Analysis and control of integro-differential Volterra equations with delays

We present a novel approach to address integro-differential systems incorporating state, input, and output delays. Our approach leverages product spaces and employs a boundary perturbation technique. Initially, we focus on state-delay equations, wherein we introduce a variation of constants formula for the mild solution. Additionally, we establish spectral properties using a characteristic equation. Subsequently, we extend our analysis to integro-differential systems affected by state, input, and output delays. Notably, we demonstrate the equivalence between a such delay system and a regular free-delay system within the Salamon–Weiss framework. This equivalence sheds valuable insights on the nature of the integro-differential system under consideration.