Upper and lower bounds for the maximal Lyapunov exponent of singularly perturbed linear switching systems

TL;DR

This paper investigates the stability of singularly perturbed linear switching systems by deriving bounds for the maximal Lyapunov exponent, considering the complex interaction between perturbation parameters and switching rates.

Contribution

It introduces a method to characterize auxiliary systems that bound the Lyapunov exponent's asymptotics as the perturbation parameter approaches zero.

Findings

Derived bounds for the maximal Lyapunov exponent.

Provided insights into stability conditions for singularly perturbed systems.

Analyzed the interplay between perturbation size and switching rate.

Abstract

In this paper we consider the problem of determining the stability properties, and in particular assessing the exponential stability, of a singularly perturbed linear switching system. One of the challenges of this problem arises from the intricate interplay between the small parameter of singular perturbation and the rate of switching, as both tend to zero. Our approach consists in characterizing suitable auxiliary linear systems that provide lower and upper bounds for the asymptotics of the maximal Lyapunov exponent of the linear switching system as the parameter of the singular perturbation tends to zero.