Robustness of Polynomial Stability with Respect to Sampling

We provide a partially affirmative answer to the following question on robustness of polynomial stability with respect to sampling: ``Suppose that a continuous-time state-feedback controller achieves the polynomial stability of the infinite-dimensional linear system. We apply an idealized sampler and a zero-order hold to a feedback loop around the controller. Then, is the sampled-data system strongly stable for all sufficiently small sampling periods? Furthermore, is the polynomial decay of the continuous-time system transferred to the sampled-data system under sufficiently fast sampling?'' The generator of the open-loop system is assumed to be a Riesz-spectral operator whose eigenvalues are not on the imaginary axis but may approach it asymptotically. We provide conditions for strong stability to be preserved under fast sampling. Moreover, we estimate the decay rate of the state of the sampled-data system with a smooth initial state and a sufficiently small sampling period.


Introduction
We study the robustness of polynomial stability with respect to sampling. To state our problem precisely, we consider the following sampled-data system with sampling period τ > 0: where A with domain D(A) is the generator of a C 0 -semigroup (T (t)) t≥0 on a Hilbert space X, and the control operator B : C → X and the feedback operator F : X → C are bounded linear operators. We assume that the C 0 -semigroup (T BF (t)) t≥0 generated by A + BF is polynomially stable with parameter α > 0, which means that sup t≥0 ∥T BF (t)∥ < ∞, the spectrum of A + BF is contained in the open left half-plane, and for all x ∈ D(A + BF ) = D(A), ∥T BF (t)x∥ = o(t −1/α ) as t → ∞, i.e., for any ε > 0, there exists t 0 > 0 such that for all t ≥ t 0 , ∥T BF (t)x∥ ≤ ε t 1/α . By density of D(A) in X, we see that under this assumption, (T BF (t)) t≥0 is strongly stable, that is, lim t→∞ ∥T BF (t)x 0 ∥ = 0 for all x 0 ∈ X. Intuitively, as the sampling period τ > 0 goes to zero, the sampled-data control input (1b) becomes closer to the continuous-time control input given by u(t) = F x(t) for t ≥ 0. Therefore, the following two questions arise: a) Is the sampled-data system (1) with sufficiently small sampling period τ > 0 strongly stable in the sense that lim t→∞ x(t) = 0 for every initial state x 0 ∈ X? b) Does the state x of the sampled-data system (1) decay polynomially for x 0 ∈ D(A) and sufficiently small τ > 0 as the orbit T BF (t)x 0 ? We provide a partially affirmative answer to these questions in this paper. The effect of sampling on systems can be regarded as a kind of structured perturbation. In this sense, the issue in the questions above is robustness analysis of polynomial stability with respect to sampling.
For finite-dimensional linear systems, it is well known that the closed-loop stability is preserved under fast sampling. However, the robustness of stability with respect to sampling is not guaranteed for all infinitedimensional linear systems; see [30]. It has been shown in [18,31] that if (T BF (t)) t≥0 is exponentially stable, that is, there exist constants M ≥ 1 and ω > 0 such that ∥T BF (t)∥ ≤ M e −ωt for all t ≥ 0, then the sampleddata system also has the same property of exponential stability for a sufficiently small sampling period τ > 0. Exponential stability is a strong property, which can be seen from the fact that exponential stability is robust under small bounded perturbations and even some classes of unbounded perturbations as shown, e.g., in [21,28]. Exploiting the advantages of exponential stability, the robustness analysis developed in [18,31] allows unbounded control operators mapping the input space into a space larger than the state space, called an extrapolation space. On the other hand, the robustness of strong stability with respect to sampling has been studied in [38], where the control operator needs an extra boundedness property related to the continuous spectrum of A. The reason for imposing this boundedness property is that strong stability is a rather delicate property that is highly sensitive to perturbations; see [26,27,29] for the robustness of strong stability of C 0 -semigroups (in the absence of polynomial stability).
Exponential stability leads to uniformly quantified asymptotic behaviors of semigroup orbits for all initial values from the unit ball of the state space. This is a desirable property from the viewpoint of many applications. Nevertheless, exponential stability may be unachievable in control problems, for example, involving wave equations or beam equations. Although strong stability can be achieved in some of those problems, it is a qualitative notion of stability unlike exponential stability, and we do not obtain any information on decay rates of semigroup orbits from strong stability itself. Polynomial stability is an important subclass of semi-uniform stability, which lies between the above two extreme types of semigroup stability, exponential stability and strong stability, and guarantees semi-uniform decay rates for semigroup orbits with initial values in the domain of the generator. Various results on polynomial stability, and more generally semi-uniform stability, have been obtained such as characterizations of decay rates by resolvent estimates on the imaginary axis iR [3,4,6,17,32] and robustness to perturbations [22][23][24][25]29]. We also refer to [7] 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 [8,20,33,34] (see also the survey article [5]) and the Cayley transform of a semigroup generator [39]. However, to the author's knowledge, robustness analysis with respect to sampling has not been well established for polynomial stability.
