Strong Stationarity Conditions for the Optimal Control of a Cahn–Hilliard–Navier–Stokes System

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 variational inequality of fourth order and the Navier–Stokes equation. The existence of solutions to the primal system and of optimal controls is established. The Lipschitz continuity of the constraint mapping is derived and used to characterize the directional derivative of the constraint mapping via a system of variational inequalities and partial differential equations. Finally, strong stationarity conditions are presented following an approach from Mignot and Puel.