Построение сингулярного множества функции оптимального результата в классе пространственных задач управления по быстродействию: случай целевого множества с положительной гауссовой кривизной границы

We study the problem of constructing a non-smooth solution for a class of spatial time-optimal control problems in the case of a three-dimensional non-convex target set $M$ with a smooth boundary $S.$ A singular set (the so-called scattering surface) is constructed, on which the optimal result function loses smoothness. For an analytical description of the singularities of the solution, pseudo vertices are introduced, which are characteristic points of the surface $S,$ which are responsible for the occurrence of singularities. The extreme points of the scattering surface, which define its boundary, are studied. A formula is found for the extreme points of the singular set in the case when the pseudo vertices are elliptical points of the surface $S.$ Necessary conditions for the existence of pseudo vertices are obtained in terms of the curvature of the normal section $S.$ An example of constructing a solution to the speed control problem based on the obtained theoretical results is given.