Exponential stability and numerical treatment for piezoelectric beams with magnetic effect

Ramos, Anderson J.A.; Gonçalves, Cledson S.L.; Corrêa Neto, Silvério S.

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 1, pp. 255-274.

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.

Classification : 35L53, 65M06
Mots clés : Piezoelectric beams, magnetic effect, exponential decay, finite-difference discretization

[1] D.S. Almeida Júnior, A.J.A. Ramos and M.L. Santos, Observability inequality for the finite-difference semidiscretization of the 1-d coupled wave equations. Adv. Comput. Math. 41 (2015) 105–130. | DOI | MR

[2] H.T. Banks, K. Ito and C. Wang, 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. | MR | Zbl

[3] A.J. Brunner, M. Barbezat, C. Huber and P.H. Flüeler, 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] C.Y.K. Chee, L. Tong and G.P. Steven, A review on the modelling of piezoelectric sensors and actuators incorporated in intelligent structures. J. Intel. Mat. Syst. Str. 9 (1998) 1–17.

[5] P. Destuynder, I. Legrain, L. Castel and N. Richard, 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] J.A. Infante and E. Zuazua, Boundary observability for the space semi-discretizations of the 1-d wave equation. ESAIM: M2AN 33 (1999) 407–438. | DOI | Numdam | MR | Zbl

[7] L.S. Joseph and S. Lefschetz, Stability by Liapunov’s Direct Method With Applications. Academic Press, New York, USA (1961). | Zbl

[8] R.D. Krieg, On the behavior of a numerical approximation to the rotatory inertia and transverse shear plate. J. Appl. Mech. 40 (1973) 977–982. | DOI | Zbl

[9] J.L. Lions, Exact controllability, stabilization and perturbations for distributed parameter systems. SIAM Rev. 30 (1988) 1–68. | DOI | MR | Zbl

[10] B. Miara and , Santos, Energy decay in piezoelectric systems. Appl. Anal. Int. J. 88 (2009) 947–960. | DOI | MR | Zbl

[11] M. Miletic and A. Arnold, A piezoelectric Euler-Bernoulli beam with dynamic boundary control: stability and dissipative FEM. Acta Appl. Math. 138 (2015) 241–277. | DOI | MR | Zbl

[12] K.A. Morris and A.Ö. Özer, Strong stabilization of piezoelectric beams with magnetic effects, in Proc. of 52nd IEEE Conference on Decision & Control. (2013) 3014–3019. | DOI

[13] K.A. Morris and A.Ö. Özer, Modeling and stabilizability of voltage-actuated piezoelectric beams with magnetic effects. SIAM J. Control Optim. 52 (2014) 2371–2398. | DOI | MR | Zbl

[14] A.Ö. Özer, Further stabilization and exact observability results for voltage-actuated piezoelectric beams with magnetic effects. Math. Control Signals Syst. 27 (2015) 219–244. | DOI | MR | Zbl

[15] A.Ö. Özer, 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. | DOI | MR | Zbl

[16] D.W. Pohl, Dynamic piezoeletric translation devices. Rev. Sci. Instrum. 41 (1987) 54–57. | DOI

[17] R.D. Richtmyer and K.W. Morton, Difference Methods for Initial Value Problems, 2nd edn. Wiley (1967). | MR | Zbl

[18] J.E.M. Rivera, Energy decay rates in linear thermoelasticity. Funkc. Ekvac. 35 (1992) 19–30. | MR | Zbl

[19] N.N. Rogacheva, The Theory of Piezoelectric Shells and Plates. CRC Press, Boca Raton, FL, USA (1994).

[20] N. Rouche, P. Habets and M. Laloy, Stability theory by Liapunov’s direct method. Appl. Math. Sci. 22 (1977) 19–30. | MR | Zbl

[21] Y.V.K. Sadasiva Rao and B.C. Nakra, Vibrations of unsymmetrical sandwich beams and plates with viscoelastic cores. J. Sound Vibr. 34 (1974) 309–326. | DOI | Zbl

[22] H.F. Sare, B. Miara and M.L. Santos, A note on analyticity to piezoelectric systems. Math. Methods Appl. Sci. 35 (2012) 2157–2165. | DOI | MR | Zbl

[23] G.D. Smith, Numerical Solutions of Partial Differential Equations: Finite Difference Methods. New York, USA (1985). | MR | Zbl

[24] R.C. Smith, Smart Material Systems: Model Development. SIAM, PHL, USA (2005). | DOI | MR | Zbl

[25] W. Strauss and L. Vazquez, Numerical solution of a nonlinear Klein-Gordon equation. J. Comput. Phys. 28 (1974) 271–278. | DOI | MR | Zbl

[26] L.T. Tebou and E. Zuazua, Uniform boundary stabilization of the finite difference space discretization of the 1-d wave equation. Adv. Comput. Math. 26 (2007) 337–365. | DOI | MR | Zbl

[27] H.F. Tiersten, Linear Piezoelectric Plate Vibrations. Plenum Press, New York, USA (1969).

[28] H.S. Tzou, Piezoelectric Shells, Vol. 19 of Solid Mech. Appl. Kluwer Academic, The Netherlands (1993).

[29] J.M. Wang and B.Z. Guo, On the stability of swelling porous elastic soils with fluid saturation by one internal damping. IMA J. Appl. Math. 71 (2006) 565–582. | DOI | MR | Zbl

[30] J. Yang, An Introduction to the Theory of Piezoelectricity. Springer (2005). | Zbl

[31] T.J. Yeh, H.R. Feng and L.S. Wen, An integrated physical model that characterizes creep and hysteresis in piezoelectric actuators. Simul. Model. Pract. Theory Sci. Direct. 16 (1998) 93–110. | DOI