Construction of a singular set of the optimal result function in the class of spatial problems of speed control: the case of a target set with positive Gaussian curvature of the boundary.

A. A. Uspenskii   ; P. D. Lebedev

Sibirskie èlektronnye matematičeskie izvestiâ, Tome 21 (2024) no. 1, pp. 513-525

Résumé

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.

Détail

Keywords: control problem, optimal result function, scattering surface, singular set, curvature, normal, pseudovertex.

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