**Mathematical Surveys and Monographs**

Volume: 136;
2007;
426 pp;
Softcover

MSC: Primary 93;
Secondary 35; 76

Print ISBN: 978-0-8218-4918-7

Product Code: SURV/136.S

List Price: $99.00

Individual Member Price: $79.20

**Electronic ISBN: 978-1-4704-1363-7
Product Code: SURV/136.S.E**

List Price: $99.00

Individual Member Price: $79.20

# Control and Nonlinearity

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*Jean-Michel Coron*

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.

#### Table of Contents

# Table of Contents

## Control and Nonlinearity

- Contents v6 free
- Preface ix10 free
- Part 1. Controllability of linear control systems 116 free
- Chapter 1. Finite-dimensional linear control systems 318
- Chapter 2. Linear partial differential equations 2338
- 2.1. Transport equation 2439
- 2.2. Korteweg-de Vries equation 3853
- 2.3. Abstract linear control systems 5166
- 2.4. Wave equation 6782
- 2.5. Heat equation 7691
- 2.6. A one-dimensional Schrödinger equation 95110
- 2.7. Singular optimal control: A linear 1-D parabolic-hyperbolic example 99114
- 2.8. Bibliographical complements 118133

- Part 2. Controllability of nonlinear control systems 121136
- Chapter 3. Controllability of nonlinear systems in finite dimension 125140
- Chapter 4. Linearized control systems and fixed-point methods 159174
- Chapter 5. Iterated Lie brackets 181196
- Chapter 6. Return method 187202
- Chapter 7. Quasi-static deformations 223238
- Chapter 8. Power series expansion 235250
- Chapter 9. Previous methods applied to a Schrödinger equation 247262

- Part 3. Stabilization 271286
- Chapter 10. Linear control systems in finite dimension and applications to nonlinear control systems 275290
- Chapter 11. Stabilization of nonlinear control systems in finite dimension 287302
- Chapter 12. Feedback design tools 313328
- Chapter 13. Applications to some partial differential equations 347362
- 13.1. Gramian and rapid exponential stabilization 347362
- 13.2. Stabilization of a rotating body-beam without damping 351366
- 13.3. Null asymptotic stabilizability of the 2-D Euler control system 356371
- 13.4. A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws 361376

- Appendix A. Elementary results on semigroups of linear operators 373388
- Appendix B. Degree theory 379394
- Bibliography 397412
- List of symbols 421436
- Index 423438

#### Readership

Graduate students and research mathematicians interested in control theory and nonlinear partial differential equations.

#### Reviews

This well written book by Jean-Michel Coron, one of the world leading experts in the field, enables the reader to enter the difficult subject and to understand the most important methods used here. A book like this was urgently needed.

-- Zentralblatt MATH

Throughout of the book, the author gives extensive historical comments and he mentions a great number of additional results.

-- Mathematical Reviews