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doi: 10.3934/eect.2021015

Dynamics of piezoelectric beams with magnetic effects and delay term

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

3. 

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

* Corresponding author: Mirelson M. Freitas

Received  February 2020 Revised  December 2020 Published  April 2021

In this paper, we consider a piezoelectric beams system with magnetic effects and delay term. We study its long-time behavior through the associated dynamical system. We prove that the system is gradient and asymptotically smooth, which as a consequence, implies the existence of a global attractor, which is characterized as unstable manifold of the set of stationary solutions. We also get the quasi-stability of the system by establishing a stabilizability estimate and therefore obtain the finite fractal dimension of the global attractor.

Citation: Mirelson M. Freitas, Anderson J. A. Ramos, Manoel J. Dos Santos, Jamille L.L. Almeida. Dynamics of piezoelectric beams with magnetic effects and delay term. Evolution Equations & Control Theory, doi: 10.3934/eect.2021015
References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations. Studies in Mathematics and its Applications, North-Holland, Amsterdam, 1992.  Google Scholar

[2]

J. M. Balthazar, R. T. Rocha, R. M. F. L. Brasil, A. M. Tusset, B. R. de Pontes and M. Silveira, Mode Saturation, Mode Coupling and Energy Harvesting From Ambient Vibration in a Portal Frame Structure, Volume 8: 26th Conference on Mechanical Vibration and Noise, American Society of Mechanical Engineers, 2014. Google Scholar

[3]

A. R. A. Barbosa and T. F. Ma, Long-time dynamics of an extensible plate equation with thermal memory, Journal of Mathematical Analysis and Applications, 416 (2014), 143-165.  doi: 10.1016/j.jmaa.2014.02.042.  Google Scholar

[4]

C. BricaultC. PézeratM. ColletA. PyskirP. PerrardG. Matten and V. Romero-García, Multimodal reduction of acoustic radiation of thin plates by using a single piezoelectric patch with a negative capacitance shunt, Applied Acoustics, 145 (2019), 320-327.  doi: 10.1016/j.apacoust.2018.10.016.  Google Scholar

[5]

M. ChenH. ChenX. MaG. JinT. YeY. Zhang and Z. Liu, The isogeometric free vibration and transient response of functionally graded piezoelectric curved beam with elastic restraints, Results in Physics, 11 (2018), 712-725.  doi: 10.1016/j.rinp.2018.10.019.  Google Scholar

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I. Chueshov, Introduction to the Theory of Infinite-dimensional Dissipative Systems, AKTA, Kharkiv, 1999.  Google Scholar

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I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912.  Google Scholar

[8]

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.  doi: 10.1016/j.eml.2016.05.015.  Google Scholar

[9]

M. F. Daqaq, R. Masana, A. Erturk and D. D. Quinn, On the role of nonlinearities in vibratory energy harvesting: A critical review and discussion, Applied Mechanics Reviews, 66 (2014), 040801. doi: 10.1115/1.4026278.  Google Scholar

[10]

R. Datko, Representation of solutions and stability of linear differential-difference equations in a Banach space, Journal of Differential Equations, 29 (1978), 105-166.  doi: 10.1016/0022-0396(78)90043-8.  Google Scholar

[11]

R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM Journal on Control and Optimization, 24 (1986), 152–156. doi: 10.1137/0324007.  Google Scholar

[12]

R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM Journal on Control and Optimization, 26 (1988), 697–713. doi: 10.1137/0326040.  Google Scholar

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R. Datko, Two questions concerning the boundary control of certain elastic systems, Journal of Differential Equations, 92 (1991), 27-44.  doi: 10.1016/0022-0396(91)90062-E.  Google Scholar

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R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Numérice pour les Sciences et las Techniques, Masson, Paris, 1987.  Google Scholar

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L. H. FatoriM. A. J. Silva and V. Narciso, Quasi-stability property and attractors for a semilinear Timoshenko system, Discrete and Continuous Dynamical Systems, 36 (2016), 6117-6132.  doi: 10.3934/dcds.2016067.  Google Scholar

