  We study the convergence problem of mean-field control theory in the presence
of state constraints and non-degenerate idiosyncratic noise. Our main result is
the convergence of the value functions associated to stochastic control
problems for many interacting particles subject to symmetric, almost-sure
constraints toward the value function of a control problem of mean-field type,
set on the space of probability measures. The key step of the proof is to show
that admissible controls for the limit problem can be turned into admissible
controls for the $N$-particle problem up to a correction which vanishes as the
number of particles increases. The rest of the proof relies on compactness
methods. We also provide optimality conditions for the mean-field problem and
discuss the regularity of the optimal controls. Finally we present some
applications and connections with large deviations for weakly interacting
particle systems.
