On the Controllability for Second Order Hyperbolic Equations in Curved Polygons

Bastos, Waldemar D.; Spezamiglio, Adalberto

doi:10.5540/tema.2007.08.02.0169

Cited by 2 publications

(2 citation statements)

References 6 publications

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“…In order to establish the local decay of energy for the solutions of (2.1) we need a technical result which was presented in Bastos and Spezamiglio [6]. Since we will refer to details of its proof, we will include it here.…”

Section: (22)mentioning

confidence: 98%

On exact boundary controllability for linearly coupled wave equations

Bastos

Spezamiglio

Raposo

2011

Journal of Mathematical Analysis and Applications

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In this paper we study exact boundary controllability for a system of two linear wave equations coupled by lower order terms. We obtain square integrable control of Neuman type for initial state with finite energy, in nonsmooth domains of the plane. PreliminariesLet Ω ⊂ R 2 be a bounded simply connected domain with piecewise C ∞ boundary ∂Ω. It is assumed that ∂Ω has no cuspid points and that Ω lays in one side of ∂Ω. We will refer to this type of domain as curvilinear polygon. The exterior unit normal vector η is defined almost everywhere in ∂Ω. Points of R 2 × R are denoted by (x 1 , x 2 , t), denotes the Laplacian operator in the spatial variable x = (x 1 , x 2 ) ∈ R 2 and subscripts are used for partial derivatives. Consider the systemwith T > 0, α, β, γ , δ real constants and ∂Ω = Γ 0 ∪ Γ 1 is a decomposition of the boundary where the parts Γ 0 and Γ 1 have disjoint interiors but their closure may have common isolated points. Many problems in structural dynamics are modeled by this type of system. For instance, the equations (1.2) describe the transverse free vibrations of an elastically connected double-membrane compound system. In this case u and v are the displacements of two vibrating membranes measured from their equilibrium positions. The coupling terms ±α(u − v) represent forces imposed by the coupling elastic layer. If the membranes are held fixed on the portion Γ 0 of the * Corresponding author.

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Section: (22)mentioning

confidence: 98%

On exact boundary controllability for linearly coupled wave equations

Bastos

Spezamiglio

Raposo

2011

Journal of Mathematical Analysis and Applications

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“…It is worth mentioning other papers in connection with the techniques developed in [12] and [30] as, for instance [2], [26]. In [2] the authors study the exact controllability for a class of hyperbolic linear partial differential equation with coefficients constants, which includes the Klein-Gordon equation, by considering piecewise smooth domains on the plane and boundary control of Robin type acting on the whole boundary. In [26] the authors study the local asymptotic behavior of the solutions of the linear Klein-Gordon equation in a piecewise smooth domain Ω.…”

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confidence: 99%

Boundary Control for a Generalized Wave Equation -- Revisiting Russell's Method of Control

Astudillo,

Cavalcanti,

Cavalcanti

et al. 2019

Preprint

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In this work we study the exact boundary controllability of a generalized wave equation in a nonsmooth domain with a nontrapping obstacle. In the more general case, this work contemplates the boundary control of a transmission problem admitting several zones of transmission. The result is obtained using the technique developed by David Russell, taking advantage of the local energy decay for the problem, obtained through the Scattering Theory as used by Vodev, combined with a powerful trace Theorem due to Tataru.

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On the Controllability for Second Order Hyperbolic Equations in Curved Polygons

Cited by 2 publications

References 6 publications

On exact boundary controllability for linearly coupled wave equations

On exact boundary controllability for linearly coupled wave equations

Boundary Control for a Generalized Wave Equation -- Revisiting Russell's Method of Control

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