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

TL;DR

This paper develops a finite-dimensional feedback control method to stabilize the d-dimensional Boussinesq system near an unstable equilibrium in low regularity Besov spaces, using Carleman estimate-based inverse theory.

Contribution

It introduces a novel stabilization approach for the Boussinesq system employing explicit, minimal, and reduced-dimension controls in critical Besov spaces, leveraging Carleman estimates for inverse problems.

Findings

Successfully stabilizes the Boussinesq system near unstable equilibrium

Constructs explicit finite-dimensional feedback controls

Utilizes Carleman estimates for inverse theorems in stabilization

Abstract

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}.