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JOURNALS // Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports] // Archive

Sib. Èlektron. Mat. Izv., 2024 Volume 21, Issue 1, Pages 513–525 (Mi semr1700)

Differentical equations, dynamical systems and optimal control

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

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, S. Kovalevskaya str., 16, Yekaterinburg, 620990, Russia

Abstract: 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.

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

UDC: 514.752.43, 517.977.55

MSC: 53A05, 49M41

Received May 23, 2022, published June 23, 2024

DOI: doi.org/10.33048/semi.2024.21.037



© Steklov Math. Inst. of RAS, 2025