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Lyapunov-Like Conditions for Tight Exit Probability Bounds through Comparison Theorems for SDEs

Nilsson, Petter and Ames, Aaron D. (2020) Lyapunov-Like Conditions for Tight Exit Probability Bounds through Comparison Theorems for SDEs. In: 2020 American Control Conference (ACC). IEEE , Piscataway, NJ, pp. 5175-5181. ISBN 9781538682661. https://resolver.caltech.edu/CaltechAUTHORS:20200730-143943668

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Abstract

Computing upper bounds on exit probabilities—the probability that a system reaches certain "bad" sets—may assist decision-making in control of stochastic systems. Existing analytical bounds for systems described by stochastic differential equations are quite loose, especially for low-probability events, which limits their applicability in practical situations. In this paper we analyze why existing bounds are loose, and conclude that it is a fundamental issue with the underlying techniques based on martingale inequalities. As an alternative, we give comparison results for stochastic differential equations that via a Lyapunov-like function allow exit probabilities of an n-dimensional system to be upper-bounded by an exit probability of a one-dimensional Ornstein-Uhlenbeck process. Even though no closed-form expression is known for the latter, it depends on three or four parameters and can be a priori tabulated for applications. We extend these ideas to the controlled setting and state a stochastic analogue of control barrier functions. The bounds are illustrated on numerical examples and are shown to be much tighter than those based on martingale inequalities.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
https://doi.org/10.23919/acc45564.2020.9147414DOIArticle
ORCID:
AuthorORCID
Nilsson, Petter0000-0001-8748-6936
Ames, Aaron D.0000-0003-0848-3177
Additional Information:© 2020 AACC.
Record Number:CaltechAUTHORS:20200730-143943668
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20200730-143943668
Official Citation:P. Nilsson and A. D. Ames, "Lyapunov-Like Conditions for Tight Exit Probability Bounds through Comparison Theorems for SDEs," 2020 American Control Conference (ACC), Denver, CO, USA, 2020, pp. 5175-5181, doi: 10.23919/ACC45564.2020.9147414
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:104671
Collection:CaltechAUTHORS
Deposited By: George Porter
Deposited On:31 Jul 2020 14:06
Last Modified:20 Nov 2020 23:56

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