Delay Equations, the Left-Shift Operator and the Infinite Dimensional Root Locus

The distributed parameter root locus is considered and a relation between simple delay equations and the left-shift operator is developed. This gives a rigorous explanation of the s-plane behaviour of delay systems and shows that the classical root locus starting on the open-loop poles of the system can become bands swept out by connected components of the spectrum of the system operator in the infinite-dimensional case