Geodesic
On Volterra functional operator games on a~given set
Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 3 (2011) no. 1, pp. 91-117.

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper is devoted to obtaining the sufficient conditions of $\varepsilon$-equilibrium in the sense of piecewise program strategies in antagonistic games associated with nonlinear controlled functional operator equations and cost functional of a general enough form. The concept of piecewise program strategies is defined on the base of a concept of Volterra set chain for operators involved in the equations controlled by the opponent players. The reduction of controlled distributed parameter systems to an equation of the type under study is illustrated by examples.
Keywords: functional operator game, nonlinear functional operator equations, piecewise program strategies, $\varepsilon$-equilibrium.
Mots-clés : Volterra set chain 
@article{MGTA_2011_3_1_a4,
     author = {Andrey V. Chernov},
     title = {On {Volterra} functional operator games on a~given set},
     journal = {Matemati\v{c}eska\^a teori\^a igr i e\"e prilo\v{z}eni\^a},
     pages = {91--117},
     publisher = {mathdoc},
     volume = {3},
     number = {1},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MGTA_2011_3_1_a4/}
}
TY  - JOUR
AU  - Andrey V. Chernov
TI  - On Volterra functional operator games on a~given set
JO  - Matematičeskaâ teoriâ igr i eë priloženiâ
PY  - 2011
SP  - 91
EP  - 117
VL  - 3
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MGTA_2011_3_1_a4/
LA  - ru
ID  - MGTA_2011_3_1_a4
ER  - 
%0 Journal Article
%A Andrey V. Chernov
%T On Volterra functional operator games on a~given set
%J Matematičeskaâ teoriâ igr i eë priloženiâ
%D 2011
%P 91-117
%V 3
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MGTA_2011_3_1_a4/
%G ru
%F MGTA_2011_3_1_a4
Andrey V. Chernov. On Volterra functional operator games on a~given set. Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 3 (2011) no. 1, pp. 91-117. http://geodesic.mathdoc.fr/item/MGTA_2011_3_1_a4/

[1] Blagodatskikh V. I., “O differentsiruemosti reshenii po nachalnym usloviyam”, Differents. uravneniya, 9:1 (1973), 2136–2140

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

[3] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1976 | MR

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

[5] Natanson I. P., Teoriya funktsii veschestvennoi peremennoi, Nauka, M., 1974 | MR

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

[7] Polityukov V. P., “O metode monotonizatsii nelineinykh uravnenii v banakhovom prostranstve”, Matem. zametki, 44:6 (1988), 814–822 | MR | Zbl

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

[9] Sirazetdinov T. K., Optimizatsiya sistem s raspredelennymi parametrami, Nauka, M., 1977 | MR

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

[11] Stepanov V. V., Kurs differentsialnykh uravnenii, GITTL, M., 1953

[12] Sumin V. I., “Funktsionalno-operatornye volterrovy uravneniya v teorii optimalnogo upravleniya raspredelennymi sistemami”, Dokl. AN SSSR, 305:5 (1989), 1056–1059 | MR | Zbl

[13] Sumin V. I., “Ob obosnovanii gradientnykh metodov dlya raspredelennykh zadach optimalnogo upravleniya”, Zh. vychisl. matem. i matem. fiz., 30:1 (1990), 3–21 | MR | Zbl

[14] Sumin V. I., “O funktsionalnykh volterrovykh uravneniyakh”, Izv. vuzov. Matematika, 1995, no. 9, 67–77 | MR | Zbl

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

[16] Chernov A. V., Volterrovy operatornye uravneniya i ikh primenenie v teorii optimizatsii giperbolicheskikh sistem, Dis. $\dots$ kand. f.-m. n., NNGU, N. Novgorod, 2000

[17] Chernov A. V., “O totalnom sokhranenii globalnoi razreshimosti funktsionalno-operatornykh uravnenii”, Vestnik Nizhegorodskogo universiteta im. N. I. Lobachevskogo, 2009, no. 3, 130–137 | MR

[18] Chernov A. V., “O volterrovykh funktsionalno-operatornykh igrakh”, Mat. modelirovanie i kraevye zadachi, Trudy Sedmoi Vserossiiskoi nauchnoi konferentsii s mezhdunarodnym uchastiem. Chast 2, SamGTU, Samara, 2010, 289–291

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

[20] 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

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

[22] Shvarts L., Analiz, v. 1, Mir, M., 1972

[23] 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

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