Maximum Principles for Optimal Control Problems with Differential Inclusions

IF 2.2 2区数学 Q2 AUTOMATION & CONTROL SYSTEMS SIAM Journal on Control and Optimization Pub Date : 2024-01-23 DOI:10.1137/22m1540740

A. D. Ioffe

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Abstract

SIAM Journal on Control and Optimization, Volume 62, Issue 1, Page 271-296, February 2024.
Abstract. There are three different forms of adjoint inclusions that appear in the most advanced necessary optimality conditions for optimal control problems involving differential inclusions: Euler–Lagrange inclusion (with partial convexification) [A. D. Ioffe, J. Optim. Theory Appl., 182 (2019), pp. 285–309], fully convexified Hamiltonian inclusion [F. H. Clarke, Mem. Amer. Math. Soc., 173 (2005), 816], and partially convexified Hamiltonian inclusion [P. D. Loewen and R. T. Rockafellar, SIAM J. Control Optim., 34 (1996), pp. 1496–1511], [A. D. Ioffe, Trans. Amer. Math. Soc., 349 (1997), pp. 2871–2900], [R. B. Vinter, SIAM J. Control Optim., 52 (2014), pp. 1237–1250] (for convex-valued differential inclusions in the first two references). This paper addresses all three types of necessary conditions for problems with (in general) nonconvex-valued differential inclusions. The first of the two main theorems, with the Euler–Lagrange inclusion, is equivalent to the main result of [A. D. Ioffe, J. Optim. Theory Appl., 182 (2019), pp. 285–309] but proved in a substantially different and much more direct way. The second theorem contains conditions that guarantee necessity of both types of Hamiltonian conditions. It seems to be the first result of such a sort that covers differential inclusions with possibly unbounded values and contains the most recent results of [F. H. Clarke, Mem. Amer. Math. Soc., 173 (2005), 816] and [R. B. Vinter, SIAM J. Control Optim., 52 (2014), pp. 1237–1250] as particular cases. And again, the proof of the theorem is based on a substantially different approach.

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带微分夹杂的最优控制问题的最大原则

SIAM 控制与优化期刊》第 62 卷第 1 期第 271-296 页，2024 年 2 月。摘要。在涉及微分夹杂的最优控制问题的最先进的必要最优性条件中，出现了三种不同形式的邻接夹杂：欧拉-拉格朗日包含（部分凸化）[A. D. Ioffe, J. Optim. Theory Appl.D. Loewen 和 R. T. Rockafellar，SIAM J. Control Optim.，34 (1996)，第 1496-1511 页]，[A. D. Ioffe，Trans. Amer. Math. Soc.，349 (1997)，第 2871-2900 页]，[R. B. Vinter，SIAM J. Control Optim.，52 (2014)，第 1237-1250 页]（前两个参考文献中的凸值微分夹杂）。本文论述了（一般情况下）非凸值微分夹杂问题的所有三类必要条件。两个主要定理中的第一个，即欧拉-拉格朗日包含，等同于 [A. D. Ioffe, J. Optim. Theory Appl.第二个定理包含保证两类汉密尔顿条件必要性的条件。它似乎是第一个涵盖可能无界值的微分夹杂的此类结果，并包含[F. H. Clarke, Mem. Amer. Math. Soc., 173 (2005), 816] 和[R. B. Vinter, SIAM J. Control Optim., 52 (2014), pp.同样，该定理的证明基于一种本质上不同的方法。

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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.