Stability and convergence properties of forced inﬁnite-dimensional discrete-time Lur’e systems

Incremental stability and convergence properties for forced, inﬁnite-dimensional, discrete-time Lur’e systems are addressed. Lur’e systems have a linear and nonlinear component and arise as the feedback interconnection of a linear control system and a static nonlinearity. Discrete-time Lur’e systems arise in, for example, sampled-data control and integro-diﬀerence models. We provide conditions, reminiscent of classical absolute stability criteria, which are suﬃcient for a range of incremental stability properties and input-to-state stability (ISS). Consequences of our results include suﬃcient conditions for the converging-input converging-state (CICS) property, and convergence to periodic solutions under periodic forcing.


Introduction
In systems and control theory, feedback interconnections comprising a linear system in the forward path and a static nonlinearity in the feedback path, as shown in Figure 1.1, are commonly referred to as Lur'e systems. In this paper, we investigate certain stability and convergence properties of forced, infinite-dimensional, discrete-time Lur'e systems. Our focus is centred around incremental stability notions, input-to-state stability (ISS) and converging-input convergingstate (CICS) properties. The concept of ISS first appeared in 1989 in [45] and is a stability concept pertaining to the states of (possibly nonlinear) control systems subject to external or exogenous inputs. It ensures boundedness of the state in terms of the initial states and inputs, respectively, generalising the familiar additive estimate for some γ ∈ (0, 1) and Γ > 0, for the state x of the exponentially stable discrete-time linear system x(t + 1) = Ax(t) + u(t), x(0) = x 0 , ∀ t ∈ Z + , with input u. Importantly, the constants γ and Γ are independent of x 0 and u. Since its inception, much attention has been devoted to ISS with numerous papers on the subject including, but not restricted to, [10,19,20,21,22,23,24,46,48,49]. The reader is referred to [8,47] for overviews of key input-to-state stability ideas.
A related concept is incremental input-to-state stability (δISS), which is simply an incremental version of the ISS concept and ensures boundedness of the difference of two state trajectories in terms of the difference of the initial conditions and the difference of the inputs. Evidently, for linear systems the notions of δISS and ISS coincide. The paper [1] constructs a suite of Lyapunov methods for δISS for finite-dimensional, continuous-time nonlinear control systems.
The study of the stability properties of Lur'e systems constitutes absolute stability theory which seeks to conclude stability of the feedback system shown in Figure 1.1 through the interplay of frequency domain properties of the linear system Σ and the boundedness or sector properties of the nonlinearity f . Absolute stability theory is typically concerned with the development of criteria for global asymptotic stability of an equilibrium (typically zero) of unforced Lur'e systems [25,27], or for L 2 -stability in an input-output setting [50]. Classical absolute stability results include the circle criterion, the (complexified) Aizerman conjecture, and the Popov criterion [17,25,50].
As is well-known, global asymptotic or exponential stability of an equilibrium of an unforced nonlinear system does, in general, not guarantee any stability or boundedness properties of the system in the presence of forcing. A recent line of enquiry [2,3,4,19,20,42,43,44] has been investigating to what extent classical absolute stability criteria can be modified to ensure ISS and state convergence properties of forced Lur'e systems. Indeed, a key finding has been that existing absolute stability criteria, under slightly stronger assumptions, are sufficient for ISS in many cases. The papers [2,19,20,42,43,44] derive, in finite-dimensional settings, sufficient conditions for ISS which are reminiscent of the complexified Aizerman conjecture and the circle criterion. Furthermore, ISS properties underpin the paper [4], which considers the converginginput converging-state (CICS) property for finite-dimensional, continuous-time Lur'e systems.
Here we consider incremental ISS notions for forced infinite-dimensional discrete-time Lur'e systems. Our main result, Theorem 3.2, presents sufficient conditions for δISS in terms of a "nonlinear incremental ball condition" inspired by the complexified Aizerman conjecture. We appeal to exponential weighting and small-gain arguments to obtain a special type of δISS, that being, exponential δISS. We subsequently utilise δISS to obtain several different stability and convergence results. In particular, Corollary 3.3 provides an ISS result, Corollary 3.7 provides sufficient conditions for δISS which are reminiscent of the circle criterion, Theorem 4.3 gives sufficient conditions for the CICS property and Theorem 4.8 presents sufficient conditions for the existence of, and convergence to, periodic solutions under periodic forcing. At the time of writing, the study of ISS for infinite-dimensional, continuous-time control systems is an emerging research area, with papers including [9,16,18,34,35,37,38], but to the best of our knowledge, there is no existing literature that overlaps with the current paper.
A motivation for the present study is its applicability in numerous areas. One such application is to infinite-dimensional sampled-data systems (see, for example, [28,29,30,40]). The sampled-data systems considered here are obtained from the feedback interconnection of a continuoustime infinite-dimensional system and a static nonlinearity using sample-and hold-operations. Theorem 6.1 provides conditions which guarantee that if the continuous-time feedback system is ISS, then the corresponding sampled-data system is ISS provided the sampling period is sufficiently small. A second class of examples arises in ecological modelling, and are so-called integral projection models (IPMs) [6,12,13], which are integro-difference models typically used for populations partitioned according to a continuous variable such as size or weight. The modelling assumption that there are both linear and nonlinear vital rates means that IPMs often naturally lead a Lur'e system structure. In Example 7.2, we demonstrate that, under natural assumptions, the theory developed in Sections 3 and 4 applies to a forced IPM for the plant Platte thistle, based on the model found in [39].
The paper is structured as follows. In Section 2 we gather relevant preliminary results regarding linear infinite-dimensional discrete-time systems. Section 3 contains the main results germane to ISS and δISS. Then, in Section 4 we utilise these results to yield convergence properties. Section 5 and Section 6 comprise applications of our earlier results in the form of 'four-block' systems and sampled-data systems, respectively. Finally, Section 7 contains detailed discussions of two examples.
