Lyapunov Type Theorems for Exponential Stability of Linear Skew-Product Three-Parameter Semiﬂows with Discrete Time

: For linear skew-product three-parameter semiﬂows with discrete time acting on an arbitrary Hilbert space, we obtain a complete characterization of exponential stability in terms of the existence of appropriate Lyapunov functions. As a nontrivial application of our work, we prove that the notion of an exponential stability persists under sufﬁciently small linear perturbations.


Introduction
The main objective of this paper is to obtain a complete characterization of exponential stability for linear skew-product semiflows with discrete time acting on an arbitrary Hilbert space in terms of the existence of appropriate Lyapunov functions. We then use this characterization to prove that the notion of an exponential stability persists under sufficiently small linear perturbations.
We stress that the use of Lyapunov functions in the study of the stability of trajectories in the theories of differential equations and dynamical systems has a long history that goes back to the landmark work of Lyapunov [1]. For some early contributions to the theory, we refer to books by LaSalle and Lefschetz [2], Hahn [3] and Bhatia and Szegö [4]. For the first contributions dealing with infinite-dimensional dynamics, we refer to the work of Daleckij and Krein [5].
In the context of nonautonomous dynamics, the relationship between exponential dichotomies and the existence of appropriate Lyapunov functions was first considered by Maizel [6]. His results were further developed by Coppel [7,8] as well as Muldowney [9]. We note that these results considered only the case of continuous time. To the best of our knowledge, the first contributions in the case of discrete time are due to Papaschinopoulos [10]. In the recent years, there has been a renewed interest in this topic. More precisely, various characterizations of nonuniform exponential behaviour in terms of Lyapunov functions were obtained (see [11][12][13]). In addition, the authors have obtained first results in the context of infinite-dimensional dynamics [14] (see also [15]) which lead to further developments [16][17][18]. Finally, for some related results in the context of ergodic theory, we refer to [19] and references therein.
The purpose of this paper is to show that techniques we developed in our previous work [14] can be used to obtain Lyapunov-type characterization of exponential stability for a very general type of nonautonomous dynamics. More precisely, we consider the so-called linear skew-product three-parameter semiflows. This notion was introduced by Megan and Stoica [20] and includes various previously studied notions as a particular case (see Examples 1 and 2).
Finally, we would like to mention that Lyapunov type characterizations of exponential stability are certainly not the only tool used to study stability of nonautonomous dynamics. Indeed, there is a vast Example 3. Let σ be a continuous three-parameter flow on a metric space Θ. Furthermore, take a map A : Θ → B(X) such that θ → A(θ)x is continuous for each x ∈ X. For (θ, n) ∈ Θ × Z and x ∈ X, let us consider a Cauchy problem y m+1 = A(σ(θ, m, n))y m m ≥ n, y n = x.
We now introduce the notion of exponential stability.

Some Auxiliary Results
We also recall some useful results established by Daleckij and Krein [5].

Lemma 1.
Assume that H is a Hilbert space and that T is a bounded operator on H. Furthermore, suppose that the spectrum of T does not cover the whole unit circle S 1 . Then every self-adjoint operator bounded operator W on H with the property that there exists δ > 0 such that is invertible.
We will also use the following result (also taken from [5]).

Lemma 2.
Assume that H is a Hilbert space and that T is a bounded operator on H. Furthermore, assume that there exists an invertible, self-adjoint and bounded linear operator W on H such that (2) holds for some δ > 0.
Then, the spectrum of T does not intersect S 1 and there exist δ > 0 satisfying Moreover, if W ≥ 0 (that is, Wx, x ≥ 0 for x ∈ H) then the spectrum of T is contained in {z ∈ C : |z| < 1}.

