Kernel-representations and realizations of delay-differential systems

We investigate linear time-invariant delay-differential systems within the behavioral approach. A suitable ring of operators including polynomial delay differential operators with commensurate point-delays but also some specific distributed delay operators is introduced. Within this class of operators it is shown that ARMA systems do always admit a kernel representation. Furthermore, we characterize those operator matrices, which are realizable as a first-order system. In this case, the polynomial model of Fuhrmann (1981) is shown to be a realization in the behavioral sense