Exponential stability and partial averaging.
type: Item
instance of: scholarly article
title: Exponential stability and partial averaging.
zbMATH Open document ID: 1035.34040
DOI: 10.1016/S0022-247X(03)00290-7
author: I. Maizurna
author: Götz Grammel
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-08-25T00:00:00Z
review text: The initial value problem for the system \(\dot{x}(t)=\varepsilon f(\varepsilon t,t, x(t))\), \(x(t_{0})=x_{0}\), is considered, where \(\varepsilon >0\) is a small parameter. The corresponding partially averaged system is \(\dot{z}(t)=\varepsilon f^{0}(\varepsilon t,z(t))\), \(z(t_{0})=x_{0}\), with \(f^{0}(s,x)=\lim_{T\rightarrow \infty}\frac{1}{T}\int_{0}^{T}f(s,t,x)\,dt\). Under certain uniformity conditions, it is shown that exponential stability of the averaged system is equivalent to exponential stability of the perturbed system for small values of the perturbation parameter \(\varepsilon\). Explicit estimates on both the approximation of single trajectories and the order of the exponential decay are obtained.
Mathematics Subject Classification ID: 34C29
Mathematics Subject Classification ID: 34D10
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 34E15
zbMATH DE Number: 1970525
zbMATH Keywords: ordinary differential equations
zbMATH Keywords: exponential stability
zbMATH Keywords: averaging
zbMATH Keywords: perturbations
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1970525