doi: 10.3934/mcrf.2021056
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

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. 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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   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. 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.  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. 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.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.   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. 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.  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

[1]

Gustavo Alberto Perla Menzala, Julian Moises Sejje Suárez. A thermo piezoelectric model: Exponential decay of the total energy. Discrete & Continuous Dynamical Systems, 2013, 33 (11&12) : 5273-5292. doi: 10.3934/dcds.2013.33.5273

[2]

Kim Dang Phung. Decay of solutions of the wave equation with localized nonlinear damping and trapped rays. Mathematical Control & Related Fields, 2011, 1 (2) : 251-265. doi: 10.3934/mcrf.2011.1.251

[3]

Aowen Kong, Carlos Nonato, Wenjun Liu, Manoel Jeremias dos Santos, Carlos Raposo. Equivalence between exponential stabilization and observability inequality for magnetic effected piezoelectric beams with time-varying delay and time-dependent weights. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021168

[4]

Mirelson M. Freitas, Anderson J. A. Ramos, Manoel J. Dos Santos, Jamille L.L. Almeida. Dynamics of piezoelectric beams with magnetic effects and delay term. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021015

[5]

Takeshi Taniguchi. Exponential boundary stabilization for nonlinear wave equations with localized damping and nonlinear boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1571-1585. doi: 10.3934/cpaa.2017075

[6]

A. F. Almeida, M. M. Cavalcanti, J. P. Zanchetta. Exponential decay for the coupled Klein-Gordon-Schrödinger equations with locally distributed damping. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2039-2061. doi: 10.3934/cpaa.2018097

[7]

Irena Lasiecka, Roberto Triggiani. Heat--structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, fractional powers. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1515-1543. doi: 10.3934/cpaa.2016001

[8]

Jing Zhang. The analyticity and exponential decay of a Stokes-wave coupling system with viscoelastic damping in the variational framework. Evolution Equations & Control Theory, 2017, 6 (1) : 135-154. doi: 10.3934/eect.2017008

[9]

Vo Anh Khoa, Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms. Evolution Equations & Control Theory, 2019, 8 (2) : 359-395. doi: 10.3934/eect.2019019

[10]

Gustavo Alberto Perla Menzala, Julian Moises Sejje Suárez. On the exponential stabilization of a thermo piezoelectric/piezomagnetic system. Evolution Equations & Control Theory, 2012, 1 (2) : 315-336. doi: 10.3934/eect.2012.1.315

[11]

Zayd Hajjej, Mohammad Al-Gharabli, Salim Messaoudi. Stability of a suspension bridge with a localized structural damping. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021089

[12]

Chunxiang Zhao, Chunyan Zhao, Chengkui Zhong. The global attractor for a class of extensible beams with nonlocal weak damping. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 935-955. doi: 10.3934/dcdsb.2019197

[13]

Marcio Antonio Jorge da Silva, Vando Narciso. Long-time dynamics for a class of extensible beams with nonlocal nonlinear damping*. Evolution Equations & Control Theory, 2017, 6 (3) : 437-470. doi: 10.3934/eect.2017023

[14]

Peng Sun. Exponential decay of Lebesgue numbers. Discrete & Continuous Dynamical Systems, 2012, 32 (10) : 3773-3785. doi: 10.3934/dcds.2012.32.3773

[15]

Manoel J. Dos Santos, João C. P. Fortes, Marcos L. Cardoso. Exponential stability for a piezoelectric beam with a magnetic effect and past history. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021283

[16]

To Fu Ma, Paulo Nicanor Seminario-Huertas. Attractors for semilinear wave equations with localized damping and external forces. Communications on Pure & Applied Analysis, 2020, 19 (4) : 2219-2233. doi: 10.3934/cpaa.2020097

[17]

Aníbal Rodríguez-Bernal, Enrique Zuazua. Parabolic singular limit of a wave equation with localized boundary damping. Discrete & Continuous Dynamical Systems, 1995, 1 (3) : 303-346. doi: 10.3934/dcds.1995.1.303

[18]

Louis Tebou. Stabilization of some elastodynamic systems with localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems, 2016, 36 (12) : 7117-7136. doi: 10.3934/dcds.2016110

[19]

Denis Mercier, Virginie Régnier. Decay rate of the Timoshenko system with one boundary damping. Evolution Equations & Control Theory, 2019, 8 (2) : 423-445. doi: 10.3934/eect.2019021

[20]

Moez Daoulatli. Rates of decay for the wave systems with time dependent damping. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 407-443. doi: 10.3934/dcds.2011.31.407

2020 Impact Factor: 1.284

Metrics

  • PDF downloads (83)
  • HTML views (42)
  • Cited by (0)

[Back to Top]