On controllability and observability of time delay systems

This paper deals with controllability and observability properties of time delay systems in the state space
R^{n} \times L^{p}
. In particular, we prove the equivalence of spectral controllability and approximate null-controllability. Moreover, it is shown that the necessary, condition for approximate
F-
controllability-obtained recently by Manitius-is also sufficient, and a verifiable and matrix type criterion for
F
-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
^T
is continuously finally observable.