Please use this identifier to cite or link to this item: http://hdl.handle.net/10995/102688
Title: Construction of scattering curves in one class of time-optimal control problems with leaps of a target set boundary curvature
Построение рассеивающих кривых в одном классе задач быстродействия при скачках кривизны границы целевого множества
Authors: Lebedev, P. D.
Uspenskii, A. A.
Issue Date: 2020
Publisher: Udmurt State University
Citation: Lebedev P. D. Construction of scattering curves in one class of time-optimal control problems with leaps of a target set boundary curvature / P. D. Lebedev, A. A. Uspenskii. — DOI 10.35634/2226-3594-2020-55-07 // Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta. — 2020. — Vol. 55. — P. 93-112.
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. © 2020 Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta. All rights reserved.
Keywords: CURVATURE
DISPERSING LINE
HAMILTON–JACOBI EQUATION
PSEUDO VERTEX
SINGULAR SET
TANGENT
TIME-OPTIMAL PROBLEM
URI: http://hdl.handle.net/10995/102688
Access: info:eu-repo/semantics/openAccess
SCOPUS ID: 85093905071
PURE ID: 13405132
ISSN: 22263594
DOI: 10.35634/2226-3594-2020-55-07
metadata.dc.description.sponsorship: 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).
Appears in Collections:Научные публикации, проиндексированные в SCOPUS и WoS CC

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