Geodesic
Nonlinear positional differential game in the class of mixed strategies
Informatics and Automation, Mathematical control theory and differential equations, Tome 277 (2012), pp. 144-151 
A. A. Krasovskii; A. N. Krasovskii. Nonlinear positional differential game in the class of mixed strategies. Informatics and Automation, Mathematical control theory and differential equations, Tome 277 (2012), pp. 144-151. http://geodesic.mathdoc.fr/item/TRSPY_2012_277_a8/
@article{TRSPY_2012_277_a8,
     author = {A. A. Krasovskii and A. N. Krasovskii},
     title = {Nonlinear positional differential game in the class of mixed strategies},
     journal = {Informatics and Automation},
     pages = {144--151},
     year = {2012},
     volume = {277},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2012_277_a8/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The feedback control problem is considered for a nonlinear dynamic system under lack of information on disturbances. The problem on minmax-maxmin of the guaranteed result for a given positional quality index is formalized as an antagonistic two-player differential game in the framework of the concept developed in the Sverdlovsk (now Yekaterinburg) school on the theory of differential games. The problem is solved in the class of mixed strategies. The existence of the value of the game and a saddle point is established. The solution to the problem is based on the application of appropriate leader models, the method of extremal shift to the accompanying points and the method of upper convex hulls. The results of the study are applied to a realistic control model. The simulation outputs are presented.

[1] Aizeks R., Differentsialnye igry, Mir, M., 1967 | MR

[2] Bellman R., Dinamicheskoe programmirovanie, Izd-vo inostr. lit., M., 1960 | MR

[3] Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mischenko E.F., Matematicheskaya teoriya optimalnykh protsessov, Fizmatgiz, M., 1961

[4] Mischenko E.F., “O nekotorykh igrovykh zadachakh presledovaniya i ukloneniya ot vstrechi”, Avtomatika i telemekhanika, 1972, no. 9, 24–30

[5] Krasovskii N.N., Subbotin A.I., Pozitsionnye differentsialnye igry, Nauka, M., 1974 | MR | Zbl

[6] Kurzhanskii A.B., Upravlenie i nablyudenie v usloviyakh neopredelennosti, Nauka, M., 1977 | MR | Zbl

[7] Osipov Yu.S., Kryazhimskii A.V., Inverse problems for ordinary differential equations: Dynamical solutions, Gordon and Breach, Amsterdam, 1995 | MR | Zbl

[8] Krasovskii A.N., Krasovskii N.N., Control under lack of information, Birkhäuser, Boston, 1994 | Zbl

[9] Krasovskii A.N., “O pozitsionnom minimaksnom upravlenii”, PMM, 44:4 (1980), 602–610 | MR

[10] Mak-Kinsi Dzh., Vvedenie v teoriyu igr, Fizmatlit, M., 1960

[11] Liptser R.Sh., Shiryaev A.N., Statistika sluchainykh protsessov: Nelineinaya filtratsiya i smezhnye voprosy, Nauka, M., 1974 | MR | Zbl

[12] Krasovskii A.N., Choi Y.S., Stochastic control with the leaders-stabilizers, Preprint, In-t matematiki i mekhaniki UrO RAN, Ekaterinburg, 2001