  This paper is concerned with a distributed optimal control problem for a
nonlocal phase field model of Cahn-Hilliard type, which is a nonlocal version
of a model for two-species phase segregation on an atomic lattice under the
presence of diffusion. The local model has been investigated in a series of
papers by P. Podio-Guidugli and the present authors; the nonlocal model studied
here consists of a highly nonlinear parabolic equation coupled to an ordinary
differential inclusion of subdifferential type. The inclusion originates from a
free energy containing the indicator function of the interval in which the
order parameter of the phase segregation attains its values. It also contains a
nonlocal term modeling long-range interactions. Due to the strong nonlinear
couplings between the state variables (which even involve products with time
derivatives), the analysis of the state system is difficult. In addition, the
presence of the differential inclusion is the reason that standard arguments of
optimal control theory cannot be applied to guarantee the existence of Lagrange
multipliers. In this paper, we employ recent results proved for smooth
logarithmic potentials and perform a so-called `deep quench' approximation to
establish existence and first-order necessary optimality conditions for the
nonsmooth case of the double obstacle potential.
