Are delay-differential systems generically controllable?

Abstract

In this paper, we study controllability properties of time-invariant linear delay-differential (d-d) systems with a single delay in the pseudo-state. We adopt a topological and geometric point of view and derive a formula for the distance of an approximately controllable system from uncontrollability. We demonstrate by examples that approximately controllable d-d systems may have zero distance from uncontrollability. Thus, the set of approximately controllable d-d systems is not open in the parameter space. To remedy this anomaly, we propose a slight strengthening of the controllability concept and introduce the well-posed property of strict controllability. We show that strictly controllable d-d systems form an open dense subset in the parameter space. Moreover, we show that a d-d system is strictly controllable if and only if it has a positive distance from uncontrollability. Finally, we prove that the d-d systems that are not strictly controllable form a closed set of Lebesgue measure zero and can be represented as the union of a proper algebraic variety and a proper local analytic variety in the parameter space.