This text explores the state-of-the-art in the rapidly developing theory of impulse control and introduces the theory of singular space-time transformations, a new method for studying shock mechanical systems. Two approaches in the theory of impulse control are presented: The first, more traditional approach defines the impulsive action as a discontinuity of phase coordinates depending on the current time, the state preceding the action, and its magnitude. The second requires the use of modern methods for describing dynamical systems - differential equations with measures. The impulse is treated as an idealization of a very short action of high magnitude, which produces an almost abrupt change of phase coordinates. The relation between these two approaches is also discussed, and several applications, both traditional and emerging, are considered.

This text is intended for graduate students and researchers in control engineering and optimal control theory for dynamical systems. Readers are assumed to be familiar with the theory of ODEs, optimal control, and functional analysis, though an appendix is included that covers many of the necessary mathematical concepts.

Auteur

Boris Miller is the Principal research fellow of the Laboratory of Models and Algorithms for Image Processing of the A.A. Kharkevich Institute for Information Transmission Problems of the Russian Academy of Sciences (IITP RAS). He is currently a member of the editorial boards of Automation and Remote Control and The Journal of Information Processes.

Evgeny Rubinovich is the principal research fellow of the Incomplete Data Control Laboratory of the V.A.Trapeznikov Institute of Control Sciences of the Russian Academy of Sciences (ICS RUS). He is also Professor in the Department of Embedded Control Systems, The Moscow Institute of Physics and Technology (MIPT). Professor Rubinovich is currently Deputy Chief Editor of Automation and Remote Control and a member of the editorial board of Control Sciences.