On some relations between accretive, positive, and pseudocontractive operators and passivity results in Hilbert spaces and nonlinear dynamic systems
type: Item
instance of: scholarly article
title: On some relations between accretive, positive, and pseudocontractive operators and passivity results in Hilbert spaces and nonlinear dynamic systems
zbMATH Open document ID: 1410.47036
DOI: 10.1155/2017/1497867
published in: Discrete Dynamics in Nature and Society
publication date: 2018-09-27T00:00:00Z
review text: Summary: This paper investigates some parallel relations between the operators \(\left(I - G\right)\) and \(G\) in Hilbert spaces in such a way that the pseudocontractivity, asymptotic pseudocontractivity, and asymptotic pseudocontractivity in the intermediate sense of one of them are equivalent to the accretivity, asymptotic accretivity, and asymptotic accretivity in the intermediate sense of the other operator. If the operators are self-adjoint then the obtained accretivity-type properties are also passivity-type properties. Such properties are very relevant in stability theory since they refer to global stability properties of passive feed-forward, in general, nonlinear, and time-varying controlled systems controlled via feedback by elements in a very general class of passive, in general, nonlinear, and time-varying controllers. These results allow the direct generalization of passivity results in controlled dynamic systems to wide classes of tandems of controlled systems and their controllers, described by \(G\)-operators, and their parallel interpretations as pseudocontractive properties of their counterpart \(\left(I - G\right)\)-operators. Some of the obtained results are also directly related to input-passivity, output-passivity, and hyperstability properties in controlled dynamic systems. Some illustrative examples are also given in the framework of dynamic systems described by extended square-integrable input and output signals.
Mathematics Subject Classification ID: 47N70
Mathematics Subject Classification ID: 93C10
Mathematics Subject Classification ID: 93D10
Mathematics Subject Classification ID: 47H06
Mathematics Subject Classification ID: 47H09
zbMATH DE Number: 6944803
zbMATH Keywords: accretive operators
zbMATH Keywords: positive operators
zbMATH Keywords: pseudocontractive operators
zbMATH Keywords: passivity
zbMATH Keywords: hyperstability
zbMATH Keywords: nonlinear dynamical systems
Wikidata QID: Q57674623
author: M. De la Sen
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2017/1497867
OpenAlex ID: W2761365507
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description: scientific article; zbMATH DE number 6944803