Stability of perturbed \(n\)-dimensional Volterra differential equations
type: Item
instance of: scholarly article
title: Stability of perturbed \(n\)-dimensional Volterra differential equations
zbMATH Open document ID: 1213.45010
author: Xiao Chang
author: Rui-Qi Wang
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2011-03-09T00:00:00Z
review text: It is shown that the zero solution of the perturbed Volterra integrodifferential equation \[ \dot x(t) = A(t) x(t) + \int_0^t C(t,s) x(s)ds + f(t,x(t)) \] with \(f(t,0) = 0\) is asymptotically stable if there exists a non-negative function \(b(t)\) and a positive definite matrix \(B\) such that \(\| f(t,x(t))\|\leq b(t)\| x(t)\|\) and \[ {1 \over 2} \lambda_M^*(t) \lambda_m^{1/2}(B) + \lambda_M^{1/2}(B) \left( \int_0^\infty \left\| C(u,t) \right\| du + b(t) \right) \leq 0, \] where \(\lambda_M^*\) is the biggest eigenvalue of \(BA(t) + A(t)^TB\), and \(\lambda_m(B)\) and \(\lambda_M(B)\) are the smallest and largest eigenvalues of \(B\), respectively. Uniform stability is established under two different sets of additional assumptions.
Mathematics Subject Classification ID: 45M10
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 45G10
zbMATH DE Number: 5862795
zbMATH Keywords: uniform stability
zbMATH Keywords: asymptotic stability
zbMATH Keywords: perturbed Volterra integrodifferential equation
reviewed by: Olof Staffans
MaRDI profile type: Publication
full work available at URL: j.na.2010.10.038
OpenAlex ID: W1971693615
cites work: Volterra integral and differential equations
cites work: Q4025612
cites work: Q4514434
cites work: Q5445574
cites work: Q5562844
DOI: 10.1016/J.NA.2010.10.038
description: scientific article; zbMATH DE number 5862795