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Q. Feng, Y. Liang and G. Song, Real-time monitoring of early-age concrete strength using piezoceramic-based smart aggregates, Journal of Aerospace Engineering, 32 (2019), 04018115. doi: 10.1061/(ASCE)AS.1943-5525.0000939.  Google Scholar

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A. G. HaddowA. D. S. Barr and D. T. Mook, Theoretical and experimental study of modal interaction in a two-degree-of-freedom structure, Journal of Sound and Vibration, 97 (1984), 451-473.  doi: 10.1016/0022-460X(84)90272-4.  Google Scholar

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J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar

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

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

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H. Y. Hu and Z. H. Wang, Dynamics of Controlled Mechanical Systems with Delayed Feedback, Springer: Berlin/Heidelbeg, Germany, 2002. doi: 10.1007/978-3-662-05030-9.  Google Scholar

[22]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long Time Dynamics, Springer Monographs in Mathematics, New York, 2010. doi: 10.1007/978-0-387-87712-9.  Google Scholar

[23]

I. IliukJ. M. BalthazarA. M. TussetJ. R. C. PiqueiraB. R. PontesJ. L. P. Felix and Á. M. Bueno, Application of passive control to energy harvester efficiency using a nonideal portal frame structural support system, Journal of Intelligent Material Systems and Structures, 25 (2013), 417-429.  doi: 10.1177/1045389X13500570.  Google Scholar

[24]

I. IliukJ. M. BalthazarA.M. TussetJ. R. C. PiqueiraB. R. PontesJ. L. P. Felix and Á. M. Bueno, A non-ideal portal frame energy harvester controlled using a pendulum, The European Physical Journal Special Topics, 222 (2013), 1575-1586.  doi: 10.1140/epjst/e2013-01946-4.  Google Scholar

[25]

I. Iliuk, R. M. L. R. da Fonseca Brasil, J. M. Balthazar, A. M. Tusset, V. Piccirillo and J. R. C. Piqueira, Potential application in energy harvesting of intermodal energy exchange in a frame: FEM analysis, International Journal of Structural Stability and Dynamics, 14 (2014), 1440027. doi: 10.1142/S0219455414400276.  Google Scholar

[26]

N. Jalili, Piezoelectric-Based Vibration Control, Springer US, 2010. doi: 10.1007/978-1-4419-0070-8.  Google Scholar

[27]

T. Jiang, Y. Zhang, L. Wang, L. Zhang and G. Song, Monitoring fatigue damage of modular bridge expansion joints using piezoceramic transducers, Sensors, 18 (2018), 3973. doi: 10.3390/s18113973.  Google Scholar

[28]

Y. Y. Lim, Z. S. Tang and S. T. Smith, Piezoelectric-based monitoring of the curing of structural adhesives: a novel experimental study, Smart Materials and Structures, 28 (2018), 015016. doi: 10.1088/1361-665X/aaeea4.  Google Scholar

[29]

M. Ling, J. Cao, Q. Li and J. Zhuang, Design, pseudostatic model, and PVDF-based motion sensing of a piezo-actuated XYZ flexure manipulator, IEEE/ASME Transactions on Mechatronics, 23 (2018), 2837–2848. doi: 10.1109/TMECH.2018.2871371.  Google Scholar

[30]

T. F. Ma and R. N. Monteiro, Singular limit and long-time dynamics of bresse systems, SIAM Journal on Mathematical Analysis, 49 (2017), 2468-2495.  doi: 10.1137/15M1039894.  Google Scholar

[31]

P. Malatkar, Nonlinear Vibrations of Cantilever Beams and Plates, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 2003. Google Scholar

[32]

P. Malatkar and A. H. Nayfeh, Steady-State dynamics of a linear structure weakly coupled to an essentially nonlinear oscillator, Nonlinear Dynamics, 47 (2006), 167-179.  doi: 10.1007/s11071-006-9066-4.  Google Scholar

