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

Mirelson M. Freitas; Anderson J. A. Ramos; Manoel J. Dos Santos; Eraldo R. N. Fonseca

doi:10.3934/cpaa.2021129

Article Contents

2021, Volume 20, Issue 11: 3745-3765. Doi: 10.3934/cpaa.2021129 open access

This issue Previous Article Normalized solutions of supercritical nonlinear fractional Schrödinger equation with potential Next Article Existence and multiplicity for Hamilton-Jacobi-Bellman equation

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
^* Corresponding author

Received: February 28, 2021

Revised: May 31, 2021

Early access: July 2021

Published: November 2021

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

Abstract Full Text(HTML) Related Papers Cited by

Abstract

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.

Keywords:

Mathematics Subject Classification: Primary: 35B40, 35B41; Secondary: 37L30.

Citation:

FullText(HTML)

Related Papers

Cited by

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. Dagdeviren, P. Joe, O. L. Tuzman, K. Park, K. J. Lee, Y Shi, Y. 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. Efendiev, A. 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. 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 (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. Langa, A. 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. Ma, R. 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.