Local exact controllability to the trajectories of the convective Brinkman-Forchheimer equations

In this article, we discuss the local exact controllability to trajectories of the following convective Brinkman-Forchheimer (CBF) equations (or damped Navier-Stokes equations) defined in a bounded domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$ ($d=2,3$) with smooth boundary: \begin{align*} \frac{\partial\boldsymbol{u}}{\partial t}-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{2}\boldsymbol{u}+\nabla p=\boldsymbol{f}+\boldsymbol{\vartheta}, \ \ \ \nabla\cdot\boldsymbol{u}=0, \end{align*} where the control $\mathit{\vartheta }$ is distributed in a subdomain $\omega \subset \mathrm{\Omega }$, and the parameters $\alpha ,\beta ,\mu >0$ are constants. We first present global Carleman estimates and observability inequality for the adjoint problem of a linearized version of CBF equations by using a global Carleman estimate for the Stokes system. This allows us to obtain its null controllability at any time $T>0$. We then use the inverse mapping theorem to deduce local results concerning the exact controllability to the trajectories of CBF equations.