December  2012, 1(2): 315-336. doi: 10.3934/eect.2012.1.315

On the exponential stabilization of a thermo piezoelectric/piezomagnetic system

1. 

National Laboratory of Scientific Computation, LNCC/MCT, Av. Getulio Vargas 333, Quitandinha, Petrópolis, RJ, 25651-070, Brazil, Brazil

Received  November 2011 Revised  May 2012 Published  October 2012

This paper is motivated by a piezoelectric/piezomagnetic phenomenon in the presence of thermal effects. The evolution system we consider is linear and coupled between one hyperbolic , two elliptic and one parabolic equation. We show the equivalence between ``the exponential decay of the total energy of our system" and an ``observability inequality for an anisotropic elastic wave system" assuming that a geometric condition is satisfied. This geometric condition ensures that the elliptic operator associated with the mechanical part of our system has no eigenfunctions $ \Psi $ such that the divergence div (Λ $ \Psi $ ) = 0 in $\Omega$ where $ Λ $ denotes the thermal expansion tensor.
Citation: Gustavo Alberto Perla Menzala, Julian Moises Sejje Suárez. On the exponential stabilization of a thermo piezoelectric/piezomagnetic system. Evolution Equations & Control Theory, 2012, 1 (2) : 315-336. doi: 10.3934/eect.2012.1.315
References:
[1]

V. I. Alshits, A. N. Darinskii and J. Lothe, On the Existence of Surface Waves in Half-Infinite Anisotropic Elastic Media with Piezoelectric and Piezomagnetic Properties, Wave Motion, 16 (1992), 265-283. Google Scholar

[2]

K. Ammari and S. Nicaise, Stabilization of a piezoelectric system, Asymptotic Analysis, 73 (2011), 125-146.  Google Scholar

[3]

I. Babuska, Error bounds for finite element method, Numerishe Mathematik, 16 (1971), 322-333.  Google Scholar

[4]

P. G. Ciarlet, "Mathematical Elasticity, Vols I and II," North-Holland, Amsterdam, Vol I(1988), Vol II (1997).  Google Scholar

[5]

C. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal, 29 (1968), 241-271.  Google Scholar

[6]

P. Destuynder and M. Salaun, A mixed finite element for shell model with free edge boundary conditions Part I. The mixed variational formulation, Comput. Methods Appl. Mech. Engrg., 120 (1995), 195-217.  Google Scholar

[7]

H. Funakubo, "Shape Memory Alloys," Translated from the Japanese by J. B. Kennedy, Gordon and Breach Science Publishers, New York, 1984. Google Scholar

[8]

D. Henry, O. Lopes and A. Perissinotto, On the essential spectrum of a semigroup of thermoelasticity, Nonlinear Anal. TMA, 21 (1993), 65-75.  Google Scholar

[9]

D. Iessan, On some theorems in Thermopiezoelectricity, J. Thermal Stresses, 12 (1989), 209-223.  Google Scholar

[10]

B. Kapitonov, B. Miara and G. Perla Menzala, Stabilization of a layered Piezoelectric 3-D body by boundary dissipation, ESAIM, Control Optimization and Calculus of Variations, 12 (2006), 198-215.  Google Scholar

[11]

B. Kapitonov, B. Miara and G. Perla Menzala, Boundary observation and exact control of a quasi electrostatic piezoelectric system in multilayered media, SIAM, J. Control and Optim., 46 (2007), 1080-1097.  Google Scholar

[12]

I. Lasiecka and B. Miara, Exact controllability of a 3D piezoelectric body, C. R. Math. Acad. Sci. Paris, 347 (2009), 167-172.  Google Scholar

[13]

G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Rational Mech. Anal, 148 (1999), 179-231.  Google Scholar

[14]

J. Y. Li, Uniqueness theorem and reciprocity theorem for the linear thermo-electro-magneto-elasticity, The Quarterly Journal of Mechanics and Applied Mathematics, 56 (2003), 35-43.  Google Scholar

