Positivity of fundamental matrix and exponential stability of delay differential system
type: Item
instance of: scholarly article
title: Positivity of fundamental matrix and exponential stability of delay differential system
zbMATH Open document ID: 1470.34165
DOI: 10.1155/2014/490816
author: Roman B. Shklyar
author: Mikhail Gitman
author: Valery Stolbov
author: A. Domoshnitsky
published in: Abstract and Applied Analysis
publication date: 2019-02-14T00:00:00Z
review text: Summary: The classical Wazewski theorem established that nonpositivity of all nondiagonal elements \(p_{i j}\) (\(i \neq j\), \(i, j = 1, \ldots, n\)) is necessary and sufficient for nonnegativity of the fundamental (Cauchy) matrix and consequently for applicability of the Chaplygin approach of approximate integration for system of linear ordinary differential equations \(x_i' \left(t\right) + \sum_{j = 1}^n p_{i j} \left(t\right) x_j \left(t\right) = f_i \left(t\right)\), \( i = 1, \ldots, n \). Results on nonnegativity of the Cauchy matrix for system of delay differential equations \(x_i' \left(t\right) + \sum_{j = 1}^n p_{i j} \left(t\right) x_j \left(h_{i j} \left(t\right)\right) = f_i \left(t\right)\), \( i = 1, \ldots, n\), which were based on nonpositivity of all diagonal elements, were presented in the previous works. Then examples, which demonstrated that nonpositivity of nondiagonal coefficients \(p_{i j}\) is not necessary for systems of delay equations, were found. In this paper first sufficient results about nonnegativity of the Cauchy matrix of the delay system without this assumption are proven. A necessary condition of nonnegativity of the Cauchy matrix is proposed. On the basis of these results on nonnegativity of the Cauchy matrix, necessary and sufficient conditions of the exponential stability of the delay system are obtained.
Mathematics Subject Classification ID: 34K06
Mathematics Subject Classification ID: 34K20
zbMATH DE Number: 7022478
Wikidata QID: Q59038717
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2014/490816
OpenAlex ID: W2102002810
cites work: Q5810108
cites work: Q5569338
cites work: Q4470325
cites work: Q4332950
cites work: Q5801485
cites work: Q3469433
cites work: On boundary value problems for systems of linear functional differential equations
cites work: Q3767640
cites work: Nonnegativity of the Cauchy matrix and exponential stability of a neutral type system of functional differential equations
cites work: Interaction between oscillations and global asymptotic stability in delay differential equations
cites work: Delay independent stability for additive neural networks
cites work: Diagonal dominance and harmless off-diagonal delays
cites work: Nonoscillation Theory of Functional Differential Equations with Applications
cites work: Q3378910
cites work: Q3348347
cites work: ON POSITIVITY OF SEVERAL COMPONENTS OF SOLUTION VECTOR FOR SYSTEMS OF LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS
cites work: About maximum principles for one of the components of solution vector and stability for systems of linear delay differential equations
cites work: On Nonoscillation of Systems of Delay Equations
cites work: Q3606283
description: scientific article; zbMATH DE number 7022478