Boundary Stabilization of a Degenerate Euler-Bernoulli Beam under Axial Force and Time Delay
Ben Bakary Junior Siriki, Adama Coulibaly
TL;DR
The paper tackles the challenge of stabilizing a degenerate Euler-Bernoulli beam under an axial force and a boundary control with time delay. It develops an augmented Hilbert space framework based on weighted Sobolev spaces and proves well-posedness via m-dissipative operator theory, then achieves exponential stability through a novel Lyapunov functional that incorporates weighted integrals and a delay term. The work extends prior results by simultaneously addressing degeneracy, axial loading, and boundary delays, delivering a precise decay rate and a robust analysis suitable for complex distributed systems. These contributions provide a rigorous foundation for the design and analysis of stabilization strategies in degenerate, delay-affected beam-like structures, with potential numerical validation to follow.
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 $τ> 0$. 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.