Higher Prolongations of Control Affine Systems: Absolute Stability and Generalized Recurrence

Abstract

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.