On the Stability of Timoshenko System With Foundation From the Classical and Second Spectrum Perspectives

ABSTRACT Timoshenko system with its various dissipative mechanisms has been thoroughly examined in the literature. In this paper, we consider the Timoshenko beam model resting on Winkler foundation coupled with Kelvin–Voigt damping. For the classical case, we establish the well‐posedness of the system using ‐semigroup theory. Without assuming the equal wave speeds condition, we prove that the ‐semigroup established with the fully viscoelastic system, where both the bending moment and the shear stress exhibit viscoelastic damping, is analytic. Consequently, exponential stability is acquired. Otherwise, the partial viscoelastic systems are not exponentially stable, no matter the value of the coefficients. Then, the two systems are shown to be polynomially stable using Borichev and Tomilov's theorem. Then, we investigate the case where the system is free of second spectrum, as we showed the well‐posedness of the system using Faedo–Galerkin technique. Furthermore, exponential stability was proved using the energy method. Finally, a numerical scheme is proposed and analyzed, and numerical simulations are provided to illustrate and validate the theoretical findings, confirming the predicted decay rates.