Stability analysis of a partially damped suspension bridge by friction

We consider a suspension bridge system of a given length where Timoshenko's theory models the deck.We use semigroup theory.We obtain the existence and uniqueness of the solution by the Lumer-Phillips theorem.For asymptotic behavior, we give under particular constraints a necessary and sufficient condition to guarantee exponential stability.However, regardless of the relationship between the system's coefficients, in a specific case, we lack exponential decay; in this case, we demonstrate the polynomial decay and the optimality of this rate.