On the relationship between stochastic turnpike and dissipativity notions

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Schießl, Jonas ; Baumann, Michael Heinrich ; Faulwasser, Timm ; Grüne, Lars:
On the relationship between stochastic turnpike and dissipativity notions.
In: IEEE Transactions on Automatic Control. (23 November 2024) .
ISSN 1558-2523
DOI: https://doi.org/10.1109/TAC.2024.3504352

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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 Lëvy-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.