To study the robustness of polynomial stability with respect to sampling, this paper continues and expands the robustness analysis developed in [38]. We assume as in [38] that A is a Riesz-spactral operator given by where (λ n ) n∈N are distinct complex numbers not on iR, (ϕ n ) n∈N forms a Riesz basis in X, and (ψ n ) n∈N is a biorthogonal sequence to (ϕ n ) n∈N ; see Section 2.2 for the details of Riesz-spactral operators. We restrict our attention to the situation where only a finite number of the eigenvalues (λ n ) n∈N are in the set Ω a := {λ ∈ C : Re λ > −ω} ∩ C \ λ ∈ C \ R : Re λ ≤ −Υ | Im λ| α for some ω, α, Υ > 0. In contrast, it is assumed in [38] that the set Ω b := {λ ∈ C \ {0} : Re λ > −ω, | arg λ| < π/2 + ϑ} contains only finitely many eigenvalues for some ω > 0 and 0 < ϑ ≤ π/2. Figure 1 illustrates the sets Ω a and Ω b . In our setting, the continuous spectrum of A has empty intersection with iR unlike the setting of [38], but the spectrum of A may approach iR asymptotically. In other words, the resolvent of A has a singularity at zero in [38], whereas, loosely speaking, the resolvent restricted to iR has a singularity at infinity in this study because the resolvent grows to infinity on iR. Therefore, the type of non-exponential stability we consider in this paper is different from that in [38]. It has been shown in [38] that only strong stability is preserved under fast sampling. Here we investigate the quantitative behavior of the state of the sampled-data system in addition to strong stability.
Another important difference from [38] is an assumption on the control operator B and the feedback operator F . Let b, f ∈ X and let B, F be written as Bu = bu for all u ∈ C and F x = ⟨x, f ⟩ for all x ∈ X. In this paper, (a) Set Ωa considered in this paper. [38]. Figure 1. Comparison of sets that contain only finitely many eigenvalues.
On the other hand, it is assumed in [38] |⟨x, ψ n ⟩| 2 |λ n | 2 < ∞ and f ∈ X. Under the assumption we make in this paper, B and F have the parameters β and γ for design flexibility, which increases the applicability of the proposed robustness analysis. The bounded linear operator ∆(τ ) on X defined by plays a key role in the analysis of robustness with respect to sampling. In fact, the sampled-data system (1) is strongly stable if and only if the discrete semigroup (∆(τ ) k ) k∈N is strongly stable, i.e., for all x 0 ∈ X. In [38], the sufficient condition for strong stability obtained in the Arendt-Batty-Lyubich-Vũ theorem [1,19] is used in order to show that (∆(τ ) k ) k∈N is strongly stable. This sufficient condition requires that the intersection of the spectrum of ∆(τ ) and the unit circle be countable, but the system we consider does not have this property in general. Instead of the Arendt-Batty-Lyubich-Vũ theorem, we here employ the characterization of strong stability by an integral condition on resolvents developed in [36]. Let 0 < δ ≤ α/2, where α > 0 is the constant for the set Ω a . We give an integral condition on resolvents under which the orbit ∆(τ ) k x 0 with x 0 ∈ D δ satisfies as k → ∞. Using this integral condition, we show that the state x of the sampled-data system (1) with sufficiently small sampling period τ > 0 satisfies Considering the open-loop case F = 0, we see that t −δ/α in the estimate (2) cannot be replaced by functions with better decay rates. It is still unknown whether the logarithmic factor √ log t in the case δ = α/2 may be removed. The paper is organized as follows. Section 2 contains preliminaries on polynomial stability of C 0 -semigroups and Riesz-spectral operators. In Section 3, we present the main result and introduce the discretized system for its proof. Section 4 is devoted to resolvent conditions for stability. To apply these conditions to the discretized system, we investigate the spectrum of ∆(τ ) in Section 5. Section 6 completes the proof of the main result with the help of the resolvent conditions for stability. To illustrate the theoretical result, we study a wave equation in Section 7. The conclusion is given in Section 8.
