On the Planning Problem for the Mean Field Games System

We consider the planning problem for a class of mean field games, consisting in a coupled system of a Hamilton-Jacobi-Bellman equation for the value function u and a Fokker-Planck equation for the density m of the players, whereas one wishes to drive the density of players from the given initial configuration to a target one at time T through the optimal decisions of the agents. Assuming that the coupling F(x,m) in the cost criterion is monotone with respect to m, and that the Hamiltonian has some growth bounded below and above by quadratic functions, we prove the existence of a weak solution to the system with prescribed initial and terminal conditions m (0), m (1) (positive and smooth) for the density m. This is also a special case of an exact controllability result for the Fokker-Planck equation through some optimal transport field.