Geodesic
Stability iterations and an evasion problem with a constraint on the number of switchings
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 2, pp. 285-302 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

For an approach-evasion differential game, we consider a variant of the method of program iterations called stability iterations. A connection is established between the iterative procedure and the solution of an evasion problem with a constraint on the number of switchings: the stability iterations define the successful solvability set of the problem. It is proved that the evasion is possible if and only if the strict evasion is possible (i.e., the evasion with respect to neighborhoods of sets defining the approach-evasion game). We specify a representation of the strategies that guarantee the evasion with a constraint on the number of switchings. These strategies are defined as triplets whose elements are a multidimensional positional control strategy, a correction strategy realized as a mapping that takes a game position to a nonanticipating multifunctional on the trajectory space and defines the choice of the switching times, and a positive integer that satisfies the constraints on the number of switchings and specifies the number of switchings of the control. It is important that we use nonanticipating multifunctionals as a tool for generating the controls of the evading player. The paper is in line with the research carried out by N.N.Krasovskii's school on control theory and the theory of differential games.
Keywords: nonanticipating multifunctional, stability operator, correction strategy. 
@article{TIMM_2017_23_2_a23,
     author = {A. G. Chentsov},
     title = {Stability iterations and an evasion problem with a constraint on the number of switchings},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {285--302},
     year = {2017},
     volume = {23},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2017_23_2_a23/}
}
TY  - JOUR
AU  - A. G. Chentsov
TI  - Stability iterations and an evasion problem with a constraint on the number of switchings
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2017
SP  - 285
EP  - 302
VL  - 23
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TIMM_2017_23_2_a23/
LA  - ru
ID  - TIMM_2017_23_2_a23
ER  - 
%0 Journal Article
%A A. G. Chentsov
%T Stability iterations and an evasion problem with a constraint on the number of switchings
%J Trudy Instituta matematiki i mehaniki
%D 2017
%P 285-302
%V 23
%N 2
%U http://geodesic.mathdoc.fr/item/TIMM_2017_23_2_a23/
%G ru
%F TIMM_2017_23_2_a23
A. G. Chentsov. Stability iterations and an evasion problem with a constraint on the number of switchings. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 23 (2017) no. 2, pp. 285-302. http://geodesic.mathdoc.fr/item/TIMM_2017_23_2_a23/

[1] Aizeks R., Differentsialnye igry, Mir, M., 1967, 480 pp. | MR

[2] Krasovskii N.N., Subbotin A.I., Pozitsionnye differentsialnye igry, Nauka, M., 1974, 456 pp. | MR

[3] Krasovskii N.N., Upravlenie dinamicheskoi sistemoi. Zadacha o minimume garantirovannogo rezultata, Nauka, M., 1985, 516 pp. | MR

[4] Chentsov A.G., “O strukture odnoi igrovoi zadachi sblizheniya”, Dokl. AN SSSR, 224:6 (1975), 1272–1275 | MR | Zbl

[5] Chentsov A.G., “K igrovoi zadache navedeniya”, Dokl. AN SSSR, 226:1 (1976), 73–76 | MR | Zbl

[6] Chistyakov S.V., “K resheniyu igrovykh zadach presledovaniya”, Prikl. matematika i mekhanika, 41:5 (1977), 825–832 | MR

[7] Ukhobotov V.I., “Postroenie stabilnogo mosta dlya odnogo klassa lineinykh igr”, Prikl. matematika i mekhanika, 41:2 (1977), 358–364 | MR

[8] Chentsov A.G., Metod programmnykh iteratsii dlya differentsialnoi igry sblizheniya-ukloneniya, Dep. v VINITI, No1933-79, In-t matematiki i mekhaniki UNTs AN SSSR, Sverdlovsk, 1979, 102 pp.

[9] Subbotin A.I., Chentsov A.G., Optimizatsiya garantii v zadachakh upravleniya, Nauka, M., 1981, 286 pp. | MR

[10] Krasovskii N.N., Subbotin A.I., “Alternativa dlya igrovoi zadachi sblizheniya”, Prikl. matematika i mekhanika, 34:6 (1970), 1005–1022

[11] Chentsov A.G., O zadache upravleniya s ogranichennym chislom pereklyuchenii, Dep. v VINITI, No4942-B87, UPI im. S.M. Kirova, Sverdlovsk, 1987, 44 pp.

[12] Chentsov A.G., O differentsialnykh igrakh s ogranicheniem na chislo korrektsii,2, Dep. v VINITI, No5406-80, In-t matematiki i mekhaniki UNTs AN SSSR, Sverdlovsk, 1980, 55 pp. | Zbl

[13] Kryazhimskii A.V., “K teorii pozitsionnykh differentsialnykh igr sblizheniya-ukloneniya”, Dokl. AN SSSR, 239:4 (1978), 779–782 | MR

[14] Kuratovskii K., Mostovskii A., Teoriya mnozhestv, Mir, M., 1970, 416 pp. | MR

[15] Dedonne Zh., Osnovy sovremennogo analiza, Mir, M., 1964, 430 pp.

[16] Billingsli P., Skhodimost veroyatnostnykh mer, Nauka, M., 1977, 352 pp. | MR

[17] Danford N., Shvarts Dzh.T., Lineinye operatory. Obschaya teoriya, Izd-vo inostr. lit., M., 1962, 895 pp. | MR

[18] Chentsov A.G., Morina S.I., Extensions and relaxations, Kluwer Acad. Publ., Dordrecht;Boston;London, 2002, 408 pp. | MR | Zbl

[19] Chentsov A.G., Elementy konechno-additivnoi teorii mery, v. I, UGTU-UPI, Ekaterinburg, 2009, 389 pp.

[20] Neve Zh., Matematicheskie osnovy teorii veroyatnostei, Mir, M., 1969, 309 pp. | MR

[21] Chentsov A.G., “Metod programmnykh iteratsii v igrovoi zadache navedeniya”, Tr. In-ta matematiki i mekhaniki UrO RAN, 22:2 (2016), 304–321 | DOI | MR