# 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 and Continuous Dynamical Systems, 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), 125-146. [2] I. Babuska, Error bounds for finite element method, Numerishe Mathematik, 16 (1971), 322-333. [3] P. G. Ciarlet, "Mathematical Elasticity, Vols I and II," North-Holland, Amsterdam, I, 1988, II, 1997. [4] C. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal., 29 (1968), 241-271. doi: 10.1007/BF00276727. [5] H. Funakubo, "Ed. Shape Memory Alloys," Translated from the Japanese by J. B. Kennedy, Gordon and Breach Science Publishers, New York, 1984. [6] D. Henry, O. Lopes and A. Perissinotto, On the essential spectrum of a semigroup of thermoelasticity, Nonlinear Anal. TMA, 21 (1993), 65-75. doi: 10.1016/0362-546X(93)90178-U. [7] D. Iessan, On some theorems in Thermopiezoelectricity, J. Thermal Stresses, 12 (1989), 209-223. doi: 10.1080/01495738908961962. [8] 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. doi: 10.1051/cocv:2005028. [9] 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. doi: 10.1137/050629884. [10] G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Rational Mech. Anal., 148 (1999), 179-231. doi: 10.1007/s002050050160. [11] J. L. Lions, "Contrôlabilité Exacte, Stabilization et Perturbations de Systèmes Distribués," Tome 1, Contrôlabilité Exacte, Masson 1988. [12] B. Miara and M. Lima, Energy decay in piezoelectric systems, Applicable Analysis, 88 (2009), 947-960. doi: 10.1080/00036810903042166. [13] R. D. Mindlin, Equations of high frequency vibrations of thermopiezoelectric crystal plates, International Journal of Solid Structures, 10 (1974), 625-637. doi: 10.1016/0020-7683(74)90047-X. [14] I. Müller, Six lectures in shape memory, Centre Recherches Mathématiques, CRM, Proceedings and Lectures Notes, 13 (1998). [15] S. Nicaise, Stability and controllability of the electromagneto-elastic system, Post. Math., 60 (2003), 73-80. [16] W. Nowacki, Some general theorems of thermopiezoelectricity, J. Thermal Stresses, 1 (1978), 171-182. doi: 10.1080/01495737808926940. [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), Brasil. (in Portuguese) [18] J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed. Applicata, (IV), CXLVI (1987), 65-96. doi: 10.1007/BF01762360. [19] R. C. Smith, "Smart Material Systems. Model Development," SIAM, Frontiers in Applied Mathematics, 2005. doi: 10.1137/1.9780898717471. [20] A. V. Srinivasan and D. M. McFarland, "Smart Structures: Analysis and Design," Cambridge University Press, Cambridge, UK, 2001. [21] K. Uchino, "Piezoelectric Actuators and Ultrasonic Motors," Kluwer Academic Publishers, Boston, 1997. doi: 10.1007/978-1-4613-1463-9. [22] E. Zuazua, Controllability of the linear system of thermoelasticity, J. Math. Pures Appl., 74 (1995), 291-315.

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##### References:
 [1] K. Ammari and S. Nicaise, Stabilization of a piezoelectric system, Asymptotic Analysis, 73 (2011), 125-146. [2] I. Babuska, Error bounds for finite element method, Numerishe Mathematik, 16 (1971), 322-333. [3] P. G. Ciarlet, "Mathematical Elasticity, Vols I and II," North-Holland, Amsterdam, I, 1988, II, 1997. [4] C. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal., 29 (1968), 241-271. doi: 10.1007/BF00276727. [5] H. Funakubo, "Ed. Shape Memory Alloys," Translated from the Japanese by J. B. Kennedy, Gordon and Breach Science Publishers, New York, 1984. [6] D. Henry, O. Lopes and A. Perissinotto, On the essential spectrum of a semigroup of thermoelasticity, Nonlinear Anal. TMA, 21 (1993), 65-75. doi: 10.1016/0362-546X(93)90178-U. [7] D. Iessan, On some theorems in Thermopiezoelectricity, J. Thermal Stresses, 12 (1989), 209-223. doi: 10.1080/01495738908961962. [8] 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. doi: 10.1051/cocv:2005028. [9] 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. doi: 10.1137/050629884. [10] G. Lebeau and E. Zuazua, Decay rates for the three-dimensional linear system of thermoelasticity, Arch. Rational Mech. Anal., 148 (1999), 179-231. doi: 10.1007/s002050050160. [11] J. L. Lions, "Contrôlabilité Exacte, Stabilization et Perturbations de Systèmes Distribués," Tome 1, Contrôlabilité Exacte, Masson 1988. [12] B. Miara and M. Lima, Energy decay in piezoelectric systems, Applicable Analysis, 88 (2009), 947-960. doi: 10.1080/00036810903042166. [13] R. D. Mindlin, Equations of high frequency vibrations of thermopiezoelectric crystal plates, International Journal of Solid Structures, 10 (1974), 625-637. doi: 10.1016/0020-7683(74)90047-X. [14] I. Müller, Six lectures in shape memory, Centre Recherches Mathématiques, CRM, Proceedings and Lectures Notes, 13 (1998). [15] S. Nicaise, Stability and controllability of the electromagneto-elastic system, Post. Math., 60 (2003), 73-80. [16] W. Nowacki, Some general theorems of thermopiezoelectricity, J. Thermal Stresses, 1 (1978), 171-182. doi: 10.1080/01495737808926940. [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), Brasil. (in Portuguese) [18] J. Simon, Compact sets in the space $L^p(0,T;B)$, Annali di Matematica Pura ed. Applicata, (IV), CXLVI (1987), 65-96. doi: 10.1007/BF01762360. [19] R. C. Smith, "Smart Material Systems. Model Development," SIAM, Frontiers in Applied Mathematics, 2005. doi: 10.1137/1.9780898717471. [20] A. V. Srinivasan and D. M. McFarland, "Smart Structures: Analysis and Design," Cambridge University Press, Cambridge, UK, 2001. [21] K. Uchino, "Piezoelectric Actuators and Ultrasonic Motors," Kluwer Academic Publishers, Boston, 1997. doi: 10.1007/978-1-4613-1463-9. [22] E. Zuazua, Controllability of the linear system of thermoelasticity, J. Math. Pures Appl., 74 (1995), 291-315.
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