Introduction to the theory of positional differentional games of systems with aftereffect (based on the $i$-smooth analisys methodology

A. V. Kim

Vestnik rossijskih universitetov. Matematika, Tome 29 (2024) no. 147, pp. 268-295

Résumé

Although the foundations of the theory of positional differential games of systems with aftereffect described by functional differential equations (FDE) were developed back in the 1970s by N. N. Krasovsky, Yu. S. Osipov and A. V. Kryazhimsky, there are still no works that, similar to [N. N. Krasovsky, A. I. Subbotin. Positional Differential Games. Moscow: Nauka, 1974, 457 p.] (hereinafter referred to as [KS]), would represent a “complete” theory of positional differential games with aftereffect. The paper presents an approach to constructively transferring all the results of the book [KS] to systems with aftereffect. This approach allows us to present the theory of positional differential games of systems with aftereffect in the same constructive and complete form as for the finite-dimensional case in [KS]. The approach is based on the methodology of $i$-smooth analysis. The obtained results of the theory of positional differential games of systems with aftereffect are completely analogous to the corresponding results of the finite-dimensional Krasovsky–Subbotin theory.

Détail

Keywords: differential games, functional differential equations, $i$-smooth analysis

Bibliographie

[1] N. N. Krasovsky, A. I. Subbotin, Positional Differential Games, Nauka Publ., Moscow, 1974, 524 (In Russian)

[2] N. N. Krasovsky, Game problems about meeting of movements, Nauka Publ., Moscow, 1970 (In Russian)

[3] N. N. Krasovskii, Yu. S. Osipov, “Linear differential-difference games”, Dokl. Akad. Nauk SSSR, 197:4 (1971), 777–780 (In Russian)

[4] N. N. Krasovsky, Yu. S. Osipov, “Control problems with incomplete information”, Mechanics of Solids, 1973, no. 4, 5–14 (In Russian)

[5] Yu. S. Osipov, “K teorii differentsialnykh igr sistem s posledeistviem”, Prikladnaya matematika i mekhanika, 35:2 (1971), 300–311

[6] Yu. S. Osipov, “Differentsialnaya igra navedeniya dlya sistem s posledeistviem”, Prikladnaya matematika i mekhanika, 35:1 (1971), 123–131

[7] N. Yu. Lukoyanov, “Funktsionalnye uravneniya tipa Gamiltona–Yakobi i differentsialnye igry s nasledstvennoi informatsiei”, Doklady RAN, 371:4 (2000), 457–461

[8] N. Yu. Lukoyanov, A. R. Plaksin, “O strategiyakh ekstremalnogo sdviga v sistemakh s zapazdyvaniem”, Tr. IMM UrO RAN, 27, no. 2, 2021, 150–161

[9] N. Yu. Lukoyanov, A. R. Plaksin, “Neravenstva dlya subgradientov funktsionala tseny v differentsialnykh igrakh dlya sistem s zapazdyvaniem”, Doklady RAN. Matematika, informatika, protsessy upravleniya, 490 (2020), 91–94 | DOI

[10] M. I. Gomoyunov, N. Yu. Lukoyanov, “On linear-quadratic differential games for fractional-order systems”, Mathematical Game Theory and its Applications, 15:2 (2023), 18–32 (In Russian)

[11] M. I. Gomoyunov, N. Yu. Lukoyanov, “Differentsialnye igry v sistemakh drobnogo poryadka: neravenstva dlya proizvodnykh funktsionala tseny po napravleniyam”, Optimalnoe upravlenie i differentsialnye igry, Sbornik statei, Trudy MIAN, 315, MIAN, M., 2021, 74–94 | DOI

[12] N. Yu. Lukoyanov, A. R. Plaksin, “K teorii pozitsionnykh differentsialnykh igr dlya sistem neitralnogo tipa”, Tr. IMM UrO RAN, 25, no. 3, 2019, 118–128 ; N. Yu. Lukoyanov, A. R. Plaksin, “On the theory of positional differential games for neutral-type systems”, Proc. Steklov Inst. Math. (Suppl.), 309:suppl. 1 (2020), S83–S92 | DOI

[13] M. I. Gomoyunov, N. Yu. Lukoyanov, A. R. Plaksin, “Ob approksimatsii minimaksnykh reshenii funktsionalnykh uravnenii Gamiltona–Yakobi dlya sistem s zapazdyvaniem”, Vypusk posvyaschen 70-letnemu yubileyu Aleksandra Georgievicha Chentsova, Tr. IMM UrO RAN, 24, no. 1, 2018, 53–62 ; M. I. Gomoyunov, N. Yu. Lukoyanov, A. R. Plaksin, “Approximation of minimax solutions of Hamilton—Jacobi functional equations for time-delay systems”, Proc. Steklov Inst. Math. (Suppl.), 304:suppl. 1 (2019), S68–S75 | DOI

[14] Yu. S. Osipov, “Differential games of systems with aftereffect”, Dokl. Akad. Nauk SSSR, 196:4 (1971), 779–782 (In Russian)

[15] Yu. S. Osipov, Problems of the Theory of Differential-Difference Games, diss. ... Doctor of Physical and Mathematical Sciences, Leningrad, 1971, 300 pp. (In Russian)

[16] A. V. Kryazhimsky, “Differential-difference game of evasion from a functional goal”, Izv. AN SSSR. Tech. Cybernetics, 4 (1973), 71–79 (In Russian)

[17] A. V. Kryazhimskii, V. I. Maksimov, “Approksimatsiya lineinykh differentsialno-raznostnykh igr”, Prikladnaya matematika i mekhanika, 42:2 (1978), 202–209

[18] A. V. Kim, $i$-Smooth Analysis and Functional-Differential Equations, IMM UB RAS, Ekaterinburg, 1996, 234 pp. (In Russian)

[19] A. V. Kim, $i$-Smooth Analysis. Theory and Applications, Scrivener Publishing, New Jersey, 2015, 270 pp.

[20] A. V. Kim, “On application of the methodology $i$-smooth analysis to elaboration of numerical methods for solving functional differential equations”, Vestnik rossiyskikh universitetov. Matematika = Russian Universities Reports. Mathematics, 26:133 (2021), 26–34 (In Russian)

[21] A. N. Kolmogorov, S. V. Fomin, Elements of the Theory of Functions and Functional Analysis, Moscow, Nauka Publ., 1976 (In Russian)