Controllability of Schrödinger equation with a nonlocal term. (6th February 2014)

• Record Type: Journal Article
• Authors: De Leo, Mariano; Sánchez Fernández de la Vega, Constanza; Rial, Diego
• Is Part Of: ESAIM. Volume 20:Number 1(2014:Jan.)
• Page Start: 23
• Page End: 41
• Publication Date: 2014-02-06
• Subjects: Nonlinear Schrödinger–Poisson, -- Hartree potential, -- constant electric field, -- internal controllability; System analysis -- Periodicals; Calculus of variations -- Periodicals; Mathematical analysis -- Periodicals; Mathematical optimization -- Periodicals; Control theory -- Periodicals; 515.64
• DOI: 10.1051/cocv/2013052
• Languages: English
• ISSNs: 1292-8119

Abstract

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.