Geometric Control Theory

International Journal of Electrical Engineering Education, Jul 1998 by Bell, D J

Geometric Control Theory: V. JURDJEVIC (Cambridge University Press, 1997, 492 pp., L60, $79.95, hardback)

The author of this book has been a major player in the development of the geometric approach to nonlinear control systems over the last twenty-five years. Unlike other texts which have appeared on the same subject, this book incorporates the geometric description of optimal control problems within the framework of vector fields and Lie algebras. Indeed, over half of the fourteen chapters are devoted to optimal control.

The book is a graduate text and the reader is assumed to have a knowledge of mathematics to engineering degree level. A fine feature of this text is the way the author illustrates the concepts through well known examples from classical mechanics. Readers with a training on traditional British applied mathematics will feel particularly at home with such problems as spheres rolling without slipping on horizontal planes or on other spheres, and rotation of a rigid body about its centre of gravity. Most concepts are defined as they arise in the text although the idea of a mathematical group is assumed known.

The first six chapters discuss reachable sets and controllability and contain much the same material as can be found in other standard references on geometric control theory. However, one of the main attractions of the present book is the wide discussion on optimal control over the remaining eight chapters. These latter chapters, in particular, deal with optimal problems on Lie groups and the maximum principle for variational problems on manifolds. Contributions from the calculus of variations, mechanics, and geometry are all incorporated into optimal control theory and make these chapters really outstanding. Variational problems on Lie groups show clear connections between Kirchhoff 's elastic problem and the heavy top problem. The geometric setting leads to an easy derivation of the equations for the latter problem. Symmetry, integrals of motion, and Hamilton-Jacobi theory all have a place in this very comprehensive exposition on optimal control.

No concessions are made to the non-mathematically inclined reader; full mathematical proofs are given. For example, the proof of the orbit theorem is given over three and a half pages (the idea of an Hausdorff topological space is here assumed without comment). However, the 'Notes and sources' given at the end of each chapter are most helpful in guiding one to further reading. Lots of problems are distributed throughout the text but no answers or solutions are offered.

In summary, the effort taken to study this latest text on the geometric approach to nonlinear control systems is time well spent. It should find a place on the reading lists for all graduate courses which contain this aspect of control theory.

D. J. BELL Department of Mathematics, UMIST