Analysis and control of integro-differential Volterra equations with delays

TL;DR

This paper develops a semigroup framework for analyzing linear integro-differential systems with delays, establishing well-posedness, spectral properties, and their classification as distributed infinite-dimensional systems.

Contribution

Introduces a semigroup approach to delay integro-differential equations, providing new well-posedness results, a variation of constants formula, and spectral analysis, linking them to regular linear systems.

Findings

Proved well-posedness of delayed integro-differential systems.

Derived a new variation of constants formula for solutions.

Showed systems form a subclass of distributed infinite-dimensional regular linear systems.

Abstract

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.