# American Institute of Mathematical Sciences

doi: 10.3934/dcdsb.2021168
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## Equivalence between exponential stabilization and observability inequality for magnetic effected piezoelectric beams with time-varying delay and time-dependent weights

 1 School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, China 2 Department of Mathematics, Federal University of Bahia, Salvador, BA, Brazil 4 Faculty of Exact Sciences and Technology, Federal University of Pará, Manoel de Abreu Street, s/n, 68440-000, Abaetetuba, Pará, Brazil 5 Department of Mathematics, Federal University of São João del-Rei, São João del-Rei, 36307-352, Minas Gerais, Brazil

* Corresponding authors: wjliu@nuist.edu.cn (W. Liu), jeremias@ufpa.br (M. Santos)

† These authors contributed equally to this work

Received  March 2021 Revised  April 2021 Early access June 2021

This paper is concerned with system of magnetic effected piezoelectric beams with interior time-varying delay and time-dependent weights, in which the beam is clamped at the two side points subject to a single distributed state feedback controller with a time-varying delay. Under appropriate assumptions on the time-varying delay term and time-dependent weights, we obtain exponential stability estimates by using the multiplicative technique, and prove the equivalence between stabilization and observability.

Citation: 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, doi: 10.3934/dcdsb.2021168
##### References:
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Cady, Piezoelectricity: An Introduction to the Theory and Applications of Electrical Phenomena in Crystals, Dover Publications, New York, 1964. Google Scholar [6] M. Chen and W. Liu and W. Zhou, Existence and general stabilization of the Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms, Advances in Nonlinear Analysis, 7 (2018), 547-569.  doi: 10.1515/anona-2016-0085.  Google Scholar [7] D. Damjanovic, Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics, Reports on Progress in Physics, 61 (1999), 1267-1324.  doi: 10.1088/0034-4885/61/9/002.  Google Scholar [8] G. Davi and A. Milazzo, Multidomain boundary integral formulation for piezoelectric materials fracture mechanics, International Journal of Solids and Structures, 38 (2001), 7065-7078.  doi: 10.1016/S0020-7683(00)00416-9.  Google Scholar [9] J. M. Dietl, A. M. Wickenheiser and E. Garcia, A Timoshenko beam model for cantilevered piezoelectric energy harvesters, Smart Materials and Structures, 19 (2010), 547-569.  doi: 10.1088/0964-1726/19/5/055018.  Google Scholar [10] B. Feng and X. G. Yang, Long-time dynamics for a nonlinear Timoshenko system with delay, Applicable Analysis, 96 (2017), 606-625.  doi: 10.1080/00036811.2016.1148139.  Google Scholar [11] M. M. Freitas, A. J. A. Ramos, A. Özer and D. S. Almeida Júnior, Long-time dynamics for a fractional piezoelectric system with magnetic effects and Fourier's law, Journal of Differential Equations, 280 (2021), 891-927.  doi: 10.1016/j.jde.2021.01.030.  Google Scholar [12] C. Galassi, M. Dinescu, K. Uchino and M. Sayer, Piezoelectric materials: Advances in science, technology and applications, Nato Science Partnership Subseries 3, Springer, Berlin, 2000. doi: 10.1007/978-94-011-4094-2.  Google Scholar [13] A. 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Chen, Long-time behavior for a thermoelastic microbeam problem with time delay and the Coleman-Gurtin thermal law, Acta Mathematica Scientia, 41 (2021), 609-632.  doi: 10.1007/s10473-021-0220-3.  Google Scholar [19] W. Liu and M. Chen, Well-posedness and exponential decay for a porous thermoelastic system with second sound and a time-varying delay term in the internal feedback, Continuum Mechanics and Thermodynamics, 29 (2017), 731-746.  doi: 10.1007/s00161-017-0556-z.  Google Scholar [20] W. Liu and H. Zhuang, Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback, Discrete and Continuous Dynamical Systems-Series B, 26 (2021), 907-942.  doi: 10.3934/dcdsb.2020147.  Google Scholar [21] S. A. Messaoudi and W. Al-Khulaifi, General and optimal decay for a viscoelastic equation with boundary feedback, Topological Methods in Nonlinear Analysis, 51 (2018), 413-427.  doi: 10.12775/tmna.2017.066.  