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.
Classification : 93B05, 30E05, 35Q41
Mots clés : null-controllability, Schrödinger equation, complex Ginzburg-Landau equation, moment problem, biorthogonal, vanishing viscosity
@article{COCV_2012__18_1_277_0,
author = {Micu, Sorin and Roven\c{t}a, Ionel},
title = {Uniform controllability of the linear one dimensional {Schr\"odinger} equation with vanishing viscosity},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {277--293},
publisher = {EDP-Sciences},
volume = {18},
number = {1},
year = {2012},
doi = {10.1051/cocv/2010055},
zbl = {1242.93019},
mrnumber = {2887936},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2010055/}
}
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 PY - 2012 DA - 2012/// SP - 277 EP - 293 VL - 18 IS - 1 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2010055/ UR - https://zbmath.org/?q=an%3A1242.93019 UR - https://www.ams.org/mathscinet-getitem?mr=2887936 UR - https://doi.org/10.1051/cocv/2010055 DO - 10.1051/cocv/2010055 LA - en ID - COCV_2012__18_1_277_0 ER -
Micu, Sorin; Rovenţa, Ionel. Uniform controllability of the linear one dimensional Schrödinger equation with vanishing viscosity. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 1, pp. 277-293. doi : 10.1051/cocv/2010055. http://www.numdam.org/articles/10.1051/cocv/2010055/
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