The Dynamics of Control

1 Introduction.- 2 Dynamics, Perturbations, and Control.- 2.1 Perturbations of Complex behavior.- 2.2 Approximation of Complex Systems.- 2.3 Generic Behavior of Perturbations.- 2.4 Stability Boundaries and Multistability.- 2.5 Reachability in Control Systems.- 2.6 Linear and Nonlinear Stability Radii.- 2.7 Stabilization of Bilinear Systems.- 2.8 The Lyapunov Spectrum of Matrices.- I Global Theory.- 3 Control Sets.- 4 Control Flows and Limit behavior.- II Linearization Theory.- 5 Linear Flows on Vector Bundles.- 6 Bilinear Systems on Vector Bundles.- 7 Linearization at a Singular Point.- III Applications.- 8 One-Dimensional Control Systems.- 9 Examples for Global behavior.- 10 Examples for the Spectrum.- 11 Stability Radii and Robust Stability.- 12 Open and Closed Loop Stabilization.- 13 Dynamics of Perturbations.- IV Appendices.- A Geometric Control Theory.- A.1 Differentiable Manifolds and Vector Fields.- A.2 Basic Definitions for Control Systems.- A.3 The Orbit Theorem.- A.4 Local Accessibility.- A.5 Notes.- B Dynamical Systems.- B.1 Vector Bundles.- B.2 Morse Decompositions, Attractors, Chains.- B.3 Ergodic Theory.- B.4 Notes.- C Numerical Computation of Orbits.- C.1 Orbits and Approximately Invariant Sets.- C.2 Computing Approximately Invariant Sets.- C.3 Computation via Time Optimal Control.- C.4 Notes.- D.1 Problem Formulation and Main Results.- D.2 Discounted and Average Functional.- D.3 Approximation of the Spectrum.- D.4 The Hamilton-Jacobi-Bellman Equation.- D.5 Discounted Optimal Control Problems.- D.6 Notes.