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Exponential stabilization of fully dynamic and electrostatic piezoelectric beams with delayed distributed damping feedback
Zeitschrift für angewandte Mathematik und Physik ( IF 2 ) Pub Date : 2021-01-11 , DOI: 10.1007/s00033-020-01457-8
A. J. A. Ramos , A. Ö. Özer , M. M. Freitas , D. S. Almeida Júnior , J. D. Martins

Fully dynamic system of equations for a single piezoelectric beam strongly couples the mechanical (longitudinal) vibrations with the total charge distribution across the beam. Unlike the electrostatic (or quasi-static) assumption of Maxwell’s equations, the hyperbolic-type charge equations have been recently shown to affect the stabilizability of the high-frequency vibrational modes if one considers only a single boundary controller; voltage at the electrodes of the beam. In this paper, we consider viscously damped beam equations and a single distributed state feedback controller with a delay. The effect of the delay in the feedback is investigated for the overall exponential stabilizability dynamics of the piezoelectric beam equations. First, the equations of motion in the state-space formulation are shown to be well-posed by the semigroup theory. Next, an energy approach by the Lyapunov theory is utilized to prove that the exponential stability is retained only if the coefficient of the delayed feedback is strictly less than the coefficient of the state feedback. Finally, the results are compared to the ones of the electrostatic case.



中文翻译:

具有延迟分布阻尼反馈的全动态和静电压电梁的指数稳定

单个压电梁的方程组的全动态系统将机械(纵向)振动与整个梁上的总电荷分布紧密耦合。与麦克斯韦方程的静电(或准静态)假设不同,最近的研究表明,如果仅考虑单个边界控制器,则双曲线型电荷方程会影响高频振动模式的稳定性。束电极上的电压。在本文中,我们考虑了粘性阻尼梁方程和具有延迟的单个分布状态反馈控制器。对于压电梁方程的整体指数稳定性,研究了反馈中延迟的影响。首先,状态空间公式中的运动方程被半群理论证明是正确的。接下来,利用李雅普诺夫理论的能量方法证明只有在延迟反馈的系数严格小于状态反馈的系数的情况下,才能保持指数稳定性。最后,将结果与静电情况下的结果进行比较。

更新日期:2021-01-12
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