Novel Controllability Conditions for a Class of Singularly-Perturbed Systems with Small State Delays

A singularly-perturbed linear time-dependent controlled system with small pointwise and distributed delays in the state variable is considered. Two simpler parameter-free systems, the slow and fast ones, can be associated with the original system. It was established in the literature that the Euclidean space controllability of the original system, valid for all sufficiently small values of the parameter of singular perturbations, follows from the controllability properties of the slow and fast systems. It also was established that such a connection between the controllability properties of the original system and the slow and fast systems is correct, in general, only in one direction. Namely, the controllability of the slow and fast systems provides the controllability of the original system, while the controllability of the original system not always yields the controllability of both the slow and fast systems. In this paper, we consider the original system such that the respective fast system is uncontrollable, meaning that the previously established controllability conditions are not applicable to this original system. In this case, novel parameter-free sufficient conditions for the Euclidean space controllability of the original system, robust with respect to the small parameter of singular perturbations, are derived. Illustrative examples are presented.