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Free, publiclyaccessible full text available April 1, 2025

Abstract An optimal, complete, continuous theory of the Luenberger dynamic compensator (or state estimator or state observer) is obtained for the recently studied class of heatstructure interaction partial differential equation (PDE) models, with structure subject to high KelvinVoigt damping, and feedback control exercised either at the interface between the two media or else at the external boundary of the physical domain in three different settings. It is a first, full investigation that opens the door to numerous and far reaching subsequent work. They will include physically relevant
fluid structure models, with wave or platestructures, possibly without KelvinVoigt damping, as explicitly noted in the text, all the way to achieving the ultimate discrete numerical theory, so critical in applications. While the general setting is functional analytic, delicate PDEenergy estimates dictate how to define the interface/boundary feedback control in each of the three cases. 
Abstract The MooreGibsonThompson [MGT] dynamics is considered. This third order in time evolution arises within the context of acoustic wave propagation with applications in high frequency ultrasound technology. The optimal boundary feedback control is constructed in order to have online regulation. The above requires wellposedness of the associated Algebraic Riccati Equation. The paper by Lasiecka and Triggiani (2022) recently contributed a comprehensive study of the Optimal Control Problem for the MGTthird order dynamics with boundary control, over an infinite timehorizon. A critical missing point in such a study is the issue of uniqueness (within a specific class) of the corresponding highly nonstandard Algebraic Riccati Equation. The present note resolves this problem in the positive, thus completing the study of Lasiecka and Triggiani (2022) with the final goal of having on line feedback control, which is also optimal.more » « less

We consider the linear third order (in time) PDE known as the SMGTJequation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control
. Optimal interior and boundary regularity results were given in [\begin{document}$ g $\end{document} 1 ], after [41 ], when , which, moreover, in the canonical case\begin{document}$ g \in L^2(0, T;L^2(\Gamma)) \equiv L^2(\Sigma) $\end{document} , were expressed by the wellknown explicit representation formulae of the wave equation in terms of cosine/sine operators [\begin{document}$ \gamma = 0 $\end{document} 19 ], [17 ], [24 ,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether or\begin{document}$ \gamma = 0 $\end{document} , since\begin{document}$ 0 \neq \gamma \in L^{\infty}(\Omega) $\end{document} is responsible only for lower order terms. Here we exploit such cosine operator basedexplicit representation formulae to provide optimal interior and boundary regularity results with\begin{document}$ \gamma \neq 0 $\end{document} "smoother" than\begin{document}$ g $\end{document} , qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [\begin{document}$ L^2(\Sigma) $\end{document} 17 ]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [22 ], [23 ], [37 ] for control smoother than , and [\begin{document}$ L^2(0, T;L^2(\Gamma)) $\end{document} 44 ] for control less regular in space than . In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [\begin{document}$ L^2(\Gamma) $\end{document} 42 ], [24 ,Section 9.8.2]. 
null (Ed.)Abstract We consider the 𝑑dimensional 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 𝑑dimensional 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, finitedimensional 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 ) (d1) , to include necessarily its 𝑑th component, and 𝑣 will be of dimension 1.The resulting space of wellposedness 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 overdetermined adjoint static problems play a critical role in the constructive solution.Their proof rests on Carlemantype estimates, a topic pioneered by M. V. Klibanov since the early 80s.more » « less