Analysis of the exponential stability of a beam-string-beam transmission problem with local damping on the string

We consider a beam-string-beam problem, where two undamped beams are coupled to a locally damped string (first by a damping of frictional type and then by one of Kelvin–Voigt (K-V) type) by transmission conditions. We show that for this structure, the dissipation produced by the frictional part is strong enough to produce an exponential decay of the solution no matter how small is its size, but the semigroup associated with the system is not exponentially stable when we apply a some localized K-V damping on the string, given by a suitable discontinuous function and certain condition under the length of the parts of the structure. However, when we impose hinged conditions on the beams on the interface points, the dissipation produced by a some localized K-V damping, given by a suitable continuous function, is strong enough to produce an exponential decay of the system solution. For the proof of these three results we use a frequency-domain method of semigroup theory, where in the proofs of exponential stability we also use a contradiction argument.