Insights into the multiplicity-induced-dominancy for scalar delay-differential equations with two delays

TL;DR

This paper investigates the multiplicity-induced-dominancy (MID) property in scalar delay-differential equations with two delays, extending the understanding from single-delay systems and proposing a new technique based on crossing imaginary roots.

Contribution

It extends the MID property analysis to two-delay systems and introduces a crossing imaginary roots method to establish MID in this context.

Findings

MID property holds for the studied two-delay scalar delay-differential equation.

The crossing imaginary roots technique effectively establishes MID where the argument principle faces limitations.

The paper provides insights into the spectral behavior of multi-delay systems.

Abstract

It has been observed in recent works that, for several classes of linear time-invariant time-delay systems of retarded or neutral type with a single delay, if a root of its characteristic equation attains its maximal multiplicity, then this root is the rightmost spectral value, and hence it determines the exponential behavior of the system, a property usually referred to as multiplicity-induced-dominancy (MID). In this paper, we investigate the MID property for one of the simplest cases of systems with two delays, a scalar delay-differential equation of first order with two delayed terms of order zero. We discuss the standard approach based on the argument principle for establishing the MID property for single-delay systems and some of its limitations in the case of our simple system with two delays, before proposing a technique based on crossing imaginary roots that allows to conclude that the MID property holds in our setting.