Volume 28, 2022
|Number of page(s)||26|
|Published online||29 June 2022|
Long time dynamics and upper semi-continuity of attractors for piezoelectric beams with nonlinear boundary feedback*, **, ***
Federal University of Pará, Raimundo Santana Street s/n, Salinópolis PA, 68721-000, Brazil
2 Department of Mathematics, Western Kentucky University, Bowling Green, KY 42101, USA
3 Federal University of Pará, Raimundo Santana Street s/n, Salinópolis PA, 68721-000, Brazil
**** Corresponding author: firstname.lastname@example.org
Accepted: 27 April 2022
A system of boundary-controlled piezoelectric beam equations, accounting for the interactions between mechanical vibrations and the fully-dynamic electromagnetic fields, is considered. Even though electrostatic and quasi-static electromagnetic field approximations of Maxwell’s equations are sufficient for most models of piezoelectric systems, where the magnetic permeability is completely discarded, the PDE model considered here retains the pronounced wave behavior of electromagnetic fields to accurately describe the dynamics for the most piezoelectric acoustic devices. It is also crucial to investigate whether the closed-loop dynamics of the fully-dynamic piezoelectric beam equations, with nonlinear state feedback and nonlinear external sources, is close to the one described by the electrostatic/quasi-static equations, when the magnetic permeability μ is small. Therefore, the asymptotic behavior is analyzed for the fully-dynamic model at first. The existence of global attractors with finite fractal dimension and the existence of exponential attractors are proved. Finally, the upper-semicontinuity of attractors with respect to magnetic permeability to the ones of the electrostatic/quasi-static beam equations is shown.
Mathematics Subject Classification: 93D20 / 35Q74 / 74K10 / 37L30 / 47H20 / 35L70
Key words: Global attractors / nonlinear boundary dissipation / exponential attractors / attractor upper-semicontinuity / piezoelectric beam / electrostatic / Maxwell’s equations
A.Ö. Özer gratefully acknowledges the financial support of the National Science Foundation under Cooperative Agreement No. 1849213.
© The authors. Published by EDP Sciences, SMAI 2022
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