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

Abstract

We consider the d-dimensional Boussinesq system defined on a sufficiently smooth bounded domain and subject to a pair Formula Presented of controls localized on Formula Presented. Here, v is a scalar Dirichlet boundary control for the thermal equation, acting on an arbitrarily small connected portion Formula Presented of the boundary Formula Presented. Instead, u is a d-dimensional internal control for the fluid equation acting on an arbitrarily small collar w supported by Formula Presented. 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 Formula Presented localized on Formula Presented. In addition, they will be minimal in number and of reduced dimension; more precisely, u will be of dimension Formula Presented, to include necessarily its d-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 Formula Presented for Formula Presented) and a corresponding Besov space for the thermal component, Formula Presented. 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.

Publication Title

Journal of Inverse and Ill-Posed Problems

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