Control strategies for the Fokker−Planck equation

¹ Institute for Mathematics and Scientific Computing, Karl-Franzens-Universität Graz, Heinrichstr. 36, 8010 Graz, Austria
² Institute for Mathematics and Scientfiic Computing, Karl-Franzens-Universität, Heinrichstr. 36, 8010 Graz, Austria and Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstraße 69, 4040 Linz, Austria
karl.kunisch@uni-graz.at

^a Corresponding author: tobias.breiten@uni-graz.at; laurent.pfeiffer@uni-graz.at

Received: 6 July 2016
Revised: 8 June 2017
Accepted: 13 June 2017

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.