The Dynamics of Control Fritz Colonius and Wolfgang Kliemann, Birkhauser Press (2000), price $99, ISBN 0817636838.
Reviewer: V. Zharnitsky, University of Illinois, Urbana-Champaign USA.

The Dynamics of Control provides a non traditional approach to the subjects of Dynamical Systems theory and Control theory. It is probably the first book where such notions as Morse decomposition, chain recurrence from dynamical systems theory are systematically used for control systems. Previous books treating dynamics and control usually consider the two as separate subjects.

The main text consists of the three parts: Global Theory, Linearization, and Applications. There is a short instructive introduction and an appendix where the necessary background can be found for virtually any reader (chapter on dynamical systems for which might be useful for control theorists and a chapter on geometric control theory that will be helpful for the readers who are less familiar with control theory). Given that the goal is to bring together the two communities, such self contained appendix is very helpful.

In the informal introduction eight basic problems at the interface of dynamics and control are formulated, which are later developed throughout the book. (The problems are formulated somewhat implicitly. It takes some effort to realize what they are).

The book contains many examples which are helpful in mastering the material. There is even a special section on applications (chapters 8-13). Many examples go all the way from abstract notions to getting the actual numbers (obtained either numerically or analytically). For readers familiar with dynamical systems theory it would be interesting to see such classical examples as Saddle-Node and Pitchfork bifurcations, Lorentz , and Bogdanov-Takens systems presented in the context of control systems.

In Summary, this book provides a fresh look at classical dynamics problems and is a source of new problems for dynamics community. Also, the systematic use of dynamical systems theory as applied to control systems should be interesting to control theorists.