Control Properties for Second-Order Hyperbolic Systems in Anisotropic Cases with Applications in Inhomogeneous and Anisotropic Elastodynamic Systems

Shang, Yunxia; Li, Shumin

doi:10.1137/17m114203x

Cited by 4 publications

(2 citation statements)

References 26 publications

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“…For homogeneous nonisotropic elastic wave equation, see [3,7,8,18,24,31,6,54,55,56,60]. For the inhomogeneous elastic wave equation, see [9,34,37,42,51].…”

Section: Introductionmentioning

confidence: 99%

Stabilization of the critical and subcritical semilinear inhomogeneous and anisotropic elastic wave equation

Ning,

Yang,

Wang

2020

Preprint

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We prove exponential decay of the critical and subcritical semilinear inhomogeneous and anisotropic elastic wave equation with locally distributed damping on bounded domain. One novelty compared to previous results, is to give a checkable condition of the inhomogeneous and anisotropic medias. Another novelty is to establish a framework to study the stability of the damped semilinear inhomogeneous and anisotropic elastic wave equation, which is hard to apply Carleman estimates to deal with. We develop the Morawetz estimates and the compactness-uniqueness arguments for the semiliear elastic wave equation to prove the unique continuation, observability inequality and stabilization result.It is pointing that our proof is different from the classical method (See Dehman et al.[15], Joly et al. [26] and Zuazua [59]), which succeeds for the subcritical semilinear wave equation and fails for the critical semilinear wave equation.

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“…For homogeneous nonisotropic elastic wave equation, see [3,7,8,18,24,31,6,54,55,56,60]. For the inhomogeneous elastic wave equation, see [9,34,37,42,51].…”

Section: Introductionmentioning

confidence: 99%

Stabilization of the critical and subcritical semilinear inhomogeneous and anisotropic elastic wave equation

Ning,

Yang,

Wang

2020

Preprint

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“…Meanwhile, stable observability relies on a observability inequality of the elastic case, which is equivalent to the exact controllability of the elastic case. In [1,24], the authors just proved the exact controllability of anisotropic for the homogeneous elastodynamic system and the inhomogeneous case. In our case, we need to establish a Carleman estimate to state the observability inequality and the stable observability for elastic case, and then give stability results for density.…”

mentioning

confidence: 99%

Stability for Time-domain Elastic Wave Equations

Chen¹,

Gao²,

Ji³

et al. 2023

Preprint

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This paper is concerned with the inverse scattering problem involving the timedomain elastic wave equations in a bounded d-dimensional domain. First, an explicit reconstruction formula for the density is established by means of the Dirichlet-to-Neumann operator. The reconstruction is mainly based on the modified boundary control method and complex geometric optics solutions for the elastic wave. Next, the stable observability is obtained by a Carleman estimate. Finally, the stability for the density is presented by the connect operator.

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Internal Observability, Controllability and Stabilization of the Inhomogeneous and Anisotropic Elastic Wave Equation

2020

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Control Properties for Second-Order Hyperbolic Systems in Anisotropic Cases with Applications in Inhomogeneous and Anisotropic Elastodynamic Systems

Cited by 4 publications

References 26 publications

Stabilization of the critical and subcritical semilinear inhomogeneous and anisotropic elastic wave equation

Stabilization of the critical and subcritical semilinear inhomogeneous and anisotropic elastic wave equation

Stability for Time-domain Elastic Wave Equations

Internal Observability, Controllability and Stabilization of the Inhomogeneous and Anisotropic Elastic Wave Equation

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