Google Scholar [22] S. A. Messaoudi, A. Fareh and N. Doudi, Well posedness and exponential stability in a wave equation with a strong damping and a strong delay, Journal of Mathematical Physics, 57 (2016), 13pp. doi: 10.1063/1.4966551.  Google Scholar [23] K. A. Morris and A. Özer, Strong stabilization of piezoelectric beams with magnetic effects, in 52nd IEEE Conference on Decision and Control, 2013, 3014–3019. doi: 10.1109/CDC.2013.6760341.  Google Scholar [24] K. A. Morris and A. Özer, Modeling and stabilizability of voltage-actuated piezoelectric beams with magnetic effects, SIAM Journal on Control and Optimization, 52 (2014), 2371-2398.  doi: 10.1137/130918319.  Google Scholar [25] S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electronic Journal of Differential Equations, 2011 (2011), 20pp.  Google Scholar [26] S. Nicaise and J. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete and Continuous Dynamical Systems-Series S, 2 (2009), 559-581.  doi: 10.3934/dcdss.2009.2.559.  Google Scholar [27] C. Nonato, M. J. {Dos Santos}, C. Raposo, Dynamics of Timoshenko system with time-varying weight and time-varying delay, Discrete and Continuous Dynamical Systems-Series B, in press. doi: 10.3934/dcdsb.2021053.  Google Scholar [28] C. Nonato, C. Raposo and B. Feng, Exponential stability for a thermoelastic laminated beam with nonlinear weights and time-varying delay, Asymptotic Analysis, in press. doi: 10.3233/ASY-201668.  Google Scholar [29] R. L. Oliveira and H. P. Oquendo, Stability and instability results for coupled waves with delay term, Journal of Mathematical Physics, 61 (2020), 13pp. doi: 10.1063/1.5144987.  Google Scholar [30] A. Pazy, Semigroups of Linear Operators 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 [31] G. Poulin-Vittrant, C. Oshman, C. Opoku, A. S. Dahiya, N. Camara, D. Alquier, Hu e, L. -P. T. H and M. Lethiecq, Fabrication and characterization of ZnO nanowire-based piezoelectric nanogenerators for low frequency mechanical energy harvesting, Physics Procedia, 70 (2015), 909-913.  doi: 10.1016/j.phpro.2015.08.188.  Google Scholar [32] 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, Zeitschrift Für Angewandte Mathematik Und Physik, 70 (2019), 14pp. doi: 10.1007/s00033-019-1106-2.  Google Scholar [33] A. J. A. Ramos, A. Özer, M. M. Freitas, D. S. Almeida Jr. 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), 26pp. doi: 10.1007/s00033-020-01457-8.  Google Scholar [34] A. J. A. Ramos, C. S. L. Gon\c{c}alves and S. S. Corrêa 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.  Google Scholar [35] C. Raposo, J. A. D. Chuquipoma, J. A. J. Avila and M. L. Santos, Exponential decay and numerical solution for a Timoshenko system with delay term in the internal feedback, International Journal of Analysis and Applications, 3 (2013), 1-13.   Google Scholar [36] Z. Sabbagh, A. Khemmoudj, M. Ferhat and M. Abdelli, Existence of global solutions and decay estimates for a viscoelastic Petrovsky equation with internal distributed delay, Rendiconti del Circolo Matematico di Palermo Series 2, 68 (2019), 477-498.  doi: 10.1007/s12215-018-0373-7.  Google Scholar [37] B. Said-Houari and Y. Laskri, A stability result of a Timoshenko system with a delay term in the internal feedback, Applied Mathematics and Computation, 217 (2010), 2857-2869.  doi: 10.1016/j.amc.2010.08.021.  Google Scholar [38] P. Wang and J. Hao, Asymptotic stability of memory-type Euler-Bernoulli plate with variable coefficients and time delay, Journal of Systems Science and Complexity, 32 (2019), 1375-1392.  doi: 10.1007/s11424-018-7370-y.  Google Scholar [39] H. J. Xiang and Z. F. Shi, Static analysis for multi-layered piezoelectric cantilevers, International Journal of Solids and Structures, 45 (2008), 113-128.  doi: 10.1016/j.ijsolstr.2007.07.022.  Google Scholar [40] J. Yang, A Review of a few topics in piezoelectricity, Applied Mechanics Reviewes, 59 (2006), 335-345.  doi: 10.1115/1.2345378.  Google Scholar [41] Y. Zheng, W. Liu and Y. Liu, Equivalence between internal observability and exponential stabilization for suspension bridge problem, Ricerche di Matematica, in press. doi: 10.1007/s11587-021-00566-4.  Google Scholar [42] F. Zhu, M. B. Ward, J. F. Li and S. J. Milne, Core-shell grain structures and ferroelectric properties of Na$_{0.5}$K$_{0.5}$NbO$_3$-LiTaO$_3$-BiScO$_3$ piezoelectric ceramics, Data in Brief, 4 (2015), 34-39.  doi: 10.1016/j.dib.2015.04.002.  Google Scholar

