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Nonlinear Systems Analysis (Classics in Applied Mathematics, 42): Mathukumalli Vidyasagar

Table of Contents

Preface

Note to the Reader

1 Introduction 1

2 Nonlinear Differential Equations 6

2.1 Mathematical Preliminaries 6

2.2 Induced Norms and Matrix Measures 19

2.3 Contraction Mapping Theorem 27

2.4 Nonlinear Differential Equations 33

2.5 Solution Estimates 46

3 Second-Order Systems 53

3.1 Preliminaries 53

3.2 Linearization Method 57

3.3 Periodic Solutions 67

3.4 Two Analytical Approximation Methods 79

4 Approximate Analysis Methods 88

4.1 Describing Functions 88

4.2 Periodic Solutions: Rigorous Arguments 109

4.3 Singular Perturbations 127

5 Lyapunov Stability 135

5.1 Stability Definitions 135

5.2 Some Preliminaries 147

5.3 Lyapunov's Linearization Method 157

5.4 Stability of Linear Systems 193

5.5 Lyapunov's Linearization Method 209

5.6 The Lur'e Problem 219

5.7 Converse Theorems 235

5.8 Applications of Converse Theorems 246

5.9 Discrete-Time Systems 264

6 Input-Output Stability 270

6.1 L [Subscript p] Spaces and their Extensions 271

6.2 Definitions of Input-Output Stability 277

6.3 Relationships Between I/O and Lyapunov Stability 284

6.4 Open-Loop Stability of Linear Systems 292

6.5 Linear Time-Invariant Feedback Systems 309

6.6 Time-Varying and/or Nonlinear Systems 337

6.7 Discrete-Time Systems 365

7 Differential Geometric Methods 376

7.1 Basics of Differential Geometry 377

7.2 Distributions, Frobenius Theorem 392

7.3 Reachability and Observability 399

7.4 Feedback Linearization: Single-Input Case 427

7.5 Feedback Linearization: Multi-Input Case 438

7.6 Input-Output Linearization 456

7.7 Stabilization of Linearizable Systems 464

A. Prevalence of Differential Equations With Unique Solutions 469

B. Proof of the Kalman-Yacubovitch Lemma 474

C. Proof of the Frobenius Theorem 476

References 486

Index 493

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