Notation and terminology. We denote by N 0 the set of non-negative integers. For ω ∈ R and r > 0, we write Note that we denote the open right half-plane by C 0 , while C + and C + are commonly used in the literature. For α, Υ > 0, define The closure of a subset Ω of C and the complex conjugate of λ ∈ C are denoted by Ω and λ, respectively. For real-valued functions f, g on J ⊂ R, we write if there exist M > 0 and t 0 ∈ J such that f (t) ≤ M g(t) for all t ≥ t 0 , and similarly, if for any ε > 0, there exists t 0 ∈ J such that f (t) ≤ εg(t) for all t ≥ t 0 . Let X and Y be Banach spaces. For a linear operator A : X → Y , we denote by D(A) and ran(A) the domain and the range of A, respectively. The space of all bounded linear operators from X to Y is denoted by L(X, Y ), and we write L(X) := L(X, X). For a linear operator A : D(A) ⊂ X → X, we denote by σ(A) and ρ(A) the spectrum and the resolvent set of A, respectively. We write R(λ, A) := (λI − A) −1 for λ ∈ ρ(A). For a subset S of X and a linear operator A : A C 0 -semigroup (T (t)) t≥0 on a Banach space X is called uniformly bounded if sup t≥0 ∥T (t)∥ < ∞ and strongly stable if lim t→∞ T (t)x = 0 for all x ∈ X. By a discrete semigroup on X, we mean a family (∆ k ) k∈N of operators, where ∆ ∈ L(X). A discrete semigroup (∆ k ) k∈N on X is called power bounded if sup k∈N ∥∆ k ∥ < ∞ and strongly stable if lim k→∞ ∆ k x = 0 fo all x ∈ X.
An inner product on a Hilbert space is denoted by ⟨·, ·⟩. For Hilbert spaces Z and W , let A * denote the Hilbert space adjoint of a densely defined linear operator A : D(A) ⊂ Z → W .

Preliminaries
In this section, we review the definition and some important properties of polynomially stable C 0 -semigroups and Riesz-spectral operators.
2.1. Polynomially stable C 0 -semigroups. We start by recalling the definition of polynomially stable C 0semigroups. In Definition 3.2 of [3], polynomial stability of C 0 -semigroups does not include uniform boundedness, but here we define polynomial stability to include uniform boundedness as in Definition 1.2 of semi-uniform stability in [4].
Definition 2.1. A C 0 -semigroup (T (t)) t≥0 on a Banach space X generated by A is polynomially stable with parameter α > 0 if the following three conditions are satisfied: Let A be the generator of a uniformly bounded C 0 -semigroup (T (t)) t≥0 on a Hilbert space. Then −A and −A * are sectorial in the sense of Chapter 2 of [15]. In particular, if (T (t)) t≥0 is a polynomially stable C 0semigroup on a Hilbert space, then A and A * are invertible, and hence the fractional powers (−A) α and (−A * ) α are well defined for all α ∈ R. We refer, e.g., to Chapter 3 of [15] and Section II.5.3 of [11] for the details of fractional powers.
We use the following characterizations for polynomial decay of a C 0 -semigroup on a Hilbert space. The proof can be found in Lemma 2.3 and Theorem 2.4 of [6]. See also Lemma 2.3 of [39] for the result on the decay rate of an individual orbit.
Theorem 2.2. Let (T (t)) t≥0 be a uniformly bounded C 0 -semigroup on a Hilbert space X with generator A such that iR ⊂ ρ(A). For fixed α, δ > 0, the following statements are equivalent: The following estimate given in Lemma 4 of [25] is useful in the robustness analysis of polynomial stability. Lemma 2.3. Let A be the generator of a polynomially stable C 0 -semigroup with parameter α > 0 on a Hilbert space X. Let β, γ ≥ 0 satisfy β + γ ≥ α and let U be a Banach space. There exists a constant M ≥ 1 such that if B ∈ L(U, X) and F ∈ L(X, U ) satisfy ran(B) ⊂ D((−A) β ) and ran(F * ) ⊂ D((−A * ) γ ), then 2.2. Riesz-spectral operators. Next we recall the definition of Riesz-spectral operators and briefly state their most relevant properties. We refer the reader to Section 3.2 of [9], Section 2.4 of [37], and Chapter 2 of [14] for more details.
Definition 2.4 (Definition 3.2.6 of [9]). Let X be a Hilbert space and let A : D(A) ⊂ X → X be a closed linear operator with simple eigenvalues (λ n ) n∈N and corresponding eigenvectors (ϕ n ) n∈N . We say that A is a Riesz-spectral operator if the following two conditions are satisfied: a) (ϕ n ) n∈N is a Riesz basis; and b) The set of eigenvalues {λ n : n ∈ N} has at most finitely many accumulation points.
