Appears in collection : Journée sous-riemannienne 2017

Optimal Control, Differential Games, Mean Field Games, and Pontryagin and Hamilton-Jacobi equations on probabilities The talk will be a short introduction to the emerging topic of Mean Field Games in connection with optimal control and differential games. I will present what is in general a Mean-Field Game and how to translate it into a coupled system of a forward continuity (or Fokker-Planck) equation on the density of players and of a backward Hamilton-Jacobi equation on their value function. Then I will focus on the case where this system has a variational origin and I will explain how this system actually corresponds to Pontryagin’s maximum principle in the space of measures... wait, wait : if thys system, which includes a Hamilton-Jacobi equation, is Pontryagin, what is the Hamilton-Jacobi equation in this case? the talk will finish evoking some answers to this question, for control problem in the space of measures and also for differential games, thus arriving up to the so-called “Master equation” introduced by P.-L. Lions, which will be sketched.