Ştefana-Lucia Aniţa
Octav Mayer Institute of Mathematics of the Romanian Academy, Iaşi, Romania

It is well known that a Fokker-Planck equation models the dynamics of the probability density function of the solution to the related stochastic differential equation. This suggests that a possible approach of a stochastic optimal control problem would be to reduce its study to the one of a deterministic optimal control problem related to a Fokker-Planck equation.

Our talk concerns an optimal control problem $(P)$ associated to a nonlinear Fokker-Planck equation. As mentioned, this problem is deeply related to a stochastic optimal control problem $(P_s)$ for a McKean-Vlasov equation. The existence of an optimal control is obtained for the deterministic problem $(P)$. The existence of an optimal control is established and necessary optimality conditions are derived for a penalized optimal control problem $(P_h)$ related to a backward Euler approximation of the nonlinear Fokker-Planck equation (with a constant discretization step $h$). Using a passing-to-the-limit-like argument one derives the necessary optimality conditions for problem $(P)$.