##
**Control theory from the geometric viewpoint.**
*(English)*
Zbl 1062.93001

Encyclopaedia of Mathematical Sciences 87. Control Theory and Optimization II. Berlin: Springer (ISBN 3-540-21019-9/hbk). xiv, 412 p. (2004).

This monograph on geometric control theory, is the new reference text for researchers and graduate students in mathematical control theory who are interested in differential-geometric methods. These methods are particularly useful for control problems formulated on manifolds or on Lie groups, as it happens very often, for instance, in robotics and in quantum control. The book, written by leader experts in the subject, is full of details that are not published elsewhere. It is translated also in Russian.

The first chapter is devoted to standard material in differential geometry and control systems.

The second chapter presents the chronological calculus, which is a non-standard language developed by Agrachev and Gamkrelidze to deal with flows of ODEs having an explicit dependence on the time. It permits to transform nonlinear objects living on a finite-dimensional manifold (e.g. points, vector fields, flows etc.) into linear operators acting on an infinite dimensional space, namely, the space of smooth functions on the manifold. Chronological calculus, albeit sometimes difficult at a first glance for neophytes, allows to simplify deeply many of the proofs appearing in the remaining of the book and is a very interesting subject in itself.

Chapters 3 and 4 deal with linear systems and with state linearizability. Chapters 5,6,7,8 are devoted to the problem of controllability, having as cornerstone the orbit theorem by Hermann-Nagano (analytic case) and Sussmann (smooth case). An application to the rotation of the rigid body is presented in Chapter 6.

Chapter 9 is about feedback and state equivalence of control systems.

The core of the book lies in the remaining chapters, where Optimal Control is presented. The proof of Pontryagin Maximum Principle has a highly geometrical taste and is written directly for systems on manifolds. Some non-trivial applications (e.g., linear time-optimal problems, linear-quadratic problems, left invariant problems on Lie groups) are also given. One chapter is devoted to the Hamilton-Jacobi equation and to dynamic programming.

The last part is mainly original material of the authors and is about second order optimality conditions, Hessian of the end-point mapping, and Jacobi curves. Such material is presented here in an unified setting, which is very useful for researchers in the field. The powerful results obtained in a general framework in these chapters have not been yet exploited to their full extent on specific systems.

The topics addressed include: curvature of control systems, reduction of control affine systems to nonlinear systems of smaller dimension, rolling bodies.

The first chapter is devoted to standard material in differential geometry and control systems.

The second chapter presents the chronological calculus, which is a non-standard language developed by Agrachev and Gamkrelidze to deal with flows of ODEs having an explicit dependence on the time. It permits to transform nonlinear objects living on a finite-dimensional manifold (e.g. points, vector fields, flows etc.) into linear operators acting on an infinite dimensional space, namely, the space of smooth functions on the manifold. Chronological calculus, albeit sometimes difficult at a first glance for neophytes, allows to simplify deeply many of the proofs appearing in the remaining of the book and is a very interesting subject in itself.

Chapters 3 and 4 deal with linear systems and with state linearizability. Chapters 5,6,7,8 are devoted to the problem of controllability, having as cornerstone the orbit theorem by Hermann-Nagano (analytic case) and Sussmann (smooth case). An application to the rotation of the rigid body is presented in Chapter 6.

Chapter 9 is about feedback and state equivalence of control systems.

The core of the book lies in the remaining chapters, where Optimal Control is presented. The proof of Pontryagin Maximum Principle has a highly geometrical taste and is written directly for systems on manifolds. Some non-trivial applications (e.g., linear time-optimal problems, linear-quadratic problems, left invariant problems on Lie groups) are also given. One chapter is devoted to the Hamilton-Jacobi equation and to dynamic programming.

The last part is mainly original material of the authors and is about second order optimality conditions, Hessian of the end-point mapping, and Jacobi curves. Such material is presented here in an unified setting, which is very useful for researchers in the field. The powerful results obtained in a general framework in these chapters have not been yet exploited to their full extent on specific systems.

The topics addressed include: curvature of control systems, reduction of control affine systems to nonlinear systems of smaller dimension, rolling bodies.

Reviewer: Ugo Boscain (Paris)

### MSC:

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

93B03 | Attainable sets, reachability |

93B05 | Controllability |

93B17 | Transformations |

93B29 | Differential-geometric methods in systems theory (MSC2000) |

49K15 | Optimality conditions for problems involving ordinary differential equations |

58A30 | Vector distributions (subbundles of the tangent bundles) |

70E18 | Motion of a rigid body in contact with a solid surface |

70G45 | Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics |

70H20 | Hamilton-Jacobi equations in mechanics |

70F25 | Nonholonomic systems related to the dynamics of a system of particles |