Controlling a Nonlinear Fokker–Planck Equation via Inputs with Nonlocal Action

This paper concerns an optimal control problem (P) associated to a nonlinear Fokker–Planck equation via inputs with nonlocal action. The Fokker–Planck equation describes the dynamics of the probability density of a population under a control that produces a repellent vector field which displaces the population. Actually, problem (P) asks to optimally displace a population via the repellent action produced by the control. The problem is deeply related to a stochastic optimal control problem \((P_S)\) $(P_S)$ 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 \((P_h)\) $(P_h)$ 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 (as \(h\rightarrow 0\) $h\to 0$ ) one derives the necessary optimality conditions for problem (P). Some possible extensions are discussed as well.