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

Gökçe Dİlek Küçük; Gabil Yagub; Ercan Çelİk

doi:10.3934/dcdss.2019033

Article Contents

2019, Volume 12, Issue 3: 503-512. Doi: 10.3934/dcdss.2019033

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

Received: February 28, 2017

Revised: June 30, 2017

Published: October 2018

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Abstract

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 $H$.

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Mathematics Subject Classification: Primary: 49N05, 34A12.

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