Stabilization of an Euler-Bernoulli Beam with Distributed Damping Under Time Delays in the Boundary

In this paper, we investigate the exponential stability of an Euler-Bernoulli beam system with distributed damping subjected to a time-delay in the boundary. At first, applying the semigroup theory of bounded linear operators we prove the well posedness of the system. And then we give the exponential stability analysis of the system by constructing an appropriate Lyapunov function. Different from the earlier results, we use the damping coefficient α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha $\end{document} and delay coefficient β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\beta $\end{document} together with the parameters of the system to give a description of the stability region. The simulation are presented to prove the effectiveness of this results.