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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][148][149][150][151][152][153][154][155][156][157][158][159][160][161][162][163][164] 1989) and Weiss (Isr J Math 65:17-43, 1989). Finally, we apply our abstract framework to an example from population dynamics.

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The aim of this paper is to prove that a class of distributed parameter systems governed by neutral FDEs provides regular linear systems. Employing the well established theory of representation, transfer function and feedback of these later we then give new representations of the state and the output function of the neutral systems.