Reconstruction of controls in parabolic systems by Tikhonov's method with nonsmooth stabilizers

A. I. Korotkii ; D. O. Mikhailova

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 16 (2010) no. 4, pp. 211-227

Résumé

The problem of reconstructing a priori unknown controls in parabolic systems by the results of approximate a posteriori observations of the motion of this system is considered. It is proposed to solve this problem by Tikhonov's method with a stabilizer containing the total time variation of the varying control. The use of such a nondifferentiable stabilizer allows one to obtain in some cases more precise results than the approximation of the desired control in Lebesgue spaces. In particular, it becomes possible to establish the piecewise uniform convergence of regularized approximations, which can be used for the numerical reconstruction of the subtle structure of the desired control. Results of numerical experiments are presented.

Détail

Keywords: control parabolic system, inverse problem of dynamics, Tikhonov's regularization method, classical variation, subgradient.
Mots-clés : piecewise uniform convergence

Bibliographie

[1] Tikhonov A.N., Arsenin V.Ya., Metody resheniya nekorrektnykh zadach, Nauka, M., 1979, 288 pp.

[2] Ivanov V.K., Vasin V.V., Tanana V.P., Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya, Nauka, M., 1978, 206 pp.

[3] Lavrentev M.M., Romanov V.G., Shishatskii S.P., Nekorrektnye zadachi matematicheskoi fiziki i analiza, Nauka, M., 1980, 288 pp.

[4] Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mischenko E.F., Matematicheskaya teoriya optimalnykh protsessov, Nauka, M., 1976, 392 pp.

[5] Krasovskii N.N., Teoriya upravleniya dvizheniem, Nauka, M., 1968, 476 pp.

[6] Krasovskii N.N., Subbotin A.I., Pozitsionnye differentsialnye igry, Nauka, M., 1974, 456 pp.

[7] Osipov Yu.S., Kryazhimskii A.V., Inverse problems of ordinary differential equations: dynamical solutions, Gordon and Breach, L., 1995, 625 pp. | MR | Zbl

[8] Osipov Yu.S., Vasilev F.P., Potapov M.M., Osnovy metoda dinamicheskoi regulyarizatsii, Izd-vo MGU, M., 1999, 237 pp.

[9] Kurzhanskii A.B., Upravlenie i nablyudenie v usloviyakh neopredelennosti, Nauka, M., 1977, 392 pp.

[10] Chernousko F.L., Melikyan A.A., Igrovye zadachi upravleniya i poiska, Nauka, M., 1978, 270 pp.

[11] Varga Dzh., Optimalnoe upravlenie differentsialnymi i funktsionalnymi uravneniyami, Nauka, M., 1977, 623 pp.

[12] Maksimov V.I., Zadachi dinamicheskogo vosstanovleniya vkhodov beskonechnomernykh sistem, UrO RAN, Ekaterinburg, 2000, 304 pp.

[13] Krutko P.D., Obratnye zadachi dinamiki upravlyaemykh sistem. Nelineinye modeli, Nauka, M., 1988, 332 pp.

[14] Ageev A.L., “Regulyarizatsiya nelineinykh operatornykh uravnenii na klasse razryvnykh funktsii”, Zhurn. vychisl. matematiki i mat. fiziki, 20:4 (1980), 819–836 | MR

[15] Vasin V.V., Serezhnikova T.I., “Ob odnom algoritme resheniya uravneniya Fredgolma - Stiltesa”, Izv. vuzov. Matematika, 2001, no. 4, 3–10

[16] Vasin V.V., Serezhnikova T.I., “Dvukhetapnyi metod approksimatsii negladkikh reshenii i vosstanovlenie zashumlennogo izobrazheniya”, Avtomatika i telemekhanika, 2004, no. 2, 126–135

[17] Vasin V.V., “Regulyarizatsiya i diskretnaya approksimatsiya nekorrektnykh zadach v prostranstve funktsii ogranichennoi variatsii”, Dokl. RAN, 376:1 (2001), 11–14 | MR | Zbl

[18] Vasin V.V., “Regularization and iterativ approximation for linear ill-posed problems in the space of function of bounded variation”, Proc. Steklov Inst. Math. Suppl. 1, 2002, 225–239 | MR