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K. Morris and A. O. Ozer, Strong Stabilization of Piezoelectric Beams with Magnetic Effects, 52nd IEEE Conference on Decision and Control, 2013. doi: 10.1109/CDC.2013.6760341.  Google Scholar

[36]

K. A. Morris and A. O. Ozer, 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

[37]

S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations, 21 (2008), 935-958.   Google Scholar

[38]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM Journal on Control and Optimization, 45 (2006), 1561–1585. doi: 10.1137/060648891.  Google Scholar

[39]

S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with delay, ESAIM: Control, Optimisation and Calculus of Variations, 16 (2009), 420–456. doi: 10.1051/cocv/2009007.  Google Scholar

[40]

A. O. Ozer, 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.  Google Scholar

[41]

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

[42]

A. Presas, Y. Luo and Z. Wang, D. Valentin and M. Egusquiza, A review of PZT patches applications in submerged systems, Sensors, 18 (2018), 2251. doi: 10.3390/s18072251.  Google Scholar

[43]

J. Przybylski and G. Gasiorski, Nonlinear vibrations of elastic beam with piezoelectric actuators, Journal of Sound and Vibration, 437 (2018), 150-165.  doi: 10.1016/j.jsv.2018.09.005.  Google Scholar

[44]

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.  Google Scholar

[45]

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 Für Angewandte Mathematik und Physik, 70 (2019), Paper No. 60, 14 pp. doi: 10.1007/s00033-019-1106-2.  Google Scholar

[46]

A. J. A. RamosA. O. OzerM. M. FreitasD. 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), 1-15.  doi: 10.1007/s00033-020-01457-8.  Google Scholar

[47]

G. Rega, Nonlinear interactions: Analytical, computational, and experimental methods, Wiley Series in Nonlinear Science Wiley, New York 2000. 760 pp, Meccanica, 35 (2000), 583-586.   Google Scholar

[48]

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

[49]

P. Shivashankar and S. B. Kandagal, Characterization of elastic and electromechanical nonlinearities in piezoceramic plate actuators from vibrations of a piezoelectric-beam, Mechanical Systems and Signal Processing, 116 (2019), 624-640.  doi: 10.1016/j.ymssp.2018.06.063.  Google Scholar

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show all references

References:
[1]

A. V. Babin and M. I. Vishik, Attractors of Evolution Equations. Studies in Mathematics and its Applications, North-Holland, Amsterdam, 1992.  Google Scholar

[2]

J. M. Balthazar, R. T. Rocha, R. M. F. L. Brasil, A. M. Tusset, B. R. de Pontes and M. Silveira, Mode Saturation, Mode Coupling and Energy Harvesting From Ambient Vibration in a Portal Frame Structure, Volume 8: 26th Conference on Mechanical Vibration and Noise, American Society of Mechanical Engineers, 2014. Google Scholar

[3]

A. R. A. Barbosa and T. F. Ma, Long-time dynamics of an extensible plate equation with thermal memory, Journal of Mathematical Analysis and Applications, 416 (2014), 143-165.  doi: 10.1016/j.jmaa.2014.02.042.  Google Scholar

[4]

C. BricaultC. PézeratM. ColletA. PyskirP. PerrardG. Matten and V. Romero-García, Multimodal reduction of acoustic radiation of thin plates by using a single piezoelectric patch with a negative capacitance shunt, Applied Acoustics, 145 (2019), 320-327.  doi: 10.1016/j.apacoust.2018.10.016.  Google Scholar

[5]

M. ChenH. ChenX. MaG. JinT. YeY. Zhang and Z. Liu, The isogeometric free vibration and transient response of functionally graded piezoelectric curved beam with elastic restraints, Results in Physics, 11 (2018), 712-725.  doi: 10.1016/j.rinp.2018.10.019.  Google Scholar

[6]

I. Chueshov, Introduction to the Theory of Infinite-dimensional Dissipative Systems, AKTA, Kharkiv, 1999.  Google Scholar

[7]

I. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., 195 (2008), viii+183 pp. doi: 10.1090/memo/0912.  Google Scholar

