Stability of difference equations with impulsive actions at the instants of time dependent on the state vector
type: Item
instance of: scholarly article
title: Stability of difference equations with impulsive actions at the instants of time dependent on the state vector
zbMATH Open document ID: 0933.93062
author: R. Z. Abdullin
published in: Automation and Remote Control
publication date: 1999-06-29T00:00:00Z
review text: A dynamic system with jumps of the state vector is given by the difference equations \[ x(n+1)= F(x(n)),\quad n+1\neq n_i, \] \[ x(n_i)= x(n_i+ 0)= x(n_i- 0)+ \varphi(x(n_i- 0)),\;x(n_0)= x_0, \] where \(x(n)\in S\subset R^k\), \(n\in \{0,1,\dots\}\); \(n_i\), \(i= 0,1,\dots\) are instants of impulsive inputs and \(F\), \(\varphi: S\to R^k\) are some continuous functions. The instants of impulsive inputs are defined by the transitions of solutions through some hypersurfaces in an extended phase space. Under some additional assumptions, the author establishes the conditions for uniform asymptotic stability of the zero solution of linear and nonlinear systems, instability of nonlinear systems, uniform asymptotic stability of the zero solution of nonperturbed systems under impulsive perturbations and the Chetaev \((\lambda,A, n_0, T)\)-stability [cf. the author, Autom. Remote Control 56, No. 1, Pt. 2, 88-95 (1995); translation from Avtom. Telemekh. 1995, No. 1, 107-116 (1995; Zbl 0932.93060)].
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 93C55
Mathematics Subject Classification ID: 39A11
zbMATH DE Number: 1263103
zbMATH Keywords: discrete-time system
zbMATH Keywords: impulsive inputs
zbMATH Keywords: asymptotic stability
reviewed by: Ryszarda Rempała
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1263103