On uniform exponential stability of exponentially bounded evolution families
type: Item
instance of: scholarly article
title: On uniform exponential stability of exponentially bounded evolution families
zbMATH Open document ID: 1165.47027
author: A. D. R. Choudary
author: Constantin Buşe
author: Silvestru Sever Dragomir
author: Manuela-Suzy Prajea
published in: Integral Equations and Operator Theory
publication date: 2008-08-21T00:00:00Z
review text: The authors deduce new characterizations for the uniform exponential stability of evolution families in terms of rearrangement Banach function spaces. By applying their methods, the authors obtain stability conditions both for semigroups and for periodic evolution families. Let \(X\) be a Banach space, let \(J \in \{{\mathbb R}, {\mathbb R}_+\}\) and \(\Delta_J := \{(t, s) \in J \times J : t\geq s\}\). One of the main results of the paper states that if \(U=\{U(t, s) : (t, s) \in \Delta_J\}\) is an exponentially bounded evolution family on \(X\) and \(E(J)\) is a rearrangement invariant Banach function space over \(J\) such that \(\lim_{r \to \infty} \|\chi_{[0, r]}(\cdot)\|_{E^\prime(J)} = \infty,\) then the following statements are equivalent: {\parindent7mm \begin{itemize}\item[(i)] the family \(U\) is uniformly exponentially stable; \item[(ii)] there exists a positive constant \(M^\prime(J)\) such that, for each \((s, t) \in \Delta_J\) and each \(x^* \in X^*\), the map \(\chi_{[s, t]}(\cdot)\;\|U(t, \cdot)^*x^*\|\) defines an element of the space \(E^\prime(J)\) and \[ \sup_{(t, s) \in \Delta_J}\|\chi_{[s, t]}(\cdot)\;\|U(t, \cdot)^*x^*\|\;\|_{E^\prime(J)} \leq M^\prime(J)\;\|x^*\|. \] \end{itemize}}
reviewed by: Bogdan Sasu
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 35B35
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 35R20
zbMATH DE Number: 5314951
zbMATH Keywords: rearrangement function space
zbMATH Keywords: evolution families of bounded linear operators
zbMATH Keywords: uniform exponential stability
MaRDI profile type: Publication
full work available at URL: s00020-008-1592-7
OpenAlex ID: W2063620534
DOI: 10.1007/S00020-008-1592-7
description: scientific article; zbMATH DE number 5314951