Almost Global Stochastic Stability

Creators: van Handel, Ramon

Abstract

We develop a method to prove almost global stability of stochastic differential equations in the sense that almost every initial point (with respect to the Lebesgue measure) is asymptotically attracted to the origin with unit probability. The method can be viewed as a dual to Lyapunov's second method for stochastic differential equations and extends the deterministic result of [A. Rantzer, Syst. Control Lett., 42 (2001), pp. 161–168]. The result can also be used in certain cases to find stabilizing controllers for stochastic nonlinear systems using convex optimization. The main technical tool is the theory of stochastic flows of diffeomorphisms.

Additional Information

©2006 Society for Industrial and Applied Mathematics Received by the editors November 13, 2004; accepted for publication (in revised form) April 3, 2006; published electronically October 3, 2006. This work was supported by the ARO under grant DAAD19-03-1-0073. This work was performed in Hideo Mabuchi's group, and the author gratefully acknowledges his support. The author would like to thank Anders Rantzer, Luc Bouten, Stephen Prajna, Paige Randall, and especially Houman Owhadi for insightful discussions and comments. The author is particularly thankful to an anonymous referee for pointing out a gap in the proofs and for his careful reading of the manuscript, which has significantly improved the presentation.

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