Stabilizing stationary linear discrete systems: Minimal and balanced expansions in any real base
type: Item
instance of: scholarly article
title: Stabilizing stationary linear discrete systems: Minimal and balanced expansions in any real base
zbMATH Open document ID: 0910.93065
author: Dominique Luzeaux
author: Bernard Beauzamy
published in: Journal of Mathematical Analysis and Applications
publication date: 1999-04-19T00:00:00Z
review text: Given the real dynamical system: \[ x_{k+1} =ax_k+bu_k,\;u_{k+ 1} =u_k+v_k, \] \(v_k\in\{-m, \dots, +m\}\), where \(x_k\), \(u_k\) and \(v_k\) are scalars, and by using an expansion in base a with coefficients in a finite interval of \(\mathbb{Z}\) containing 0 and a balanced expansion for which the sums of the digits of any initial segment are uniformly bounded, one derives the following main result. If \(m\geq a-1\) the given system can be stabilized for any value of \(a>1\). Conversely, the condition \(m\geq [a]-1\) is necessary for stabilization for all values of \(a\). If a belongs to a particular set of reals (a certain union of intervals), the condition \(m\geq a-1\) is necessary and sufficient for stabilization.
Mathematics Subject Classification ID: 93D15
Mathematics Subject Classification ID: 93C55
Mathematics Subject Classification ID: 93C99
zbMATH DE Number: 1215758
zbMATH Keywords: linear discrete systems
zbMATH Keywords: balanced expansion
zbMATH Keywords: stabilization
reviewed by: Mihail Voicu
MaRDI profile type: Publication
full work available at URL: jmaa.1998.5990
OpenAlex ID: W2080378149
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DOI: 10.1006/JMAA.1998.5990
description: scientific article; zbMATH DE number 1215758