# American Institute of Mathematical Sciences

November  2013, 33(11&12): 5273-5292. doi: 10.3934/dcds.2013.33.5273

## A thermo piezoelectric model: Exponential decay of the total energy

 1 National Laboratory of Scientific Computation, LNCC/MCT, Av. Getulio Vargas 333, Quitandinha, Petrópolis, RJ, 25651-070, Brazil 2 National Laboratory of Scientific Computation, LNCC/MCT, Av. Getulio Vargas 333, Quitandinha, Petrópolis, RJ, CEP 25651-070, Brazil

Received  October 2011 Revised  April 2012 Published  May 2013

We consider a linear evolution model describing a piezoelectric phenomenon under thermal effects as suggested by R. Mindlin [13] and W. Nowacki [16]. We prove the equivalence between exponential decay of the total energy and an observability inequality for an anisotropic elastic wave system. Our strategy is to use a decoupling method to reduce the problem to an equivalent observability inequality for an anisotropic elastic wave system and assume a condition which guarantees that the corresponding elliptic operator has no eigenfunctions with null divergence.
Citation: Gustavo Alberto Perla Menzala, Julian Moises Sejje Suárez. A thermo piezoelectric model: Exponential decay of the total energy. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 5273-5292. doi: 10.3934/dcds.2013.33.5273
##### References:
 [1] K. Ammari and S. Nicaise, Stabilization of a piezoelectric system,, Asymptotic Analysis, 73 (2011), 125. Google Scholar [2] I. Babuska, Error bounds for finite element method,, Numerishe Mathematik, 16 (1971), 322. Google Scholar [3] P. G. Ciarlet, "Mathematical Elasticity, Vols I and II,", North-Holland, I (1988). Google Scholar [4] C. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity,, Arch. Rational Mech. Anal., 29 (1968), 241. doi: 10.1007/BF00276727. Google Scholar [5] H. Funakubo, "Ed. Shape Memory Alloys,", Translated from the Japanese by J. B. Kennedy, (1984). Google Scholar [6] D. Henry, O. Lopes and A. Perissinotto, On the essential spectrum of a semigroup of thermoelasticity,, Nonlinear Anal. TMA, 21 (1993), 65. doi: 10.1016/0362-546X(93)90178-U. Google Scholar [7] D. Iessan, On some theorems in Thermopiezoelectricity,, J. Thermal Stresses, 12 (1989), 209. doi: 10.1080/01495738908961962. Google Scholar [8] B. Kapitonov, B. Miara and G. Perla Menzala, Stabilization of a layered Piezoelectric 3-D body by boundary dissipation,, ESAIM, 12 (2006), 198. doi: 10.1051/cocv:2005028. Google Scholar [9] 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. doi: 10.1137/050629884. Google Scholar [10] G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity,, Arch. Rational Mech. Anal., 148 (1999), 179. doi: 10.1007/s002050050160. Google Scholar [11] J. L. Lions, "Contrôlabilité Exacte, Stabilization et Perturbations de Systèmes Distribués,", Tome 1, (1988). Google Scholar [12] B. Miara and M. Lima, Energy decay in piezoelectric systems,, Applicable Analysis, 88 (2009), 947. doi: 10.1080/00036810903042166. Google Scholar [13] R. D. Mindlin, Equations of high frequency vibrations of thermopiezoelectric crystal plates,, International Journal of Solid Structures, 10 (1974), 625. doi: 10.1016/0020-7683(74)90047-X. Google Scholar [14] I. Müller, Six lectures in shape memory,, Centre Recherches Mathématiques, 13 (1998). Google Scholar [15] S. Nicaise, Stability and controllability of the electromagneto-elastic system,, Post. Math., 60 (2003), 73. Google Scholar [16] W. Nowacki, Some general theorems of thermopiezoelectricity,, J. Thermal Stresses, 1 (1978), 171. doi: 10.1080/01495737808926940. Google Scholar [17] J. M. Sejje Suárez, Modelagem de fenômenos termopiezoelétricos: Analise assintótica e Simulação Numérica,, Tese de Doutorado (2011) Laboratório Nacional de Computação Cientifica (LNCC-MCT), (2011). Google Scholar [18] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Annali di Matematica Pura ed. Applicata, CXLVI (1987), 65. doi: 10.1007/BF01762360. Google Scholar [19] R. C. Smith, "Smart Material Systems. Model Development,", SIAM, (2005). doi: 10.1137/1.9780898717471. Google Scholar [20] A. V. Srinivasan and D. M. McFarland, "Smart Structures: Analysis and Design,", Cambridge University Press, (2001). Google Scholar [21] K. Uchino, "Piezoelectric Actuators and Ultrasonic Motors,", Kluwer Academic Publishers, (1997). doi: 10.1007/978-1-4613-1463-9. Google Scholar [22] E. Zuazua, Controllability of the linear system of thermoelasticity,, J. Math. Pures Appl., 74 (1995), 291. Google Scholar

