Asymptotic Behavior of Systems involving Delays: Preliminary Results

TL;DR

This paper explores the asymptotic behavior of systems with delays by analyzing eigensolutions of limiting and perturbed functional differential equations, providing insights into stability and delay effects.

Contribution

It introduces a Perron type theorem for comparing eigensolutions of delayed differential equations and applies it to stability analysis of perturbed systems.

Findings

Established a relation between eigensolutions of limiting and perturbed equations

Provided criteria for global stability based on characteristic zeros

Extended analysis to systems with point and distributed delays

Abstract

This paper investigates the relations between the particular eigensolutions of a limiting functional differential equation of any order, which is the nominal (unperturbed) linear autonomous differential equations, and the associate ones of the corresponding perturbed functional differential equation. Both differential equations involve point and distributed delayed dynamics. The proofs are based on a Perron type theorem for functional equations so that the comparison is governed by the real part of a dominant zero of the characteristic equation of the nominal differential equation. The obtained results are also applied to investigate the global stability of the perturbed equation based on that of its corresponding limiting equation.