Let A be a Riesz-spectral operator on a Hilbert space X with simple eigenvalues (λ n ) n∈N and corresponding eigenvectors (ϕ n ) n∈N . Let (ψ n ) n∈N be the eigenvectors of the adjoint A * corresponding to the eigenvalues (λ n ) n∈N . Then (ψ n ) n∈N can be suitable scaled so that (ϕ n ) n∈N and (ψ n ) n∈N are biorthogonal, i.e., ⟨ϕ n , ψ m ⟩ = 1 if n = m 0 otherwise.
A sequence biorthogonal to a Riesz basis in X is unique and also forms a Riesz basis in X. Throughout this paper, we set the sequence (ψ n ) n∈N of the eigenvectors of the adjoint A * so that (ψ n ) n∈N are biorthogonal to (ϕ n ) n∈N . Every x ∈ X can be represented uniquely by ⟨x, ϕ n ⟩ψ n .
Moreover, there exist constants M a , M b > 0 such that for all x ∈ X, We shall frequently use these inequalities without comment. The Riesz-spectral operator A has the following representation: The spectrum of the Riesz-spectral operator A is the closure of its point spectrum, that is, σ(A) = {λ n : n ∈ N}. For λ ∈ ρ(A), the resolvent R(λ, A) is given by The Riesz-spectral operator A generates a C 0 -semigroup on X if and only if sup n∈N Re λ n < ∞, and the C 0 -semigroup (T (t)) t≥0 generated by A can be written as e tλn ⟨x, ψ n ⟩ϕ n for all x ∈ X and t ≥ 0. The adjoint A * is also a Riesz-spectral operator and is represented as Moreover, the C 0 -semigroup generated by A * is given by (T (t) * ) t≥0 . To make assumptions on the ranges of the control operator B ∈ L(C, X) and the adjoint of the feedback operator F ∈ L(X, C), we use the following subsets with parameters β, γ ≥ 0:

Stability of sampled-data systems
In this section, we present the system under consideration and state the main result. We also introduce the discretized system as the first step of its proof.
3.1. Main result. Let X be a Hilbert space, and consider the following sampled-data system with state space X and input space C:ẋ where x(t) ∈ X is the state, u(t) ∈ C is the control input, τ > 0 is the sampling period, A : D(A) ⊂ X → X is the generator of a C 0 -semigroup (T (t)) t≥0 on X, B ∈ L(C, X) is the control operator, and F ∈ L(X, C) is the feedback operator. To state the main result, we make the following assumption on the sampled-data system (5).
Assumption 3.2. Let A be a Riesz-spectral operator on a Hilbert space X with simple eigenvalues (λ n ) n∈N and corresponding eigenvectors (ϕ n ) n∈N . Let (ψ n ) n∈N be the eigenvectors of A * such that (ϕ n ) n∈N and (ψ n ) n∈N are biorthogonal. Let the control operator B ∈ L(C, X) and the feedback operator F ∈ L(X, C) be represented as for some b, f ∈ X. Assume that the operators A, B, and F satisfy the following conditions: (A1) There exist constants ω, α, Υ > 0 such that C −ω ∩ (C \ Ω α,Υ ) has only finite elements of (λ n ) n∈N . (A2) {λ n : n ∈ N} ∩ iR = ∅. (A3) A + BF generates a polynomially stable C 0 -semigroup (T BF (t)) t≥0 with parameter α on X. (A4) There exist constants β, γ ≥ 0 such that b ∈ D β , f ∈ D γ * , and one of the following conditions holds: (i) β, γ ∈ N 0 and β + γ ≥ α.
By (A1), the Riesz-spectral operator A generates a C 0 -semigroup (T (t)) t≥0 . Since σ(A) = {λ n : n ∈ N}, it follows that σ(A) ∩ iR = ∅ under (A1) and (A2). The eigenvalues (λ n ) n∈N may approach iR asymptotically, and an upper bound of the asymptotic rate is represented by the parameter α given in (A1). Note that when lim ℓ→∞ Re λ n ℓ = 0 for some subsequence (λ n ℓ ) ℓ∈N , there does not exist a feedback operator F ∈ L(X, C) such that the C 0 -semigroup (T BF (t)) t≥0 generated by A + BF is exponentially stable; see, e.g., Theorem 8.2.3 of [9]. We assume by (A4) that B and F have stronger boundedness properties related to the parameter α than the standard boundedness properties B ∈ L(C, X) and F ∈ L(X, C). Assumptions similar to (A4) are placed to perturbation operators in the robustness analysis of polynomial stability developed in [22][23][24]. Note that not all of (A1)-(A4) are imposed in every result. In fact, (A2) is not used in Section 5 except for Lemma 5.2, while (A3) is not imposed in Sections 6.1 and 6.2.
