A Berger-Wang formula for impulsive linear switched systems
Yacine Chitour, Jamal Daafouz, Ihab Haidar, Paolo Mason, Mario Sigalotti
TL;DR
This work develops a spectral Berger–Wang framework for impulsive linear switched systems by embedding them into a weighted discrete-time switched-system setting. It introduces a precise decomposition of the maximal Lyapunov exponent into a sum of contributions from continuous-time dynamics and impulse-induced switching, and proves a Berger–Wang-type formula that equates the exponential-growth rate with the maximum of a spectral-radius term and a Lyapunov exponent term. The main result yields a spectral characterization of exponential stability: the system is ES if and only if the maximal Lyapunov exponent is negative, and EU if and only if it is positive. The analysis provides continuity results for the maximal Lyapunov exponent under perturbations of the switching data and clarifies when growth is driven by unswitched trajectories, with implications for numerical spectral methods in switched systems.
Abstract
This paper addresses a class of impulsive systems defined by a mix of continuous-time and discrete-time switched linear dynamics. We first analyze a related class of weighted discrete-time switched systems for which we establish a Berger--Wang-type result. An analogous result is then derived for impulsive systems and subsequently used to characterize their exponential stability through a spectral approach, thereby extending existing results in switched-systems theory.