Mean-field optimal control as Gamma-limit of finite agent controls

This paper focuses on the role of a government of a large population of interacting agents as a mean field optimal control problem derived from deterministic finite agent dynamics. The control problems are constrained by a PDE of continuity-type without diffusion, governing the dynamics of the probability distribution of the agent population. We derive existence of optimal controls in a measure-theoretical setting as natural limits of finite agent optimal controls without any assumption on the regularity of control competitors. In particular, we prove the consistency of mean-field optimal controls with corresponding underlying finite agent ones. The results follow from a $\Gamma$-convergence argument constructed over the mean-field limit, which stems from leveraging the superposition principle.