Necessary conditions of asymptotic stability for unilateral dynamical systems
type: Item
instance of: scholarly article
title: Necessary conditions of asymptotic stability for unilateral dynamical systems
zbMATH Open document ID: 1068.49004
author: Daniel Goeleven
author: B. Brogliato
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2005-06-13T00:00:00Z
full work available at URL: RR-5171.pdf
review text: The authors present necessary conditions of asymptotic stability of the trivial stationary solution for a general class of unilateral dynamical systems involving the subdifferential \(\partial \varphi\) of a proper, convex, lower semicontinuous function \(\varphi:R^n\to\;R\cup\{+\infty\}\) and a nonlinear term expressed by means of a continuous operator \(F\) on \(R^n\). Precisely, defining the mapping \(\Lambda(x)=x-P_\varphi(x-F(x))\), where \(P_\varphi=(id+\partial\varphi)^{-1}\), it is shown that if \(O\) is an isolated zero of \(\Lambda\) and is asymptotically stable, then there exists \(\rho_0>0\) such that \(\deg(\Lambda,B_\rho,O)=1\), \(\forall \rho\in (0,\rho_0]\). Here \(\deg(\Lambda,B_\rho,O)\) denotes the Brouwer degree of \(\Lambda\) with respect to the open ball \(B_\rho=\{x\in R^n:\;\| x\| <\rho\}\) and \(O\). A special attention is payed to the situation where \(\varphi\) is the indicator function of a closed, convex set, and in particular to the complementarity dynamical systems. Several examples and applications are discussed.
Mathematics Subject Classification ID: 49J40
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 35K85
zbMATH DE Number: 2175669
zbMATH Keywords: variational inequalities
zbMATH Keywords: differential inclusions
zbMATH Keywords: topological degree
zbMATH Keywords: stability theory for unilateral dynamical systems
reviewed by: Dumitru Motreanu
MaRDI profile type: Publication
OpenAlex ID: W2121096840
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DOI: 10.1016/J.NA.2005.01.037
description: scientific article; zbMATH DE number 2175669