The pointwise feedback relation for linear dynamical systems
type: Item
instance of: scholarly article
title: The pointwise feedback relation for linear dynamical systems
zbMATH Open document ID: 0932.93018
DOI: 10.1016/S0024-3795(98)00020-2
published in: Linear Algebra and its Applications
publication date: 2000-02-28T00:00:00Z
review text: \(R\) denotes a commutative ring. If \(\Sigma\) is an \(m\)-input \(n\)-dimensional linear dynamical system and \(\mathbf p\) is a prime ideal, then \(\Sigma (\mathbf p)\) denotes the natural extension of \(\Sigma\) to \(k(\mathbf p),\) the residual field of \(R\) at \(\mathbf p\). Two systems \(\Sigma\) and \(\Sigma '\) over \(R\) are called pointwise feedback equivalent if \(\Sigma (\mathbf p)\) and \(\Sigma ' (\mathbf p)\) are feedback equivalent over \(k(\mathbf p)\) for all prime ideals \(\mathbf p\) of \(R\). A set of invariants are constructed for the equivalence class of \(\Sigma\) by the action of the feedback group. An example is given to show that these invariants are not sufficient to specify the feedback equivalence class of \(\Sigma\) in general, but it is proved that the feedback class of a reachable linear dynamical system over a field is characterized by the constructed invariants. In fact, the set of Kronecker indices may be deduced from these invariants in this case. Turning to pointwise feedback equivalence, a characterization is found in terms of the set of the constructed invariants and proved to be of topological nature. The relationship between feedback equivalence and pointwise feedback equivalence is then investigated. The main result states that a ring \(R\) is absolutely flat if and only if the feedback relation and the pointwise feedback relation are equivalent.
Mathematics Subject Classification ID: 93B25
Mathematics Subject Classification ID: 93B52
Mathematics Subject Classification ID: 93C05
zbMATH DE Number: 1355343
zbMATH Keywords: feedback
zbMATH Keywords: pointwise feedback
zbMATH Keywords: systems over rings
zbMATH Keywords: absolutely flat ring
zbMATH Keywords: linear system
zbMATH Keywords: prime ideal
zbMATH Keywords: feedback group
zbMATH Keywords: invariants
zbMATH Keywords: Kronecker indices
author: Miguel V. Carriegos
author: Jose Angel Hermida Alonso
author: T. Sánchez-Giralda
reviewed by: Frieda Theron
MaRDI profile type: Publication
cites work: Q5597890
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Wikidata QID: Q126843370
description: scientific article; zbMATH DE number 1355343