Approximate controllability of the linearized Boussinesq system in a two dimensional channel

Abstract

In this paper, we prove an approximate controllability result for the linearized Boussinesq system around a fluid at rest, in a two dimensional channel, when the control acts only on the temperature, through the upper boundary. This result can be seen as a first step to obtain an open loop stabilization result of the nonlinear Boussinesq system, in the spirit of the article [CE19] by Chowdhury, Ervedoza concerning the Navier Stokes equations. The proof relies on the well-known Fattorini criterion, i.e. we show an unique continuation property for the adjoint system, by expanding the solution in Fourier series and using ordinary differential equations arguments. More precisely, we prove that the spectrum of the adjoint operator splits into two parts corresponding respectively to the Stokes eigenvalues and the Dirichlet Laplacian eigenvalues. Whereas the second part can be treated easily thanks to the wellknown form of the eigenfunctions, the first part requires to show that a matrix of size three, depending analytically of the parameters of the problem, is "generically" invertible.