KL-stability for a class of hybrid dynamical systems

Abstract

This article studies KL-stability (the stability expressed by KL-class function) for a class of hybrid dynamical systems (HDS). The notions ofKLK-property andKL-stability are proposed for HDS with respect to the hybrid-event-Time. The KL-stability, which is based on K or L property of the continuous flow, the discrete jump, and the event in an HDS, extends theKLL-stability and the event-stability reported in the literature for HDS. The relationships between KLK-property and KL-stability are established via introducing the hybrid dwell-Time condition (HDT). The HDT generalizes the average dwell-Time condition in the literature. For an HDS with KLK-property consisting of stabilizing L-property and destabilizing K-property, it is shown that there exists a common HDT under which the HDS will achieve KL-stability. Thus HDT may help to derive some easily tested conditions for HDS to achieve uniform asymptotic stability. Moreover, a criterion ofKL-stability is derived by using the multiple Lyapunov-like functions. Examples are given to illustrate the obtained theoretical results.