Spiral-Like Extremals near a Singular Surface in a Rocket Control Problem

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.