发表机构
Kyushu University; Japan Aerospace Exploration Agency (JAXA); University of Colorado Boulder（九州大学; 日本宇宙航空研究开发机构; 科罗拉多大学博尔德分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文揭示了哈密顿平衡点最优控制中Jordan块导致线性阶无分岔，并在三次阶产生周期轨道，通过摆和Hill三体问题验证。
AI 中文摘要
与Pontryagin最小原理相关的哈密顿系统的平衡点表现出自然谱的精确加倍，并且在简单的配对条件下，每个简单的纯虚特征值处都有一个Jordan块。因此，经典的Lyapunov中心定理不适用于增广系统，且没有非零最优控制的周期轨道在线性阶分岔。我们一般性地建立了这一机制，并表明最优控制诱导的周期族在Lindstedt–Poincaré展开的三次阶出现。该机制以摆的闭式形式说明，并对Hill受限三体问题的平面$L_2$平衡点进行数值评估，其中三阶近似与独立计算的周期轨道族进行了验证。
英文摘要
Equilibria of the Hamiltonian system associated with Pontryagin's minimum principle exhibit an exact doubling of the natural spectrum and, under a simple pairing condition, a Jordan block at every simple purely imaginary eigenvalue. Consequently, the classical Lyapunov Center Theorem does not apply to the augmented system, and no periodic orbit with nonzero optimal control bifurcates at linear order. We establish this mechanism in general and show that an optimal-control-induced periodic family emerges at cubic order in a Lindstedt--Poincaré expansion. The mechanism is illustrated in closed form for the pendulum and evaluated numerically for the planar $L_2$ equilibrium of Hill's restricted three-body problem, where the third-order approximation is validated against an independently computed family of periodic orbits.
