Optimal nonlinearity control of Schrödinger equation

Kai Wang; Dun Zhao; Binhua Feng

doi:10.3934/eect.2018016

Article Contents

2018, Volume 7, Issue 2: 317-334. Doi: 10.3934/eect.2018016

This issue Previous Article Asymptotic behavior of a hierarchical size-structured population model Next Article

Optimal nonlinearity control of Schrödinger equation

1.
School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China
2.
Department of Mathematic, Northwest Normal University, Lanzhou 730070, China

* Corresponding author: Dun Zhao
* Corresponding author: Dun Zhao

Received: June 2017

Revised: February 2018

Published: May 2018

This work is supported by the NSFC under grants No. 11475073, No. 11325417 and No. 11601435.

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Abstract

We study the optimal nonlinearity control problem for the nonlinear Schrödinger equation $iu_{t} = -\triangle u+V(x)u+h(t)|u|^α u$ , which is originated from the Fechbach resonance management in Bose-Einstein condensates and the nonlinearity management in nonlinear optics. Based on the global well-posedness of the equation for $0<α<\frac{4}{N}$ , we show the existence of the optimal control. The Fréchet differentiability of the objective functional is proved, and the first order optimality system for $N≤ 3$ is presented.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 49J20.

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