show all references

##### References:
 [1] H. Y. S. Al-Zahrani, J. Pal, M. A. Migliorato, G. Tse and D. Yu, Piezoelectric field enhancement in III-V core-shell nanowires, Nano Energy, 14 (2015), 382-391.  doi: 10.1016/j.nanoen.2014.11.046.  Google Scholar [2] V. Barros and C. Nonato and C. Raposo, Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights, Electronic Research Archive, 28 (2020), 205-220.  doi: 10.3934/era.2020014.  Google Scholar [3] A. Benaissa, A. Benguessoum and S. A. Messaoudi, Energy decay of solutions for a wave equation with a constant weak delay and a weak internal feedback, Electronic Journal of Qualitative Theory of Differential Equations, 2014 (2014), 13 pp. doi: 10.14232/ejqtde.2014.1.11.  Google Scholar [4] A. Blanguernon and F. Léné and M. Bernadou, Active control of a beam using a piezoceramic element, Smart Materials and Structures, 8 (1999), 116-124.  doi: 10.1088/0964-1726/8/1/013.  Google Scholar [5] W. G. Cady, Piezoelectricity: An Introduction to the Theory and Applications of Electrical Phenomena in Crystals, Dover Publications, New York, 1964. Google Scholar [6] M. Chen and W. Liu and W. Zhou, Existence and general stabilization of the Timoshenko system of thermo-viscoelasticity of type III with frictional damping and delay terms, Advances in Nonlinear Analysis, 7 (2018), 547-569.  doi: 10.1515/anona-2016-0085.  Google Scholar [7] D. Damjanovic, Ferroelectric, dielectric and piezoelectric properties of ferroelectric thin films and ceramics, Reports on Progress in Physics, 61 (1999), 1267-1324.  doi: 10.1088/0034-4885/61/9/002.  Google Scholar [8] G. Davi and A. Milazzo, Multidomain boundary integral formulation for piezoelectric materials fracture mechanics, International Journal of Solids and Structures, 38 (2001), 7065-7078.  doi: 10.1016/S0020-7683(00)00416-9.  Google Scholar [9] J. M. Dietl, A. M. Wickenheiser and E. Garcia, A Timoshenko beam model for cantilevered piezoelectric energy harvesters, Smart Materials and Structures, 19 (2010), 547-569.  doi: 10.1088/0964-1726/19/5/055018.  Google Scholar [10] B. Feng and X. G. Yang, Long-time dynamics for a nonlinear Timoshenko system with delay, Applicable Analysis, 96 (2017), 606-625.  doi: 10.1080/00036811.2016.1148139.  Google Scholar [11] M. M. Freitas, A. J. A. Ramos, A. Özer and D. S. Almeida Júnior, Long-time dynamics for a fractional piezoelectric system with magnetic effects and Fourier's law, Journal of Differential Equations, 280 (2021), 891-927.  doi: 10.1016/j.jde.2021.01.030.  Google Scholar [12] C. Galassi, M. Dinescu, K. Uchino and M. Sayer, Piezoelectric materials: Advances in science, technology and applications, Nato Science Partnership Subseries 3, Springer, Berlin, 2000. doi: 10.1007/978-94-011-4094-2.  Google Scholar [13] A. Haraux, Two remarks on hyperbolic dissipative problems, Research Notes in Mathematics Pitman, 122 (1985), 161-179.   Google Scholar [14] H. Kawai, The Piezoelectricity of poly (vinylidene Fluoride), Japanese Journal of Applied Physics, 8 (1969), 975-976.  doi: 10.1143/JJAP.8.975.  Google Scholar [15] T. Kato, Linear and Quasi-Linear Equations of Evolution of Hyperbolic Type, Summer Sch., 72, Springer, Heidelberg, 2011,125–191. doi: 10.1007/978-3-642-11105-1_4.  Google Scholar [16] M. Kirane and B. Said-Houari and M. N. Anwar, Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks, Communications on Pure and Applied Analysis, 10 (2011), 667-686.  doi: 10.3934/cpaa.2011.10.667.  Google Scholar [17] G. Liu and L. Diao, Energy decay of the solution for a weak viscoelastic equation with a time-varying delay, Acta Applicandae Mathematicae, 155 (2018), 9-19.  doi: 10.1007/s10440-017-0142-1.  Google Scholar [18] W. Liu and D. Chen and Z. Chen, Long-time behavior for a thermoelastic microbeam problem with time delay and the Coleman-Gurtin thermal law, Acta Mathematica Scientia, 41 (2021), 609-632.  