# Obstruction to the billinear control of the Gross-Pitaevskii equation: an example with an unbounded potential

## TL;DR

This paper presents an example demonstrating that controllability obstructions in bilinear control extend from linear to certain nonlinear dynamics, even with unbounded control potentials, highlighting limitations in control strategies for such systems.

## Contribution

It provides a specific nonlinear example where the known linear controllability obstruction persists despite unbounded control potentials.

## Key findings

- Obstruction to controllability applies to certain nonlinear systems.
- Unbounded control potentials do not guarantee controllability.
- Extends classical linear results to nonlinear dynamics with unbounded controls.

## Abstract

In 1982, Ball, Marsden, and Slemrod proved an obstruction to the controllability of linear dynamics with a bounded bilinear control term. This note presents an example of nonlinear dynamics with respect to the state for which this obstruction still holds while the control potential is not bounded.