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##### References:
 [1] K. Ammari and S. Nicaise, Stabilization of a piezoelectric system,, Asymptotic Analysis, 73 (2011), 125. Google Scholar [2] I. Babuska, Error bounds for finite element method,, Numerishe Mathematik, 16 (1971), 322. Google Scholar [3] P. G. Ciarlet, "Mathematical Elasticity, Vols I and II,", North-Holland, I (1988). Google Scholar [4] C. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity,, Arch. Rational Mech. Anal., 29 (1968), 241. doi: 10.1007/BF00276727. Google Scholar [5] H. Funakubo, "Ed. Shape Memory Alloys,", Translated from the Japanese by J. B. Kennedy, (1984). Google Scholar [6] D. Henry, O. Lopes and A. Perissinotto, On the essential spectrum of a semigroup of thermoelasticity,, Nonlinear Anal. TMA, 21 (1993), 65. doi: 10.1016/0362-546X(93)90178-U. Google Scholar [7] D. Iessan, On some theorems in Thermopiezoelectricity,, J. Thermal Stresses, 12 (1989), 209. doi: 10.1080/01495738908961962. Google Scholar [8] B. Kapitonov, B. Miara and G. Perla Menzala, Stabilization of a layered Piezoelectric 3-D body by boundary dissipation,, ESAIM, 12 (2006), 198. doi: 10.1051/cocv:2005028. Google Scholar [9] 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. doi: 10.1137/050629884. Google Scholar [10] G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity,, Arch. Rational Mech. Anal., 148 (1999), 179. doi: 10.1007/s002050050160. Google Scholar [11] J. L. Lions, "Contrôlabilité Exacte, Stabilization et Perturbations de Systèmes Distribués,", Tome 1, (1988). Google Scholar [12] B. Miara and M. Lima, Energy decay in piezoelectric systems,, Applicable Analysis, 88 (2009), 947. doi: 10.1080/00036810903042166. Google Scholar [13] R. D. Mindlin, Equations of high frequency vibrations of thermopiezoelectric crystal plates,, International Journal of Solid Structures, 10 (1974), 625. doi: 10.1016/0020-7683(74)90047-X. Google Scholar [14] I. Müller, Six lectures in shape memory,, Centre Recherches Mathématiques, 13 (1998). Google Scholar [15] S. Nicaise, Stability and controllability of the electromagneto-elastic system,, Post. Math., 60 (2003), 73. Google Scholar [16] W. Nowacki, Some general theorems of thermopiezoelectricity,, J. Thermal Stresses, 1 (1978), 171. doi: 10.1080/01495737808926940. Google Scholar [17] J. M. Sejje Suárez, Modelagem de fenômenos termopiezoelétricos: Analise assintótica e Simulação Numérica,, Tese de Doutorado (2011) Laboratório Nacional de Computação Cientifica (LNCC-MCT), (2011). Google Scholar [18] J. Simon, Compact sets in the space $L^p(0,T;B)$,, Annali di Matematica Pura ed. Applicata, CXLVI (1987), 65. doi: 10.1007/BF01762360. Google Scholar [19] R. C. Smith, "Smart Material Systems. Model Development,", SIAM, (2005). doi: 10.1137/1.9780898717471. Google Scholar [20] A. V. Srinivasan and D. M. McFarland, "Smart Structures: Analysis and Design,", Cambridge University Press, (2001). Google Scholar [21] K. Uchino, "Piezoelectric Actuators and Ultrasonic Motors,", Kluwer Academic Publishers, (1997). doi: 10.1007/978-1-4613-1463-9. Google Scholar [22] E. Zuazua, Controllability of the linear system of thermoelasticity,, J. Math. Pures Appl., 74 (1995), 291. Google Scholar
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