Abstract:
For the development of analytical and numerical algorithms for constructing nonsmooth solutions of optimal control problems, procedures are proposed for constructing scattering curves for a single class of control velocity problems. We consider the reduction problems for a minimal time of solutions of a dynamical system with a circular velocity vectogram for the case where the target set is generally nonconvex, and its boundary has points at which the curvature smoothness is violated. These points are referred to as pseudovertices, the characteristic points of the target set, which are responsible for the occurrence of the singularity of the optimal result function. When forming a proper reparameterization (in this case, taking into account the geometry of the velocity vector diagram) of the arc of the boundary of the target set containing a pseudovertex, the scattering curve is constructed as an integral curve. Moreover, the initial conditions of the corresponding Cauchy problem are determined by the properties of the pseudovertex. One of the numerical characteristics of the pseudovertex, the pseudovertex marker, determines the initial velocity of the material point describing a smooth portion of the scattering curve. This approach to the identification and construction (in analytical or numerical form) of singular curves was previously substantiated for a number of cases of a target boundary that are different in the order of smoothness. It should be emphasized that the case considered in this paper is the most specific, in particular, because of the revealed connection between the dynamic problem and the problem of polynomial algebra. It is proved that the pseudovertex marker is the nonpositive root of some third-order polynomial whose coefficients are determined by the one-sided derivatives of curvatures of the pseudovertex of the target set. The effectiveness of the developed theoretical methods and numerical procedures is illustrated by specific examples.
Keywords:
velocity problem, dispersing curve, bisector of a set, pseudovertex, optimal result function, curvature.
Citation:
P. D. Lebedev, A. A. Uspenskii, “Construction of a solution to a velocity problem in the case of violation of the smoothness of the curvature of the target set boundary”, Izv. IMI UdGU, 53 (2019), 98–114
\Bibitem{LebUsp19}
\by P.~D.~Lebedev, A.~A.~Uspenskii
\paper Construction of a solution to a velocity problem in the case of violation of the smoothness of the curvature of the target set boundary
\jour Izv. IMI UdGU
\yr 2019
\vol 53
\pages 98--114
\mathnet{http://mi.mathnet.ru/iimi374}
\crossref{https://doi.org/10.20537/2226-3594-2019-53-09}
\elib{https://elibrary.ru/item.asp?id=38503202}
Linking options:
https://www.mathnet.ru/eng/iimi374
https://www.mathnet.ru/eng/iimi/v53/p98
This publication is cited in the following 6 articles:
P. D. Lebedev, A. A. Uspenskii, “Metod Nyutona pri postroenii singulyarnogo mnozhestva minimaksnogo resheniya v odnom klasse kraevykh zadach dlya uravnenii Gamiltona — Yakobi”, Chelyab. fiz.-matem. zhurn., 9:1 (2024), 63–76
P. D. Lebedev, A. A. Uspenskii, “Numerical-analytic construction of a generalized solution to the eikonal equation in the plane case”, Sb. Math., 215:9 (2024), 1224–1248
A. A. Uspenskii, P. D. Lebedev, “On singularity structure of minimax solution to Dirichlet problem for eikonal type equation with discontinuous curvature of boundary of boundary set”, Ufa Math. J., 13:3 (2021), 126–151
P. D. Lebedev, A. A. Uspenskii, “Postroenie rasseivayuschikh krivykh v odnom klasse zadach bystrodeistviya pri skachkakh krivizny granitsy tselevogo mnozhestva”, Izv. IMI UdGU, 55 (2020), 93–112
P. D. Lebedev, A. A. Uspenskii, “Elementy analiticheskogo konstruktora reshenii v klasse zadach upravleniya po bystrodeistviyu s tselevym mnozhestvom s razryvnoi kriviznoi granitsy”, Vestnik rossiiskikh universitetov. Matematika, 25:132 (2020), 370–386
A. A. Uspenskii, P. D. Lebedev, “Svoistva nestatsionarnykh psevdovershin kraevogo mnozhestva pri razryve gladkosti krivizny ego granitsy v zadache Dirikhle dlya uravneniya tipa eikonala”, Sib. elektron. matem. izv., 17 (2020), 2028–2044
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