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

Sorin Micu; Ionel Rovenţa

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 1, page 277-293
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Micu, Sorin, and Rovenţa, Ionel. "Uniform controllability of the linear one dimensional Schrödinger equation with vanishing viscosity." ESAIM: Control, Optimisation and Calculus of Variations 18.1 (2012): 277-293. <http://eudml.org/doc/276372>.

@article{Micu2012,
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. },
author = {Micu, Sorin, Rovenţa, Ionel},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Null-controllability; Schrödinger equation; complex Ginzburg-Landau equation; moment problem; biorthogonal; vanishing viscosity; null-controllability; biorthogonal system},
language = {eng},
month = {2},
number = {1},
pages = {277-293},
publisher = {EDP Sciences},
title = {Uniform controllability of the linear one dimensional Schrödinger equation with vanishing viscosity},
url = {http://eudml.org/doc/276372},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Micu, Sorin
AU - Rovenţa, Ionel
TI - Uniform controllability of the linear one dimensional Schrödinger equation with vanishing viscosity
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/2//
PB - EDP Sciences
VL - 18
IS - 1
SP - 277
EP - 293
AB - 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.
LA - eng
KW - Null-controllability; Schrödinger equation; complex Ginzburg-Landau equation; moment problem; biorthogonal; vanishing viscosity; null-controllability; biorthogonal system
UR - http://eudml.org/doc/276372
ER -

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