New-type stability theorem for stochastic functional differential equations with application to SFDSs with distributed delays

Abstract

In this paper, a new-type stability theorem for stochastic functional differential equations (SFDEs) is established, which is not a direct copy of the basic stability theorem for deterministic functional differential equations (DFDEs). By the new-type stability theorem, one can use the most simple Lyapunov functions and employ the equations repeatedly to deal with the delayed terms encountered conveniently and to carry out stability criteria for the equations. Based on the theorem, a practical stability theorem in accordance with the Lyapunov function method is also established, and then the asymptotic stability of SFDEs with distributed delays in the diffusive terms is investigated and a stability criterion for SFDSs is obtained, which is described by algebraic matrix equations. Finally, an example is given to illustrate the effectiveness of our method and results.

Keywords:

• SFDSs
• time-delays
• Lyapunov functions
• stability theorem
• matrix equations

Acknowledgements

The authors gratefully acknowledge the associate editor and reviewer’s comments. The authors would like to thank the Natural Science Foundation of Guangdong Province under grant 10251064101000008, the National Natural Science Foundation of China under grant 61273126, 60874114 and the Fundamental Research Funds for the Central Universities 2012ZM0059 for their financial support.

Additional information

Notes on contributors

Xueyan Zhao

Xueyan Zhao was born in 1984. Now she is a PhD student at South China University of Technology, Guangzhou. Her research interests are stability and stabilisation of stochastic systems and non-linear systems.

Feiqi Deng

Feiqi Deng was born in 1962. He received the PhD degree in control theory and control engineering from South China University of Technology, Guangzhou, in June 1997. Since October 1999, he has been a professor with South China University of Technology and the director of the Systems Engineering Institute of the university. Now he is serving as the chair of the IEEE SMC Guangzhou Chapter, an associate editor-in-chief of the Journal of South China University of Technology and a member of the editorial boards of the following journals: Control Theory and Applications, Journal of Systems Engineering and Electronics and Journal of Systems Engineering. His main research interests include stability, stabilisation and robust control theory of complex systems, including time-delay systems, non-linear systems and stochastic systems.