American Institute of Mathematical Sciences

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

On the exponential stabilization of a thermo piezoelectric/piezomagnetic system

Gustavo Alberto Perla Menzala 1, and  Julian Moises Sejje Suárez 1,

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.
Keywords: decoupling method., energy decay, Piezoelectric/piezomagnetic materials, thermal effects.
Mathematics Subject Classification: 35B40, 74H4.
Citation: Gustavo Alberto Perla Menzala, Julian Moises Sejje Suárez. On the exponential stabilization of a thermo piezoelectric/piezomagnetic system. Evolution Equations and 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.

[2]

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

[3]

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

[4]

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

[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.

[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.

[7]

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

[8]

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

[9]

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

[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.

[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.

[12]

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

[13]

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

[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.

[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.

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[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).

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

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.

[2]

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

[3]

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

[4]

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

[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.

[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.

[7]

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

[8]

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

[9]

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

[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.

[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.

[12]

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

[13]

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

[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.

[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.

[16]

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

[17]

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

[18]

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

[19]

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

[20]

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

[21]

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

[22]

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

[23]

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

[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).

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

[1]

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
[2]

Mirelson M. Freitas, Anderson J. A. Ramos, Manoel J. Dos Santos, Eraldo R. N. Fonseca. Attractors and pullback dynamics for non-autonomous piezoelectric system with magnetic and thermal effects. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3745-3765. doi: 10.3934/cpaa.2021129
[3]

Michela Eleuteri, Luca Lussardi. Thermal control of a rate-independent model for permanent inelastic effects in shape memory materials. Evolution Equations and Control Theory, 2014, 3 (3) : 411-427. doi: 10.3934/eect.2014.3.411
[4]

Mirelson M. Freitas, Anderson J. A. Ramos, Manoel J. Dos Santos, Jamille L.L. Almeida. Dynamics of piezoelectric beams with magnetic effects and delay term. Evolution Equations and Control Theory, 2022, 11 (2) : 583-603. doi: 10.3934/eect.2021015
[5]

Sandra Carillo. Materials with memory: Free energies & solution exponential decay. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1235-1248. doi: 10.3934/cpaa.2010.9.1235
[6]

Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184
[7]

Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196
[8]

Michela Eleuteri, Luca Lussardi, Ulisse Stefanelli. A rate-independent model for permanent inelastic effects in shape memory materials. Networks and Heterogeneous Media, 2011, 6 (1) : 145-165. doi: 10.3934/nhm.2011.6.145
[9]

Haolei Wang, Lei Zhang. Energy minimization and preconditioning in the simulation of athermal granular materials in two dimensions. Electronic Research Archive, 2020, 28 (1) : 405-421. doi: 10.3934/era.2020023
[10]

Tomasz Komorowski, Stefano Olla, Marielle Simon. Macroscopic evolution of mechanical and thermal energy in a harmonic chain with random flip of velocities. Kinetic and Related Models, 2018, 11 (3) : 615-645. doi: 10.3934/krm.2018026
[11]

Willy Sarlet, Tom Mestdag. Compatibility aspects of the method of phase synchronization for decoupling linear second-order differential equations. Journal of Geometric Mechanics, 2022, 14 (1) : 91-104. doi: 10.3934/jgm.2021019
[12]

Mohammed Aassila. On energy decay rate for linear damped systems. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 851-864. doi: 10.3934/dcds.2002.8.851
[13]

Rachid Assel, Mohamed Ghazel. Energy decay for the damped wave equation on an unbounded network. Evolution Equations and Control Theory, 2018, 7 (3) : 335-351. doi: 10.3934/eect.2018017
[14]

Bopeng Rao. Optimal energy decay rate in a damped Rayleigh beam. Discrete and Continuous Dynamical Systems, 1998, 4 (4) : 721-734. doi: 10.3934/dcds.1998.4.721
[15]

Gustavo Alberto Perla Menzala, Julian Moises Sejje Suárez. On the one-dimensional version of the dynamical Marguerre-Vlasov system with thermal effects. Conference Publications, 2009, 2009 (Special) : 536-547. doi: 10.3934/proc.2009.2009.536
[16]

Irena Lasiecka, To Fu Ma, Rodrigo Nunes Monteiro. Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions. Discrete and Continuous Dynamical Systems - B, 2018, 23 (3) : 1037-1072. doi: 10.3934/dcdsb.2018141
[17]

Gilbert Peralta. Uniform exponential stability of a fluid-plate interaction model due to thermal effects. Evolution Equations and Control Theory, 2020, 9 (1) : 39-60. doi: 10.3934/eect.2020016
[18]

Asim Aziz, Wasim Jamshed, Yasir Ali, Moniba Shams. Heat transfer and entropy analysis of Maxwell hybrid nanofluid including effects of inclined magnetic field, Joule heating and thermal radiation. Discrete and Continuous Dynamical Systems - S, 2020, 13 (10) : 2667-2690. doi: 10.3934/dcdss.2020142
[19]

Brenton LeMesurier. Modeling thermal effects on nonlinear wave motion in biopolymers by a stochastic discrete nonlinear Schrödinger equation with phase damping. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 317-327. doi: 10.3934/dcdss.2008.1.317
[20]

Xiaoxiao Li, Yingjing Shi, Rui Li, Shida Cao. Energy management method for an unpowered landing. Journal of Industrial and Management Optimization, 2022, 18 (2) : 825-841. doi: 10.3934/jimo.2020180

2021 Impact Factor: 1.169

Tools

Metrics

  • PDF downloads (89)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy