In this paper, we prove the existence of controls insensitizing the $L^2$-norm of the solution of the Boussinesq system. The novelty here is that no control is used on the temperature equation. Furthermore, the control acting on the fluid equation can be chosen to have one vanishing component. It is well known that the insensitizing control problem is equivalent to a null controllability result for a cascade system, which is obtained thanks to a suitable Carleman estimate for the adjoint of the linearized system and an inverse mapping theorem. The particular form of the adjoint equation will allow us to obtain the null controllability of the linearized system.
"Insensitizing controls for the Boussinesq system with no control on the temperature equation." Adv. Differential Equations 22 (3/4) 235 - 258, March/April 2017.