# Remarks on local controllability for the Boussinesq system with Navier boundary condition

## TL;DR

This paper investigates the local controllability of the Boussinesq system with Navier boundary conditions, showing that in 2D only one control is needed, while in 3D two controls are necessary.

## Contribution

It establishes local exact controllability results for the Boussinesq system with nonlinear Navier-slip boundary conditions, highlighting differences between 2D and 3D cases.

## Key findings

- In 2D, local controllability achieved with a single control.
- In 3D, two scalar controls are required for controllability.
- Results extend controllability theory to systems with Navier boundary conditions.

## Abstract

This note deals with the local exact controllability to a particular class of trajectories for the Boussinesq system with nonlinear Navier-slip boundary conditions and internal controls having vanishing components. Briefly speaking, in two dimensions, the local exact controllability property is obtained using only one control in the heat equation, meanwhile two scalar controls are required in three dimensions.