Key problems in the extension of module-behaviour duality
type: Item
instance of: scholarly article
title: Key problems in the extension of module-behaviour duality
zbMATH Open document ID: 1096.93015
DOI: 10.1016/S0024-3795(01)00532-8
author: J. Wood
published in: Linear Algebra and its Applications
publication date: 2002-08-28T00:00:00Z
review text: The duality for linear constant coefficient partial differential equations between behaviours and finitely generated modules over the operator ring is a very powerful tool linking equation structure to dynamic behaviour. This duality is critically dependent on the choice of signal space. In the paper under review the author discusses two key algebraic problems which form an obstacle to the extension of this theory to general signal spaces. The first of these is the so-called Willems closure problem, which limits the ability of system equations to directly describe the system. The second is the elimination problem, the general solution of which depends upon an algebraic property (injectivity) of the signal space. He demonstrates the importance of these problems in the module-behaviour framework, and some of the useful consequences of a full or partial solution. The issues here are of particular relevance to the extension of the current duality theory for behaviours defined by linear partial differential equations from the case of constant to non-constant coefficients.
Mathematics Subject Classification ID: 93C05
Mathematics Subject Classification ID: 13A99
Mathematics Subject Classification ID: 16D99
Mathematics Subject Classification ID: 93B25
zbMATH DE Number: 1790305
zbMATH Keywords: linear systems
zbMATH Keywords: multidimensional systems
zbMATH Keywords: behaviors
zbMATH Keywords: module theory
zbMATH Keywords: Willems closure
zbMATH Keywords: elimination problem
zbMATH Keywords: image representations
reviewed by: Nikolai I. Osetinski
MaRDI profile type: Publication
cites work: Finite poles and zeros of linear systems: An intrinsic approach
cites work: Q4692705
cites work: Q5590963
cites work: Q4317713
cites work: Direct methods for primary decomposition
cites work: Some basic structural properties of generalized linear systems
cites work: Q4010430
cites work: A remark on Willems' trajectory characterization of linear controllability
cites work: Flatness and defect of non-linear systems: introductory theory and examples
cites work: Continuous time-varying linear systems
cites work: Q3156401
cites work: Gröbner bases and primary decomposition of polynomial ideals
cites work: A Behavioral Approach To Delay-Differential Systems
cites work: Q3994888
cites work: System equivalence for AR-systems over rings -- with an application to delay-differential systems
cites work: Closed subspaces, polynomial operators in the shift, and ARMA representations
cites work: Q4224184
cites work: Finite-dimensional time-invariant linear dynamical systems: Algebraic theory
cites work: Q5516276
cites work: Divisible Modules
cites work: Multidimensional constant linear systems
cites work: Variations on the fundamental principle for linear systems of partial differential and difference equations with constant coefficients
cites work: Q5585729
cites work: A Behavioral Approach to Control of Distributed Systems
cites work: Generalized Bézout identity.
cites work: Algebraic analysis of linear multidimensional control systems
cites work: Formal elimination for multidimensional systems and application to control theory.
cites work: A functorial approach to the behaviour of multidimensional control systems
cites work: Q3882615
cites work: Q3244672
cites work: The Nullstellensatz for systems of PDE
cites work: Q4049223
cites work: Paradigms and puzzles in the theory of dynamical systems
cites work: The contact problem for linear continuous-time dynamical systems: a geometric approach
cites work: Modules and behaviours in nD systems theory
cites work: Controllable and autonomous nD linear systems
cites work: Notes on the definition of behavioural controllability
cites work: Primeness of multivariate polynomial matrices
cites work: Extension modules in behavioral linear systems theory
description: scientific article; zbMATH DE number 1790305