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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 and Related Fields, doi: 10.3934/mcrf.2021056
References:
[1]

C. BardosG. 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.

[2]

A. J. BrunnerM. BarbezatC. 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. 

[3]

C. Y. K. CheeL. 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. 

[4]

G. ChenS. A. FullingF. 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.

[5]

J. EstebanF. 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. 

[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.

[7]

F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. 

[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.

[9]

P. Martinez, Decay of solutions of the wave equation with a local highly degenerate dissipation, Asymptot. Anal., 19 (1999), 1-17. 

[10]

B. Miara and M. L. Santos, Energy decay in piezoelectric systems, Appl. Anal., 88 (2009), 947-960.  doi: 10.1080/00036810903042166.

[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.

[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.

[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.

[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.

[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.

[16]

J. Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.

[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.

[18]

A. J. A. RamosC. 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.

[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.

[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.

[21]

E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. Pures Appl., 70 (1991), 513-529. 

show all references

References:
[1]

C. BardosG. 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.

[2]

A. J. BrunnerM. BarbezatC. 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. 

[3]

C. Y. K. CheeL. 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. 

[4]

G. ChenS. A. FullingF. 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.

[5]

J. EstebanF. 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. 

[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.

[7]

F. L. Huang, Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. 

[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.

[9]

P. Martinez, Decay of solutions of the wave equation with a local highly degenerate dissipation, Asymptot. Anal., 19 (1999), 1-17. 

[10]

B. Miara and M. L. Santos, Energy decay in piezoelectric systems, Appl. Anal., 88 (2009), 947-960.  doi: 10.1080/00036810903042166.

[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.

[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.

[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.

[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.

[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.

[16]

J. Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.  doi: 10.2307/1999112.

[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.

[18]

A. J. A. RamosC. 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.

[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.

[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.

[21]

E. Zuazua, Exponential decay for the semilinear wave equation with localized damping in unbounded domains, J. Math. Pures Appl., 70 (1991), 513-529. 

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