For normed spaces V and W, we let L(V, W ) denote the normed space of bounded linear operators from V to W and set L(V ) := L(V, V ). Recall that an operator A ∈ L(V ) is exponentially (or power) stable if the spectral radius of A is strictly less than 1. It is well known that A ∈ L(V ) is exponentially stable if, and only if, there exist M ≥ 1 and µ ∈ (0, 1) such that In addition, the infimum of all µ > 0 satisfying (1.1) for some M ≥ 1, is equal to the spectral radius of A. For L ∈ L(V, W ) and r > 0, we let denote the open ball of radius r, centred at L. Throughout, for given normed spaces V and W we set For a Banach space W and α > 0, we define the Hardy space with norm given by g H ∞ α := sup z∈Eα g(z) .
For ease of notation, we define H ∞ (W ) := H ∞ 1 (W ) . For t ∈ R, we define ⌊t⌋ to be the greatest integer less than or equal to t and ⌈t⌉ to be the smallest integer greater than or equal to t, that is, the floor and ceiling of t, respectively. For a given τ ∈ Z + , we define τ := {0, 1, . . . , τ }, τ := {τ, τ + 1, . . .} and the left-shift operator Λ τ by (Λ τ v)(t) := v(t + τ ) for every t ∈ Z + and every v : Z + → V . For v : Z + → V and t ∈ Z + , we set We denote the set of functions from Z + → V by V Z + and the set of continuous functions R + → V by C(R + , V ). Finally, for v e ∈ V , we will abuse notation and interchangeably write v e to denote an element of V and the constant function Z + → V with value v e .

Preliminaries
To begin with, we present some preliminary results regarding the following linear difference equation Here X and V are complex Banach spaces and U and Y are complex Hilbert spaces. The variables x and y in (2.1) are called the state and output, respectively, and u and v are inputs. Occasionally, it will be convenient to identify the linear system (2.1) and the sextuple (A, B, B e , C, D, D e ) and to refer to the linear system (A, B, B e , C, D, D e ). For ease of notation, we set Σ := (A, B, B e , C, D, D e ) ∈ L.
Before continuing, it is worth noting that (2.1) encompasses other seemingly more general linear systems. For instance, the linear system where v 1 ∈ X Z + and v 2 ∈ Y Z + , is a special case of (2.1) with V = X × Y , B e = I 0 , We record some definitions associated with (2.1). First, we define the behaviour of (2.1) as and set G(z) = C(zI − A) −1 B + D, a L(U, Y )-valued function of the complex variable z, the so-called transfer function of (2.1) from u to y. If µ denotes the exponential growth constant of A, then G ∈ H ∞ α (L(U, Y )) for all α > µ, meaning that G is bounded and holomorphic on the exterior of any open disc in C centred at 0 with radius greater than µ. We say that Σ ∈ L is stabilisable (respectively detectable) if (A, B, C, D) is (exponentially) stabilisable (respectively detectable).
We define the set of admissible feedback operators by For L ∈ A(D), we set We denote by G L the transfer function of (A L , B L , C L , D L ), that is, where the second equality follows easily from (2.2). For given L ∈ A(D), the operators A L , B L , C L and D L arise by applying the feedback u = Ly +ũ to (2.1) whereũ ∈ U Z + . The transfer function fromũ to y is G L . Finally, we define the set of (complex) stabilising feedback operators We next state three lemmas which underpin our development. The first lemma is a discrete-time version of [15,Proposition 5.6]. We omit the proof, since it is similar to that in [15].
The following lemma relates B lin to the behaviour of a certain feedback system, a process often called 'loop shifting' in control theory. The proof is relegated to Appendix A.
Lemma 2.2. Let Σ ∈ L and L ∈ A(D). The quadruple (u, v, x, y) is in B lin if, and only if, (u, v, x, y) satisfies We set Σ L := (A L , B L , B e + B L LD e , C L , D L , (I − DL) −1 D e ) ∈ L.
We now state the third lemma, the proof of which is elementary, and is therefore omitted.
Lemma 2.3. Let Σ ∈ L and assume that A is exponentially stable. Then there exist c 1 , c 2 , c 3 > 0 such that, for every (u, v, x, y) ∈ B lin , we have The nonlinear control systems considered in the current paper are given by the interconnection of (2.1) with the nonlinear feedback u = f (y + w) for some f : Y → U , where w ∈ Y Z + is an output disturbance (see Figure 2.1).
Namely, we study We note that in the case that I − Df is invertible, then (2.3) may be expressed more succinctly as We define the behaviour of (2.3) as .
Associated with (2.3) is the following initial-value-problem.
For a given x 0 ∈ X, v ∈ V Z + and w ∈ Y Z + , we say that (x, y) ∈ X Z + × Y Z + is a solution of (2.5) if x(0) = x 0 and (v, w, x, y) ∈ B. It is straightforward to prove that if the map I − Df is surjective, then, for a given x 0 ∈ X, v ∈ V Z + and w ∈ Y Z + , solutions to (2.5) exist. It is also straightforward to prove that if I − Df is injective, then, for a given x 0 ∈ X, v ∈ V Z + and w ∈ Y Z + , there is at most one solution of (2.5). We note that both of these properties are evidently satisfied if D = 0. The following example demonstrates that each of the previous conclusions need not hold if injectivity or surjectivity of I − Df are respectively dropped.
ii) Let f (s) = 2s − s 2 − s 3 , for all s ∈ C. Note that it is easy to verify that there exist multiple solutions to (2.6) with x 0 = 1 and v = w = 0. ♦

Exponential incremental stability
In this section, we recall notions of exponential input-to-state stability, define notions of exponential incremental input-to-state stability and present a condition which guarantees that the Lur'e system (2.3) is exponentially incrementally input-to-state/output stable. Throughout the following definitions we let Σ ∈ L and f : Y → U . A quadruple (v e , w e , x e , y e ) ∈ V × Y × X × Y is called an equilibrium quadruple of (2.3) if (v e , w e , x e , y e ) ∈ B. An equilibrium quadruple (v e , w e , x e , y e ) is said to be exponentially input-to-state stable (ISS) if there exist c > 0 and a ∈ (0, 1) such that, for all (v, w, x, y) ∈ B we have Further, an equilibrium quadruple is said to be exponentially input-to-state/output stable (ISOS) if there exist c > 0 and a ∈ (0, 1) such that, for all (v, w, x, y) ∈ B, (3.1) holds and We say that (2.3) is exponentially ISS (respectively, ISOS) if (0, 0, 0, 0) is an exponentially ISS (respectively, ISOS) equilibrium quadruple of (2.3).
