  In this paper, we study the decay rate of the Cayley transform of the
generator of a polynomially stable $C_0$-semigroup. To estimate the decay rate
of the Cayley transform, we develop an integral condition on resolvents for
polynomial stability. Using this integral condition, we relate polynomial
stability to Lyapunov equations. We also study the robustness analysis of
polynomial stability against a certain class of structured perturbations.
