# American Institute of Mathematical Sciences

June  2019, 12(3): 503-512. doi: 10.3934/dcdss.2019033

## On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation

 1 Iǧdır University, Faculty of Science and Art, Department of Mathematics, Iǧdır, Turkey 2 Kafkas University, Faculty of Science and Art, Department of Mathematics, Kars, Turkey 3 Atatürk University, Faculty of Science, Department of Mathematics, Erzurum, Turkey

* Corresponding author: gokce.kucuk@igdir.edu.tr

Received  March 2017 Revised  July 2017 Published  September 2018

In this paper, an optimal control problem for Schrödinger equation with complex coefficient which contains gradient is examined. A theorem is given that states the existence and uniqueness of the solution of the initial-boundary value problem for Schrödinger equation. Then for the solution of the optimal control problem, two different cases are investigated. Firstly, it is shown that the optimal control problem has a unique solution for $α >0$ on a dense subset $G$ on the space $H$ which contains the measurable square integrable functions on $\left(0,l\right)$ and secondly the optimal control problem has at least one solution for any $α ≥ 0$ on the space $H$.

Citation: Gökçe Dİlek Küçük, Gabil Yagub, Ercan Çelİk. On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 503-512. doi: 10.3934/dcdss.2019033
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##### References:
 [1] Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 [2] Kai Wang, Dun Zhao, Binhua Feng. Optimal nonlinearity control of Schrödinger equation. Evolution Equations & Control Theory, 2018, 7 (2) : 317-334. doi: 10.3934/eect.2018016 [3] Camille Laurent. Internal control of the Schrödinger equation. Mathematical Control & Related Fields, 2014, 4 (2) : 161-186. doi: 10.3934/mcrf.2014.4.161 [4] Binhua Feng, Xiangxia Yuan. On the Cauchy problem for the Schrödinger-Hartree equation. Evolution Equations & Control Theory, 2015, 4 (4) : 431-445. doi: 10.3934/eect.2015.4.431 [5] Binhua Feng, Dun Zhao. On the Cauchy problem for the XFEL Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4171-4186. doi: 10.3934/dcdsb.2018131 [6] Christian Léonard. A survey of the Schrödinger problem and some of its connections with optimal transport. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1533-1574. doi: 10.3934/dcds.2014.34.1533 [7] Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control & Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011 [8] Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems & Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469 [9] Hiroyuki Hirayama, Mamoru Okamoto. Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 6943-6974. doi: 10.3934/dcds.2016102 [10] Phan Van Tin. On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021028 [11] Qing Tang. On an optimal control problem of time-fractional advection-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 761-779. doi: 10.3934/dcdsb.2019266 [12] Peter I. Kogut. On approximation of an optimal boundary control problem for linear elliptic equation with unbounded coefficients. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2105-2133. doi: 10.3934/dcds.2014.34.2105 [13] Max Gunzburger, Sung-Dae Yang, Wenxiang Zhu. Analysis and discretization of an optimal control problem for the forced Fisher equation. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 569-587. doi: 10.3934/dcdsb.2007.8.569 [14] Hang-Chin Lai, Jin-Chirng Lee, Shuh-Jye Chern. A variational problem and optimal control. Journal of Industrial & Management Optimization, 2011, 7 (4) : 967-975. doi: 10.3934/jimo.2011.7.967 [15] Xiaorui Wang, Genqi Xu, Hao Chen. Uniform stabilization of 1-D Schrödinger equation with internal difference-type control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021022 [16] Umberto Biccari. Internal control for a non-local Schrödinger equation involving the fractional Laplace operator. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021014 [17] Claude Bardos, François Golse, Peter Markowich, Thierry Paul. On the classical limit of the Schrödinger equation. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5689-5709. doi: 10.3934/dcds.2015.35.5689 [18] D.G. deFigueiredo, Yanheng Ding. Solutions of a nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 563-584. doi: 10.3934/dcds.2002.8.563 [19] Frank Wusterhausen. Schrödinger equation with noise on the boundary. Conference Publications, 2013, 2013 (special) : 791-796. doi: 10.3934/proc.2013.2013.791 [20] Shuai Zhang, Shaopeng Xu. The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3367-3385. doi: 10.3934/cpaa.2020149

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