Optimal boundary control of a system describing thermal convection

A. I. Korotkii ; D. A. Kovtunov

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 1, pp. 76-101

Résumé

A problem of optimal boundary control of thermal sources for a stationary model of natural thermal convection of a high-viscosity inhomogeneous incompressible fluid in the Boussinesq approximation is investigated. Some conditions for the solvability of the problem are given; necessary and sufficient optimality conditions are specified. Optimality conditions and the corresponding conjugate problems defining the gradient of the quality functional are written for a number of particular cases of the functional. Procedures for the numerical finding of an optimal control based on gradient methods are described. The results of numerical experiments are given.

Détail

Keywords: optimal control, thermal convection, gradient method.

Bibliographie

[1] Gad-el-Hak M., “Flow control”, Appl. Mech. Rev., 42:10 (1989), 261–393 | DOI

[2] Alifanov O. M., Obratnye zadachi teploobmena, Mashinostroenie, M., 1988, 277 pp.

[3] Marchuk G. I., Matematicheskoe modelirovanie v probleme okruzhayuschei sredy, Nauka, M., 1982, 319 pp. | MR

[4] Verte L. A., Magnitnaya gidrodinamika v metallurgii, Nauka, M., 1975, 288 pp.

[5] Gelfgat Yu. M., Lielausis O. A., Scherbinin E. V., Zhidkii metall pod vozdeistviem elektromagnitnykh sil, Zinatne, Riga, 1976, 248 pp.

[6] Glukhikh V. A., Tananaev A. V., Kirillov I. R., Magnitnaya gidrodinamika v yadernoi energetike, Nauka, M., 1987, 264 pp.

[7] Lavrentev I. V., “Zhidkometallicheskie sistemy termoyadernykh reaktorov-tokomakov”, Magnitnaya gidrodinamika, 1990, no. 2, 105–124 | MR

[8] Shashkov Yu. N., Vyraschivanie monokristallov metodom vytyagivaniya, Nauka, M., 1982, 312 pp.

[9] Muller G., Convection and inhomogeneties in crystal growth from the melt, Springer-Verlag, Berlin–Heidelberg, 1988, 150 pp.

[10] Gunzburger M. D., Flow control, Springer-Verlag, New York–Berlin–Heidelberg, 1995, 381 pp. | MR | Zbl

[11] Alekseev G. V., Tereshko D. A., Analiz i optimizatsiya v gidrodinamike vyazkoi zhidkosti, Dalnauka, Vladivostok, 2008, 364 pp.

[12] Ioffe A. D., Tikhomirov V. M., Teoriya ekstremalnykh zadach, Nauka, M., 1974, 479 pp. | MR | Zbl

[13] Vasilev F. P., Metody optimizatsii, Faktorial Press, M., 2002, 824 pp.

[14] Sea Zh., Optimizatsiya. Teoriya i algoritmy, Mir, M., 1973, 244 pp.

[15] Polak E., Chislennye metody otimizatsii. Edinyi podkhod, Mir, M., 1974, 376 pp. | MR | Zbl

[16] Klark F., Optimizatsiya i negladkii analiz, Nauka, M., 1988, 280 pp. | MR | Zbl

[17] Demyanov V. F., Rubinov A. M., Osnovy negladkogo analiza i kvazidifferentsialnoe ischislenie, Nauka, M., 1990, 432 pp. | MR

[18] Lions Zh.-L., Optimalnoe upravlenie sistemami, opisyvaemymi uravneniyami s chastnymi proizvodnymi, Mir, M., 1972, 414 pp. | MR | Zbl

[19] Fursikov A. V., Optimalnoe upravlenie raspredelennymi sistemami. Teoriya i prilozheniya, Nauchnaya kniga, Novosibirsk, 1999, 352 pp. | Zbl

[20] Litvinov V. G., Optimizatsiya v ellipticheskikh granichnykh zadachakh s prilozheniyami k mekhanike, Nauka, M., 1987, 368 pp. | MR

[21] Korotkii A. I., Kovtunov D. A., “O razreshimosti statsionarnykh zadach estestvennoi teplovoi konvektsii vysokovyazkoi zhidkosti”, Tr. In-ta matematiki i mekhaniki UrO RAN, 14, no. 1, 2008, 61–73

[22] Kovtunov D. A., “Razreshimost statsionarnoi zadachi teplovoi konvektsii vysokovyazkoi zhidkosti”, Differents. uravneniya, 45:1 (2009), 74–85 | Zbl

[23] Korotkii A. I., Kovtunov D. A., “Rekonstruktsiya granichnykh rezhimov”, Teoriya upravleniya i teoriya obobschennykh reshenii uravnenii Gamiltona–Yakobi, tr. Mezhdunar. seminara, posvyaschennogo 60-letiyu akad. A. I. Subbotina, v. 2, Izd-vo UrGU, Ekaterinburg, 2006, 82–91

