# American Institute of Mathematical Sciences

doi: 10.3934/mcrf.2021056
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## Piezoelectric beams with magnetic effect and localized damping

 1 Département de Mathématiques et Informatique, Faculté Polydisciplinaire de Safi, Université Cadi Ayyad, Maroc 2 Department of Mathematics, College of Sciences, University of Sharjah, P.O.Box 27272, Sharjah, UAE 3 Institute of Exact and Natural Sciences, Doctoral Program in Mathematics. Federal University of Pará, Augusto Corrêa Street, Number 01, 66075-110, Belém-PA-Brazil

* Corresponding author: Abdelaziz Soufyane

Received  April 2021 Revised  September 2021 Early access December 2021

Fund Project: The second author is supported by University of Sharjah, grant # 1802144069. The third author is supported by University of Sharjah, CNPq Grant # 303026/2018-9

In this work we are considering a one-dimensional dissipative system of piezoelectric beams with magnetic effect and localized damping. We prove that the system is exponential stable using a damping mechanism acting only on one component and on a small part of the beam.

Citation: Mounir Afilal, Abdelaziz Soufyane, Mauro de Lima Santos. Piezoelectric beams with magnetic effect and localized damping. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021056
##### References:
 [1] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.  Google Scholar [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 (2005), 561-567.   Google Scholar [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 (1998), 3-19.   Google Scholar [4] G. Chen, S. A. Fulling, F. J. Narcowich and S. Sun, Exponential decay of energy of evolution equations with locally distributed damping, SIAM J. Appl. Math., 51 (1991), 266-301.  doi: 10.1137/0151015.  Google Scholar [5] J. Esteban, F. Lalande and C. A. Rogers, Theoretical modeling of wave localization due to material damping, In: Smart Structures and Materials 1996: Smart Structures and Integrated Systems, 2717 (1996), 332-340.   Google Scholar [6] L. F. Ho, Exact controllability of the one dimensional wave equation with locally distributed control, SIAM J. Control Optim., 28 (1990), 733-748.  doi: 10.1137/0328043.  Google Scholar [7] F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.   Google Scholar [8] K. Liu and Z. Liu, Exponential decay of energy of vibrating strings with local viscoelasticity, Z. Angew. Math. Phys., 53 (2002), 265-280.  doi: 10.1007/s00033-002-8155-6.  Google Scholar [9] P. Martinez, Decay of solutions of the wave equation with a local highly degenerate dissipation, Asymptot. Anal., 19 (1999), 1-17.   Google Scholar [10] B. Miara and M. L. Santos, Energy decay in piezoelectric systems, Appl. Anal., 88 (2009), 947-960.  doi: 10.1080/00036810903042166.  Google Scholar [11] K. A. Morris and A. O. Ozer, Strong stabilization of piezoelectric beams with magnetic effects, In 52nd IEEE Conference on Decision and Control, (2013), 3014–3019. Google Scholar [12] 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: 10.1137/130918319.  Google Scholar [13] A. Ö. Özer, Further stabilization and exact observability results for voltage-actuated piezoelectric beams with magnetic effects, Math. Control Signals Systems, 27 (2015), 219-244.  doi: 10.1007/s00498-015-0139-0.  Google Scholar [14] A. O. Özer and W. Horner, Uniform boundary observability of finite difference approximations of non-compactly coupled piezoelectric beam equations. Applicable Analysis, Applicable Analysis, (2021), 1–22. Google Scholar [15] A. Pazy, Semigoups of Linear Operator and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [16] J. Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.  Google Scholar [17] A. J. A. Ramos, M. M. Freitas, D. S. Almeida Jr., 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 (2019), Paper No. 60, 14 pp. doi: 10.1007/s00033-019-1106-2.  Google Scholar [18] A. J. A. Ramos, C. S. L. Gonçalves and S. S. Correa Neto, Exponential stability and numerical treatment for piezoelectric beams with magnetic effect, ESAIM Math. Model. Numer. Anal., 52 (2018), 255-274.  doi: 10.1051/m2an/2018004.  Google Scholar [19] 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 (2021), Paper No. 166, 12 pp. doi: 10.1007/s00033-021-01593-9.  Google Scholar [20] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations, 15 (1990), 205-235.  doi: 10.1080/03605309908820684.  Google Scholar [21] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. Pures Appl., 70 (1991), 513-529.   Google Scholar

show all references

##### References:
 [1] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.  Google Scholar [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 (2005), 561-567.   Google Scholar [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 (1998), 3-19.   Google Scholar [4] G. Chen, S. A. Fulling, F. J. Narcowich and S. Sun, Exponential decay of energy of evolution equations with locally distributed damping, SIAM J. Appl. Math., 51 (1991), 266-301.  doi: 10.1137/0151015.  Google Scholar [5] J. Esteban, F. Lalande and C. A. Rogers, Theoretical modeling of wave localization due to material damping, In: Smart Structures and Materials 1996: Smart Structures and Integrated Systems, 2717 (1996), 332-340.   Google Scholar [6] L. F. Ho, Exact controllability of the one dimensional wave equation with locally distributed control, SIAM J. Control Optim., 28 (1990), 733-748.  doi: 10.1137/0328043.  Google Scholar [7] F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56.   Google Scholar [8] K. Liu and Z. Liu, Exponential decay of energy of vibrating strings with local viscoelasticity, Z. Angew. Math. Phys., 53 (2002), 265-280.  doi: 10.1007/s00033-002-8155-6.  Google Scholar [9] P. Martinez, Decay of solutions of the wave equation with a local highly degenerate dissipation, Asymptot. Anal., 19 (1999), 1-17.   Google Scholar [10] B. Miara and M. L. Santos, Energy decay in piezoelectric systems, Appl. Anal., 88 (2009), 947-960.  doi: 10.1080/00036810903042166.  Google Scholar [11] K. A. Morris and A. O. Ozer, Strong stabilization of piezoelectric beams with magnetic effects, In 52nd IEEE Conference on Decision and Control, (2013), 3014–3019. Google Scholar [12] 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: 10.1137/130918319.  Google Scholar [13] A. Ö. Özer, Further stabilization and exact observability results for voltage-actuated piezoelectric beams with magnetic effects, Math. Control Signals Systems, 27 (2015), 219-244.  doi: 10.1007/s00498-015-0139-0.  Google Scholar [14] A. O. Özer and W. Horner, Uniform boundary observability of finite difference approximations of non-compactly coupled piezoelectric beam equations. Applicable Analysis, Applicable Analysis, (2021), 1–22. Google Scholar [15] A. Pazy, Semigoups of Linear Operator and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.  Google Scholar [16] J. Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.  Google Scholar [17] A. J. A. Ramos, M. M. Freitas, D. S. Almeida Jr., 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 (2019), Paper No. 60, 14 pp. doi: 10.1007/s00033-019-1106-2.  Google Scholar [18] A. J. A. Ramos, C. S. L. Gonçalves and S. S. Correa Neto, Exponential stability and numerical treatment for piezoelectric beams with magnetic effect, ESAIM Math. Model. Numer. Anal., 52 (2018), 255-274.  doi: 10.1051/m2an/2018004.  Google Scholar [19] 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 (2021), Paper No. 166, 12 pp. doi: 10.1007/s00033-021-01593-9.  Google Scholar [20] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations, 15 (1990), 205-235.  doi: 10.1080/03605309908820684.  Google Scholar [21] E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. Pures Appl., 70 (1991), 513-529.   Google Scholar
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