  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.
