Optimal control of the Fokker-Planck equation under state constraints in the Wasserstein space

We analyze a problem of optimal control of the Fokker-Planck equation with state constraints in the Wasserstein space of probability measures. Our main result is to derive optimality conditions in the form of a Mean Field Game system of partial differential equations completed with an exclusion condition. As a by-product we obtain the Lipschitz regularity of optimal controls.