This talk is concerned with the optimal control of McKean-Vlasov equations, which has been knowing a surge of interest since the emergence of the mean-field game theory, a decade ago by J.M. Lasry and P.L Lions. Such problem is originally motivated from large population stochastic control in mean-field interaction, and finds various applications in economy, finance, or social sciences for modelling motion of socially interacting individuals and herd behavior. It is also relevant for dealing with intermittence questions arising typically in risk management.
I will present the dynamic programming approach for the control of general McKean-Vlasov dynamics, and introduce in particular the recent mathematical tools that have been developed in this context : differentiability in the Wasserstein space of probability measures, Itô formula along a flow of probability measures and Master Bellman equation. We shall then focus on the important class of linear-quadratic McKean-Vlasov (LQMcKV) control problem, which provides a major source for examples and applications, and illustrate our results with some examples.