AbstractIn this paper, we study the decay rate of the Cayley transform of the generator of a polynomially stable $$C_0$$ 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 robustness of polynomial stability for a certain class of structured perturbations.