On the admissibility of retarded delay systems

TL;DR

This paper analyzes the admissibility of control operators in retarded delay systems modeled as Hilbert space dynamical systems, using perturbation theorems and semigroup theory to establish well-posedness.

Contribution

It provides a novel framework for assessing admissibility in retarded delay systems through the application of the Miyadera--Voigt theorem and Weiss conjecture.

Findings

Retarded delay systems can be represented as well-posed abstract Cauchy problems.

The solution semigroup is initially log-concave and bounded.

The approach links delay system admissibility to semigroup perturbation theory.

Abstract

We investigate a Hilbert space dynamical system of the form $\dot{z}(t)=Az(t)+A_1z(t-\tau)+Bu(t)$, where $A$ generates a semigroup of contractions and $A_1$ is a bounded operator, in order to determine whether the operator $B$ is admissible. Our approach is based on the Miyadera--Voigt perturbation theorem and the Weiss conjecture on admissibility of control operators for contraction semigroups. We demonstrate that the retarded delay system can be represented as a well-posed abstract Cauchy problem with a solution formed by an initially log-concave bounded semigroup.