Optimal control of a nonlinear kinetic Fokker-Planck equation
Tobias Breiten, Karl Kunisch
TL;DR
This work tackles a nonlinear, nonlocal kinetic Fokker-Planck equation with distance-dependent fluctuations and an unbounded velocity-control operator, placing the study in a hypocoercive setting. The authors develop a framework based on admissible control operators and semigroup/variational techniques to establish local existence of mild solutions near the Maxwellian $\\mu$ and to prove the existence (and partial uniqueness) of locally optimal controls for a quadratic tracking cost. Central to the analysis are Lipschitz estimates for the nonlinearities $h_1$ and $h_2$, a fixed-point argument, and a careful treatment of the linearized system together with admissibility of $\\overline{R}^{1/2}$. The results provide a rigorous foundation for optimal control of nonlinear kinetic equations in settings where standard coercivity is absent, with potential implications for mean-field models and related hypocoercive PDEs, and point to future work on optimality conditions, infinite-horizon problems, and numerical discretization.
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.