Behavioral Controllability of Delay-Differential Systems

In this paper we will prove that the system described by the delay-differential equation $R(d/dt,\Delta)w=0$ (with $\Delta$ the unit delay operator) is controllable if and only if the rank of $R(\lambda,e^{-\lambda})$ is constant for all $\lambda \in {\bkC}$. This condition is compared with the existing results obtained both by the analytic approach and by the algebraic approach to delay-differential systems.