The following theorem is the main result of this paper, which shows that polynomial stability is robust with respect to sampling. Theorem 3.3. If Assumption 3.2 is satisfied, then there exists τ * > 0 such that the following statements hold for all τ ∈ (0, τ * ): a) The sampled-data system (5) is strongly stable. b) Let 0 < δ ≤ α/2, and assume that δ ≤ 1 or β ≥ α. Then, for every initial state x 0 ∈ D δ , the state x of the sampled-data system (5) satisfies Let α, δ > 0 and consider the case F = 0 and λ n = −1/n α + in for n ∈ N. Then Assumption 3.2 holds for all B ∈ L(C, X), where the constants β and γ in (A4) are chosen such that β = 0 and γ > α. We also have ∥T (t)(−A) −δ ∥ = O(t −δ/α ) as t → ∞, and this decay rate is optimal in the sense that lim inf t→∞ t δ/α ∥T (t)(−A) −δ ∥ > 0. Therefore, one cannot replace t −δ/α in the estimate (7) by functions with better decay rates. Whether the logarithmic correction term √ log t for the case δ = α/2 may be omitted remains open.
The assumption δ ≤ 1 implies D(A) ⊂ D δ . On the other hand, the assumption β ≥ α leads to the uniform boundedness of ∥R(z, T (τ ))S(τ )∥ on an annulus {z ∈ C : 1 < |z| < 1 + ε} with some sufficiently small ε > 0 for a fixed τ > 0, where S(τ ) ∈ L(C, X) is defined by We will employ these assumptions in Section 6.2. The proof of Theorem 3.3 is divided into several steps. In the next subsection, we prove the equivalence between the stability of the sampled-data system (5) and that of the discretized system. Section 4 is devoted to resolvent conditions for the stability of discrete semigroups on Hilbert spaces. To apply these resolvent conditions, in Section 5, we investigate the spectrum of the operator that represents the dynamics of the discretized system. In Section 6, we complete the proof of Theorem 3.3, by using the resolvent conditions presented in Section 4.
Then the state x of the sampled-data system (5) satisfies (9) x (k + 1)τ = ∆(τ )x(kτ ) for all k ∈ N 0 , which we call the discretized system. To prove Theorem 3.3, it suffices by the next result to investigate the discrete semigroup (∆(τ ) k ) k∈N .
Proposition 3.4. Let A be the generator of a C 0 -semigroup (T (t)) t≥0 on a Banach space X. Let B ∈ L(C, X) and F ∈ L(X, C). The following statements hold for a fixed τ > 0: a) The sampled-data system (5) is strongly stable if and only if the discrete semigroup (∆(τ ) k ) k∈N is strongly stable. b) Let g : (0, ∞) → R, and suppose that there exist constants k 0 ∈ N and M 1 , M 2 > 0 such that for all k ∈ N with k ≥ k 0 and all s ∈ [0, τ ), Then the state x of the sampled-data system (5) with initial state x 0 ∈ X satisfies Proof. The statement a) has been proved in Proposition 2.2 in [38], and therefore we show only the statement b). Assume that (11) holds for the state x of the sampled-data system (5) with initial state x 0 ∈ X. Take ε > 0. There exists t 1 > 0 such that for all t ≥ t 1 , Choose k 1 ∈ N so that k 1 ≥ k 0 and k 1 τ ≥ t 1 . By (9) and (10), we have that for all k ≥ k 1 . Hence, (12) holds. Conversely, assume that x 0 ∈ X satisfies (12), and take ε > 0. We have that Then ∥x(kτ + s)∥ ≤ K∥x(kτ )∥ for all k ∈ N 0 and s ∈ [0, τ ). By assumption, there exists k 2 ∈ N such that for all k ≥ k 2 , Combining this estimate and (9), we obtain for all k ≥ k 2 and s ∈ [0, τ ). It follows from (10) that for all k ≥ max{k 0 , k 2 } and s ∈ [0, τ ). Thus, we obtain (11). □ We immediately see that the condition (10) holds for g(t) := t −δ with δ > 0. The function g defined by also satisfies the condition (10). In fact, we obtain

Resolvent conditions for stability of discrete semigroups
First, we review resolvent characterizations of power boundedness and strong stability of discrete semigroups on Hilbert spaces. A resolvent characterization of power bounded discrete semigroups has been obtained in Theorem II.1.12 of [10], which is an analogue of the characterization of uniformly bounded C 0 -semigroups due to [13,35]. Moreover, a resolvent characterization of strongly stable discrete semigroups has been developed in Theorem 3.11 of [36]; see also Theorem II.2.23 of [10].
