Startseite Variation method solving of the inverse problems for Schrödinger-type equation
Artikel
Lizenziert
Nicht lizenziert Erfordert eine Authentifizierung

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

  • Arif Mir Calal Pashaev , Asaf Dashdamir Iskenderov EMAIL logo , Qabil Yavar Yaqubov und Matanet Asaf Musaeva
Veröffentlicht/Copyright: 8. Oktober 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. 10.1137/0313008Suche 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.2005.04.006Suche in Google Scholar

[3] M. Edelstein, Farthest points of sets in uniformly convex Banach spaces, Israel J. Math. 4 (1966), 171–176. 10.1007/BF02760075Suche in Google Scholar

[4] M. Goebel, On existence of optimal control, Math. Nachr. 93 (1979), 67–73. 10.1002/mana.19790930106Suche 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. Suche 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. Suche 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. Suche 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. 10.32523/2306-6172-2014-2-2-102-121Suche in Google Scholar

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

[11] O. A. Ladyzhenskaya, Boundary-Value Problems of Mathematical Physics, Izdat. “Nauka”, Moscow, 1973. Suche 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. Suche 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. 10.1063/1.3062347Suche in Google Scholar

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

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

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

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

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

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

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

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

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

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

Heruntergeladen am 30.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/jiip-2020-0095/html
Button zum nach oben scrollen