Stackelberg exact controllability for the Boussinesq system

We consider a Stackelberg control strategy applied to the Boussinesq system. More precisely, we act on this system with a hierarchy of two controls. The aim of the "leader" control is the null-controllability property whereas the objective of the "follower" control is to keep the state close to a given trajectory. By solving first the optimal control problem associated with the follower control, we are lead to show the null-controllability property of a system coupling a forward with a backward Boussinesq type systems. Our main result states that for an adequate weighted functional for the optimal control problem, this coupled system is locally null-controllable. To show this result, we first study the adjoint system of the linearized system and obtain a weighted observability estimate by combining several Carleman estimates and an adequate decomposition for the heat and the Stokes system.