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

ISBN-13: 9783031301322 / Angielski

Wouter Jongeneel;Emmanuel Moulay

Opis

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.

Spis treści

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 Introduction to 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’ski˘ı 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 

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 

 

O autorze

Wouter Jongeneel received the M.Sc. degree in Systems and Control from Delft University of Technology in 2019 and is currently pursuing the Ph.D. degree in Electrical Engineering at the École Polytechnique Fédérale de Lausanne (EPFL). His research is centered around the interplay of control, dynamical systems, optimization, and topology.  

Emmanuel Moulay received his M.S. degree in mathematics from the University of Lille, France, in 2002. He obtained his Ph.D. degree in automatic control in 2005 from the Ecole Centrale de Lille and his habilitation from the University of Poitiers in 2014. He joined the CNRS as a research scientist at the Ecole Centrale de Nantes in 2006 and moved at the XLIM institute of the University of Poitiers in 2009. He is also an adjunct professor in the department of Electrical and Computer Engineering at the University of Nebraska–Lincoln since 2020. His main interests are in control theory and its practical applications.

He has published over 100 research articles in international journals and conferences and worked on the topological obstructions to stability and stabilization; he has also published 2 book chapters in the Springer book series Lecture Notes in Control and Information Sciences.