In this paper, we consider a one-dimensional dissipative system of piezoelectric beams with magnetic effect, inspired by the model studied by Morris and Özer (Proc. of 52nd IEEE Conference on Decision & Control (2013) 3014–3019). Our main interest is to analyze the issues relating to exponential stability of the total energy of the continuous problem and reproduce a numerical counterpart in a totally discrete domain, which preserves the important decay property of the numerical energy.
Keywords: Piezoelectric beams, magnetic effect, exponential decay, finite-difference discretization
Ramos, Anderson J.A. 1 ; Gonçalves, Cledson S.L. 1 ; Corrêa Neto, Silvério S. 1
@article{M2AN_2018__52_1_255_0,
author = {Ramos, Anderson J.A. and Gon\c{c}alves, Cledson S.L. and Corr\^ea Neto, Silv\'erio S.},
title = {Exponential stability and numerical treatment for piezoelectric beams with magnetic effect},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {255--274},
year = {2018},
publisher = {EDP Sciences},
volume = {52},
number = {1},
doi = {10.1051/m2an/2018004},
zbl = {1398.35232},
mrnumber = {3808160},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2018004/}
}
TY - JOUR AU - Ramos, Anderson J.A. AU - Gonçalves, Cledson S.L. AU - Corrêa Neto, Silvério S. TI - Exponential stability and numerical treatment for piezoelectric beams with magnetic effect JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 255 EP - 274 VL - 52 IS - 1 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2018004/ DO - 10.1051/m2an/2018004 LA - en ID - M2AN_2018__52_1_255_0 ER -
%0 Journal Article %A Ramos, Anderson J.A. %A Gonçalves, Cledson S.L. %A Corrêa Neto, Silvério S. %T Exponential stability and numerical treatment for piezoelectric beams with magnetic effect %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 255-274 %V 52 %N 1 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2018004/ %R 10.1051/m2an/2018004 %G en %F M2AN_2018__52_1_255_0
Ramos, Anderson J.A.; Gonçalves, Cledson S.L.; Corrêa Neto, Silvério S. Exponential stability and numerical treatment for piezoelectric beams with magnetic effect. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 1, pp. 255-274. doi: 10.1051/m2an/2018004
[1] , and , Observability inequality for the finite-difference semidiscretization of the 1-d coupled wave equations. Adv. Comput. Math. 41 (2015) 105–130. | MR | DOI
[2] , and , Exponentially stable approximations of weakly damped wave equations, in Estimation and Control of Distributed Parameter Systems. Vol. 100 of International Series in Numerical Analysis. Birkhauser, Basel (1991) 1–33. | Zbl | MR
[3] , , and , The potential of active fiber composites made from piezoelectric fibers for actuating and sensing applications in structural health monitoring. Mat. Struct. 38 (2005) 561–567. | DOI
[4] , and , A review on the modelling of piezoelectric sensors and actuators incorporated in intelligent structures. J. Intel. Mat. Syst. Str. 9 (1998) 1–17.
[5] , , and , Theoretical, numerical and experimental discussion of the use of piezoelectric devices for control-structure interaction. Eur. J. Mech. A. Solids. 11 (1992) 181–213. | MR
[6] and , Boundary observability for the space semi-discretizations of the 1-d wave equation. ESAIM: M2AN 33 (1999) 407–438. | Zbl | MR | Numdam | DOI
[7] and , Stability by Liapunov’s Direct Method With Applications. Academic Press, New York, USA (1961). | Zbl
[8] , On the behavior of a numerical approximation to the rotatory inertia and transverse shear plate. J. Appl. Mech. 40 (1973) 977–982. | Zbl | DOI
[9] , Exact controllability, stabilization and perturbations for distributed parameter systems. SIAM Rev. 30 (1988) 1–68. | Zbl | MR | DOI
[10] and , Santos, Energy decay in piezoelectric systems. Appl. Anal. Int. J. 88 (2009) 947–960. | Zbl | MR | DOI
[11] and , A piezoelectric Euler-Bernoulli beam with dynamic boundary control: stability and dissipative FEM. Acta Appl. Math. 138 (2015) 241–277. | Zbl | MR | DOI
[12] and , Strong stabilization of piezoelectric beams with magnetic effects, in Proc. of 52nd IEEE Conference on Decision & Control. (2013) 3014–3019. | DOI
[13] and , Modeling and stabilizability of voltage-actuated piezoelectric beams with magnetic effects. SIAM J. Control Optim. 52 (2014) 2371–2398. | Zbl | MR | DOI
[14] , Further stabilization and exact observability results for voltage-actuated piezoelectric beams with magnetic effects. Math. Control Signals Syst. 27 (2015) 219–244. | Zbl | MR | DOI
[15] , Modeling and control results for an active constrained layered (ACL) beam actuated by two voltage sources with/without magnetic effects. IEEE Trans. Autom. Control 62 (2017) 6445–6450. | Zbl | MR | DOI
[16] , Dynamic piezoeletric translation devices. Rev. Sci. Instrum. 41 (1987) 54–57. | DOI
[17] and , Difference Methods for Initial Value Problems, 2nd edn. Wiley (1967). | Zbl | MR
[18] , Energy decay rates in linear thermoelasticity. Funkc. Ekvac. 35 (1992) 19–30. | Zbl | MR
[19] , The Theory of Piezoelectric Shells and Plates. CRC Press, Boca Raton, FL, USA (1994).
[20] , and , Stability theory by Liapunov’s direct method. Appl. Math. Sci. 22 (1977) 19–30. | Zbl | MR
[21] and , Vibrations of unsymmetrical sandwich beams and plates with viscoelastic cores. J. Sound Vibr. 34 (1974) 309–326. | Zbl | DOI
[22] , and , A note on analyticity to piezoelectric systems. Math. Methods Appl. Sci. 35 (2012) 2157–2165. | Zbl | MR | DOI
[23] , Numerical Solutions of Partial Differential Equations: Finite Difference Methods. New York, USA (1985). | Zbl | MR
[24] , Smart Material Systems: Model Development. SIAM, PHL, USA (2005). | Zbl | MR | DOI
[25] and , Numerical solution of a nonlinear Klein-Gordon equation. J. Comput. Phys. 28 (1974) 271–278. | Zbl | MR | DOI
[26] and , Uniform boundary stabilization of the finite difference space discretization of the 1-d wave equation. Adv. Comput. Math. 26 (2007) 337–365. | Zbl | MR | DOI
[27] , Linear Piezoelectric Plate Vibrations. Plenum Press, New York, USA (1969).
[28] , Piezoelectric Shells, Vol. 19 of Solid Mech. Appl. Kluwer Academic, The Netherlands (1993).
[29] and , On the stability of swelling porous elastic soils with fluid saturation by one internal damping. IMA J. Appl. Math. 71 (2006) 565–582. | Zbl | MR | DOI
[30] , An Introduction to the Theory of Piezoelectricity. Springer (2005). | Zbl
[31] , and , An integrated physical model that characterizes creep and hysteresis in piezoelectric actuators. Simul. Model. Pract. Theory Sci. Direct. 16 (1998) 93–110. | DOI
Cité par Sources :
