Finite-Time Stability and Instability of Nonlinear Impulsive Systems

. In this paper, the ﬁnite-time stability and instability are studied for nonlinear impulsive systems. There are mainly four concerns. 1) For the system with stabilizing impulses, a Lyapunov theorem on global ﬁnite-time stability is presented. 2) When the system without impulsive effects is globally ﬁnite-time stable (GFTS) and the settling time is continuous at the origin, it is proved that it is still GFTS over any class of impulse sequences, if the mixed impulsive jumps satisfy some mild conditions. 3) For systems with destabilizing impulses, it is shown that to be ﬁnite-time stable, the destabilizing impulses should not occur too frequently, otherwise, the origin of the impulsive system is ﬁnite-time instable, which are formulated by average dwell time (ADT) conditions respectively. 4) A theorem on ﬁnite-time instability is provided for system with stabilizing impulses. For each GFTS theorem of impulsive systems con-sidered in this paper, the upper boundedness of settling time is given, which depends on the initial value and impulsive effects. Some numerical examples are given to illus-trate the theoretical analysis.