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Geometric Control and Non-holonomic Mechanics
Edited by: V. Jurdjevic and R. W. Sharpe, University of Toronto, ON, Canada
A co-publication of the AMS and Canadian Mathematical Society.
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Conference Proceedings, Canadian Mathematical Society
1998; 239 pp; softcover
Volume: 25
ISBN-10: 0-8218-0795-1
ISBN-13: 978-0-8218-0795-8
List Price: US$60
Member Price: US$48
Order Code: CMSAMS/25
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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.

Table of Contents

  • V. Jurdjevic -- Lie determined systems and optimal problems with symmetries
  • A. A. Agrachev, El C. El-A., and J. P. Gauthier -- Sub-Riemannian metrics on \({\mathbb{R}^3}\)
  • B. Bonnard and M. Chyba -- Sub-Riemannian 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. Monroy-Pérez -- Three dimensional non-Euclidean 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. Martínez-García -- Geometry and structure in the control of linear time invariant systems
  • Index
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