Asymptotic decay rates from the growth properties of Lyapunov functions near singular attractors
type: Item
instance of: scholarly article
title: Asymptotic decay rates from the growth properties of Lyapunov functions near singular attractors
zbMATH Open document ID: 0632.34051
DOI: 10.1016/0022-247X(87)90161-2
author: B. George
published in: Journal of Mathematical Analysis and Applications
publication date: 1987-01-01T00:00:00Z
review text: Asymptotic stability of the origin and estimates on the rate of decay to the zero solution of the autonomous differential equation (1) \(\dot x=G(x)\), \(x\in X\), where X is a real Hilbert space, can be obtained from a Lyapunov function \(F\in C^ 1(X,R)\) such that: \(a\| x\|^ k\leq F(x)\leq b\| x\|^ k\), \(<\nabla F(x),G(x)>\leq -c\| x\|^{k+m}.\) Similar estimates are derived from a somewhat different set of conditions, namely (2) \(F(0)=0\), \(F(x)\geq a\| x\|^ k\), (3) \(\| \nabla F(x)\| \geq bF^{\ell}(x)\), \(<\nabla F(x),G(x)>\leq - c\| \nabla F(x)\|^ m.\) Lyapunov functions of this type arise naturally in systems such as \(\dot x=-H(x)\nabla F(x)\), where H(x) is a bounded self-adjoint positive definite linear operator. Such an equation generalizes the continuous steepest descent equation for F. The author explores circumstances under which (2) implies (3). In particular a detailed study is made in case (i) \(F(0)=0\), \(F(x)>a\| x\|^ 4\); (ii) the quadratic part of F is positive definite on its range i.e. \[ \exists \lambda_ 2>0,\quad \forall y\in (Ker \nabla_ 2F(0))^{\perp},\quad <y,\nabla_ 2F(0)y>\geq \lambda_ 2\| y\|^ 2; \] (iii) \(<\nabla F(x),G(x)>\leq -c\| \nabla F(x)\|^ m\), \(m>4/3\). For such systems, the origin is asymptotically stable and \(\| x(t)\| =O(t^{-1/(3m-4)})\) as \(t\to \infty\). Examples for \(X={\mathbb{R}}\) or \({\mathbb{R}}^ 2\) are discussed. They motivate the theory.
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 4027940
zbMATH Keywords: first order differential equation
zbMATH Keywords: Asymptotic stability
zbMATH Keywords: estimates
zbMATH Keywords: rate of decay
zbMATH Keywords: autonomous differential equation
zbMATH Keywords: Lyapunov function
zbMATH Keywords: Examples
reviewed by: Patrick Habets
MaRDI profile type: Publication
cites work: Convergence Rates for Newton’s Method at Singular Points
cites work: Q5730313
cites work: Q3845518
cites work: Stability theory by Liapunov's direct method
full work available at URL: 0022-247x(87)90161-2
OpenAlex ID: W2170826047
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description: scientific article; zbMATH DE number 4027940