### 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

#### Abstract

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.

#### Keywords

Time-optimal problem, Dispersing surface, Bisector, Pseudo-vertex, Extreme point, Curvature, Singular set, Frenet-Serret frame (TNB frame)

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