[8]

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.  doi: 10.1016/j.eml.2016.05.015.  Google Scholar

[9]

M. F. Daqaq, R. Masana, A. Erturk and D. D. Quinn, On the role of nonlinearities in vibratory energy harvesting: A critical review and discussion, Applied Mechanics Reviews, 66 (2014), 040801. doi: 10.1115/1.4026278.  Google Scholar

[10]

R. Datko, Representation of solutions and stability of linear differential-difference equations in a Banach space, Journal of Differential Equations, 29 (1978), 105-166.  doi: 10.1016/0022-0396(78)90043-8.  Google Scholar

[11]

R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM Journal on Control and Optimization, 24 (1986), 152–156. doi: 10.1137/0324007.  Google Scholar

[12]

R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM Journal on Control and Optimization, 26 (1988), 697–713. doi: 10.1137/0326040.  Google Scholar

[13]

R. Datko, Two questions concerning the boundary control of certain elastic systems, Journal of Differential Equations, 92 (1991), 27-44.  doi: 10.1016/0022-0396(91)90062-E.  Google Scholar

[14]

R. Dautray and J.-L. Lions, Analyse Mathématique et Calcul Numérice pour les Sciences et las Techniques, Masson, Paris, 1987.  Google Scholar

[15]

L. H. FatoriM. A. J. Silva and V. Narciso, Quasi-stability property and attractors for a semilinear Timoshenko system, Discrete and Continuous Dynamical Systems, 36 (2016), 6117-6132.  doi: 10.3934/dcds.2016067.  Google Scholar

[16]

Q. Feng, Y. Liang and G. Song, Real-time monitoring of early-age concrete strength using piezoceramic-based smart aggregates, Journal of Aerospace Engineering, 32 (2019), 04018115. doi: 10.1061/(ASCE)AS.1943-5525.0000939.  Google Scholar

[17]

A. G. HaddowA. D. S. Barr and D. T. Mook, Theoretical and experimental study of modal interaction in a two-degree-of-freedom structure, Journal of Sound and Vibration, 97 (1984), 451-473.  doi: 10.1016/0022-460X(84)90272-4.  Google Scholar

[18]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, 25. American Mathematical Society, Providence, RI, 1988. doi: 10.1090/surv/025.  Google Scholar

[19]

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

[20]

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

[21]

H. Y. Hu and Z. H. Wang, Dynamics of Controlled Mechanical Systems with Delayed Feedback, Springer: Berlin/Heidelbeg, Germany, 2002. doi: 10.1007/978-3-662-05030-9.  Google Scholar

[22]

I. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-posedness and Long Time Dynamics, Springer Monographs in Mathematics, New York, 2010. doi: 10.1007/978-0-387-87712-9.  Google Scholar

[23]

I. IliukJ. M. BalthazarA. M. TussetJ. R. C. PiqueiraB. R. PontesJ. L. P. Felix and Á. M. Bueno, Application of passive control to energy harvester efficiency using a nonideal portal frame structural support system, Journal of Intelligent Material Systems and Structures, 25 (2013), 417-429.  doi: 10.1177/1045389X13500570.  Google Scholar

[24]

I. IliukJ. M. BalthazarA.M. TussetJ. R. C. PiqueiraB. R. PontesJ. L. P. Felix and Á. M. Bueno, A non-ideal portal frame energy harvester controlled using a pendulum, The European Physical Journal Special Topics, 222 (2013), 1575-1586.  doi: 10.1140/epjst/e2013-01946-4.  Google Scholar

[25]

I. Iliuk, R. M. L. R. da Fonseca Brasil, J. M. Balthazar, A. M. Tusset, V. Piccirillo and J. R. C. Piqueira, Potential application in energy harvesting of intermodal energy exchange in a frame: FEM analysis, International Journal of Structural Stability and Dynamics, 14 (2014), 1440027. doi: 10.1142/S0219455414400276.  Google Scholar

[26]