[24] Korotkii A. I., Kovtunov D. A., “Rekonstruktsiya granichnykh rezhimov v obratnoi zadache teplovoi konvektsii vysokovyazkoi zhidkosti”, Tr. In-ta matematiki i mekhaniki UrO RAN, 12, no. 2, 2006, 88–97 | MR | Zbl

[25] Korotkii A. I., “Granichnoe optimalnoe upravlenie sistemoi, opisyvayuschei teplovuyu konvektsiyu vysokovyazkoi neodnorodnoi neszhimaemoi zhidkosti”, Differentsialnye uravneniya i topologiya, tez. dokl. Mezhdunar. konf., posvyaschennoi 100-letiyu so dnya rozhdeniya L. S. Pontryagina, Izd-vo VMK MGU, M., 2008, 353 pp.

[26] Alekseev G. V., “Razreshimost statsionarnykh zadach granichnogo upravleniya dlya uravneniya teplovoi konvektsii”, Sib. mat. zhurn., 39:5 (1998), 982–998 | MR | Zbl

[27] Alekseev G. V., Smyshlyaev A. B., Tereshko D. A., “Razreshimost kraevoi zadachi dlya statsionarnykh uravnenii teplomassoperenosa pri smeshannykh kraevykh usloviyakh”, Zhurn. vychisl. matematiki i mat. fiziki, 43:1 (2003), 66–80 | MR

[28] Alekseev G. V., “Razreshimost obratnykh ekstremalnykh zadach dlya statsionarnykh uravnenii teplomassoperenosa”, Sib. mat. zhurn., 42:5 (2001), 971–991 | MR | Zbl

[29] Alekseev G. V., “Razreshimost kraevoi zadachi dlya statsionarnoi modeli magnitnoi gidrodinamiki vyazkoi teploprovodnoi zhidkosti”, Sib. zhurn. industr. matem., 9:1 (2006), 13–27 | MR

[30] Chandrasekhar S., Hydrodynamic and hydromagnetic stability, Dover Publications, New York, 1981, 654 pp. | MR

[31] Landau L. D., Lifshits E. M., Teoreticheskaya fizika, v. 6, Gidrodinamika, Fizmatlit, M., 2001, 736 pp. | MR

[32] Ladyzhenskaya O. A., Matematicheskie voprosy dinamiki vyazkoi neszhimaemoi zhidkosti, Fizmatgiz, M., 1961, 203 pp.

[33] Antontsev S. N., Kazhikhov A. V., Monakhov V. N., Kraevye zadachi mekhaniki neodnorodnykh zhidkostei, Nauka, Novosibirsk, 1983, 320 pp. | MR | Zbl

[34] Adams R. A., Sobolev spaces, Acad. Press, New York, 1975, 268 pp. | MR | Zbl

[35] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973, 576 pp. | MR

[36] Mikhailov V. P., Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1976, 392 pp. | MR | Zbl

[37] Sobolev S. L., Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, Nauka, M., 1988, 336 pp. | MR

[38] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973, 408 pp. | MR

[39] Smirnov V. I., Kurs vysshei matematiki, v. 5, Fizmatlit, M., 1959, 657 pp.

[40] Mikhlin S. G., Lineinye uravneniya v chastnykh proizvodnykh, Vysshaya shkola, M., 1977, 432 pp. | MR

[41] Korn G., Korn T., Spravochnik po matematike, Nauka, M., 1968, 832 pp. | MR

[42] Korotkii A. I., “Zavisimost reshenii ellipticheskikh uravnenii ot koeffitsientov i prilozhenie k korrektnosti zadach optimalnogo upravleniya”, Kachestvennye voprosy teorii differentsialnykh uravnenii i upravlyaemykh sistem, UrO AN SSSR, Sverdlovsk, 1988, 20–33 | MR

[43] Korotkii A. I., Pryamye i obratnye zadachi dinamiki upravlyaemykh sistem s raspredelennymi parametrami, Dis. $\dots$ d-ra fiz.-mat. nauk, IMM UrO RAN, Ekaterinburg, 1993, 331 pp.

[44] Kellogg R. B., Osborn J. E., “A regularity result for the Stokes problem in a convex polygon”, J. Funct. Anal., 21:4 (1976), 397–431 | DOI | MR | Zbl

[45] Dauge M., “Stationary Stokes and Navier–Stokes systems on two- or three-dimensional domains with corners. Part I: Linearized equations”, SIAM J. Math. Anal., 20:1 (1989), 74–97 | DOI | MR | Zbl

[46] Brown R. M., Shen Z., “Estimates for the Stokes operator in Lipschitz domains”, Indiana Univ. Math. J., 44:4 (1995), 1183–1206 | DOI | MR | Zbl

[47] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1972, 496 pp. | MR