Geodesic
Optimization in a nonlinear parabolic system with a control in the coefficients
Sbornik. Mathematics, Tome 81 (1995) no. 2, pp. 533-543 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Two extremal problems are considered for a system described by a nonlinear equation of parabolic type with a control in the coefficients. Since the nonlinearity of the system is sufficiently strong, it is difficult to establish differentiability of the state function with respect to the control, and this is an obstacle to getting optimality conditions by the standard methods. The low regularity of the optimality criteria does not permit use of the penalty method. The apparatus of extended differentiability is used to overcome the difficulties arising. It is proved that extended differentiability holds for the state function with respect to the control, and this leads to the necessary optimality conditions. 
@article{SM_1995_81_2_a11,
     author = {S. Ya. Serovaǐskiǐ},
     title = {Optimization in a~nonlinear parabolic system with a~control in the~coefficients},
     journal = {Sbornik. Mathematics},
     pages = {533--543},
     year = {1995},
     volume = {81},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1995_81_2_a11/}
}
TY  - JOUR
AU  - S. Ya. Serovaǐskiǐ
TI  - Optimization in a nonlinear parabolic system with a control in the coefficients
JO  - Sbornik. Mathematics
PY  - 1995
SP  - 533
EP  - 543
VL  - 81
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1995_81_2_a11/
LA  - en
ID  - SM_1995_81_2_a11
ER  - 
%0 Journal Article
%A S. Ya. Serovaǐskiǐ
%T Optimization in a nonlinear parabolic system with a control in the coefficients
%J Sbornik. Mathematics
%D 1995
%P 533-543
%V 81
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1995_81_2_a11/
%G en
%F SM_1995_81_2_a11
S. Ya. Serovaǐskiǐ. Optimization in a nonlinear parabolic system with a control in the coefficients. Sbornik. Mathematics, Tome 81 (1995) no. 2, pp. 533-543. http://geodesic.mathdoc.fr/item/SM_1995_81_2_a11/

[1] Iskenderov A. D., Tagiev R. K., “Zadacha optimizatsii s upravleniem v koeffitsientakh dlya parabolicheskikh uravnenii”, Differents. uravneniya, 19:8 (1983), 1324–1334 | MR | Zbl

[2] Sokolovwsky J., “On optimal control of parabolic systems”, Lect. Notes Comp. Sci., 1, 1976, 623–635

[3] Zolezzi T., “Necessary conditions for optimal control of elliptic or parabolic problems”, SIAM J. Control, 10:2 (1972), 594–607 | DOI | MR | Zbl

[4] Ahmed K. L., Teo K. L., “Optimization control of systems governed by a class of nonlinear evolutional equations in a reflexive Banach space”, J. Optim. Theory Appl., 28:1 (1978), 57–81 | DOI | MR

[5] Barbu V., “Necessary conditions for control problems governed by nonlinear partial equations”, Res. Notes Math., 60, 1981, 19–47 | MR

[6] Friedman A., “Nonlinear optimal control problems for parabolic equations”, SIAM J. Contr. Optim., 22:5 (1984), 805–816 | DOI | MR | Zbl

[7] Wolfersdorf L., “Optimal control for processing governed by Midley nonlinear differential equations of parabolic type”, ZAMM, 56 (1976), 531–538 | DOI | MR | Zbl

[8] Matveev A. S., Yakubovich V. A., “Optimalnoe upravlenie nekotorymi sistemami s raspredelennymi parametrami”, Sib. matem. zhurn., 19:5 (1978), 1109–1140 | MR | Zbl

[9] Lions Zh.–L., Upravlenie singulyarnymi raspredelennymi sistemami, Nauka, M., 1987 | MR

[10] Serovaiskii S. Ya., “Neobkhodimye i dostatochnye usloviya optimalnosti dlya sistemy, opisyvaemoi nelineinym uravneniem ellipticheskogo tipa”, Sib. matem. zhurn., 32:3 (1991), 141–150 | MR | Zbl

[11] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR

[12] Gaevskii Kh., Greger K., Zakharias K., Nelineinye operatornye uravneniya i operatornye differentsialnye uravneniya, Mir, M., 1978 | MR

[13] Lions Zh.-L., Optimalnoe upravlenie sistemami, opisyvaemymi uravneniyami s chastnymi proizvodnymi, Mir, M., 1972 | MR

[14] Kantorovich L. V., Akilov G. P., Funktsionalnyi analiz i ego prilozheniya, Nauka, M., 1977 | MR | Zbl

[15] Serovaiskii S. Ya., “Kvazisopryazhennye sistemy i neobkhodimye usloviya optimalnosti v nelineinykh beskonechnomernykh sistemakh”, Izv. vuzov. Matematika, 1989, no. 4, 61–69 | MR

[16] Serovaiskii S. Ya., “Rasshirennaya differentsiruemost neyavnoi funktsii v prostranstvakh bez normy”, Izv. vuzov. Matematika, 1991, no. 12, 55–63 | MR | Zbl