Stability theorems for nonautonomous functional differential equations by Lyapunov functionals
type: Item
instance of: scholarly article
title: Stability theorems for nonautonomous functional differential equations by Lyapunov functionals
zbMATH Open document ID: 0677.34060
author: László Hatvani
author: T. A. Burton
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1989-01-01T00:00:00Z
review text: We consider a system of functional differential equations (1) \(x'(t)=F(t,x_ t)\) having finite delay h and satisfying \(F(t,0)=0\). The object is to give conditions on Lyapunov functionals to ensure stability without asking that F(t,\(\phi)\) be bounded for \(\phi\) bounded. We begin by surveying seven classical examples in which we note that an \(L^ 2\)- norm of the solution x(t) frequently appears in the derivative of the Lyapunov functional, but that investigators ignore it in favor of a pointwise norm on x(t). We show that the \(L^ 2\)-norm can be much more useful than the pointwise norm. A measurable function \(\eta\) : \(R\to R\) is said to be integrally positive with parameter \(\delta >0\) (\(\eta\in IP(\delta))\) if whenever \(\{t_ i\}\) and \(\{\delta_ i\}\) satisfy \(t_ i+\delta_ i<t_{i+1}\), \(\delta_ i\geq \delta\), then \(\sum^{\infty}_{i=1}\int^{t_ i+\delta_ i}_{t_ i}\eta (t)dt=\infty.\) Let \(W_ i\) denote a nonnegative increasing function with \(W_ i(0)=0\) and let \(\|| \phi \||\) denote the \(L^ 2\)-norm of \(\phi\). The following is a typical result. Theorem. Let \(\eta \in IP(h_ 1)\) with \(0<h_ 1<h\) and suppose there is a continuous functional V(t,\(\phi)\) satisfying \((i)\quad W_ 1(| \phi (0)|)\leq V(t,\phi)\leq W_ 2(| \phi (0)| +W_ 3(\|| \phi \||)\) and \((ii)\quad V'_{(1)}(t,x_ t)\leq -\eta (t)W_ 4(\|| x_ t\||.\) Then \(x=0\) is asymptotically stable. It is interesting to note that this result fails for ODE's.
Mathematics Subject Classification ID: 34K20
Mathematics Subject Classification ID: 34D20
zbMATH DE Number: 4109180
zbMATH Keywords: Lyapunov functionals
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178227868
OpenAlex ID: W2083113110
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DOI: 10.2748/TMJ/1178227868
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description: scientific article; zbMATH DE number 4109180