Uniform decay rates for the energy of Timoshenko system with the arbitrary speeds of propagation and localized nonlinear damping

We consider the Timoshenko model for vibrating beams under effect of two nonlinear and localized frictional damping mechanisms acting on the transverse displacement and on the rotational angle. We prove that the damping placed on an arbitrarily small support, unquantitized at the origin and without assuming equal speeds of propagation of waves, leads to uniform decay rates (asymptotic in time) for the energy function. This result removes the necessity (as long as both transverse displacements and rotational angles are minimally damped) of the assumption on equal speeds which has been imposed in the prior literature. The proof of this result relies on the method introduced in Daloutli et al. (Discret Contin Dyn Syst 2(1):67–94, 2009), which reduces the nonlinear stabilization to the observability inequality established for the associated linear problem. The latter is important on its own rights within the context of internal and localized controllability/observability of free linear systems.