Optimal control problem for Stokes system: Asymptotic analysis via unfolding method in a perforated domain

TL;DR

This paper analyzes the asymptotic behavior of an optimal control problem constrained by the stationary Stokes equations in a perforated domain, using the unfolding method to establish convergence to a limit problem.

Contribution

It introduces an asymptotic analysis framework for the Stokes OCP in perforated domains using the unfolding method, including convergence results for solutions and cost functionals.

Findings

Convergence of solutions from perforated to non-perforated domain

Establishment of limit optimal control problem

Convergence of the associated cost functional

Abstract

This article's subject matter is the study of the asymptotic analysis of the optimal control problem (OCP) constrained by the stationary Stokes equations in a periodically perforated domain. We subject the interior region of it with distributive controls. The Stokes operator considered involves the oscillating coefficients for the state equations. We characterize the optimal control and, upon employing the method of periodic unfolding, establish the convergence of the solutions of the considered OCP to the solutions of the limit OCP governed by stationary Stokes equations over a non-perforated domain. The convergence of the cost functional is also established.