Optimal control of a semidiscrete Cahn-Hilliard-Navier-Stokes system with non-matched fluid densities
Submission Status: | in press |
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- 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.