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arXiv · 2005.13377

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

Thomas Berger
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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.

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Thomas Berger. 2020-05-27. Funnel control of the Fokker-Planck equation for a multi-dimensional Ornstein-Uhlenbeck process. https://arxiv.org/abs/2005.13377

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