The following example demonstrates a situation where (2.3) is exponentially ISS but not exponentially ISOS.
is exponentially stable, Since BC = 0 = BD and A is exponentially stable, it is easily checked that (2.3) is exponentially ISS. We shall now show that (2.3) is not exponentially ISOS. To this end, first let x 0 ∈ R 2 and let x := (x 1 , x 2 ) T ∈ (R 2 ) Z + be such that By setting w := (−x 1 , 0) T ∈ (R 2 ) Z + and y := (y 1 , 0) T ∈ (R 2 ) Z + for arbitrary y 1 ∈ R Z + , we see that and, since BC = 0 = BD and Bw = 0, Hence, (0, w, x, y) ∈ B. This holds for any y 1 ∈ R Z + and so (2.3) is not ISOS. ♦ For a non-empty subset S ⊆ Y, we define the following sub-behaviour of (2.3) and note that B Y = B.
We say that (2.3) is exponentially incrementally input-to-state stable (δISS) with respect to the non-empty sets S 1 , S 2 ⊆ Y if there exist c > 0 and a ∈ (0, 1) such that, for all (v 1 , w 1 , x 1 , y 1 ) ∈ B S 1 and for all (v 2 , w 2 , x 2 , y 2 ) ∈ B S 2 we have Further, (2.3) is exponentially incrementally input-to-state/output stable (δISOS) with respect to the sets S 1 and S 2 , if there exist c > 0 and a ∈ (0, 1) such that, for all (v 1 , w 1 , x 1 , y 1 ) ∈ B S 1 and for all (v 2 , w 2 , x 2 , y 2 ) ∈ B S 2 , (3.2) holds and In the case that f (0) = 0, if (2.3) is δISS (respectively, δISOS) with respect to S 1 := Y and S 2 := {0}, then, trivially, (2.3) is also ISS (respectively, ISOS). The Lur'e system (2.3) is said to be exponentially δISS or exponentially δISOS if S 1 = S 2 = Y in the above respective definitions. Trivially, exponential δISOS with respect to S 1 and S 2 implies exponential δISS with respect to the same sets. The following theorem is the main result of this section.
Theorem 3.2. Let Σ ∈ L be stabilisable and detectable and let S 1 , S 2 ⊆ Y be non-empty. Assume that r > 0 and K ∈ L(Y, U ) satisfy B(K, r) ⊆ S(G) and that there exists δ ∈ (0, r) such that Then the following hold.
As an immediate consequence of Theorem 3.2, by taking q = ∞, we obtain the following corollary.   Proof of Theorem 3.2. The proof uses a combination of small-gain and exponential weighting arguments. Since Σ is stabilisable and detectable, it follows that Σ K is as well. Moreover, since G K ∈ H ∞ (L(U, Y )), it follows that A K is exponentially stable, with exponential growth constant µ ∈ (0, 1). Let α ∈ (µ, 1) so that G K ∈ H ∞ α (L(U, Y )) and consider G K on the closed annulus A := {z ∈ C : β ≤ |z| ≤ 1}, where β ∈ (α, 1). Owing to the continuity of G K on E α , G K is uniformly continuous on A. Thus, there exists γ ∈ (0, 1 − β) such that for all s 1 , s 2 ∈ A with |s 1 − s 2 | < γ, we have To prove statement (i), set ω : By the choice of ρ, we have that and thus ρA K is exponentially stable.
Remark 3.4. (a) By inspecting the above proof, we are able to see that Theorem 3.2 holds true if X and V are real Banach spaces and Y and U are real Hilbert spaces, provided that the complex ball condition where Y c and U c denote the complexifications of Y and U , respectively. The same can be said of the rest of the results in sections 3, 4, 6 and 5.
(b) For later purposes, it will be useful to consider Theorem 3.2 in the (rather degenerate) situation wherein G = 0. If G = 0, then G K = 0 for all K ∈ L(Y, U ) and by Lemma 2.1, it follows that B(K, r) ⊆ S(G) for all r > 0. Consequently, in the case wherein G = 0, the conclusions of Theorem 3.2 hold, provided that there exists K ∈ L(Y, U ) such that We next present a corollary to Theorem 3.2 which is reminiscent of the circle criterion. To this end, for α ∈ (0, 1], we denote by H * α (L(U, Y )) the set of functions H : E α → L(U, Y ) which are holomorphic on E α , with the exception of isolated singularities, that is, poles and essential singularities. We always assume that removable singularities have been removed via holomorphic extension. For convenience, we set H * (L(U, Y )) := H * 1 (L(U, Y )). Let H ∈ H * α (L(U )). We define Σ H ⊆ E α to be the set of isolated singularities of H. The function H is said to be positive real if The following lemma is an analogue to [15,Proposition 3.3], which concerns positive real functions on the right-half complex plane, for positive real functions on the exterior of the unit disc. We relegate the proof to Appendix A. We shall also require the following technical lemma, the proof of which is identical, mutatis mutandis, to that of [15,Corollary 2.3].
Corollary 3.7. Let Σ ∈ L be stabilisable and detectable, S ⊆ Y non-empty and K 1 , The following proof is in part inspired by a method outlined in the proof of [15, Theorem 6.8].
