Finite-time stability via GKL-functions for impulsive dynamical systems

Abstract

This paper studies the finite-time stability via $\mathrm{GKL}$-functions ($\mathrm{GKL}$-FTS) for impulsive dynamical systems (IDS). The notions of $\mathrm{GKL}$-functions, $\mathrm{GKL}$-FTS, and event-$\mathrm{GKL}$-FTS are proposed for IDS. The $\mathrm{GKL}$-FTS is a type of well-defined finite-time stability which is expressed via $\mathrm{GKL}$-functions. The $\mathrm{GKL}$-FTS is decomposed into specific types through the decomposition of $\mathrm{GKL}$-functions. By establishing the comparison principles of FTS including $\mathrm{GKL}$-FTS and event-$\mathrm{GKL}$-FTS, and by using the decompositions of $\mathrm{GKL}$-functions, the criteria on $\mathrm{GKL}$-FTS and event-$\mathrm{GKL}$-FTS are derived for IDS. And with the help of the decompositions of $\mathrm{GKL}$-FTS, the settling time of the $\mathrm{GKL}$-FTS is effectively calculated. Moreover, two types of specific $\mathrm{GKL}$-FTS with fixed settling time, i.e., $\mathrm{GKL}$-FTS via resetting state to zero, and event-$\mathrm{GKL}$-FTS via Zeno behaviour, are provided. And four examples with numerical simulations are presented to demonstrate the effectiveness of the results. It is shown that the $\mathrm{GKL}$-FTS criteria are less conservative in relaxing the FTS conditions of IDS in the literature. And specific effects of impulses on FTS are given in these $\mathrm{GKL}$-FTS criteria, including that an unstable continuous system may obtain FTS under a finite number of impulses.