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Variation method solving of the inverse problems for Schrödinger-type equation

  • Arif Mir Calal Pashaev , Asaf Dashdamir Iskenderov EMAIL logo , Qabil Yavar Yaqubov and Matanet Asaf Musaeva
Published/Copyright: October 8, 2020

Abstract

A variation method for solving the inverse problems of determining of the complex quantum potential in a nonlinear non-stationary Schrödinger-type equation with final and boundary observations is considered. The existence and uniqueness theorem of the solution of the variation formulation of the inverse problem is proved, the continuity and continuous differentiability of the quality criterion are established, the formula for its gradient is found, the necessary condition of optimality is proved and iterative regularization of the solution is indicated.

MSC 2010: 49N45; 47J06

Dedicated to 75th years of Professor A. Q. Yaqola


References

[1] J. Baranger and R. Temam, Nonconvex optimization problems depending on a parameter, SIAM J. Control 13 (1975), 146–152. 10.1137/0313008Search in Google Scholar

[2] L. Baudouin, O. Kavian and J.-P. Puel, Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control, J. Differential Equations 216 (2005), no. 1, 188–222. 10.1016/j.jde.2005.04.006Search in Google Scholar

[3] M. Edelstein, Farthest points of sets in uniformly convex Banach spaces, Israel J. Math. 4 (1966), 171–176. 10.1007/BF02760075Search in Google Scholar

[4] M. Goebel, On existence of optimal control, Math. Nachr. 93 (1979), 67–73. 10.1002/mana.19790930106Search in Google Scholar

[5] A. D. Iskenderov, Variational formulations of multidimensional inverse problems of mathematical physics, Dokl. Akad. Nauk SSSR 274 (1984), no. 3, 531–533. Search in Google Scholar

[6] A. D. Iskenderov and G. Y. Yagubov, A variational method for solving an inverse problem of determining the quantum mechanical potential, Dokl. Akad. Nauk SSSR 303 (1988), no. 5, 1044–1048. Search in Google Scholar

[7] A. D. Iskenderov and G. Y. Yagubov, Optimal control of nonlinear quantum-mechanical systems, Avtomat. i Telemekh. (1989), no. 12, 27–38. Search in Google Scholar

[8] A. D. Iskenderov, G. Y. Yagubov, N. S. Ibragimov and Y. N. Aksoy, Variation formulation of the inverse problem of determining the complex-coefficient of equation of quasi optics, Eurasian J. Math. Comput. Appl. 2 (2014), 102–121. 10.32523/2306-6172-2014-2-2-102-121Search in Google Scholar

[9] A. D. Iskenderov, G. Y. Yagubov and M. A. Musaeva, The Identification of Quantum Mechanics Potentials, Chashyoglu, Baku, 2012. Search in Google Scholar

[10] V. K. Ivanov, V. V. Vasin and V. P. Tanana, The Theory of Linear Ill-Posed Problems and its Applications, “Nauka”, Moscow, 1978. Search in Google Scholar

[11] O. A. Ladyzhenskaya, Boundary-Value Problems of Mathematical Physics, Izdat. “Nauka”, Moscow, 1973. Search in Google Scholar

[12] O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type, Izdat. “Nauka”, Moscow, 1967. Search in Google Scholar

[13] L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory. Course of Theoretical Physics. Vol. 3, Pergamon Press, London, 1958. 10.1063/1.3062347Search in Google Scholar

[14] M. M. Lavrentiev, V. G. Romanov and M. P. Shishatsky, Incorrect Problems of Mathematical Physics and Analysis, Nauka, Moscow, 1980. Search in Google Scholar

[15] J.-L. Lions, Inhomogeneous Boundary Value Problems and Their Applications, Mir, Moscow, 1971. 10.1007/978-3-642-65217-2Search in Google Scholar

[16] J.-L. Lions, Optimal Control of Systems Described by Partial Differential Equations, Mir, Moscow, 1972. 10.1007/978-3-642-65024-6Search in Google Scholar

[17] M. A. Musaeva, Variation Methods for Determining the Quantum Potential, Izdat. Elm ve tehsil, Baku, 2019. Search in Google Scholar

[18] A. M. Pashaev, Development of Methods and Devices for Non-Contact Measurement of Semiconductor Material Parameters in Field of High and Ultra-High Frequencies, Edit. ASA, Baku, 1966. Search in Google Scholar

[19] V. G. Romanov, Inverse Problems of Mathematical Physics, “Nauka”, Moscow, 1984. Search in Google Scholar

[20] A. N. Tikhonov and V. Y. Arsenin, Methods for Solving of Ill-Posed Problems, “Nauke”, Moscow, 1979. Search in Google Scholar

[21] A. N. Tikhonov, V. Y. Arsenin and A. A. Timonov, Mathematical Problems of Computed Tomography, “Nauka”, Moscow, 1987. Search in Google Scholar

[22] A. N. Tikhonov, A. S. Leonov and A. G. Yagola, Nonlinear Incorrect Problems, Nauka, Moscow, 1995. 10.1515/9783110883237.505Search in Google Scholar

[23] O. V. Vasiliev, Optimization Methods, World Federation, Atlanta, 1996. Search in Google Scholar

[24] G. Y. Yagubov and M. A. Musaeva, On an identification problem for the nonlinear Schrödinger equation, Differ. Equ. 33 (1997), 1691–1698. Search in Google Scholar

[25] K. Yosida, Functional Analysis, “Mir”, Moscow, 1967. 10.1007/978-3-662-11791-0Search in Google Scholar

Received: 2020-08-06
Revised: 2020-08-25
Accepted: 2020-08-29
Published Online: 2020-10-08
Published in Print: 2023-12-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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