AI 中文总结
该研究从动力系统视角，结合庞特里亚金最大值原理，将控制目标转化为辛几何迷向子流形，关联反馈可控性与不变流形奇点，通过低维示例阐述控制与动力系统的联系。
AI 中文摘要
控制理论与动力系统是紧密交织的领域。庞特里亚金最大值原理提供了一种紧密联系，它通过将控制系统提升为（哈密顿）动力系统，为综合反馈控制律提供了一种构造性方法。反馈律随后可被编码为该诱导动力系统的不变流形——前提是这些流形可微分同胚地投影回基础控制系统。尽管对许多控制系统而言反馈镇定是不可能的，但上述过程仍可进行。在这种情况下，不变流形不再可微分同胚地投影，从而导致焦散的出现。本文通过将控制中的目标集转化为辛几何中的迷向子流形，并将反馈可控性与诱导不变流形的奇点关联起来，对上述联系进行了概述。文中包含多个低维示例以阐明该理论。
英文摘要
Control theory and dynamical systems are closely intertwined fields. Pontryagin's maximum principal offers a strong connection by providing a constructive way to synthesize feedback control laws via lifting the control system to a (Hamiltonian) dynamical system. Feedback laws can then be encoded as invariant manifolds of this induced dynamical system - under the condition that these manifolds project diffeomorphically back to the base control system. While feedback stabilization is impossible for many control systems, the above procedure can still be carried out. In this setting, the invariant manifold no longer projects diffeomorphically which results in the emergence of caustics. This paper offers an overview of the above connection by translating target sets from controls to isotropic submanifolds in symplectic geometry and associates feedback controllability to singularities of the induced invariant manifolds. Multiple low-dimensional examples are included to elucidate the theory.
Comments22 pages, 11 figures
