次黎曼测地线的刻划

Characterization for Sub-Riemannian Geodesics

次黎曼测地线问题是变分学中的一个有约束的Lagrange问题,但在变分的过程中有个难点——如何推出约束条件“γ(′t)∈D,在[a,b]上几乎处处成立”的解析式。该文从最优控制论的角度,将次黎曼测地线问题转化为一个最优控制问题,将约束条件γ′(t)∈D转化为k∑i=1ui(t)Xi(t)-γ(′t)=0,进而得到了次黎曼Ham iltonianH(q,p)=12g-1(p|D,p|D)的具体形式。同时将奇异曲线刻划为非正规极值曲线的投影。

The problem of sub-Riemannian geodesics is a Lagrange problem with constraint. How to describe the constraint condition ＂γ＇（t） ∈ D, a. e. [a,b]＂ is a difficult problem. In this paper we turn the problem of sub-Riemannian geodesics into an optimal control problem, express the constraint to ∑i=1^ku＇（t）Xi （ t ） -γ＇ （t） = 0 , and sub-Riemannian Hamihonian H（ q ,p） =1/2g^-1（p｜D,p｜D）in detail. We also characterize the singular curves into abnormal extreme curves.