Proof. We define L : Hence L * L is bounded away from 0 and, by combining this with the self-adjointness of L * L, we have that L * L is invertible. We define L # := (L * L) −1 L * and let Q := LL # . It is clear that Therefore, Q is the orthogonal projection onto (ker L # ) ⊥ along ker L # . Utilising this with (3.22) gives that Moreover, since L # is bounded away from 0 on im L, there exists ν > 0 such that Hence, combining this with (3.23) yields and so Next, on the one hand we compute that and, on the other, that Invoking the positive realness of (I − K 2 G)(I − K 1 G) −1 along with Lemma 3.6, employing the expressions (3.25) and (3.26) yields that In addition, evidently LG and thus, by Lemma 2.1, we see that Finally, let (v, w, x, y) ∈ B and note that, since I = L # L,

Convergence properties
In this section we use Theorem 3.2 to establish convergence properties of the state x and output y of the Lur'e system (2.3) when subject to converging or periodic inputs v and w.

The converging-input converging-state property
Here we give conditions under which the Lur'e system exhibits the so-called converging-input converging-state property. We say that, for Σ ∈ L and f : Y → U , the discrete-time Lur'e system (2.3) has the converging-input converging- We note that some authors (see, for example, [47]) use the term CICS for the special case wherein v ∞ = 0, w ∞ = 0 and x ∞ = 0.
Our main result of this section is Theorem 4.3, from which we obtain sufficient conditions for the CICS property in Corollary 4.6. We comment that if w in (2.3) is perceived to be an output disturbance to the system, then convergence of w is not an assumption which will be generically satisfied. Hence, in addition to considering the CICS property, we also develop a result for bounded but not necessarily convergent w, see Corollary 4.7.
Let K ∈ S(G) and define the map (4.1) For ease of notation in the sequel, for given ξ ∈ Y , we write F −1 K (ξ) to denote the inverse image of the singleton {ξ} under F K , instead of the more cumbersome F −1 K ({ξ}). Moreover, we denote the cardinality of F −1 K (ξ) by #F −1 K (ξ). To facilitate the proofs of the main results in this section, it is useful to state two lemmas, the proofs of which may be found in Appendix A.
Lemma 4.1. Let Σ ∈ L, S ⊆ Y be non-empty, K ∈ S(G), F K be given by (4.1), and assume that γ := 1/ G K H ∞ < ∞ and Then the following statements hold.
If there exists δ > 0 such that f and K satisfy (3.3) with r = γ and S 1 = S 2 = Y , then (ii) F K is globally Lipschitz and bijective; (iii) the inverse F −1 K is globally Lipschitz.
Although we assume that γ < ∞ in Lemma 4.1, if actually G K H ∞ = 0, then F K is the identity map which is trivially globally Lipschitz and bijective.
Lemma 4.2. Let Σ ∈ L be stabilisable and detectable, v ∞ ∈ V and w ∞ ∈ Y. Assume that K ∈ S(G) and is nonempty. Let z ∞ ∈ T K and define y ∞ := z ∞ − w ∞ and Then
The formulae in (4.3), (4.4) and (4.5) are motivated by the desire to solve the steady-state equations where v ∞ and w ∞ are given, for x ∞ and y ∞ to yield an equilibrium quadruple (v ∞ , w ∞ , x ∞ , y ∞ ). However, I − A need not be invertible, and so G(1) not well defined, hence the inclusion of the loop-shifting term K. In the simple case wherein U = V , B e = B, D e = D, K = 0 and w ∞ = 0, the condition that and, in this case, (4.4) and (4.5) respectively read We now state the main theorem in this section.
Before proving Theorem 4.3, we provide some commentary.
(b) Assumption (3.3) with r = γ and S 1 = S 2 = Y may be rewritten as which trivially implies (4.2) with S = Y , and is itself equivalent to the function ξ → f (ξ) − Kξ being globally Lipschitz with Lipschitz constant smaller than γ. In this case, arguments similar to those used in the proof of Lemma 4.1 show that the map I −D K (f −K) is bijective and hence, by using Lemma 2.2, for all x 0 ∈ X, v ∈ V Z + and all w ∈ Y Z + , the initial-value problem (2.5) has a unique solution.
(c) Under the assumptions of Theorem 4.3, a consequence of Lemma 4.1 is that the "steady state gain maps" The convergence property provided by Theorem 4.3 is uniform in the following sense: given a set of inputs V ⊆ V Z + × Y Z + which is equi-convergent to (v ∞ , w ∞ ) and κ > 0, the set of solutions is equi-convergent to (x ∞ , y ∞ ). ♦ Proof of Theorem 4.3. First, statement (i) of Lemma 4.1 yields that #T K = 1. Using Lemma 4.2 gives that (v ∞ , w ∞ , x ∞ , y ∞ ) is an equilibrium quadruple of (2.3) and since y ∞ + w ∞ ∈ S, we have that (v ∞ , w ∞ , x ∞ , y ∞ ) ∈ B S . We invoke statement (ii) of Theorem 3.2, with q = ∞, to obtain c, d > 0 and a ∈ (0, 1) such that for all (v, w, x, y) ∈ B and all t ∈ Z + , we have and Let (v, w, x, y) ∈ B and fix t ∈ Z + . Note that (4.8) and (4.9) hold for from the time-invariance property (2.4). In light of the identity ⌈t/2⌉ + ⌊t/2⌋ = t, it follows that Appealing to (4.8) again yields Finally, using the properties of the ceiling and floor functions we arrive at Starting instead from (4.9) and proceeding in the same manner, we obtain a similar estimate for y(t) − y ∞ . Thus, we obtain (4.6) after estimating and relabelling the constants appropriately.  As previously mentioned at the start of this section, if w in (2.3) is considered to be an output disturbance, then it may be unreasonable to expect convergence of w. The next result is an immediate corollary to Theorem 4.3 and yields that asymptotic 'closeness' of the state and output of (2.3) to the equilibrium components x ∞ and y ∞ , respectively, is linearly bounded by w ℓ ∞ .
Proof. The claim follows from (4.6), the time-invariance property (2.4), and a standard timeinvariance argument.

