Absence of Exponential Stability and Polynomial Stabilization in a Class of Beam Models with Tip Rotary Inertia

TL;DR

This paper analyzes how boundary dissipation affects the stability of Euler-Bernoulli beam models with tip rotary inertia, showing that hybrid dissipation does not induce exponential decay and only causes slow polynomial decay.

Contribution

It demonstrates that hybrid dissipation at the boundary does not improve exponential stability in beam models with tip rotary inertia.

Findings

Hybrid dissipation does not induce exponential decay.

When alone, it causes a slow $t^{-1/2}$ decay.

It does not affect decay when combined with other mechanisms.

Abstract

We investigate the impact of dissipative dynamic boundary conditions applied at one end of a beam, analyzing their influence on model stability within the Euler-Bernoulli framework. Our primary finding is that hybrid dissipation does not alter the decay characteristics of the original model. We examine two scenarios: first, when hybrid dissipation is the sole dissipative mechanism, and second, when it complements other dissipative mechanisms. In the first case, we demonstrate that hybrid dissipation fails to induce exponential decay, instead producing a slow decay rate of $t^{-1/2}$ for large $t$. In the second case, when acting as a complementary mechanism, hybrid dissipation neither enhances nor diminishes the decay behavior of the original model.