On Dissipativity of the Fokker-Planck Equation for the Ornstein-Uhlenbeck Process

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Fleig, Arthur ; Grüne, Lars:
On Dissipativity of the Fokker-Planck Equation for the Ornstein-Uhlenbeck Process.
Bayreuth , 2018 . - 7 S.

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Abstract

We study conditions for stability and near optimal behavior of the closed loop generated by Model Predictive Control for tracking Gaussian probability density functions associated with linear stochastic processes. More precisely, we analyze whether the corresponding optimal control problems are strictly dissipative. This is the key property required to infer statements about stability and performance of the closed loop system when tracking so-called unreachable setpoints, in which case a nonnegative cost is induced at the desired state. For verifying strict dissipativity, the choice of the so-called storage function is crucial. We focus on linear ones due to their close connection to the Lagrange function. The Ornstein-Uhlenbeck process serves as a prototype for our analysis, in which we show the limits of linear storage functions and present nonlinear alternatives, thus providing structural insight into dissipativity in case of bilinear system dynamics.