Periodic inputs
For given τ ∈ N and normed space W , we say that We say that (v, w, x, y) ∈ B is τ -periodic if each of the functions v, w, x and y is τ -periodic.
Theorem 4.8. Let τ ∈ N and let v p ∈ V Z + and w p ∈ Y Z + be τ -periodic. If the assumptions of Theorem 3.2 hold with S 1 = S 2 = Y , then there exist a unique τ -periodic trajectory (v p , w p , x p , y p ) ∈ B and κ > 1 such that Proof. The proof is in part inspired by that of [1,Proposition 4.4]. The hypotheses of Theorem 3.2 hold and so by statement (ii) of that result with q = ∞, it follows that there exist c > 0 and θ ∈ (0, 1) such that (3.6) holds for all (v 1 , w 1 , x 1 , y 1 ), (v 2 , w 2 , x 2 , y 2 ) ∈ B. An application of statement (ii) of Lemma 4.1 gives that F K is bijective and so, see Remark 4.4 (b), for each v e ∈ V and w e ∈ Y, there exist (unique) x e ∈ X and y e ∈ Y such that (v e , w e , x e , y e ) is an equilibrium quadruple of the Lur'e system (2.3). Let (v p , w p , x, y) ∈ B. Invoking (3.6) with (v p , w p , x, y) and (v e , w e , x e , y e ), we see that there exists µ > 0 such that hence showing that x and y are bounded. Moreover, since (Λ σ v p , Λ σ w p , Λ σ x, Λ σ y) ∈ B for every σ ∈ Z + and Λ σ v p = Λ σ+kτ v p and Λ σ w p = Λ σ+kτ w p for every k, σ ∈ Z + , statement (ii) of Theorem 3.2 ensures that there exist c > 0 and θ ∈ (0, 1) such that Thus, for all t, n, m ∈ Z + with m ≥ n, we have Therefore, (Λ nτ x) n∈Z + and (Λ nτ y) n∈Z + are Cauchy sequences in ℓ ∞ (X) and ℓ ∞ (Y ), respectively. We denote their respective limits by x p and y p . The calculation shows that x p is τ -periodic. The τ -periodicity of y p is proven similarly. We proceed to show that (v p , w p , x p , y p ) ∈ B. Indeed, for all n ∈ Z + , we have that (4.12) The estimate (3.3) gives that f is continuous and hence, by taking the limit as n → ∞ in (4.11) and (4.12), we yield that Moreover, let κ ∈ (1, 1/θ) and invoke (3.6) to obtain Finally, to establish uniqueness of (v p , w p , x p , y p ), assume that (v p , w p ,x p ,ỹ p ) ∈ B is also τperiodic. Then (4.10) implies that x p =x p and y p =ỹ p , hence completing the proof.

Application to four-block Lur'e systems
In the following, we demonstrate how the results of earlier sections apply to the related class of so-called "four-block" Lur'e systems which are informally described by the block diagram arrangement in Figure 5.1, where Σ = (A, B, B e , C, D, D e ) ∈ L and the signal y is given by In this section we use superscripts to denote decompositions of signals, as opposed to subscripts which have been used to distinguish trajectories in the context of incremental stability. The motivation for studying the four-block setting is that there may be outputs which are of interest, but not used for feedback (denoted y 1 in Figure 5 Throughout this section, we assume that the output space Y is of the form Y = Y 1 × Y 2 , where Y 1 and Y 2 are complex Hilbert spaces, and we define the maps P j : Y → Y j , y 1 y 2 → y j , j = 1, 2 . To fix notation, we assume that (only) the component y 2 := P 2 y of (2.3) is used for feedback purposes, giving rise to the Lur'e system where C j := P j C, D j := P j D, D j e = P j D e . In this section, we set Σ j := (A, B, B e , P j C, P j D, P j D e ) and we denote the behaviour of (5.1) byB, that is, As before, we write G(z) = C(zI − A) −1 B + D.
Our main result of this section states that the conclusions of Theorem 3.2 apply to (5.1) provided the linear system Σ 2 and f satisfy the assumptions of Theorem 3.2.
Corollary 5.1. Let Σ ∈ L, let S 1 , S 2 ⊆ Y 2 be non-empty. Assume that Σ 2 is stabilisable and detectable, r > 0 and K 2 ∈ L(Y 2 , U ) satisfy B(K 2 , r) ⊆ S(P 2 G) and that there exists δ ∈ (0, r) such that (3.3) holds with K and Y replaced by K 2 and Y 2 , respectively. Then the conclusions of Theorem 3.2 hold for the Lur'e system (5.1).
Proof. In the following, we shall only prove that statement (i) of Theorem 3.2 holds for (5.1), since the proof of statement (ii) for (5.1) is similar. We shall consider the Lur'e system which is obtained from (5.1) by applying P 2 to the output equation. Note that the Lur'e system (5.2) is of the form (2.3) with Y , C, D, D e and y replaced by Y 2 , C 2 , D 2 , D 2 e and y 2 , respectively.
We close the current section by remarking that the various results presented in Sections 3 and 4 for the forced Lur'e system (2.3) also have obvious extensions to the four-block settings considered here, namely system (5.1). For brevity and to avoid repetition, we do not give formal statements of these results.

Application to sampled-data systems
In this section we provide an application of Theorem 3.2 in the form of an ISS result for a class of forced, infinite-dimensional sampled-data control systems.
Let A be the generator of a strongly continuous semigroup on X, denoted by (T(t)) t≥0 , B ∈ L(U, X) and C ∈ L(X, Y ), and consider the following continuous-time, infinite-dimensional linear systemẋ = Ax + Bu + v, x(0) = x 0 ∈ X, y = Cx . (6.1) As usual, x and y in (6.1) denote the state and output, and u and v are inputs, with the former being available for feedback purposes.
Throughout this section, we assume that • X, U and Y are Hilbert spaces, with U and Y finite-dimensional; • the pair (A, B) is (exponentially) stabilisable, that is, there exists F ∈ L(X, U ) such that the strongly continuous semigroup generated by A + BF is exponentially stable; • the pair (C, A) is (exponentially) detectable, that is, there exists H ∈ L(Y, X) such that the strongly continuous semigroup generated by A + HC is exponentially stable.
