Exponential convergence and instability of solutions of uncertain dynamical systems
type: Item
instance of: scholarly article
title: Exponential convergence and instability of solutions of uncertain dynamical systems
zbMATH Open document ID: 1120.34037
author: Yu. A. Martynyuk-Chernienko
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2007-03-09T00:00:00Z
review text: A system of ordinary differential equations \[ \dot x= f(t,x,\alpha),\quad x\in\mathbb{R}^n \] with initial conditions \(x(t_0)= x_0\) is considered in this paper. The parameter \(\alpha\) represents deterministic uncertainties; it is assumed to belong to a compact set \({\mathcal S}\subset\mathbb{R}^d\). Let \(r(\alpha)> 0\) be a function such that \[ \lim_{\|\alpha\|\to 0} r(\alpha)= r_0\quad\text{and}\quad \lim_{\|\alpha\|\to\infty} r(\alpha)= +\infty. \] The aim of the paper is to provide conditions for either exponential stability or instability of the moving set \(A(r)= \{x\in\mathbb{R}^n:\| x\|= r(\alpha)\}\) for each \(\alpha\in{\mathcal S}\). The approach is based on the method of matrix-valued Lyapunov functions \[ V(t,x,\eta)= \eta^T U(t,x)\eta, \] where \(U\) is a matrix-valued continuously differentiable function and \(\eta\) is a vector. The results are illustrated by examples.
reviewed by: Andrea Bacciotti
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 93C41
zbMATH DE Number: 5132538
zbMATH Keywords: uncertain system
zbMATH Keywords: matrix-valued Lyapunov function
MaRDI profile type: Publication
full work available at URL: j.na.2006.01.018
OpenAlex ID: W2052413798
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DOI: 10.1016/J.NA.2006.01.018
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description: scientific article; zbMATH DE number 5132538