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

TL;DR

This paper develops a framework for controlling nonlinear Fokker-Planck equations with nonlocal inputs, establishing existence and optimality conditions through approximation and limit processes, linking to stochastic control problems.

Contribution

It introduces a novel approach to optimal control of nonlinear Fokker-Planck equations, including existence proofs and derivation of necessary optimality conditions via discretization.

Findings

Existence of optimal controls for the nonlinear Fokker-Planck problem.

Derivation of necessary optimality conditions through penalized approximations.

Connection between deterministic control and stochastic McKean-Vlasov control.

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 $(P)$.