Let ω(T) be the exponential growth constant of T, that is, For a fixed sampling period τ > 0, we define the sampling operator S : and the (zero order) hold operator H as which maps U Z + into the set of step-functions mapping [0, ∞) to U . We shall consider the forced sampled-data Lur'e system arising from the feedback interconnection of (6.1) and the nonlinear sampled-data output feedback control where w ∈ Y Z + is an output disturbance and f : Y → U with f (0) = 0. Thus, for given x 0 ∈ X, v ∈ L ∞ loc (R + , X) and w ∈ Y Z + , we consider the initial-value probleṁ We say that x ∈ C(R + , X) is a (mild) solution to (6.3) if x satisfies x(0) = x 0 and It is clear that, for all x 0 ∈ X, v ∈ L ∞ loc (R + , X) and w ∈ Y Z + , there exists a unique solution of (6.3). Note that if x 0 = 0, v = 0 and w = 0, then 0 is a solution of (6.3), as f (0) = 0.
The sampled-data Lur'e system (6.3) is said to be exponentially input-to-state stable (ISS) if there exist constants c, γ > 0 such that, for all initial states x 0 ∈ X, all inputs v ∈ L ∞ loc (R + , X) and all output disturbances w ∈ Y Z + , the solution x of (6.3) satisfies The following theorem gives a sufficient condition for exponential ISS of (6.3). Theorem 6.1. Assume that K ∈ S c (H) and where r < 1/ sup s∈C 0 H K (s) . Then there exists τ * > 0 such that (6.3) is exponentially ISS for all τ ∈ (0, τ * ).
We note that under the assumptions of Theorem 6.1, it follows from [16,Theorem 4.1] that the continuous-time Lur'e systeṁ is exponentially ISS. Theorem 6.1 shows that exponential ISS is inherited by the sample-hold discretization (6.2) of the continuous-time system (6.6), provided the sampling period is sufficiently small.
To facilitate the proof of Theorem 6.1, we state a technical lemma. To this end, for τ > 0, we set and, for L ∈ L(Y, U ) and r > 0, we let Lemma 6.2. Let r > 0 and K ∈ L(Y, U ) and assume that B cl (K, r) ⊆ S c (H). Then there exists τ * > 0 such that for all L ∈ B cl (K, r) and every τ ∈ (0, τ * ), the operator A L τ is exponentially stable.
To avoid disruption of the flow of the presentation, the proof of the lemma is placed at the end of this section.

T(s)
and note that, for all k ∈ Z + , Hence, there exists c 2 > 0 such that It remains to use the discrete-time estimate (6.8) to bound the state x over all times. To this end, note that for all k ∈ Z + and all t ∈ (0, τ ], Appealing to (6.5), we estimate Taking norms in (6.9) and substituting in (6.10) and (6.11) yields that, for all k ∈ Z + and all t ∈ (0, τ ], The claim now follows in light of the above inequality and (6.8).
Proof of Lemma 6.2. The proof is a refinement of that of [29,Theorem 3.1]. For F ∈ L(Y, U ), we let T F denote the strongly continuous semigroup generated by A + BF C. By hypothesis, B cl (K, r) ⊆ S c (H), (A, B) is stabilisable and (C, A) is detectable, and so, by [7, Theorem 7.32], for each F ∈ B cl (K, r), there exist ω F < 0 and M F ≥ 1 such that T F (t) ≤ M F e ω F t for all t ≥ 0. We seek to show that there exists ω < 0 and 1 ≤ M < ∞ such that To this end, note that for each F ∈ B cl (K, r), there exists ε F > 0 such that and thus, by [36, Theorem 1.1, Chapter 3], The balls B(F, ε F ) form an open cover of B cl (K, r) and, since U and Y are finite dimensional, B cl (K, r) is compact. Hence, there exist finitely many F 1 , . . . , F n ∈ B cl (K, r), ε 1 , . . . , ε n ∈ (0, ∞) and ω 1 , . . . , ω n ∈ (−∞, 0) such that B cl (K, r) ⊆ ∪ n i=1 B(F i , ε i ) and . . , n}.
Next, we claim that for all ε > 0, there exists T > 0 such that To prove (6.13), we will show that for all ε > 0 and all F ∈ B cl (K, r), there exist r F > 0 and T F > 0 such that and then use another compactness argument. To this end, fix ε > 0 and let F ∈ B cl (K, r). Since U and Y are finite dimensional and C is bounded, it follows that F C ∈ L(X, U ) is a compact operator. Furthermore, as X is a Hilbert space, [ Therefore, we invoke [29, Lemma 2.1] to yield that Choose r F > 0 such that and letT F > 0 be such that We invoke [36, Corollary 1.3, Chapter 3] to obtain LetT F > 0 be such that Setting T F := min{T F ,T F }, it follows that, for all t ∈ [0, T F ] and all L ∈ B(F, r F ), Hence, for all ε > 0 and for all F ∈ B cl (K, r), there exists r F > 0 and T F > 0 such that A compactness argument similar to that establishing (6.12) can now be used to prove that for all ε > 0, there exists T > 0 such that (6.13) holds.
Finally, we seek to use (6.12) and (6.13) to yield the existence of τ * > 0 such that A L τ is discrete-time exponentially stable for all L ∈ B cl (K, r) and every τ ∈ (0, τ * ). To that end, fix L ∈ B cl (K, r). The variation-of-parameters formula for perturbed semigroups [36, equation (1.2), page 77] gives, for all τ ≥ 0 and all x ∈ X, where P τ x := τ 0 T(τ − s)BLC(I − T L (s))xds for all x ∈ X. As in [29, Theorem 3.1], let us introduce a new norm on X given by where ω < 0 is as in (6.12). Note that where M ≥ 1 is as in (6.12). For all x ∈ X and all t ≥ 0, we have Therefore, For G ∈ L(X), let |G| denote the operator norm of G induced by the new norm, that is, Combining (6.14) with (6.15), (6.16) and the inequality e ωτ ≤ 1 + ωτ e ωτ , ∀ τ ∈ R + , we obtain that Combining this with (6.13) shows that, for fixed δ ∈ (0, −ω), there exists τ * > 0 (independent of L ∈ B cl (K, r)) such that Finally, invoking (6.15), we obtain that, for all τ ∈ (0, τ * ) and all n ∈ Z + , In light of (6.17), the above inequality yields the exponential stability of A L τ for all L ∈ B cl (K, r) and all τ ∈ (0, τ * ), completing the proof.

