Stochastic input-to-state stability for impulsive switched stochastic nonlinear systems with multiple jumps

ABSTRACT

In this study, we investigate the stochastic input-to-state stability (SISS) of impulsive switched stochastic nonlinear systems. In this model, the impulse jumps are component multiple maps that depend on time. Thus the model differs from traditional impulsive systems with single impulse between two adjacent switching times. We provide sufficient conditions in three cases with the SISS system by using the Lyapunov function and average impulsive interval approach. The destabilising impulses cannot destroy the SISS properties if the impulses do not occur too frequently when all the subsystems that control the continuous dynamics are SISS. In other words, the average impulsive interval satisfies a lower bound restraint. Conversely, when all subsystems that control the continuous dynamics are not SISS, impulses can contribute to stabilising the system in the SISS sense when the average impulsive interval satisfies an upper bound. Then, we investigate the SISS property of impulsive switched stochastic nonlinear systems with some subsystems that are not SISS under certain conditions such that the property remains obtained. Finally, we show three examples to demonstrate the validity of the main result.

KEYWORDS:

• Impulsive switched system
• stochastic input-to-state stability
• multiple jumps
• unstable subsystems

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the National Natural Science Foundation (NNSF) of China under Grants 61403228, 61673242, and 61673243, Natural Science Foundation of Shandong Province under Grants ZR2017QF013 and ZR2019MF067, and the Project of Taishan Scholar of Shandong Province of China.

Notes on contributors

Lijun Gao

Lijun Gao was born in Shandong, China. She received the Ph.D. degree in applied mathematics from Qufu Normal University, China, in 2017. She has been with the Department of Automation, Qufu Normal University of China, since 2017, where she is currently a professor. Her current research interests include switched systems, Markov jumps systems, and impulsive control.

Meng Zhang

Meng Zhang received the B.S. degree in applied mathematics from Dezhou University, China, in 2014. She is currently pursuing the Ph.D. degree in statistics at School of Mathematical Sciences and Institute of Finance and Statistics, Nanjing Normal University. Her research interests include switched systems, stochastic control systems.

Xiuming Yao

Xiuming Yao was born in Heze, Shandong, China in 1981. She received the B.E. degree in measurement, control technology and instrument from North China Electric Power University, Baoding, China, in 2005, and the Ph.D. degree in control science and control engineering from Harbin Institute of Technology, Harbin, China, in 2010. From September 2010 to March 2011, she was a Research Associate in the School of Computing and Mathematics, University of Western Sydney, Australia. From May 2011 to December 2014, she did her postdoctoral research in the National Key Laboratory on Aircraft Control Technology, Beihang, University. Her current research interests include train operation control systems for high speed railways, pedestrian dynamics, Markovian jump hybrid systems and anti-disturbance control.