The exponential stability of a coupled hyperbolic/parabolicsystem arising in structural acoustics

Avalos, George

doi:10.1155/s1085337596000103

Cited by 61 publications

(89 citation statements)

References 15 publications

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Section: Statement Of the Problem Main Resultsmentioning

confidence: 99%

“…The case α = 1; hence D 0 = 0; β = 0, in (1.2c)- (1.2d) below (see [1,3,16]). In the aforementioned references, the original model consisted of a coupled system of two partial differential equations (PDEs) of different type-a hyperbolic PDE and a parabolic-like PDE-which are strongly coupled at the interface (represented by the flexible wall Γ 0 ).…”

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confidence: 99%

“…Introduction; singular estimates for coupled PDE systems. The present paper sets itself along that line of research established by [1,3]-and later streamlined and simplified in the exposition of [16]-that is focused on mathematical properties of an accepted acoustic chamber model [8,31], subject to unbounded control action on its flexible wall.We consider a generalization of an established structural acoustic model. This consists of a coupled system of two partial differential equations of different types-a hyperbolic PDE acting within the acoustic chamber and a paraboliclike PDE acting on the elastic wall of the chamber-which are strongly coupled at their common interface.…”

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confidence: 99%

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Singular estimates and uniform stability of coupled systems ofhyperbolic/parabolic PDEs

Bucci

Lasiecka

Triggiani

2002

Abstract and Applied Analysis

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Statement of the problem. Main results1.1. Introduction; singular estimates for coupled PDE systems. The present paper sets itself along that line of research established by [1,3]-and later streamlined and simplified in the exposition of [16]-that is focused on mathematical properties of an accepted acoustic chamber model [8,31], subject to unbounded control action on its flexible wall.We consider a generalization of an established structural acoustic model. This consists of a coupled system of two partial differential equations of different types-a hyperbolic PDE acting within the acoustic chamber and a paraboliclike PDE acting on the elastic wall of the chamber-which are strongly coupled at their common interface. Unlike prior literature, we allow the parameter α-which measures the strength of damping in the plate-like component-to run over the entire range 1/2 ≤ α ≤ 1 of analyticity (parabolicity) of its free dynamics. Prior literature considered only the limit case α = 1 ("Kelvin Voight damping"). However, the limit case α = 1/2 ("structural, square root damping") is at least equally important; indeed, even more so in applications. Entirely new phenomena arise over the case α = 1 as α decreases from 1 to 1/2, which we describe below. Indeed, as α decreases from 1 to 1/2, we seek to achieve two critical control-theoretic properties of the overall coupled system: (i) a quantitatively precise singular estimate for the operator e At B as t 0 and, simultaneously, (ii) a uniform (exponential) stability result of the free dynamics operator e At (which is not analytic, of course). Here, B is the unbounded control operator: it typically contains derivatives of Dirac measures supported on points or curves of the elastic wall of the chamber. It turns out that goals (i) and (ii) are in conflict with one another, as α decreases from 1 to 1/2. Indeed, as the damping strength α of the plate (parabolic) component decreases from 1 to 1/2, a progressively stronger

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Section: Statement Of the Problem Main Resultsmentioning

confidence: 99%

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confidence: 99%

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confidence: 99%

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Singular estimates and uniform stability of coupled systems ofhyperbolic/parabolic PDEs

Bucci

Lasiecka

Triggiani

2002

Abstract and Applied Analysis

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“…This motivates one to consider the damped coupled model (1.1) where the dissipation is exercised through the bending moment of the Kirchoff equation. This model differs in a critical way from other models recently studied in noise reduction problems (see [1], [3]) in that the elastic dynamics of the moving floor is more realistically represented by a hyperbolic Kirchoff equation, rather than by a structurally damped Euler-Bernoulli equation with so-called Kelvin-Voight damping, as in the past models.…”

Section: Figure (Ii)mentioning

confidence: 99%

“…The results existing in the literature on the stabilization of structural acoustic models refer to those where the floor is strongly damped by means of structural damping (see [1], [2], [7]). In the case of structural damping present in the model, the component of the uncoupled system corresponding to the Euler-Bernoulli equation represents an analytic semigroup ( [22], [6]).…”

Section: Figure (Ii)mentioning

confidence: 99%

Uniform stabilization of a coupled structural acoustic system byboundary dissipation

Çamurdan

1998

Abstract and Applied Analysis

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Abstract. We consider a coupled PDE system arising in noise reduction problems. In a two dimensional chamber, the acoustic pressure (unwanted noise) is represented by a hyperbolic wave equation. The floor of the chamber is subject to the action of piezo-ceramic patches (smart materials). The goal is to reduce the acoustic pressure by means of the vibrations of the floor which is modelled by a hyperbolic Kirchoff equation. These two hyperbolic equations are coupled by appropriate trace operators. This overall model differs from those previously studied in the literature in that the elastic chamber floor is here more realistically modeled by a hyperbolic Kirchoff equation, rather than by a parabolic Euler-Bernoulli equation with Kelvin-Voight structural damping, as in past literature. Thus, the hyperbolic/parabolic coupled system of past literature is replaced here by a hyperbolic/hyperbolic coupled model. The main result of this paper is a uniform stabilization of the coupled PDE system by a (physically appealing) boundary dissipation.

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Stability analysis of coupled structural acoustics PDE models under thermal effects and with no additional dissipation

Avalos

Geredeli

2018

Mathematische Nachrichten

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In this study we consider a coupled system of partial differential equations (PDE's) which describes a certain structural acoustics interaction. One component of this PDE system is a wave equation, which serves to model the interior acoustic wave medium within a given three dimensional chamber Ω. This acoustic wave equation is coupled on a boundary interface Γ 0 to a two dimensional system of thermoelasticity: this thermoelastic PDE is composed in part of a structural beam or plate equation, which governs the vibrations of flexible wall portion Γ 0 of the chamber Ω. Moreover, this elastic dynamics is coupled to a heat equation which also evolves on Γ 0 , and which imparts a thermal damping onto the entire structural acoustic system. As we said, the interaction between the wave and thermoelastic PDE components takes place on the boundary interface Γ 0 , and involves coupling boundary terms which are above the level of finite energy. We analyze the stability properties of this coupled structural acoustics PDE model, in the absence of any additive feedback dissipation on the hard walls Γ 1 of the boundary Ω = Γ 0 ∪ Γ 1 . Under a certain geometric assumption on Γ 1 , an assumption which has appeared in the literature in connection with structural acoustic flow, and which allows for the invocation of a recently derived microlocal boundary trace estimate, we show that classical solutions of this thermally damped structural acoustics PDE decay uniformly to zero, with a rational rate of decay. K E Y W O R D Scoupled systems, equations of mixed type, partial differential equations, uniform stability M S C ( 2 0 1 0 ) 35M13, 93D20 1 0, (Γ 0 ) =

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The exponential stability of a coupled hyperbolic/parabolicsystem arising in structural acoustics

Cited by 61 publications

References 15 publications

Singular estimates and uniform stability of coupled systems ofhyperbolic/parabolic PDEs

Singular estimates and uniform stability of coupled systems ofhyperbolic/parabolic PDEs

Uniform stabilization of a coupled structural acoustic system byboundary dissipation

Stability analysis of coupled structural acoustics PDE models under thermal effects and with no additional dissipation

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