Geodesic
Algorithm for numerical method of solution of the optimal control problem for semilinear Sobolev type models on basis of decomposition method
Journal of computational and engineering mathematics, Tome 2 (2015) no. 3, pp. 43-59

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

The question of finding the numerical solutions of optimal control problems in mathematical models based on the degenerate semilinear equations of mathematical physics is one of the most important. It is so because of a large number of applications of such problems. In the case of non-linear state equation the search for the numerical solution of optimal control problem becomes more difficult. A lot of initial-boundary value problems for the equations and the systems of equations which are not resolved with respect to time derivative are considered in the framework of abstract Sobolev type equations that make up the vast field of non-classical equations of mathematical physics. We are interested in the optimal control problem to solutions of the Showalter - Sidorov problem for the semilinear Sobolev type equation. The article proposes to use a decomposition method, allowing to linearize a non-linear equation, and the penalty method, which allows to find an approximate solution of the problem. Our numerical method is based on the Galerkin method, and the method of decomposition. Using it we find approximate solutions of optimal control problem for the Hoff mathematical model and the generalized mathematical model of deformation of the structure of I-beams.
Keywords: the Sobolev type equation, optimal control, the Showalter – Sidorov problem, the Galerkin method, decomposition method. 
N. A. Manakova. Algorithm for numerical method of solution of the optimal control problem for semilinear Sobolev type models on basis of decomposition method. Journal of computational and engineering mathematics, Tome 2 (2015) no. 3, pp. 43-59. http://geodesic.mathdoc.fr/item/JCEM_2015_2_3_a4/
@article{JCEM_2015_2_3_a4,
     author = {N. A. Manakova},
     title = {Algorithm for numerical method of solution of the optimal control problem for semilinear {Sobolev} type models on basis of decomposition method},
     journal = {Journal of computational and engineering mathematics},
     pages = {43--59},
     year = {2015},
     volume = {2},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JCEM_2015_2_3_a4/}
}
TY  - JOUR
AU  - N. A. Manakova
TI  - Algorithm for numerical method of solution of the optimal control problem for semilinear Sobolev type models on basis of decomposition method
JO  - Journal of computational and engineering mathematics
PY  - 2015
SP  - 43
EP  - 59
VL  - 2
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/JCEM_2015_2_3_a4/
LA  - en
ID  - JCEM_2015_2_3_a4
ER  - 
%0 Journal Article
%A N. A. Manakova
%T Algorithm for numerical method of solution of the optimal control problem for semilinear Sobolev type models on basis of decomposition method
%J Journal of computational and engineering mathematics
%D 2015
%P 43-59
%V 2
%N 3
%U http://geodesic.mathdoc.fr/item/JCEM_2015_2_3_a4/
%G en
%F JCEM_2015_2_3_a4

[1] N. Sidorov, B. Loginov, A. Sinithsyn, M. Falaleev, Shmidt Methods in Nonlinear Analysis and Applications, Kluwer Academic Publishers, Dordrecht–Boston–London, 2002, 548 pp.

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

[3] G. V. Demidenko, “$L_p$-Theory of Boundary Value Problems for Sobolev Type Equaitons”, Partial Differential Equations, Banach Center Publications, 27, 1992, 101–109 | MR | Zbl

[4] A. G. Sveshnikov, A. B. Alshin, M. O. Korpusov, Yu. D. Pletner, Lineinye i nelineinye uravneniya sobolevskogo tipa, FIZMATLIT, M., 2007, 736 pp.

