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Izv. IMI UdGU, 2020, Volume 55, Pages 93–112 (Mi iimi393)  

MATHEMATICS

Construction of scattering curves in one class of time-optimal control problems with leaps of a target set boundary curvature

P. D. Lebedevab, A. A. Uspenskiiab

a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi, 16, Yekaterinburg, 620219, Russia
b Ural Federal University, ul. Mira, 19, Yekaterinburg, 620002, Russia

Abstract: We consider a time-optimal control problem on the plane with a circular vectogram of velocities and a non-convex target set with a boundary having a finite number of points of discontinuity of curvature. We study the problem of identifying and constructing scattering curves that form a singular set of the optimal result function in the case when the points of discontinuity of curvature have one-sided curvatures of different signs. It is shown that these points belong to pseudo-vertices that are characteristic points of the boundary of the target set, which are responsible for the generation of branches of a singular set. The structure of scattering curves and the optimal trajectories starting from them, which fall in the neighborhood of the pseudo-vertex, is investigated. A characteristic feature of the case under study is revealed, consisting in the fact that one pseudo-vertex can generate two different branches of a singular set. The equation of the tangent to the smoothness points of the scattering curve is derived. A scheme is proposed for constructing a singular set, based on the construction of integral curves for first-order differential equations in normal form, the right-hand sides of which are determined by the geometry of the boundary of the target in neighborhoods of the pseudo-vertices. The results obtained are illustrated by the example of solving the control problem when the target set is a one-dimensional manifold.

Keywords: time-optimal problem, dispersing line, curvature, tangent, Hamilton–Jacobi equation, singular set, pseudo vertex.

Funding Agency Grant Number
Russian Foundation for Basic Research 18-01-00221_a
18-01-00264_а
This work was funded by the Russian Foundation for Basic Research (Theorems 3.1 and 3.3 were proved by P. D. Lebedev with the support of the project no. 18–01–00221; Theorem 3.2 was proved by A. A. Uspenskii with the support of the project no. 18–01–00264).


DOI: https://doi.org/10.35634/2226-3594-2020-55-07

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Bibliographic databases:

UDC: 517.977
MSC: 35A18, 35F16, 35D40, 58K70
Received: 27.02.2020

Citation: P. D. Lebedev, A. A. Uspenskii, “Construction of scattering curves in one class of time-optimal control problems with leaps of a target set boundary curvature”, Izv. IMI UdGU, 55 (2020), 93–112

Citation in format AMSBIB
\Bibitem{LebUsp20}
\by P.~D.~Lebedev, A.~A.~Uspenskii
\paper Construction of scattering curves in one class of time-optimal control problems with leaps of a target set boundary curvature
\jour Izv. IMI UdGU
\yr 2020
\vol 55
\pages 93--112
\mathnet{http://mi.mathnet.ru/iimi393}
\crossref{https://doi.org/10.35634/2226-3594-2020-55-07}


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