Finite difference approximations of optimal control problems for semilinear elliptic equations with discontinuous coefficients and solutions

F. V. Lubyshev

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 52 (2012) no. 8, pp. 1378-1399

Résumé

Mathematical formulation of nonlinear optimal control problems for semilinear elliptic equations with discontinuous coefficients and discontinuous solutions are examined. Finite difference approximations of optimization problems are constructed, and the approximation error is estimated with respect to the state and the cost functional. Weak convergence of the approximations with respect to the control is proved. The approximations are regularized using Tikhonov regularization.

Bibliographie

[1] Vasilev F. P., Metody optimizatsii, Faktorial Press, M., 2002

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

[3] Lure K. A., Optimalnoe upravlenie v zadachakh matematicheskoi fiziki, Nauka, M., 1975 | MR

[4] Litvinov V. G., Optimizatsiya v ellipticheskikh granichnykh zadachakh s prilozheniyami k mekhanike, Nauka, M., 1987 | MR

[5] Raitum U. E., Zadachi optimalnogo upravleniya dlya ellipticheskikh uravnenii, Zinatne, Riga, 1989 | MR | Zbl

[6] Egorov A. I., Optimalnoe upravlenie teplovymi i diffuzionnymi protsessami, Nauka, M., 1978 | MR

[7] Ishmukhametov A. Z., Voprosy ustoichivosti i approksimatsii zadach optimalnogo upravleniya, VTs RAN, M., 1999

[8] Ishmukhametov A. Z., Voprosy ustoichivosti i approksimatsii zadach optimalnogo upravleniya sistemami s raspredelennymi parametrami, VTs RAN, M., 2001

[9] Potapov M. M., Approksimatsiya ekstremalnykh zadach v matematicheskoi fizike (giperbolicheskie uravneniya), Izd-vo MGU, M., 1985

[10] Lubyshev F. V., Raznostnye approksimatsii zadach optimalnogo upravleniya sistemami, opisyvaemymi uravneniyami v chastnykh proizvodnykh, BashGu, Ufa, 1999

[11] Samarskii A. A., Andreev V. B., Raznostnye metody dlya ellipticheskikh uravnenii, Nauka, M., 1976 | MR | Zbl

[12] Samarskii A. A., Vabischevich P. N., Vychislitelnaya teploperedacha, Knizhnyi dom “Librokom”, M., 2009

[13] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1989 | MR

[14] Tsurko V. A., “O tochnosti raznostnykh skhem dlya parabolicheskikh uravnenii s razryvnym resheniem”, Differents. ur-niya, 36:7 (2000), 986–992 | MR | Zbl

[15] Tsurko V. A., “Raznostnye metody dlya zadach konvektsii-diffuzii s razryvnymi koeffitsientami i resheniyami”, Differents. ur-niya, 41:2 (2005), 274–280 | MR | Zbl

[16] Lubyshev F. V., “Tochnost raznostnykh approksimatsii i regulyarizatsiya zadach optimalnogo upravleniya dlya ellipticheskogo uravneniya s upravleniyami v koeffitsientakh”, Zh. vychisl. matem. i matem. fiz., 29:9 (1989), 1431–1433 | Zbl

[17] Lubyshev F. V., “Approksimatsiya i regulyarizatsiya zadach optimalnogo upravleniya dlya nesamosopryazhennogo ellipticheskogo uravneniya s peremennymi koeffitsientami”, Zh. vychisl. matem. i matem. fiz., 31:1 (1991), 17–30 | MR | Zbl

[18] Lubyshev F. V., “Approksimatsiya i regulyarizatsiya zadach optimalnogo upravleniya koeffitsientami parabolicheskikh uravnenii”, Zh. vychisl. matem. i matem. fiz., 33:8 (1993), 1166–1183 | MR | Zbl

[19] Lubyshev F. V., “Approksimatsiya i regulyarizatsiya zadach optimalnogo upravleniya dlya parabolicheskikh uravnenii s upravleniyami v koeffitsientakh”, Dokl. RAN, 349:5 (1996), 598–602 | MR | Zbl

[20] Lubyshev F. V., Fairuzov M. E., “Approksimatsiya i regulyarizatsiya zadach optimalnogo upravleniya dlya kvazilineinykh ellipticheskikh uravnenii”, Zh. vychisl. matem. i matem. fiz., 41:8 (2001), 1148–1164 | MR | Zbl

[21] Lubyshev F. V., Manapova A. R., “O nekotorykh zadachakh optimalnogo upravleniya i ikh raznostnykh approksimatsiyakh i regulyarizatsii dlya kvazilineinykh ellipticheskikh uravnenii s upravleniyami v koeffitsientakh”, Zh. vychisl. matem. i matem. fiz., 47:3 (2007), 376–396 | MR | Zbl

[22] Zarubin V. S., Inzhenernye metody resheniya zadach teploprovodnosti, Energoatomizdat, M., 1983

[23] Belyaev N. M., Ryadno A. A., Metody nestatsionarnoi teploprovodnosti, Vysshaya shkola, M., 1978

[24] Muchnik G. F., Rubashov I. B., Metody teorii teploobmena, v. 1, Teploprovodnost, Vyssh. shkola, M., 1970

[25] Kartashov E. M., Analiticheskie metody v teorii teploprovodnosti tverdykh tel, Vyssh. shkola, M., 1985

[26] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Izd-vo SO AN SSSR, Novosibirsk, 1962

[27] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973 | MR

[28] Gaevskii Kh., Greger K., Zakharias K., Nelineinye operatornye uravneniya i operatornye differentsialnye uravneniya, Mir, M., 1978 | MR

[29] Kufner A., Fuchik S., Nelineinye differentsialnye uravneniya, Nauka, M., 1988 | MR

[30] Gilbarg D., Trudinger N., Ellipticheskie differentsialnye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Nauka, M., 1989 | MR | Zbl

[31] Kinderlerer D., Stampakkya G., Vvedenie v variatsionnye neravenstva i ikh prilozheniya, Mir, M., 1983 | MR

[32] Rektoris K., Variatsionnye metody v matematicheskoi fizike i tekhnike, Mir, M., 1985 | MR | Zbl

[33] Brauder F. E., Materialy k sovmestnomu sovetsko-amerikanskomu simpoziumu po uravneniyam s chastnymi proizvodnymi, Novosibirsk, 1963

[34] Drenska N. T., “Tochnost chislennykh algoritmov dlya odnomernoi zadachi ob ostyvanii metalla v formakh”, Vestn. Mosk. un-ta. Ser. 15. Vychisl. matem. i kibernetika, 1981, no. 4, 15–21 | MR | Zbl

[35] Samarskii A. A., Lazarov R. D., Makarov V. L., Raznostnye skhemy dlya differentsialnykh uravnenii s obobschennymi resheniyami, Vyssh. shkola, M., 1987

[36] Tikhonov A. N., Arsenin V. Ya., Metody resheniya nekorrektnykh zadach, Nauka, M., 1986 | MR | Zbl