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

$1$-dimensional multi-agent optimal control with aggregation and distance constraints: qualitative properties and mean-field limit