Optimal Control of a Cahn-Hilliard-Navier-Stokes Model with State Constraints

We investigate in this article the Pontryagin's maximum principle for a class of control problems associated with a coupled Cahn-Hilliard-Navier-Stokes model in a two dimensional bounded domain. The model consists of the Navier-Stokes equations for the velocity v, coupled with a Cahn-Hilliard model for the order (phase) parameter φ. The optimal problems involve a state constraint similar to that considered by G. Wang [Optimal controls of 3-dimensional Navier-Stokes equations with state constraints, SIAM J. Control Optim. 41(2) (2002) 583--606]. We derive the Pontryagin's maximum principle for the control problems assuming that a solution exists. Let us note that the coupling between the Navier-Stokes and the Cahn-Hilliard systems makes the analysis of the control problem more involved.