  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 a
PDE of continuity-type, governing the dynamics of the probability distribution
of the agent population. We show the existence of mean field optimal controls
both in the stochastic and deterministic setting. We derive rigorously the
first order optimality conditions useful for numerical computation of mean
field optimal controls. We introduce a novel approximating hierarchy of
sub-optimal controls based on a Boltzmann approach, whose computation requires
a very moderate numerical complexity with respect to the one of the optimal
control. We provide numerical experiments for models in opinion formation
comparing the behavior of the control hierarchy.
