The central exponent and asymptotic stability of linear differential algebraic equations of index~1
type: Item
instance of: scholarly article
title: The central exponent and asymptotic stability of linear differential algebraic equations of index~1
zbMATH Open document ID: 1126.34004
author: Hoang Nam
published in: Vietnam Journal of Mathematics
publication date: 2007-01-31T00:00:00Z
review text: The author considers perturbed time-varying linear differential algebraic equations (DAEs) of the form \[ A(t)\dot{x}(t) + B(t)x(t) + f(t,x(t))=0,\quad t\geq 0, \] where \(A\), \(B\) are bounded and continuous matrix-valued functions and \(f\) is a nonlinear perturbation. Throughout the paper it is assumed that the DAE has index one which -- roughly speaking -- means that the algebraic conditions can be solved and the DAE can be reduced to an ordinary differential equation (ODE). First, the central exponent of the unperturbed DAE (i.e.\ \(f\equiv 0\)) is defined. The central exponent can be viewed as an exponential growth rate of the solutions of the unperturbed DAE. It is then shown that the central exponent of the unperturbed DAE is equal to the central exponent of the inherent ODE. Finally, the influence of the perturbation on the central exponent of the DAE is studied and sufficient conditions for the perturbation \(f\) are given which ensure that the solution of the perturbed DAE is exponentially stable. The paper is well structured and nicely written.
reviewed by: Stephan Trenn
Mathematics Subject Classification ID: 34A09
Mathematics Subject Classification ID: 34C11
Mathematics Subject Classification ID: 34E10
Mathematics Subject Classification ID: 34D08
zbMATH DE Number: 5120080
zbMATH Keywords: time-varying linear DAE
zbMATH Keywords: index 1
zbMATH Keywords: perturbations
zbMATH Keywords: asymptotic stability
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5120080