Small-time local control of a Schr\"odinger equation: a negative and a positive quadratic result

TL;DR

This paper investigates the small-time local controllability of a bilinear Schrödinger equation near its ground state, providing both a negative quadratic obstruction and a positive quadratic controllability result using Fourier-based methods.

Contribution

It introduces the first positive quadratic controllability result for a physical PDE with a single scalar control and develops new tools for analyzing quadratic forms with non-regular kernels.

Findings

Identifies a PDE instance of Sussmann's quadratic obstruction.

Establishes positive controllability at quadratic order for a Schrödinger equation.

Develops Fourier-based techniques for non-regular quadratic kernel analysis.

Abstract

We study the small-time local controllability (STLC) of a bilinear Schr\"odinger equation with Neumann boundary conditions near its ground state. We focus on the degenerate case where the linearized system is not controllable, necessitating a second-order analysis. We prove two complementary results. The negative result provides a new PDE instance of Sussmann's classical quadratic obstruction, corresponding to a non-vanishing Lie bracket. The positive result appears to be the first to establish STLC at the quadratic order for a physical PDE with a single scalar control. Both proofs rely on a Fourier-based approach, which is crucial because the integral kernel of the second-order term lacks the regularity required by standard integration-by-parts arguments. Along the way, we develop tools valid in a more general setting to analyze such quadratic forms. In particular, we prove results that allow for the multiplication of a kernel by a modulation function.