On the exponential decay of the Euler-Bernoulli beam with boundary energy dissipation

TL;DR

This paper investigates how boundary energy dissipation with a memory effect influences the exponential decay of energy in a clamped-free Euler-Bernoulli beam, establishing conditions for rapid energy decay.

Contribution

It proves that exponential decay of the memory kernel is both necessary and sufficient for exponential energy decay in the beam.

Findings

Exponential decay of the memory kernel ensures exponential energy decay.

Boundary control with memory affects the asymptotic behavior of the beam.

Necessary and sufficient conditions for energy decay are established.

Abstract

We study the asymptotic behavior of the Euler-Bernoulli beam which is clamped at one end and free at the other end. We apply a boundary control with memory at the free end of the beam and prove that the "exponential decay" of the memory kernel is a necessary and sufficient condition for the exponential decay of the energy.