Decay rates for Timoshenko beam system with suspenders and arbitrary nonlinear localized damping

In this article, we analyze a vibrating Timoshenko beam system with suspension in a one-dimensional bounded domain, under nonlinear localized internal damping mechanisms affecting all three wave equations. Specifically, we prove that damping applied to an arbitrarily small interval with positive measure, no matter how small, is effective. We also establish the existence and uniqueness of solutions using nonlinear semigroup theory and determine some rates of decay for these solutions without assuming any relationship between the coefficients. Additionally, we prove a result concerning internal observability for the conservative system to achieve the aforementioned asymptotic behavior.