[5] A. A. Zamyshlyaeva, Lineinye uravneniya sobolevskogo tipa vysokogo poryadka, Izd. tsentr YuUrGU, Chelyabinsk, 2012, 107 pp. | MR

[6] M. A. Sagadeeva, Dikhotomii reshenii lineinykh uravnenii sobolevskogo tipa, Izd. tsentr YuUrGU, Chelyabinsk, 2012, 107 pp. | MR

[7] R. E. Showalter, “The Sobolev Equation”, Applicable Analysis, 5:2 (1975), 81–89 | DOI | MR

[8] G. A. Sviridyuk, A. A. Efremov, “Optimalnoe upravlenie lineinymi uravneniyami tipa Soboleva s otnositelno p-sektorialnymi operatorami”, Differentsialnye uravneniya, 31:11 (1995), 1912–1919 | MR | Zbl

[9] N. A. Manakova, A. G. Dylkov, “Optimalnoe upravlenie resheniyami nachalno-konechnoi zadachi dlya lineinoi modeli Khoffa”, Matematicheskie zametki, 94:2 (2013), 225–236 | DOI | MR | Zbl

[10] A. A. Zamyshlyaeva, O. N. Tsyplenkova, “Optimalnoe upravlenie resheniyami zadachi Shouoltera - Sidorova - Dirikhle dlya uravneniya Bussineska - Lyava”, Differentsialnye uravneniya, 49:11 (2013), 1390–1398 | MR | Zbl

[11] A. V. Keller, “Chislennoe reshenie zadachi optimalnogo upravleniya vyrozhdennoi lineinoi sistemoi uravnenii s nachalnymi usloviyami Shouoltera - Sidorova”, Vestnik YuUrGU. Seriya: Matematicheskoe modelirovanie i programmirovanie, 2008, no. 27 (127), 50–56 | Zbl

[12] A. V. Keller, M. A. Sagadeeva, “Chislennoe reshenie zadach optimalnogo i zhestkogo upravleniya dlya odnoi nestatsionarnoi sistemy leontevskogo tipa”, Nauchnye vedomosti Belgorodskogo gosudarstvennogo universiteta. Seriya: Matematika. Fizika, 32:19 (2013), 57–66

[13] A. L. Shestakov, G. A. Sviridyuk, “Optimalnoe izmerenie dinamicheski iskazhennykh signalov”, Vestnik YuUrGU. Seriya: Matematicheskoe modelirovanie i programmirovanie, 2011, no. 17 (234), 70–75 | Zbl

[14] A. L. Shestakov, A. V. Keller, E. I. Nazarova, “Chislennoe reshenie zadachi optimalnogo izmereniya”, Avtomatika i telemekhanika, 2012, no. 1, 107–115 | Zbl

[15] G. A. Sviridyuk, S. A. Zagrebina, “Zadacha Shouoltera - Sidorova kak fenomen uravnenii sobolevskogo tipa”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya: Matematika, 3:1 (2010), 51–72 | MR

[16] S. A. Zagrebina, “Nachalno-konechnye zadachi dlya neklassicheskikh modelei matematicheskoi fiziki”, Vestnik YuUrGU. Seriya: Matematicheskoe modelirovanie i programmirovanie, 6:2 (2013), 5–24 | Zbl

[17] J.-L. Lions, Contrôle optimal de systémes gouvernés par des équations aux dérivées partielles, Dunod, Paris, 1968, 367 pp. | MR | MR

[18] G. A. Sviridyuk, N. A. Manakova, “Zadacha optimalnogo upravleniya dlya uravneniya Khoffa”, Sibirskii zhurnal industrialnoi matematiki, 8:2 (2005), 144–151 | MR

[19] N. A. Manakova, “An optimal control to solutions of the Showalter – Sidorov problem for the Hoff model on the geometrical graph”, J. Comp. Eng. Math., 1:1 (2014), 26–33 | MR | Zbl

[20] G. A. Sviridyuk, “Odna zadacha dlya obobschennogo filtratsionnogo uravneniya Bussineska”, Izvestiya vuzov. Matematika, 33:2 (1989), 62–73 | MR | Zbl

[21] N. A. Manakova, “Metod dekompozitsii v zadache optimalnogo upravleniya dlya polulineinykh modelei sobolevskogo tipa”, Vestnik YuUrGU. Seriya: Matematicheskoe modelirovanie i programmirovanie, 8:2 (2015), 133–137 | Zbl