  In this paper we consider a nonlinear system of PDEs coupling the viscous
Cahn-Hilliard-Oono equation with dynamic boundary conditions enjoying a similar
structure on the boundary. After proving well-posedness of the corresponding
initial boundary value problem, we study an associated optimal control problem
related to a tracking-type cost functional, proving first-order necessary
conditions of optimality. The controls enter the system in the form of a
distributed and a boundary source. We can account for general potentials in the
bulk and in the boundary part under the common assumption that the boundary
potential is dominant with respect to the bulk one. For example, the regular
quartic potential as well as the logarithmic potential can be considered in our
analysis.

Analysis of PDEs