Metrization and Liapunov functions. I
type: Item
instance of: scholarly article
title: Metrization and Liapunov functions. I
zbMATH Open document ID: 0548.54028
DOI: 10.1007/BF01956781
published in: Acta Mathematica Hungarica
publication date: 1983-01-01T00:00:00Z
review text: Let (X,d) be a metric space and \(T_ t:X\to X\) (\(t\in {\mathbb{R}})\) be a dynamical system. Let M, \(\emptyset\neq M\subset X\), be an asymptotically stable closed invariant set and A(M) be its region of attraction. It is well known that there exists a continuous function \(V:A(M)\to {\mathbb{R}}\) satisfying the following conditions: \((i)\quad V(x)=0\) (\(x\in M)\) and \(V(x)>0\quad (x\in A(M)\backslash M);\) (ii) \(V(T_ tx)\to 0\) as \(t\to +\infty\) (\(x\in A(M))\); \((iii)\quad V(x)\geq d(x,M)\) (\(x\in A(M))\). Theorem 1. Assume that there is a retraction \(r:A(M)\to M\) such that for any \(\{y_ n\}^{\infty}_{n=1}\subset A(M),\) \(x\in M\) the conditions \(d(y_ n,M)\to 0\) and \(r(y_ n)\to X\) imply \(y_ n\to x\). Then there exists a metric \(\rho\) on X such that \(\rho| M\times M=d| M\times M\) and \(V(x)=\rho (x,M)=\rho (x,r(x))\) for \(x\in A(M)\). It is also shown that an analogous assertion holds for Morse-Smale flows on compact oriented surfaces.
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 54F05
Mathematics Subject Classification ID: 30D99
zbMATH DE Number: 3874198
zbMATH Keywords: monotone Lyapunov function
zbMATH Keywords: asymptotically stable closed invariant set
zbMATH Keywords: region of attraction
zbMATH Keywords: retraction
zbMATH Keywords: Morse-Smale flows
zbMATH Keywords: compact oriented surfaces
author: Barnabas M. Garay
MaRDI profile type: Publication
cites work: Extending a metric
cites work: Q5813401
cites work: Q3932062
cites work: Differential Topology
cites work: Q3226265
cites work: Global Structure of Ordinary Differential Equations in the Plane
cites work: Structural stability on two-dimensional manifolds
cites work: Q5595759
cites work: Differentiable dynamical systems
cites work: Lyapunov functions and isolating blocks
full work available at URL: https://doi.org/10.1007/bf01956781
OpenAlex ID: W4255798556
Recommended article: Metrization and Liapunov functions. V
Recommended article: Metrization and Liapunov functions. II, III
Recommended article: Q3806358
Recommended article: On dynamical systems in metric spaces: Metrics as Liapunov functions
Recommended article: Q4672829
Recommended article: Q3312989
Recommended article: Q3713170
Recommended article: Lyapunov functions and attractors in arbitrary metric spaces
Recommended article: Metrizable extensions of dynamical systems from function algebras
Recommended article: On the construction of Lyapunov functions. I
description: scientific article; zbMATH DE number 3874198