Control Strategies for the Fokker-Planck Equation

TL;DR

This paper develops control strategies for the Fokker-Planck equation using a projection-based approach, deriving feedback laws and stabilization results with numerical illustrations.

Contribution

It introduces a novel operator theoretic framework and control laws for the Fokker-Planck equation, including Riccati and Lyapunov equations, with stability analysis.

Findings

Derived projected Riccati and Lyapunov equations.

Proved well-posedness and stabilization of closed-loop systems.

Numerical results demonstrate control effectiveness in a double well potential.

Abstract

Using a projection-based decoupling of the Fokker-Planck equation, control strategies that allow to speed up the convergence to the stationary distribution are investigated. By means of an operator theoretic framework for a bilinear control system, two different feedback control laws are proposed. Projected Riccati and Lyapunov equations are derived and properties of the associated solutions are given. The well-posedness of the closed loop systems is shown and local and global stabilization results, respectively, are obtained. An essential tool in the construction of the controls is the choice of appropriate control shape functions. Results for a two dimensional double well potential illustrate the theoretical findings in a numerical setup.