November  2021, 20(11): 3745-3765. doi: 10.3934/cpaa.2021129

Attractors and pullback dynamics for non-autonomous piezoelectric system with magnetic and thermal effects

1. 

Faculty of Mathematics, Federal University of Pará, Raimundo Santana Street, S/N, 68721-000, Salinópolis, PA, Brazil

2. 

Faculty of Exact Sciences and Technology, Federal University of Pará, Manoel de Abre Street, S/N, 68440-000, Abaetetuba, PA, Brazil

* Corresponding author

Received  March 2021 Revised  June 2021 Published  November 2021 Early access  July 2021

Fund Project: A. J. A. Ramos is supported by Grant 310729/2019-0

This paper is concerned with the study of the pullback dynamics of a piezoelectric system with magnetic and thermal effects and subjected to small perturbations of non-autonomous external forces with a parameter $ \epsilon $. The existence of pullback exponential attractors and the existence of pullback attractors for the associated non-autonomous dynamical system are proved. Finally, the upper-semicontinuity of pullback attractors as $ \epsilon\to0 $ is shown.

Citation: Mirelson M. Freitas, Anderson J. A. Ramos, Manoel J. Dos Santos, Eraldo R. N. Fonseca. Attractors and pullback dynamics for non-autonomous piezoelectric system with magnetic and thermal effects. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3745-3765. doi: 10.3934/cpaa.2021129
References:
[1]

A. N. Carvalho, J. A. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer-Verlag, New York, 2013. doi: 10.1007/978-1-4614-4581-4.

[2]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: theoretical results, Commun. Pure Appl. Anal., 12 (2013), 3047-3071.  doi: 10.3934/cpaa.2013.12.3047.

[3]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: properties and applications, Commun. Pure Appl. Anal., 13 (2014), 1141-1165.  doi: 10.3934/cpaa.2014.13.1141.

[4]

R. Czaja, Pullback exponential attractors with admissible exponential growth in the past, Nonlinear Anal., 104 (2014), 90-108.  doi: 10.1016/j.na.2014.03.020.

[5]

R. Czaja and M. Efendiev, Pullback exponential attractors for nonautonomous equations Part I: semilinear parabolic problems, J. Math. Anal. Appl., 381 (2011), 748-765.  doi: 10.1016/j.jmaa.2011.03.053.

[6]

R. Czaja and P. Mar'in-Rubio, Pullback exponential attractors for parabolic equations with dynamical boundary conditions, Taiwan. J. Math., 21 (2017), 819-839.  doi: 10.11650/tjm/7862.

[7]

C. DagdevirenP. JoeO. L. TuzmanK. ParkK. J. LeeY ShiY. Huang and J. A. Rogers, Recent progress in flexible and stretchable piezoelectric devices for mechanical energy harvesting, sensing and actuation, Extreme Mechanics Letters, 9 (2016), 269-281. 

[8]

M. EfendievA. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, Proc. Royal Soc. Edinburg A., 135 (2005), 703-730.  doi: 10.1017/S030821050000408X.

[9]

M. M. FreitasA. J. A. RamosA. 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 (2021), 891-927.  doi: 10.1016/j.jde.2021.01.030.

[10]

A. Haraux, Une remarque sur la stabilisation de certains systemes du deuxieme ordre en temps, Portugaliae mathematica, 46 (1989), 245-258. 

[11]

I.R. Henderson, Piezoelectric Ceramics: Principles and Applications, APC International, Pennsylvania, USA, 2002.

[12]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Soc, Vol. 176, 2011. doi: 10.1090/surv/176.

[13]

J. A. LangaA. Miranville and J. Real, Pullback exponential attractors, Discret. Contin. Dyn. Syst., 26 (2010), 1329-1357.  doi: 10.3934/dcds.2010.26.1329.

[14]

T. F. MaR. N. Monteiro and A. C. Pereira, Pullback Dynamics of Non-autonomous Timoshenko Systems, Appl. Math. Optim., 80 (2019), 391-413.  doi: 10.1007/s00245-017-9469-2.

[15]

J. C. Maxwell, A Dynamical Theory of the Electromagnetic Field, Philos. Trans. R. Soc.Lond., 1865.

[16] J. C. Maxwell, A Treatise on Electricity and Magnetism, Cambridge University Press, 2009. 
[17]

K. Morris and A. Ö. Özer, Strong Stabilization of Piezoelectric Beams with Magnetic Effects, 52nd IEEE Conference on Decision and Control, 2013.

[18]

K. A. Morris and A. Ö. Özer, Modeling and Stabilizability of Voltage-Actuated Piezoelectric Beams with Magnetic Effects, SIAM J. Contr. Optim., 52 (2014), 2371–2398. doi: 10.1137/130918319.

[19]

A. Ö. Özer, Further stabilization and exact observability results for voltage-actuated piezoelectric beams with magnetic effects, Mathematics of Control, Signals, and Systems, 27 (2015), 219-244.  doi: 10.1007/s00498-015-0139-0.

[20]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[21]

A. J. A. Ramos, C. S. L. Gonçalves and S. S. C. Neto, Exponential stability and numerical treatment for piezoelectric beams with magnetic effect, ESAIM: Mathematical Modelling and Numerical Analysis, 52 (2018), 255–274. doi: 10.1051/m2an/2018004.

