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This paper is concerned with the internal distributed control problem for the 1D Schrödinger equation, i ut(x,t) = -uxx+α(x) u+m(u) u, that arises in quantum semiconductor models. Here m(u) is a non local Hartree-type nonlinearity stemming from the coupling with the 1D Poisson equation, and α(x) is a regular function with linear growth at infinity, including constant electric fields. By means of both the Hilbert Uniqueness Method and the contraction mapping theorem it is shown that for initial and target states belonging to a suitable small neighborhood of the origin, and for distributed controls supported outside of a fixed compact interval, the model equation is controllable. Moreover, it is shown that, for distributed controls with compact support, the exact controllability problem is not possible.
@article{COCV_2014__20_1_23_0,
author = {De Leo, Mariano and S\'anchez Fern\'andez de la Vega, Constanza and Rial, Diego},
title = {Controllability of {Schr\"odinger} equation with a nonlocal term},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {23--41},
publisher = {EDP-Sciences},
volume = {20},
number = {1},
year = {2014},
doi = {10.1051/cocv/2013052},
mrnumber = {3182689},
zbl = {1282.93052},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013052/}
}
TY - JOUR AU - De Leo, Mariano AU - Sánchez Fernández de la Vega, Constanza AU - Rial, Diego TI - Controllability of Schrödinger equation with a nonlocal term JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 23 EP - 41 VL - 20 IS - 1 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013052/ DO - 10.1051/cocv/2013052 LA - en ID - COCV_2014__20_1_23_0 ER -
%0 Journal Article %A De Leo, Mariano %A Sánchez Fernández de la Vega, Constanza %A Rial, Diego %T Controllability of Schrödinger equation with a nonlocal term %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 23-41 %V 20 %N 1 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2013052/ %R 10.1051/cocv/2013052 %G en %F COCV_2014__20_1_23_0
De Leo, Mariano; Sánchez Fernández de la Vega, Constanza; Rial, Diego. Controllability of Schrödinger equation with a nonlocal term. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 1, pp. 23-41. doi: 10.1051/cocv/2013052
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