On linearly perturbed linear systems
type: Item
instance of: scholarly article
title: On linearly perturbed linear systems
zbMATH Open document ID: 0595.34060
DOI: 10.1016/0022-247X(85)90243-4
published in: Journal of Mathematical Analysis and Applications
publication date: 1985-01-01T00:00:00Z
review text: The author studies linear systems of differential or difference equations \(x'(t)=A(t)x(t),\) and the linearly perturbed system \(y'(t)=(A(t)+B(t))y(t),\) where the perturbation matrix B(t) is bounded in the operator norm. The main results express the growth constants of the perturbed equation in terms of the growth constants of the unperturbed equation and a bound on the matrix B(t). Assuming that solutions of the unperturbed equation satisfy an estimate of the form \(\| x(t)\| \leq \gamma \| x(s)\| e^{\alpha (t-s)},\) it is shown that solutions of the perturbed equation satisfy the estimate \(\| y(t)\| \leq \gamma e^{(\alpha +\beta \gamma)(t-s)},\) where \(\beta\) is an upper bound for \(\| B\|\) on the given interval, which may be unbounded. The results obtained improve those of Vidyasagar, Boyarsky, and Vannelli in the following aspects: First, it is only assumed that B is bounded, whereas previous work assumed that B(t)\(\to 0\) as \(t\to \infty\). Second a more precise estimate on the rate of growth of solutions is obtained, and finally the methods are elementary. The proof is by an application of Gronwall's inequality, and both the differential and difference equation cases are treated similarly.
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34D05
zbMATH DE Number: 3957705
zbMATH Keywords: first order differential equation
zbMATH Keywords: growth constants
zbMATH Keywords: Gronwall's inequality
author: Bernd Aulbach
reviewed by: Allan L. Edelson
MaRDI profile type: Publication
cites work: Continuous and discrete dynamics near manifolds of equilibria
cites work: Q5822590
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description: scientific article; zbMATH DE number 3957705