Controllability of schrödinger equation with a nonlocal term

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...

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Autores principales: De Leo, M., Sánchez Fernández De La Vega, C., Rial, D.
Formato: JOUR
Materias:
Constant electric field
Hartree potential
Internal controllability
Nonlinear Schrödinger-Poisson
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_12928119_v20_n1_p23_DeLeo
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
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spelling todo:paper_12928119_v20_n1_p23_DeLeo2023-10-03T16:09:15Z Controllability of schrödinger equation with a nonlocal term De Leo, M. Sánchez Fernández De La Vega, C. Rial, D. Constant electric field Hartree potential Internal controllability Nonlinear Schrödinger-Poisson 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. © 2014 EDP Sciences, SMAI. Fil:Rial, D. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales; Argentina. JOUR info:eu-repo/semantics/openAccess http://creativecommons.org/licenses/by/2.5/ar http://hdl.handle.net/20.500.12110/paper_12928119_v20_n1_p23_DeLeo
institution Universidad de Buenos Aires
institution_str I-28
repository_str R-134
collection Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA)
topic Constant electric field
Hartree potential
Internal controllability
Nonlinear Schrödinger-Poisson
spellingShingle Constant electric field
Hartree potential
Internal controllability
Nonlinear Schrödinger-Poisson
De Leo, M.
Sánchez Fernández De La Vega, C.
Rial, D.
Controllability of schrödinger equation with a nonlocal term
topic_facet Constant electric field
Hartree potential
Internal controllability
Nonlinear Schrödinger-Poisson
description 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. © 2014 EDP Sciences, SMAI.
format JOUR
author De Leo, M.
Sánchez Fernández De La Vega, C.
Rial, D.
author_facet De Leo, M.
Sánchez Fernández De La Vega, C.
Rial, D.
author_sort De Leo, M.
title Controllability of schrödinger equation with a nonlocal term
title_short Controllability of schrödinger equation with a nonlocal term
title_full Controllability of schrödinger equation with a nonlocal term
title_fullStr Controllability of schrödinger equation with a nonlocal term
title_full_unstemmed Controllability of schrödinger equation with a nonlocal term
title_sort controllability of schrödinger equation with a nonlocal term
url http://hdl.handle.net/20.500.12110/paper_12928119_v20_n1_p23_DeLeo
work_keys_str_mv AT deleom controllabilityofschrodingerequationwithanonlocalterm
AT sanchezfernandezdelavegac controllabilityofschrodingerequationwithanonlocalterm
AT riald controllabilityofschrodingerequationwithanonlocalterm
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