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 robustness of polynomial stability for a certain class of structured perturbations.

中文翻译：

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多项式稳定 $$C_0$$C 0 -半群的生成器的凯莱变换

在本文中，我们研究了多项式稳定\(C_0\)-半群的生成器的凯莱变换的衰减率。为了估计 Cayley 变换的衰减率，我们开发了关于多项式稳定性的求解器的积分条件。使用这个积分条件，我们将多项式稳定性与李雅普诺夫方程联系起来。我们还研究了某类结构化扰动的多项式稳定性的鲁棒性。