Existence and optimal control for periodic parabolic equations with nonlocal terms

Motivated by models which have been proposed for some problems in mathematical biology and fisheries management and elsewhere, we consider a nonlinear periodic parabolic problem and an associated cost functional J. A key feature of our problem is the presence of a nonlocal term which—as we show by direct example—renders the standard mono-tonicity methods invalid. We therefore employ topological methods to deal both with existence of solutions and of minima of J over the Control set. Some considerations are also presented on related systems and on the question of uniqueness.