Theorem 4.1. Let X be a Hilbert space and let ∆ ∈ L(X) satisfy E 1 ⊂ ρ(∆). Then the following statements hold: Next, we investigate a resolvent condition on the rate of decay for discrete semigroups on Hilbert spaces. To this end, the following equalities given in Lemma II.1.11 of [10] are useful. Lemma 4.2. Let X be a Banach space and let ∆ ∈ L(X) with spectral radius r(∆). Then for all k ∈ N and r > r(∆).
We state the discrete analogue of Lemma 3.2 in [39]. Proposition 4.3. Let (∆ k ) k∈N be a power bounded discrete semigroup on a Hilbert space X. The following statements hold for a fixed x ∈ X: where by Parseval's equality for vector-valued functions, which can be proved by the scalar-valued Parseval's equality as done for Plancherel's theorem in Section 1.8 of [2]. Noting that the first equality in Lemma 4.2 is true also for k = 0, we have for each positive integer k, On the other hand, Cauchy's integral theorem implies that for each non-positive integer k, Therefore, we have from the equality (15) that First suppose that 0 < δ < 1/2. Then the estimate (14) yields Since ε > 0 was arbitrary, the desired conclusion (13) holds for 0 < δ < 1/2. Next we consider the case δ = 1/2. By the well-known formula for the first polylogarithm (see, e.g., p. 3 of [16]), we obtain ∞ k=1 1 kr 2k = log Therefore, the estimate (14) gives This proves that (13) holds for δ = 1/2. b) By Lemma 4.2 and the Cauchy-Schwartz inequality, we have that for all x, y ∈ X, r > 1, and k ∈ N, Take r 1 > 1. Since (∆ k ) k∈N is power bounded, Theorem 4.1.a) and the uniform boundedness principle imply that there exists a constant M 1 > 0 such that for all y ∈ X and r ∈ (1, r 1 ). Hence for all x ∈ X and r ∈ (1, r 1 ). Put r := 1 + 1 k+1 . Then r k+2 (k + 1)(r − 1) Moreover, we obtain Hence, there exists k 1 ∈ N such that for all k ≥ k 1 , Combining this estimate with (13), we obtain ∥∆ k x∥ = o(k −δ ) as k → ∞ for 0 < δ < 1/2 and

Spectrum and sampling
To apply Theorem 4.1 to the discretized system (9), we have to show that E 1 ⊂ ρ(∆(τ )) is satisfied. The aim of this section is to prove the following theorem.
First, we apply a spectral decomposition for A. Next, we prove the inclusion This estimate for the continuous-time system leads to an analogous estimate for the discretized system, i.e., a lower bound of |1−F (zI −T (τ )) −1 S(τ )| on ρ(T (τ ))∩E 1 . Finally, the desired inclusion E 1 ⊂ ρ(∆(τ )) is proved. 5.1. Spectral decomposition. We start by applying a spectral decomposition for A under (A1). A more general version of spectral decompositions for unbounded operators can be found in Lemma 2.4.7 of [9] and Proposition IV.1.16 of [11].
Since only finite elements of (λ n ) n∈N are in C −ω ∩ (C \ Ω α,Υ ), there exists a smooth, positively oriented, and simple closed curve Φ in ρ(A) containing σ(A) ∩ C 0 in its interior and σ(A) ∩ (C \ C 0 ) in its exterior. The operator (16) Π : is a projection on X and yields the decomposition We have that dim X + < ∞. Moreover, X + and X − are T (t)-invariant for all t ≥ 0. Define by changing the order of (λ n ) n∈N if necessary. By construction, we obtain N b ≥ N a . The series expansions of A + and A − are given by Then (T + (t)) t≥0 and (T − (t)) t≥0 are C 0 -semigroups with generators A + and A − , respectively. The adjoint Π * is also a projection on X and yields a spectral decomposition for A * . We define From (6), we have that Lemma 5.2. If (A1) and (A2) hold, then the C 0 -semigroups (T − (t)) t≥0 and (T − * (t)) t≥0 constructed as above are polynomially stable with parameter α.

Inclusion
Let a Riesz-spectral operator A on a Hilbert space X generate a C 0 -semigroup. Let B ∈ L(C, X) and F ∈ L(X, C) be such that A + BF is the generator of a uniformly bounded C 0 -semigroup. Then, the fractional power (−A − BF ) β is well defined for every β > 0. 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 [39] for the proof. Lemma 5.3. Let X be a Banach space and let V ∈ L(X). Suppose that A and A + V are the generators of exponentially stable C 0 -semigroups on X.
We investigate the relation between D(A n ) and D((A + BF ) n ) for n ∈ N.
