Controllability of Schrödinger equation with a nonlocal term

¹ Instituto de Ciencias, Universidad Nacional de General Sarmiento, J.M. Gutiérrez 1150 (1613) Los Polvorines, Buenos Aires, Argentina
mdeleo@ungs.edu.ar
² IMAS – CONICET and Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Ciudad Universitaria, Pabellón I (1428) Buenos Aires, Argentina
csfvega@dm.uba.ar; drial@dm.uba.ar

Received: 19 March 2012
Revised: 18 December 2012

Abstract

This paper is concerned with the internal distributed control problem for the 1D Schrödinger equation, i u_t(x,t) = −u_xx+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.