Asymptotic Behavior of Imaginary Zeros of Linear Systems with Commensurate Delays

This paper addresses the problem of asymptotic stability of linear time-delay systems with commensurate delays. We study the asymptotic behavior of the critical characteristic zeros of such systems on the imaginary axis. This behavior determines whether the imaginary zeros cross from one half plane into another, and hence plays a critical role in determining the stability of a time-delay system. We consider time-delay systems given in both state-space form and as a quasipolynomial. Our results reveal that in the former case the zero asymptotic behavior can be characterized by solving a simple eigenvalue problem, and in the latter case, by computing the derivatives of the quasipolynomial. To perform such an analysis, we make use of an operator perturbation approach