Lemma 5.4. Let A be a linear operator on a Banach space X and let F ∈ L(X, C). Define B ∈ L(C, X) by Bu := bu for u ∈ C, where b ∈ X. Then the following assertion holds for all n ∈ N: If x ∈ D(A n ) and b ∈ D(A n−1 ), then x ∈ D((A + BF ) n ) and where q m := F (A + BF ) n−m−1 x ∈ C for m = 0, . . . , n − 1.
Proof. We prove the assertion by induction. In the case n = 1, x ∈ D(A) satisfies x ∈ D(A + BF ) and (A + BF )x = Ax + (F x)b. Now, assume that the assertion holds for some n ∈ N. Let x ∈ D(A n+1 ) and b ∈ D(A n ). Then A n x ∈ D(A) and for all m = 0, . . . , n − 1. This and the inductive assumption imply and hence x ∈ D((A + BF ) n+1 ). Moreover, Thus, (19) holds when n is replaced by n + 1. □ Combining Lemmas 5.3 and 5.4, we obtain the following result.
Lemma 5.7. If (A1), (A3), and (A4) hold, then there exists ε > 0 such that Moreover, a simple calculation shows that for all λ ∈ ρ(A) ∩ ρ(A + BF ). Hence Lemma 5.6 implies that there exists ε > 0 such that |1 − F R(λ, A)B| > ε for all λ ∈ ρ(A) ∩ C 0 . □ The estimate on the continuous-time system obtained in Lemma 5.7 leads to an analogous estimate on the discretized system as in the robustness analysis of exponential stability [31] and strong stability [38]. To show this, we use the series expansion of R(z, T (τ ))S(τ ) under (A1), where S(τ ) is defined by (8). If 0 ∈ ρ(A), then S(τ ) is written as and hence the series expansion of R(z, T (τ ))S(τ ) is given by for z ∈ ρ(T (τ )). If 0 ̸ ∈ ρ(A), then 0 is a simple eigenvalue of A under (A1). Let n 0 ∈ N satisfy λ n0 = 0. Analogously, we obtain e τ λn − 1 z − e τ λn · ⟨b, ψ n ⟩ λ n ϕ n for z ∈ ρ(T (τ )). Recall that N b ∈ N is chosen so that (18) holds. Therefore, λ n ̸ = 0 for all n ≥ N b . For each n ≥ N b , the nth term of the series expansion of R(z, T (τ ))S(τ ) satisfies the following estimate, which is obtained from arguments similar to those in the proofs of Theorem 2.1 in [31] and Lemma 3.8 in [38].
The proof of Lemma 5.9 is based on the approximation approach developed in the proof of Theorem 2.1 of [31] for the preservation of exponential stability under sampling. We decompose the transfer functions G(λ) := F (λI − A) −1 B and H τ (z) := F (zI − T (τ )) −1 S(τ ) into finite-dimensional truncations and infinitedimensional tails with approximation order N ∈ N: where, for simplicity of notation, we assume that 0 ̸ ∈ {λ n : n ∈ N}. The main idea of the approximation approach in [31] is twofold. First, we prove that the infinite-dimensional tails become arbitrarily small as N increases. Next, we show that if τ > 0 is sufficiently small, then the finite-dimensional truncations with a fixed N ∈ N are close (except near the unstable poles) under the relationship z = e τ λ of the variable λ in the continuous-time setting and the variable z in the discrete-time setting. For the infinite-dimensional tails, a treatment different from the previous studies [31,38] is required due to the geometric property of the eigenvalues of the generator A and the conditions on the control operator B and the feedback operator F . On the other hand, the analysis of the finite-dimensional truncations has no difficulty arising from polynomial stability. Hence, to the finite-dimensional truncations, one can apply the arguments developed in the proof of Theorem 2.1 of [31] with only minor modifications; see also the proof of Lemma 3.8 of [38].
Proof of Lemma 5.9.
Step 1: Let N b ∈ N be such that (18) holds. We show that for all ε > 0, there exists for all N ≥ N c 0 . As in the spectral decomposition described in Section 5.1, there exists a smooth, positively oriented, and simple closed curve Φ b in ρ(A) containing {λ n : 1 ≤ n ≤ N b − 1} in its interior and σ(A) \ {λ n : 1 ≤ n ≤ N b − 1} in its exterior. Define the projection Π b on X by As in Lemma 5.2, (T − b (t)) t≥0 is a polynomially stable C 0 -semigroup with parameter α on X − b . We denote by A − b the generator of (T − b (t)) t≥0 . Theorem 2.2 implies that (18), there exists a constant κ > 0 such that |λ n | ≥ κ for all n ≥ N b . The Cauchy-Schwartz inequality implies that for all N ≥ N b and λ ∈ C 0 , Since b ∈ D β and f ∈ D γ * , we obtain Hence, for all ε > 0, there exists N c 0 ≥ N b such that (38) holds.

