SIAM Journal on Control and Optimization, Volume 58, Issue 6, Page 3019-3040, January 2020.
This manuscript deals with higher prolongations and higher prolongational limit sets of control affine systems. The higher prolongations extend the positive semiorbit and determine a great number of stability concepts, indexed by the ordinal numbers. The highest stability is named absolute stability and can be characterized by a continuous Lyapunov functional. It is proved that compact positively invariant uniform attractors and positive semiorbits of dispersive control systems are absolutely stable sets. The higher prolongational limit sets determine the generalized recurrence, which extends the recursive concepts of Poincaré recurrence and nonwandering points. The notion of chain prolongation is introduced in order to discuss the generalized recurrence. The main result shows that the chain prolongation is the largest extension of the positive semiorbit, and then the chain recurrence is the more general concept of recurrence.

中文翻译：

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控制仿射系统的更高延展：绝对稳定性和广义递归

SIAM控制与优化杂志，第58卷，第6期，第3019-3040页，2020年1月。
该手稿涉及控制仿射系统的更高延伸和更高延伸极限集。较高的延伸范围扩展了正半轨道，并确定了许多由序数索引的稳定性概念。最高的稳定性称为绝对稳定性，并且可以通过连续的Lyapunov函数来表征。证明了紧的正不变一致吸引子和色散控制系统的正半轨道是绝对稳定的集合。较高的延伸极限集确定了广义的递归，这扩展了Poincaré递归和非漂移点的递归概念。为了讨论广义复发，引入了链延长的概念。