Combined algorithms for constructing a solution to the time-optimal problem in three-dimensional space based on the selection of extreme points of the scattering surface

A class of time-optimal control problems in three-dimensional space with a spherical velocity vector is considered. A smooth regular curve $\Gamma$ is chosen as the target set. We distinguish pseudo-vertices that are characteristic points on $\Gamma$ and responsible for the appearance of a singularity in the function of the optimal result. We reveal analytical relationships between pseudo-vertices and extreme points of a singular set belonging to the family of bisectors. The found analytical representation for the extreme points of the bisector is taken as the basis for numerical algorithms for constructing a singular set. The effectiveness of the developed approach for solving non-smooth dynamic problems is illustrated by an example of numerical-analytical construction of resolving structures for the time-optimal control problem.