On the relationship between stochastic turnpike and dissipativity notions

TL;DR

This paper explores the connection between stochastic dissipativity and turnpike properties in discrete-time nonlinear optimal control, extending classic concepts to stochastic settings and analyzing their relationships using various metrics.

Contribution

It introduces new stochastic dissipativity notions and turnpike properties, extending classical control theory to stochastic variables, measures, and moments, and analyzes their interrelations.

Findings

Extended dissipativity notions to $L^r$ random variables and probability measures.

Established links between dissipativity and various stochastic turnpike properties.

Analyzed the role of metrics like Wasserstein in stochastic control analysis.

Abstract

In this paper, we introduce and study different dissipativity notions and different turnpike properties for discrete-time stochastic nonlinear optimal control problems. The proposed stochastic dissipativity notions extend the classic notion of Jan C. Willems to $L^r$ random variables and to probability measures. Our stochastic turnpike properties range from a formulation for random variables via turnpike phenomena in probability and in probability measures to the turnpike property for the moments. Moreover, we investigate how different metrics (such as Wasserstein or L\'evy-Prokhorov) can be leveraged in the analysis. Our results are built upon stationarity concepts in distribution and in random variables and on the formulation of the stochastic optimal control problem as a finite-horizon Markov decision process. We investigate how the proposed dissipativity notions connect to the various stochastic turnpike properties and we work out the link between different forms of dissipativity.