On stability for generalized linear differential equations and applications to impulsive systems

TL;DR

This paper investigates stability concepts for generalized linear differential equations, providing new characterizations and demonstrating their application to impulsive and periodic systems, establishing equivalences among different stability notions.

Contribution

It introduces a comprehensive framework for stability analysis of GLDEs, including new definitions, characterizations, and their equivalence to exponential stability.

Findings

Stability notions characterized via transition matrix bounds

Application to impulsive and periodic systems demonstrated

Equivalence of stability definitions established

Abstract

In this paper, we are interested in investigating notions of stability for generalized linear differential equations (GLDEs). Initially, we propose and revisit several definitions of stability and provide a complete characterisation of them in terms of upper bounds and asymptotic behaviour of the transition matrix. In addition, we illustrate our stability results for GLDEs to linear periodic systems and linear impulsive differential equations. Finally, we prove that the well known definitions of uniform asymptotic stability and variational asymptotic stability are equivalent to the global uniform exponential stability introduced in this article.