论文标题
新的二阶最优条件下
New second-order optimality conditions in sub-Riemannian Geometry
论文作者
论文摘要
我们研究了亚riemannian测量问题的扩展终点图的二阶扩展的几何形状。将几何现实转化为方程式，我们在亚riemannian几何形状中得出了新的二阶必要最佳条件。特别是，我们找到了异常亚riemannian Geodesics速度的颂歌。它允许将异常的最小化器分为两个类别，我们建议将其称为2师范和2-抗极端的极端。在2级正常情况下，上述ode完全确定了曲线的速度，而在2-迫切情况下，在某些或所有点都无法确定速度。随着提出的结果的一些增强，应该有可能证明所有2正常的极端物质的规律性（2-敏锐的情况似乎需要研究高阶条件），从而迈出了解决亚养蜂异常地理位置的平稳性问题的一步。 作为副产品，我们提出了GOH条件的新推导。我们还证明，假设比[Boarotto，Monti，Palmurella，2020]中使用的假设弱，以得出三阶的GOH条件，这意味着异常的极端异常的$ C^2 $规律性。
We study the geometry of the second-order expansion of the extended end-point map for the sub-Riemannian geodesic problem. Translating the geometric reality into equations we derive new second-order necessary optimality conditions in sub-Riemannian Geometry. In particular, we find an ODE for velocity of an abnormal sub-Riemannian geodesics. It allows to divide abnormal minimizers into two classes, which we propose to call 2-normal and 2-abnormal extremals. In the 2-normal case the above ODE completely determines the velocity of a curve, while in the 2-abnormal case the velocity is undetermined at some, or at all points. With some enhancement of the presented results it should be possible to prove the regularity of all 2-normal extremals (the 2-abnormal case seems to require study of higher-order conditions) thus making a step towards solving the problem of smoothness of sub-Riemannian abnormal geodesics. As a by-product we present a new derivation of Goh conditions. We also prove that the assumptions weaker than these used in [Boarotto, Monti, Palmurella, 2020] to derive third-order Goh conditions, imply piece-wise-$C^2$ regularity of an abnormal extremal.