[15]

J. Y. Li and M. L. Dunn, Micromechanics of magnetoelectroelastic composite materials: Average fields and effective behavior, Journal of Intelligent Material Systems and Structures, 9 (1998), 404-416. Google Scholar

[16]

J. L. Lions, "Contrôlabilité Exacte, Stabilization et Perturbations de Systémes Distribués," Tome 1 Contrôlabilité exacte, Masson, 1988.  Google Scholar

[17]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Values Problems and Applications," Volume I, Springer-Verlag Berlin Heidelberg, New York, 1972.  Google Scholar

[18]

G. P. Menzala and J. S. Suárez, On a thermopiezoelectric model: Exponential decay of the total energy,, (Submitted for publication)., ().   Google Scholar

[19]

D. Mercier and S. Nicaise, Existence, uniqueness, and regularity results for piezoelectric systems, SIAM J. MATH. ANAL., 37 (2005), 651-672.  Google Scholar

[20]

B. Miara and M. L. Santos, Energy decay in piezoelectric systems, Applicable Analysis, 88 (2009), 947-960.  Google Scholar

[21]

R. D. Mindlin, Equations of high frequency vibrations of thermopiezoelectric crystal plates, International Journal of Solid Structures, 10 (1974), 625-637. Google Scholar

[22]

S. Nicaise, Stability and controllability of the electromagneto-elastic system, Portugalial. Math., 60 (2003), 73-80.  Google Scholar

[23]

W. Nowacki, Some general theorems of thermopiezoelectricity, J. Thermal Stresses, 1 (1978), 171-182. Google Scholar

[24]

J. M. Sejje Suárez, "Modelling of Thermopiezoelectric Phenomenon: Asymptotic Analysis and Numerical Simulation," Doctoral thesis, 2011. National Laboratory of Scientific Computation (LNCC$|$MCT), Brazil, (in Portuguese). Google Scholar

[25]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura et. Applicata, (IV), CXLVI (1987), 65-96.  Google Scholar

[26]

R. C. Smith, "Smart Material Systems. Model development," SIAM, Frontiers in Applied Mathematics, 2005.  Google Scholar

[27]

A. V. Srinivasan and D. M. McFarland, "Smart Structures: Analysis and Design," Cambridge University Press, Cambridge, UK, 2001. Google Scholar

[28]

K. Uchino, "Piezoelectric Actuators and Ultrasonic Motors," Kluwer Academic Publishers, Boston, 1997. Google Scholar

[29]

E. Zuazua, Controllability of the linear system of thermoelasticity, J. Math. Pures Appl., 74 (1995), 291-315.  Google Scholar

show all references

References:
[1]

V. I. Alshits, A. N. Darinskii and J. Lothe, On the Existence of Surface Waves in Half-Infinite Anisotropic Elastic Media with Piezoelectric and Piezomagnetic Properties, Wave Motion, 16 (1992), 265-283. Google Scholar

[2]

K. Ammari and S. Nicaise, Stabilization of a piezoelectric system, Asymptotic Analysis, 73 (2011), 125-146.  Google Scholar

[3]

I. Babuska, Error bounds for finite element method, Numerishe Mathematik, 16 (1971), 322-333.  Google Scholar

[4]

P. G. Ciarlet, "Mathematical Elasticity, Vols I and II," North-Holland, Amsterdam, Vol I(1988), Vol II (1997).  Google Scholar

[5]

C. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal, 29 (1968), 241-271.  Google Scholar

[6]

P. Destuynder and M. Salaun, A mixed finite element for shell model with free edge boundary conditions Part I. The mixed variational formulation, Comput. Methods Appl. Mech. Engrg., 120 (1995), 195-217.  Google Scholar

[7]

H. Funakubo, "Shape Memory Alloys," Translated from the Japanese by J. B. Kennedy, Gordon and Breach Science Publishers, New York, 1984. Google Scholar

[8]

