In this paper, we consider the minimum time problem for a space rocket whose dynamics is given by a control-affine system with drift. The admissible control set is a disc. We study extremals in the neighbourhood of singular points of the second order. Our approach is based on applying the method of a descending system of Poisson brackets and the Zelikin – Borisov method for resolution of singularities to the Hamiltonian system of Pontryagin’s maximum principle. We show that in the neighbourhood of any singular point there is a family of spiral-like solutions of the Hamiltonian system that enter the singular point in a finite time, while the control performs an infinite number of rotations around the circle.

在本文中，我们考虑了由具有漂移的控制仿射系统给出其动力学的太空火箭的最短时间问题。可接受的控制集是一个圆盘。我们研究二阶奇点附近的极值。我们的方法是基于应用泊松括号降序系统的方法和 Zelikin – Borisov 方法来解决 Pontryagin 最大原理的哈密顿系统的奇点。我们表明，在任何奇异点的邻域中，存在哈密顿系统的一系列螺旋状解，它们在有限时间内进入奇异点，而控制系统围绕圆进行无限次旋转。