A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations

Tebou, Louis

doi:10.1051/cocv:2007066

Cited by 13 publications

(5 citation statements)

References 32 publications

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“…As can easily be checked in the literature, most of the problems discussed in this framework deal with viscous damping of the form ay t or ag( y t ), e.g. [3][4][5][6]9,10,12,16,[18][19][20][21][22]. When it comes to the case of locally distributed viscoelastic damping of Kelvin-Voigt type, the literature is poorly documented; only a few papers discuss this particular topic, e.g.…”

Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

A constructive method for the stabilization of the wave equation with localized Kelvin–Voigt damping

Tebou

2012

Comptes Rendus. Mathématique

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Section: Introduction and Statement Of The Main Resultsmentioning

confidence: 99%

A constructive method for the stabilization of the wave equation with localized Kelvin–Voigt damping

Tebou

2012

Comptes Rendus. Mathématique

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“…For multi-dimensional problems, the analysis of the exponential decay rate cannot be performed by spectral analysis methods and it requires of tools such as microlocal analysis (see the Appendix 2 of [17] by Bardos-Lebeau-Rauch and [1] where the exponential decay is proved by microlocal tools under the so-called Geometric Control Condition), multiplier techniques ( [19], [27], [28]), and Carleman inequalities [23].…”

Section: Preliminariesmentioning

confidence: 99%

Boundary Stabilization of Numerical Approximations of the 1-D Variable Coefficients Wave Equation: A Numerical Viscosity Approach

Marica

Zuazua

2014

Lecture Notes in Computational Science and Engineering

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Abstract. In this paper, we consider the boundary stabilization problem associated to the 1 − d wave equation with both variable density and diffusion coefficients and to its finite difference semi-discretizations. It is well-known that, for the finite difference semi-discretization of the constant coefficients wave equation on uniform meshes [24] or on some non-uniform meshes [20], the discrete decay rate fails to be uniform with respect to the mesh-size parameter. We prove that, under suitable regularity assumptions on the coefficients and after adding an appropriate artificial viscosity to the numerical scheme, the decay rate is uniform as the mesh-size tends to zero. This extends previous results in [24] on the constant coefficient wave equation. The methodology of proof consists in applying the classical multiplier technique at the discrete level, with a multiplier adapted to the variable coefficients. PreliminariesLet us consider the following initial boundary value problem associated to the 1 − d wave equation with variable coefficients and with a damping mechanism acting on the right endpoint of the space interval:Here we have taken the dissipative boundary condition σ(1)v x (1, t) + v t (1, t) = 0, but similar results can be proved for more general feedback terms as, for instance,with k > 0 and k = 1. In fact, problem (1.1) with the second dissipative condition can be reduced to (1.1) by scaling the time variable.The energy corresponding to the solution of (1.1),obeys the following dissipation law:When the variable coefficients ρ and σ belong to the BV (0, 1) class of functions with bounded variation, the following stabilization property holds, ensuring the exponential decay of the energy E ρ,σ (v(·, t), v t (·, t))The main idea of this work originated when the first author has visited BCAM in December 2012.

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“…Starting from Theorem 3.2, one can show the following stabilization result for system (3.25) ( [42]).…”

Section: (318)mentioning

confidence: 99%

A Unified Controllability/Observability Theory for Some Stochastic and Deterministic Partial Differential Equations

Zhang¹

2011

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)

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The purpose of this paper is to present a universal approach to the study of controllability/observability problems for infinite dimensional systems governed by some stochastic/deterministic partial differential equations. The crucial analytic tool is a class of fundamental weighted identities for stochastic/deterministic partial differential operators, via which one can derive the desired global Carleman estimates. This method can also give a unified treatment of the stabilization, global unique continuation, and inverse problems for some stochastic/deterministic partial differential equations. (2000). Primary 93B05; Secondary 35Q93, 93B07. Mathematics Subject Classificationrelationship exist between the controllability/observability theories for these two equations of different nature. Especially, it would be quite interesting to establish, in some sense and to some extend, a unified controllability/observability theory for parabolic equations and hyperbolic equations. This problem was initially studied by D.L. Russell ([39]).The main purpose of this paper is to present the author's and his collaborators' works with an effort towards a unified controllability/observability theory for stochastic/deterministic PDEs. The crucial analytic tool we employ is a class of elementary pointwise weighted identities for partial differential operators. Starting from these identities, we develop a unified approach, based on global Carleman estimate. This universal approach not only deduces the known controllability/observability results (that have been derived before via Carleman estimates) for the linear parabolic, hyperbolic, Schrödinger and plate equations, but also provides new/sharp results on controllability/observability, global unique continuation, stabilization and inverse problems for some stochastic/deterministic linear/nonlinear PDEs.The rest of this paper is organized as follows. Section 2 analyzes the main differences between the existing controllability/observability theories for parabolic equations and hyperbolic equations. Sections 3 and 4 address, among others, the unified treatment of the controllability/observability problem for deterministic PDEs and stochastic PDEs, respectively.

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A Carleman estimates based approach for the stabilization of some locally damped semilinear hyperbolic equations

Abstract: Abstract. First

Cited by 13 publications

References 32 publications

A constructive method for the stabilization of the wave equation with localized Kelvin–Voigt damping

A constructive method for the stabilization of the wave equation with localized Kelvin–Voigt damping

Boundary Stabilization of Numerical Approximations of the 1-D Variable Coefficients Wave Equation: A Numerical Viscosity Approach

A Unified Controllability/Observability Theory for Some Stochastic and Deterministic Partial Differential Equations

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