The optimal feedback control problem for Voigt model with variable density

V. G. Zvyagin ; M. V. Turbin

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 4 (2020), pp. 93-98.

Résumé

The work is devoted to the study of weak solvability of the optimal feedback control problem for the Voigt model with variable density. The proof is based on the approximation-topological approach. First, the solvability of the approximation problem is proved, and then, based on a priori estimates that are independent of the approximation parameter, it is shown that from the sequence of solutions of the approximation problem, a subsequence converging to the solution of the original problem can be chosen. Then it is shown that among weak solutions to the problem there is at least one that gives a minimum to a given cost functional.

Détail

Keywords: optimal control problem, feedback, weak solution, approximation-topological approach, nonhomogeneous fluid, variable-density fluid.
Mots-clés : Voigt model

Bibliographie

[1] Filippov A.F., “O nekotorykh voprosakh teorii optimalnogo regulirovaniya”, Vestn. Moskovsk. un-ta. Ser. matem., mekhan., astronomiya, fiz., khimiya, 2 (1959), 25–32 | Zbl

[2] Aubin J.-P., Cellina A., Differential Inclusions, Springer-Verlag, Berlin, 1984 | MR | Zbl

[3] Kamenskii M., Obukhovskii V., Zecca P., Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces, de Gruyter, Berlin–New York, 2001 | MR | Zbl

[4] Zvyagin V., Obukhivskii V., Zvyagin A., “On inclusions with multivalued operators and their applications to some optimization problems”, J. Fixed Theory and Appl., 16:1–2 (2014), 27–82 | DOI | MR | Zbl

[5] Zvyagin V.G., Zvyagin A.V., Turbin M.V., “Optimalnoe upravlenie s obratnoi svyazyu dlya modeli Bingama s periodicheskimi usloviyami po prostranstvennym peremennym”, Zap. nauchn. sem. POMI, 477, 2018, 54–86

[6] Zvyagin V.G., Turbin M.V., “Optimal Feedback Control in the Mathematical Model of Low Concentrated Aqueous Polymer Solutions”, J. Optim. Theory and Appl., 148:1 (2011), 146–163 | DOI | MR | Zbl

[7] Plotnikov P.I., Turbin M.V., Ustyuzhaninova A.S., “Teorema suschestvovaniya slabogo resheniya zadachi optimalnogo upravleniya s obratnoi svyazyu dlya modifitsirovannoi modeli Kelvina–Foigta slabo kontsentrirovannykh vodnykh rastvorov polimerov”, Dokl. Akademii Nauk, 488:2 (2019), 133–136 | DOI | Zbl

[8] Zvyagin V.G., Turbin M.V., “Optimalnoe upravlenie s obratnoi svyazyu dvizheniem sredy Bingama s periodicheskimi usloviyami po prostranstvennym peremennym”, Dokl. Akademii Nauk, 485:2 (2019), 139–141 | DOI | Zbl

[9] Kazhikhov A.V., “Razreshimost nachalno-kraevoi zadachi dlya uravnenii dvizheniya neodnorodnoi vyazkoi neszhimaemoi zhidkosti”, DAN SSSR, 216:5 (1974), 1008–1010 | Zbl

[10] Lions P.-L., Mathematical topics in fluid mechanics, v. 1, Incompressible Models, Clarendon Press, Oxford, 1996 | MR | Zbl

[11] Simon J., “Nonhomogeneous viscous incompressible fluids: existence of velocity, density and pressure”, SIAM J. Math. Anal., 21:5 (1990), 1093–1117 | DOI | MR | Zbl

[12] Antontsev S.N., de Oliveira H.B., Khompysh Kh., “Existence and large time behavior for generalized Kelvin-Voigt equations governing nonhomogeneous and incompressible fluids”, IOP Conf. Series: J. Physics: Conf. Ser., 1268 (2019), 012008 | DOI | MR

[13] Ladyzhenskaya O.A., Matematicheskie voprosy dinamiki vyazkoi neszhimaemoi zhidkosti, GIFML, M., 1961

[14] Solonnikov V.A., “Ob otsenkakh tenzorov Grina dlya nekotorykh granichnykh zadach”, DAN SSSR, 130:5 (1960), 988–991 | MR | Zbl

[15] Vorovich I.I., Yudovich V.I., “Statsionarnye techeniya vyazkoi neszhimaemoi zhidkosti”, Matem. sb., 53:4 (1961), 393–428

[16] Zvyagin V.G., Turbin M.V., Matematicheskie voprosy gidrodinamiki vyazkouprugikh sred, KRASAND (URSS), M., 2012

[17] Fursikov A.V., Optimalnoe upravlenie raspredelennymi sistemami. Teoriya i prilozheniya, Nauchn. kn., Novosibirsk, 1999