# Geometric Control and Non-holonomic Mechanics

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*V. Jurdjevic; R. W. Sharpe*

A co-publication of the AMS and Canadian Mathematical Society

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 sub-Riemannian geomtery and provides a generic
classification of singularities for two-dimensional distributions of
contact type in a three-dimensional 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 non-holonomic 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.