Exponential Stability of a Degenerate Euler-Bernoulli Beam with Axial Force and Delayed Boundary Control
Ben Bakary Junior Siriki, Adama Coulibaly
TL;DR
The paper analyzes a degenerate Euler-Bernoulli beam with a nonuniform axial force and delayed boundary control, establishing well-posedness in a weighted Sobolev framework and proving uniform exponential stability. By reformulating the problem as an abstract Cauchy problem and constructing a Lyapunov functional that combines the energy with a cross-term, the authors derive explicit decay rates and show the energy decays as $E(t) \le E(0) e^{1 - t/M}$ for some $M>0$ independent of initial data. A key requirement is the dominance condition $\\kappa_1 > |\\kappa_2|$ (with a delay-robust bound on auxiliary parameters), enabling the semigroup generation via the Lümer-Phillips theorem and the stability analysis via energy estimates. The results extend stability theory for complex distributed-parameter systems with degeneracy and delay, offering a rigorous framework for robust boundary control of aging or defect-prone beams.
Abstract
This work investigates the global exponential stabilization of a degenerate Euler-Bernoulli beam subjected to a non uniform axial force and a delayed feedback control. First, we establish the well-posedness of the system by constructing an appropriate energy space in weighted Sobolev settings. Using Lümer-Phillips theorem, we prove that the linear operator associated with the problem generates a $\mathcal{C}_0$-semigroup of contractions. Second, we establish the uniform exponential stability of the system. By constructing a novel Lyapunov functional incorporating weighted integral terms, we demonstrate that the energy's system exponentially decays to zero and derive a precise decay rate estimate. This work provides a significant extension to the stability theory for complex distributed parameter systems.