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

Distributed optimal control of a nonstandard nonlocal phase field system with double obstacle potential