  The convective Brinkman-Forchheimer (CBF) equations describe the motion of
incompressible viscous fluid through a rigid, homogeneous, isotropic, porous
medium. In this work, we consider some distributed optimal control problems
like total energy minimization, minimization of enstrophy, etc governed by the
two dimensional CBF equations with the absorption exponent $r=1,2$ and $3$. We
show the existence of an optimal solution and the first order necessary
conditions of optimality for such optimal control problems in terms of the
Euler-Lagrange system. Furthermore, for the case $r=3$, we show the second
order necessary and sufficient conditions of optimality. We also investigate an
another control problem which is similar to that of the data assimilation
problems in meteorology of obtaining unknown initial data, when the system
under consideration is 2D CBF equations, using optimal control techniques.
