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We consider the -dimensional Boussinesq system defined on a sufficiently smooth bounded domain and subject to a pair{v,u}\{v,\boldsymbol{u}\}of controls localized on{Γ~,ω}\{\widetilde{\Gamma},\omega\}. Here,  is a scalar Dirichlet boundary control for the thermal equation, acting on an arbitrarily small connected portionΓ~\widetilde{\Gamma}of the boundaryΓ=∂⁡Ω\Gamma=\partial\Omega. Instead,  is a -dimensional internal control for the fluid equation acting on an arbitrarily small collar  supported byΓ~\widetilde{\Gamma}. 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,u}\{v,\boldsymbol{u}\}localized on{Γ~,ω}\{\widetilde{\Gamma},\omega\}. In addition, they will be minimal in number and of reduced dimension; more precisely,  will be of dimension(d-1)(d-1), to include necessarily its -th component, and  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 toL3⁢(Ω)\boldsymbol{L}^{3}(\Omega)ford=3d=3) and a corresponding Besov space for the thermal component,q&gt;dq&gt;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 80s.

Finite-dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory