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Lyapunov-Like Solutions to Stability Problems for Discrete-Time Systems on Asymptotically Contractive Sets

Authors
  • Bouyekhf, R.1
  • Gruyitch, Ly. T.1
  • 1 Ecole Nationale d'Ingénieurs, Laboratoire dAutomatique, Mécatronique, Productique et Systémique, Espace Bartholdi, Belfort Technopôle, Belfort, 90016, France , Belfort
Type
Published Article
Journal
Nonlinear Dynamics
Publisher
Springer-Verlag
Publication Date
Feb 01, 1999
Volume
18
Issue
2
Pages
107–127
Identifiers
DOI: 10.1023/A:1008310800491
Source
Springer Nature
Keywords
License
Yellow

Abstract

This paper presents new criteria for stability properties of discrete-time non-stationary systems. The criteria are based on the concept of asymptotically contractive sets. As a result, general necessary conditions are established for asymptotic stability of the zero equilibrium state, the instantaneous asymptotic stability domain of which can be either time-invariant or time-varying and then possibly asymptotically contractive. It is shown that the classical Lyapunov stability conditions including the invariance principle by LaSalle cannot be applied to the stability test as soon as the system instantaneous domain of asymptotic stability is asymptotically contractive. In order to investigate asymptotic stability of the zero state in such a case novel criteria are established. Under the criteria the total first time difference of a system Lyapunov function may be non-positive only and still can guarantee asymptotic stability of the zero state. The results are illustrated by examples.

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