  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 an additional difficulty for
the analysis. The nonlinearities in the bulk and surface free energies are of
logarithmic type, which entails that the thermodynamic forces driving the phase
separation process may become singular. We show existence for the optimal
control problem under investigation, prove the Fr\'echet differentiability of
the associated control-to-state mapping in suitable Banach spaces and derive
the first-order necessary optimality conditions in terms of a variational
inequality and the associated adjoint system. Due to the strong nonlinear
couplings between state variables and control, the corresponding proofs require
a considerable analytical effort.
