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Autonomy, Controllability, and Extension Modules
Author(s) -
Zerz Eva
Publication year - 2003
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200310210
Subject(s) - controllability , mathematics , extension (predicate logic) , algebra over a field , polynomial , context (archaeology) , algebraic number , duality (order theory) , linear system , pure mathematics , constant coefficients , differential (mechanical device) , discrete mathematics , computer science , mathematical analysis , paleontology , engineering , biology , programming language , aerospace engineering
Abstract The so‐called behavioral approach to systems theory, developed by Willems, provides a unified framework for the mathematical treatment of linear systems. In the behavioral context, a linear system is nothing but the solution space of a linear system of (partial) difference or differential equations. For simplicity, the coefficients are supposed to be constant. Oberst proved a duality theorem that builds upon an earlier result of Palamodov. It says that for certain signal spaces of interest, e.g., the smooth functions or the distributions, the solutions spaces of linear systems of partial differential equations, are dual to certain polynomial modules associated to them. Then the solution space and the module contain the same information, and algebraic properties of the module translate to analytic properties of the solution space. Powerful tools from commutative algebra may be used to derive them. As a prominent example, we study two properties that lie at the very heart of systems and control theory: autonomy and controllability. We summarize the characterizations given by several authors, and unify them in the language of extension modules, an algebraic concept which yields a full classification of these systems theoretic notions.