由微分方程定义的边界约束的横截性条件
Transversality Conditions for Boundary Constraints Defined by Differential Equations
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中文总结 AI 辅助
本文针对边界条件由微分方程定义的最优控制问题，建立通用初始与最终时刻横截性条件，引入协调/非协调时钟时间等概念，为相关轨迹优化问题提供理论基础。
中文摘要 AI 辅助
当最优控制问题的边界条件由微分方程定义时，对应的横截性条件是什么？这个看似奇特的问题源于N体系统中的轨迹优化问题，但其本质更具基础性，不仅适用于天体动力学问题，还可推广到一般的不可积动力系统。本文的主要贡献是，针对边界条件由带辅助条件的微分方程定义的最优控制问题，建立了通用的初始时刻和最终时刻横截性条件，其中包含了微分边界条件的数学定义，这是本文所构建基础的一部分。为支撑这些新基础，本文引入了协调/非协调时钟时间和弱伴随余向量的概念。在非协调时钟时间的情况下，新的横截性条件揭示了一种特殊情形：弱伴随余向量与边界微分方程的向量场正交，该条件与经典的关于端点流形的正交性表述存在显著差异。本文所推导的定理具有通用性，这些定理在三体问题若干情形中的应用将在其他论文中阐述。
英文摘要
What are the transversality conditions for an optimal control problem when the boundary conditions are defined by differential equations? This seemingly bizarre question is motivated by trajectory optimization problems in the $N$-body system. The question, however, is more fundamental and goes beyond problems in astrodynamics to nonintegrable dynamical systems in general. The main contribution of this paper is the development of generic initial- and final-time transversality conditions for optimal control problems whose boundary conditions are defined in terms of differential equations with side conditions. The mathematical definition of differential boundary conditions are part of the foundations developed in this paper. To support the new fundamentals, the concept of coordinated/uncoordinated clock times and weak adjoint covectors are introduced. In the case of uncoordinated clock times, the new transversality conditions reveal that there exists a special situation where a weak adjoint covector is orthogonal to the vector field of the boundary differential equation. This condition is sharply different from the classical statement of orthogonality with respect to the endpoint manifold. The theorems developed in this paper are generic. An application of the theorems to several cases in the three-body problem are described in separate papers.
发表机构
- Naval Postgraduate School（海军研究生院）
机构由 AI 辅助整理，请以论文原文为准。
