  In this paper, we investigate a distributed optimal control problem for a
convective viscous Cahn-Hilliard system with dynamic boundary conditions. Such
systems govern phase separation processes between two phases taking place in an
incompressible fluid in a container and, at the same time, on the container
boundary. The cost functional is of standard tracking type, while the control
is exerted by the velocity of the fluid in the bulk. In this way, the coupling
between the state (given by the associated order parameter and chemical
potential) and control variables in the governing system of nonlinear partial
differential equations is bilinear, which presents a difficulty for the
analysis. In contrast to the previous paper arXiv:1709.02335 [math.AP] by the
same authors, the bulk and surface free energies are of double obstacle type,
which renders the state constraint nondifferentiable. It is well known that for
such cases standard constraint qualifications are not satisfied so that
standard methods do not apply to yield the existence of Lagrange multipliers.
In this paper, we overcome this difficulty by taking advantage of results
established in the quoted paper for logarithmic nonlinearities, using a
so-called `deep quench approximation'. We derive results concerning the
existence of optimal controls and the first-order necessary optimality
conditions in terms of a variational inequality and the associated adjoint
system.
