Spectrum of Monodromy Operator for a Time-Delay System With Application to Stability Analysis

This note studies the spectral properties of monodromy operators, which play an important role in stability analysis of linear time-invariant time-delay feedback systems. The note 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. This implies that stability of time-delay systems is independent of the underlying function spaces. It is further shown that the operator spectrum is continuous at monodromy operators, which justifies the spectrum computation of the monodromy operator through its approximation by any sort of tractable operators. A numerical study relevant to the theoretical development is provided and a practical implication of our theoretical study is suggested.