Invariance Principles for [math]-Brownian-Motion-Driven Stochastic Differential Equations and Their Applications to [math]-Stochastic Control

IF 2.2 2区数学 Q2 AUTOMATION & CONTROL SYSTEMS SIAM Journal on Control and Optimization Pub Date : 2024-05-29 DOI:10.1137/23m1564936

Xiaoxiao Peng, Shijie Zhou, Wei Lin, Xuerong Mao

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Abstract

SIAM Journal on Control and Optimization, Volume 62, Issue 3, Page 1569-1589, June 2024.
Abstract. The G-Brownian-motion-driven stochastic differential equations (G-SDEs) as well as the G-expectation, which were seminally proposed by Peng and his colleagues, have been extensively applied to describing a particular kind of uncertainty arising in real-world systems modeling. Mathematically depicting long-time and limit behaviors of the solution produced by G-SDEs is beneficial to understanding the mechanisms of system’s evolution. Here, we develop a new G-semimartingale convergence theorem and further establish a new invariance principle for investigating the long-time behaviors emergent in G-SDEs. We also validate the uniqueness and the global existence of the solution of G-SDEs whose vector fields are only locally Lipschitzian with a linear upper bound. To demonstrate the broad applicability of our analytically established results, we investigate its application to achieving G-stochastic control in a few representative dynamical systems.

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数学]-布朗运动驱动的随机微分方程的不变性原理及其在[数学]-随机控制中的应用

SIAM 控制与优化期刊》第 62 卷第 3 期第 1569-1589 页，2024 年 6 月。摘要。G-布朗运动驱动随机微分方程（G-SDEs）以及G-期望（G-expectation）是由彭晓赤和他的同事们从理论上提出的，已被广泛应用于描述现实世界系统建模中出现的一种特殊的不确定性。用数学方法描述 G-SDEs 所产生的解的长期和极限行为有利于理解系统的演化机制。在此，我们提出了一个新的 G-semartingale 收敛定理，并进一步建立了一个新的不变性原理，用于研究 G-SDE 中出现的长期行为。我们还以线性上界验证了矢量场仅为局部 Lipschitzian 的 G-SDE 解的唯一性和全局存在性。为了证明我们通过分析得出的结果的广泛适用性，我们研究了它在几个代表性动力学系统中实现 G-随机控制的应用。

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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.