EXACT CONTROLLABILITY OF FOKKER-PLANCK EQUATIONS AND MCKEAN-VLASOV SDEs*

In this work we design an explicit feedback controller u = \Phi (p) for the exact con-trollability problem to the nonlinear Fokker-Planck equation pt -\Delta\beta(p) + div(up) = 0 in (0, T) \times o, p (0) = p0, p(T) = p1, with reflecting boundary conditions (V \beta(p) -up) \cdot n = 0 on \partialo. Here, o \subset Rd, 1 \leq d, is a bounded, open set with smooth boundary \partialo and \beta is a smooth monotonically increasing function. If p0 and p1 are probability densities, one obtains in particular the exact controllability for McKean-Vlasov stochastic differential equations on o with reflecting (impenetrable) barrier.