Work

Small-Time Controllability for the Nonlinear Schrödinger Equation on \({\mathbb{R}^N}\) via Bilinear Electromagnetic Fields

Year: 2024

Type: article

Abstract: . We address the small-time controllability problem for a nonlinear Schrödinger equation ( NLS ) on [Formula: see text] in the presence of magnetic and electric external fields. We choose a particular framework where the equation becomes [Formula: see text]. Here, the control operators are defined by the zeroth Hermite function [Formula: see text] and the momentum operator [Formula: see text]. In detail, we study when it is possible to control the dynamics of NLS as fast as desired via sufficiently large control signals [Formula: see text] and [Formula: see text]. We first show the existence of a family of quantum states for which this property is verified: this extends to [Formula: see text] the validity of a small-time control property recently shown on [Formula: see text] by the first author and Nersesyan, and on [Formula: see text] by Chambrion and the second author. Second, by considering some specific states belonging to this family, as a physical consequence we show the capability of controlling arbitrary changes of energy in bounded regions of the quantum system, in time zero. Our results are proved by exploiting the idea that the nonlinear term in NLS is only a perturbation of the linear problem when the time is as small as desired. The core of the proof, then, is the controllability of the bilinear equation which is tackled by using specific noncommutativity properties of infinite-dimensional propagators.

Source: SIAM Journal on Control and Optimization

DOI: 10.1137/23m1590305

Authors: Alessandro Duca, Eugenio Pozzoli