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In this paper, we study the decay rate of the Cayley transform of thegenerator of a polynomially stable $C_0$-semigroup. To estimate the decay rateof the Cayley transform, for polynomial stability. Using this integralcondition, we relate polynomial stability to Lyapunov equations. We also studyrobustness of polynomial stability for a certain class of structuredperturbations.

The Cayley transform of the generator of a polynomially stable $C_0$-semigroup