L2-Tracking of Gaussian Distributions via Model Predictive Control for the Fokker–Planck Equation

  • Arthur Fleig
  • Lars Grüne


This paper presents the first results for the stability analysis of Model Predictive Control schemes applied to the Fokker–Planck equation for tracking probability density functions. The analysis is carried out for linear dynamics and Gaussian distributions, where the distance to the desired reference is measured in the L2-norm. We present results for general such systems with and without control penalization. Refined results are given for the special case of the Ornstein–Uhlenbeck process. Some of the results establish stability for the shortest possible (discrete time) optimization horizon N = 2.


Model Predictive Control Fokker–Planck equation Probability density function Stochastic optimal control 

Mathematics Subject Classification (2010)

35Q84 35Q93 49N35 60G15 93C15 



The authors wish to thank Tobias Damm for very helpful suggestions and the referees for their valuable comments that helped to improve the manuscript.

Funding information

This work was supported by the German Research Foundation (DFG) (grant number GR 1569/15-1).


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Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.University of BayreuthBayreuthGermany

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