一系列二维控制亚Finsler问题的凸三角解
Extremals for a series of sub-Finsler problems with 2-dimensional control via convex trigonometry
摘要：
本文考虑一类二维控制位于任意凸紧集$Ω$上的最优控制问题。所考虑的问题是很好地研究的情况下，当$Ω$是一个单位盘，但几乎没有研究任意$Ω$。我们推导出这些问题的极值在一般情况下，通过使用机械凸三角，这使我们能够做到这一点相同和独立的形状$Ω$。本文讨论了Lobachevsky双曲平面上的Finsler问题，所有么模三维李群（SU（2），SL（2），SE（2），SH（2））上的左不变子Finsler问题，距离函数由$Ω$给出的平面上的滚球问题，Lobachevsky双曲平面上的Finsler问题和Lobachevsky双曲平面上的左不变子Finsler问题中的测地线。（iv）一系列“游艇问题”，推广了Euler弹性问题、Markov-Dubins问题、Reeds-Shepp问题和一个新的亚黎曼问题;（v）平面动力学问题。
英文摘要：
We consider a series of optimal control problems with 2-dimensional control lying in an arbitrary convex compact set $Ω$. The considered problems are well studied for the case when $Ω$ is a unit disc, but barely studied for arbitrary $Ω$. We derive extremals to these problems in general case by using machinery of convex trigonometry, which allows us to do this identically and independently on the shape of $Ω$. The paper describes geodesics in (i) the Finsler problem on the Lobachevsky hyperbolic plane; (ii) left-invariant sub-Finsler problems on all unimodular 3D Lie groups (SU(2), SL(2), SE(2), SH(2)); (iii) the problem of rolling ball on a plane with distance function given by $Ω$; (iv) a series of "yacht problems" generalizing Euler's elastic problem, Markov-Dubins problem, Reeds-Shepp problem and a new sub-Riemannian problem on SE(2); and (v) the plane dynamic motion problem.
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