Optimal Control of a Higher-Order Cahn-Hilliard Equation Coupled with Brinkman Equation

TL;DR

This paper develops an optimal control framework for a coupled sixth-order Cahn-Hilliard and Brinkman system, incorporating sparsity-promoting $L^1$-norm terms, and derives first-order optimality conditions for both differentiable and nondifferentiable cases.

Contribution

It introduces a novel optimal control approach for a higher-order Cahn-Hilliard-Brinkman system with sparsity constraints, extending existing methods to nondifferentiable cost functionals.

Findings

Derived first-order optimality conditions for the control problem.

Established conditions for both differentiable and nondifferentiable cost functionals.

Analyzed the impact of sparsity-promoting $L^1$-norm on control solutions.

Abstract

In this work, we investigate optimal control of a Brinkman equation couple with sixth-order Cahn-Hilliard equation. The Cahn-Hilliard equation is endowed with a source term accounting for mass exchange and the velocity equation contains a non divergence-free forcing term, which act as distributed control variable. We consider the aforementioned system with constant mobility, viscosity and nonlinearity of double-well shape is regular. The cost functional of the optimal control problem contains a nondifferentiable term like the $L^1$-norm with sparsity constant $\kappa$, which leads to sparsity of optimal controls. We study the first order necessary optimality condition for both the case $\kappa=0$ and $\kappa>0.$ When the cost functional is differentiable, first order necessary optimality conditions are characterized by Lagrange multiplier method and for nondifferentiable case we have used the idea of Casas and Tr\"oltzsch from the paper (Math. control Relat. Fields, 10(3):527-546, 2020).