A Lebeau--Robbiano approach to the controllability of the Boussinesq system with a reduced number of controls

In this paper, we prove the local null controllability of the Boussinesq system in dimensions two and three with a reduced number of localized controls. More precisely, the controls act on the temperature equation and on only $N-2$ components of the velocity equation. Our proof is based on a spectral approach in the spirit of the Lebeau–Robbiano method. The main difficulty comes from the coupled structure of the system. The velocity and temperature equations are governed by the Stokes operator and the Dirichlet Laplacian, respectively. Their natural spectral decompositions are different, and there is no common spectral localization naturally adapted to the coupled system. Therefore, the usual componentwise spectral argument cannot be applied directly. We show that the cascade structure of the linearized system makes it possible to overcome this difficulty. The main ingredient is a mixed observability estimate in which only the Stokes component is spectrally localized, while the heat component is treated without any frequency restriction. Combining this estimate with the dissipation of the high Stokes frequencies allows us to carry out a Lebeau–Robbiano iteration for the linearized system. The resulting linear controllability estimate is transferred to the nonlinear Boussinesq system through a time-iteration argument. As an important consequence, the controls have a small-time cost bounded by $C\exp (C\mathrm{/}T)$, which recovers the expected parabolic order while preserving the reduced number of controls.