Nonlinear small-gain theorems for input-to-state stability of infinite interconnections

We consider infinite heterogeneous networks, consisting of input-to-state stable subsystems of possibly infinite dimension. We show that the network is input-to-state stable, provided that the gain operator satisfies a certain small-gain condition. We show that for finite networks of nonlinear systems this condition is equivalent to the so-called strong small-gain condition of the gain operator (and thus our results extend available results for finite networks), and for infinite networks with a linear gain operator they correspond to the condition that the spectral radius of the gain operator is less than one. We provide efficient criteria for input-to-state stability of infinite networks with linear gains, governed by linear and homogeneous gain operators, respectively.


Introduction
We live in a hyperconnected world, where the size of networks and the number of connections between their components are rapidly growing. Emerging technologies such as the Internet of Things, Cloud Computing, 5G communication, and so on make this trend even more distinct. Such complex networked systems include smart grids, connected vehicles, swarm robotics, and smart cities in which the participating agents may be plugged in and out from the network at any time.
The unknown and possibly time-varying size of such networks poses new challenges for stability analysis and control design. One of the promising approaches to this problem is to over-approximate the network by an infinite network, and perform We start by introducing a general class of infinite-dimensional control systems, which includes many classes of evolution PDEs, time-delay systems, ODEs, infinite switched systems, etc. Next, we introduce the concept of infinite interconnections for systems of this class, extending the framework developed in [27,37]. Theorems 6.1 and 6.4 are our small-gain results for uniform global stability of infinite networks. They use the monotone bounded invertibility (MBI) property of the gain operator , which is equivalent for finite networks (see Proposition 7.12) to the strong small-gain condition, employed in the small-gain analysis of finite networks in [18,Thm. 8] and [37]. The proof of this result is based on the proof of the corresponding result for finite networks, see [18,Thm. 8].
Theorems 6.2 and 6.5 are our ISS small-gain results for infinite networks in semi-maximum and summation formulation, which state that an infinite network consisting of ISS systems is ISS provided that the discrete-time system induced by the gain operator has the so-called monotone limit (MLIM) property. This property concerns the input-to-state behavior of the discrete-time control system x(k + 1) ≤ Γ (x(k)) + u(k) induced by the gain operator Γ and is implied by ISS of this system for monotonically decreasing solutions and in turn implies the monotone bounded invertibility property.
In Section 7, we analyze the MBI and MLIM properties, which are employed in the small-gain analysis of infinite networks of ISS systems. In Section 7.1, we characterize the MBI property in terms of the uniform small-gain condition, which is a uniform version of the classical small-gain condition Γ (x) ≥ x for all x ≥ 0. In Section 7.2, we relate the uniform small-gain condition to the strong and robust strong small-gain conditions, which have already been exploited in the small-gain analysis of finite [18,16] and infinite [14] networks. In Section 7.3, we show in Proposition 7.12 that the uniform and strong small-gain conditions as well as the MBI and MLIM properties are equivalent for finite-dimensional nonlinear systems, if the gain operator is of summation or max-type. As a consequence of Proposition 7.12, we see that our results extend those of [37], and thus also the classical small-gain theorems for finitely many finite-dimensional systems from [18], even with minimal regularity assumptions on the interconnection (we require well-posedness and the BIC property only).
In Appendix A, we derive a characterization of exponential ISS (eISS) for discretetime systems with a generating and normal cone, induced by homogeneous of degree one and subadditive operators (Proposition A.1). We apply this and recent results in [19], to show in Proposition 7.16 that for linear infinite-dimensional systems with a generating and normal cone the MBI, MLIM and the uniform small-gain condition all are equivalent to the spectral small-gain condition (saying that the spectral radius of the gain operator is less than one).
Finally, in Appendix B, we study relations between various uniform and non-uniform small-gain conditions for max-form gain operators with nonlinear gains, which are of particular importance in small-gain theory. Following [14], we study also the properties of the strong transitive closure of the gain operator. We use these properties to show (in Proposition 7.17) the equivalence of the MBI property, the MLIM property, and the existence of a path of strict decay for the max-form gain operator with linear gains. The results of that section are important for the development of linear and nonlinear Lyapunov-based small-gain theorems for infinite networks.
Propositions 7.16, 7.17 and A.1 are useful, in particular, to obtain efficient smallgain theorems for infinite networks with linear gains, see Corollaries 6.3, 6.6.

