# American Institute of Mathematical Sciences

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. Google Scholar [2] K. Ammari and S. Nicaise, Stabilization of a piezoelectric system,, Asymptotic Analysis, 73 (2011), 125. Google Scholar [3] I. Babuska, Error bounds for finite element method,, Numerishe Mathematik, 16 (1971), 322. Google Scholar [4] P. G. Ciarlet, "Mathematical Elasticity, Vols I and II,", North-Holland, (1988). 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. 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. Google Scholar [7] H. Funakubo, "Shape Memory Alloys,", Translated from the Japanese by J. B. Kennedy, (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. Google Scholar [9] D. Iessan, On some theorems in Thermopiezoelectricity,, J. Thermal Stresses, 12 (1989), 209. Google Scholar [10] B. Kapitonov, B. Miara and G. Perla Menzala, Stabilization of a layered Piezoelectric 3-D body by boundary dissipation,, ESAIM, 12 (2006), 198. 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, 46 (2007), 1080. Google Scholar [12] I. Lasiecka and B. Miara, Exact controllability of a 3D piezoelectric body,, C. R. Math. Acad. Sci. Paris, 347 (2009), 167. 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. 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. 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. Google Scholar [16] J. L. Lions, "Contrôlabilité Exacte, Stabilization et Perturbations de Systémes Distribués,", Tome 1 Contrôlabilité exacte, (1988). Google Scholar [17] J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Values Problems and Applications,", Volume I, (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. Google Scholar [20] B. Miara and M. L. Santos, Energy decay in piezoelectric systems,, Applicable Analysis, 88 (2009), 947. Google Scholar [21] R. D. Mindlin, Equations of high frequency vibrations of thermopiezoelectric crystal plates,, International Journal of Solid Structures, 10 (1974), 625. Google Scholar [22] S. Nicaise, Stability and controllability of the electromagneto-elastic system,, Portugalial. Math., 60 (2003), 73. Google Scholar [23] W. Nowacki, Some general theorems of thermopiezoelectricity,, J. Thermal Stresses, 1 (1978), 171. Google Scholar [24] J. M. Sejje Suárez, "Modelling of Thermopiezoelectric Phenomenon: Asymptotic Analysis and Numerical Simulation,", Doctoral thesis, (2011). Google Scholar [25] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Annali di Matematica Pura et. Applicata, CXLVI (1987), 65. Google Scholar [26] R. C. Smith, "Smart Material Systems. Model development,", SIAM, (2005). Google Scholar [27] A. V. Srinivasan and D. M. McFarland, "Smart Structures: Analysis and Design,", Cambridge University Press, (2001). Google Scholar [28] K. Uchino, "Piezoelectric Actuators and Ultrasonic Motors,", Kluwer Academic Publishers, (1997). Google Scholar [29] E. Zuazua, Controllability of the linear system of thermoelasticity,, J. Math. Pures Appl., 74 (1995), 291. Google Scholar

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##### 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. Google Scholar [2] K. Ammari and S. Nicaise, Stabilization of a piezoelectric system,, Asymptotic Analysis, 73 (2011), 125. Google Scholar [3] I. Babuska, Error bounds for finite element method,, Numerishe Mathematik, 16 (1971), 322. Google Scholar [4] P. G. Ciarlet, "Mathematical Elasticity, Vols I and II,", North-Holland, (1988). 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. 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. Google Scholar [7] H. Funakubo, "Shape Memory Alloys,", Translated from the Japanese by J. B. Kennedy, (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. Google Scholar [9] D. Iessan, On some theorems in Thermopiezoelectricity,, J. Thermal Stresses, 12 (1989), 209. Google Scholar [10] B. Kapitonov, B. Miara and G. Perla Menzala, Stabilization of a layered Piezoelectric 3-D body by boundary dissipation,, ESAIM, 12 (2006), 198. 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, 46 (2007), 1080. Google Scholar [12] I. Lasiecka and B. Miara, Exact controllability of a 3D piezoelectric body,, C. R. Math. Acad. Sci. Paris, 347 (2009), 167. 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. 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. 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. Google Scholar [16] J. L. Lions, "Contrôlabilité Exacte, Stabilization et Perturbations de Systémes Distribués,", Tome 1 Contrôlabilité exacte, (1988). Google Scholar [17] J. L. Lions and E. Magenes, "Non-Homogeneous Boundary Values Problems and Applications,", Volume I, (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. Google Scholar [20] B. Miara and M. L. Santos, Energy decay in piezoelectric systems,, Applicable Analysis, 88 (2009), 947. Google Scholar [21] R. D. Mindlin, Equations of high frequency vibrations of thermopiezoelectric crystal plates,, International Journal of Solid Structures, 10 (1974), 625. Google Scholar [22] S. Nicaise, Stability and controllability of the electromagneto-elastic system,, Portugalial. Math., 60 (2003), 73. Google Scholar [23] W. Nowacki, Some general theorems of thermopiezoelectricity,, J. Thermal Stresses, 1 (1978), 171. Google Scholar [24] J. M. Sejje Suárez, "Modelling of Thermopiezoelectric Phenomenon: Asymptotic Analysis and Numerical Simulation,", Doctoral thesis, (2011). Google Scholar [25] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Annali di Matematica Pura et. Applicata, CXLVI (1987), 65. Google Scholar [26] R. C. Smith, "Smart Material Systems. Model development,", SIAM, (2005). Google Scholar [27] A. V. Srinivasan and D. M. McFarland, "Smart Structures: Analysis and Design,", Cambridge University Press, (2001). Google Scholar [28] K. Uchino, "Piezoelectric Actuators and Ultrasonic Motors,", Kluwer Academic Publishers, (1997). Google Scholar [29] E. Zuazua, Controllability of the linear system of thermoelasticity,, J. Math. Pures Appl., 74 (1995), 291. Google Scholar
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