TY - JOUR

T1 - Positivity of fundamental matrix and exponential stability of delay differential system

AU - Domoshnitsky, Alexander

AU - Shklyar, Roman

AU - Gitman, Mikhail

AU - Stolbov, Valery

N1 - Publisher Copyright:
© 2014 Alexander Domoshnitsky et al.

PY - 2014

Y1 - 2014

N2 - The classical Wazewski theorem established that nonpositivity of all nondiagonal elements p i j (i ≠ j, i, j = 1,., 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 ′ t + j = 1 n p i j t x j t = f i t, i = 1,., n. Results on nonnegativity of the Cauchy matrix for system of delay differential equations x i ′ t + j = 1 n p i j t x j h i j t = f i t, i = 1,., 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.

AB - The classical Wazewski theorem established that nonpositivity of all nondiagonal elements p i j (i ≠ j, i, j = 1,., 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 ′ t + j = 1 n p i j t x j t = f i t, i = 1,., n. Results on nonnegativity of the Cauchy matrix for system of delay differential equations x i ′ t + j = 1 n p i j t x j h i j t = f i t, i = 1,., 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.

UR - http://www.scopus.com/inward/record.url?scp=84934986800&partnerID=8YFLogxK

U2 - 10.1155/2014/490816

DO - 10.1155/2014/490816

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AN - SCOPUS:84934986800

SN - 1085-3375

VL - 2014

JO - Abstract and Applied Analysis

JF - Abstract and Applied Analysis

M1 - 490816

ER -