Optimal Control of a Semidiscrete Cahn-Hilliard-Navier-Stokes System with Non-Matched Fluid Densities

TL;DR

This paper develops a framework for the optimal control of a coupled Cahn-Hilliard/Navier-Stokes system with variable densities, using time-discretization, energy estimates, and regularization techniques to establish existence and optimality conditions.

Contribution

It introduces a novel approach for controlling a complex coupled system with double-obstacle potential, including regularization and limit analysis for optimality conditions.

Findings

Existence of solutions for primal and regularized systems

Derivation of first-order optimality conditions

Establishment of a stationarity system for the original problem

Abstract

This paper is concerned with the distributed optimal control of a time-discrete Cahn--Hilliard/Navier--Stokes system with variable densities. It focuses on the double-obstacle potential which yields an optimal control problem for a family of coupled systems in each time instance of a variational inequality of fourth order and the Navier--Stokes equation. By proposing a suitable time-discretization, energy estimates are proved and the existence of solutions to the primal system and of optimal controls is established for the original problem as well as for a family of regularized problems. The latter correspond to Moreau--Yosida type approximations of the double-obstacle potential. The consistency of these approximations is shown and first order optimality conditions for the regularized problems are derived. Through a limit process, a stationarity system for the original problem is established which is related to a function space version of C-stationarity.