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Recent Developments in Stability Theory for Stochastic Hybrid Inclusions

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Feedback Stabilization of Controlled Dynamical Systems

Part of the book series: Lecture Notes in Control and Information Sciences ((LNCIS,volume 473))

Abstract

Stochastic hybrid systems (SHS) combine continuous evolution, instantaneous jumps, and random inputs that affect each type of evolution. Various types of SHS have been studied for over three decades and can be used to model many interesting systems in science and engineering. The most recent developments regarding SHS focus on models that permit nonunique solutions, perhaps thereby modeling the effect of an adversarial input on the system dynamics, and robustness properties, which can again be linked to the effect of adversaries. We call such systems “stochastic hybrid inclusions” (SHI). Using, as a departure point, developments over the past ten years on modeling, sequential compactness of the solution space, and robustness of stability for non-stochastic hybrid systems, a comprehensive modeling framework for SHI is being developed. The ultimate goal is an extensive, robust stability theory for SHI. In this paper, we review recent results that have been obtained in this direction, describing a solution concept for a class of SHI, defining stability notions like recurrence and asymptotic stability in probability, stating equivalent characterizations (involving uniformity and robustness) that follow from a sequential compactness result, providing Lyapunov-based necessary and sufficient conditions for these properties, and describing relaxed sufficient conditions that are based on an invariance-like principle.

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Notes

  1. 1.

    For more information, see [30, pp. 154–155].

  2. 2.

    See [30, Definition 5.4].

  3. 3.

    See [30, Definition 5.14].

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Acknowledgements

The research reported in this chapter has been supported by the US Air Force Office of Scientific Research through grants FA9550-12-1-0127 and FA9550-15-1-0155, and by the US National Science Foundation through grants ECCS-1232035 and ECCS-1508757.

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  1. ECE Department, University of California, Santa Barbara, CA, 93106-9560, USA

    Andrew R. Teel

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  1. Centre Automatique et Systèmes CAS, MINES ParisTech, PSL Research University, Centre Automatique et Systèmes CAS, Paris Cedex 06, France

    Nicolas Petit

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Teel, A.R. (2017). Recent Developments in Stability Theory for Stochastic Hybrid Inclusions. In: Petit, N. (eds) Feedback Stabilization of Controlled Dynamical Systems. Lecture Notes in Control and Information Sciences, vol 473. Springer, Cham. https://doi.org/10.1007/978-3-319-51298-3_13

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  • DOI: https://doi.org/10.1007/978-3-319-51298-3_13

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