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

Abstract

In 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.

Robustness of Polynomial Stability with respect to Sampling

“…The proof can be found in Lemma 2.3 and Theorem 2.4 of [4]. See also Lemma 2.3 of [37] for the result on the decay rate of an individual orbit.…”

“…We state the analogue of Proposition 3.1 (b) and (c) in [37] for discrete semigroups. Proposition 4.3.…”

“…We will show that if ran(B) ⊂ D β for some β > 0, then D β ⊂ D((−A − BF ) β ) holds for all β ∈ [0, β). To this end, the following result is useful; see Lemma 5.4 of [37] for the proof. Lemma 5.3.…”

“…We also refer to [5] for an overview of semi-uniform stability. A discrete version of semi-uniform stability has been investigated in the context of the quantified Katznelson-Tzafriri theorem [6,18,31,32] and the Cayley transform of a generator [37]. However, to the author's knowledge, robustness analysis with respect sampling has not been well established for polynomial stability.…”

Decay Rate of $\exp(A^{-1}t)A^{-1}$ on a Hilbert Space and the Crank-Nicolson Scheme with Smooth Initial Data

“…Define A d := A d (2) = (I + A)(I − A) −1 . It has been shown that in Example 4.6 of [24] that the optimal decay rate of A n d A −β is n −β/3 for all β > 0. Hence the norm estimate (3.11) has an example for which the decay rate cannot be improved.…”