  In this paper we consider an optimal control problem for a large population
of interacting agents with deterministic dynamics, aggregating potential and
constraints on reciprocal distances, in dimension 1. We study existence and
qualitative properties of periodic in time optimal trajectories of the finite
agents optimal control problem, with particular interest on the compactness of
the solutions' support and on the saturation of the distance constraint.
Moreover, we prove, through a $\Gamma$-convergence result, the consistency of
the mean-field optimal control problem with density constraints with the
corresponding underlying finite agent one and we deduce some qualitative
results for the time periodic equilibria of the limit problem.
