Geodesic
Variational statement of a coefficient inverse problem for a multidimensional parabolic equation
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 212 (2022), pp. 92-99.

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

In this paper, we consider the variational statement of an inverse problem of determining the leading coefficient of a multidimensional parabolic equation with nonlocal conditions. The leading coefficient of the equation playing the role of a control function is an element of the Sobolev space. The objective functional is based on the overdetermination condition, which can be interpreted as setting the weighted average value of the solution of the equation considered with respect to the time variable. The well-posedness of the problem in the weak topology of the control space is examined, the Fréchet differentiability of the objective functional is proved, and a necessary optimality condition is obtained.
Keywords: inverse problem, integral boundary condition, well-posedness, necessary optimality condition.
Mots-clés : parabolic equation 
@article{INTO_2022_212_a9,
     author = {R. K. Tagiyev and Sh. I. Maharramli},
     title = {Variational statement of a coefficient inverse problem for a multidimensional parabolic equation},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {92--99},
     publisher = {mathdoc},
     volume = {212},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2022_212_a9/}
}
TY  - JOUR
AU  - R. K. Tagiyev
AU  - Sh. I. Maharramli
TI  - Variational statement of a coefficient inverse problem for a multidimensional parabolic equation
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2022
SP  - 92
EP  - 99
VL  - 212
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2022_212_a9/
LA  - ru
ID  - INTO_2022_212_a9
ER  - 
%0 Journal Article
%A R. K. Tagiyev
%A Sh. I. Maharramli
%T Variational statement of a coefficient inverse problem for a multidimensional parabolic equation
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2022
%P 92-99
%V 212
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2022_212_a9/
%G ru
%F INTO_2022_212_a9
R. K. Tagiyev; Sh. I. Maharramli. Variational statement of a coefficient inverse problem for a multidimensional parabolic equation. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 212 (2022), pp. 92-99. http://geodesic.mathdoc.fr/item/INTO_2022_212_a9/

[1] Alifanov O. A., Artyukhin E. A., Rumyantsev S. V., Ekstremalnye metody resheniya nekorrektnykh zadach, Nauka, M., 1988 | MR

[2] Vasilev F. P., Metody resheniya ekstremalnykh zadach, Nauka, M., 1981 | MR

[3] Iskenderov A. D., “O variatsionnykh postanovkakh mnogomernykh obratnykh zadach matematicheskoi fiziki”, Dokl. AN SSSR., 274:3 (1984), 531–533 | MR | Zbl

[4] Kabanikhin S. I., Obratnye i nekorrektnye zadachi, Sib. nauch. izd-vo, Novosibirsk, 2009

[5] Kabanikhin S. I., Dairbaeva G., “Obratnaya zadacha nakhozhdeniya koeffitsienta uravneniya teploprovodnosti”, Tr. Mezhdunar. konf. «Obratnye i nekorrektnye zadachi matematicheskoi fiziki», posv. 75-letiyu akad. M. M. Lavrenteva (Novosibirsk, 2007), IM SO RAN, Novosibirsk, 2007, 1–5 | MR

[6] Kostin A. B., “Vosstanovlenie koeffitsienta pered $u_t$ v uravnenii teploprovodnosti po usloviyu nelokalnogo nablyudeniya po vremeni”, Zh. vychisl. mat. mat. fiz., 55:1 (2015), 89–104 | Zbl

[7] Ladyzhenskaya O. A., Solonnikov V. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967 | MR

[8] Prilepko A. I., Kostin A. B., Solovev V. V., “Obratnye zadachi nakhozhdeniya istochnika i koeffitsientov dlya ellipticheskikh i parabolicheskikh uravnenii v prostranstvakh Geldera i Soboleva”, Sib. zh. chist. prikl. mat., 17:3 (2017), 67–85 | Zbl

[9] Tagiev R. K., “Optimalnoe upravlenie koeffitsientami v parabolicheskikh sistemakh”, Differ. uravn., 45:10 (2009), 1492–1501 | MR | Zbl

[10] Tagiev R. K., Kasumov R. A., “Ob optimizatsionnoi postanovke koeffitsientnoi obratnoi zadachi dlya parabolicheskogo uravneniya s dopolnitelnym integralnym usloviem”, Vestn. Tomsk. gos. un-ta. Mat. mekh., 2017, no. 45, 49–59

[11] Tagiev R. K., Magerramli Sh. I., “Variatsionnaya postanovka odnoi obratnoi zadachi dlya parabolicheskogo uravneniya s integralnymi usloviyami”, Vestn. Yuzhno-Ural. gos. un-ta. Cer. Mat. Mekh. Fiz., 12:3 (2020), 34–40 | Zbl

[12] Tagiev R. K., Magerramli Sh. I., “O razreshimosti nachalno-kraevoi zadachi dlya odnomernogo lineinogo parabolicheskogo uravneniya s integralnym granichnym usloviem”, Vestn. Bakinsk. un-ta. Ser. fiz.-mat. nauk., 2019, no. 2, 17–26

[13] Tikhonov A. N., “O reshenii nekorrektno postavlennykh zadach i metode regulyarizatsii”, Dokl. AN SSSR., 151:3 (1963), 501–504 | Zbl

[14] Iskenderov A. D., Tagiyev R. K., “Variational method of solving the problem of identification of the coefficients of a quasilinear parabolic problem”, Proc. 7th Int. Conf. “Inverse Problems: Modelling and Simulation” (IMPS-2014) (May 26–31, 2014, Ölüdeniz, Fethiye, Turkey), 2014, 31