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

*2023, Volume 28, Number 2, pp. 148-161*

Author(s):

**Ronzhina M. I., Manita L. A.**

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.

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