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Abstract We consider thed-dimensional MagnetoHydroDynamics (MHD) system defined on a sufficiently smooth bounded domain,$$d = 2,3$$ $d=2,3$with homogeneous boundary conditions, and subject to external sources assumed to cause instability. The initial conditions for both fluid and magnetic equations are taken of low regularity. We then seek to uniformly stabilize such MHD system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of explicitly constructed, static, feedback controls, which are localized on an arbitrarily small interior subdomain. In additional, they will be minimal in number. The resulting space of well-posedness and stabilization is a suitable product space$$\displaystyle \widetilde{\textbf{B}}^{2- ^{2}\!/_{p}}_{q,p}(\Omega )\times \widetilde{\textbf{B}}^{2- ^{2}\!/_{p}}_{q,p}(\Omega ), \, 1< p < \frac{2q}{2q-1}, \, q > d,$$ ${\tilde{B}}_{q,p}^{2{-}^2{/}_p}\left(\Omega \right)\times {\tilde{B}}_{q,p}^{2{-}^2{/}_p}\left(\Omega \right),1<p<\frac{2q}{2q-1},q>d,$of tight Besov spaces for the fluid velocity component and the magnetic field component (each “close” to$$\textbf{L}^3(\Omega )$$ $L^3\left(\Omega \right)$for$$d = 3$$ $d=3$). Showing maximal$$L^p$$ $L^p$-regularity up to$$T = \infty $$ $T=\infty$for the feedback stabilized linear system is critical for the analysis of well-posedness and stabilization of the feedback nonlinear problem. 

We provide maximal 𝐿𝑝-regularity up to the level 𝑇 < ∞ or 𝑇 = ∞ of an abstract evolution equation in Banach space, which captures boundary closed-loop parabolic systems, defined on a bounded multidimensional domain, with finitely many boundary control vectors and finitely many boundary sensors/actuators. Illustrations given include classical parabolic equations as well as Navier-Stokes equations in 𝐿𝑝(Ω) or 𝐿𝑞 𝜎(Ω), respectively. 

Abstract We consider the 𝑑-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, 𝑣 is a scalar Dirichlet boundary control for the thermal equation, acting on an arbitrarily small connected portion Γ ~ \widetilde{\Gamma} of the boundary Γ = ∂ ⁡ Ω \Gamma=\partial\Omega .Instead, 𝒖 is a 𝑑-dimensional internal control for the fluid equation acting on an arbitrarily small collar 𝜔 supported by Γ ~ \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, 𝒖 will be of dimension ( d - 1 ) (d-1) , 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 L 3 ⁢ ( Ω ) \boldsymbol{L}^{3}(\Omega) for d = 3 d=3 ) and a corresponding Besov space for the thermal component, q > d 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 80s.