doi: 10.1007/s10473-021-0220-3.  Google Scholar [19] W. Liu and M. Chen, Well-posedness and exponential decay for a porous thermoelastic system with second sound and a time-varying delay term in the internal feedback, Continuum Mechanics and Thermodynamics, 29 (2017), 731-746.  doi: 10.1007/s00161-017-0556-z.  Google Scholar [20] W. Liu and H. Zhuang, Global attractor for a suspension bridge problem with a nonlinear delay term in the internal feedback, Discrete and Continuous Dynamical Systems-Series B, 26 (2021), 907-942.  doi: 10.3934/dcdsb.2020147.  Google Scholar [21] S. A. Messaoudi and W. Al-Khulaifi, General and optimal decay for a viscoelastic equation with boundary feedback, Topological Methods in Nonlinear Analysis, 51 (2018), 413-427.  doi: 10.12775/tmna.2017.066.  Google Scholar [22] S. A. Messaoudi, A. Fareh and N. Doudi, Well posedness and exponential stability in a wave equation with a strong damping and a strong delay, Journal of Mathematical Physics, 57 (2016), 13pp. doi: 10.1063/1.4966551.  Google Scholar [23] K. A. Morris and A. Özer, Strong stabilization of piezoelectric beams with magnetic effects, in 52nd IEEE Conference on Decision and Control, 2013, 3014–3019. doi: 10.1109/CDC.2013.6760341.  Google Scholar [24] K. A. Morris and A. Özer, Modeling and stabilizability of voltage-actuated piezoelectric beams with magnetic effects, SIAM Journal on Control and Optimization, 52 (2014), 2371-2398.  doi: 10.1137/130918319.  Google Scholar [25] S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electronic Journal of Differential Equations, 2011 (2011), 20pp.  Google Scholar [26] S. Nicaise and J. Valein and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete and Continuous Dynamical Systems-Series S, 2 (2009), 559-581.  doi: 10.3934/dcdss.2009.2.559.  Google Scholar [27] C. Nonato, M. J. {Dos Santos}, C. Raposo, Dynamics of Timoshenko system with time-varying weight and time-varying delay, Discrete and Continuous Dynamical Systems-Series B, in press. doi: 10.3934/dcdsb.2021053.  Google Scholar [28] C. Nonato, C. Raposo and B. Feng, Exponential stability for a thermoelastic laminated beam with nonlinear weights and time-varying delay, Asymptotic Analysis, in press. doi: 10.3233/ASY-201668.  Google Scholar [29] R. L. Oliveira and H. P. Oquendo, Stability and instability results for coupled waves with delay term, Journal of Mathematical Physics, 61 (2020), 13pp. doi: 10.1063/1.5144987.  Google Scholar [30] A. Pazy, Semigroups of Linear Operators 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 [31] G. Poulin-Vittrant, C. Oshman, C. Opoku, A. S. Dahiya, N. Camara, D. Alquier, Hu e, L. -P. T. H and M. Lethiecq, Fabrication and characterization of ZnO nanowire-based piezoelectric nanogenerators for low frequency mechanical energy harvesting, Physics Procedia, 70 (2015), 909-913.  doi: 10.1016/j.phpro.2015.08.188.  Google Scholar [32] 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, Zeitschrift Für Angewandte Mathematik Und Physik, 70 (2019), 14pp. doi: 10.1007/s00033-019-1106-2.  Google Scholar [33] A. J. A. Ramos, A. Özer, M. M. Freitas, D. S. Almeida Jr. 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), 26pp. doi: 10.1007/s00033-020-01457-8.  Google Scholar [34] A. J. A. Ramos, C. S. L. Gon\c{c}alves and S. S. Corrêa 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.  Google Scholar [35] C. Raposo, J. A. D. Chuquipoma, J. A. J. Avila and M. L. Santos, Exponential decay and numerical solution for a Timoshenko system with delay term in the internal feedback, International Journal of Analysis and Applications, 3 (2013), 1-13.   Google Scholar [36] Z. Sabbagh, A. Khemmoudj, M. Ferhat and M. 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