Some examples of linear systems with delays

The controllability notion, well known for finite dimensional systems, has received a wide variety of extensions for systems with delays. An interpretation was recently given to most of the widely used definitions using a module theoretic framework. Since choosing between all of these mathematical notions on theoretical grounds is certainly a difficult task, we try, by inspecting several practical examples, to search for a notion that would appear quite frequently. We show that such a definition exists and seems to be new. We have named it δ-freeness, a typical case of which is, two finite dimensional controllable systems interconnected with a delay. Some examples will be examined, including a wind tunnel model for the control of the Mach number, a two-reflector antenna and a flexible rod with a mass.