Improving the Convergence Rates for the Kinetic Fokker-Planck Equation by Optimal Control

TL;DR

This paper explores how optimal control of the confinement potential can enhance the convergence rates of the kinetic Fokker-Planck equation without changing its invariant measure, using advanced control and numerical methods.

Contribution

It introduces a novel control framework for the kinetic Fokker-Planck equation that improves convergence analysis through an abstract bilinear control system and optimal control strategies.

Findings

Existence of a unique solution to the controlled system.

Feasibility and existence of solutions to the optimal control problem under certain conditions.

Numerical results demonstrating improved convergence rates with the proposed control approach.

Abstract

The long time behavior and detailed convergence analysis of Langevin equations has received increased attention over the last years. Difficulties arise from a lack of coercivity, usually termed hypocoercivity, of the underlying kinetic Fokker-Planck operator which is a consequence of the partially deterministic nature of a second order stochastic differential equation. In this manuscript, the effect of controlling the confinement potential without altering the original invariant measure is investigated. This leads to an abstract bilinear control system with an unbounded but infinite-time admissible control operator which, by means of an artificial diffusion approach, is shown to possess a unique solution. The compactness of the underlying semigroup is further used to define an infinite-horizon optimal control problem on an appropriately reduced state space. Under smallness assumptions on the initial data, feasibility of and existence of a solution to the optimal control problem are discussed. Numerical results based on a local approximation based on a shifted Riccati equation illustrate the theoretical findings.