强勒让德正则性与异常曲线的短时分析
Strong Legendre regularity and short-time analysis of abnormal curves
浏览论文内容
中文总结 AI 辅助
本文研究满足戈赫条件与强广义勒让德条件的严格异常次黎曼极值曲线，证明其控制光滑、为短时间长度极小化曲线，且相关估计具一致性，未作流形维数等额外假设。
中文摘要 AI 辅助
我们研究满足戈赫（Goh）条件和强（即 coercive，强制）广义勒让德条件的严格异常次黎曼极值曲线，该条件仅施加于参考控制的勒贝格点。我们证明此类控制是光滑的，端点映射的投影扩展海森矩阵在短时间内于容许变分集合上是一致强制的，且异常余向量能定量分离参考弧的端点与非线性端点映射下容许集合的像。由此可得参考弧是刚性的，且在短时间内为长度极小化曲线：所有具有相同端点且长度不大于参考弧的容许曲线均为参考弧的重新参数化。特别地，在恒速控制中，参考控制是唯一的长度极小化曲线。所有估计在容许集合上是一致的，且未对流形的维数、分布的增长向量或端点映射微分的余秩作任何假设。
英文摘要
We study strictly abnormal sub-Riemannian extremals satisfying the Goh condition and a strong (i.e., coercive) generalized Legendre condition, imposed only at the Lebesgue points of the reference control. We prove that such a control is smooth, that the projected extended Hessian of the end-point map is uniformly coercive on the admissible set of variations for short times, and that the abnormal covector gives a quantitative separation of the end-point of the reference arc from the image of the admissible set under the nonlinear end-point map. As a consequence the reference arc is rigid and, for short times, it is a length minimizer: every admissible curve with the same end-points and no greater length is a reparametrization of the reference arc. In particular, among controls of constant speed the reference control is the unique length minimizer. All the estimates are uniform on the admissible set, and no assumption is made on the dimension of the manifold, on the growth vector of the distribution, or on the corank of the differential of the end-point map.
