P.D. Lebedev, A.A. Uspenskii. On the analytical construction of solutions for one class of time-optimal control problems with nonconvex target set ... P. 128-140

A time-optimal control problem with a circular velocity vectogram is considered. For one class of nonconvex planar target sets such that a part of their boundary coincides with a line segment, conditions are found that allow one to construct branches of singular (scattering) curves in analytical form. Explicit formulas are obtained for pseudovertices, i.e., singular points of the boundary of the target set generating branches of the singular set. An analytical relation is revealed between the endpoints of different optimal trajectories that have the same initial conditions on the singular set and hit the target set in a neighborhood of a pseudovertex. Formulas are found for the extreme points of branches of the singular set. The developed approaches to the exact construction of nonsmooth solutions of dynamic control problems are illustrated with examples.

Keywords: scattering curve, pseudovertex, mapping, curvature

Received April 31, 2021

Revised May 31, 2021

Accepted June 7, 2021

Funding Agency: P.D. Lebedev’s research is supported by the Russian Science Foundation (project no. 19-11-00105).

Pavel Dmitrievich Lebedev, Cand. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: pleb@yandex.ru

Aleksandr Aleksandrovich Uspenskii, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: uspen@imm.uran.ru

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Cite this article as: P.D. Lebedev, A.A. Uspenskii. On the analytical construction of solutions for one class of time-optimal control problems with nonconvex target set, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2021, vol. 27, no. 3, pp. 128–140.