Stability of a tree-shaped network of strings and beams

TL;DR

This paper investigates the stability properties of a tree-shaped network of elastic strings and beams, demonstrating exponential decay under certain conditions and polynomial decay otherwise, using frequency domain methods.

Contribution

It introduces a novel analysis of stability for complex tree-shaped elastic networks with feedbacks, highlighting conditions for exponential and polynomial energy decay.

Findings

System is asymptotically stable

Energy decays exponentially if no beam follows a string from root to leaves

Energy decays polynomially if such a beam exists

Abstract

In this paper we study the stability of a tree-shaped network of elastic strings and beams with some feedbacks at the ends. The whole system is asymptotically stable. Moreover, the energy of the solution decay exponentially to zero if there is no beam following a string ( from the root to the leaves) and decay polynomially if not. We use a frequency domain method from the semigroups theory.