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The strong asymptotic stabilization of 3D hyperbolic dynamics is achieved by a damped 2D elastic structure. The model is a Neumann wave-type equation with low regularity coupling conditions given in terms of a nonlinear von Karman plate. This problem is motivated by the elimination of aeroelastic instability (sustained oscillations of bridges, airfoils, etc.) in engineering applications. Empirical observations indicate that the subsonic wave-plate system converges to equilibria. Classical approaches which decouple the plate and wave dynamics have fallen short. Here, we operate on the model as it appears in the engineering literature with no regularization and achieve stabilization by microlocalizing the Neumann boundary data for the wave equation (given through the plate dynamics). We observe a compensation by the plate dynamics precisely where the regularity of the 3D Neumann wave is compromised (in the characteristic sector). 

We consider the linearized third order SMGTJ equation defined on a sufficiently smooth boundary domain in and subject to either Dirichlet or Neumann rough boundary control. Filling a void in the literature, we present a direct general system approach based on the vector state solution {position, velocity, acceleration}. It yields, in both cases, an explicit representation formula: input solution, based on the s.c. group generator of the boundary homogeneous problem and corresponding elliptic Dirichlet or Neumann map. It is close to, but also distinctly and critically different from, the abstract variation of parameter formula that arises in more traditional boundary control problems for PDEs L‐T.6. Through a duality argument based on this explicit formula, we provide a new proof of the optimal regularity theory: boundary control {position, velocity, acceleration} with low regularity boundary control, square integrable in time and space.

 

Abstract The Moore-Gibson-Thompson [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 on-line 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 MGT-third order dynamics with boundary control, over an infinite time-horizon. A critical missing point in such a study is the issue of uniqueness (within a specific class) of the corresponding highly non-standard 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. 

Boundary feedback stabilization of a critical third–order (in time) semilinear Jordan–Moore–Gibson–Thompson (JMGT) is considered. The word critical here refers to the usual case where media–damping effects are non–existent or non–measurable and therefore cannot be relied upon for stabilization purposes. Motivated by modeling aspects in high-intensity focused ultrasound (HIFU) technology, the boundary feedback under consideration is supported only on a portion of the boundary. At the same time, the remaining part is undissipated and subject to Neumann/Robin boundary conditions. As such, unlike Dirichlet, it fails to satisfy the Lopatinski condition, a fact which compromises tangential regularity on the boundary [37]. In such a configuration, the analysis of uniform stabilization from the boundary becomes subtle and requires careful geometric considerations and microlocal analysis estimates. The nonlinear effects in the model demand construction of suitably small solutions which are invariant under the dynamics. The assumed smallness of the initial data is required only at the lowest energy level topology, which is sufficient to construct sufficiently smooth solutions to the nonlinear model.

 

The Jordan–Moore–Gibson–Thompson (JMGT) equation is a well-established and recently widely studied model for nonlinear acoustics (NLA). It is a third–order (in time) semilinear Partial Differential Equation (PDE) with a distinctive feature of predicting the propagation of ultrasound waves at finite speed. This is due to the heat phenomenon known as second sound which leads to hyperbolic heat-wave propagation. In this paper, we consider the problem in the so called "critical" case, where free dynamics is unstable. In order to stabilize, we shall use boundary feedback controls supported on a portion of the boundary only. Since the remaining part of the boundary is not "controlled", and the imposed boundary conditions of Neumann type fail to saitsfy Lopatinski condition, several mathematical issues typical for mixed problems within the context o boundary stabilizability arise. To resolve these, special geometric constructs along with sharp trace estimates will be developed. The imposed geometric conditions are motivated by the geometry that is suitable for modeling the problem of controlling (from the boundary) the acoustic pressure involved in medical treatments such as lithotripsy, thermotherapy, sonochemistry, or any other procedure involving High Intensity Focused Ultrasound (HIFU).

 

We consider the linear third order (in time) PDE known as the SMGTJ-equation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control \begin{document}$ g $\end{document}. Optimal interior and boundary regularity results were given in [1], after [41], when \begin{document}$ g \in L^2(0, T;L^2(\Gamma)) \equiv L^2(\Sigma) $\end{document}, which, moreover, in the canonical case \begin{document}$ \gamma = 0 $\end{document}, were expressed by the well-known explicit representation formulae of the wave equation in terms of cosine/sine operators [19], [17], [24,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether \begin{document}$ \gamma = 0 $\end{document} or \begin{document}$ 0 \neq \gamma \in L^{\infty}(\Omega) $\end{document}, since \begin{document}$ \gamma \neq 0 $\end{document} is responsible only for lower order terms. Here we exploit such cosine operator based-explicit representation formulae to provide optimal interior and boundary regularity results with \begin{document}$ g $\end{document} "smoother" than \begin{document}$ L^2(\Sigma) $\end{document}, qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [17]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [22], [23], [37] for control smoother than \begin{document}$ L^2(0, T;L^2(\Gamma)) $\end{document}, and [44] for control less regular in space than \begin{document}$ L^2(\Gamma) $\end{document}. 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 [42], [24,Section 9.8.2].

 

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.