Asymptotic stability for time-variant systems and observability: Uniform and nonuniform criteria
type: Item
instance of: scholarly article
title: Asymptotic stability for time-variant systems and observability: Uniform and nonuniform criteria
zbMATH Open document ID: 0899.93030
DOI: 10.1007/BF02741883
published in: MCSS. Mathematics of Control, Signals, and Systems
publication date: 1998-11-17T00:00:00Z
review text: The paper deals with stability and asymptotic stability of time-varying systems of ordinary differential equations \[ \dot x= f(x,t). \] It is well known that the origin is uniformly stable if there is a Lyapunov function \(V\) which is positive definite, decrescent, and whose derivative \(\dot V\) is negative semidefinite. The authors are interested in conditions which guarantee asymptotic stability, but which are weaker than the usual negative definiteness assumption about \(\dot V\). It is shown that this problem can be reinterpreted as an observability problem, the observation map \(h\) being related to \(V\) by \(| h(x,t)|^2= -\dot V(x, t)\). The observability achievements require positiveness of the integrals \(\int^{t+ T}_t| h|^2\) for some interval length \(T\). For certain classes of systems, by this way it is possible to reduce the stability analysis to systems of simpler form. Some illustrative examples are discussed in the last section.
Mathematics Subject Classification ID: 93D20
Mathematics Subject Classification ID: 93C99
Mathematics Subject Classification ID: 93B07
Mathematics Subject Classification ID: 34D08
zbMATH DE Number: 1161158
zbMATH Keywords: asymptotic stability
zbMATH Keywords: time-varying systems
zbMATH Keywords: observability
author: D. Aeyels
author: Rodolphe Sepulchre
author: Joan Peuteman
reviewed by: Andrea Bacciotti
MaRDI profile type: Publication
cites work: Q4856491
cites work: Asymptotic stability of nonautonomous systems by Liapunov's direct method
cites work: Q4846575
cites work: Exponential stability of linear equations arising in adaptive identification
cites work: New Results in Linear System Stability
cites work: Uniform asymptotic stability via the limiting equations
cites work: Stability, observability and invariance
cites work: Global stability properties by means of limiting equations
cites work: Q3281164
cites work: Q5574288
cites work: Q3849137
cites work: Nonlinear systems
cites work: Asymptotic Stability of Systems: Results Involving the System Topology
cites work: On the Uniform Asymptotic Stability of Certain Linear Nonautonomous Differential Equations
cites work: Persistent excitation in adaptive systems
cites work: Stability theory by Liapunov's direct method
cites work: Q5627959
full work available at URL: https://doi.org/10.1007/bf02741883
OpenAlex ID: W2152065011
description: scientific article; zbMATH DE number 1161158