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

Zhang Cheng-ke; Xu Meng; Yang Lu

doi:10.12052/gdutxb.230010

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

更新时间：2026-05-27

- Nash Differential Game for Discrete-time Markov Jump System with Partially Unknown Transition Probabilities
- Vol. 41, Issue 2, Pages: 129-138(2024)
- 作者机构：
  
  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
- 作者简介：
- 基金信息：
- DOI：10.12052/gdutxb.230010
  CLC： F224
- Online First：25 March 2024，
  
  Published：2024
- 稿件说明：
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张成科, 徐萌, 杨璐. 转移概率部分未知的离散时间Markov跳变系统Nash微分博弈[J]. 广东工业大学学报, 2024, 41(2): 129-138.

Zhang Cheng-ke, Xu Meng, Yang Lu. Nash Differential Game for Discrete-time Markov Jump System with Partially Unknown Transition Probabilities[J]. 2024, 41(2): 129-138.
张成科, 徐萌, 杨璐. 转移概率部分未知的离散时间Markov跳变系统Nash微分博弈[J]. 广东工业大学学报, 2024, 41(2): 129-138. DOI： 10.12052/gdutxb.230010.

Zhang Cheng-ke, Xu Meng, Yang Lu. Nash Differential Game for Discrete-time Markov Jump System with Partially Unknown Transition Probabilities[J]. 2024, 41(2): 129-138. DOI： 10.12052/gdutxb.230010.

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

$\varepsilon -$

img text_id='M1' class='formula-img' style='display:none;' src='gdgydxxb-41-2-129_M1.jpg'>

次优控制问题，借助自由连接权矩阵和配方法，得到了

$ \varepsilon - $

img text_id='M2' class='formula-img' style='display:none;' src='gdgydxxb-41-2-129_M2.jpg'>

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

$ \varepsilon - $

img text_id='M2' class='formula-img' style='display:none;' src='gdgydxxb-41-2-129_M2.jpg'>

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

$ \varepsilon - $

img text_id='M2' class='formula-img' style='display:none;' src='gdgydxxb-41-2-129_

M2.jpg'>

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。

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

$ \varepsilon - $

img text_id='M5' class='formula-img' style='display:none;' src='gdgydxxb-41-2-129_M5.jpg'>

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

$ \varepsilon - $

img text_id='M5' class='formula-img' style='display:none;' src='gdgydxxb-41-2-129_M5.jpg'>

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

$ \varepsilon - $

img text_id='M5' class='formula-img' style='display:none;' src='gdgydxxb-41-2-129_M5.jpg'>

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

$ \varepsilon - $

img text_id='M5' class='formula-img' style='display:none;' src='gdgydxxb-

41-2-129_M5.jpg'>

suboptimal Nash equilibrium strategies. Finally

the numerical examples are provided to demonstrate the validity of the main conclusions.

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Keywords

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