OPTIMAL CONTROL OF A SEMIDISCRETE CAHN-HILLIARD-NAVIER-STOKES SYSTEM WITH NONMATCHED FLUID DENSITIES

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 family of coupled systems in each time instant of a variational inequality of fourth order and the Navier-Stokes equation. By proposing a suitable time-discretization, energy estimates are proved, and the existence of solutions to the primal system and of optimal controls is established for the original problem as well as for a family of regularized problems. The latter correspond to Moreau-Yosida-type approximations of the double-obstacle potential. The consistency of these approximations is shown, and first-order optimality conditions for the regularized problems are derived. Through a limit process with respect to the regularization parameter, a stationarity system for the original problem is established. The resulting system corresponds to a function space version of C-stationarity which is a special notion of stationarity for mathematical programs with equilibrium constraints.