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doi: 10.3934/dcdsb.2021283
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Exponential stability for a piezoelectric beam with a magnetic effect and past history

1. 

Faculty of Exact Sciences and Technology, Federal University of Pará, Abaetetuba, PA, 68440-000, Brazil

2. 

Faculty of Mathematics, Federal University of the South and Southeast of Pará, Marabá, PA, 68500-000, Brazil

3. 

Faculty of Mathematics, Federal University of Pará, Salinópolis, PA, 68721-000, Brazil

*Corresponding author

Received  May 2021 Revised  October 2021 Early access November 2021

Solutions for systems consisting of coupled wave equations, one of them with past history, may present different behaviors due to the type of coupling. In this paper, the issue of exponential stability for a piezoelectric beam with magnetic effect and past history is analyzed. In the work is proved that the past history term acting on the longitudinal motion equation is sufficient to cause the exponential decay of the semigroup associated with the system, independent of any relation involving the model coefficients.

Citation: 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 and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021283
References:
[1]

R. G. C. Almeida and M. L. Santos, Lack of exponential decay of a coupled system of wave equations with memory, Nonlinear Anal. Real World Appl., 12 (2011), 1023-1032.  doi: 10.1016/j.nonrwa.2010.08.025.

[2]

F. Ammar-KhodjaA. BenabdallahJ. E. Muñoz Rivera and R. Racke, Energy decay for timoshenko systems of memory type, J. Differential Equations, 194 (2003), 82-115.  doi: 10.1016/S0022-0396(03)00185-2.

[3]

H. T. Banks, R. C. Smith and Y. Wang, Smart Material Structures: Modeling, Estimation and Control, Wiley-Masson Series Research in Applied Mathematics, Wiley, 1996.

[4]

S. M. S. CordeiroR. F. C. Lobato and C. A. Raposo, Optimal polynomial decay for a coupled system of wave with past history, Open Journal of Mathematical Analysis, 4 (2020), 49-59.  doi: 10.30538/psrp-oma2020.0052.

[5]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.

[6]

M. J. Dos SantosR. F. C. LobatoS. M. S. Cordeiro and A. C. B. Dos Santos, Quasi-stability and attractors for a nonlinear coupled wave system with memory, Boll. Unione Mat. Ital., 14 (2021), 279-321.  doi: 10.1007/s40574-020-00258-1.

[7]

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

[8]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, Evolution Equations, Semigroups and Functional Analysis, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 50 (2000), 155-178. 

[9]

Y. GuoM. A. RammahaS. SakuntasathienE. S. Titi and D. Toundykov, Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping, J. Differential Equations, 257 (2014), 3778-3812.  doi: 10.1016/j.jde.2014.07.009.

[10]

S. Hansen, Analysis of a plate with a localized piezoelectric patch, In Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171), 3 (1998), 2952-2957.

[11]

B. KapitonovB. Miara and G. P. Menzala, Stabilization of a layered piezoelectric 3-d body by boundary dissipation, ESAIM: COCV, 12 (2006), 198-215.  doi: 10.1051/cocv:2005028.

[12]

M. Kirane and B. Said-Houari, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Zeitschrift für angewandte Mathematik und Physik, 62 (2011), 1065–1082. doi: 10.1007/s00033-011-0145-0.

[13]

V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.

[14]

A. Kong, C. Nonato, W. Liu, M. J. Dos Santos and C. 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.

[15]

S. A. Messaoudi and B. Said-Houari, Uniform decay in a timoshenko-type system with past history, J. Math. Anal. Appl., 360 (2009), 459-475.  doi: 10.1016/j.jmaa.2009.06.064.

[16]

K. Morris and A. O. Özer, Strong stabilization of piezoelectric beams with magnetic effects, 52nd IEEE Conference on Decision and Control, (2013), 3014–3019. doi: 10.1109/CDC.2013.6760341.

[17]

K. A. Morris and A. O. Özer, Modeling and stabilizability of voltage-actuated piezoelectric beams with magnetic effects, SIAM J. Control Optim., 52 (2014), 2371-2398.  doi: 10.1137/130918319.

[18]

J. E. Muñoz Rivera and H. D. Fernández Sare, Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482-502.  doi: 10.1016/j.jmaa.2007.07.012.

[19]

J. E. Muñoz Rivera and R. Barreto, Decay rates of solutions to thermoviscoelastic plates with memory, IMA J. Appl. Math., 60 (1998), 263-283.  doi: 10.1093/imamat/60.3.263.

[20]

A. O. Özer and W. Horner, Uniform boundary observability of finite difference approximations of non-compactly coupled piezoelectric beam equations, Applicable Analysis, 0 (2021), 1-22.  doi: 10.1080/00036811.2021.1936508.

[21]

P. X. PamplonaJ. E. Muñoz Rivera and R. Quintanilla, On the decay of solutions for porous-elastic systems with history, J. Math. Anal. Appl., 379 (2011), 682-705.  doi: 10.1016/j.jmaa.2011.01.045.

[22]

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

[23]

A. J. A. RamosC. S. L. G. calves and S. S. Corrêa 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.

[24]

A. J. A. Ramos, A. Ö. Özer, M. M. Freitas, D. S. A. Júnior and J. D. Martins, Exponential stabilization of fully dynamic and electrostatic piezoelectric beams with delayed distributed damping feedback,, Zeitschrift für angewandte Mathematik und Physik, 72 (2021), 26. doi: 10.1007/s00033-020-01457-8.

