# Controllability of the 1D Schr\"odinger equation using flatness

## TL;DR

This paper demonstrates the exact controllability of the 1D free Schrödinger equation using a flatness approach, providing explicit controls through a parametrization by derivatives of a flat output.

## Contribution

It introduces a direct flatness-based method to achieve explicit and regular controls for the Schrödinger equation's controllability.

## Key findings

- Explicit control functions are derived for the Schrödinger equation.
- The method achieves controllability in the energy space.
- Controls are parametrized by derivatives of a flat output.

## Abstract

We derive in a direct way the exact controllability of the 1D free Schr\"odinger equation with Dirichlet boundary control. We use the so-called flatness approach, which consists in parametrizing the solution and the control by the derivatives of a "flat output". This provides an explicit and very regular control achieving the exact controllability in the energy space.