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Tytuł artykułu

A Note on Optimal Control Problem Governed by Schrödinger Equation

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
In this work, we present some results showing the controllability of the linear Schrödinger equation with complex potentials. Firstly we investigate the existence and uniqueness theorem for solution of the considered problem. Then we find the gradient of the cost functional with the help of Hamilton-Pontryagin functions. Finally we state a necessary condition in the form of variational inequality for the optimal solution using this gradient.
Wydawca

Czasopismo
Rocznik
Tom
13
Numer
1
Opis fizyczny
Daty
otrzymano
2015-09-28
zaakceptowano
2015-11-26
online
2015-12-31
Twórcy
autor
  • Department of Mathematics,
    Ağrı İbrahim Çeçen Universty,Faculty of Science and
    Art,Ağrı,Turkey
autor
  • Department of Mathematics, Ataturk University,
    Faculty of Science, Erzurum, Turkey
  • Department of Mathematics, Kafkas University,
    Faculty of Science and Art, Kars, Turkey
Bibliografia
  • [1] A. D. Iskenderov, G. Ya. Yagubov, A variational method for solvingthe inverse problem of determining the quantum- mechanicalpotential, Soviet Math. Dokl.(English Trans.). AMS 38(1989)
  • [2] A. D. Iskenderov, G. Ya. Yagubov, Optimal control of nonlinearquantum-mechanical systems, Automatica and Telemechanic12(1989) 27-38
  • [3] A. D. Iskenderov, Definition of a potential in Schrödingers’ nonstationaryequation. In: Problemi moton. Modelşrovania andopmolno go upravleva, Bakü, 2001, pp.6-36 (in Russian)
  • [4] M. Goebel, On existence of optimal control, Math. Nachr. 1979.Vol.93, pp.67-73
  • [5] J. L. Lions, Optimal Control of Systems Governed by Partial DifferentialEquations, Springer, Berlin, 1971
  • [6] E. Zuazua, Remarks on the controllability of the Schrödingerequation, Quantum control: mathematical and numerical challenges,2003, Vol.33, pp. 193-211
  • [7] E. Zuazua, An introduction to the controllability of partial differentialequations, Quelques questions de théorie du contrôle,2004,
  • [8] E. Zuazua, Controllability of partial differential equations andits semi-discrete approximations, Discrete and continuous dynamicalsystems, 2004, 8 (2), 469-517
  • [9] N.Y. Aksoy, B. Yildiz, H. Yetiskin, Variational problem with complexcoeflcient of a nonlinear Schrödinger equation Proceedingsof the Indian Academy of Sciences:Mathematical Sciences,2012, 122 (3), pp. 469-484
  • [10] Y. Koçak, E. Çelik, Optimal control problem for stationary quasiopticequations, Boundary Value Problems 2012, 2012:151
  • [11] Y. Koçak, M.A. Dokuyucu, E. Çelik, Well-Posedness of OptimalControl Problem for the Schrödinger Equations with ComplexPotential, International Journal of Mathematics and Computation,2015, 26 (4), 11-16
  • [12] G.Ya. Yagubov, N.S. Ibrahimov, Optimal control problem fornonstationary quasi optic equation, Problems of mathematicalmodeling and optimal control, Baku, 2001, pp. 49-57 (in Russian).
  • [13] H. yetişkin., M. Subaşı., On the optimal control problem forSchrödinger equation with complex potential, Applied Mathematicsand Computation, 2010, 216(7), pp. 1896–1902
  • [14] G.Ya. Yagubov, M.A. Musayeva, On the identification problemfor nonlinear Schrödinger equation, Differentsial’niye uravneniya3(12) (1997) 1691–1698 (in Russian)
  • [15] M. Koksal, M.E. Koksal, Commutativity of Linear Time-varyingDifferential Systems with Non-zero Initial Conditions: A Reviewand Some New Extensions,Mathematical Problems in Engineering,2011 (2011) Article Number: 678575, 1-25, 2
  • [16] M. Koksal, M.E. Koksal, Commutativity of Cascade ConnectedDiscrete Time Linear Time-Varying Systems, Transactions of theInstitute of Measurement and Control, 2015, 37 (5) 615-622
  • [17] M.A. Vorontsov, V. I. Shmalgauzen, Principles of adaptive optics.Moscow, Izdatel’stvo Nauka, 1985, 336 p. (in Russian)
  • [18] K. Yosida, Functional analysis. Springer, 1980
  • [19] O.A. Ladyzhanskaya, V.A. Sollonnikov, N.M. Uraltseva, Linearand Quasi- Linear Equations of Parabolic Type, Translation ofMathematical Monographs. AMS, Rhode Island, 1968
  • [20] N.S. Ibrahimov, Solubility of initial-boundary value problemsfor linear stationary equation of quasi optic. Journal of QafqazUniversity. 2010, No:29, pp. 61-70 (in Russian)
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.-psjd-doi-10_1515_phys-2015-0051
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