Geodesic
Relaxation of the game problem of guidance connected with alternative in guidance-evasion differential game
Vestnik rossijskih universitetov. Matematika, Tome 25 (2020) no. 130, pp. 196-244
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Differential game (DG) of guidance-evasion for a finite time interval is considered; as parameters, the target set (TS) and the set defining phase constraints (PC) are used. Player I interested in realization of guidance with TS under validity PC uses set-valued quasistrategies (nonanticipating strategies) and Player II having opposite target uses strategies with nonanticipating choice of correction instants and finite numbers of such instants. On informative level, the setting corresponds to alternative theorem of N. N. Krasovskii and A. I. Subbotin. For position not belonging to solvability set of Player I, determination of the least size of neighborhoods for set-parameters under that Player I guarantees guidance (under weakened conditions) is interested. In article, this scheme is supplemented by priority elements in questions of TS attainment and PC validity; this is realized by special parameter defining relation for sizes of corresponding neighborhoods. Under these conditions, a function of the least size of TS neighborhood is defined by procedure used program iteration method for two variants. The above-mentioned function is fixed point for one of two used “program” operators. Special type of the quality functional for which values of the above-mentioned function coincide with values of the minimax-maximin games is established.
Keywords: differential game, quasistrategy, program iteration method. 
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A. G. Chentsov. Relaxation of the game problem of guidance connected with alternative in guidance-evasion differential game. Vestnik rossijskih universitetov. Matematika, Tome 25 (2020) no. 130, pp. 196-244. http://geodesic.mathdoc.fr/item/VTAMU_2020_25_130_a7/

[1] R. Ayzeks, Differentsial’nyye igry, Mir Publ., Moscow, 1967 (In Russian)

[2] N. N. Krasovskiy, A. I. Subbotin, “Al’ternativa dlya igrovoy zadachi sblizheniya”, Prikladnaya matematika i mekhanika, 34:6 (1970), 1005–1022 (In Russian)

[3] N. N. Krasovskiy, A. I. Subbotin, Pozitsionnyye differentsial’nyye igry, Moscow, Nauka Publ., 1974 (In Russian) | MR

[4] N. N. Krasovskii, Game Problems About Meeting Motions, Fizmatlit Publ., Moscow, 1970 (In Russian)

[5] A. V. Kryazhimskiy, “On the theory of positional differential games of convergence-evasion”, Dokl. Akad. Nauk SSSR, 239:4 (1978), 779–782 (In Russian)

[6] A. I. Subbotin, Minimaksnyye neravenstva i uravneniya Gamil’tona-Yakobi, Nauka Publ., Moscow, 1991 (In Russian)

[7] A. I. Subbotin, Generalized Solutions of First-Order PDES. The Dynamical Optimization Perspective, Birkhäuser, Boston-Basel-Berlin, 1995 | MR

[8] A. I. Subbotin, Obobshchennyye resheniya uravneniy v chastnykh proizvodnykh pervogo poryadka. Perspektivy dinamicheskoy optimizatsii, Institut komp’yuternykh isledovaniy, Moscow-Izhevsk, 2003 (In Russian) | MR

[9] A. I. Subbotin, “Ob odnom svoistve subdifferentsiala”, Matem. sb., 182:9 (1991), 1315–1330; A. I. Subbotin, “On a property of the subdifferential”, Math. USSR-Sb., 74:1 (1993), 63–78 | MR | Zbl

[10] A. G. Chentsov, “The structure of a certain game-theoretic approach problem”, Dokl. Akad. Nauk SSSR, 224:6 (1975), 1272–1275 (In Russian) | Zbl

[11] A. G. Chentsov, “On a game problem of guidance with information memory”, Dokl. Akad. Nauk SSSR, 227:2 (1976), 306–308 (In Russian)

[12] A. G. Chentsov, “Ob igrovoi zadache sblizheniya v zadannyi moment vremeni”, Matem. sb., 99(141):3 (1976), 394–420

[13] A. G. Chentsov, “Ob igrovoi zadache sblizheniya k zadannomu momentu vremeni”, Izv. AN SSSR. Ser. matem., 42:2 (1978), 455–467

[14] V. I. Ukhobotov, “Postroyeniye stabil’nogo mosta dlya odnogo klassa lineynykh igr”, Prikladnaya matematika i mekhanika, 41:2 (1977), 358–364 (In Russian) | MR

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

[16] A. I. Subbotin, A. G. Chentsov, “Iteratsionnaya protsedura postroyeniya minimaksnykh i vyazkostnykh resheniy”, Doklady Akademii nauk, 348:6 (1996), 736–739 (In Russian) | MR | Zbl

[17] A. G. Chentsov, D. M. Khachai, “Relaxation of a differential game of approach-evasion and iterative methods”, Trudy Inst. Mat. i Mekh. UrO RAN, 24, no. 4, 2018, 246–269 (In Russian)

[18] A. G. Chentsov, D. M. Khachay, “Program Iterations Method and Relaxation of a Pursuit-Evasion Differential Game”, Advanced Control Techniques in Complex Engineering Systems: Theory and Applications, v. 203, Studies in Systems, Decision and Control, 2019, 129–161 | DOI | Zbl

[19] A. I. Subbotin, A. G. Chentsov, Optimizatsiya garantii v zadachakh upravleniya, Moscow, Nauka Publ., 1977 (In Russian)

[20] A. G. Chentsov, “Metod programmnykh iteratsii v igrovoi zadache navedeniya”, Tr. IMM UrO RAN, 22, no. 2, 2016, 304–321; A. G. Chentsov, “The program iteration method in a game problem of guidance”, Proc. Steklov Inst. Math. (Suppl.), 297:suppl. 1 (2017), 43–61 | MR

[21] K. Kuratovskiy, A. Mostovskiy, Teoriya mnozhestv, Moscow, Mir Publ., 1970 (In Russian)

[22] J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1977 | MR

[23] A. G. Chentsov, S. I. Morina, Extensions and Relaxations, Kluwer Acad. Publ., Dordrecht–Boston–London, 2002 | MR | Zbl

[24] N. Danford, Dzh. T. Shvarts, Lineynyye Operatory. Obshchaya Teoriya, Izd-vo inostr. lit., Moscow, 1962 (In Russian)

[25] P. Billingsli, Skhodimost’ Veroyatnostnykh Mer, Moscow, Nauka Publ., 1977 (In Russian)

[26] R. Engel’king, Obshchaya Topologiya, Mir Publ., Moscow, 1986 (In Russian) | MR

[27] V. I. Bogachev, Osnovy teorii mery, v. 2, NITS “Regulyarnaya i Khaoticheskaya Dinamika”, Moscow–Izhevsk, 2003 (In Russian)

[28] V. I. Bogachev, Slabaya Skhodimost’ mer, Institut komp’yuternykh issledovaniy, Moscow–Izhevsk, 2016 (In Russian) | MR

[29] A. G. Chentsov, “Stability iterations and an evasion problem with a constraint on the number of switchings”, Trudy Inst. Mat. i Mekh. UrO RAN, 23, no. 2, 2017, 285–302 (In Russian)

[30] Zh. D’yedonne, Osnovy Sovremennogo Analiza, Mir Publ., Moscow, 1964 (In Russian)

[31] A. G. Chentsov, Metod programmnykh iteratsiy dlya differentsial’noy igry sblizheniya-ukloneniya, Dep. v VINITI, 1933-79, Ural’skiy politekhnicheskiy institut im. S. M. Kirova, Sverdlovsk, 1979 (In Russian)