Maria Tomäs-Rodriguez and Stephen R Banks

Linear, Time-varying Approximations to Nonlinear Dynamical Systems with Applications in Control and Optimization

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Contents

1 Introduction to Nonlinear Systems 1 1.1 Overview 1 1.2 Existence and Uniqueness 2 1.3 Logistic Systems 3 1.4 Control of Nonlinear Systems 4 1.5 Vector Fields on Manifolds 5 1.6 Nonlinear Partial Differential Equations 6 1.7 Conclusions and Outline of the Book 8 References 9

2 Linear Approximations to Nonlinear Dynamical Systems 11 2.1 Introduction 11 2.2 Linear, Time-varying Approximations 12 2.3 The Lorenz Attractor 16 2.4 Convergence Rate 17 2.5 Influence of the Initial Conditions on the Convergence 20 2.6 Notes on Different Configurations 22 2.7 Comparison with the Classical Linearisation Method 23 2.8 Conclusions 26 References 27

3 The Structure and Stability of Linear, Time-varying Systems 29 3.1 Introduction 29 3.2 Existence and Uniqueness 29 3.3 Explicit Solutions 32 3.4 Stability Theory 46 3.5 Lyapunov Exponents and Oseledec's Theorem 51

X Contents

3.6 Exponential Dichotomy and the Sacker-Sell Spectrum 57 3.7 Conclusions 59 References 60

4 General Spectral Theory of Nonlinear Systems 61 4.1 Introduction 61 4.2 A Frequency-domain Theory of Nonlinear Systems 61 4.3 Exponential Dichotomies 70 4.4 Conclusions 73 References 74

5 Spectral Assignment in Linear, Time-varying Systems . . . 75 5.1 Introduction 75 5.2 Pole Placement for Linear, Time-invariant Systems 77 5.3 Pole Placement for Linear, Time-varying Systems 79 5.4 Generalisation to Nonlinear Systems 89 5.5 Application to F-8 Crusader Aircraft 94 5.6 Conclusions 97 References 98

6 Optimal Control 101 6.1 Introduction 101 6.2 Calculus of Variations and Classical Linear

Quadratic Control 101 6.3 Nonlinear Control Problems 106 6.4 Examples 109 6.5 The Hamilton-Jacobi-Bellman Equation, Viscosity

Solutions and Optimality 114 6.6 Characteristics of the Hamilton-Jacobi Equation 117 6.7 Conclusions 120 References 121

7 Sliding Mode Control for Nonlinear Systems 123 7.1 Introduction 123 7.2 Sliding Mode Control for Linear Time-invariant Systems . . . 124 7.3 Sliding Mode Control for Linear Time-varying Systems 125 7.4 Generalisation to Nonlinear Systems 129 7.5 Conclusions 137 References 139

Contents XI

8 Fixed Point Theory and Induction 141 8.1 Introduction 141 8.2 Fixed Point Theory 141 8.3 Stability of Systems 145 8.4 Periodic Solutions 147 8.5 Conclusions 149 References 150

9 Nonlinear Partial Differential Equations 151 9.1 Introduction 151 9.2 A Moving Boundary Problem 152 9.3 Solution of the Unforced System 153 9.4 The Control Problem 155 9.5 Solitons and Boundary Control 161 9.6 Conclusions 167 References 167

10 Lie Algebraic Methods 169 10.1 Introduction 169 10.2 The Lie Algebra of a Differential Equation 170 10.3 Lie Groups and the Solution of the System 174 10.4 Solvable Systems 177 10.5 The Killing Form and Invariant Spaces 179 10.6 Compact Lie Algebras 185 10.7 Modal Control 190 10.8 Conclusions 194 References 194

11 Global Analysis on Manifolds 195 11.1 Introduction 195 11.2 Dynamical Systems on Manifolds 196 11.3 Local Reconstruction of Systems 197 11.4 Smooth Transition Between Operating Conditions 199 11.5 From Local to Global 201 11.6 Smale Theory 203 11.7 Two-dimensional Manifolds 205 11.8 Three-dimensional Manifolds 208 11.9 Four-dimensional Manifolds 212 11.10 Conclusions 215 References 216

12 Summary, Conclusions and Prospects for Development 219 12.1 Introduction 219 12.2 Travelling Wave Solutions in Nonlinear Lattice Differential

Equations 219

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12.3 Travelling Waves 220 12.4 An Approach to the Solution 221 12.5 A Separation Theorem for Nonlinear Systems 222 12.6 Conclusions 227 References 227

A Linear Algebra 229 A.l Vector Spaces 229 A.2 Linear Dependence and Bases 231 A.3 Subspaces and Quotient Spaces 233 A.4 Eigenspaces and the Jordan Form 234 References 237

В Lie Algebras 239 B.l Elementary Theory 239 B.2 Cartan Decompositions of Semi-simple Lie Algebras 241 B.3 Root Systems and Classification of Simple Lie Algebras . . . . 244 B.4 Compact Lie Algebras 254 References 255

С Differential Geometry 257 C.l Differentiable Manifolds 257 C.2 Tangent Spaces 258 C.3 Vector Bundles 259 C.4 Exterior Algebra and de Rham Cohomology 260 C.5 Degree and Index 261 C.6 Connections and Curvature 264 C.7 Characteristic Classes 267 References 269

D Functional Analysis 271 D.l Banach and Hilbert Spaces 271 D.2 Examples 274 D.3 Theory of Operators 275 D.4 Spectral Theory 277 D.5 Distribution Theory 280 D.6 Sobolev Spaces 285 D.7 Partial Differential Equations 287 D.8 Semigroup Theory 289 D.9 The Contraction Mapping and Implicit Function

Theorems 292 References 293

Index 295