Stability of Time-Delay Systems

The importance of time delays in control is now well recognized in a wide range of applications (transportation systems, communication networks, teleoperation systems, etc.). However, even if some fundamental results were obtained 40 years ago during the early space explorations (Bellman & Cooke, 1963; Morse, 1976; Kamen, 1978; Manitius & Triggiani, 1978), time-delay systems have actually been the subject of intensive research works since the nineties. This is probably due to the new emerging applications in engineering (such as network controlled systems) combined with new theoretical results, that allowed one to solve some open problems (decoupling problems, stabilization, robustness, etc.) and to give less and less conservative results. Of course applications motivate the need of theory, which in return makes the control applications possible. As a preamble, let us briefly recall that a time-delay system may be described by different models such as functional differential equations (Hale, 1977), infinite dimensional systems over operators (Curtain & Pritchard, 1978), or ring models (Morse, 1976). Tools for each class of systems may then allow one to derive results for time-delay systems.