Un lien entre le coÛt des contrÔles rapides pour l'équation de la chaleur 1-D et l'uniforme contrÔlabilité d'une équation de transport-diffusion 1-D

In this Note, we explain how results on the cost of the null-controllability of the one-dimensional heat equation in small time can be used to bound from above the cost of the null-controllability of a one-dimensional transport-diffusion equation in the vanishing viscosity limit. We improve previous results about the minimal time needed to obtain the exponential decrease of the cost of the control and explain what would provide the usual conjecture concerning the cost of fast controls for the heat equation.