N. Jalili, Piezoelectric-Based Vibration Control, Springer US, 2010. doi: 10.1007/978-1-4419-0070-8.  Google Scholar

[27]

T. Jiang, Y. Zhang, L. Wang, L. Zhang and G. Song, Monitoring fatigue damage of modular bridge expansion joints using piezoceramic transducers, Sensors, 18 (2018), 3973. doi: 10.3390/s18113973.  Google Scholar

[28]

Y. Y. Lim, Z. S. Tang and S. T. Smith, Piezoelectric-based monitoring of the curing of structural adhesives: a novel experimental study, Smart Materials and Structures, 28 (2018), 015016. doi: 10.1088/1361-665X/aaeea4.  Google Scholar

[29]

M. Ling, J. Cao, Q. Li and J. Zhuang, Design, pseudostatic model, and PVDF-based motion sensing of a piezo-actuated XYZ flexure manipulator, IEEE/ASME Transactions on Mechatronics, 23 (2018), 2837–2848. doi: 10.1109/TMECH.2018.2871371.  Google Scholar

[30]

T. F. Ma and R. N. Monteiro, Singular limit and long-time dynamics of bresse systems, SIAM Journal on Mathematical Analysis, 49 (2017), 2468-2495.  doi: 10.1137/15M1039894.  Google Scholar

[31]

P. Malatkar, Nonlinear Vibrations of Cantilever Beams and Plates, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, 2003. Google Scholar

[32]

P. Malatkar and A. H. Nayfeh, Steady-State dynamics of a linear structure weakly coupled to an essentially nonlinear oscillator, Nonlinear Dynamics, 47 (2006), 167-179.  doi: 10.1007/s11071-006-9066-4.  Google Scholar

[33] J. C. Maxwell, A Dynamical Theory of the Electromagnetic Field, Scottish Academic Press, Edinburgh, 1982.   Google Scholar
[34] J. C. Maxwell, A Treatise on Electricity and Magnetism, The Clarendon Press, Oxford University Press, New York, 1998.   Google Scholar
[35]

K. Morris and A. O. Ozer, Strong Stabilization of Piezoelectric Beams with Magnetic Effects, 52nd IEEE Conference on Decision and Control, 2013. doi: 10.1109/CDC.2013.6760341.  Google Scholar

[36]

K. A. Morris and A. O. Ozer, 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

[37]

S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations, 21 (2008), 935-958.   Google Scholar

[38]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM Journal on Control and Optimization, 45 (2006), 1561–1585. doi: 10.1137/060648891.  Google Scholar

[39]

S. Nicaise and J. Valein, Stabilization of second order evolution equations with unbounded feedback with delay, ESAIM: Control, Optimisation and Calculus of Variations, 16 (2009), 420–456. doi: 10.1051/cocv/2009007.  Google Scholar

[40]

A. O. Ozer, 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.  Google Scholar

[41]

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

[42]

A. Presas, Y. Luo and Z. Wang, D. Valentin and M. Egusquiza, A review of PZT patches applications in submerged systems, Sensors, 18 (2018), 2251. doi: 10.3390/s18072251.  Google Scholar

[43]

J. Przybylski and G. Gasiorski, Nonlinear vibrations of elastic beam with piezoelectric actuators, Journal of Sound and Vibration, 437 (2018), 150-165.  doi: 10.1016/j.jsv.2018.09.005.  Google Scholar

[44]

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.  Google Scholar

[45]

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 Für Angewandte Mathematik und Physik, 70 (2019), Paper No. 60, 14 pp. doi: 10.1007/s00033-019-1106-2.  Google Scholar

[46]

A. J. A. RamosA. O. OzerM. M. FreitasD. 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), 1-15.  doi: 10.1007/s00033-020-01457-8.  Google Scholar

[47]

G. Rega, Nonlinear interactions: Analytical, computational, and experimental methods, Wiley Series in Nonlinear Science Wiley, New York 2000. 760 pp, Meccanica, 35 (2000), 583-586.   Google Scholar

[48]

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

[49]

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