Extremals in the vicinity of a second-order singular surface in the problem of rocket control.

We will consider the time minimizing problem a space rocket. The rocket dynamics is given by a system with drift, which is affine on control. In this problem, the control is two-dimensional and changes in the unit circle. We study extremals in a neighborhood of singular points of the second order. We prove that in the neighborhood of any singular extremal of the second order, there exist non-singular extremals in the form of logarithmic spirals that join the singular one in a finite time, while the corresponding control performs a countable number of revolutions along the unit circle. The proof is based on the application of the method of the descending system of Poisson brackets and the Zelikin-Borisov method of resolving the singularities of the Hamiltonian system of the Pontryagin maximum principle.