Optimal control and non-zero-sum differential game for Hurwicz model considering uncertain dynamic systems with multiple input delays

ABSTRACT

Uncertainty theory is a field in axiomatic mathematics committed to disposing of belief degrees. By dint of uncertain theory and Hurwicz criterion, this article mainly addresses optimal control and non-zero-sum differential game of uncertain delay dynamic systems, which are depicted as a sort of uncertain differential equation with multiple input delays. Employing the technology of dynamic programming, the optimality principle is put forward and the optimality equation is formulated simultaneously to deal with the optimal control problem. In addition, an equilibrium equation is derived to solve the Nash equilibrium for the multi-player non-zero-sum uncertain differential game on the strength of the proposed optimality equation. An example is devised to illustrate the availability of the results in the end.

KEYWORDS:

• Optimal control
• non-zero-sum game
• multiple input delays
• Hurwicz criterion
• uncertainty theory

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 was supported by the Chongqing Talent Program [grant no. cstc2021ycjhbgzxm0107], the Group Building Scientific Innovation Project for Universities in Chongqing [grant no. CXQT21021] and the Joint Training Base Construction Project for Graduate Students in Chongqing [grant no. JDLHPYJD2021016].

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.