Asymptotic behavior of dynamical and control systems under perturbation and discretization
type: Item
instance of: scholarly article
title: Asymptotic behavior of dynamical and control systems under perturbation and discretization
zbMATH Open document ID: 0991.37001
author: Lars Grüne
published in: Lecture Notes in Mathematics
publication date: 2002-06-04T00:00:00Z
full work available at URL: t1783.htm
review text: This monograph (based on the author's habilitation thesis) contributes to the numerical analysis of dynamical systems and control systems. It lies in the intersection of perturbation theory of dynamical systems, numerical dynamics, control theory and (dynamic) game theory. Concepts and methods of all these areas are interwoven. The general theme is the behavior of attracting sets and attractors under discretization. This contribution is not, as many others, based on a priori structural assumptions like hyperbolicity. This assumption is already violated for most relevant dynamical systems from applications; if controls or perturbations are present, they are restricted to small amplitudes. Thus, although it is possible (and also interesting) to develop those theories, their relevance, however, is very restricted. Grüne uses the classical method of comparison functions for stability analysis, going back to W. Hahns work. They allow for a quantitative description of stability properties, and in the last few years, they have witnessed a renaissance in control theory. He considers two main classes of problems: systems with internal disturbances and controlled systems. In order to understand the behavior of numerical approximations to these systems, further (external) disturbances are introduced, which are of two different types: for systems with internal disturbances, Grüne takes time dependent disturbances; for control systems he considers external disturbances, generated by nonanticipatory strategies, a concept from game theory. In the first case, so-called strong concepts are obtained, in the second case, weak concepts. Nonanticipatory perturbations make it possible to model numerical errors for controlled systems. After the introduction and the preparatory Chapter 2 (mainly on the considered systems classes), Chapters 3 and 4 study attracting sets. Here the new concept of dynamical Input-to-State-Stability is central. In Chapter 5 the connection between time and space…
reviewed by: Fritz Colonius
Mathematics Subject Classification ID: 37-02
Mathematics Subject Classification ID: 93-02
Mathematics Subject Classification ID: 65P40
Mathematics Subject Classification ID: 37B25
Mathematics Subject Classification ID: 93D30
Mathematics Subject Classification ID: 93B05
Mathematics Subject Classification ID: 93B35
Mathematics Subject Classification ID: 49L25
zbMATH DE Number: 1749051
zbMATH Keywords: dynamical systems
zbMATH Keywords: control systems
zbMATH Keywords: discretization
describes a project that uses: ode23
MaRDI profile type: Publication
OpenAlex ID: W1532162288
DOI: 10.1007/B83677
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description: scientific article; zbMATH DE number 1749051