Controllability of Schroedinger equation with a nonlocal term

TL;DR

This paper investigates the controllability of a 1D Schrödinger equation with a nonlocal Hartree-type nonlinearity, demonstrating controllability under certain conditions and limitations with compactly supported controls.

Contribution

It establishes controllability results for a nonlinear Schrödinger equation with nonlocal terms using Hilbert Uniqueness and fixed point methods, highlighting conditions for success and failure.

Findings

Controllability is achieved with controls outside a fixed compact interval.

Exact controllability is not possible with controls of compact support.

The results apply to initial and target states near the origin.

Abstract

This paper is concerned with the internal distributed control problem for the 1D Schroedinger equation, $i\,u_t(x,t)=-u_{xx}+\alpha(x)\,u+m(u)\,u,$ 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 $\alpha(x)$ is a regular function with linear growth at infinity, including constant electric fields. By means of both the Hilbert Uniqueness Method and the Schauder's fixed point 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.