Characteristic matrices for linear periodic delay differential equations

TL;DR

This paper analyzes the limitations of existing characteristic matrices for linear periodic delay differential equations and proposes a modified construction that mitigates pole-related issues to improve stability analysis.

Contribution

It identifies pole-related issues in existing characteristic matrices and introduces a generalized construction that reduces pole effects near the origin.

Findings

Original matrices can have problematic poles affecting stability analysis

Modified matrices push poles into a small neighborhood of the origin

Enhanced matrices improve the reliability of stability determination

Abstract

Szalai et al. (SIAM J. on Sci. Comp. 28(4), 2006) gave a general construction for characteristic matrices for systems of linear delay-differential equations with periodic coefficients. First, we show that matrices constructed in this way can have a discrete set of poles in the complex plane, which may possibly obstruct their use when determining the stability of the linear system. Then we modify and generalize the original construction such that the poles get pushed into a small neighborhood of the origin of the complex plane.