Preliminaries
Notation. We write R for the real numbers, Z for the integers, and N = {1, 2, 3, . . .} for the natural numbers. R + and Z + denote the sets of nonnegative reals and integers, respectively.
For a normed linear space (W, · W ) and any r > 0, we write B r,W := {w ∈ W : w W < r} (the open ball of radius r around 0 in W ). By B r,W we denote the corresponding closed ball. If the space W is clear from the context, we simply write B r and B r , respectively. For any nonempty set S ⊂ W and any x ∈ W , we denote the distance from x to S by dist(x, S) := inf y∈S x − y W .
For a set U , we let U R+ denote the space of all maps from R + to U . For a nonempty set J ⊂ R + , we denote by w J the sup-norm of a bounded function w : J → W , i.e., w J = sup s∈J w(s) W . Given a nonempty index set I, we write ℓ ∞ (I) for the Banach space of all functions x : I → R with x ℓ∞(I) := sup i∈I |x(i)| < ∞. Moreover, ℓ ∞ (I) + := {x ∈ ℓ ∞ (I) : x(i) ≥ 0, ∀i ∈ I}. We write 1 for the vector in ℓ ∞ (I) + whose components are all equal to 1. If I = N, we simply write ℓ ∞ and ℓ + ∞ , respectively. By e i , i ∈ I, we denote the i-th unit vector in ℓ ∞ (I).
Throughout the paper, all considered vector spaces are vector spaces over R.
Ordered vector spaces and positive operators. In the following, X always denotes a real vector space. For two sets A, B ⊂ X, we write A + B = {a + b : a ∈ A, b ∈ B}, −A = {−a : a ∈ A}, and R + · A = {r · a : a ∈ A, r ∈ R + }.
Recall that a partial order on a set X is a relation on X which is reflexive, transitive and antisymmetric. A subset X + ⊂ X is called a (positive) cone in X if (i) X + ∩ (−X + ) = {0}, (ii) R + · X + ⊂ X + , and (iii) X + + X + ⊂ X + . A cone X + introduces a partial order "≤" on X via x ≤ y whenever y − x ∈ X + .
The pair (X, X + ) is also called an ordered vector space. If X is a Banach space and the cone X + is closed, we call (X, X + ) an ordered Banach space. In this case, the cone X + is called generating if X + + (−X + ) = X. Clearly, a cone X + is generating if and only if X + spans X. If the cone X + is generating, then by [1,Thm. 2.37] there exists a constant M > 0 such that every x ∈ X can be decomposed as The norm in X is called monotone if for any In this case, one can always find an equivalent norm which is monotone [1,Thm. 2.38].
Let (X, X + ) and (Y, Y + ) be ordered vector spaces. We say that a map f :

