Optimal Control of a Navier–Stokes–Cahn–Hilliard System for Membrane-fluid Interaction

We consider an optimal control problem for a two-dimensional Navier–Stokes–Cahn–Hilliard system arising in the modeling of fluid-membrane interaction. The fluid dynamics is governed by the incompressible Navier–Stokes equations, which are nonlinearly coupled with a sixth-order Cahn–Hilliard type equation representing the deformation of a flexible membrane through a phase-field variable. Building on the previously established existence and uniqueness of global strong solutions for the coupled system, we introduce an external forcing term acting on the fluid as the control variable. Then we seek to minimize a tracking-type cost functional, demonstrating the existence of an optimal control and deriving the associated first-order necessary optimality conditions. A key issue is to establish sufficient regularity for solutions of the adjoint system, which is crucial for the rigorous derivation of optimality conditions in the fluid dynamic setting.