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2002 | Buch

1. Introduction: Dynamics, Perturbation and Discretization Kapitel lesen Erstes Kapitel lesen

Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization

verfasst von: Lars Grüne

Verlag: Springer Berlin Heidelberg

Buchreihe : Lecture Notes in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter
1. Introduction: Dynamics, Perturbation and Discretization
Abstract
  • 1.1 Starting Point
  • 1.2 Different Approaches
  • 1.3 Basic Idea
  • 1.4 Outline of the Results
  • 1.5 Open Questions and Future Research
Lars Grüne
2. Setup and Preliminaries
Abstract
  • 2.1 Continuous Time Systems
  • 2.2 Discrete Time Systems
  • 2.3 Sets, Distances and Limits
Lars Grüne
3. Strongly Attracting Sets
Abstract
  • 3.1 Strong Attraction
  • 3.2 Robustness Concepts
  • 3.3 Geometric Characterizations
  • 3.4 Relation between Robustness Concepts
  • 3.5 Lyapunov Function Characterization
  • 3.6 Stability of Robustness Concepts
  • 3.7 Inflated Systems
  • 3.8 Discrete and Continuous Time Systems
Lars Grüne
4. Weakly Attracting Sets
Abstract
  • 4.1 Weak Attraction
  • 4.2 Robustness Concepts
  • 4.3 Geometric Characterizations
  • 4.4 Relation between Robustness Concepts
  • 4.5 Lyapunov Function Characterization
  • 4.6 Stability of Robustness Concepts
  • 4.7 Inflated Systems
  • 4.8 Discrete and Continuous Time Systems
Lars Grüne
5. Relation between Discretization and Perturbation
Abstract
  • 5.1 Time Discretization: Theoretical Framework
  • 5.2 Time Discretization: Numerical Schemes
  • 5.3 Space Discretization
Lars Grüne
6. Discretizations of Attracting Sets
Abstract
  • 6.1 Strongly Attracting Sets
  • 6.2 Strong Attractors
  • 6.3 Subdivision Algorithm
  • 6.4 Weakly Attracting Sets
Lars Grüne
7. Domains of Attraction
Abstract
  • 7.1 Definitions and Basic Properties
  • 7.2 Zubov’s Method
  • 7.3 Robustness for Domains of Attraction
  • 7.4 Domains of Attraction under One-Step Discretizations
  • 7.5 Subdivision Algorithm
  • 7.6 Numerical Approximation of Zubov’s Method
  • 7.7 Reachable Sets
Lars Grüne
Appendix A: Viscosity Solutions
Abstract
  • A.1 Definition
  • A.2 Optimality Principles
Lars Grüne
Appendix B: Comparison Functions
Abstract
  • B.1 Definition
  • B.2 Approximation by Smooth Functions
Lars Grüne
Appendix C: Numerical Examples
Abstract
  • C.1 Subdivision Algorithm: Test Examples
  • C.2 Subdivision Algorithm: Further Examples
  • C.3 Zubov’s Method
Lars Grüne
Notation
Abstract
Abstract not available
Lars Grüne
References
Abstract
Abstract not available
Lars Grüne
Index
Abstract
Abstract not available
Lars Grüne
Backmatter
Metadaten
Titel
Asymptotic Behavior of Dynamical and Control Systems under Perturbation and Discretization
verfasst von
Lars Grüne
Copyright-Jahr
2002
Verlag
Springer Berlin Heidelberg
Electronic ISBN
978-3-540-36784-0
Print ISBN
978-3-540-43391-0
DOI
https://doi.org/10.1007/b83677