Geodesic
The optimal control problem for the model of dynamics of weakly viscoelastic fluid
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 7 (2015) no. 3, pp. 22-29 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In this article we study the optimal control of solutions of the Dirichlet–Showalter–Sidorov problem for the system of equations of Kelvin–Voight zero order fluid motion, which is called a system of Oskolkov equations. The case of the degenerate equation is considered. Existence of global in time weak generalized solution of the model in the space of solenoidal functions is proved. The existence of optimal control of weak generalized solutions of Showalter–Sidorov problem for abstract semilinear Sobolev type equation is shown. The obtained abstract results are applied to the Oskolkov model.
Keywords: the system of Oskolkov equations; the optimal control problem; Sobolev type equations. 
@article{VYURM_2015_7_3_a3,
     author = {N. A. Manakova},
     title = {The optimal control problem for the model of dynamics of weakly viscoelastic fluid},
     journal = {Vestnik \^U\v{z}no-Uralʹskogo gosudarstvennogo universiteta. Seri\^a, Matematika, mehanika, fizika},
     pages = {22--29},
     year = {2015},
     volume = {7},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VYURM_2015_7_3_a3/}
}
TY  - JOUR
AU  - N. A. Manakova
TI  - The optimal control problem for the model of dynamics of weakly viscoelastic fluid
JO  - Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika
PY  - 2015
SP  - 22
EP  - 29
VL  - 7
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VYURM_2015_7_3_a3/
LA  - ru
ID  - VYURM_2015_7_3_a3
ER  - 
%0 Journal Article
%A N. A. Manakova
%T The optimal control problem for the model of dynamics of weakly viscoelastic fluid
%J Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika
%D 2015
%P 22-29
%V 7
%N 3
%U http://geodesic.mathdoc.fr/item/VYURM_2015_7_3_a3/
%G ru
%F VYURM_2015_7_3_a3
N. A. Manakova. The optimal control problem for the model of dynamics of weakly viscoelastic fluid. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 7 (2015) no. 3, pp. 22-29. http://geodesic.mathdoc.fr/item/VYURM_2015_7_3_a3/

[1] Oskolkov A. P., “Some nonstationary linear and quasilinear systems occurring in the investigation of the motion of viscous fluids”, J. of Soviet Mathematics, 10:2 (1976), 299–335 | MR | Zbl

[2] Oskolkov A. P., “Initial-boundary value problems for equations of motion of Kelvin–Voight fluids and Oldroid fluids”, Proceedings of the Steklov Institute of Mathematics, 179, 1989, 137–182 | MR | Zbl

[3] Sviridyuk G. A., “A model of dynamics of incompressible viskoelastic liquid”, Russian Mathematics (Izvestiya VUZ. Matematika), 38:1 (1994), 59–68 | MR | MR | Zbl

[4] Sviridyuk G. A., Sukacheva T. G., “On the solvability of a nonstationary problem describing the dynamics of an incompressible viscoelastic fluid”, Mathematical Notes, 63:3 (1998), 388–395 | MR | Zbl

[5] Sukacheva T. G., “On the solvability of the nonstationary problem of the dynamics of an incompressible viscoelastic fluid of the Kelvin–Voigt nonzero”, Russian Mathematics (Izvestiya VUZ. Matematika), 43:3 (1998), 44–51 | MR | MR

[6] Korpusov M. O., Sveshnikov A. G., “Blow-up of Oskolkov's system of equations”, Sbornik: Mathematics, 200:4 (2009), 549–572 | MR | Zbl

[7] G. A. Sviridyuk, V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht–Boston–Koln–Tokyo, 2003, 216 pp. | MR | Zbl

[8] Sviridyuk G. A., Zagrebina S. A., “The Showalter-Sidorov problem as a phenomena of the Sobolev type equations”, The Bulletin of Irkutsk State University. Series: Mathematics, 3:1 (2010), 104–125 (in Russ.) | MR | Zbl

[9] Keller A. V., “Numerical solution of the optimal control problem for degenerate linear system of equations with Showalter–Sidorov initial conditions”, Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 27(127) (2008), 50–56 (in Russ.) | Zbl

[10] Sviridyuk G. A., Manakova N. A., “An optimal control problem for the Hoff equation”, Journal of Applied and Industrial Mathematics, 1:2 (2007), 247–253 | MR | MR

[11] Sviridyuk G. A., “A singular system of ordinary differential equations”, Differentsial'nye Uravneniya, 23:9 (1987), 1637–1639 (in Russ.) | MR | Zbl

[12] Ladyzhenskaya O. A., The mathematical theory of viscous incompressible flow, Gordon and Breach, N.Y.–London–Paris, 1969, 234 pp. | MR | MR | Zbl

[13] Temam R., Navier–Stokes equations. Theory and numerical analysis, Stud. Math. Appl., 2, North-Holland, Amsterdam–N.Y., 1979 | MR | MR | Zbl

[14] Solonnikov V. A., Linear elliptic systems. Lecture notes, LGU, L., 1979 (in Russ.)