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
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
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
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
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