On stability of integral manifolds of dynamic systems
type: Item
instance of: scholarly article
title: On stability of integral manifolds of dynamic systems
zbMATH Open document ID: 0817.93061
author: N. A. Bobylev
author: Vladimir N. Burkov
author: A. Yu. Zalozhnev
published in: Automation and Remote Control
publication date: 1995-02-20T00:00:00Z
review text: Lyapunov's direct method for local asymptotic stability of a non-compact integral manifold \(M\) of the system: \(\dot x= f(x)\) with \(x\in {\mathfrak R}^ n\), is investigated. If locally, there exists a \(C^ 1\) function \(x\mapsto V(x)\) which vanishes on \(M\), which is bounded below and above by functions monotonically increasing from zero with the distance from \(x\) to \(M\) and which satisfies: \[ \dot V< -\alpha(d(x, M), | x|)| f(x)|, \] where \(\alpha\), \(C^ 0\) and positive, satisfies (in order to handle the non-compacity of \(M\)), \(\int^ \infty_ 0 \inf_{u\geq w}(\alpha(u, v))dv= \infty\) for any small positive \(w\), then the distance from a solution to \(M\) tends to zero for large times and appropriate initial conditions. The result is adapted and applied to a model of a consumer market which assumes that individual productivity is a function of the value of money only. The manifold of equilibria relating wages and price index is asymptotically stable.
Mathematics Subject Classification ID: 93D20
Mathematics Subject Classification ID: 93D30
Mathematics Subject Classification ID: 91B26
Mathematics Subject Classification ID: 91B74
zbMATH DE Number: 710810
zbMATH Keywords: local asymptotic stability
zbMATH Keywords: non-compact
zbMATH Keywords: time-invariant
reviewed by: A. Akutowicz
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 710810