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Uniform controllability of the linear one dimensional Schrödinger equation with vanishing viscosity

Published online by Cambridge University Press:  19 January 2011

Sorin Micu  and
Ionel Rovenţa
Sorin Micu
Affiliation:
Facultatea de Matematica si Informatica, Universitatea din Craiova, 200585 Craiova, Romania. sd_micu@yahoo.com, roventaionel@yahoo.com
Ionel Rovenţa
Affiliation:
Facultatea de Matematica si Informatica, Universitatea din Craiova, 200585 Craiova, Romania. sd_micu@yahoo.com, roventaionel@yahoo.com
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Abstract

This article considers the linear 1-d Schrödinger equation in (0,π) perturbed by a vanishing viscosity term depending on a small parameter ε > 0. We study the boundary controllability properties of this perturbed equation and the behavior of its boundary controls vε as ε goes to zero. It is shown that, for any time T sufficiently large but independent of ε and for each initial datum in H−1(0,π), there exists a uniformly bounded family of controls (vε)ε in L2(0, T) acting on the extremity x = π. Any weak limit of this family is a control for the Schrödinger equation.

Keywords

Null-controllabilitySchrödinger equationcomplex Ginzburg-Landau equationmoment problembiorthogonalvanishing viscosity
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 1 , January 2012 , pp. 277 - 293
Copyright
© EDP Sciences, SMAI, 2010

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