zbMATH — the first resource for mathematics

Theory of chattering control with applications to astronautics, robotics, economics, and engineering. (English) Zbl 0820.70003
The book is an interesting and modern study of some problems in control theory in the language of differential geometry. It is devoted to exploration of the phenomenon “chattering” (optimal control with an infinite number of switches on a finite-time interval).
After an introduction to chattering problem with special emphasis on a heuristic approach, a detailed analysis of Fuller’s problem is made both for historical reasons and for treating the problem as a simple model for creating the whole theory.
The main theorems of the book are: a theorem on chattering bundles, a theorem on Lagrange manifolds, the chattering optimality theorem, and a theorem on regular projection. Further, the book deals with the ubiquity of Fuller’s phenomenon; the results of I. Kupka are outlined and it is proved that, in general, the codimension of chattering is not greater than 7. The higher order singular modes are treated.
The sixth chapter is devoted to the applications of the chattering theory to problems in mechanics, robot control, space navigation, mathematical economics, etc. The authors’ theory has a complete form in the case of singular modes of second order for systems with single input.
Finally, the perspectives of study of chattering modes for control systems with several inputs are considered. Inspite of the high theoretical level of the book, it is appropriate for mathematically educated control engineers.

70-02 Research exposition (monographs, survey articles) pertaining to mechanics of particles and systems
70Q05 Control of mechanical systems
70B15 Kinematics of mechanisms and robots
93C85 Automated systems (robots, etc.) in control theory
74M05 Control, switches and devices (“smart materials”) in solid mechanics