The exact boundaries of the stability domains of linear differential equations with distributed delay
type: Item
instance of: scholarly article
title: The exact boundaries of the stability domains of linear differential equations with distributed delay
zbMATH Open document ID: 1181.34073
DOI: 10.3103/S1066369X08070037
author: V. V. Malygina
published in: Russian Mathematics
publication date: 2009-10-26T00:00:00Z
review text: This paper addresses the following linear differential equation with distributed varying delay \[ \dot{x}(t) + \int_{t-r(t)}^t x(s) ds = f(t), \quad t\in \mathbb{R}_+;\qquad x(\xi)=0, \text{ for } \xi<0. \tag{1} \] Stability criteria similar to criteria introduced by \textit{A.~D.~Myshkis} [Mat. Sbor. 28, No. 3, 641--658 (1951; Zbl 0054.03901)] for equations with concentrated varying delay are studied. The main result is the following theorem. Theorem. Let \(0<\inf_{t\in \mathbb{R}_+} r(t) \leq \sup_{t\in \mathbb{R}_+} r(t) <2\). Then equation (1) is exponentially stable. Stability criteria for some generalization (when integral part of equation (1) has a parameter) of equation (1) are also formulated.
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 34K06
Mathematics Subject Classification ID: 47G20
zbMATH DE Number: 5621363
zbMATH Keywords: functional differential equations
zbMATH Keywords: integral-differential equations
zbMATH Keywords: distributed delay
zbMATH Keywords: stability
reviewed by: Denis Sidorov
MaRDI profile type: Publication
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full work available at URL: s1066369x08070037
OpenAlex ID: W2079826445
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description: scientific article; zbMATH DE number 5621363