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In this paper, we investigate a distributed optimal control problem for aconvective viscous Cahn-Hilliard system with dynamic boundary conditions. Suchsystems govern phase separation processes between two phases taking place in anincompressible fluid in a container and, at the same time, on the containerboundary. The cost functional is of standard tracking type, while the controlis exerted by the velocity of the fluid in the bulk. In this way, the couplingbetween the state (given by the associated order parameter and chemicalpotential) and control variables in the governing system of nonlinear partialdifferential equations is bilinear, which presents a difficulty for theanalysis. In contrast to the previous paper arXiv:1709.02335 [math.AP] by thesame authors, the bulk and surface free energies are of double obstacle type,which renders the state constraint nondifferentiable. It is well known that forsuch cases standard constraint qualifications are not satisfied so thatstandard methods do not apply to yield the existence of Lagrange multipliers.In this paper, we overcome this difficulty by taking advantage of resultsestablished in the quoted paper for logarithmic nonlinearities, using aso-called `deep quench approximation'. We derive results concerning theexistence of optimal controls and the first-order necessary optimalityconditions in terms of a variational inequality and the associated adjointsystem.

Optimal velocity control of a convective Cahn-Hilliard system with double obstacles and dynamic boundary conditions: a `deep quench' approach