Asymptotic Behavior of the Transmission Problem for the Timoshenko System With Distributed Delay Terms

In this work, we investigate a transmission problem for the Timoshenko system with distributed delay terms acting on the rotation-angle equations of a beam. The model describes a structure composed of two different materials connected at an interface, which introduces transmission conditions that complicate the stability analysis. Distributed delays are incorporated into the internal feedback laws associated with the variables \(v_i\) , which may lead to destabilizing effects if they are not properly balanced with non-delayed damping mechanisms. Under suitable assumptions on the relative weights of the delayed and non-delayed feedback terms, as well as on the wave propagation speeds, we first establish the well-posedness of the system by using semigroup theory. In the case of equal propagation speeds, we construct a suitable Lyapunov functional and apply the energy method to derive an exponential decay estimate for the total energy of the system. This result shows that the combined action of distributed delay and non-delayed feedback is capable of guaranteeing uniform stability, provided that the delay contribution remains sufficiently small. Finally, we highlight the importance of the equal wave speed condition, which plays a crucial role in ensuring exponential convergence to equilibrium.