A study on the spectrum of monodromy operator for a time-delay system

Abstract

This paper studies the spectral properties of mon-odromy operators, which play an important role in stability analysis of linear time-invariant time-delay feedback systems. The paper is motivated by the fact that this operator can actually be defined naturally on four spaces, where the difference stems from different choices for the function space on which the infinite-dimensional state of such a time-delay system is assumed to take its value. It is first shown that the spectrum of the monodromy operator is independent of the spaces on which it is defined. It is further shown that the operator spectrum is continuous at monodromy operators, which is a crucial fundamental fact in justifying the spectrum computation of the monodromy operator through its approximation by any sort of tractable operators.