The piezoelectric beam model is investigated with memory, magnetic effect, and time-varying delay, which describes the conservation law of piezoelectric viscoelastic beam in a magnetic field. Based on the global well-posedness demonstrated via Kato’s variable norm technique, the exponential stability of the considered piezoelectric beam system can be achieved by using some delicate energy estimates on transport terms for delay and memory combined with multiplier techniques.
AlabauF.CannarsaP.SforzaD. (2008). Decay estimates for second order evolution equations with memory. Journal of Functional Analysis, 254(5), 1342–1372.
2.
BrunnerA. J.BarbezatM.HuberC.FlüelerP. H. (2005). The potential of active fiber composites made from piezoelectric fibers for actuating and sensing applications in structural health monitoring. Materials and Structures, 38, 561–567.
3.
CavalcantiM. M.FatoriL. H.MaT. F. (2016). Attractors for wave equations with degenerate memory. Journal of Differential Equations, 260(1), 56–83.
4.
CheeC. Y. K.TongL.StevenG. P. (1998). A review on the modelling of piezoelectric sensors and actuators incorporated in intelligent structures. Journal of Intelligent Material Systems and Structures, 9(1), 3–19.
5.
ChepyzhovV. V.PataV. (2006). Some remarks on stability of semigroups arising from linear viscoelasticity. Asymptotic Analysis, 46(3–4), 251–273.
6.
DafermosC. M. (1970). Asymptotic stability in viscoelasticity. Archive for Rational Mechanics and Analysis, 37, 297–308.
7.
DagdevirenC. YangB. D. SuY. TranP. L. JoeP. AndersonE. XiaJ. DoraiswamyV. DehdashtiB. FengX. LuB. PostonR. KhalpeyZ. GhaffariR. HuangY. SlepianMJ.RogersJ. A. (2014). Conformal piezoelectric energy harvesting and storage from motions of the heart, lung, and diaphragm. Proceedings of the National Academy of Sciences of the United States of America, 111(5), 1927–1932.
8.
ErturkA.InmanD. J. (2008). A distributed parameter electromechanical model for cantilevered piezoelectric energy harvesters. Journal of Vibration and Acoustics, 130(4), 1257–1261.
9.
GrasselliM.PataV. (2002). Uniform attractors of nonautonomous dynamical systems with memory. Progress in Nonlinear Differential Equations and Their Applications, 50, 155–178.
10.
HuM.YangX.YuanJ. (2024). Stability and dynamics for Lamé system with degenerate memory and time-varying delay. Applied Mathematics and Optimization, 89(14), 34.
11.
KatoT. (1985). Abstract differential equations and nonlinear mixed problems. Publications of the scuola normale superiore. Edizioni della Normale Pisa.
12.
KatoT. (2011). Linear and quasi-linear equations of evolution of hyperbolic type. C.I.M.E. Summer Schools, 72, 125–191.
13.
KiraneM.Said-HouariB.AnwarM. N. (2011). Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Communications on Pure and Applied Analysis, 10(2), 667–686.
14.
LiY.HanZ.XuG. (2023). Stabilization of nonlinear non-uniform piezoelectric beam with time-varying delay in distributed control input. Journal of Differential Equations, 377, 38–70.
15.
LiuZ.ZhengS. (1996). On the exponential stability of linear viscoelasticity and thermoviscoelasticity. Quarterly of Applied Mathematics, 54(1), 21–31.
16.
MorrisK.ÖzerA. Ö. (2013). Strong stabilization of piezoelectric beams with magnetic effects. In 52nd IEEE conference on decision and control (pp. 3014–3019).
17.
MorrisK.ÖzerA. Ö. (2014). Modeling and stabilizability of voltage-actuated piezoelectric beams with magnetic effects. SIAM Journal on Control and Optimization, 52(4), 2371–2398.
18.
NicaiseS.PignottiC. (2006). 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(5), 1561–1585.
19.
ÖzerA. Ö. (2015). Further stabilization and exact observability results for voltage-actuated piezoelectric beams with magnetic effects. Mathematics of Control, Signals, and Systems, 27(2), 219–244.
20.
PataV. (2006). Exponential stability in linear viscoelasticity. Quarterly of Applied Mathematics, 64(3), 499–513.
21.
PataV.ZucchiA. (2001). Attractors for a damped hyperbolic equation with linear memory. Advances in Mathematical Sciences and Applications, 11(2), 505–529.
22.
PazyA. (1983). Semigroups of linear operators and applications to partial differential equations. Springer.
23.
PengJ.XiangM.LiL.SunH.WangX. (2019a). Time-delayed feedback control of piezoelectric elastic beams under superharmonic and subharmonic excitations. Applied Sciences, 9(8), 1557.
24.
PengJ.ZhangG.XiangM.SunH.WangX.XieX. (2019b). Vibration control for the nonlinear resonant response of a piezoelectric elastic beam via time-delayed feedback. Smart Materials and Structures, 28(9), 095010.
25.
PohlD. W. (1987). Dynamic piezoelectric translation devices. The Review of Scientific Instruments, 58(1), 54–57.
26.
RamosA. J. A.FreitasM. M.Almeida JúniorD. S.JesusS. S.MouraT. R. S. (2019). Equivalence between exponential stabilization and boundary observability for piezoelectric beams with magnetic effect. Zeitschrift für Angewandte Mathematik und Physik, 70(2), 14.
27.
RamosA. J. A.GoncalvesC. S. L.Corrêa NetoS. S. (2018). Exponential stability and numerical treatment for piezoelectric beams with magnetic effect. ESAIM: Mathematical Modelling and Numerical Analysis, 52(1), 255–274.
28.
RamosA. J. A.ÖzerA. Ö.FreitasM. M.Almeida JúniorD. S.MartinsJ. D. (2021). Exponential stabilization of fully dynamic and electrostatic piezoelectric beams with delayed distributed damping feedback. Zeitschrift für Angewandte Mathematik und Physik, 72(1), 15.
29.
TrefethenL. N. (2000). Spectral methods in MATLAB. SIAM.
30.
VinogradovA. M.SchmidtV. H.TuthillG. F.BohannanG. W. (2004). Damping and electromechanical energy losses in the piezoelectric polymer PVDF. Mechanics of Materials, 36(10), 1007–1016.
31.
YangJ. (2005). An introduction to the theory of piezoelectricity. Springer.
32.
YangX.ZhangJ.WangS. (2020). Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay. Discrete and Continuous Dynamical Systems, 40(3), 1493–1515.
33.
YehT. J.FengH. R.WenL. S. (2008). An integrated physical model that characterizes creep and hysteresis in piezoelectric actuators. Simulation Modelling Practice and Theory, 16(1), 93–110.