[25]

J. Yang, An Introduction to the Theory of Piezoelectricity, Advances in Mechanics and Mathematics, Springer, 2005. doi: 10.1007/978-3-030-03137-4.

[26]

S. Zheng, Nonlinear Evolution Equations, Chapman and Hall/CRC, 2004. doi: 10.1201/9780203492222.

show all references

References:
[1]

R. G. C. Almeida and M. L. Santos, Lack of exponential decay of a coupled system of wave equations with memory, Nonlinear Anal. Real World Appl., 12 (2011), 1023-1032.  doi: 10.1016/j.nonrwa.2010.08.025.

[2]

F. Ammar-KhodjaA. BenabdallahJ. E. Muñoz Rivera and R. Racke, Energy decay for timoshenko systems of memory type, J. Differential Equations, 194 (2003), 82-115.  doi: 10.1016/S0022-0396(03)00185-2.

[3]

H. T. Banks, R. C. Smith and Y. Wang, Smart Material Structures: Modeling, Estimation and Control, Wiley-Masson Series Research in Applied Mathematics, Wiley, 1996.

[4]

S. M. S. CordeiroR. F. C. Lobato and C. A. Raposo, Optimal polynomial decay for a coupled system of wave with past history, Open Journal of Mathematical Analysis, 4 (2020), 49-59.  doi: 10.30538/psrp-oma2020.0052.

[5]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.

[6]

M. J. Dos SantosR. F. C. LobatoS. M. S. Cordeiro and A. C. B. Dos Santos, Quasi-stability and attractors for a nonlinear coupled wave system with memory, Boll. Unione Mat. Ital., 14 (2021), 279-321.  doi: 10.1007/s40574-020-00258-1.

[7]

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

[8]

M. Grasselli and V. Pata, Uniform attractors of nonautonomous dynamical systems with memory, Evolution Equations, Semigroups and Functional Analysis, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 50 (2000), 155-178. 

[9]

Y. GuoM. A. RammahaS. SakuntasathienE. S. Titi and D. Toundykov, Hadamard well-posedness for a hyperbolic equation of viscoelasticity with supercritical sources and damping, J. Differential Equations, 257 (2014), 3778-3812.  doi: 10.1016/j.jde.2014.07.009.

[10]

S. Hansen, Analysis of a plate with a localized piezoelectric patch, In Proceedings of the 37th IEEE Conference on Decision and Control (Cat. No.98CH36171), 3 (1998), 2952-2957.

[11]

B. KapitonovB. Miara and G. P. Menzala, Stabilization of a layered piezoelectric 3-d body by boundary dissipation, ESAIM: COCV, 12 (2006), 198-215.  doi: 10.1051/cocv:2005028.

[12]

M. Kirane and B. Said-Houari, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Zeitschrift für angewandte Mathematik und Physik, 62 (2011), 1065–1082. doi: 10.1007/s00033-011-0145-0.

[13]

V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994.

[14]

A. Kong, C. Nonato, W. Liu, M. J. Dos Santos and C. 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.

[15]

S. A. Messaoudi and B. Said-Houari, Uniform decay in a timoshenko-type system with past history, J. Math. Anal. Appl., 360 (2009), 459-475.  doi: 10.1016/j.jmaa.2009.06.064.

[16]

K. Morris and A. O. Özer, Strong stabilization of piezoelectric beams with magnetic effects, 52nd IEEE Conference on Decision and Control, (2013), 3014–3019. doi: 10.1109/CDC.2013.6760341.

[17]

K. A. Morris and A. O. Özer, Modeling and stabilizability of voltage-actuated piezoelectric beams with magnetic effects, SIAM J. Control Optim., 52 (2014), 2371-2398.  doi: 10.1137/130918319.

[18]

J. E. Muñoz Rivera and H. D. Fernández Sare, Stability of Timoshenko systems with past history, J. Math. Anal. Appl., 339 (2008), 482-502.  doi: 10.1016/j.jmaa.2007.07.012.

[19]

J. E. Muñoz Rivera and R. Barreto, Decay rates of solutions to thermoviscoelastic plates with memory, IMA J. Appl. Math., 60 (1998), 263-283.  doi: 10.1093/imamat/60.3.263.

[20]

A. O. Özer and W. Horner, Uniform boundary observability of finite difference approximations of non-compactly coupled piezoelectric beam equations, Applicable Analysis, 0 (2021), 1-22.  doi: 10.1080/00036811.2021.1936508.

[21]

P. X. PamplonaJ. E. Muñoz Rivera and R. Quintanilla, On the decay of solutions for porous-elastic systems with history, J. Math. Anal. Appl., 379 (2011), 682-705.  doi: 10.1016/j.jmaa.2011.01.045.

[22]

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

[23]

A. J. A. RamosC. S. L. G. calves and S. S. Corrêa 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.

[24]

A. J. A. Ramos, A. Ö. Özer, M. M. Freitas, D. S. A. Júnior and J. D. Martins, Exponential stabilization of fully dynamic and electrostatic piezoelectric beams with delayed distributed damping feedback,, Zeitschrift für angewandte Mathematik und Physik, 72 (2021), 26. doi: 10.1007/s00033-020-01457-8.

[25]

J. Yang, An Introduction to the Theory of Piezoelectricity, Advances in Mechanics and Mathematics, Springer, 2005. doi: 10.1007/978-3-030-03137-4.

[26]

S. Zheng, Nonlinear Evolution Equations, Chapman and Hall/CRC, 2004. doi: 10.1201/9780203492222.

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