When will zeros of time-delay systems cross imaginary axis?

A time-delay system may or may not be stable for different periods of delay. When will then a delay system be stable or unstable, and for what ranges of delay? This paper attempts to answer these questions. We show that by finding a set of critical delay values, for which the system´s characteristic quasipolynomial has zeros on the imaginary axis, it is possible to determine its stability in the full range of the delay parameter by characterizing the analytical behaviors of the zeros. This characterization is facilitated by an operator perturbation approach, which is both conceptually attractive and computationally efficient. The entire procedure, which first identifies the critical zeros on the imaginary axis and next determines whether the zeros cross the imaginary axis, requires only solving a generalized eigenvalue problem.