Nash equilibrium and bang-bang property for the non-zero-sum differential game of multi-player uncertain systems with Hurwicz criterion

Abstract

How to select the optimisation criterion of the performance function is a pivotal research topic for the uncertain differential game. The Hurwicz criterion is a flexible optimisation measure to allocate some weight between optimism and pessimism to achieve balance for both of them. Based on uncertainty theory, this paper considers the Nash equilibrium and bang-bang property for the multi-player non-zero-sum uncertain differential game model by applying the Hurwicz criterion to optimise the performance function. To coordinate onto the Nash equilibrium, an equilibrium equation is derived. In addition, the application of bang-bang optimal control in the multi-player game problem of the uncertain linear system is investigated. Finally, an example is shown to validate the results obtained.

Keywords:

• Nash equilibrium
• bang-bang control
• differential game
• uncertainty theory
• Hurwicz criterion

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The authors confirm that the data supporting the findings of this study are available within the article.

Additional information

Funding

This work is supported by the Group Building Scientific Innovation Project for Universities in Chongqing [grant no. CXQT21021].

Notes on contributors

Xi Li

Xi Li was born in 1992. She received the B.S. degree in Applied Mathematics in 2015 and the M.S. degree in Management in 2018, and she is working toward the Ph.D. degree in Management Science and Engineering, all from Chongqing Jiaotong University, Chongqing, China. Her current research interest is the differential game theory of uncertain systems.

Qiankun Song

Qiankun Song was born in 1964. He received the B.S. degree in Mathematics in 1986 from Sichuan Normal University, Chengdu, China, the M.S. degree in Applied Mathematics in 1996 from Northwestern Polytechnical University, Xi'an, China, and the Ph. D. degree in Applied Mathematics in 2010 from Sichuan University, Chengdu, China.From July 1986 to December 2000, he was with Department of Mathematics, Sichuan University of Science and Engineering, Sichuan, China. From January 2001 to June 2006, he was with the Department of Mathematics, Huzhou University, Zhejiang, China. In July 2006, he moved to the Department of Mathematics, Chongqing Jiaotong University, Chongqing, China. He is currently a Professor at Chongqing Jiaotong University.He is currently serving as an Editorial Board Member for Neurocomputing, Neural Processing Letters, Systems Science and Control Engineering, Journal of Applied Mathematics, British Journal of Mathematics & Computer Science, ISRN Applied Mathematics, Asian Journal of Mathematics and Computer Research, and a reviewer for Mathematical Reviews.He is the author or coauthor of more than 80 journal papers and two edited books. His current research interests include stability theory of neural networks and chaos synchronisation.

Yurong Liu

Yurong Liu received his B.S. degree in Mathematics from Suzhou University, Suzhou, China, in 1986, the M.S. degree in Applied Mathematics from Nanjing University of Science and Technology, Nanjing, China, in 1989, and the Ph.D. degree in Applied Mathematics from Suzhou University, Suzhou, China, in 2000. Dr. Liu is currently a Professor in the Department of Mathematics at Yangzhou University, China. He has published more than 50 papers in refereed international journals. His current interests include neural networks, complex networks, nonlinear dynamics, time-delay systems, multiagent systems, and chaotic dynamics.

Fuad E. Alsaadi

Fuad E. Alsaadi received the B.S. and M.Sc. degrees in electronic and communication from King Abdulaziz University, Jeddah, Saudi Arabia, in 1996 and 2002. He then received the Ph.D. degree in Optical Wireless Communication Systems from the University of Leeds, Leeds, UK, in 2011. Between 1996 and 2005, he worked in Jeddah as a communication instructor in the College of Electronics & Communication. He was a lecturer in the Faculty of Engineering in King Abdulaziz University, Jeddah, Saudi Arabia in 2005. He is currently an assistant professor of the Electrical and Computer Engineering Department within the Faculty of Engineering, King Abdulaziz University, Jeddah, Saudi Arabia. He published widely in the top IEEE communications conferences and journals and has received the Carter award, University of Leeds for the best PhD. He has research interests in optical systems and networks, signal processing, synchronisation and systems design.