Differential games in a Banach space on a fixed chain

Andrey V. Chernov

Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 12 (2020) no. 3, pp. 89-118.

Résumé

The paper is devoted to obtaining the sufficient conditions for existence of $\varepsilon$-equilibrium in the sense of piecewise program strategies in antagonistic games associated with nonlinear non-autonomous controlled differential equation in a Banach space and cost functional of a general enough form. The concept of piecewise program strategies in such a game is defined on the base of a concept of Volterra set chain for a right-hand operator of the corresponding integral equation controlled by the opponent players and according to a given partition of the time segment. As example we consider the game associated with a nonlinear pseudoparabolic partial differential equation governing the evolution of electric field in a semiconductor.

Détail

Keywords: differential game, nonlinear differential equation in a Banach space, piecewise program strategies, $\varepsilon$-equilibrium.
Mots-clés : Volterra set chain

Bibliographie

[1] Bonch-Bruevich V. L., Kalashnikov S. G., Fizika poluprovodnikov, Nauka, M., 1990

[2] Vasilev F. P., “O dvoistvennosti v lineinykh zadachakh upravleniya i nablyudeniya”, Differents. uravneniya, 31:11 (1995), 1893–1900 | MR | Zbl

[3] Vlasenko A. A., Chikrii A. A., “Ob odnoi differentsialnoi igre v sisteme s raspredelennymi parametrami”, Tr. IMM UrO RAN, 20, no. 4, 2014, 71–80

[4] Vlasenko A. A., Rutkas A. G., Chikrii A. A., “O differentsialnoi igre v abstraktnoi parabolicheskoi sisteme”, Tr. IMM UrO RAN, 21, no. 2, 2015, 26–40

[5] Gaevskii Kh., Grëger K., Zakharias K., Nelineinye operatornye uravneniya i operatornye differentsialnye uravneniya, Mir, M., 1978

[6] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz, Nauka, M., 1984

[7] Karchevskii M. M., Pavlova M. F., Uravneniya matematicheskoi fiziki. Dopolnitelnye glavy, KGU, Kazan, 2012

[8] Korpusov M. O., Sveshnikov A. G., “Razrushenie reshenii silno nelineinykh uravnenii psevdoparabolicheskogo tipa”, Sovremennaya matematika i ee prilozheniya, 40, 2006, 3–138

[9] Korpusov M. O., “Usloviya globalnoi razreshimosti nachalno-kraevoi zadachi dlya nelineinogo uravneniya psevdoparabolicheskogo tipa”, Differents. uravneniya, 41:5 (2005), 678–685 | MR | Zbl

[10] Korpusov M. O., Sveshnikov A. G., “Trekhmernye nelineinye evolyutsionnye uravneniya psevdoparabolicheskogo tipa v zadachakh matematicheskoi fiziki”, Zh. vychisl. matem. i matem. fiz., 43:12 (2003), 1835–1869 | MR | Zbl

[11] Lyusternik L. A., Sobolev V. I., Elementy funktsionalnogo analiza, Nauka, M., 1965 | MR

[12] Osipov Yu. S., “K teorii differentsialnykh igr v sistemakh s raspredelennymi parametrami”, DAN SSSR, 223:6 (1975), 1314–1317 | MR | Zbl

[13] Petrosyan L.A, Zenkevich N. A., Semina E. A., Teoriya igr, Vysshaya shkola, M., 1998

[14] Satimov N. Yu., Tukhtasinov M., “Ob uklonenii ot vstrechi v odnom klasse raspredelennykh upravlyaemykh sistem”, Matem. zametki, 97:5 (2015), 749–760 | MR | Zbl

[15] Sokolov S. V., “O reshenii zadachi differentsialnoi igry dlya raspredelennykh dinamicheskikh sistem”, Problemy upravleniya i informatiki, 157:1 (2004), 71–77

[16] Furman A. S., “O stratifikatsii ob'emnogo zaryada pri perekhodnykh protsessakh v poluprovodnikakh”, Fizika tverdogo tela, 28:7 (1986), 2083–2090

[17] Chernov A. V., “O volterrovykh funktsionalno-operatornykh igrakh na zadannom mnozhestve”, MTIiP, 3:1 (2011), 91–117 | MR | Zbl

[18] Chernov A. V., “O suschestvovanii $\varepsilon$-ravnovesiya v volterrovykh funktsionalno-operatornykh igrakh bez diskriminatsii”, MTIiP, 4:1 (2012), 74–92 | MR | Zbl

[19] Chernov A. V., “Ob $\varepsilon$-ravnovesii v beskoalitsionnykh funktsionalno-operatornykh igrakh so mnogimi uchastnikami”, Trudy Instituta matematiki i mekhaniki UrO RAN, 19, no. 1, 2013, 316–328

[20] Chernov A. V., “O suschestvovanii $\varepsilon$-ravnovesiya v differentsialnykh igrakh, svyazannykh s ellipticheskimi uravneniyami, upravlyaemymi mnogimi igrokami”, MTIiP, 6:1 (2014), 91–115 | MR | Zbl

[21] Chernov A. V., “O suschestvovanii ravnovesiya po Neshu v differentsialnoi igre, svyazannoi s ellipticheskimi uravneniyami: monotonnyi sluchai”, MTIiP, 7:3 (2015), 48–78 | Zbl

[22] Chernousko F. L., “Granichnye upravleniya v sistemakh s raspredelennymi parametrami”, Prikl. matematika i mekhanika, 56:5 (1992), 810–826 | MR | Zbl

[23] Berkovitz L. D., “The existence of value and saddle point in games of fixed duration”, SIAM J. Control Optim., 23 (1985), 172–196 | DOI | MR | Zbl

[24] Ibragimov G., Alias I. A., Waziri U., Ja'afaru A. B., “Differential game of optimal pursuit for an infinite system of differential equations”, Bull. Malays. Math. Sci. Soc., 2(42):1 (2019), 391–403 | DOI | MR | Zbl

[25] Il'in V.A., Tikhomirov V. V., “The wave equation with a boundary control at both endpoints and the complete vibration damping problem”, Differ. Equations, 35:5 (1999), 697–708 | MR | Zbl

[26] Lions J.-L., “Exact controllability, stabilization and perturbations for distributed systems”, SIAM Rev., 30:1 (1988), 1–68 | MR | Zbl

[27] Ramaswamy M., Shaiju A. J., “Construction of approximate saddle-point strategies for differential games in a Hilbert space”, J. Optim. Theory Appl., 141 (2009), 349–370 | MR | Zbl

[28] Roxin E. O., “Differential games with partial differential equations”, Differential Games Appl., Proc. Workshop (Enschede, 1977), Lect. Notes Control Inf. Sci., 3, 1977, 186–204 | MR | Zbl

[29] Troeva M., “Existence of equilibrium point for noncooperative differential game in Banach space”, AIP Conf. Proc., 1479, 2012, 1234–1237