Exponential stabilization of fully dynamic and electrostatic piezoelectric beams with delayed distributed damping feedback

Abstract

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.

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Funding

A. J. A. Ramos thanks the CNPq for financial support through the projects “Asymptotic stabilization and numerical treatment for carbon nanotubes” (CNPq Grant 310729/2019-0).

A. Ö. Özer gratefully acknowledges the financial support of the KY NSF grant (#1514712-1) and the RCAP (#20-8038) grant of Western Kentucky University for this research.

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Correspondence to A. Ö. Özer.

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Ramos, A.J.A., Özer, A.Ö., Freitas, M.M. et al. Exponential stabilization of fully dynamic and electrostatic piezoelectric beams with delayed distributed damping feedback. Z. Angew. Math. Phys. 72, 26 (2021). https://doi.org/10.1007/s00033-020-01457-8

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Keywords

  • Fully dynamic
  • Electrostatic
  • Maxwell’s equations
  • Piezoelectric beam
  • Exponential stability
  • Time delayed control
  • Distributed feedback control

Mathematics Subject Classification

  • 35Q60
  • 35Q93
  • 74F15
  • 35Q74
  • 93B52