  In this article, the following controlled convective Brinkman-Forchheimer
extended Darcy (CBFeD) system is considered in a $d$-dimensional torus:
  \begin{align*}
  \frac{\partial\boldsymbol{y}}{\partial t}-\mu
\Delta\boldsymbol{y}+(\boldsymbol{y}\cdot\nabla)\boldsymbol{y}+\alpha\boldsymbol{y}+\beta\vert
\boldsymbol{y}\vert ^{r-1}\boldsymbol{y}+\gamma\vert \boldsymbol{y}\vert
^{q-1}\boldsymbol{y}+\nabla p=\boldsymbol{g}+\boldsymbol{u},\
\nabla\cdot\boldsymbol{y}=0,
  \end{align*}
  where $d\in\{2,3\}$, $\mu,\alpha,\beta>0$, $\gamma\in\mathbb{R}$,
$r,q\in[1,\infty)$ with $r>q\geq 1$. We prove the exponential stabilization of
CBFeD system by finite- and infinite-dimensional feedback controllers. The
solvability of the controlled problem is achieved by using the abstract theory
of $m$-accretive operators and density arguments. As an application of the
above solvability result, by using infinite-dimensional feedback controllers,
we demonstrate exponential stability results such that the solution preserves
an invariance condition for a given closed and convex set. By utilizing the
unique continuation property of controllability for finite-dimensional systems,
we construct a finite-dimensional feedback controller which exponentially
stabilizes CBFeD system locally, where the control is localized in a smaller
subdomain. Furthermore, we establish the local exponential stability of CBFeD
system via proportional controllers.
