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

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.