Geodesic
Variational method of solving a coefficient inverse problem for an elliptic equation
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 10 (2018) no. 1, pp. 12-20 Cet article a éte moissonné depuis la source Math-Net.Ru

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One of the main types of inverse problems for equations with partial derivatives are the problems in which coefficients of equations or included values have to be determined based on some additional information. Such problems are called coefficient inverse problems for equations with partial derivatives. Inverse problems for equations with partial derivatives can be set in a variational form, i. e. like problems of optimal control by corresponding systems. Variational setting of one coefficient inverse problem for a two-dimensional elliptic equation with additional integral condition is considered. At that, the control function gets included in the coefficient when solving the equation of state, and is an element of a space of quadric totalized functions in the sense of Lebeg. Objective functional is set on the basis of an additional integral condition. Boundary conditions for equation of the state are mixed, i.e. the second boundary condition is given in one part of the boundary, and the first boundary condition is given in another part. Solving the boundary problem at each fixed control coefficient intends a generalized solution from the Sobolev space. The questions of correctness of the considered coefficient inverse problem in variational setting are studied. It is proved that the considered problem is correctly set in the weak topology of control functions’ space. I. e. the multitude of optimal controls is nonvacuous and weakly compact; and any minimizing sequence of the problem weakly converges to the multitude of optimal controls. Besides, differentiability of objective functional in the sense of Frechet is proved, and a formula for its gradient is obtained. The necessary optimum condition in the form of variational inequality is determined.
Mots-clés : elliptic equation
Keywords: inverse problem, integral condition, variational method. 
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R. K. Tagiev; R. S. Kasymova. Variational method of solving a coefficient inverse problem for an elliptic equation. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 10 (2018) no. 1, pp. 12-20. http://geodesic.mathdoc.fr/item/VYURM_2018_10_1_a1/

[1] Tikhonov A. N., DAN SSSR, 151:3 (1963), 501–504 (in Russ.) | Zbl

[2] Iskenderov A. D., DAN SSSR, 274:3 (1984), 531–533 (in Russ.) | Zbl

[3] Alifanov O. A., Artiukhin E. A., Rumiantsev S. V., Extreme methods for solving ill-posed problems, Nauka Publ., M., 1988, 285 pp. (in Russ.)

[4] A.L. Karchevsky, “Properties the misfit functional for a nonlinear one-dimensional coefficient hyperbolic inverse problem”, Journal of Inverse and Ill-Posed Problems, 5:2 (1997), 139–163 | DOI | MR | Zbl

[5] Kabanikhin S. I., Iskakov K. T., “Justification of the Steepest Descent Method for the Integral Statement of an Inverse Problem for a Hyperbolic Equation”, Siberian Mathematical Journal, 42:3 (2001), 478–494 | DOI | MR | Zbl

[6] Tagiev R. K., “A Variational Method for Solving the Inverse Problem of Determining the Coefficients of Elliptic Equations” (Azerbaijan, Sumgait, May 5–6, 2003), International Conference “Inverse Problems of Theoretical and Mathematical Physics”, 29–31 (in Russ.)

[7] Iskenderov A. D., Gamidov R. A., “Optimal identification of coefficients of elliptic equations”, Automation and Remote Control, 72:12 (2011), 2553–2562 | DOI | MR | Zbl

[8] A.D. Iskenderov, R.K. Tagiyev, “Variational method solving the problem of identification of the coefficients of quasilinear parabolic problem”, The 7th International Conference «Inverse Problems: modelling and Simulation» (IPMS-2014) (May 26–31, 2014), 31 | Zbl

[9] Tagiyev R. K., Kasumov R. A., “On the optimization formulation of the coefficient inverse problem for a parabolic equation with an additional integral condition”, Bulletin of Tomsk State University. Mathematics and mechanics, 2017, no. 45, 49–59 | DOI

[10] Ladyzhenskaya O. A., Ural'tseva N. N., Nauka Publ., M., 1973, 576 pp. (in Russ.)

[11] Ladyzhenskaya O. A., Solonnikov V. A., Ural'tseva N. N., Nauka Publ., M., 1967, 736 pp. (in Russ.)

[12] Vasil'ev F.P., Metody resheniya ekstremal'nykh zadach, Nauka Publ., M., 1981, 400 pp. (in Russ.)