Mean square finite-time boundary stabilisation and H∞ boundary control for stochastic reaction-diffusion systems

ABSTRACT

In this paper, the boundary control problem of stochastic reaction-diffusion systems (SRDSs) is studied. First, a distribution controller is designed, and a sufficient condition is established to achieve mean square finite-time stability. Then a boundary controller is proposed, and a criterion is obtained for mean square finite-time stability by using Poincaré's inequality and Gronwall's inequality. When the system is subject to external noise, a boundary $H_{\mathrm{\infty }}$ controller is presented for the systems with an $H_{\mathrm{\infty }}$ performance. Furthermore, a boundary controller is presented to ensure robust mean square finite-time stability of linear uncertain SRDSs. Numerical examples illustrate the validity of the theoretical results.

KEYWORDS:

• Stochastic reaction-diffusion systems
• boundary control
• finite-time stability
• $H_{\mathrm{\infty }}$ control

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Kai-Ning Wu http://orcid.org/0000-0003-4122-9832

Additional information

Notes on contributors

Xiao-Zhen Liu

Xiao-Zhen Liu received the B.S. and M.S. degrees in mathematics from the Harbin Institute of Technology at Weihai, China, in 2015 and 2017 respectively. She is currently pursuing the Ph.D. degree in mathematics from the Harbin Institute of Technology, Harbin, China. Her research interests include stochastic control and partial differential systems.

Kai-Ning Wu

Kai-Ning Wu received the B.S., M.S., and Ph. D. degrees from the Harbin Institute of Technology, Harbin, Helongjiang, China, in 2003, 2005 and 2009 respectively, all in mathematics. He is currently with the Department of Mathematics, Harbin Institute of Technology at Weihai, China, as an associate professor. His research interests include robust control, stochastic control and delay differential systems.

Weihai Zhang

Weihai Zhang received his M.S. degree from Hangzhou University (currently, Zhejiang University), and the Ph.D. degree from Zhejiang University, Hangzhou, China, in 1994 and 1998, respectively. He is currently a Professor of Shandong University of Science and Technology. Dr Zhang is a Taishan Scholar of Shandong Province of China, and serves as an Associate Editor for Asian Journal of Control, International Journal of Fuzzy Systems and Journal of the Franklin Institute. Dr Zhang is a Senior Member of IEEE and a Member of Technical Committee on Control Theory of Chinese Association of Automation. Dr Zhang has published more than 120 peer reviewed journal papers and one monograph (Zhang, W., Xie, L., & Chen, B. S., Stochastic H₂/H_∞ control: A Nash game approach, Boca Raton, FL: CRC Press, 2017). Dr Zhang has won the second prize of the Ministry of Education of the People's Republic of China twice. His research interests include linear and nonlinear stochastic optimal control, mean-field systems, robust H_∞ control, stochastic stability and stabilisation, multi-objective optimisation, and fuzzy adaptive control.