Theoretical and numerical study of the exponential stability of a double Euler-Bernoulli beam system

Abstract

In this paper, we theoretically and numerically study the exponential stability of a system of two elastic beams coupled through velocity terms, modeled by the classical Euler-Bernoulli theory. We aim to achieve exponential stabilization of the full coupled system using a single boundary damping mechanism applied to only one of the beam equations. We begin by proving the well-posedness of the problem using semigroup theory. Then, employing the multiplier method, we establish the uniform exponential decay of the solution. For numerical approximation, we use a finite element discretization in space and an implicit Euler scheme in time. We derive a discrete stability property and a priori error estimates, proving the linear convergence of the approximations under suitable regularity assumptions. Finally, some numerical experiments are performed to validate the numerical convergence of the scheme, analyze the behavior of the solutions and the discrete energy decay, and examine the influence of the coupling parameter on the solutions and the discrete energy.