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Stability of the Timoshenko Beam Equation with One Weakly Degenerate Local Kelvin-Voigt Damping

Ruijuan Liu, Qiong Zhang

Abstract

We consider the Timoshenko beam equation with locally distributed Kelvin-Voigt damping, which affects either the shear stress or the bending moment. The damping coefficient exhibits a singularity, causing its derivative to be discontinuous. By using the frequency domain method and multiplier technique, we prove that the associated semigroup is polynomial stability. Specifically, regardless of whether the local Kelvin-Voigt damping acts on the shear stress or the bending moment, the system decays polynomially with rate $t^{-\frac{1}{2}}$.

Stability of the Timoshenko Beam Equation with One Weakly Degenerate Local Kelvin-Voigt Damping

Abstract

We consider the Timoshenko beam equation with locally distributed Kelvin-Voigt damping, which affects either the shear stress or the bending moment. The damping coefficient exhibits a singularity, causing its derivative to be discontinuous. By using the frequency domain method and multiplier technique, we prove that the associated semigroup is polynomial stability. Specifically, regardless of whether the local Kelvin-Voigt damping acts on the shear stress or the bending moment, the system decays polynomially with rate .

Paper Structure

This paper contains 6 sections, 9 theorems, 73 equations.

Table of Contents

  1. Introduction
  2. Main Results and Preliminaries
  3. Polynomial Stability Analysis
  4. The case $a_1(x)>0,\;a_2(x)=0$
  5. The case $a_1(x)=0,\;a_2(x)>0$
  6. Conclusion

Key Result

Lemma 2.1

Assume coefficient functions $D_{1}$ and $D_{2}$ are continuous, nonnegative and satisfy H1-H2. Then, the operator ${\cal A}$ generates a C$_0$-semigroup of contractions $e^{t{\cal A}}$ on ${\cal H}$, and ${\hbox{i}}\mathbb{R}\subset \rho({\cal A})$, the resolvent set of ${\cal A}.$$\blacktrianglele

Theorems & Definitions (9)

  • Lemma 2.1
  • Lemma 2.2
  • Theorem 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6