  We analyze the problem of controlling a multi-agent system with additive
white noise through parsimonious interventions on a selected subset of the
agents (leaders). For such a controlled system with a SDE constraint, we
introduce a rigorous limit process towards an infinite dimensional optimal
control problem constrained by the coupling of a system of ODE for the leaders
with a McKean-Vlasov-type SDE, governing the dynamics of the prototypical
follower. The latter is, under some assumptions on the distribution of the
initial data, equivalent with a (nonlinear parabolic) PDE-ODE system. The
derivation of the limit mean-field optimal control problem is achieved by
linking the mean-field limit of the governing equations together with the
$\Gamma$-limit of the cost functionals for the finite dimensional problems.