D. Henry, O. Lopes and A. Perissinotto, On the essential spectrum of a semigroup of thermoelasticity, Nonlinear Anal. TMA, 21 (1993), 65-75.  Google Scholar

[9]

D. Iessan, On some theorems in Thermopiezoelectricity, J. Thermal Stresses, 12 (1989), 209-223.  Google Scholar

[10]

B. Kapitonov, B. Miara and G. Perla Menzala, Stabilization of a layered Piezoelectric 3-D body by boundary dissipation, ESAIM, Control Optimization and Calculus of Variations, 12 (2006), 198-215.  Google Scholar

[11]

B. Kapitonov, B. Miara and G. Perla Menzala, Boundary observation and exact control of a quasi electrostatic piezoelectric system in multilayered media, SIAM, J. Control and Optim., 46 (2007), 1080-1097.  Google Scholar

[12]

I. Lasiecka and B. Miara, Exact controllability of a 3D piezoelectric body, C. R. Math. Acad. Sci. Paris, 347 (2009), 167-172.  Google Scholar

[13]

G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Rational Mech. Anal, 148 (1999), 179-231.  Google Scholar

[14]

J. Y. Li, Uniqueness theorem and reciprocity theorem for the linear thermo-electro-magneto-elasticity, The Quarterly Journal of Mechanics and Applied Mathematics, 56 (2003), 35-43.  Google Scholar

[15]

J. Y. Li and M. L. Dunn, Micromechanics of magnetoelectroelastic composite materials: Average fields and effective behavior, Journal of Intelligent Material Systems and Structures, 9 (1998), 404-416. Google Scholar

[16]

J. L. Lions, "Contrôlabilité Exacte, Stabilization et Perturbations de Systémes Distribués," Tome 1 Contrôlabilité exacte, Masson, 1988.  Google Scholar

[17]

J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Values Problems and Applications," Volume I, Springer-Verlag Berlin Heidelberg, New York, 1972.  Google Scholar

[18]

G. P. Menzala and J. S. Suárez, On a thermopiezoelectric model: Exponential decay of the total energy,, (Submitted for publication)., ().   Google Scholar

[19]

D. Mercier and S. Nicaise, Existence, uniqueness, and regularity results for piezoelectric systems, SIAM J. MATH. ANAL., 37 (2005), 651-672.  Google Scholar

[20]

B. Miara and M. L. Santos, Energy decay in piezoelectric systems, Applicable Analysis, 88 (2009), 947-960.  Google Scholar

[21]

R. D. Mindlin, Equations of high frequency vibrations of thermopiezoelectric crystal plates, International Journal of Solid Structures, 10 (1974), 625-637. Google Scholar

[22]

S. Nicaise, Stability and controllability of the electromagneto-elastic system, Portugalial. Math., 60 (2003), 73-80.  Google Scholar

[23]

W. Nowacki, Some general theorems of thermopiezoelectricity, J. Thermal Stresses, 1 (1978), 171-182. Google Scholar

[24]

J. M. Sejje Suárez, "Modelling of Thermopiezoelectric Phenomenon: Asymptotic Analysis and Numerical Simulation," Doctoral thesis, 2011. National Laboratory of Scientific Computation (LNCC$|$MCT), Brazil, (in Portuguese). Google Scholar

[25]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura et. Applicata, (IV), CXLVI (1987), 65-96.  Google Scholar

[26]

R. C. Smith, "Smart Material Systems. Model development," SIAM, Frontiers in Applied Mathematics, 2005.  Google Scholar

[27]

A. V. Srinivasan and D. M. McFarland, "Smart Structures: Analysis and Design," Cambridge University Press, Cambridge, UK, 2001. Google Scholar

[28]

K. Uchino, "Piezoelectric Actuators and Ultrasonic Motors," Kluwer Academic Publishers, Boston, 1997. Google Scholar

[29]

E. Zuazua, Controllability of the linear system of thermoelasticity, J. Math. Pures Appl., 74 (1995), 291-315.  Google Scholar

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