Exponential stability of a damped beam-string-beam transmission problem

TL;DR

This paper proves exponential stability for a coupled beam-string-beam system with damping, showing that frictional damping ensures decay regardless of initial size, using energy and frequency domain methods.

Contribution

It establishes exponential stability for a transmission problem with damping, employing novel energy and frequency domain techniques, and proves higher regularity of solutions.

Findings

Frictional damping guarantees exponential decay regardless of initial size.

Energy method is used for damped-damped-damped case.

Frequency domain method with contradiction and multiplier techniques is used for undamped-damped-undamped case.

Abstract

We consider a beam-string-beam transmission problem, where two structurally damped or undamped beams are coupled with a frictionally damped string by transmission conditions. We show that for this type of structure, the dissipation produced by the frictional part is strong enough to produce exponential decay of the solution no matter how small is its size: for the exponential stability in the damped-damped-damped situation we use energy method and in the undamped-damped-undamped situation we use a frequency domain method from the semigroups theory, which combines a contradiction argument with the multiplier technique to carry out a special analysis for the resolvent. Additionally, we show that the solution first defined by the weak formulation, in fact, has higher Sobolev space regularity.