Stability of multi-dimensional nonlinear piezoelectric beam with viscoelastic infinite memory

Zhang, H. E; Xu, Genqi; Han, Zhong‐Jie

doi:10.1007/s00033-022-01790-0

Cited by 5 publications

(5 citation statements)

References 25 publications

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“…Much attention has been devoted to the dynamical behavior of the system (1.4) with various controllers, see ref. [10][11][12][13][14] and references therein. Zhang et al [14] proved the strongly coupled system with a friction-type infinite memory term only in one equation can be indirectly stabilized polynomially based on frequency domain method and verified the optimality of the decay rete by detailed spectral analysis.…”

Section: Introductionmentioning

confidence: 99%

“…[10][11][12][13][14] and references therein. Zhang et al [14] proved the strongly coupled system with a friction-type infinite memory term only in one equation can be indirectly stabilized polynomially based on frequency domain method and verified the optimality of the decay rete by detailed spectral analysis. Let us recall some works on piezoelectric beam systems with thermal effects [15][16][17].…”

Section: Introductionmentioning

confidence: 99%

“…Much attention has been devoted to the dynamical behavior of the system () with various controllers, see ref. [10–14] and references therein. Zhang et al.…”

Section: Introductionmentioning

confidence: 99%

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Exponential stability of piezoelectric beams with delay and second sound

Lv,

Wang,

Yang

2024

Z Angew Math Mech

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In this paper, we consider a fully‐dynamic piezoelectric beam model subjected to a magnetic effect, where the heat flux is given by Cattaneo's law. It is well known that, in the absence of delay, the dissipation produced by the heat conduction is strong enough to make the piezoelectric beams exponentially stable. However, time delay effects may destroy this behavior. Here, we show the existence and uniqueness of solutions through the semigroup theory. Furthermore, under a smallness condition on the delay, we prove an exponential stability result via establishing the appropriate Lyapunov functional. Finally, we numerically illustrate the asymptotic behavior of the solution.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Exponential stability of piezoelectric beams with delay and second sound

Lv,

Wang,

Yang

2024

Z Angew Math Mech

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“…It is worth mentioning that imposing linear internal friction damping to any equation within the system in Ramos et al [21] also obtained the same stability by constructing Lyapunov functionals. Recently, Zhang et al [22] considered a nonlinear fully dynamic piezoelectric beam system with viscoelastic infinite memory damping, which replaced frictional damping:…”

Section: Introductionmentioning

confidence: 99%

({, \begin{matrix} left left \\ array ρ v_{t t} - α △ v + γ β △ p + \int_{0}^{\infty} g (s) △ v (x, t - s) d s = f_{1} (v, p) & array in Ω \times R^{+}, \\ array μ p_{t t} - β △ p + γ β △ v = f_{2} (v, p) & array in Ω \times R^{+}, \\ array v (x, t) = p (x, t) = 0 & array in Γ_{0} \times R^{+}, \\ array α \frac{\partial v}{\partial n} - γ β \frac{\partial p}{\partial n} = β \frac{\partial p}{\partial n} - γ β \frac{\partial v}{\partial n} = 0 & array in Γ_{1} \times R^{+}, \\ array v (x, 0) = v_{0} (x), v_{t} (x, 0) = v_{1} (x), p (x, 0) = p_{0} \end{matrix})

…”

Section: Introductionmentioning

confidence: 99%

New general decay results for a fully dynamic piezoelectric beam system with fractional delays under memory boundary controls

Hao,

Yang

2023

Math Methods in App Sciences

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In this paper, we investigate the general stability results of a fully dynamic piezoelectric beam system with internal fractional delays under memory boundary controls. Firstly, by introducing two new equations and using two Volterra's inverse operators to deal with fractional delay terms and boundary memory terms, respectively, a new equivalent system is obtained. Secondly, by combining Faedo–Galerkin approximations with the compactness argument, we prove the local existence and uniqueness of weak solution. Finally, under a broader class of kernel functions, by constructing Lyapunov functionals and new equations and applying some technical lemmas, we establish the optimal and explicit energy decay results for two cases, namely, initial values and without imposing initial values equal to 0, that is, . It is worth pointing out that we focus more on the analysis of , because the stability results of are special cases of . This is the first time to analyze the stability of strong coupling problem by combining internal fractional delay and boundary viscoelastic damping. The obtained general decay results are not necessarily exponential or polynomial, which generalize the relevant results in the existing literature.

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