A note on the representation of delay-differential equations as time-delay feedback systems

Stimulated by the monodromy operator approach to time-delay systems (TDSs) developed recently, this paper studies the conversion problems of delay-differential equations (DDEs) into the representation as time-delay feedback systems. We give two conversion methods including the conversion of initial conditions, where we show that each of them corresponds to a different definition for the solutions of DDEs, in general. The study is actually closely related to the subtle behaviors of the solutions of DDEs under discontinuous initial conditions, and simple examples illustrating such subtleties as well as the validity of the conversion methods are also provided.