On \(t_{\infty}\) quasi-similarity of linear systems (*)
type: Item
instance of: scholarly article
title: On \(t_{\infty}\) quasi-similarity of linear systems (*)
zbMATH Open document ID: 0634.34007
DOI: 10.1007/BF01766598
author: Willaim F. Trench
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 1985-01-01T00:00:00Z
review text: Let the linear homogeneous system (1) \(x'=A(t)x\), (t\(\geq a)\) have certain properties in connection with stability and asymptotic behaviour of solutions. The old problem of finding conditions connecting the matrix A and B which imply that the second system (2) \(y'=B(t)y\), (t\(\geq a)\) has the same or related properties, is considered. This problem has been investigated e.g. by Conti who defined \(t_{\infty}\) similarity of systems (1) and (2) and showed that it is an equivalence relation which preserves uniform and strict stability. Here this definition is weakened by imposing less stringent integrability conditions, some in terms of perhaps conditionally convergent improper integrals, on the matrix function relating A and B. Integrability conditions of this kind are rather unusual [see e.g. the author's paper in Proc. Amer. Math. Soc. 87, 263-270 (1983; Zbl 0514.34031)]. The extended relation \(t_{\infty}\) quasi-similarity, is not symmetric or transitive. However, it is shown that if (2) is \(t_{\infty}\) quasi-similar to (1) and (1) is uniformly, uniformly asymptotically, or strictly stable, then so is (2). Results are also given concerning linear asymptotic equilibrium of (2) in the case where (1) is strictly stable or has linear asymptotic equilibrium. (Based on the author's summary.)
reviewed by: J.Vosmanský
Mathematics Subject Classification ID: 34A30
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34D10
zbMATH DE Number: 4032114
zbMATH Keywords: asymptotic equilibrium
zbMATH Keywords: asymptotic behaviour of solutions
zbMATH Keywords: \(t_{\infty }\) similarity
zbMATH Keywords: \(t_{\infty }\) quasi-similarity
MaRDI profile type: Publication
cites work: Q3848373
cites work: Q3228799
cites work: Asymptotic behavior of solutions of linear ordinary differential equations
cites work: Asymptotic integration of a nonhomogeneous differential equation with integrable coefficients
cites work: Q5335550
cites work: Systems of Differential Equations Subject to Mild Integral Conditions
cites work: Asymptotics of Differential Systems with Deviating Arguments
cites work: On a Theorem of Bocher in the Theory of Ordinary Linear Differential Equations
full work available at URL: https://doi.org/10.1007/bf01766598
OpenAlex ID: W2075199545
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description: scientific article; zbMATH DE number 4032114