##
**Geometric control theory.**
*(English)*
Zbl 0940.93005

Cambridge Studies in Advanced Mathematics. 52. Cambridge: Cambridge Univ. Press. xviii, 492 p. (1997).

This book on Geometric Control Theory stands at a crossroad for differential geometry, mathematical physics, and optimal control. The interdisciplinary character of this subject demands a wide range of mathematical topics as well as good geometric insights. This book provides an important reference for graduate students and mathematicians whose work involves both the concepts and the language of control and systems theory. The geometric language chosen to present the main results lends itself to clear and intuitive interpretations and makes the book also accessible to physicists and engineers. Despite the technical nature of the subject matter, the book is well written, almost self-contained, and easy to read. It starts with an excellent introduction, each chapter is introduced by a summary, and a bibliography note at the end of each chapter gives a brief historical background of the topics discussed. However, some additional sources should have been provided to the reader.

The book is divided into two parts. The first part on “reachable sets and controllability”, comprises six chapters, the second part on “optimal control”, comprises 8 chapters. Geometric control theory arises from the introduction of concepts, structures and techniques from differential geometry into classical control theory. The combined use of concepts such as Lie algebras of vector fields, integral manifolds, distributions, Lie groups, homogeneous spaces, symplectic structures, etc., led to new results on several important control-theoretical questions. Controllability and optimal control are the primary areas of interest in this book. In this geometric setting, a control system is a family of vector fields, on a differentiable manifold, parameterized by controls. A trajectory of such a system is then a continuous curve obtained by joining up finitely many segments of integral curves of vector fields in the family. The first part of the book deals with qualitative properties of the reachable sets defined by families of vector fields and establishes the theoretical foundation for a study of control systems on differentiable manifolds. This geometric approach justifies an introductory discussion of differentiable manifolds and vector fields, and the algebraic structure given by the Lie bracket operation, the basic theoretical tool in geometric control theory.

The second part of the book highlights the innovative ideas that geometric control theory brought to the classical calculus of variations and focuses on the interconnections of optimal control with problems of geometry and mechanics. A natural connection between mechanics and optimal control is through the Maximum Principle which yields from an optimal control problem a Hamiltonian system. The vector-field view produces invariant formulations of the Maximum Principle on manifolds. The book also dedicates particular attention to optimal control on Lie groups and symmetric spaces, which give rise to Hamiltonian systems with symmetries and thus with a great deal of structure, including the existence of conservation laws. To reinforce the connection between mechanics and control theory, the author also discusses in detail the control of certain mechanical systems.

This book is a welcome addition to a rather meager literature in the field.

The book is divided into two parts. The first part on “reachable sets and controllability”, comprises six chapters, the second part on “optimal control”, comprises 8 chapters. Geometric control theory arises from the introduction of concepts, structures and techniques from differential geometry into classical control theory. The combined use of concepts such as Lie algebras of vector fields, integral manifolds, distributions, Lie groups, homogeneous spaces, symplectic structures, etc., led to new results on several important control-theoretical questions. Controllability and optimal control are the primary areas of interest in this book. In this geometric setting, a control system is a family of vector fields, on a differentiable manifold, parameterized by controls. A trajectory of such a system is then a continuous curve obtained by joining up finitely many segments of integral curves of vector fields in the family. The first part of the book deals with qualitative properties of the reachable sets defined by families of vector fields and establishes the theoretical foundation for a study of control systems on differentiable manifolds. This geometric approach justifies an introductory discussion of differentiable manifolds and vector fields, and the algebraic structure given by the Lie bracket operation, the basic theoretical tool in geometric control theory.

The second part of the book highlights the innovative ideas that geometric control theory brought to the classical calculus of variations and focuses on the interconnections of optimal control with problems of geometry and mechanics. A natural connection between mechanics and optimal control is through the Maximum Principle which yields from an optimal control problem a Hamiltonian system. The vector-field view produces invariant formulations of the Maximum Principle on manifolds. The book also dedicates particular attention to optimal control on Lie groups and symmetric spaces, which give rise to Hamiltonian systems with symmetries and thus with a great deal of structure, including the existence of conservation laws. To reinforce the connection between mechanics and control theory, the author also discusses in detail the control of certain mechanical systems.

This book is a welcome addition to a rather meager literature in the field.

Reviewer: M.F.Silva Leite (Coimbra)

### MSC:

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

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

70Q05 | Control of mechanical systems |

70H33 | Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics |

93B05 | Controllability |

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