Some kinds of Lyapunov functional in stability theory of RFDE
type: Item
instance of: scholarly article
title: Some kinds of Lyapunov functional in stability theory of RFDE
zbMATH Open document ID: 0837.34077
DOI: 10.1007/BF02013157
author: Xuetang Du
published in: Acta Mathematicae Applicatae Sinica. English Series
publication date: 1995-09-26T00:00:00Z
review text: By means of Lyapunov functionals the uniform asymptotic stability of the zero solution of one of the following types of retarded functional differential equations is studied: \[ \dot x(t) = A_0 x(t) + \sum^N_{k = 1} A_k x(t - T_k), \tag{1} \] \[ \dot x(t) = A_0 x(t) + Bx \bigl( t - \tau (t) \bigr) + \int^t_{t - \tau (t)} C(t,s) x(s) ds, \tag{2} \] \[ \dot x(t) = - f \bigl( t,x(t) \bigr) + g \bigl( ,x(t - \tau) \bigr), \tag{3} \] \[ \dot x(t) = - f \bigl( t,x(t) \bigr) + g \bigl( t,x (t - \tau) \bigr) + \int^t_{t - \tau (t)} h \bigl( t,s,x(s) \bigr) ds. \tag{4} \] In these equations \(x \in R^n\) and in particular (1) can be written as \[ \dot x(t) = \sum^n_{j = 1} a^{(0)}_{ij} x_j(t) + \sum^N_{k = 1} \sum^n_{j = 1} a_{ij}^{(k)} x_j (t - T_k), \quad i = 1,2, \dots, n. \tag{5} \] Some easily checked sufficient conditions for the uniform asymptotic stability of the zero solution are obtained. The following Theorem being one of these results gives an idea of what type they are: Theorem. If there exist positive constants \(\alpha_k\) \((k = 1,2, \dots, N)\), \(\mu, \nu\) and \(d_i\) \((i = 1,2, \dots, n)\) satisfying the inequalities \[ d_i a_{ii}^{(0)} + \sum^n_{j = 1,j \neq j} d_j |a^{(0)}_{ji} |, \quad \mu - \sum^N_{k = 1} \alpha_k \geq \nu, \] \[ \alpha_k - \beta \max_{1 \leq j \leq n} \sum^n_{i = 1} |a_{ij}^{(k)} |\geq 0 \quad \left( \text{here } \beta = \max_{1 \leq j \leq n} \{d_j\} \right) \] for \(i = 1,2, \dots, n\) and \(k = 1,2, \dots, N\), then the zero solution of (5) is uniformly asymptotically stable.
reviewed by: Ivan Ginchev
Mathematics Subject Classification ID: 34K20
zbMATH DE Number: 801091
zbMATH Keywords: uniform asymptotic stability
zbMATH Keywords: retarded functional differential equations
MaRDI profile type: Publication
cites work: On the Lyapunov functional in the theory of stability of systems with time-lag
cites work: Volterra integral and differential equations
cites work: Stability and periodic solutions of ordinary and functional differential equations
cites work: Theory of functional differential equations. 2nd ed
cites work: Q3820977
description: scientific article; zbMATH DE number 801091