SIAM Journal on Applied Dynamical Systems, Volume 20, Issue 2, Page 784-825, January 2021.
We establish versions of Conley's (i) fundamental theorem and (ii) decomposition theorem for a broad class of hybrid dynamical systems. The hybrid version of (i) asserts that a globally defined hybrid complete Lyapunov function exists for every hybrid system in this class. Motivated by mechanics and control settings where physical or engineered events cause abrupt changes in a system's governing dynamics, our results apply to a large class of Lagrangian hybrid systems (with impacts) studied extensively in the robotics literature. Viewed formally, these results generalize those of Conley and Franks for continuous-time and discrete-time dynamical systems, respectively, on metric spaces. However, we furnish specific examples illustrating how our statement of sufficient conditions represents merely an early step in the longer project of establishing which formal assumptions can and cannot endow hybrid systems models with the topologically well-characterized partitions of limit behavior that make Conley's theory so valuable in those classical settings.

中文翻译：

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一类混合系统的康利基本定理

SIAM应用动力系统杂志，第20卷，第2期，第784-825页，2021年1月。
我们为广泛的混合动力系统建立了Conley（i）基本定理和（ii）分解定理的版本。（i）的混合版本断言，此类中的每个混合系统都存在一个全局定义的混合完整Lyapunov函数。受物理和工程事件导致系统控制动力学突然变化的力学和控制设置的激励，我们的结果适用于在机器人学文献中进行了广泛研究的一大类拉格朗日混合系统（有影响）。从形式上看，这些结果概括了Conley和Franks在度量空间上分别针对连续时间和离散时间动力系统的结果。然而，