[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, Zeitschrift fü angewandte Mathematik und Physik, 70 (2019), 14pp. doi: 10.1007/s00033-019-1106-2.

[23]

A. J. A. Ramos, A. Ö. Özer, M. M. Freitas, D. S. Almeida and J. D. Martins, Exponential stabilitization of fully dynamic and electrostatic piezoelectric beams with delayed distributed damping feedback, Zeitschrift für angewandte Mathematik und Physik, 72 (2021), 15pp. doi: 10.1007/s00033-020-01457-8.

[24]

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 (2006), 337-365.  doi: 10.1007/s10444-004-7629-9.

show all references

References:
[1]

A. N. Carvalho, J. A. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Springer-Verlag, New York, 2013. doi: 10.1007/978-1-4614-4581-4.

[2]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: theoretical results, Commun. Pure Appl. Anal., 12 (2013), 3047-3071.  doi: 10.3934/cpaa.2013.12.3047.

[3]

A. N. Carvalho and S. Sonner, Pullback exponential attractors for evolution processes in Banach spaces: properties and applications, Commun. Pure Appl. Anal., 13 (2014), 1141-1165.  doi: 10.3934/cpaa.2014.13.1141.

[4]

R. Czaja, Pullback exponential attractors with admissible exponential growth in the past, Nonlinear Anal., 104 (2014), 90-108.  doi: 10.1016/j.na.2014.03.020.

[5]

R. Czaja and M. Efendiev, Pullback exponential attractors for nonautonomous equations Part I: semilinear parabolic problems, J. Math. Anal. Appl., 381 (2011), 748-765.  doi: 10.1016/j.jmaa.2011.03.053.

[6]

R. Czaja and P. Mar'in-Rubio, Pullback exponential attractors for parabolic equations with dynamical boundary conditions, Taiwan. J. Math., 21 (2017), 819-839.  doi: 10.11650/tjm/7862.

[7]

C. DagdevirenP. JoeO. L. TuzmanK. ParkK. J. LeeY ShiY. Huang and J. A. Rogers, Recent progress in flexible and stretchable piezoelectric devices for mechanical energy harvesting, sensing and actuation, Extreme Mechanics Letters, 9 (2016), 269-281. 

[8]

M. EfendievA. Miranville and S. Zelik, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, Proc. Royal Soc. Edinburg A., 135 (2005), 703-730.  doi: 10.1017/S030821050000408X.

[9]

M. M. FreitasA. J. A. RamosA. 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 (2021), 891-927.  doi: 10.1016/j.jde.2021.01.030.

[10]

A. Haraux, Une remarque sur la stabilisation de certains systemes du deuxieme ordre en temps, Portugaliae mathematica, 46 (1989), 245-258. 

[11]

I.R. Henderson, Piezoelectric Ceramics: Principles and Applications, APC International, Pennsylvania, USA, 2002.

[12]

P. E. Kloeden and M. Rasmussen, Nonautonomous Dynamical Systems, American Mathematical Soc, Vol. 176, 2011. doi: 10.1090/surv/176.

[13]

J. A. LangaA. Miranville and J. Real, Pullback exponential attractors, Discret. Contin. Dyn. Syst., 26 (2010), 1329-1357.  doi: 10.3934/dcds.2010.26.1329.

[14]

T. F. MaR. N. Monteiro and A. C. Pereira, Pullback Dynamics of Non-autonomous Timoshenko Systems, Appl. Math. Optim., 80 (2019), 391-413.  doi: 10.1007/s00245-017-9469-2.

[15]

J. C. Maxwell, A Dynamical Theory of the Electromagnetic Field, Philos. Trans. R. Soc.Lond., 1865.

[16] J. C. Maxwell, A Treatise on Electricity and Magnetism, Cambridge University Press, 2009. 
[17]

K. Morris and A. Ö. Özer, Strong Stabilization of Piezoelectric Beams with Magnetic Effects, 52nd IEEE Conference on Decision and Control, 2013.

[18]

K. A. Morris and A. Ö. Özer, Modeling and Stabilizability of Voltage-Actuated Piezoelectric Beams with Magnetic Effects, SIAM J. Contr. Optim., 52 (2014), 2371–2398. doi: 10.1137/130918319.

[19]

A. Ö. Özer, Further stabilization and exact observability results for voltage-actuated piezoelectric beams with magnetic effects, Mathematics of Control, Signals, and Systems, 27 (2015), 219-244.  doi: 10.1007/s00498-015-0139-0.

[20]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[21]

A. J. A. Ramos, C. S. L. Gonçalves and S. S. C. Neto, Exponential stability and numerical treatment for piezoelectric beams with magnetic effect, ESAIM: Mathematical Modelling and Numerical Analysis, 52 (2018), 255–274. doi: 10.1051/m2an/2018004.

[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, Zeitschrift fü angewandte Mathematik und Physik, 70 (2019), 14pp. doi: 10.1007/s00033-019-1106-2.

[23]

A. J. A. Ramos, A. Ö. Özer, M. M. Freitas, D. S. Almeida and J. D. Martins, Exponential stabilitization of fully dynamic and electrostatic piezoelectric beams with delayed distributed damping feedback, Zeitschrift für angewandte Mathematik und Physik, 72 (2021), 15pp. doi: 10.1007/s00033-020-01457-8.

[24]

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 (2006), 337-365.  doi: 10.1007/s10444-004-7629-9.

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