Topological method for controllability

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.