TY - JOUR A1 - Keil, Tobias A1 - Hintermüller, Michael A1 - Wegner, Donat T1 - Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system with non-matched fluid densities JF - SIAM Journal on Control and Optimization N2 - 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 MPECs. Y1 - 2017 VL - 55 SP - 1954 EP - 1989 ER -