Mean field control hierarchy

In this paper we model the role of a government of a large population as a mean field optimal control problem. Such control problems are constrainted by aPDE of continuity-type, governing the dynamics of the probability distributionof the agent population. We show the existence of mean field optimal controlsboth in the stochastic and deterministic setting. We derive rigorously thefirst order optimality conditions useful for numerical computation of meanfield optimal controls. We introduce a novel approximating hierarchy ofsub-optimal controls based on a Boltzmann approach, whose computation requiresa very moderate numerical complexity with respect to the one of the optimalcontrol. We provide numerical experiments for models in opinion formationcomparing the behavior of the control hierarchy.