Razumikhin-type theorems on stability of stochastic retarded systems

Abstract

The Razumikhin-type theorems have been well developed and are very effective for the stability of deterministic retarded dynamic systems. Chang (1984) studied the Razumikhin-type stability conditions for stochastic retarded functional differential equations (SRFDEs). But the stability is not in a global sense. Recently, by utilising the Razumikhin-type argument, Mao (1997a) provided some useful criteria for exponential stability of stochastic retarded functional differential equations. However, the important general asymptotic stability has not been considered. The aim of this article is to establish Razumikhin-type theorems on asymptotic stability of stochastic retarded systems, which is a generalisation of the result obtained by Mao. Moreover, generalised exponential stability is considered for better estimate of Lyapunov exponent of time-variant systems. In the conclusion, this article is to show the underlying principle of the Razumikhin-type theorems on stability of stochastic retarded functional differential equations. With this idea, it is not difficult to apply Razumikhin-type techniques to stochastic retarded systems.

Keywords:

• stochastic systems
• retarded systems
• Razumikhin-type theorem
• asymptotic stability
• generalised exponential stability

Acknowledgements

The authors gratefully acknowledge the associate editor and the reviewers' comments. The authors would like to thank Professor X. Mao for his interesting seminars and very helpful literature.