自由终止时间扫掠过程问题的必要条件

中文摘要 AI 辅助

本文推导了由扫掠过程动力学控制的自由终止时间最优控制问题的必要最优性条件。与经典最优控制问题相比，状态方程中法锥项的存在引入了关于状态变量的不连续性，并阻碍了最优控制理论中通常采用的标准Lipschitz正则性假设。因此，轨迹分析和最优性条件的表述都需要精细的工具。我们处理具有一般端点约束和时间相关数据的问题，同时考虑动力学在时间上Lipschitz连续的情况，以及更具挑战性的仅可测时间依赖情形。特别关注与自由终止时间变量相关的附加横截条件。在Lipschitz情形下，这些条件通过一个有界变差函数表达，该函数几乎处处与最大化Hamiltonian一致。在可测情形下，我们表明可以通过文献中最近引入的最大化Hamiltonian的下本质值和上本质值概念获得合适的解释，从而得到加强的横截条件。这些条件还包含一个附加项，反映协态弧与最优端点处移动集法锥之间的相互作用。我们的方法结合了适当的扰动技术和针对扫掠过程定制的惩罚方法。移动集被建模为满足适当约束规范的有限个不等式定义集的交集，包括正线性无关条件和相关Gram矩阵的对角占优性质。我们还引入了约束规范条件的局部版本。我们的结果适用于广泛的问题类别，包括涉及无界移动集（如多面体）的问题。所得最优性系统将经典自由终止时间问题的近期结果扩展并强化到扫掠过程的非Lipschitz、不连续框架中。本文获得的横截条件的重要性通过两个简单示例加以说明。

英文摘要

This paper derives necessary optimality conditions for free end-time optimal control problems governed by sweeping process dynamics. In contrast to classical optimal control problems, the presence of the normal cone term in the state equation introduces discontinuities with respect to the state variable and prevents the standard Lipschitz regularity assumptions typically used in optimal control theory. As a result, both the analysis of trajectories and the formulation of optimality conditions require refined tools. We address problems with general endpoint constraints and time-dependent data, considering both the case in which the dynamics are Lipschitz continuous in time and the more challenging setting of merely measurable time dependence. Special attention is devoted to the additional transversality conditions associated with the free end-time variable. In the Lipschitz case, these conditions are expressed in terms of a function of bounded variation that coincides almost everywhere with the maximized Hamiltonian. In the measurable case, we show that a suitable interpretation can be obtained via the notions of sub- and super-essential values of the maximized Hamiltonian, recently introduced in the literature, leading to strengthened transversality conditions. These conditions also involve an additional term reflecting the interaction between the costate arc and the normal cone to the moving set at the optimal endpoints. Our approach combines suitable perturbation techniques with a penalization method tailored to sweeping processes. The moving set is modelled as a finite intersection of inequality-defined sets satisfying suitable constraint qualifications, including a positive linear independence condition and a diagonal dominance property of the associated Gramian matrix. We also introduce a local version of the constraint qualification condition. Our results apply to a broad class of problems, including those involving unbounded moving sets such as polyhedra. The resulting optimality system extends and sharpens recent results for classical free end-time problems to the non-Lipschitz, discontinuous framework of sweeping processes. The relevance of the transversality conditions obtained in this article is illustrated through two simple examples.

发表机构

• Univ Brest（布雷斯特大学）
• CNRS（法国国家科学研究中心）

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