Analysis and control of integro-differential Volterra equations with delays

The purpose of this paper is to introduce a semigroup approach to linear integro-differential systems with delays in state, control and observation parts. On the one hand, we use product spaces to reformulate state-delay integro-differential equations to a standard Cauchy problem and then use a perturbation technique (feedback) to prove the well-posendess of the problem, a new variation of constants formula for the solution as well as some spectral properties. On the other hand, we use the obtained results to prove that integro-differential systems with delays in state, control and observation parts form a subclass of distributed infinite-dimensional regular linear systems in the Salamon-Weiss sense.