Study of the Stability Properties for a General Shape of Damped Euler–Bernoulli Beams under Linear Boundary Conditions

Abstract

We study in this paper a general shape of damped Euler–Bernoulli beams with variable coefficients. Our main goal is to generalize several works already done on damped Euler–Bernoulli beams. We start by studying the spectral properties of a particular case of the system, and then we determine asymptotic expressions that generalize those obtained by other authors. At last, by adopting well-known techniques, we establish the Riesz basis property of the system in the general case, and the exponential stability of the system is obtained under certain conditions relating to the feedback coefficients and the sign of the internal damping on the interval studied of length $1$ .