  We introduce the concept of {\it mean-field optimal control} which is the
rigorous limit process connecting finite dimensional optimal control problems
with ODE constraints modeling multi-agent interactions to an infinite
dimensional optimal control problem with a constraint given by a PDE of
Vlasov-type, governing the dynamics of the probability distribution of
interacting agents. While in the classical mean-field theory one studies the
behavior of a large number of small individuals {\it freely interacting} with
each other, by simplifying the effect of all the other individuals on any given
individual by a single averaged effect, we address the situation where the
individuals are actually influenced also by an external {\it policy maker}, and
we propagate its effect for the number $N$ of individuals going to infinity. On
the one hand, from a modeling point of view, we take into account also that the
policy maker is constrained to act according to optimal strategies promoting
its most parsimonious interaction with the group of individuals. This will be
realized by considering cost functionals including $L^1$-norm terms penalizing
a broadly distributed control of the group, while promoting its sparsity. On
the other hand, from the analysis point of view, and for the sake of
generality, we consider broader classes of convex control penalizations. In
order to develop this new concept of limit rigorously, we need to carefully
combine the classical concept of mean-field limit, connecting the finite
dimensional system of ODE describing the dynamics of each individual of the
group to the PDE describing the dynamics of the respective probability
distribution, with the well-known concept of $\Gamma$-convergence to show that
optimal strategies for the finite dimensional problems converge to optimal
strategies of the infinite dimensional problem.
