Optimal control and zero‐sum game subject to differential equations with Liu processes and random matrices

Chen, Xin; Zhu, Yuanguo

doi:10.1002/oca.3098

Cited by 2 publications

(1 citation statement)

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“…Liu's pioneering work [17] in 2013 laid the foundation for exploring uncertain stochastic hybrid systems [18,19] based on subadditive measures, followed by subsequent research by Fei et al [20] in 2014, introducing uncertain stochastic hybrid optimal control and the equation of optimality via uncertain stochastic hybrid differential equations. With the help of uncertain stochastic optimal control, there has been a growing body of literature focusing on uncertain stochastic hybrid differential game systems [21][22][23][24]. Unlike the models and methods mentioned earlier, this work is the beginning of linear quadratic uncertain stochastic hybrid differential games in finite and infinite horizons.…”

Section: Introductionmentioning

confidence: 99%

Saddle-Point Equilibrium Strategy for Linear Quadratic Uncertain Stochastic Hybrid Differential Games Based on Subadditive Measures

Jia,

2024

Mathematics

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This paper describes a kind of linear quadratic uncertain stochastic hybrid differential game system grounded in the framework of subadditive measures, in which the system dynamics are described by a hybrid differential equation with Wiener–Liu noise and the performance index function is quadratic. Firstly, we introduce the concept of hybrid differential games and establish the Max–Min Lemma for the two-player zero-sum game scenario. Next, we discuss the analysis of saddle-point equilibrium strategies for linear quadratic hybrid differential games, addressing both finite and infinite time horizons. Through the incorporation of a generalized Riccati differential equation (GRDE) and guided by the principles of the Itô–Liu formula, we prove that that solving the GRDE is crucial and serves as both a sufficient and necessary precondition for identifying equilibrium strategies within a finite horizon. In addition, we also acquire the explicit formulations of equilibrium strategies in closed forms, alongside determining the optimal values of the cost function. Through the adoption of a generalized Riccati equation (GRE) and applying a similar approach to that used for the finite horizon case, we establish that the ability to solve the GRE constitutes a sufficient criterion for the emergence of equilibrium strategies in scenarios extending over an infinite horizon. Moreover, we explore the dynamics of a resource extraction problem within a finite horizon and separately delve into an H∞ control problem applicable to an infinite horizon. Finally, we present the conclusions.

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Section: Introductionmentioning

confidence: 99%

Saddle-Point Equilibrium Strategy for Linear Quadratic Uncertain Stochastic Hybrid Differential Games Based on Subadditive Measures

Jia,

2024

Mathematics

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Optimal control and zero-sum differential game for stochastic uncertain systems with randomness and uncertainty

Chen,

Lu,

Shao

et al. 2024

Transactions of the Institute of Measurement and Control

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This paper presents a stochastic uncertain system characterized by a differential equation involving multiple Wiener processes and uncertain matrices. Our study delves into the realm of chance theory to address problems related to optimal control and zero-sum differential games within this stochastic uncertain framework. Based on the properties of both the Wiener processes and the uncertain matrices, we present an equation of optimality tailored for solving stochastic uncertain optimal control problems. Leveraging this equation, we derive analytical optimal solutions for both bang–bang and linear quadratic control problems. Furthermore, we extend our results to encompass the domain of zero-sum differential games, proposing equilibrium equations to identify the saddle point equilibrium in stochastic uncertain zero-sum games. The practical applicability of our theoretical framework is demonstrated through the analysis of a counterterrorism game scenario.

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Optimal control and zero‐sum game subject to differential equations with Liu processes and random matrices

Cited by 2 publications

References 44 publications

Saddle-Point Equilibrium Strategy for Linear Quadratic Uncertain Stochastic Hybrid Differential Games Based on Subadditive Measures

Saddle-Point Equilibrium Strategy for Linear Quadratic Uncertain Stochastic Hybrid Differential Games Based on Subadditive Measures

Optimal control and zero-sum differential game for stochastic uncertain systems with randomness and uncertainty

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