##
**Control and nonlinearity.**
*(English)*
Zbl 1140.93002

Mathematical Surveys and Monographs 136. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-3668-2/hbk). xiv, 426 p. (2007).

Since some years, the controllability and stabilizability of distributed parameter systems belong to the most active fields of modern control theory. Many deep results were contributed to this subject in the recent past, and as many interesting open problems will be tackled in the near future. However, the control of partial differential equations (PDEs) is a difficult matter, in particular, if nonlinearities come into play. The discussion of nonlinear systems of ordinary differential equations (ODEs) is already very technical compared with the linear control theory. In the case of PDEs, these difficulties are essentially increased by the technicalities and the diversity of such equations. In view of this, it is very difficult to keep track with the present fast development in the control of PDEs and to understand the associated techniques.

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. It is both textbook and research monography, and it also surveys the relevant state of the art. In all of its 13 chapters, the author proceeds by the same nice didactic principle: First, he explains the main idea by a fairly simple example, in general in terms of a linear or nonlinear ODE. After this step, the reader is prepared to better understand the more technical case of PDEs. Moreover, the book, its 3 parts, and all the chapters begin with a fluently written and instructive exposition of their main contents.

Part 1 of the book is devoted to the controllability linear control systems. It starts with a brief survey on classical concepts of linear finite-dimensional control theory, where also the well-known Hilbert uniqueness method (HUM) is explained for ODEs. Next, the authors turns over to linear PDEs and studies the problem of exact controllability (here to control a given state to another one) for the transport, the Korteweg-de Vries, and the wave equation. Motivated by the smoothing property of the heat equation, he introduces another concept of controllability, namely to reach a given trajectory from a given state. Also the Schrödinger equation is discussed.

Part 2 introduces three main concepts of nonlinear controllability – the return method, introduced by the author in 1992, the technique of quasistatic deformation, and the power series method. First, the control theory of nonlinear ODEs is briefly sketched, in particular the application of iterated Lie brackets. Then the controllability of nonlinear PDEs around an equilibrium is discussed under the assumption that the associated linearized equation is controllable. As examples, the Korteveg-de Vries equation, some hyperbolic equations and a nonlinear 1-D Schrödinger equation are investigated. Later, the author explains, why the application of iterated Lie brackets is less powerful for PDEs than for ODEs. Next, the three main methods mentioned above are explained and used; again, first for ODEs and next for PDEs.

Part 3 deals with stabilizability and feedback laws, mainly for the case of ODEs. In a first chapter, cases are handled, where a linearization at an equilibrium is helpful to deal with the nonlinear system. Hereafter, equations are studied, where the nonlinearity is really dominant and linearization is not able to stabilize the system. As remedies, discontinuous and time-varying feedback laws and the problem of output stabilization are sketched. Different ways to constructing stabilizing feedback laws are presented and explained by means of examples. While these topics are introduced for ODEs, the last chapter deals with applications to partial differential equations.

In two appendices, basic knowledge on semigroups of linear operators and some introduction to degree theory are provided. The references include more than 500 items and provide, together with the many hints on additional results and open problems in the former chapters, an excellent survey on the state of the art.

This excellent book can be recommended to everybody who is interested in controllability and stabilizability of linear or nonlinear partical differential equations. It is certainly obligatory for all, who aim at doing research in this field.

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. It is both textbook and research monography, and it also surveys the relevant state of the art. In all of its 13 chapters, the author proceeds by the same nice didactic principle: First, he explains the main idea by a fairly simple example, in general in terms of a linear or nonlinear ODE. After this step, the reader is prepared to better understand the more technical case of PDEs. Moreover, the book, its 3 parts, and all the chapters begin with a fluently written and instructive exposition of their main contents.

Part 1 of the book is devoted to the controllability linear control systems. It starts with a brief survey on classical concepts of linear finite-dimensional control theory, where also the well-known Hilbert uniqueness method (HUM) is explained for ODEs. Next, the authors turns over to linear PDEs and studies the problem of exact controllability (here to control a given state to another one) for the transport, the Korteweg-de Vries, and the wave equation. Motivated by the smoothing property of the heat equation, he introduces another concept of controllability, namely to reach a given trajectory from a given state. Also the Schrödinger equation is discussed.

Part 2 introduces three main concepts of nonlinear controllability – the return method, introduced by the author in 1992, the technique of quasistatic deformation, and the power series method. First, the control theory of nonlinear ODEs is briefly sketched, in particular the application of iterated Lie brackets. Then the controllability of nonlinear PDEs around an equilibrium is discussed under the assumption that the associated linearized equation is controllable. As examples, the Korteveg-de Vries equation, some hyperbolic equations and a nonlinear 1-D Schrödinger equation are investigated. Later, the author explains, why the application of iterated Lie brackets is less powerful for PDEs than for ODEs. Next, the three main methods mentioned above are explained and used; again, first for ODEs and next for PDEs.

Part 3 deals with stabilizability and feedback laws, mainly for the case of ODEs. In a first chapter, cases are handled, where a linearization at an equilibrium is helpful to deal with the nonlinear system. Hereafter, equations are studied, where the nonlinearity is really dominant and linearization is not able to stabilize the system. As remedies, discontinuous and time-varying feedback laws and the problem of output stabilization are sketched. Different ways to constructing stabilizing feedback laws are presented and explained by means of examples. While these topics are introduced for ODEs, the last chapter deals with applications to partial differential equations.

In two appendices, basic knowledge on semigroups of linear operators and some introduction to degree theory are provided. The references include more than 500 items and provide, together with the many hints on additional results and open problems in the former chapters, an excellent survey on the state of the art.

This excellent book can be recommended to everybody who is interested in controllability and stabilizability of linear or nonlinear partical differential equations. It is certainly obligatory for all, who aim at doing research in this field.

Reviewer: Fredi Tröltzsch (Berlin)

### MSC:

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

93B05 | Controllability |

49N35 | Optimal feedback synthesis |

93C20 | Control/observation systems governed by partial differential equations |

93B52 | Feedback control |

93D15 | Stabilization of systems by feedback |