Optimal Velocity Control of a Brinkman–Cahn–Hilliard System with Curvature Effects

Abstract We address an optimal control problem governed by a system coupling a Brinkman-type momentum equation for the velocity field with a sixth-order Cahn–Hilliard equation for the phase variable, incorporating curvature effects in the free energy. The control acts as a distributed velocity control, allowing for the manipulation of the flow field and, consequently, the phase separation dynamics. We establish the existence of optimal controls, prove the Fréchet differentiability of the control-to-state operator, and derive first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables. We also discuss the aspect of sparsity. Beyond its analytical novelty, this work provides a rigorous control framework for Brinkman–Cahn–Hilliard systems incorporating a curvature regularization, offering a foundation for applications in microfluidic design and controlled pattern formation.