  We consider the stochastic optimal control problem of McKean-Vlasov
stochastic differential equation where the coefficients may depend upon the
joint law of the state and control. By using feedback controls, we reformulate
the problem into a deterministic control problem with only the marginal
distribution of the process as controlled state variable, and prove that
dynamic programming principle holds in its general form. Then, by relying on
the notion of differentiability with respect to pro\-bability measures recently
introduced by P.L. Lions in [32], and a special It{\^o} formula for flows of
probability measures, we derive the (dynamic programming) Bellman equation for
mean-field stochastic control problem, and prove a veri\-fication theorem in
our McKean-Vlasov framework. We give explicit solutions to the Bellman equation
for the linear quadratic mean-field control problem, with applications to the
mean-variance portfolio selection and a systemic risk model. We also consider a
notion of lifted visc-sity solutions for the Bellman equation, and show the
viscosity property and uniqueness of the value function to the McKean-Vlasov
control problem. Finally, we consider the case of McKean-Vlasov control problem
with open-loop controls and discuss the associated dynamic programming equation
that we compare with the case of closed-loop controls.
