Distributed optimal control of a nonstandard nonlocal phase field system

We investigate a distributed optimal control problem for a nonlocal phase field modelof viscous Cahn-Hilliard type. The model constitutes a nonlocal version of a model for two-speciesphase segregation on an atomic lattice under the presence of di usion that has been studied in a seriesof papers by P. Podio-Guidugli and the present authors. The model consists of a highly nonlinearparabolic equation coupled to an ordinary di erential equation. The latter equation contains bothnonlocal and singular terms that render the analysis di cult. Standard arguments of optimal controltheory do not apply directly, although the control constraints and the cost functional are of standardtype. We show that the problem admits a solution, and we derive the first-order necessary conditionsof optimality.