Abstract
С помощью основ теории групп и алгебр Ли, их (ко)присоединенных представлений и принципа максимума Понтрягина для задачи оптимального быстродействия даны независимое обоснование методов геодезического векторного поля поиска геодезических левоинвариантных (суб)финслеровых метрик на группах Ли и поиска соответствующих локально оптимальных управлений в (суб)римановом случае, а также несколько их применений.
Highlights
After Gromov’s 1980s papers, homogeneous sub-Finsler manifolds, in particular, sub-Riemannian manifolds were actively studied [1], [15], [22], [26]
The search for geodesics of homogeneousFinsler manifolds are reduced to the case of Lie groups with left-invariantFinsler metrics
A left-invariant sub-Riemannian metric d is defined on the Lie group SH(2)
Summary
After Gromov’s 1980s papers, homogeneous sub-Finsler manifolds, in particular, sub-Riemannian manifolds were actively studied [1], [15], [22], [26]. The shortest arcs on Lie groups with left-invariant (sub)-Finsler metrics are optimal trajectories of the corresponding left-invariant time-optimal problem on Lie groups [3]. This permits to apply the Pontryagin maximum principle (PMP) for their search [13]. To find geodesics of left-invariant (sub-)Finsler metrics on Lie groups and corresponding locally optimal controls in (sub-)Riemannian case we use the geodesic vector field method (Theorems 7, 8) and an improved version of method from [8], applying (co)adjoint representations. Other researchers did not apply PMP for the time-optimal problem to find geodesics of left-invariant metrics on Lie groups
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