On $\varepsilon$-equilibrium in noncooperative functional operator $n$-person games

A. V. Chernov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 1, pp. 316-328

Résumé

The paper is devoted to the investigation of noncooperative Volterra functional operator $n$-person games. Functional operator games are understood as games associated with controlled functional operator equations. Such games provide a uniform description for a wide class of differential games associated with evolutionary partial differential equations. We introduce the concept of abstract reachable set for functional operator equations and prove its precompactness under certain assumptions. The precompactness is used to prove the existence of a Nash $\varepsilon$-equilibrium in the sense of piecewise program strategies defined by means of the Volterra property in the games under consideration. As an example of the reduction of a controlled distributed-parameter system to a functional operator equation of the considered type, a mixed-value problem for a semilinear wave equation is considered, and the proposed hypotheses are verified for this equation. An example of a corresponding game statement is given.

Détail

Keywords: functional operator $n$-person game, piecewise program strategies, $\varepsilon$-equilibrium.

Bibliographie

[1] Kleimenov A. F., “Universalnoe reshenie v neantagonisticheskoi pozitsionnoi differentsialnoi igre s vektornymi kriteriyami”, Tr. In-ta matematiki i mekhaniki UrO RAN, 1, 1992, 97–105 | MR | Zbl

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

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

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

[5] 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

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

[7] Satimov N. Yu., Tukhtasinov M., “O nekotorykh igrovykh zadachakh v upravlyaemykh evolyutsionnykh uravneniyakh pervogo poryadka”, Differents. uravneniya, 41:8 (2005), 1114–1121 | MR | Zbl

[8] Chernov A. V., “O volterrovykh funktsionalno-operatornykh igrakh na zadannom mnozhestve”, Mat. teoriya igr i ee prilozheniya, 3:1 (2011), 91–117 | Zbl

[9] Chernov A. V., “O potochechnoi otsenke raznosti reshenii upravlyaemogo funktsionalno-operatornogo uravneniya v lebegovykh prostranstvakh”, Mat. zametki, 88:2 (2010), 288–302 | DOI | MR | Zbl

[10] Chernov A. V., “Ob odnom mazhorantnom priznake totalnogo sokhraneniya globalnoi razreshimosti upravlyaemogo funktsionalno-operatornogo uravneniya”, Izv. vuzov. Matematika, 2011, no. 3, 95–107 | MR | Zbl

[11] Chernov A. V., “O mazhorantno-minorantnom priznake totalnogo sokhraneniya globalnoi razreshimosti upravlyaemogo funktsionalno-operatornogo uravneniya”, Izv. vuzov. Matematika, 2012, no. 3, 62–73 | Zbl

[12] Chernov A. V., “O dostatochnykh usloviyakh upravlyaemosti nelineinykh raspredelennykh sistem”, Zhurn. vychisl. matematiki i mat. fiziki, 52:8 (2012), 1400–1414

[13] Chernov A. V., “O skhodimosti metoda uslovnogo gradienta v raspredelennykh zadachakh optimizatsii”, Zhurn. vychisl. matematiki i mat. fiziki, 51:9 (2011), 1616–1629 | MR

[14] Chernov A. V., “O totalnom sokhranenii globalnoi razreshimosti funktsionalno-operatornykh uravnenii”, Vest. Nizhegorod. un-ta im. N. I. Lobachevskogo, 2009, no. 3, 130–137

[15] Chernov A. V., “O volterrovom obobschenii metoda monotonizatsii dlya nelineinykh funktsionalno-operatornykh uravnenii”, Vest. Udmurt. un-ta. Matematika. Mekhanika. Kompyuternye nauki, 2012, no. 2, 84–99

[16] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz, Nauka, M., 1984, 752 pp. | MR | Zbl

[17] Fedorov V. M., Kurs funktsionalnogo analiza, Lan, Spb., 2005, 352 pp.

[18] Sumin V. I., Chernov A. V., “Operatory v prostranstvakh izmerimykh funktsii: volterrovost i kvazinilpotentnost”, Differents. uravneniya, 34:10 (1998), 1402–1411 | MR | Zbl

[19] Sumin V. I., “Ob obosnovanii gradientnykh metodov dlya raspredelennykh zadach optimalnogo upravleniya”, Zhurn. vychisl. matematiki i mat. fiziki, 30:1 (1990), 3–21 | MR | Zbl

[20] Petrosyan L. A., Zenkevich N. A., Semina E. A., Teoriya igr, Vysshaya shkola, M., 1998, 304 pp. | MR | Zbl

[21] Krasnoselskii M. A., Topologicheskie metody v teorii nelineinykh integralnykh uravnenii, GITTL, M., 1956, 392 pp. | MR

[22] Mordukhovich B. Sh., Metody approksimatsii v zadachakh optimizatsii i upravleniya, Nauka, M., 1988, 360 pp. | MR | Zbl

[23] Vasin A. A., Morozov V. V., Vvedenie v teoriyu igr s prilozheniyami k ekonomike, Izd-vo MGU, M., 2003, 278 pp.

[24] Ladyzhenskaya O. A., Smeshannaya zadacha dlya giperbolicheskogo uravneniya, GITTL, M., 1953, 280 pp.