Stabilization of string–beam–string transmission systems with localized frictional damping and delay

In this paper, we consider two string–beam–string boundary transmission problems with localized frictional damping and delay. The first transmission problem is that the frictional damping and delay are in the left string equation and the right string equation is thermoelastic. The second transmission problem is that the damping and the delay are both only acting on the middle beam equation. Using the semigroup theory, we get the existence and the uniqueness of solution under suitable assumptions on the weight of the damping and the weight of the delay. Furthermore, we prove that the first transmission problem is exponentially stable and the second transmission problem is polynomially stable using the frequency domain method and the multiplier techniques.