Examples
We conclude the paper with a detailed discussion of two examples.
Example 7.1. Consider the following controlled and observed heat equation describing the temperature evolution in a unit rod Here z(ξ, t) denotes the temperature of the rod at position ξ and time t, z 0 ∈ L 2 (0, 1) is the initial temperature distribution, χ [1/2,1] is the indicator function of the interval [1/2, 1] and b e ∈ L 2 (0, 1). Further, u and v are inputs and y is the output (or observation which has a simple pole at s = 0 and so (7.1) is neither exponentially nor input-output stable. To illustrate the sampled-data control results of Section 6, we consider the following problem: find conditions which are sufficient for the sampled-data system given by (7.1) and the feedback (6.2) to be exponentially ISS.
We set b e = χ [1/4,1/2] , and define the constant and periodic input v 1 = 3 and v 2 (t) = 3 sin(2t) for all t ≥ 0, respectively. Certain simulations use the initial temperature distribution z 0 (ξ) = e −|ξ−1/2| 2 . We simulate the closed-loop feedback system (7.1) and (6.2) by performing a semidiscretization in space using a finite-element method with 31 elements, the details of which are given in Appendix B. Figures 7.3(a)-7.3(d) show plots of x(t) against t in the following situations described in Table 7.2. Simulation data are also listed in Table 7.2. To conclude the example, we comment that although taking τ = 0.25 appears to "work", in the sense that the numerical results agree with what the theory predicts, in fact the constant τ * , the existence of which is guaranteed by Theorem 6.1, could be either smaller or larger than 0.25. Determining the maximal τ * analytically or numerically is a difficult open problem. It seems that Figure 7.3(d) shows divergence when τ = 2.5, indicating that τ * < 2.5. ♦ Example 7.2. We consider a forced Integral Projection Model (IPM) for the monocarpic plant Platte thistle (Cirsium canescens), based on the model presented in [5,39]. For a recent overview of IPMs we refer the reader to [33]. Platte thistle is a perennial plant native to central North America. The IPM describes the distribution of plant size, according to the natural logarithm of the crown diameter in mm. Following [5], we assume that the continuous variable of natural logarithm of crown diameter takes minimum and maximum values given by m 1 = −0.5 and m 2 = 3.5 (so that roughly e m 1 = 0.6mm and e m 2 = 33mm), respectively, and that the timesteps correspond to years. Incorporating an additive input, the model is where η(t, ·) denotes the distribution of plant size at time-step t, with initial distribution η 0 ∈ L 1 ([m 1 , m 2 ]). In the following, our aim is to write (7.5) in the form of a forced, infinitedimensional Lur'e system (2.3) with the natural state space X = L 1 (Ω), where Ω := [m 1 , m 2 ]. Before doing this, we provide some commentary on the model (7.5).
The first term on the right hand side of the difference equation in (7.5) models survival and growth of existing plants. Here p(ξ, ζ) denotes the probability of an individual of size ζ surviving to one of size ξ in one time-step, and is assumed in [5,39] to have the structure where s(ζ) is the survival probability of an individual of size ζ, f p (ζ) is the probability that an individual of size ζ flowers, and g(ξ, ζ) is the probability of an individual of size ζ growing to size ξ, each over one time-step. We take s, f p and g as in [5, Table 2]. The term 1 − f p appears on the right-hand side of (7.6) as flowering is fatal to Platte thistle, that is, it is monocarpic.
The second term on the right hand side of the difference equation in (7.5) models reproduction and recruitment into the population. In particular, b ∈ X denotes the distribution of offspring plant size, c * x equals the total number of new seeds recruited into the population by the distribution z ∈ X in one time-step, and is given by In addition to the terms in (7.6), S(θ) denotes the number of seeds produced on average by a plant of size θ. We take b = J, where J is as in [5, Table 2], and the function S is given in [5, Table 2]. We have c * ∈ X * as θ → s(θ)f p (θ)S(θ) ∈ L ∞ (Ω). The function h in (7.5) denotes the probability of seed germination, and is a nonlinear function of the total number of seeds produced, and so seeks to model density-dependence in the seed germination probability. As such, it is assumed to be non-increasing, representing competition or crowding affects at higher seed abundances. Two situations are explored in [5]: first, h is constant with value 0.067, and; second, h is defined by h(s) = s −0.33 . We note that there is uncertainty in modelling nonlinear terms for Platte thistle, see [11], and in order to demonstrate different settings where the incremental condition (3.3) holds, we shall choose a different h below.
The third term term on the right hand side of the difference equation in (7.5) is an additive input, which may be the arrival of new plants via planned replanting schemes, or accidental movement. We assume that b e ∈ X and v ∈ (R + ) Z + , which capture the distribution and magnitude, respectively.
Combining the above, and setting x(t) = η(t, ·) for all t ∈ Z + , we see that (7.5) may be written as a forced Lur'e system, on the state-space X = L 1 (Ω), and with U = V = Y = R. Here f (s) := h(s)s for s ≥ 0 and we extend h and f to all of R by setting h(s) = f (s) = 0 for s ∈ (−∞, 0). The extension is to ensure that f is defined on the whole of Y , so that the results of the paper are applicable.
We seek to apply Theorems 3.2, 4.3 and 4.8 to (7.7) to infer various (incremental) stability and convergence notions. To simulate (7.7) we use a finite-element approximation, the details of which are given in Appendix C.
