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 $t^{-\frac{1}{2}}$.

• The paper demonstrates that the Timoshenko beam exhibits a polynomial decay rate of t^{-1/2} under weakly degenerate local Kelvin-Voigt damping.
• It employs frequency domain methods and multiplier techniques to derive resolvent estimates around the imaginary axis.
• The analysis confirms robust energy decay despite non-smooth damping coefficients, extending stability results in degenerate dissipative systems.

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

Problem Formulation and Motivation

This work studies the long-time behavior of the Timoshenko beam system under the influence of a single, locally distributed, weakly degenerate Kelvin-Voigt damping term. The model equation addresses a physically relevant regime where viscoelastic damping acts only on a subinterval of the spatial domain, and the damping coefficient is not regular—its derivative is discontinuous, vanishing on part of the interval, and possibly has singularities at the interface. This setting arises in the context of structural vibration control when damping devices or materials are nonhomogeneous or locally applied.

The system is governed by

$$\begin{aligned} \rho_1 w_{tt} - \left[\kappa_1(w' + \phi) + D_1(w' + \phi)_t \right]' &= 0, \ \rho_2 \phi_{tt} - \left( \kappa_2 \phi' + D_2 \phi_t' \right)' + \kappa_1(w' + \phi) + D_1(w' + \phi)_t &= 0, \end{aligned}$$

with homogeneous Dirichlet boundary and specified initial data. Either $D_1$ or $D_2$ is identically zero, leading to dissipation localized to either the shear or the bending component. The nonzero damping coefficient is such that $D_i(x) = 0$ on $[-1,0]$ and $D_i(x) = a_i(x)$ on $(0,1]$, with $a_i$ continuous, $C^1$ on $(0,1]$, $D_1$0 on $D_1$1, $D_1$2, and a uniform pointwise upper bound on $D_1$3 strictly less than one.

The central goal is to characterize the decay rate of energy for this Timoshenko system under these structural and regularity assumptions on the Kelvin-Voigt damping, extending and clarifying the landscape of stability properties in the degenerate, non-smooth case.

Main Theoretical Results

By employing the frequency domain method and multiplier techniques, the authors establish that the associated $D_1$4-semigroup is polynomially stable of order $D_1$5. Explicitly, for all admissible initial data, the energy satisfies

$D_1$6

This decay holds regardless of whether the localized Kelvin-Voigt damping acts in the shear or bending term, provided the degeneracy condition $D_1$7 is satisfied. The result matches the decay rate previously proven for the case of non-continuous but non-degenerate Kelvin-Voigt damping on local subintervals [Wehbe, Ghader, "A transmission problem for the Timoshenko system with one local Kelvin-Voigt damping and non-smooth coefficient at the interface," Comput. Appl. Math., 2021]. This demonstrates the robustness of polynomial stability even under further relaxation toward weakly degenerate coefficients.

Methodologically, the proof leverages an abstract characterization of polynomial decay rates for operator semigroups [Borichev-Tomilov, 2009], reducing the analysis to a resolvent estimate around the imaginary axis. A contradiction argument shows that any sequence violating a polynomial resolvent bound must converge weakly to zero—a contradiction with uniform boundedness established via careful multiplier cutoffs and Hardy-type estimates. The investigation separately handles the cases where Kelvin-Voigt damping is present in either the shear or bending moment, constructing appropriate multipliers to manage the interface singularities.

Numerical and Analytical Benchmarks

The decay order $D_1$8 obtained is independent of where the local damping is placed (shear or bending) and of the spatial degeneracy level, as long as the key assumption $D_1$9 is maintained. This is consistent with previously reported rates for nondegenerate and less regular local damping, but the present result accommodates even weaker coefficients.

The work consolidates the following classification for the Timoshenko beam with local Kelvin-Voigt damping (summarized here in compact form):

Damping configuration	Coefficient regularity	Energy decay
Both $D_2$0, $D_2$1	Uniform	$D_2$2
Both $D_2$3, $D_2$4	Weakly degenerate	$D_2$5 with $D_2$6
One $D_2$7, other weakly degenerate ($D_2$8)	Polynomially degenerate	$D_2$9
Both strongly degenerate, $D_i(x) = 0$0	Highly degenerate	(Open)

Implications and Open Problems

These results clarify the qualitative stability regime of the Timoshenko system with weakly degenerate local Kelvin-Voigt damping and fill a gap in prior literature, where only the cases of smooth or nondegenerate local damping were sharply characterized. From a practical viewpoint, this suggests that localization and weak degeneracy in damping deployment still guarantee robust sub-exponential energy decay, relevant in the engineering of structures with partial or inhomogeneous viscoelastic components.

From a control-theoretical perspective, these findings confirm that local degeneracy at the interface does not generically preclude uniform sub-exponential stabilization—although the optimality of the decay order $D_i(x) = 0$1 in this setting remains conjectural. Specifically, it is not currently known whether a sharper lower bound or even exponential decay can be approached under alternative interface regularity conditions.

Potential future directions include:

• Determining the optimality of the $D_i(x) = 0$2 decay rate for the weakly degenerate case.
• Characterizing stability for even more singular coefficients, e.g., where the upper bound in $D_i(x) = 0$3 is violated, or both coefficients are degenerate.
• Extending the analysis to multidimensional Timoshenko-type systems with localized or interface Kelvin-Voigt damping.

Conclusion

This paper provides a rigorous analysis of the polynomial stability of the Timoshenko beam equation with one weakly degenerate, locally supported Kelvin-Voigt damping term. The main result establishes that the decay rate of the system's energy cannot exceed $D_i(x) = 0$4 under the assumed degeneracy structure, extending known stability results for the Timoshenko system to broader, physically motivated scenarios with non-smooth and degenerate damping. The approach combines sharp multiplier estimates with resolvent criteria, and the findings raise further analytical and control-theoretic questions regarding optimal stabilization in degenerate dissipative structures.

Reference: "Stability of the Timoshenko Beam Equation with One Weakly Degenerate Local Kelvin-Voigt Damping" (2604.01809).