Asymptotic controllability of nonlinear Fokker-Planck equations

We discuss here the asymptotic controllability problem for the Fokker-Planck equations, rho(t) - Delta beta(rho) + div(u rho) = 0 in (0, infinity) x R-d, rho(0, x) = rho(0)(x), x is an element of R-d, that is, the existence of a feedback controller u equivalent to u(x, rho) such that lim(t ->infinity) rho(t, x) = rho(1)(x), a.e. x is an element of R-d, where rho(0), rho(1) are given probability densities and beta is an element of C-2(R) is a monotonically increasing function. In this work, it is designed such a controller u for a certain class of final states rho(1) which is identified. This problem is related to the controllability of McKean-Vlasov stochastic differential equations and the approach used here relies on the H-theorem established in (Barbu, Rockner in Indiana Univ Math J, 2020), Theorem 6.1, for nonlinear and nondegenerate Fokker-Planck equations.