Brunovsky's canonical form for linear dynamical systems over commutative rings
type: Item
instance of: scholarly article
title: Brunovsky's canonical form for linear dynamical systems over commutative rings
zbMATH Open document ID: 0843.93012
DOI: 10.1016/0024-3795(94)00062-X
author: M. Pilar Perez
author: Jose Angel Hermida Alonso
author: T. Sánchez-Giralda
published in: Linear Algebra and its Applications
publication date: 1996-08-15T00:00:00Z
review text: \textit{P. A. Brunovsky} [Kybernetika 3, 173-188 (1970; Zbl 0199.48202)]\ showed that a reachable linear dynamical system \((F, G)\) over a field \(k\) is feedback equivalent to a system \((F_c G_c)\) which has a canonical form determined completely by invariants \(k_1, \dots, k_s\), called the Kronecker indices. Examples showing that Brunovsky's classification is not valid for systems over a general commutative ring have long been known. Systems over a commutative ring \(R\) with unit that are feedback equivalent to a system \((F_c, G_c)\) (as for a field), are called Brunovsky systems here. In the main theorem of this paper, the authors succeed in characterizing the Brunovsky systems over a commutative ring \(R\) with unit such that finitely generated projective \(R\)-modules are free. The characterization is obtained in terms of the minors of the matrices \((G, FG, \dots, F^{i-1} G)\) for \(1\leq i\leq n\) and a complete set of invariants which generalizes the Kronecker indices is obtained. A determinantal method to obtain this set of invariants is included as well, yielding an alternative method to obtain the Kronecker indices in the case of a system over a field.
reviewed by: Frieda Theron
Mathematics Subject Classification ID: 93B10
Mathematics Subject Classification ID: 93B25
zbMATH DE Number: 841742
zbMATH Keywords: reachable
zbMATH Keywords: canonical form
zbMATH Keywords: invariants
zbMATH Keywords: Kronecker indices
zbMATH Keywords: Brunovsky systems
zbMATH Keywords: commutative ring
zbMATH Keywords: free
MaRDI profile type: Publication
cites work: Q3745932
cites work: Q5597890
cites work: Finitely generated projective modules and Fitting ideals
cites work: Dynamical systems and their applications: linear theory
cites work: Feedback invariants for linear dynamical systems over a principal ideal domain
cites work: Q4066911
cites work: Q4093557
cites work: Feedback invariants of linear multivariable systems
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description: scientific article; zbMATH DE number 841742