Optimal control and zero-sum differential game for Hurwicz model considering singular systems with multifactor and uncertainty

Abstract

Uncertainty theory is a new branch of mathematics concerned with the analysis of subjective indeterminacy. Based on the concept of uncertain theory, this paper considers optimal control and zero-sum differential game for multi-factor uncertain continuous-time singular systems with Hurwicz criterion, where dynamic systems are modelled by a type of uncertain differential equation driven by canonical Liu process. Using the approach of dynamic programming, the principle of optimality is proposed and then the equation of optimality is derived to resolve an uncertain control model. A two-player zero-sum uncertain differential game is further investigated based on the optimality equation above, and an equilibrium equation is presented to locate a saddle-point in the game. Finally, an example is discussed to illuminate the effectiveness of the results.

Keywords:

• Singular system
• uncertainty theory
• Hurwicz criterion
• optimal control
• zero-sum game

Disclosure statement

No potential conflict of interest was reported by the authors.

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 Team Building Project for Graduate Tutors in Chongqing [grant number JDDSTD201802] and the Group Building Scientific Innovation Project for Universities in Chongqing [grant number 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.

Zhenjiang Zhao

Zhenjiang Zhao was born in 1962. He received the B.S. degree in Mathematics in 1984 from Kashgar Teachers College, Kashgar, China, the M.S. degree in Applied Mathematics in 1998 from Okayama University, Okayama, Japan. He is currently a Professor at Huzhou Teachers College. He is the author or coauthor of more than 30 journal papers. His current research interests include stability theory of neural networks.

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, synchronization and systems design.