Brief paper: Finite-time stability and instability of stochastic nonlinear systems

Authors:
Juliang Yin;Suiyang Khoo;Zhihong Man;Xinghuo Yu
Affiliations:
Department of Statistics, Jinan University, Guangzhou, China;School of Engineering, Deakin University, VIC, Australia;School of Engineering & Industrial Sciences, Swinburne University of Technology, VIC, Australia;Platform Technologies Research Institute, RMIT University, VIC, Australia
Venue:
Automatica (Journal of IFAC)
Year:
2011

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Abstract

This paper presents a new definition of finite-time stability for stochastic nonlinear systems. This definition involves stability in probability and finite-time attractiveness in probability. An important Lyapunov theorem on finite-time stability for stochastic nonlinear systems is established. A theorem extending the stochastic Lyapunov theorem is also proved. Moreover, an example and a lemma are presented to illustrate the scope of extension. A useful inequality, extended from Bihari's inequality, is derived, which plays an important role in showing the Lyapunov theorem. Finally, a Lyapunov theorem on finite-time instability is proved, which states that almost surely globally asymptotical stability is not equivalent to finite-time stability for some stochastic systems. Two simulation examples are given to illustrate the theoretical analysis.