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

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