倒立摆最优控制中的超敏性与转枢现象：动力系统视角
Hypersensitivity and Turnpikes in Optimal Control of Inverted Pendulum: A Dynamical Systems Perspective
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中文总结 AI 辅助
从动力系统视角揭示倒立摆最优控制中超敏性源于初始伴随变量的分形结构，转枢源于退化中心流形慢动力学，逃逸通道由NHIMs形成，并导致轨迹优化数值不稳定。
中文摘要 AI 辅助
本文从动力系统的视角研究了倒立摆最优控制中的超敏性与转枢现象。我们证明，对于固定终端时间和固定终端状态的最优控制问题，（1）超敏性源于相关哈密顿动力学中初始伴随变量集合的分形结构，（2）转枢现象源于退化中心流形附近的慢动力学，（3）逃逸通道由法向双曲不变流形（NHIMs）形成。因此，初始伴随变量的微小扰动会导致性质上截然不同的极值轨迹，从而在轨迹优化中造成严重的数值不稳定性。分形结构和不变集均通过数值和解析方法进行了表征。
英文摘要
The hypersensitivity and turnpike phenomena in the optimal control of an inverted pendulum are investigated from a dynamical-systems perspective. We show that, for a fixed terminal time and a fixed terminal state optimal control problem, (1) the hypersensitivity originates from the fractal structure of the set of initial adjoint variables in the associated Hamiltonian dynamics, (2) the turnpike arises from slow dynamics in the vicinity of a degenerate center manifold, and (3) the escape channels are formed by normally hyperbolic invariant manifolds (NHIMs). As a consequence, small perturbations in the initial adjoint variables lead to qualitatively distinct extremal trajectories, resulting in severe numerical instability in trajectory optimization. Both the fractal structure and the invariant sets are characterized numerically and analytically.
发表机构
- Kyushu University（九州大学）
- Hokkaido University（北海道大学）
机构由 AI 辅助整理，请以论文原文为准。
