Exponential stability and uniform boundedness of solutions for nonautonomous periodic abstract Cauchy problems. An evolution semigroup approach
type: Item
instance of: scholarly article
title: Exponential stability and uniform boundedness of solutions for nonautonomous periodic abstract Cauchy problems. An evolution semigroup approach
zbMATH Open document ID: 1279.47063
author: Constantin Buşe
author: D. Lassoued
author: Thanh Lan Nguyen
author: Olivia Saierli
published in: Integral Equations and Operator Theory
publication date: 2013-04-15T00:00:00Z
review text: Let \(\mathcal{U}=\{U(t,s)\}_{t\geq s}\) be a strongly continuous and \(q\)-periodic evolution family on a Banach space \(X\). The authors show that, if \[ \sup_{\mu\in \mathbb R}\sup_{t\geq \tau\geq 0}\left\|\int^t_\tau e^{i\mu s}U(t,s)x\, ds\right\|<\infty \text{ for all } x\in X, \] then \[ \sup_{\mu\in \mathbb R}\sup_{t\geq 0}\left\|\int^t_\tau e^{-i\mu s}\mathcal{T}(s)f\, ds\right\|<\infty \text{ for all } f\in \mathrm{span}\{\cup_{t\geq 0}\mathcal{A}_t\}, \] where \[ \mathcal{T}(t)f(s):=\left\{ \begin{matrix} U(s,s-t)f(s-t), & s\geq t \\ 0, & s<t, \\ \end{matrix} \right. \;\;t\geq 0, \;s\in \mathbb R, \;f\in \overline{\mathrm{span}}\{\cup_{t\geq 0}\mathcal{A}_t\}, \] and \(\mathcal{A}_t\) is the set of all \(X\)-valued functions \(f\) on \(\mathbb R\) for which there exists a function \(F\in P_q(R,X)\cap AP_1(R,X)\) such that \(F(t)=0\), \(f=F|_{[t,\infty)}\) and \(f(s)=0\) for \(s<t\). Here, \(P_q(\mathbb R,X)\) denotes the \(q\)-periodic, \(X\)-valued, uniformly continuous functions on \(\mathbb R\) and \(AP_1(\mathbb R,X)\) is a certain space of almost periodic functions (cf.\ [\textit{C. Corduneanu}, Almost periodic oscillations and waves. New York, NY: Springer (2009; Zbl 1163.34002)]).
reviewed by: Jin Liang
Mathematics Subject Classification ID: 47D06
Mathematics Subject Classification ID: 47A05
Mathematics Subject Classification ID: 47A30
Mathematics Subject Classification ID: 47A10
Mathematics Subject Classification ID: 35B15
Mathematics Subject Classification ID: 35B10
zbMATH DE Number: 6153933
zbMATH Keywords: periodic evolution families
zbMATH Keywords: uniform exponential stability
zbMATH Keywords: boundedness
zbMATH Keywords: evolution semigroup
zbMATH Keywords: almost periodic functions
MaRDI profile type: Publication
full work available at URL: s00020-012-1993-5
OpenAlex ID: W1964833048
cites work: Q2716079
cites work: Connections between exponential stability and boundedness of solutions of a couple of differential time depending and periodic systems
cites work: Q3320717
cites work: Almost periodicity of parabolic evolution equations with inhomogeneous boundary values
cites work: Q4495411
cites work: Uniform stability of periodic discrete systems in Banach spaces
cites work: Individual exponential stability for evolution families of linear and bounded operators
cites work: Q4258125
cites work: Stability Radius and Internal Versus External Stability in Banach Spaces: An Evolution Semigroup Approach
cites work: Almost Periodic Oscillations and Waves
cites work: Q4272980
cites work: One-Parameter Semigroups for Linear Evolution Equations
cites work: Q3923136
cites work: On a theorem of Gearhart
cites work: Exponential stability of linear systems in Banach spaces
cites work: Q5552090
cites work: Semigroups of linear operators and applications to partial differential equations
cites work: On stability of \(C_0\)-semigroups
cites work: On nonautonomous functional differential equations
cites work: Q4951587
cites work: Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line
cites work: Q3025713
cites work: Q4513479
cites work: Q4407529
cites work: Q4800024
cites work: Individual stability of \(C_0\)-semigroups with uniformly bounded local resolvent
cites work: Asymptotic behavior of 𝐶₀-semigroups in Banach spaces
cites work: Q3799203
cites work: Mathematical control theory: an introduction
DOI: 10.1007/S00020-012-1993-5
Recommended article: A new estimation of the growth bound of a periodic evolution family on Banach spaces
Recommended article: Q3165139
Recommended article: Q5698877
Recommended article: A Theorem of Rolewicz's type in solid function spaces
Recommended article: An inequality concerning the growth bound of an evolution family and the norm of a convolution operator
Recommended article: On strong evolution inequalities for the equations \(u_{tt}= a(t) u_{xx}\)
Recommended article: On uniform N-equistability
Recommended article: Q3818634
Recommended article: Q5409551
Recommended article: Stability problems in nonautonomous linear differential equations in infinite dimensions
description: scientific article; zbMATH DE number 6153933