On controllability and observability of time delay systems

This paper deals with controllability and observability properties of time delay systems in the state space . In particular, we prove the equivalence of spectral controllability and approximate null-controllability. Moreover, it is shown that the necessary, condition for approximate controllability-obtained recently by Manitius-is also sufficient, and a verifiable and matrix type criterion for -controllability is derived for systems with commensurate delays. Finally, we introduce the dual observability notion of approximate controllability and prove that the control system Σ is exactly null-controllable if and only if the transposed delay system Σ^Tis continuously finally observable.