Nash equilibrium strategies for non-zero-sum differential games of SDEs with time-varying coefficient and infinite Markov jumps

Yueying Liu; Mengping Sun; Zhen Wang; Xiangyun Lin; Cuihua Zhang; Yueying Liu; Mengping Sun; Zhen Wang; Xiangyun Lin; Cuihua Zhang

doi:10.3934/era.2025112

Electronic Research Archive

2025, Volume 33, Issue 4: 2525-2542. doi: 10.3934/era.2025112

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Research article

Nash equilibrium strategies for non-zero-sum differential games of SDEs with time-varying coefficient and infinite Markov jumps

1.
College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
2.
Department of Automation, Institute of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China

Received: 15 January 2025 Revised: 14 April 2025 Accepted: 16 April 2025 Published: 27 April 2025

This paper mainly discusses the non-zero-sum Nash differential games for stochastic differential equations (SDEs) involving time-varying coefficient and infinite Markov jumps. First of all, a necessary and sufficient conditions for the existence of Nash equilibrium strategies is given, which turns the non-zero-sum Nash differential games into solving the equations that are composed of countable coupled generalized differential Riccati equations (CGDREs). As an application, a unified treatment is presented for $ H_{2} $, $ H_{\infty} $, and $ H_{2}/H_{\infty} $ control by the Nash game approach, which can reveal the relationship among these three problems. Furthermore, the theoretical results are used to solve a numerical example.
Citation: Yueying Liu, Mengping Sun, Zhen Wang, Xiangyun Lin, Cuihua Zhang. Nash equilibrium strategies for non-zero-sum differential games of SDEs with time-varying coefficient and infinite Markov jumps[J]. Electronic Research Archive, 2025, 33(4): 2525-2542. doi: 10.3934/era.2025112

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Abstract

This paper mainly discusses the non-zero-sum Nash differential games for stochastic differential equations (SDEs) involving time-varying coefficient and infinite Markov jumps. First of all, a necessary and sufficient conditions for the existence of Nash equilibrium strategies is given, which turns the non-zero-sum Nash differential games into solving the equations that are composed of countable coupled generalized differential Riccati equations (CGDREs). As an application, a unified treatment is presented for $ H_{2} $, $ H_{\infty} $, and $ H_{2}/H_{\infty} $ control by the Nash game approach, which can reveal the relationship among these three problems. Furthermore, the theoretical results are used to solve a numerical example.

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