The regular linear systems associated with the shift semigroups and application to control linear systems with delay.

We investigate the infinite dimensional control linear systems with delays in the state and input. We give a new variation of constants formula when the state and control delay operators are unbounded. We prove the existence of mild and classical solutions of such systems. Our approach is based on the theory of abstract and regular linear systems introduced by Salamon (Math. Syst. Theor. 21:147–164, 1989) and Weiss (Isr. J. Math. 65:17–43, 1989). Finally, we apply our abstract framework to an example from population dynamics.