Boundary Stabilization of a Degenerate Euler-Bernoulli Beam under Axial Force and Time Delay

TL;DR

This paper analyzes the boundary stabilization of a degenerate Euler-Bernoulli beam with axial force and time delay, establishing exponential decay of energy through advanced mathematical methods.

Contribution

It extends existing results by incorporating axial force and generalized control laws into the stabilization analysis of degenerate Euler-Bernoulli beams.

Findings

Established uniform exponential energy decay.

Provided decay rate estimates.

Extended previous models to include axial force and control delays.

Abstract

This paper provides a qualitative analysis of a non-uniform Euler-Bernoulli beam with degenerate flexural rigidity, subjected to axial force and boundary control with time delay $\tau > 0$. By reformulating the system as an abstract evolution problem in an augmented Hilbert space incorporating weighted Sobolev spaces, we employ semigroup theory to ensure well-posedness. Using the energy multiplier method and a non-standard Lyapunov functional featuring weighted integral terms, we establish uniform exponential energy decay and provide a precise decay rate estimate. This work extends the results of Salhi et al. \cite{salhi2025} and Siriki et al. \cite{siriki2025} by incorporating axial force and generalized control laws, including rotational velocity control. The proposed framework offers a robust approach for analyzing complex distributed systems.