Stabilization of a piezoelectric system

Serge Nicaise

Description

Stabilization of a piezoelectric system Kaïs Ammari ? and Serge Nicaise † Abstract. We consider a stabilization problem for a piezoelectric system. We prove an expo- nential stability result under some Lions geometric condition. Our method is based on an identity with multipliers that allows to show an appropriate observability estimate. Key words. elasticity system, Maxwell's system, piezoelectric system, stabilization AMS subject classification. 35A05, 35B05, 35J25, 35Q60 1 Introduction Let ? be a bounded domain of R3 with a Lipschitz boundary ?. In that domain we consider the non-stationary piezoelectric system that consists in a coupling between the elasticity sys- tem with the Maxwell equation. More precisely we analyze the following partial differential equations : (1.1) ?ij(u,E) = aijkl?kl(u)? ekijEk ? i, j = 1, 2, 3, (1.2) Di = ?ijEj + eikl?kl(u)? i = 1, 2, 3 (1.3) B = µH. The equations of equilibrium are (1.4) ∂2t ui = ∂j?ji ? i = 1, 2, 3 for the elastic displacement and (1.5) ∂tD = curlH, ∂tB = ?curlE for the electric/magnetic fields. This system models the coupling between Maxwell's system and the elastic one, in which E(x, t),H(x, t) are the electric and magnetic fields at the point x ? ? at time t, u(x, t) is the displacement field at

• maxwell's system


• tensor


• indeed using


• †université de valenciennes et du hainaut cambrésis


• semigroup theory


• equations allow


• ekij

Sujets

• Saint Nicaise
• Ammari
• Tensor
• Semigroup
• mathématiques
• mathématique
• Mathématiques

Informations

• Publié par: pefav
• Nombre de lectures: 20
• Langue: English

Extrait

Stabilization of a piezoelectric system Kaïs Ammari ∗ and Serge Nicaise †

Abstract. We consider a stabilization problem for a piezoelectric system. We prove an expo- nential stability result under some Lions geometric condition. Our method is based on an identity with multipliers that allows to show an appropriate observability estimate. Key words. elasticity system, Maxwell’s system, piezoelectric system, stabilization AMS subject classification. 35A05, 35B05, 35J25, 35Q60

1 Introduction Let Ω be a bounded domain of R 3 with a Lipschitz boundary Γ . In that domain we consider the non-stationary piezoelectric system that consists in a coupling between the elasticity sys- tem with the Maxwell equation. More precisely we analyze the following partial differential equations : (1.1) σ ij ( u, E ) = a ijkl γ kl ( u ) − e kij E k ∀ i, j = 1 , 2 , 3 ,

(1.2) D i = ε ij E j + e ikl γ kl ( u ) ∀ i = 1 , 2 , 3

(1.3) B = µH. The equations of equilibrium are (1.4) ∂ t 2 u i = ∂ j σ ji ∀ i = 1 , 2 , 3 for the elastic displacement and (1.5) ∂ t D = curlH, ∂ t B = − curlE for the electric/magnetic fields. This system models the coupling between Maxwell’s system and the elastic one, in which E ( x, t ) , H ( x, t ) are the electric and magnetic fields at the point x ∈ Ω at time t, u ( x, t ) is the displacement field at the point x ∈ Ω at time t, and γ ij ( u ) i 3 ,j =1 is the strain tensor given by 1 ∂ γ ij ( u ) =  2  ∂x u j i + ∂ ∂ x u i j  . Here σ = ( σ ij ) i 3 ,j =1 , D = ( D 1 , D 2 , D 3 ) , and B = ( B 1 , B 2 , B 3 ) are the stress tensor, electric displacement, and magnetic induction, respectively. ε, µ are the electric permittivity and ∗ Département de Mathématiques, Faculté des Sciences de Monastir, 5019 Monastir, Tunisie, email: [e-mail] † Université de Valenciennes et du Hainaut Cambrésis, LAMAV, FR CNRS 2956, Le Mont Houy, 59313 Valenciennes Cedex 9, France, email : [e-mail]

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