A stability criterion for a linear Volterra equation of convolution type
type: Item
instance of: scholarly article
title: A stability criterion for a linear Volterra equation of convolution type
zbMATH Open document ID: 0722.45004
DOI: 10.1007/BF01950942
author: J. Terjéki
published in: Acta Mathematica Hungarica
publication date: 1990-01-01T00:00:00Z
review text: The author studies the stability of the equation \(x'(t)=Ax(t)+\int^{t}_{0}B(t-s)x(s)ds,\quad t\geq 0,\) in \({\mathbb{C}}^ n\). A necessary condition for stability (and asymptotic stability) is that the characteristic equation \(\det(zI-A-\hat B(z))=0\) should have no roots with positive real part. A sufficient condition for stability is that in addition there are at most finitely many roots on the imaginary axis and that they are all simple. Here the author assumes that \(\int^{\infty}_{0}s^ 2\| B(s)\| ds<\infty\) and then shows that in this case the zero solution is stable if and only if the resolvent is bounded. In the main result the author gives a condition on the roots of the characteristic equation on the imaginary axis that is necessary and sufficient for the boundedness of the resolvent.
reviewed by: Gustaf Gripenberg
Mathematics Subject Classification ID: 45J05
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 45M10
Mathematics Subject Classification ID: 45M05
zbMATH DE Number: 4190409
zbMATH Keywords: linear Volterra equation of convolution type
zbMATH Keywords: stability
zbMATH Keywords: asymptotic stability
zbMATH Keywords: resolvent
zbMATH Keywords: characteristic equation
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4190409