  In this paper we study a distributed optimal control problem for a nonlocal
convective Cahn--Hilliard equation with degenerate mobility and singular
potential in three dimensions of space. While the cost functional is of
standard tracking type, the control problem under investigation cannot easily
be treated via standard techniques for two reasons: the state system is a
highly nonlinear system of PDEs containing singular and degenerating terms, and
the control variable, which is given by the velocity of the motion occurring in
the convective term, is nonlinearly coupled to the state variable. The latter
fact makes it necessary to state rather special regularity assumptions for the
admissible controls, which, while looking a bit nonstandard, are however quite
natural in the corresponding analytical framework. In fact, they are
indispensable prerequisites to guarantee the well-posedness of the associated
state system. In this contribution, we employ recently proved existence,
uniqueness and regularity results for the solution to the associated state
system in order to establish the existence of optimal controls and appropriate
first-order necessary optimality conditions for the optimal control problem.

Analysis of PDEs