Topological Obstructions to Stability and Stabilization: History, Recent Advances and Open Problems

✍ Scribed by Wouter Jongeneel, Emmanuel Moulay

• Publisher: Springer
• Year: 2023
• Tongue: English
• Leaves: 134
• Series: SpringerBriefs in Electrical and Computer Engineering: Control, Automation and Robotics
• Category: Library

✦ Synopsis

This open access book provides a unified overview of topological obstructions to the stability and stabilization of dynamical systems defined on manifolds and an overview that is self-contained and accessible to the control-oriented graduate student. The authors review the interplay between the topology of an attractor, its domain of attraction, and the underlying manifold that is supposed to contain these sets. They present some proofs of known results in order to highlight assumptions and to develop extensions, and they provide new results showcasing the most effective methods to cope with these obstructions to stability and stabilization. Moreover, the book shows how Borsuk’s retraction theory and the index-theoretic methodology of Krasnosel’skii and Zabreiko underlie a large fraction of currently known results. This point of view reveals important open problems, and for that reason, this book is of interest to any researcher in control, dynamical systems, topology, or related fields.

✦ Table of Contents

Preface
Acknowledgements
Contents
1 Introduction
1.1 Impetus
1.2 Historical Remarks
1.2.1 Topology
1.2.2 Dynamical Systems
1.2.3 Modern Control Theory
1.3 Case Study: Optimal Control on Lie Groups
1.4 Content and Structure
References
2 General Topology
2.1 Topological Spaces
2.2 Homotopy and Retractions
2.3 Comments on Triangulation
References
3 Differential Topology
3.1 Differentiable Structures
3.2 Submanifolds and Transversality
3.3 Bundles
3.4 Intersection and Index Theory
3.5 Poincaré–Hopf and the Bobylev–Krasnosel'skiĭ theorem
References
4 Algebraic Topology
4.1 Singular Homology
4.2 The Euler Characteristic
References
5 Dynamical Control Systems
5.1 Dynamical Systems
5.2 Lyapunov Stability Theory
5.3 Control Systems
References
6 Topological Obstructions
6.1 Obstructions to the Stabilization of Points
6.1.1 Local Obstructions
6.1.2 Global Obstructions
6.1.3 A Local Odd-Number Obstruction to Multistabilization
6.2 Obstructions to the Stabilization of Submanifolds
6.3 Obstructions to the Stabilization of Sets
6.4 Other Obstructions
References
7 Towards Accepting and Overcoming Topological Obstructions
7.1 On Accepting the Obstruction
7.2 On Time-Varying Feedback
7.3 On Discontinuous Control
7.3.1 Hybrid Control Exemplified
7.3.2 Topological Perplexity
References
8 Generalizations and Open Problems
8.1 Comments on Discrete-Time Systems and Periodic Orbits
8.2 Comments on Generalized Poincaré–Hopf Theory
8.3 A Decomposition Through Morse Theory
8.4 An Application of Lusternik–Schnirelmann Theory
8.5 Introduction to Conley Index Theory
8.6 Conclusion and Open Problems
References
Appendix Series Editor Biographies