A Tikhonov Theorem for McKean–Vlasov Two-Scale Systems and a New Application to Mean Field Optimal Control Problems

IF 2.2 2区数学 Q2 AUTOMATION & CONTROL SYSTEMS

SIAM Journal on Control and Optimization Pub Date : 2024-09-04 DOI:10.1137/22m1543070

Matteo Burzoni, Alekos Cecchin, Andrea Cosso

引用次数: 0

Abstract

SIAM Journal on Control and Optimization, Volume 62, Issue 5, Page 2475-2505, October 2024.
Abstract. We provide a new version of the Tikhonov theorem for both two-scale forward systems and also two-scale forward-backward systems of stochastic differential equations, which also covers the McKean–Vlasov case. Differently from what is usually done in the literature, we prove a type of convergence for the “fast” variable, which allows the limiting process to be discontinuous. This is relevant for the second part of the paper, where we present a new application of this theory to the approximation of the solution of mean field control problems. Towards this aim, we construct a two-scale system whose “fast” component converges to the optimal control process, while the “slow” component converges to the optimal state process. The interest in such a procedure is that it allows one to approximate the solution of the control problem, avoiding the usual step of the minimization of the Hamiltonian.

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麦金-弗拉索夫双尺度系统的提霍诺夫定理及其在均值场最优控制问题中的新应用

SIAM 控制与优化期刊》，第 62 卷第 5 期，第 2475-2505 页，2024 年 10 月。摘要。我们为随机微分方程的双尺度前向系统和双尺度前向-后向系统提供了一个新版本的 Tikhonov 定理，它也涵盖了 McKean-Vlasov 情况。与文献中的通常做法不同，我们证明了 "快速 "变量的收敛类型，它允许极限过程是不连续的。这与本文的第二部分相关，在这一部分中，我们提出了这一理论在均值场控制问题近似解法中的新应用。为此，我们构建了一个双尺度系统，其 "快速 "部分收敛于最优控制过程，而 "慢速 "部分收敛于最优状态过程。这种程序的意义在于，它允许我们近似求解控制问题，避免了最小化哈密顿的常规步骤。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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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.