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Explicit Solutions for a Series of Optimization Problems with 2-Dimensional Control via Convex Trigonometry
Doklady Mathematics ( IF 0.5 ) Pub Date : 2021-01-14 , DOI: 10.1134/s1064562420050257
A. A. Ardentov , L. V. Lokutsievskiy , Yu. L. Sachkov

Abstract

We consider a number of optimal control problems with 2-dimensional control lying in an arbitrary convex compact set \(\Omega \). Solutions to these problems are obtained using methods of convex trigonometry. The paper includes (1) geodesics in the Finsler problem on the Lobachevsky hyperbolic plane; (2) left-invariant sub-Finsler geodesics on all unimodular 3D Lie groups (\({\text{SU}}(2)\), \({\text{SL}}(2)\), \({\text{SE}}(2)\), \({\text{SH}}(2)\)); (3) the problem of a ball rolling on a plane with a distance function given by \(\Omega \); and (4) a series of “yacht problems” generalizing Euler’s elastic problem, the Markov–Dubins problem, the Reeds–Shepp problem, and a new sub-Riemannian problem on SE(2).



中文翻译:

通过凸三角函数进行二维控制的一系列优化问题的显式解决方案

摘要

我们考虑二维控制中的任意最优控制问题,它们位于任意凸紧凑集\(\ Omega \)中。使用凸三角法可以解决这些问题。本文包括(1)Lobachevsky双曲平面上Finsler问题的测地线;(2)所有单模3D Lie组(\({\ text {SU}}(2)\)\({\ text {SL}}(2)\)\({ \ text {SE}}(2)\)\({\ text {SH}}(2)\)); (3)球在具有\(\ Omega \)给出的距离函数的平面上滚动的问题; (4)一系列“游艇问题”,泛化了Euler的弹性问题,Markov-Dubins问题,Reeds-Shepp问题以及关于SE(2)的新的次黎曼问题。

更新日期:2021-01-14
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