期刊
JOURNAL OF DYNAMICAL AND CONTROL SYSTEMS
卷 12, 期 2, 页码 185-216出版社
SPRINGER/PLENUM PUBLISHERS
DOI: 10.1007/s10450-006-0380-4
关键词
Nilpotent Lie algebras; extremal curves; sub-Riemannian geodesics; exponential mapping
We study the sub-Riemannian structure determined by a left-invariant distribution of rank n on a step-2 simply-connected nilpotent Lie group G of dimension n(n + 1)/2. We describe a transitive group action that leaves invariant the sub-Riemannian structure. By using geometric optimal control theory techniques, we derive necessary conditions for the length-minimality of the sub-Riemannian geodesics. We perform an integration process for the associated Hamiltonian system that yields explicit expressions for the extremal curves and the corresponding sub-Riemannian geodesics, the obtained formulas allow the complete parametrization of the exponential mapping in terms of algebraic invariants of the problem.
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