Existence of optimal periodic strategies in a model with nonlocal spatiotemporal dispersal

In this paper, we investigate a control problem for a nonlinear integro-difffferential system incorporating both spatial and temporal nonlocal terms, subject to periodic condition in time. The equation, derived from population dynamics, describes the evolution of population density under growth, migration, and memory effects. The control set is feedback-based and spatially nonlocal, depending on a weighted integral of the state. We establish the existence of controlled trajectories that approximate the infimum or supremum of a cost functional with arbitrary precision. Moreover, under additional regularity assumptions on the bounding functions of the control set, we provide the existence of truly optimal controlled trajectories. The analysis employs topological methods in nonlinear analysis and tailored techniques to address temporal nonlocalities.