Stabilisation of nonlinear hybrid stochastic systems with time-varying delay by discrete-time feedback controls with a time delay

Abstract

In this paper, we investigate stabilisation of hybrid stochastic systems with time-varying delay by feedback control based on discrete-time state and mode observations with a time delay. By constructing a proper Lyapunov functional, the $H_{\mathrm{\infty }}$, asymptotic and exponential stability of the controlled systems are studied in the pth moment sense for p>1. Meanwhile, we obtain the upper bound on the duration τ between two consecutive state and mode observations. At last, two numerical examples are illustrated to support our theoretical results.

Keywords:

• Hybrid stochastic systems
• time-varying delay
• Lyapunov functional
• $H_{\mathrm{\infty }}$ stability
• asymptotic stability
• exponential stability
• feedback control
• discrete-time state and mode observations

Disclosure statement

No potential conflict of interest was reported by the authors.

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Additional information

Funding

The authors would like to thank for the financial support by the Fundamental Research Funds for The Central Universities and Graduate Student Innovation Fund of Donghua University, CUSF-DH-D-2021054.

Notes on contributors

Jianqiu Lu

Jianqiu Lu is a Lecturer with the Department of Statistics, Donghua University, Shanghai, China. She received the B.Sc. degree in Mathematics and Statistics at University of Strathclyde, Glasgow, UK in 2013. At the same year, she also received the B.Sc. degree in Applied Mathematics from Donghua University, Shanghai, China. She received her Ph.D. degree in 2018 at University of Strathclyde, Glasgow, UK. Her main research interests lie in the field of stochastic analysis including stochastic stability, stabilization and control.

Yun Lu

Yun Lu received the B.Sc. degree from Chengdu Normal University, Chengdu, Sichuan, China, in 2020. She started the M.Sc. degree study at Donghua University, Shanghai, China from 2020. Her research is mainly concerned with stability and stabilization of stochastic systems.

Simin Wang

Simin Wang received the B.Sc. degree in Mathematics and Applied Mathematics from South-Central University for Nationalities, Wuhan, Hubei, China in 2018. At the same year, she started her master project at Donghua University, Shanghai, China, and then, she received M.Sc. degree in 2021. She is currently working at Wuchang Shouyi University, Wuhan Hubei, China. Her research is mainly concerned with stability and stabilization of stochastic systems.

Liangjian Hu

Liangjian Hu is a Professor with the Department of Statistics, Donghua University, Shanghai, China. He received the B.Sc. degree from Anhui Normal University, Wuhu, Anhui, China, the M.Sc. degree and the Ph.D. degree from Donghua University, Shanghai, China, in 1985, 1988, and 2003, respectively. He had been a postdoctoral researcher in the Department of Electrical Engineering, National Tsinghua University, Hsin-Chu, Taiwan, and in the Department of Statistics & Modelling Science, University of Strathclyde, Glasgow, Scotland, UK. His research interests include stochastic control, fuzzy control, and stochastic differential equation with applications. Dr. Hu is the author of 4 books and more than 70 research papers.

Yanan Jiang

Yanan Jiang received the B.Sc. degree from Zhoukou Normal University, Zhoukou, Henan, China, in 2015, and the M.Sc. degree in Computational Mathmatics from Shanghai Normal University, Shanghai, China, in 2019. She is currently a doctoral student in Control Science and Engineering from Donghua University, Shanghai, China. Her research interests lie in the theory of stochastic differential equations and their application.