A note on mutual conversion between linear time-invariant delay-differential equations and time-delay feedback systems

A new approach to the stability analysis of time-delay systems (TDSs) called the monodromy operator approach has been proposed recently. The underlying assumption of such an approach is that a TDS is represented as a time-delay feedback system. Hence to apply this approach to the analysis of delay-differential equations (DDEs), one needs to convert them into such feedback systems. This paper studies the mutual conversion problems between linear time-invariant DDEs with commensurate delays and time-delay feedback systems. We consider two types of representations in each direction of conversion, and establish that each of the representations is related to a different definition of the solutions to DDEs, in general. This paper is actually closely related to the subtle behavior of the solutions of DDEs under discontinuous initial conditions.