Almost global stochastic stability

Ramon Van Handel

doi:10.1137/040618850

Almost global stochastic stability

Ramon Van Handel

Research output: Contribution to journal › Article › peer-review

26 Scopus citations

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.

Original language	English (US)
Pages (from-to)	1297-1313
Number of pages	17
Journal	SIAM Journal on Control and Optimization
Volume	45
Issue number	4
DOIs	https://doi.org/10.1137/040618850
State	Published - 2006
Externally published	Yes

All Science Journal Classification (ASJC) codes

Control and Optimization
Applied Mathematics

Keywords

Nonlinear stochastic control
Stochastic flows
Stochastic stability

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10.1137/040618850

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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.",

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T1 - Almost global stochastic stability

AU - Van Handel, Ramon

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N2 - 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.

AB - 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.

KW - Nonlinear stochastic control

KW - Stochastic flows

KW - Stochastic stability

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JO - SIAM Journal on Control and Optimization

JF - SIAM Journal on Control and Optimization

IS - 4

ER -

Almost global stochastic stability

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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