Application of resolvent conditions to discretized system
In this section, we complete the proof of the main result, Theorem 3.3. To do so, we prove that for a sufficiently small sampling period τ > 0, the operator ∆(τ ) = T (τ ) + S(τ )F satisfies the integral conditions on resolvents given in Theorem 4.1 and Proposition 4.3. We divide the resolvent R(z, ∆(τ )) into two terms, by applying the well-known Sherman-Morrison-Woodbury formula presented in the next lemma. This formula can be obtained from a straightforward calculation. Lemma 6.1. Let X and U be Banach spaces and let A : D(A) ⊂ X → X be a closed linear operator. Take B ∈ L(U, X), F ∈ L(X, U ), and λ ∈ ρ(A). If 1 ∈ ρ(F R(λ, A)B), then λ ∈ ρ(A + BF ) and Suppose that Assumption 3.2 hold. By Lemmas 5.7 and 5.9, if the sampling period τ > 0 is sufficiently small, then for all z ∈ ρ(T (τ )) ∩ E 1 , one has 1 ∈ ρ(F R(z, T (τ ))S(τ )). Hence the Sherman-Morrison-Woodbury formula presented in Lemma 6.1 yields In what follows, we separately investigate the integrals of ∥R(z, T (τ ))∥ 2 and ∥R(z, T (τ ))S(τ )F R(z, T (τ ))∥ 2 .

6.2.2.
Case γ ≥ α. For the case γ ≥ α and the case β, γ < α, we need a preliminary lemma. Note that a constant Υ 0 in the next lemma depends on the sampling period τ unlike the constants Υ 1 and Υ 2 in Lemma 5.8, because we consider the situation r ↓ 1 for a fixed τ > 0 in Proposition 6.3.
Proposition 6.8. Let A be the generator of a uniformly bounded C 0 -semigroup on a Banach space X such that 0 ∈ ρ(A). Let 0 < β < α. Then there exists a constant ς > 0 such that for all x ∈ X.
We have shown in the proof of a) that the discrete semigroup (∆(τ ) k ) k∈N is strongly stable. Therefore, it is power bounded. Proposition 4.3.b) implies that for all x ∈ D δ , as k → ∞. By Proposition 3.4.b) and the subsequent discussion, we conclude that for every initial state x 0 ∈ D δ , the state x of the sampled-data system (5) satisfies as t → ∞. x 1 x 2 ∈ X, y = y 1 y 2 ∈ X.
Put ζ := −A −1 0 ζ 1 ζ 2 ∈ X, η := 0 η 0 ∈ X, and define V ∈ L(X) by which is in the form of one-rank perturbations. The perturbed wave equation (68) is transformed into the abstract evolution equationẋ = Ax + Bu, where A := A 1 + V with domain D(A) = D(A 1 ). Assume that the perturbations ζ 1 , ζ 2 , and η 0 are chosen so that A is a Riesz-spectral operator of the form (3) and has the spectral properties (A1) and (A2). Such perturbations ζ 1 , ζ 2 , and η 0 can be constructed with minor modifications of the proof of Theorem 13 in [22]; see also Theorem 1 in [40].
We apply the spectral decomposition by the projection Π given in (16). Let N a ∈ N satisfy (17), and assume that ⟨b, ψ n ⟩ ̸ = 0 for all n = 1, . . . , N a − 1. This condition on b is satisfied if and only if there exists f 1 ∈ X + * = Π * X such that the matrix    is Hurwitz; see, e.g., Theorem 8.2.3 of [9]. Choose f 1 ∈ X + * such that the matrix given in (69) is Hurwitz, and define F 1 ∈ L(X, C) by F 1 x := ⟨x, f 1 ⟩, x ∈ X. Since (T − (t)) t≥0 is polynomially stable with parameter α, the C 0 -semigroup (T BF1 (t)) t≥0 generated by A+BF 1 has the same stability property by Theorem 9 of [25].

Conclusion
We have studied the robustness of polynomial stability with respect to sampling. The generator we consider is a Riesz-spectral operator whose eigenvalues may approach the imaginary axis asymptotically. We have presented conditions for the preservation of strong stability under fast sampling. Moreover, an estimate for the rate of decay of the state has been provided for the sampled-data system with a smooth initial state and a sufficiently small sampling period. Future work will focus on relaxing the assumption on the generator and addressing systems with multi-and infinite-dimensional input spaces.