[19] Vasin V.V., “Ustoichivaya approksimatsiya negladkikh reshenii nekorrektno postavlennykh zadach”, Dokl. RAN, 402:5 (2005), 586–589 | MR

[20] Vasin V.V., “Approksimatsiya negladkikh reshenii lineinykh nekorrektnykh zadach”, Tr. Instituta matematiki i mekhaniki UrO RAN, 12, no. 1, 2006, 64–77 | MR | Zbl

[21] Vasin V.V., Korotkii M.A., “Tikhonov regularization with nondifferentiable stabilizing functional”, J. Inverse and Ill-Posed Problems, 15:8 (2007), 853–865 | DOI | MR | Zbl

[22] Leonov A.S., “Regularization of ill-posed problems in Sobolev space $W^1_1$”, J. Inverse and Ill-Posed Problems, 13:6 (2005), 595–619 | MR | Zbl

[23] Leonov A.S., “Kusochno-ravnomernaya regulyarizatsiya nekorrektnykh zadach s razryvnymi resheniyami”, Zhurn. vychisl. matematiki i mat. fiziki, 22:3 (1982), 516–531 | MR | Zbl

[24] Tikhonov A.N., Leonov A.S., Yagola A.G., Nelineinye nekorrektnye zadachi, Nauka, M., 1995, 212 pp.

[25] Giusti E., Minimal surfaces and functions of bounded variations, Birkhauser, Basel, 1984, 239 pp. | MR | Zbl

[26] Acar R., Vogel C.R., “Analysis of bounded variation penalty method for ill-posed problems”, Inverse Problems, 10 (1994), 1217–1229 | DOI | MR | Zbl

[27] Chavent G., Kunish K., “Regularization of linear least squares problems by total bounded variation control”, Optimization and Calculus of variation, 2 (1997), 359–376 | DOI | MR | Zbl

[28] Vogel C.R., Computation methods for inverse problems, SIAM, Philadelphia, 2002, 183 pp. | MR | Zbl

[29] Korotkii M.A., “Vosstanovlenie upravlenii i parametrov metodom Tikhonova s negladkimi stabilizatorami”, Izv. vuzov. Matematika, 2009, no. 2, 76–82

[30] Korotkii M.A., “Vosstanovlenie upravlenii staticheskim i dinamicheskim metodami regulyarizatsii s negladkimi stabilizatorami”, Prikl. matematika i mekhanika, 73:1 (2009), 39–53 | MR | Zbl

[31] Korotkii M.A., Metod regulyarizatsii Tikhonova s negladkimi stabilizatorami, dis. ... kand. fiz.-mat. nauk, IMM UrO RAN, Ekaterinburg, 2009, 132 pp.

[32] Soboleva D.O., “Rekonstruktsiya upravlenii v parabolicheskikh sistemakh”, Vestn. Buryatskogo gos. un-ta. Matematika i informatika, 2010, no. 9, 59–67

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

[34] Ladyzhenskaya O.A., Solonnikov V.A., Uraltseva N.N., Lineinye i kvazilineinye urav- neniya parabolicheskogo tipa, Nauka, M., 1967, 736 pp.

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

[36] Korotkii A.I., “O korrektnosti zadach optimalnogo upravleniya parabolicheskimi i giperboli- cheskimi sistemami”, Zadachi pozitsionnogo modelirovaniya, UNTs AN SSSR, Sverdlovsk, 1986, 19–40

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

[38] Lyusternik L.A., Sobolev V.I., Elementy funktsionalnogo analiza, Nauka, M., 1965, 496 pp.

[39] Natanson I.P., Teoriya funktsii veschestvennoi peremennoi, Nauka, M., 1974, 480 pp.

[40] Vulikh B.Z., Kratkii kurs teorii funktsii veschestvennoi peremennoi., Nauka, M., 1973, 352 pp.

[41] Iosida K., Funktsionalnyi analiz, Mir, M., 1967, 624 pp.

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

[43] Ioffe A.D., Tikhomirov V.M., Teoriya ekstremalnykh zadach, Nauka, M., 1974, 480 pp.

[44] Polyak B.T., Vvedenie v optimizatsiyu, Nauka, M., 1983, 384 pp.

[45] Demyanov V.F., Vasilev V.P., Nedifferentsiruemaya optimizatsiya, Nauka, M., 1981, 384 pp.