Well-posedness and general energy decay of solutions for a nonlinear damping piezoelectric beams system with thermal and magnetic effects

Year 2023, , 1615 - 1630, 03.11.2023

Hassan Messaoudi , Abdelouaheb Ardjouni , Salah Zıtounı , Khochemane Houssem Eddine

https://doi.org/10.15672/hujms.1187356

Cited By: 1

Abstract

In this article, we study the piezoelectric beams with thermal and magnetic effects in the presence of a nonlinear damping term acting on the mechanical equation. First, we prove that the system is well-posed in the sense of semigroup theory. And by constructing a suitable Liapunov functional, we show a general decay result of the solution for the system from which the polynomial and exponential decay are only special cases. Furthermore, our result does not depend on any relationship between system parameters.

Keywords

General decay, Liapunov functional, semigroup approach, nonlinear damping, piezoelectric beams.

References

• [1] T.A. Apalara, A general decay for a weakly nonlinearly damped porous system, J. Dyn. Control Syst. 25 (3), 311-322, 2019.
• [2] 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, Materials and structures, 38 (5), 561-567, 2005.
• [3] C.Y.K. Chee, L. Tong and G.P. Steven, A review on the modelling of piezoelectric sensors and actuators incorporated in intelligent structures, Journal of intelligent material systems and structures, 9 (1), 3-19, 1998.
• [4] I. Chueshov and I. Lasiecka, Von Karman Evolution Equations: Well-posedness and Long Time Dynamics, New York, Springer, 2010.
• [5] Ph. Destuynder, I. Legrain, L. Castel and N. Richard, Theoretical, numerical and experimental discussion on the use of piezoelectric devices for control-structure inter-action, European J. Mech. A Solids 11 (2), 181-213, 1992.
• [6] H.D. Fernández Sare, B. Miara and M.L. Santos, A note on analyticity to piezoelectric systems, Math. Methods Appl. Sci. 35 (18), 2157-2165, 2012.
• [7] M.M. Freitas, A.J.A. Ramos, M.J. Dos Santos and J.L.L. Almeida, Dynamics of piezoelectric beams with magnetic effects and delay term, Evol. Equ. Control Theory 11 (2), 583-603, 2022.
• [8] M.M. Freitas, A.J.A. Ramos, A.Ö. Özer and D.S. Almeida Júnior, Long-time dynamics for a fractional piezoelectric system with magnetic effects and Fourier’s law, J. Differ. Equ. 280, 891-927, 2021.
• [9] A. Guesmia, Asymptotic stability of abstract dissipative systems with infinite memory, J. Math. Anal. Appl. 382 (2), 748-760, 2011.
• [10] H.E. Khochemane, L. Bouzettouta and S. Zitouni, General decay of a nonlinear damping porous-elastic system with past history, Ann. Univ. Ferrara Sez. VII Sci. Mat. 65 (2), 249-275, 2019.
• [11] H.E. Khochemane, S. Zitouni and L. Bouzettouta, Stability result for a nonlinear damping porous-elastic system with delay term, Nonlinear Stud. 27 (2), 487-503, 2020.
• [12] Y.Y. Lim, Z.S. Tang and S.T. Smith, Piezoelectric-based monitoring of the curing of structural adhesives: a novel experimental study, Smart Mater. Struct. 28 (1), 015016, 2018.
• [13] Y. Liu , S. Migórski, V.T. Nguyen and S. Zeng, Existence and convergence results for elastic frictional contact problem with nonmonotone subdifferential boundary cond- tions, Acta Math. Sci. 41, 1151-1168, 2021.
• [14] Z. Liu, D. Motreanu and S. Zeng, Generalized penalty and regularization method for differential variational-hemivariational inequalities, SIAM J. Optim. 31, 1158-1183, 2021.
