Optimal regularity of robustness for parameterized perturbations
type: Item
instance of: scholarly article
title: Optimal regularity of robustness for parameterized perturbations
zbMATH Open document ID: 1209.34071
author: Luís Barreira
author: Claudia Valls
published in: Bulletin des Sciences Mathématiques
publication date: 2010-10-19T00:00:00Z
review text: The authors consider a linear differential equation of the form \[ v'= [A(t)+ B(t,\lambda)]v\tag{1} \] in a Banach space, where \(A\) and \(B\) are continuous functions and \(\lambda\) is a parameter in some open subset of a Banach space. By means of a function \(\rho(t)\) a new dichotomy for (1) is introduced. It has the form \[ \begin{aligned} \| T(t,s)P(s)\|&\leq De^{a(\rho(t)- \rho(s))+ \varepsilon|\rho(s)|},\\ \| T(t,s)^{-1}Q(t)\|&\leq De^{-b(\rho(s)- \rho(t))+\varepsilon|\rho(t)|}, \end{aligned} \] for \(t\geq s\). Here, \(T(t,s)\) is the evolution operator, \(T(t,s)P(s)= P(t) T(t,s)\) for \(t,s\in J\), \(Q(t)= \text{id}- P(t)\). A notion of robustness is introduced.
reviewed by: S. Kostadinov
Mathematics Subject Classification ID: 34G10
Mathematics Subject Classification ID: 34D09
Mathematics Subject Classification ID: 34D10
zbMATH DE Number: 5802607
zbMATH Keywords: Banach space
zbMATH Keywords: growth rates
zbMATH Keywords: exponential dichotomy
MaRDI profile type: Publication
full work available at URL: j.bulsci.2009.12.003
OpenAlex ID: W2025720472
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DOI: 10.1016/J.BULSCI.2009.12.003
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description: scientific article; zbMATH DE number 5802607