On the controllability of the Boussinesq equation in low regularity

In this paper, we consider the internal control problem for the Boussinesq equation posed on the torus T. Previous results had dealt with this problem when the state space is H-2(T) x L-2(T). The main goal of this work is to improve the regularity until H-s (T) x Hs-2(T) for s >= -1/2. The exact controllability of the linearized equation is proved by using the moment method and spectral analysis. In order to get the same result for the nonlinear equation, we use a fixed point argument in Bourgain spaces.