Geodesic
The stability of solutions to extremal problems of boundary control for stationary heat convection equations
Sibirskij žurnal industrialʹnoj matematiki, Tome 13 (2010) no. 2, pp. 5-18.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study the extremal problems of boundary control for the stationary heat convection equations with Dirichlet boundary conditions on the velocity and temperature. As cost functionals we choose the mean square deviation of the required temperature field from the temperature field measured in some part of the flow region. The controls are functions involved in the Dirichlet conditions. We prove the stability of solutions under certain perturbations of both the quality functional and one of the known functions involved in the original equations of the model.
Keywords: heat convection, extremal problems, uniqueness, stability estimates. 
@article{SJIM_2010_13_2_a1,
     author = {G. V. Alekseev and A. M. Khludnev},
     title = {The stability of solutions to extremal problems of boundary control for stationary heat convection equations},
     journal = {Sibirskij \v{z}urnal industrialʹnoj matematiki},
     pages = {5--18},
     publisher = {mathdoc},
     volume = {13},
     number = {2},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJIM_2010_13_2_a1/}
}
TY  - JOUR
AU  - G. V. Alekseev
AU  - A. M. Khludnev
TI  - The stability of solutions to extremal problems of boundary control for stationary heat convection equations
JO  - Sibirskij žurnal industrialʹnoj matematiki
PY  - 2010
SP  - 5
EP  - 18
VL  - 13
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJIM_2010_13_2_a1/
LA  - ru
ID  - SJIM_2010_13_2_a1
ER  - 
%0 Journal Article
%A G. V. Alekseev
%A A. M. Khludnev
%T The stability of solutions to extremal problems of boundary control for stationary heat convection equations
%J Sibirskij žurnal industrialʹnoj matematiki
%D 2010
%P 5-18
%V 13
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJIM_2010_13_2_a1/
%G ru
%F SJIM_2010_13_2_a1
G. V. Alekseev; A. M. Khludnev. The stability of solutions to extremal problems of boundary control for stationary heat convection equations. Sibirskij žurnal industrialʹnoj matematiki, Tome 13 (2010) no. 2, pp. 5-18. http://geodesic.mathdoc.fr/item/SJIM_2010_13_2_a1/

[1] Alifanov O. M., Obratnye zadachi teploobmena, Mashinostroenie, M., 1988

[2] Georgievskii P. Yu., Levin V. A., “Upravlenie obtekaniem razlichnykh tel s pomoschyu lokalizovannogo podvoda energii v sverkhzvukovoi nabegayuschii potok”, Izv. RAN. Mekhanika zhidkosti i gaza, 2003, no. 5, 154–167 | MR

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

[4] Mohamed Gad-el-Hak, “Flow control”, Appl. Mech. Rev., 42:10 (1989), 261–293 | DOI

[5] Gunzburger M. D., Hou L., Svobodny T. P., “The approximation of boundary control problems for fluid flows with an application to control by heating and cooling”, Comput. Fluids, 22 (1993), 239–251 | DOI | MR | Zbl

[6] Ito K., Ravindran S. S., “Optimal control of thermally convected fluid flows”, SIAM J. Sci. Comput., 19:6 (1998), 1847–1869 | DOI | MR | Zbl

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

[8] Lee H.-C., Imanuvilov O. Yu., “Analysis of optimal control problems for the 2-D stationary Boussinesq equations”, J. Math. Anal. Appl., 242 (2000), 191–211 | DOI | MR | Zbl

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

[10] Alekseev G. V., “Obratnye ekstremalnye zadachi dlya statsionarnykh uravnenii teplovoi konvektsii”, Vestnik Novosibirsk. gos. un-ta. Ser. Matematika, mekhanika i informatika, 6:2 (2006), 7–31

[11] Alekseev G. V., “Edinstvennost i ustoichivost v koeffitsientnykh obratnykh ekstremalnykh zadachakh dlya statsionarnoi modeli massoperenosa”, Dokl. RAN, 416:6 (2007), 750–753 | MR | Zbl

[12] Alekseev G. V., “Koeffitsientnye obratnye ekstremalnye zadachi dlya statsionarnykh uravnenii teplomassoperenosa”, Zhurn. vychisl. matematiki i mat. fiziki, 47:6 (2007), 1055–1076

[13] Alekseev G. V., Soboleva O. V., Tereshko D. A., “Zadachi identifikatsii dlya statsionarnoi modeli massoperenosa”, Prikl. mekhanika i tekhn. fizika, 49:4 (2008), 24–35 | MR

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

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

[16] Alekseev G. V., Brizitskii R. V., “O edinstvennosti i ustoichivosti reshenii ekstremalnykh zadach dlya statsionarnykh uravnenii Nave–Stoksa”, Differents. uravneniya, 46:1 (2010), 68–79

[17] Alekseev G. V., “Zadachi upravleniya dlya statsionarnoi modeli magnitnoi gidrodinamiki vyazkoi teploprovodnoi zhidkosti”, Uspekhi mekhaniki, 2006, no. 2, 66–116

[18] Khludnev A. M., Sokolowski J., Modelling and Control in Solid Mechanics, Birkhauser, Basel–Boston–Berlin, 1997 | MR | Zbl