in a 2-dimensional domain with an oscillating boundary, the boundary OCP governed by a parabolic equation, the viscosity-dependent Stokes equation coupled with the transport equation, and the boundary OCP governed by the p-Laplacian operator in varying domains. The goal is to characterize the optimal control and find the effective behavior of optimal solutions using the finite element heterogeneous multi-scale method (FE-HMM). The study also examines the limiting behavior of optimal control and state and identifies the limit OCP with corrector results. The project will obtain numerical results using the multi-scale finite element (MsFEM) technique. The non-linearity of the solution makes finding the existence and uniqueness of optimal state and optimal control difficult, and characterization of such control is also difficult. |