This talk is divided into two parts. The first part provides a brief introduction to homogenization, a method crucial for understanding multi-scale phenomena. Homogenization has diverse applications, including composite materials, porous media, oscillating boundaries, and more. Additionally, we introduce the concept of optimal control problems.
In the second part, we delve into optimal control problems within oscillating domains. The motivation for studying such problems typically stems from the need to understand flows in channels with rough boundaries, heat transfer in winglets, and similar contexts. We present a detailed exploration of optimal control problems governed by the Stokes system in a domain with highly oscillating boundary, examining the convergence analysis of the optimal solutions and identifying the limiting optimal control problem in a fixed domain. Furthermore, we briefly discuss our findings on homogenization and optimal control problems.