The property A < 1 implies that B(0, r) ⊆ S(G) for all r where ρ 1 , ρ 2 and ρ 3 are positive parameters. Broadly, ρ 1 captures the probability of germination at low abundance, ρ 2 determines the rate of transition and ρ 3 the value at which the transition occurs. Figure 7.4 contains plots of h for several parameter values.
If (7.9) holds, then Theorem 3.2 yields that (7.7) is exponentially ISS. If the inequality holds, then f satisfies (3.3) with K = 0, S 1 = S 2 = R, and hence Theorem 3.2 yields that (7.7) is exponentially δISS. In this case, it also follows from Theorem 4.3 that (7.7) has the CICS property. Moreover, the inequality (7.10) is sufficient for the hypotheses of Theorem 4.8 to hold, which ensures that (7.7) admits a periodic trajectory when subject to periodic inputs, and that all other trajectories generated by the same periodic input asymptotically approach this trajectory.
Numerical simulations are plotted in Figure 7.6. Throughout we take and with these parameter values it can be shown that f satisfies (7.10). Additional simulation data are recorded in Table 7.5. Panels (a)-(d) respectively show: the 0-GES property; ISS; incremental stability; and asymptotically periodic response to periodic forcing. Note that panel (d) shows that, asymptotically, the responses to the same input are identical and do not depend on the initial conditions, thereby illustrating a typical aspect of ISS. ♦
Let v ∈ U be such that J v = ∅. Such a v ∈ U does exist, because otherwise Lemma 2.1 in [15] would yield that H −j = 0 for every j > 1 and so z 0 would not be a singularity, thus yielding a contradiction. Define h ∈ H * (C) by h(z) = H(z)v, v for all z ∈ E. If J v is infinite, then h has an essential singularity at z 0 and so, using the Casorati-Weierstrass theorem ([41, Theorem 10.21]), there exists z * ∈ ∆ such that Re H(z * )u, u = Re h(z * ) < 0, contradicting the positive realness of H.
Now, assume that J v is finite and set k := max J v . In this case, h has a pole of order k at z 0 and so h can be written as where h 0 = 0, g is holomorphic on ∆ ∪ {z 0 } and g(z 0 ) = 0. For sufficiently small r > 0, we have h(z 0 + re iθ ) = r −k e −ikθ (h 0 + g(z 0 + re iθ )), ∀ θ ∈ (−π, π]. Let θ 0 ∈ (−π, π] be such that Re(e −ikθ 0 h 0 ) < 0. Note that, using that g(z 0 ) = 0, we obtain, for sufficiently small r > 0, Contradicting the positive realness of H. Consequently H does not have any singularities in E.
Proof of Lemma 4.2. Setting f K (ξ) := f (ξ) − K(ξ) and invoking definitions of x ∞ and y ∞ , we obtain Noting that F K (z ∞ ) = C K (I −A K ) −1 B e +B K KD e v ∞ +(I −DK) −1 D e v ∞ + I +G K (1)K w ∞ , it is easily seen that as required. To prove that (v ∞ , w ∞ , x ∞ , y ∞ ) is an equilibrium quadruple, we note that Invoking Lemma 2.2 completes the proof.

B Sampled-data example: further details
We provide details on the numerical approximation used in Example 7.1. We first derive the weak form of (7.1), from which the finite-element approximation is computed. Given z 0 ∈ X and v ∈ L ∞ loc (R + ), let z denote a solution of (7.1). Multiplying both sides of the PDE in (7.1) by ψ ∈ H 1 (0, 1), integrating over the spatial domain and integrating by parts gives Observe that we do not need to impose any boundary conditions on the space of test functions, so we take H 1 (0, 1) as the test function space. We seek an approximate solution to (B.1) of the form where N ∈ N and φ j ∈ H 1 (0, 1) are the usual (piecewise linear) hat or tent functions over the uniform mesh on [0, 1], and the a i (t) are scalar coefficients. Therefore, setting In our numerical simulations we take N = 30. The resulting sampled-data ordinary differential equation is solved numerically using the Mathwork's MATLAB [32] command ode45, over each sampling period.

C IPM example: further details
We provide details on the numerical approximation used in Example 7.2. For notational convenience in this section we set N := {1, 2, . . . , N } for each N ∈ N. To derive a finite element approximation of the forced IPM (7.7), we first derive a weak form. For which purpose, we multiply both sides of (7.7) by ψ ∈ L 1 (Ω) and integrate over Ω to give We seek an approximate solution to (C.1) of the form x N (t, ξ) = N j=1 a j (t)φ j (ξ), ∀ t ∈ Z + , ∀ ξ ∈ Ω , (C.2) where N ∈ N, φ j are given L 1 functions and a j (t) are scalar coefficients. Substituting (C.2) into (C.1), and testing against ψ = φ i for each i ∈ N gives N j=0 Ω φ i (ξ)φ j (ξ) dξ a j (t + 1) = N j=0 Ω φ i (ξ) Ω p(ξ, ζ)φ j (ξ) dξ a j (t) Noting that Since we are seeking to approximate the L 1 (Ω) functions x(t, ·) in L 1 (Ω) (that is, we are not approximating any derivatives), we choose as finite-dimensional approximation spaces the linear span of N piecewise constant functions. Specifically, for fixed N ∈ N, we define ξ j := m 1 + j(m 2 − m 1 )/N for j ∈ {0, 1, . . . , N }, ∆ := (m 2 − m 1 )/N and An advantage of such a choice is that, as readily seen, M = I, because M ij = 0 if i = j and Consequently, (C.4) becomes z + = Dz + F f (Lz) + Jv . (C.7) Moreover, by inspection of (C.5) and (C.6), it follows that with the above choice of piecewise constant φ j , the matrices D, L, F and J in (C.6) are componentwise nonnegative.
For the simulations in Example 7.2 we use (C.7) with N = 30, and the matrices D, L, J and F are computed numerically.