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This book presents methods to study the controllability and the stabilization of nonlinear control systems in finite and infinite dimensions. The emphasis is put on specific phenomena due to nonlinearities. In particular, many examples are given where nonlinearities turn out to be essential to get controllability or stabilization. Various methods are presented to study the controllability or to construct stabilizing feedback laws. The power of these methods is illustrated by numerous examples coming from such areas as celestial mechanics, fluid mechanics, and quantum mechanics. The book is addressed to graduate students in mathematics or control theory, and to mathematicians or engineers with an interest in nonlinear control systems governed by ordinary or partial differential equations.

Jean-Michel Coron is a French mathematician, born in 1956. He first studied at École Polytechnique, where he worked on his PhD thesis advised by Haïm Brezis. Since 1992, he has studied the Control Theory of Partial Differential Equations, and which includes both control and stabilization (see his book [1] or his Scholarpedia article [2]). His results concern partial differential equations related to fluid dynamics, with emphasis on nonlinear phenomena, and part of them found applications to control channels.

He had previously worked in the field of non-linear functional analysis, where he also obtained significant results. Jean-Michel Coron was awarded numerous prizes, like the Fermat prize in 1993, the Jaffé prize in 1995 by the Académie des Sciences, and the Dargelos prize in 2002.

He was invited at the 1990 International Congress of Mathematicians (Kyoto) in the section Partial Differential Equations, and he was also invited as a plenary speaker at the 2010 International Congress of Mathematicians, Hyderabad, India.[3] He is now a Professor at the University Pierre et Marie Curie in Paris, and a Senior member of the Institut Universitaire de France.[4] Jean-Michel Coron is the husband of Claire Voisin who was also plenary speaker at the 2010 International Congress of Mathematicians.

Contents
Preface ix
Part 1. Controllability of linear control systems 1
Chapter 1. Finite-dimensional linear control systems 3
1.1. Definition of controllability 3
1.2. An integral criterion for controllability 4
1.3. Kalman's type conditions for controllability 9
1.4. The Hilbert Uniqueness Method 19
Chapter 2. Linear partial differential equations 23
2.1. Transport equation 24
2.2. Korteweg-de Vries equation 38
2.3. Abstract linear control systems 51
2.4. Wave equation 67
2.5. Heat equation 76
2.6. A one-dimensional Schrodinger equation 95
2.7. Singular optimal control: A linear l-D parabolic-hyperbolic example 99
2.8. Bibliographical complements 118
Part 2. Controllability of nonlinear control systems 121
Chapter 3. Controllability of nonlinear systems in finite dimension 125
3.1. The linear test 126
3.2. Iterated Lie brackets and the Lie algebra rank condition 129
3.3. Controllability of driftless control affine systems 134
3.4. Bad and good iterated Lie brackets 141
3.5. Global results 150
3.6. Bibliographical complements 156
Chapter 4. Linearized control systems and fixed-point methods 159
4.1. The Linear test: The regular case 159
4.2. The linear test: The case of loss of derivatives 165
4.3. Global controllability for perturbations of linear controllable systems 177
Chapter 5. Iterated Lie brackets 181
Chapter 6. Return method 187
6.1. Description of the method 187
6.2. Controllability of the Euler and Navier-Stokes equations 192
6.3. Local controllability of a 1-D tank containing a fluid modeled by the
Saint-Venant equations 203
Chapter 7. Quasi-static deformations 223
7.1. Description of the method 223
7.2. Application to a semilinear heat equation 225
Chapter 8. Power series expansion 235
8.1. Description of the method 235
8.2. Application to a Korteweg-de Vries equation 237
Chapter 9. Previous methods applied to a Schrodinger equation 247
9.1. Controllability and uncontrollability results 247
9.2. Sketch of the proof of the controllability in large time 252
9.3. Proof of the nonlocal controllability in small time 263
Part 2. Controllability of nonlinear control systems 121
Chapter 3. Controllability of nonlinear systems in finite dimension 125
3.1. The linear test 126
3.2. Iterated Lie brackets and the Lie algebra rank condition 129
3.3. Controllability of driftless control affine systems 134
3.4. Bad and good iterated Lie brackets 141
3.5. Global results 150
3.6. Bibliographical complements 156
Chapter 4. Linearized control systems and fixed-point methods 159
4.1. The Linear test: The regular case 159
4.2. The linear test: The case of loss of derivatives 165
4.3. Global controllability for perturbations of linear controllable systems 177
Chapter 5. Iterated Lie brackets 181
Chapter 6. Return method 187
6.1. Description of the method 187
6.2. Controllability of the Euler and Navier-Stokes equations 192
6.3. Local controllability of a 1-D tank containing a fluid modeled by the Saint-Venant equations 203
Chapter 7. Quasi-static deformations 223
7.1. Description of the method 223
7.2. Application to a semilinear heat equation 225
Chapter 8. Power series expansion 235
8.1. Description of the method 235
8.2. Application to a Korteweg-de Vries equation 237
Chapter 9. Previous methods applied to a Schrodinger equation 247
9.1. Controllability and uncontrollability results 247
9.2. Sketch of the proof of the controllability in large time 252
9.3. Proof of the nonlocal controllability in small time 263
Part 3. Stabilization 271
Chapter 10. Linear control systems in finite dimension and applications to nonlinear control systems 275
10.1. Pole-shifting theorem 275
10.2. Direct applications to the stabilization of finite-dimensional control systems 279
10.3. Gramian and stabilization 282
Chapter 11. Stabilization of nonlinear control systems in finite dimension 287
11.1. Obstructions to stationary feedback stabilization 288
11.2. Time-varying feedback laws 295
11.3. Output feedback stabilization 305
11.4. Discontinuous feedback laws 311
Chapter 12. Feedback design tools 313
12.1. Control Lyapunov function 313
12.2. Damping feedback laws 314
12.3. Homogeneity 328
12.4. Averaging 332
12.5. Backstepping 334
12.6. Forwarding 337
12.7. Transverse functions 340
Chapter 13. Applications to some partial differential equations 347
13.1. Gramian and rapid exponential stabilization 347
13.2. Stabilization of a rotating body-beam without damping 351
13.3. Null asymptotic stabilizability of the 2-D Euler control system 356
13.4. A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws 361
Appendix A. Elementary results on semigroups of linear operators 373
Appendix B. Degree theory 379
Bibliography 397
List of symbols 421
Index 423
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