Abstract: The talk will be devoted to the analysis of a variational problem thatdescribes the optimal evolution of a probability measure, subject todensity constraints and nonlocal potential of Riesz type. The problemarises in the study of some first order Mean Field type controlproblems, with density constraints and aggregating interactions. Undersuitable symmetry assumptions, I will discuss the existence ofperiodic orbits, and their convergence to heteroclinics, which connectdifferent ground states of the stationary problem in an infinite timehorizon. Finally, I will discuss some qualitative properties of thediscrete counterpart of the problem, which involves the optimalevolution of a finite number of particles that are subject to distanceconstraints.