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Shortest Paths for Sub-Riemannian Metrics on Rank-Two Distributions
Wensheng Liu and Héctor J. Sussmann, Rutgers University, New Brunswick, NJ
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Memoirs of the American Mathematical Society
1996; 104 pp; softcover
Volume: 118
ISBN-10: 0-8218-0404-9
ISBN-13: 978-0-8218-0404-9
List Price: US$44 Individual Members: US$26.40
Institutional Members: US\$35.20
Order Code: MEMO/118/564

This work studies length-minimizing arcs in sub-Riemannian manifolds $$(M, E, G)$$ where the metric $$G$$ is defined on a rank-two bracket-generating distribution $$E$$. The authors define a large class of abnormal extremals--the "regular" abnormal extremals--and present an analytic technique for proving their local optimality. If $$E$$ satisfies a mild additional restriction-valid in particular for all regular two-dimensional distributions and for generic two-dimensional distributions--then regular abnormal extremals are "typical," in a sense made precise in the text. So the optimality result implies that the abnormal minimizers are ubiquitous rather than exceptional.

Graduate students, mathematicians, physicists, engineers interested in geometry, optimal control theory, or the calculus of variations.

• Introduction
• Three examples
• Notational conventions and definitions
• Abnormal extremals
• Sub-Riemannian manifolds, length minimizers and extremals
• Regular abnormal extremals for rank-two distributions
• Local optimality of regular abnormal extremals
• Strict abnormality
• Some special cases
• Appendix A: The Gaveau-Brockett problem
• Appendix B: Proof of Theorem 1
• Appendix C: Local optimality of normal extremals
• Appendix D: Rigid sub-Riemannian arcs and local optimality
• Appendix E: A nonoptimality proof
• References