论文标题
次曼尼亚次控制系统的最小时间函数的部分Lipschitz
Partial Lipschitz regularity of the minimum time function for sub-Riemannian control systems
论文作者
论文摘要
在维度2或3的欧几里得空间中，我们研究了与满足Hörmander的支架产生条件的实用分析矢量磁场系统相关的最短时间问题，在该系统中，目标是非空的封闭式集合。我们表明，在维度2中，最小时间函数是局部Lipschitz连续的，而在尺寸3中，在一组量度零的补充中是Lipschitz连续的。特别是，在这两种情况下，最小时间函数都是A.E.在目标的补充方面可区分。通常，在维度3中，没有希望具有与维度2中相同的规律性结果。的确，最小时间函数在局部lipschitz连续的情况下已知示例。
In Euclidean space of dimension 2 or 3, we study a minimum time problem associated with a system of real-analytic vector fields satisfying Hörmander's bracket generating condition, where the target is a nonempty closed set. We show that, in dimension 2, the minimum time function is locally Lipschitz continuous while, in dimension 3, it is Lipschitz continuous in the complement of a set of measure zero. In particular, in both cases, the minimum time function is a.e. differentiable on the complement of the target. In dimension 3, in general, there is no hope to have the same regularity result as in dimension 2. Indeed, examples are known where the minimum time function fails to be locally Lipschitz continuous.
