Generic controllability properties for the bilinear Schr\"odinger equation

TL;DR

This paper demonstrates that approximate controllability of the bilinear Schrödinger equation is generically valid with respect to both the uncontrolled and controlled potentials, using perturbation theory and eigenfunction analysis.

Contribution

It proves the genericity of controllability conditions for the bilinear Schrödinger equation with respect to potential variations.

Findings

Controllability is generic with respect to the controlled potential W when V is fixed.

Controllability is generic with respect to the uncontrolled potential V when W is fixed and non-constant.

Analytic perturbation and eigenfunction asymptotics are key tools used.

Abstract

In [15] we proposed a set of sufficient conditions for the approximate controllability of a discrete-spectrum bilinear Schr\"odinger equation. These conditions are expressed in terms of the controlled potential and of the eigenpairs of the uncontrolled Schr\"odinger operator. The aim of this paper is to show that these conditions are generic with respect to the uncontrolled and the controlled potential, denoted respectively by $V$ and $W$. More precisely, we prove that the Schr\"odinger equation is approximately controllable generically with respect to $W$ when $V$ is fixed and also generically with respect to $V$ when $W$ is fixed and non-constant. The results are obtained by analytic perturbation arguments and through the study of asymptotic properties of eigenfunctions.