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 Izv. IMI UdGU, 2017, Volume 49, Pages 17–54 (Mi iimi338)

Iterations of stability and the evasion problem with a constraint on the number of switchings of the formed control

A. G. Chentsovab

a N. N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi, 16, Yekaterinburg, 620990, Russia
b Institute of Radioelectronics and Information Technologies, Ural Federal University, ul. Mira, 32, Yekaterinburg, 620002, Russia

Abstract: This paper is concerned with one variant of the programmed iterations method used for solving the differential game of guidance-evasion. The proposed procedure is connected with iterations on the basis of the property of stability of sets introduced by N.N. Krasovskii. A relationship is established between the resulting iteration procedure and the solution to the evasion problem under a constraint on the switching number of the formed control: the stability iterations define the set of successful solvability of the above-mentioned problem. It is proved that the guaranteed realization of evasion is possible if and only if (guaranteed) strong evasion (namely, the evasion with respect to neighborhoods of sets defining the guidance-evasion game) is realizable. A representation of strategies guaranteeing the evasion with constraints on the switching number is presented. The concrete operation of every such strategy consists in the formation of constant control extruding the trajectory from the set corresponding to the next iteration on the basis of the stability operator. The duration of operation of the above-mentioned control is defined in terms of the result of the multifunctional employment defined on the trajectory space; the values of this multifunctional are nonempty subsets of the remaining time interval. Attention is given to problems involved in the convergence, in the sense of Hausdorff metric, of fragments of sets which are realized by the iteration procedure. On this basis, conditions for convergence (in the Hausdorff metric) for the sets-iterations themselves are obtained.

Keywords: method of programmed iterations, nonanticipating multifunctional, stability operator, correction strategy.

DOI: https://doi.org/10.20537/2226-3594-2017-49-02

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Bibliographic databases:

Document Type: Article
UDC: 517.977, 519.837.3
MSC: 37N35, 65J15, 47J25, 52A01, 91A25

Citation: A. G. Chentsov, “Iterations of stability and the evasion problem with a constraint on the number of switchings of the formed control”, Izv. IMI UdGU, 49 (2017), 17–54

Citation in format AMSBIB
\Bibitem{Che17} \by A.~G.~Chentsov \paper Iterations of stability and the evasion problem with a constraint on the number of switchings of the formed control \jour Izv. IMI UdGU \yr 2017 \vol 49 \pages 17--54 \mathnet{http://mi.mathnet.ru/iimi338} \crossref{https://doi.org/10.20537/2226-3594-2017-49-02} \elib{http://elibrary.ru/item.asp?id=29357380} 

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This publication is cited in the following articles:
1. A. G. Chentsov, D. M. Khachai, “Relaksatsiya differentsialnoi igry sblizheniya-ukloneniya i metody iteratsii”, Tr. IMM UrO RAN, 24, no. 4, 2018, 246–269
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