Controlling a nonlinear Fokker-Planck equation via inputs with nonlocal action

Abstract: This paper concerns an optimal control problem $(P)$ related to a nonlinear Fokker-Planck equation. The 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 $h$). Passing to the limit ($h\rightarrow 0$) one derives the necessary optimality conditions for problem $(P)$.