• [15] Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, CRC Press, 1999.
• [16] H. Messaoudi, S. Zitouni, H.E. Khochemane and A. Ardjouni, General stability for piezoelectric beams with a nonlinear damping term, Ann. Univ. Ferrara Sez. VII Sci. Mat. 2022 (doi:10.1007/s11565-022-00443-4).
• [17] B. Miara and M.L. Santos, Energy decay in piezoelectric systems, Appl. Anal. 88 (7), 947-960, 2009.
• [18] K. Morris and A.Ö. Özer, Strong stabilization of piezoelectric beams with magnetic effects, In 52nd IEEE Conference on Decision and Control, 3014-3019, 2013.
• [19] P.X. Pamplona, J.E. Muñoz Rivera and R. Quintanilla, On the decay of solutions for porous-elastic systems with history, J. Math. Anal. Appl. 379 (2), 682-705, 2011.
• [20] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
• [21] D.W. Pohl, Dynamic piezoelectric translation devices, Rev. Sci. Instrum. 58 (1), 54- 57, 1987.
• [22] A.J.A. Ramos, M.M. Freitas, D.S. Almeida, S.S. Jesus and T.R.S. Moura, Equivalence between exponential stabilization and boundary observability for piezoelectric beams with magnetic effect, Z. Angew. Math. Phys. 70 (2), 1-14, 2019.
• [23] A.J.A. Ramos, C.S.L. Gonçalve and S.S. Corrêa Neto, Exponential stability and numerical treatment for piezoelectric beams with magnetic effect, ESAIM Math. Model. Numer. Anal. 52 (1), 255-274, 2018.
• [24] P. Shivashankar and S.B. Kandagal, Characterization of elastic and electromechanical nonlinearities in piezoceramic plate actuators from vibrations of a piezoelectric-beam, Mech. Syst. Signal Process. 116, 624-640, 2019.
• [25] A. Soufyane, M. Afilal and M.L. Santos, Energy decay for a weakly nonlinear damped piezoelectric beams with magnetic effects and a nonlinear delay term, Z. Angew. Math. Phys. 72 (4), 1-12, 2021.
• [26] H.F. Tiersten, Linear Piezoelectric Plate Vibrations: Elements of the Linear Theory of Piezoelectricity and the Vibrations Piezoelectric Plates, Springer, 2013.
• [27] K. Uchino, Chapter 1, The development of piezoelectric materials and the new perspective, In Kenji Uchino, editor, Advanced Piezoelectric Materials, Woodhead Publishing in Materials, 1-92, 2017.
• [28] J. Yang, Special Topics in the Theory of Piezoelectricity, Springer Science and Business Media, 2010.
• [29] T.J. Yeh, H. Ruo-Feng and L. Shin-Wen, An integrated physical model that characterizes creep and hysteresis in piezoelectric actuators, Simul. Model. Pract. Theory 16 (1), 93-110, 2008.
• [30] S. Zeng, S. Migórski and A.A. Khan, Nonlinear quasi-hemivariational inequalities: existence and optimal control, SIAM J. Control Optim. 59, 1246-1274, 2021.
• [31] S. Zeng, S. Migórski and Z. Liu, Well-posedness, optimal control, and sensitivity analysis for a class of differential variational-hemivariational inequalities, SIAM J. Optim. 31, 2829-2862, 2021.
• [32] S. Zitouni, A. Ardjouni, M.B. Mesmouli and R. Amiar, Well-posedness and stability of nonlinear wave equations with two boundary time-varying delays, MESA 8 (2), 147-170, 2017.
• [33] S. Zitouni, A. Ardjouni, K. Zennir and R. Amiar, Existence and exponential stability of solutions for transmission system with varying delay in R, Math. Morav. 20 (2), 143-161, 2016.
• [34] S. Zitouni, A. Ardjouni, K. Zennir and R. Amiar, Well-posedness and decay of solution for a transmission problem in the presence of infinite history and varying delay, Nonlinear Stud. 25 (2), 445-465, 2018.