  In this paper, we study an optimal boundary control problem for a model for
phase separation taking place in a spatial domain that was introduced by
Podio-Guidugli in Ric. Mat. 55 (2006), pp. 105-118. The model consists of a
strongly coupled system of nonlinear parabolic differential equations, in which
products between the unknown functions and their time derivatives occur that
are difficult to handle analytically. In contrast to the existing control
literature about this PDE system, we consider here a dynamic boundary condition
involving the Laplace-Beltrami operator for the order parameter of the system,
which models an additional nonconserving phase transition occurring on the
surface of the domain. We show the Fr\'echet differentiability of the
associated control-to-state operator in appropriate Banach spaces and derive
results on the existence of optimal controls and on first-order necessary
optimality conditions in terms of a variational inequality and the adjoint
state system.
