On a differential game in an abstract parabolic system

L. A. Vlasenko   ; A. G. Rutkas   ; A. A. Chikrii

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 21 (2015) no. 2, pp. 26-40

Résumé

We consider the game problem of approach for a system whose dynamics is described by a differential operator equation in a Hilbert space. The equation is written in an implicit form with generally non-invertible operator multiplying the derivative. It is assumed that the characteristic operator pencil corresponding to the linear part of the equation satisfies a constraint of parabolic type in a right half-plane. Using the method of resolving functionals, we obtain sufficient conditions for the approach of a dynamical vector of the system to a cylindrical terminal set. Applications to systems described by partial differential equations are considered.

Détail

Keywords: differential game, parabolic system, ergodic theorem, pseudoresolvent, generator of a semigroup, set-valued mapping, resolving functional, partial differential equation.

Bibliographie

[1] Isaacs R., Differential Games, John Wiley, New York, 1965, 480 pp. | MR | Zbl

[2] Krasovskii N.N., Subbotin A.I., Pozitsionnye differentsialnye igry, Nauka, M., 1974, 455 pp. | MR

[3] Pontryagin L.S., Izbrannye nauchnye trudy, v. 2, Nauka, M., 1988, 576 pp. | MR

[4] Subbotin A.I., Minimaksnye neravenstva i uravneniya Gamiltona - Yakobi, Nauka, M., 1991, 216 pp. | MR

[5] Subbotin A.I., Generalized solutions of first order PDEs: The dynamic optimization perspective, Birkhauser, Boston, 1995, 312 pp. | MR

[6] Subbotin A.I., Obobschennye resheniya uravnenii v chastnykh proizvodnykh pervogo poryadka: perspektivy dinamicheskoi optimizatsii, Institut kompyuter. issled., Izhevsk, 2003, 336 pp.

[7] Subbotin A.I., Chentsov A.G., Optimizatsiya garantii v zadachakh upravleniya, Nauka, M., 1981, 288 pp. | MR

[8] Subbotina N.N., “The method of characteristics for Hamilton-Jacobi equation and its applications in dynamic optimization”, J. Math. Sci., 135:3 (2006), 2955–3091 | DOI | MR

[9] Nikolskii M.S., “Ob upravlenii pri nalichii protivodeistviya”, Vestn. Mosk. un-ta, 1972, no. 1, 67–72 | MR

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

[11] Osipov Yu.S., “Pozitsionnoe upravlenie v parabolicheskikh sistemakh”, Prikl. matematika i mekhanika, 41:2 (1977), 195–201 | MR

[12] Vlasenko L.A., Chikrii A.A., “Metod razreshayuschikh funktsionalov dlya odnoi dinamicheskoi igry v sisteme tipa Soboleva”, Problemy upravleniya i informatiki, 2014, no. 4, 5–14

[13] Vlasenko L.A., Chikrii A.A., “Ob odnoi differentsialnoi igre v sisteme s raspredelennymi parametrami”, Tr. In-ta matematiki i mekhaniki UrO RAN, 20:4 (2014), 71–80 | MR

[14] Chikrii A.A., Conflict-controlled processes, Springer, Boston; London; Dordrecht, 1997, 424 pp. | MR

[15] Chikrii A.A., “Ob odnom analiticheskom metode v dinamicheskikh igrakh sblizheniya”, Tr. MIAN, 271 (2010), 76–92 | MR | Zbl

[16] Iosida K., Funktsionalnyi analiz, Mir, M., 1967, 624 pp. | MR

[17] Pazy A., Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983, 279 pp. | MR | Zbl

[18] Lions Zh.-L., Optimalnoe upravlenie sistemami, opisyvaemymi uravneniyami s chastnymi proizvodnymi, Mir, M., 1972, 415 pp. | MR

[19] Khille E., Fillips R., Funktsionalnyi analiz i polugruppy, Inostr. lit., M., 1962, 830 pp. | MR

[20] Krein S.G., Lineinye differentsialnye uravneniya v banakhovom prostranstve, Nauka, M., 1967, 464 pp. | MR

[21] Vlasenko L.A., Myshkis A.D., Rutkas A.G., “Ob odnom klasse differentsialnykh uravnenii parabolicheskogo tipa s impulsnymi vozdeistviyami”, Differents. uravneniya, 44:2 (2008), 222–231 | MR | Zbl

[22] Vlasenko L.A., Rutkas A.G., “Ob odnom klasse impulsnykh funktsionalno-differentsialnykh uravnenii s neatomarnym raznostnym operatorom”, Mat. zametki, 95:1 (2014), 37–49 | MR | Zbl

[23] Vlasenko L.A., Evolyutsionnye modeli s neyavnymi i vyrozhdennymi differentsialnymi uravneniyami, Sistemnye tekhnologii, Dnepropetrovsk, 2006, 273 pp.

[24] Khenri D., Geometricheskaya teoriya polulineinykh parabolicheskikh uravnenii., Mir, M., 1985, 367 pp. | MR

[25] Mizokhata C., Teoriya uravnenii s chastnymi proizvodnymi., Mir, M., 1977, 504 pp.

[26] Kantorovich L.V., Akilov G.P., Funktsionalnyi analiz, Nauka, M., 1977, 744 pp. | MR

[27] Aubin J.-P., Frankowska H., Set-valued analysis, Birkhauser, Boston, 1990, 461 pp. | MR | Zbl

[28] M.A. Krasnoselskii, P.P. Zabreiko, E.I. Pustylnik, P.E. Sobolevskii, Integralnye operatory v prostranstvakh summiruemykh funktsii, Nauka, M., 1966, 500 pp. | MR

[29] Varga Dzh., Optimalnoe upravlenie differentsialnymi i funktsionalnymi uravneniyami, Nauka, M., 1977, 624 pp. | MR

[30] Brekhovskikh L.M., Goncharov V.V., Vvedenie v mekhaniku sploshnykh sred (v prilozhenii k teorii voln), Nauka, M., 1982, 336 pp.