Optimal control of a nonlinear kinetic Fokker-Planck equation

TL;DR

This paper investigates an optimal control problem for a nonlinear, nonlocal kinetic Fokker-Planck equation, addressing challenges posed by hypocoercivity and unbounded control operators to establish existence of solutions and optimal controls.

Contribution

It introduces a novel approach using admissible control operators and fixed point arguments to handle the hypocoercive nonlinear Fokker-Planck equation for optimal control.

Findings

Established local existence of solutions for the nonlinear equation.

Proved the existence of optimal controls despite hypocoercivity.

Developed new techniques to handle unbounded control operators.

Abstract

A tracking type optimal control problem for a nonlinear and nonlocal kinetic Fokker-Planck equation which arises as the mean field limit of an interacting particle systems that is subject to distance dependent random fluctuations is studied. As the equation of interest is only hypocoercive and the control operator is unbounded with respect to the canonical state space, classical variational solution techniques cannot be utilized directly. Instead, the concept of admissible control operators is employed. For the underlying nonlinearities, local Lipschitz estimates are derived and subsequently used within a fixed point argument to obtain local existence of solutions. Again, due to hypocoercivity, existence of optimal controls requires non standard techniques as (compensated) compactness arguments are not readily available.