Lipschitz stability of nonlinear systems of differential equations
type: Item
instance of: scholarly article
title: Lipschitz stability of nonlinear systems of differential equations
zbMATH Open document ID: 0595.34054
DOI: 10.1016/0022-247X(86)90325-2
author: Fozi Dannan
author: Saber Elaydi
published in: Journal of Mathematical Analysis and Applications
publication date: 1986-01-01T00:00:00Z
review text: The authors define a new notion of stability - uniform Lipschitz stability (ULS) for nonlinear systems of differential equations (1) \(\dot x=f(t,x)\), where \(f\in C[J\times R^ n,R^ n]\) and \(f_ x=\partial f/\partial x\) exists and is continuous on \(J\times R^ n\), \(J=[t_ 0,\infty)\), \(t_ 0\geq 0\), \(f(t,0)=0\), and \(x(t,t_ 0,x_ 0)\) is the solution of (1) with \(x(t_ 0,t_ 0,x_ 0)=x_ 0\). Definition. The zero solution of (1) is said to be (ULS) if there exists \(M>0\) and \(\delta >0\) such that \(| x(t,t_ 0,x_ 0)| \leq M| x_ 0|,\) whenever \(| x_ 0| \leq \delta\) and \(t\geq t_ 0\geq 0\). Several criteria for the (ULS) are obtained. It is shown by examples that the (ULS) coincides with the uniform stability in the linear case, which means that the (ULS) is a nonlinear phenomenon. The relationship between (ULS) and various types of stability notions is illustrated by a diagram.
Mathematics Subject Classification ID: 34D10
zbMATH DE Number: 3957699
zbMATH Keywords: variational systems
zbMATH Keywords: first order differential equations
zbMATH Keywords: uniform Lipschitz stability
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3957699