This volume provides an integrated development of the mathematical connections between nonlinear control, dynamical systems and time-varying perturbed systems for scientists and engineers. It includes examples and definitions and is designed for use as a self-study or reference guide for all scientists and engineers.
Preface xi Introduction 1(6) Dynamics, Perturbations, and Control 7(38) Perturbations of Complex Behavior 9(6) Approximation of Complex Systems 15(4) Generic Behavior of Perturbations 19(4) Stability Boundaries and Multistability 23(2) Reachability in Control Systems 25(4) Linear and Nonlinear Stability Radii 29(5) Stabilization of Bilinear Systems 34(3) The Lyapunov Spectrum of Matrices 37(8) I Global Theory 45(92) Control Sets 47(42) Problem Formulation and Main Results 47(6) Control Sets 53(16) Relative Invariance and Multistability 69(10) Chain Control Sets 79(8) Notes 87(2) Control Flows and Limit Behavior 89(48) Problem Formulation and Main Results 90(5) The Shift Space 95(4) The Control Flow 99(10) Recurrence and Ergodic Theory 109(4) Chain Controllability and Inner Pairs 113(10) Genericity 123(3) Perturbations and Order 126(8) Notes 134(3) II Linearization Theory 137(172) Linear Flows on Vector Bundles 139(70) Problem Formulation and Main Results 139(9) Morse Decompositions 148(11) The Morse Spectrum 159(9) Morse Spectrum and Lyapunov Spectrum 168(15) Relations to Other Spectra 183(13) Invariant and Stable Manifolds 196(12) Notes 208(1) Bilinear Systems on Vector Bundles 209(46) Problem Formulation and Main Results 209(12) Floquet and Morse Spectrum 221(16) The Lyapunov Spectrum 237(5) Invariant and Stable Manifolds 242(10) Notes 252(3) Linearization at a Singular Point 255(54) Problem Formulation and Main Results 256(9) Semigroups and Lyapunov Exponents 265(16) Control Sets and Spectra 281(21) Invariant and Stable Manifolds 302(5) Notes 307(2) III Applications 309(190) One-Dimensional Control Systems 311(42) Problem Formulation and Main Results 312(6) Global Structure and Bifurcation 318(11) Linearization and Bifurcation 329(11) Feedback Control and Stabilization 340(12) Notes 352(1) Examples for Global Behavior 353(24) Stirred Tank Reactor: Global Behavior 354(6) Stirred Tank Reactor: Bifurcation 360(5) A Bacterial Respiration Model 365(2) The Takens--Bogdanov System 367(8) Notes 375(2) Examples for the Spectrum 377(30) Spectrum in Dimension Two 377(16) Numerically Computed Spectra 393(12) Notes 405(2) Stability Radii and Robust Stability 407(26) Stability Radii of Linear Systems 408(8) Stability Radii of Nonlinear Systems 416(7) Robust Stabilization of Linear Systems 423(9) Notes 432(1) Open and Closed Loop Stabilization 433(44) Feedback Stabilization of Regular Systems 434(15) Open Loop Stabilization: The Singular Case 449(12) Feedback Stabilization: The Singular Case 461(12) Notes 473(4) Dynamics of Perturbations 477(22) Persistence of Attractors and Spectra 478(6) The Lorenz Equation 484(5) A Model for Ship Roll Motion 489(8) Notes 497(2) IV Appendices 499(98) A Geometric Control Theory 501(30) A.1 Differentiable Manifolds and Vector Fields 502(3) A.2 Basic Definitions for Control Systems 505(3) A.3 The Orbit Theorem 508(17) A.4 Local Accessibility 525(5) A.5 Notes 530(1) B Dynamical Systems 531(24) B.1 Vector Bundles 531(8) B.2 Morse Decompositions, Attractors, Chains 539(12) B.3 Ergodic Theory 551(3) B.4 Notes 554(1) C Numerical Computation of Orbits 555(14) C.1 Orbits and Approximately Invariant Sets 555(6) C.2 Computing Approximately Invariant Sets 561(4) C.3 Computation via Time Optimal Control 565(2) C.4 Notes 567(2) D Computation of the Spectrum 569(28) Lars Grune D.1 Problem Formulation and Main Results 569(4) D.2 Discounted and Average Functionals 573(5) D.3 Approximation of the Spectrum 578(5) D.4 The Hamilton-Jacobi-Bellman Equation 583(4) D.5 Discounted Optimal Control Problems 587(8) D.6 Notes 595(2) Bibliography 597(24) Index 621
