# Topological method for controllability

## TL;DR

This paper introduces a topological approach to determine controllability in linear and nonlinear systems, providing new insights and methods applicable to various equations including Schrödinger, Korteweg-de Vries, and Saint-Venant.

## Contribution

It presents a novel topological method for controllability, extending its application to complex systems like Schrödinger, Korteweg-de Vries, and Saint-Venant equations.

## Key findings

- Controllability achieved for 1D Schrödinger equation.
- Method successfully applied to Korteweg-de Vries and Saint-Venant equations.
- Illustrative examples demonstrate the effectiveness of the topological approach.

## Abstract

In this work, we found a non trivial topology to achieve the controllability for linear and nonlinear system in finite or infinite time horizon. We give several examples illustrating this topologizing method for the controllability results. We obtain by the way the controllability for the one dimensional Schr\"odinger. We also apply this method to achieve the both controllability of Korteweg-de Vries and Saint-Venant equations.