Controllability of systems described by convolutional or delay-differential equations

In this paper controllability properties of linear time-invariant infinite dimensional systems described by delay-differential and by more general convolutional equations are considered. Various controllability notions which have been introduced for this class of systems in Willems' behavioral approach (1989, 1993) and in Fliess' module theoretic approach (1992), are here discussed. A characterization of spectral controllability is given, extending results that are known to hold for difference or differential equations. Two of the most important contribution of these two approaches, i.e. existence of an image representation and flatness, are compared. Finally, it is shown that a theorem, which states the equivalence of spectral controllability and the existence of an image representation, holds true for a class of delay-differential systems, including systems in state-space form. However, this result is false for generic delay-differential or convolutional systems, as an example shows.