Open loop stabilization of incompressible Navier-Stokes equations in a 2d channel using power series expansion

International audienceIn this article, we discuss the stabilization of incompressible Navier-Stokes equations in a 2d channel around a fluid at rest when the control acts only on the normal component of the upper boundary. In this case, the linearized equations are not controllable nor stabilizable at an exponential rate higher than $\nu\pi^2 /L^2$, when the channel is of width $L$ and of length $2\pi$ and $\nu$ denotes the viscosity parameter. Our main result allows to go above this threshold and reach any exponential decay rate by using the non-linear term to control the directions which are not controllable for the linearized equations. Our approach therefore relies on writing the controlled trajectory as an expansion of order two taking the form $\varepsilon \alpha + \varepsilon^2 \beta$ for some $\varepsilon >0$ small enough. This method is inspired by the previous work [18] by J.-M. Coron and E. Crépeau on the controllability of the Korteweg de Vries equations