Control theory, a synthesis of geometric theory of differential equations enriched with variational principles and the associated symplectic geometry, emerges as a new mathematical subject of interest to engineers, mathematicians, and physicists. This collection of articles focuses on several distinctive research directions having origins in mechanics and differential geometry, but driven by modern control theory. The first of these directions deals with the singularities of small balls for problems of subRiemannian geomtery and provides a generic classification of singularities for twodimensional distributions of contact type in a threedimensional ambient space. The second direction deals with invariant optimal problems on Lie groups exemplified through the problem of Dublins extended to symmetric spaces, the elastic problem of Kirchhoff and its relation to the heavy top. The results described in the book are explicit and demonstrate convincingly the power of geometric formalism. The remaining directions deal with the geometric nature of feedback analyzed through the language of fiber bundles, and the connections of geometric control to nonholonomic problems in mechanics, as exemplified through the motions of a sphere on surfaces of revolution. This book provides quick access to new research directions in geometric control theory. It also demonstrates the effectiveness of new insights and methods that control theory brings to mechanics and geometry. Titles in this series are copublished with the Canadian Mathematical Society. Members of the Canadian Mathematical Society may order at the AMS member price. Readership Graduate students, research mathematicians, engineers and physicists working in control theory. Table of Contents  V. Jurdjevic  Lie determined systems and optimal problems with symmetries
 A. A. Agrachev, El C. ElA., and J. P. Gauthier  SubRiemannian metrics on \({\mathbb{R}^3}\)
 B. Bonnard and M. Chyba  SubRiemannian geometry: the Martinet case
 D. Mittenhuber  Dubins' problem in hyperbolic space
 D. Mittenhuber  Dubins' problem in the hyperbolic plane using the open disc model
 F. MonroyPérez  Three dimensional nonEuclidean Dubins' problem
 B. Jakubczyk  Symmetries of nonlinear control systems and their symbols
 J. L. F. Chapou  The motion of a sphere on a surface of revolution: a geometric approach
 J. C. MartínezGarcía  Geometry and structure in the control of linear time invariant systems
 Index
