  The goal of this work is to obtain optimal rates for the convergence problem
in mean field control. Our analysis covers cases where the solutions to the
limiting problem may not be unique nor stable. Equivalently the value function
of the limiting problem might not be differentiable on the entire space. Our
main result is then to derive sharp rates of convergence in two distinct
regimes. When the data is sufficiently regular, we obtain rates proportional to
$N^{-1/2}$, with $N$ being the number of particles. When the data is merely
Lipschitz and semi-concave with respect to the first Wasserstein distance, we
obtain rates proportional to $N^{-2/(3d+6)}$. Noticeably, the exponent
$2/(3d+6)$ is close to $1/d$, which is the optimal rate of convergence for
uncontrolled particle systems driven by data with a similar regularity. The key
argument in our approach consists in mollifying the value function of the
limiting problem in order to produce functions that are almost classical
sub-solutions to the limiting Hamilton-Jacobi equation (which is a PDE set on
the space of probability measures). These sub-solutions can be projected onto
finite dimensional spaces and then compared with the value functions associated
with the particle systems. In the end, this comparison is used to prove the
most demanding bound in the estimates. The key challenge therein is thus to
exhibit an appropriate form of mollification. We do so by employing
sup-convolution within a convenient functional Hilbert space. To make the whole
easier, we limit ourselves to the periodic setting. We also provide some
examples to show that our results are sharp up to some extent.
