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We propose a general control framework for two-phase flows with variable densities in the diffuse interface formulation, where the distribution of the fluid components is described by a phase field. The flow is governed by the diffuse interface model proposed in [Abels, Garcke, Gr\"un, M3AS 22(3):1150013(40), 2012]. On the basis of the stable time discretization proposed in [Garcke, Hinze, Kahle, APPL NUMER MATH, 99:151--171, 2016] we derive necessary optimality conditions for the time-discrete and the fully discrete optimal control problem. We present numerical examples with distributed and boundary controls, and also consider the case, where the initial value of the phase field serves as control variable.

Optimal control of time-discrete two-phase flow driven by a diffuse-interface model