Funnel control of the Fokker-Planck equation for a multi-dimensional Ornstein-Uhlenbeck process

TL;DR

This paper investigates the application of funnel control techniques to the Fokker-Planck equation of a multi-dimensional Ornstein-Uhlenbeck process, demonstrating stability and control feasibility for a complex nonlinear PDE system.

Contribution

It introduces a novel approach to applying funnel control to a nonlinear, time-varying PDE governing probability densities of stochastic processes.

Findings

Covariance matrix of solutions converges exponentially to a constant.

Funnel control is feasible even with disturbances.

Simulation results validate the control approach.

Abstract

In this paper the feasibility of funnel control techniques for the Fokker-Planck equation corresponding to a multi-dimensional Ornstein-Uhlenbeck process on an unbounded spatial domain is explored. First, using weighted Lebesgue and Sobolev spaces, an auxiliary operator is defined via a suitable sesquilinear form. This operator is then transformed to the desired Fokker-Planck operator. We show that any mild solution of the controlled Fokker-Planck equation (which is a probability density) has a covariance matrix that exponentially converges to a constant matrix. After a simple feedforward control approach is discussed, we show feasibility of funnel control in the presence of disturbances by exploiting semigroup theory. We emphasize that the closed-loop system is a nonlinear and time-varying PDE. The results are illustrated by some simulations.