  We consider the d-dimensional Boussinesq system defined on a sufficiently
smooth bounded domain, and subject to a pair $\{ v, \boldsymbol{u} \}$ of
controls localized on $\{ \widetilde{\Gamma}, \omega \}$. Here, $v$ is a scalar
Dirichlet boundary control for the thermal equation, acting on an arbitrary
small connected portion $\widetilde{\Gamma}$ of the boundary $\Gamma = \partial
\Omega$. Instead, $\boldsymbol{u}$ is a $d$-dimensional internal control for
the fluid equation acting on an arbitrary small collar $\omega$ supported by
$\widetilde{\Gamma}$ (Fig 1). The initial conditions for both fluid and heat
equations are taken of low regularity. We then seek to uniformly stabilize such
Boussinesq system in the vicinity of an unstable equilibrium pair, in the
critical setting of correspondingly low regularity spaces, by means of an
explicitly constructed, finite dimensional feedback control pair $\{ v,
\boldsymbol{u} \}$ localized on $\{ \widetilde{\Gamma}, \omega \}$. In
addition, they will be minimal in number, and of reduced dimension: more
precisely, $\boldsymbol{u}$ will be of dimension $(d-1)$, to include
necessarily its $d$\textsuperscript{th} component, and $v$ will be of dimension
$1$. The resulting space of well-posedness and stabilization is a suitable,
tight Besov space for the fluid velocity component (close to
$\boldsymbol{L}^3(\Omega$) for $ d = 3 $) and a corresponding Besov space for
the thermal component, $ q > d $. Unique continuation inverse theorems for
suitably over determined adjoint static problems play a critical role in the
constructive solution. Their proof rests on Carleman-type estimates, a topic
pioneered by M. V. Klibanov since the early 80's, after the 1939- breakthrough
publication \cite{Car}.
