A behavioral approach to delay-differential systems

We will study linear time-invariant delay-differential systems from the behavioral point of view as it was introduced for dynamical systems by Willems [Dynam. Report., 2 (1989), pp. 171-269]. A ring H which lies between ℝ[s, z, z^-1] and ℝ(s)[z, z^-1] will be presented, whose elements can be interpreted as a generalized version of delay-differential operators on C^∞(ℝ, ℝ). In this framework, a behavior is the kernel of such an operator. Using the ring H, an algebraic characterization of inclusion, respectively, equality of the behaviors under consideration, is given. Finally, controllability of the behaviors is characterized in terms of the rank of the associated matrices. In the case of time-delay state-space systems this criterion becomes the known Hautus criterion for spectral controllability.