广东工业大学学报 ›› 2024, Vol. 41 ›› Issue (02): 129-138.doi: 10.12052/gdutxb.230010

• 综合研究 • 上一篇    

转移概率部分未知的离散时间Markov跳变系统Nash微分博弈

张成科1, 徐萌2, 杨璐3   

  1. 1. 广东工业大学 经济学院, 广东 广州 510520;
    2. 广东工业大学 管理学院, 广东 广州 510520;
    3. 广东技术师范大学 管理学院, 广东 广州 510180
  • 收稿日期:2023-01-28 出版日期:2024-03-25 发布日期:2024-04-23
  • 通信作者: 徐萌(1993-),女,博士研究生,主要研究方向为动态博弈理论及应用,E-mail:1112008001@mail2.gdut.edu.cn
  • 作者简介:张成科(1964-),男,教授,博士,博士生导师,主要研究方向为动态博弈理论及应用,E-mail:zhangck@gdut.edu.cn
  • 基金资助:
    国家自然科学基金资助项目(71571053)；国家社会科学基金后期资助暨优秀博士论文项目(21FJYB025)；广东省基础与应用基础研究基金资助项目(2023A1515012335)

Nash Differential Game for Discrete-time Markov Jump System with Partially Unknown Transition Probabilities

Zhang Cheng-ke1, Xu Meng2, Yang Lu3   

  1. 1. School of Economics, Guangdong University of Technology, Guangzhou 510520, China;
    2. School of Management, Guangdong University of Technology, Guangzhou 510520, China;
    3. School of Management, Guangdong Polytechnic Normal University, Guangzhou 510180, China
  • Received:2023-01-28 Online:2024-03-25 Published:2024-04-23

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摘要： 考虑到转移概率矩阵元素无法完全获悉，如何在转移概率部分未知的情境下研究离散时间Markov跳变系统Nash微分博弈是有待解决的问题之一，这一问题可以为转移概率部分未知的Markov跳变系统Nash微分博弈理论在管理问题上的应用提供理论支撑。基于此，本文首先研究单人博弈情形，即ε-次优控制问题，借助自由连接权矩阵和配方法，得到了ε-次优控制策略存在的充分性条件，并给出了成本函数上界的显式表达；然后延伸至双人博弈进行分析，得到了ε-次优Nash均衡策略存在的条件等价于求解双线性矩阵不等式和矩阵不等式的优化问题，并通过启发式算法求解优化问题得到ε-次优Nash均衡策略；最后通过数值算例证明了主要结论的有效性。

关键词: 离散时间Markov跳变系统, ε-次优控制, ε-次优Nash均衡

Abstract: It is noted that transition probability matrix elements cannot be fully known. How to study Nash differential game for discrete-time Markov jump system (MJS) under the condition of unknown transition probability is one of the problems to be solved. This problem can provide theoretical support for the application of Nash differential game theory in Markov jump systems with partial unknown transition probability to management problems. Based on it, the case of one-player game, which is called the ε-suboptimal control problem, is firstly studied. By using the free-connection weighting matrix and “complete square” method, the sufficient conditions for the existence of the ε-suboptimal cntrol strategy are obtained, and an explicit expression of the upper bound of the cost function is given. Then, the conditions for the existence of ε-suboptimal Nash equilibrium strategy are equivalent to solving the optimization problem, which satisfied the bilinear matrix inequalities (BMIs) and matrix inequalities. The heuristic algorithm is used to solve the optimization problem to obtain the ε-suboptimal Nash equilibrium strategies. Finally, the numerical examples are provided to demonstrate the validity of the main conclusions.

Key words: discrete time Markov jump system, ε-suboptimal control, ε-suboptimal Nash equilibrium

中图分类号: 

  • F224
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