Feedback and Open-Loop Nash Equilibria for LQ Infinite-Horizon Discrete-Time Dynamic Games

IF 2.2 2区数学 Q2 AUTOMATION & CONTROL SYSTEMS SIAM Journal on Control and Optimization Pub Date : 2024-05-13 DOI:10.1137/23m1579960

Andrea Monti, Benita Nortmann, Thulasi Mylvaganam, Mario Sassano

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Abstract

SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1417-1436, June 2024.
Abstract. We consider dynamic games defined over an infinite horizon, characterized by linear, discrete-time dynamics and quadratic cost functionals. Considering such linear-quadratic dynamic games, we focus on their solutions in terms of Nash equilibrium strategies. Both feedback (F-NE) and open-loop (OL-NE) Nash equilibrium solutions are considered. The contributions of the paper are threefold. First, our detailed study reveals some interesting structural insights in relation to F-NE solutions. Second, as a stepping stone toward our consideration of OL-NE strategies, we consider a specific infinite-horizon discrete-time (single-player) optimal control problem, wherein the dynamics are influenced by a known exogenous input and draw connections between its solution obtained via dynamic programming and Pontryagin’s minimum principle. Finally, we exploit the latter result to provide a characterization of OL-NE strategies of the class of infinite-horizon dynamic games. The results and key observations made throughout the paper are illustrated via a numerical example.

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LQ 无限视距离散时间动态博弈的反馈和开环纳什均衡器

SIAM 控制与优化期刊》第 62 卷第 3 期第 1417-1436 页，2024 年 6 月。摘要我们考虑的是定义在无限视界上的动态博弈，其特征是线性离散时间动态和二次成本函数。考虑到这种线性-二次动态博弈，我们重点研究它们在纳什均衡策略方面的解。我们同时考虑了反馈（F-NE）和开环（OL-NE）纳什均衡解。本文有三方面的贡献。首先，我们的详细研究揭示了与 F-NE 解相关的一些有趣的结构性见解。其次，作为我们考虑 OL-NE 策略的垫脚石，我们考虑了一个特定的无限视距离散时间（单人）最优控制问题，其中动态受已知外生输入的影响，并得出了通过动态编程获得的解与庞特里亚金最小原理之间的联系。最后，我们利用后一结果提供了一类无限视距动态博弈的 OL-NE 策略特征。我们将通过一个数值例子来说明本文的结果和主要观点。

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来源期刊

SIAM Journal on Control and Optimization 数学-应用数学

CiteScore

4.00

自引率

4.50%

发文量

143

审稿时长

12 months

期刊介绍： SIAM Journal on Control and Optimization (SICON) publishes original research articles on the mathematics and applications of control theory and certain parts of optimization theory. Papers considered for publication must be significant at both the mathematical level and the level of applications or potential applications. Papers containing mostly routine mathematics or those with no discernible connection to control and systems theory or optimization will not be considered for publication. From time to time, the journal will also publish authoritative surveys of important subject areas in control theory and optimization whose level of maturity permits a clear and unified exposition. The broad areas mentioned above are intended to encompass a wide range of mathematical techniques and scientific, engineering, economic, and industrial applications. These include stochastic and deterministic methods in control, estimation, and identification of systems; modeling and realization of complex control systems; the numerical analysis and related computational methodology of control processes and allied issues; and the development of mathematical theories and techniques that give new insights into old problems or provide the basis for further progress in control theory and optimization. Within the field of optimization, the journal focuses on the parts that are relevant to dynamic and control systems. Contributions to numerical methodology are also welcome in accordance with these aims, especially as related to large-scale problems and decomposition as well as to fundamental questions of convergence and approximation.