This text is an enhanced, English version of the Russian edition, published in early 2021 and is appropriate for an introductory course in geometric control theory. The concise presentation provides an accessible treatment of the subject for advanced undergraduate and graduate students in theoretical and applied mathematics, as well as to experts in classic control theory for whom geometric methods may be introduced. Theory is accompanied by characteristic examples such as stopping a train, motion of mobile robot, Euler elasticae, Dido's problem, and rolling of the sphere on the plane. Quick foundations to some recent topics of interest like control on Lie groups and sub-Riemannian geometry are included. Prerequisites include only a basic knowledge of calculus, linear algebra, and ODEs; preliminary knowledge of control theory is not assumed. The applications problems-oriented approach discusses core subjects and encourages the reader to solve related challenges independently. Highly-motivated readers can acquire working knowledge of geometric control techniques and progress to studying control problems and more comprehensive books on their own. Selected sections provide exercises to assist in deeper understanding of the material.

Yuri Sachkov is the Chief of Control Processes Research Center at Program Systems Institute, Russian Academy of Sciences, in Pereslavl-Zalessky, Russia. His research interests include optimal control theory, sub-Riemannian, sub-Finsler and sub-Lorentzian geometry, and their applications to mechanics, robotics, and vision. Prof. Sachkov is a world renown expert in geometric control theory and has authored more than 80 research papers in leading international and Russian journals. Sachkov has also authored two well-known books: (with A.A. Agrachev) Control Theory from the Geometric Viewpoint, Springer (c) 2004, and Controllability and symmetries of invariant systems on Lie groups and homogeneous spaces (in Russian), Moscow, Fizmatlit, 2007. He is a managing editor of Journal of Dynamical and Control Systems, Springer.

List of figures.- 1. Introduction.- 2. Controllability problem.- 3. Optimal control problem.- 4. Solution to optimal control problems.- 5. Conclusion.- A. Elliptic integrals, functions and equation of pendulum.- Bibliography and further reading.- Index.