P.D. Lebedev, A.A. Uspenskii. Construction of a nonsmooth solution in a time-optimal problem with a low order of smoothness of the boundary of the target set ... P. 108-119

Vol. 25, no. 1, 2019

Procedures for the construction of an optimal result function have been developed for a planar time-optimal control problem with a circular velocity vectorgram and nonconvex compact target set whose boundary has smoothness 1 or 2. Pseudovertices, which are characteristic points of the boundary of the target set defining the character of the singularity of this function, are studied. Differential dependences for smooth segments of the singular set are revealed, which allows to consider and construct them as arcs of integral curves. The necessary conditions for the existence of pseudovertices are found and formulas for the projections of points of the singular set in neighborhoods of pseudovertices are obtained. The proposed procedures are implemented in the form of computational algorithms. Their efficiency is illustrated by examples of the numerical solution of optimal-time control problems with different orders of smoothness of the boundaries of the target sets. Visualization of the results is performed.

Keywords: time-optimal problem, singular set, dispersing curve, optimal result function, pseudo-vertex, symmetry set

Received December 5, 2018

Revised February 7, 2019

Accepted February 11, 2019

Funding Agency: This work was supported by Resolution No. 211 of March 16, 2013, of the Government of the Russian Federation (agreement no. 02.A03.21.0006).

Pavel Dmitrievich Lebedev, Cand. Sci. (Phys.-Math.), Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia; Ural Federal University, Yekaterinburg, 620002 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; Ural Federal University, Yekaterinburg, 620002 Russia, e-mail: uspen@imm.uran.ru

Cite this article as: P.D. Lebedev, A.A. Uspenskii. Construction of a nonsmooth solution in a time-optimal problem with a low order of smoothness of the boundary of the target set, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 1, pp.  108-119. 

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