Main Results
The following is our first main result.
is an exponentially stable linear skew-product three-parameter semiflow over a continuous three-parameter flow σ. Then, there exists a family S (θ,n) , (θ, n) ∈ Θ × Z of bounded, self-adjoint and invertible operators on X and K, δ > 0 such that for (θ, n) ∈ Θ × Z: It follows from (1) that Hence, the first inequality (3) holds. Furthermore, we have that which implies that (4) holds with δ = 1. Set now Clearly, l 2 is a Hilbert space with respect to the scalar product for every x = (x n ) n∈Z ∈ l 2 . Hence, A (θ,n) is well-defined and bounded linear operator for each (θ, n) ∈ Θ × Z. We need the following auxiliary results.
Proof of the Lemma. Take (θ, n) ∈ Θ × Z, we define B (θ,n) : l 2 → l 2 by Obviously, B (θ,n) is a well-defined and bounded linear operator. For x = (x n ) n∈Z and y = (y n ) n∈Z in l 2 , we have that which readily implies the desired conclusion.
Proof of the Lemma. Fix (θ, n) ∈ Θ × Z. Then, for each k ∈ N and x = (x m ) m∈Z ∈ l 2 we have that for each m ∈ Z. This readily yields that (A k (θ,n) ≤ De −λk . Since k ∈ N was arbitrary we conclude that the statement of the lemma holds with t = e −λ < 1.
It follows easily from the already proved first inequality in (3) that W (θ,n) is a well-defined and bounded linear operator on l 2 . Moreover, it is easy to show that W (θ,n) is self-adjoint.
On the other hand, observe that for (θ, n) ∈ Θ × Z and x = (x n ) n∈Z ∈ l 2 , we have that for each m ∈ Z. Hence, the already proved inequality (4) (we recall that it holds with δ = 1) implies that for each (θ, n) ∈ Θ × Z. Hence, Lemmas 1 and 4 imply that W (θ,n) is invertible for every (θ, n) ∈ Θ × Z. By multiplying this identity on the right by (A (θ,n) − Id) −1 and on the left by (A * (θ,n) − Id) −1 , we obtain that Therefore, for every x ∈ l 2 . On the other hand, Combining the last two estimates, we obtain that 2 and thus for x ∈ l 2 . It follows from Lemma 4 that sup (θ,n)∈Θ×Z Hence, there exist R > 0 such that n) x for x ∈ l 2 and (θ, n) ∈ Θ ∈ Z.
Hence, sup and the proof of the lemma is completed.
Proof of the Lemma. Observe that S (θ,n) ≥ Id and thus S (θ,n) is injective. Take v ∈ X and consider y = (y m ) m∈Z ∈ l 2 given by y 0 = v and y m = 0 for m = 0. Since W (θ,n) is invertible, there exists Hence, S (θ,n) is also surjective and thus it is invertible. Moreover, Therefore, S −1 (θ,n) ≤ W −1 (θ,n) for all (θ, n) ∈ Θ × Z. Now the second inequality in (3) follows directly from the previous lemma.
It remains to establish (5). Using the same notation as in the proof of Lemma 5 we have Moreover, multiplying this equality on the left by A (θ,n) − Id and on the right by A * (θ,n) − Id yields that Hence, Observe that for each x ∈ l 2 , we have that Since there exists L > 0 such that for every x ∈ l 2 . By applying (7) for x = (x m ) m∈Z ∈ l 2 given by x m = 0 for m = 1 and where v ∈ X is arbitrary, we conclude that (5) holds with δ = 1 L > 0.
We now establish the converse of Theorem 1.
The conclusion of the lemma now readily follows.
Take now (θ, n) ∈ Θ × Z, v ∈ X and consider a sequence y = (y m ) m∈Z by It is easy to verify that Then, Lemma 7 implies that there exist C > 0 such that In particular, (9) implies that Φ(θ, k, n)v ≤ C v , for (θ, n) ∈ Θ × Z, k ≥ n and v ∈ X.
Take now θ ∈ Θ, v ∈ X and m ≥ n. Then, for each n ≤ k ≤ m we have that Summing over k and using (9), we obtain that Thus, Consequently, there exist N 0 ∈ N such that Φ(θ, m, n) ≤ e −1 , for θ ∈ Θ and m, n ∈ Z such that m − n ≥ N 0 .

Applications
In this section, we use Theorems 1 and 2 to prove that the notion of exponential stability persists under sufficiently small linear perturbations.

1.
Φ is exponentially stable; 2. there exists c > 0 such that Then, if c is sufficiently small, Ψ is also exponentially stable.

Conclusions
In this paper, we obtained a complete Lyapunov-type characterization of exponential stability for linear skew-product three-parameter semiflows with discrete time. More precisely, we proved that exponential stability can be described in terms of the existence of appropriate quadratic Lyapunov functions. We then applied these results and prove that the notion of exponential stability persists under sufficiently small linear perturbations.
Author Contributions: D.D. and C.P. contributed equally in the preparation of this paper. All authors have read and agreed to the published version of the manuscript.
Funding: This research was funded by the Croatian Science Foundation under the project IP-2019-04-1239 and by the University of Rijeka under the projects uniri-prirod-18-9 and uniri-prprirod-