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<p>In this paper, we investigate the stability of a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(1-D\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>1</mn> <mo>-</mo> <mi>D</mi> </mrow> </math></EquationSource> </InlineEquation> piezoelectric bar where the electrodes are connected through a resistor <i>R</i> carrying current <i>I</i>. Two cases are treated: With variable piezoelectric coefficients and an internal viscous damping acting on the bar, the total energy decays exponentially. With constant coefficients and no internal viscous damping, the system is polynomially stable, with an optimal decay rate of type <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(t^{-1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>t</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation>. Both results are obtained by combining the frequency domain approach with the multiplier method. The case with variable coefficients and no internal viscous damping remains an open question.</p>

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Piezoelectric beam with electrical boundary coupling: well-posedness, spectral analysis and energy decay

  • 期刊论文

  • 发表日期：

    2026年7月27日

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  • Mohammad Akil,
  • Serge Nicaise,
  • Hussein Saleh

摘要

In this paper, we investigate the stability of a \(1-D\) 1 - D piezoelectric bar where the electrodes are connected through a resistor R carrying current I. Two cases are treated: With variable piezoelectric coefficients and an internal viscous damping acting on the bar, the total energy decays exponentially. With constant coefficients and no internal viscous damping, the system is polynomially stable, with an optimal decay rate of type \(t^{-1}\) t - 1 . Both results are obtained by combining the frequency domain approach with the multiplier method. The case with variable coefficients and no internal viscous damping remains an open question.