Controllability of Schrödinger equation with a nonlocal term

Mariano De Leo; Constanza Sánchez Fernández de la Vega; Diego Rial

ESAIM: Control, Optimisation and Calculus of Variations (2014)

  • Volume: 20, Issue: 1, page 23-41
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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De Leo, Mariano, Sánchez Fernández de la Vega, Constanza, and Rial, Diego. "Controllability of Schrödinger equation with a nonlocal term." ESAIM: Control, Optimisation and Calculus of Variations 20.1 (2014): 23-41. <http://eudml.org/doc/272756>.

@article{DeLeo2014,
abstract = {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.},
author = {De Leo, Mariano, Sánchez Fernández de la Vega, Constanza, Rial, Diego},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {nonlinear Schrödinger–Poisson; Hartree potential; constant electric field; internal controllability; nonlinear Schrödinger-Poisson},
language = {eng},
number = {1},
pages = {23-41},
publisher = {EDP-Sciences},
title = {Controllability of Schrödinger equation with a nonlocal term},
url = {http://eudml.org/doc/272756},
volume = {20},
year = {2014},
}

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
PB - EDP-Sciences
VL - 20
IS - 1
SP - 23
EP - 41
AB - 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.
LA - eng
KW - nonlinear Schrödinger–Poisson; Hartree potential; constant electric field; internal controllability; nonlinear Schrödinger-Poisson
UR - http://eudml.org/doc/272756
ER -

References

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  13. [13] E. Zuazua, Remarks on the controllability of the Schrödinger equation, in Quantum Control: Mathematical and Numerical Challenges, vol. 33 of CRM Proc. Lect. Notes (2003) 193–211. MR2043529

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