Asymptotic properties of $C_0$-semigroups under perturbations

TL;DR

This paper investigates how the long-term behavior of $C_0$-semigroups is affected by Staffans-Weiss perturbations of their generators, with a focus on robustness and applications to neutral semigroups.

Contribution

It extends the understanding of the stability of $C_0$-semigroups under specific perturbations and applies results to neutral semigroup asymptotics.

Findings

Robustness of asymptotic properties under Staffans-Weiss perturbations

Characterization of perturbed semigroup behavior

Application to neutral semigroup asymptotics

Abstract

For a given $C_0$-semigroup $(T(t))_{t\geq 0}$ we consider Staffans-Weiss perturbations $(B,C)$ of its generator as studied in [1] and investigate the robustness of asymptotic properties of the perturbed $C_0$-semigroup $(T_{BC}(t))_{t\geq 0}$. As a concrete application we study the asymptotic behavior of a neutral semigroup.