Asymptotic Behavior of Dynamical and Control Systems Under Pertubation and Discretization : Lecture Notes in Mathematics - Lars Grune

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Asymptotic Behavior of Dynamical and Control Systems Under Pertubation and Discretization

By: Lars Grune, L. Grune, Lars Gr Ne

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Paperback | 1 April 2002

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248 Pages

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This book provides an approach to the study of perturbation and discretization effects on the long-time behavior of dynamical and control systems. It analyzes the impact of time and space discretizations on asymptotically stable attracting sets, attractors, asumptotically controllable sets and their respective domains of attractions and reachable sets. Combining robust stability concepts from nonlinear control theory, techniques from optimal control and differential games and methods from nonsmooth analysis, both qualitative and quantitative results are obtained and new algorithms are developed, analyzed and illustrated by examples.

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Introduction: Dynamics, Perturbation and Discretization	p. 1
Starting Point	p. 2
Different Approaches	p. 4
Basic Idea	p. 6
Outline of the Results	p. 7
Open Questions and Future Research	p. 10
Setup and Preliminaries	p. 13
Continuous Time Systems	p. 13
Discrete Time Systems	p. 19
Sets, Distances and Limits	p. 22
Strongly Attracting Sets	p. 27
Strong Attraction	p. 27
Robustness Concepts	p. 29
Geometric Characterizations	p. 34
Relation between Robustness Concepts	p. 38
Lyapunov Function Characterization	p. 43
Stability of Robustness Concepts	p. 52
Inflated Systems	p. 56
Discrete and Continuous Time Systems	p. 64
Weakly Attracting Sets	p. 69
Weak Attraction	p. 69
Robustness Concepts	p. 75
Geometric Characterizations	p. 78
Relation between Robustness Concepts	p. 82
Lyapunov Function Characterization	p. 87
Stability of Robustness Concepts	p. 96
Inflated Systems	p. 100
Discrete and Continuous Time Systems	p. 107
Relation between Discretization and Perturbation	p. 113
Time Discretization: Theoretical Framework	p. 113
Time Discretization: Numerical Schemes	p. 117
Space Discretization	p. 129
Discretizations of Attracting Sets	p. 137
Strongly Attracting Sets	p. 137
Strong Attractors	p. 142
Subdivision Algorithm	p. 148
Weakly Attracting Sets	p. 151
Domains of Attraction	p. 157
Definitions and Basic Properties	p. 157
ZubovÆs Method	p. 159
Robustness for Domains of Attraction	p. 169
Domains of Attraction under One-Step Discretizations	p. 173
Subdivision Algorithm	p. 178
Numerical Approximation of ZubovÆs Method	p. 185
Reachable Sets	p. 189
Appendices
Viscosity Solutions	p. 195
Definition	p. 195
Optimality Principles	p. 196
Comparison Functions	p. 201
Definition	p. 201
Approximation by Smooth Functions	p. 203
Numerical Examples	p. 207
Subdivision Algorithm: Test Examples	p. 207
Subdivision Algorithm: Further Examples	p. 212
ZubovÆs Method	p. 215
Notation	p. 219
References	p. 221
Index	p. 229
Table of Contents provided by Publisher. All Rights Reserved.

Asymptotic Behavior of Dynamical and Control Systems Under Pertubation and Discretization

At a Glance

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