### Convergence rates for the approximations of the solutions to algebraic Riccati Equations with unbounded coefficients: case of analytic semigroups.

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Optimization problem for a structural acoustic model with controls governed by unbounded operators on the state space is considered. This type of controls arises naturally in the context of "smart material technology". The main result of the paper provides an optimal synthesis and solvability of associated nonstandard Riccati equations. It is shown that in spite of the unboundedness of control operators, the resulting gain operators (feedbacks) are bounded on the state space. This allows to provide...

Finite element semidiscrete approximations on nonlinear dynamic shallow shell models in considered. It is shown that the algorithm leads to rates of convergence. The result presented in the paper improves upon the existing literature where the rates of convergence were derived for initial data only [19].

We consider mixed problems for Kirchhoff elastic and thermoelastic systems, subject to boundary control in the clamped Boundary Conditions B.C. (“clamped control”). If w denotes elastic displacement and θ temperature, we establish optimal regularity of {w, w_t, w_tt} in the elastic case, and of {w, w_t, w_tt, θ} in the thermoelastic case. Our results complement those presented in (Lagnese and Lions, 1988), where sharp (optimal) trace regularity results are obtained for the corresponding boundary...

We focus on the blow-up in finite time of weak solutions to the wave equation with interior and boundary nonlinear sources and dissipations. Our central interest is the relationship of the sources and damping terms to the behavior of solutions. We prove that under specific conditions relating the sources and the dissipations (namely p > m and k > m), weak solutions blow up in finite time.

The null controllability problem for a structurally damped abstract wave equation –often referred to in the literature as a structurally damped equation– is considered with a view towards obtaining optimal rates of blowup for the associated minimal energy function ${\mathcal{E}}_{min}\left(T\right)$, as terminal time $T\downarrow 0$. Key use is made of the underlying analyticity of the semigroup generated by the elastic operator $\mathcal{A}$, as well as of the explicit characterization of its domain of definition. We ultimately find that the blowup rate...

We show exact null-controllability for two models of non-classical, parabolic partial differential equations with distributed control: (i) second-order structurally damped equations, except for a limit case, where exact null controllability fails; and (ii) thermo-elastic equations with hinged boundary conditions. In both cases, the problem is solved by duality.

This note provides sharp regularity results for general, time-independent, second order, hyperbolic equations with non-homogeneous data of Neumann type.

We study the uniform stabilization problem for the Euler-Bernoulli equation defined on a smooth bounded domain of any dimension with feedback dissipative operators in various boundary conditions.

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