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 Dierential topology 3.1 Dierentiable structures 3.2 Submanifolds and transversality 3.

3 Bundles 3.4 Intersection and index theory 3.5 Poincaré-Hopf and the Bobylev-Krasnosel'skii 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.