Control systems and their stability
In this paper, we work with the following definition of a control system (which provides all the features that are necessary for a global stability analysis).
Definition 3.1 Consider a triple Σ = (X, U, φ) consisting of the following: (i) A normed vector space (X, · X ), called the state space.
(ii) A vector space U of input values and a normed vector space of inputs (U, · U ), where U is a linear subspace of U R+ . We assume that the following axioms hold: -The axiom of shift invariance: for all u ∈ U and all τ ≥ 0, the time-shifted function u(· + τ ) belongs to U with u U ≥ u(· + τ ) U . -The axiom of concatenation: for all u 1 , u 2 ∈ U and for all t > 0 the concatenation of u 1 and u 2 at time t, defined by is called the maximal domain of definition of the mapping t → φ(t, x, u), which we call a trajectory of the system.
(Σ4) The cocycle property: for all x ∈ X, u ∈ U and t, h ≥ 0 so that [0, This class of systems encompasses control systems generated by ordinary differential equations, switched systems, time-delay systems, many classes of partial differential equations, important classes of boundary control systems and many other systems.
An important property of ordinary differential equations with Lipschitz continuous right-hand sides is the possibility of extending a solution, which is bounded on a time interval [0, t), to a larger time interval [0, t + ε).
Next, we introduce the input-to-state stability property, which unifies the classical asymptotic stability concept with the input-output stability notion, and is one of the cornerstones of nonlinear control theory [35,45].
Two properties, implied by ISS, will be important in the sequel: Definition 3.6 A forward complete system Σ = (X, U, φ) has the bounded input uniform asymptotic gain (bUAG) property if there exists a γ ∈ K ∪ {0} such that for all ε, r > 0 there is a time τ = τ (ε, r) ≥ 0 for which The UGS and bUAG properties are extensions of global Lyapunov stability and uniform global attractivity to systems with inputs.
The following lemma provides a useful criterion for the input-to-state stability in terms of uniform global stability and the bUAG property (see [37,Lem. 3.7]). It is a special case of stronger ISS characterizations shown in [39] and [37,Sec. 6].
Lemma 3.7 Let Σ = (X, U, φ) be a control system with the BIC property. If Σ is UGS and has the bUAG property 1 , then Σ is ISS.

Infinite interconnections
In this subsection, we introduce (feedback) interconnections of an arbitrary number of control systems, indexed by some nonempty set I. For each i ∈ I, let (X i , · Xi ) be a normed vector space which will serve as the state space of a control system Σ i . Before we can specify the space of inputs for Σ i , we first have to construct the overall state space. In the following, we use the sequence notation (x i ) i∈I for functions with domain I. The overall state space is then defined as It is a vector space with respect to pointwise addition and scalar multiplication, and we can turn it into a normed space in the following way: The state space X is a normed space with respect to the norm If all of the spaces (X i , · Xi ) are Banach spaces, then so is (X, · X ).
The proof of the proposition is straightforward, hence we omit it.
We also define for each i ∈ I the normed vector space X =i by the same construction as above, but for the restricted index set I \ {i}. Then X =i can be identified with the closed linear subspace {(x j ) j∈I ∈ X : x i = 0} of X.
Now consider for each i ∈ I a control system of the form where PC b (R + , X =i ) is the space of all globally bounded piecewise continuous functions w : R + → X =i , with the norm w ∞ = sup t≥0 w(t) X =i . The norm on Here we assume that U ⊂ U R+ for some vector space U , and U satisfies the axioms of shift invariance and concatenation. Then, by the definition of PC b (R + , X =i ) and the norm (4.1), these axioms are also satisfied for the product space PC b (R + , X =i ) × U.
Definition 4.2 Given the control systems Σ i (i ∈ I) as above, assume that there is a map φ : D φ → X, defined on D φ ⊂ R + × X × U, such that: where φ =i (·) = (φ j (·, x, u)) j∈I\{i} for all i ∈ I. 2 We also assume that φ is maximal in the sense that no other mapφ :D φ → X withD φ ⊃ D φ exists, which satisfies all of the above properties, and coincides with φ on D φ . If the map φ is unique with above properties, and if Σ = (X, U, φ) is a control system satisfying BIC property, then Σ is called the (feedback) interconnection of the systems Σ i .
We then call X =i the space of internal input values, PC b (R + , X =i ) the space of internal inputs, and U the space of external inputs of the system Σ i . Moreover, we call Σ i the i-th subsystem of Σ. ⊳ The stability properties introduced above are defined in terms of the norm of the whole input, and this is not suitable for the consideration of coupled systems, as we are interested not only in the collective influence of all inputs on a subsystem, but in the influence of particular subsystems on a given subsystem. The next definition provides the needed flexibility.
Definition 4.3 Given the spaces (X j , · Xj ), j ∈ I, and the system Σ i for a fixed i ∈ I, we say that Σ i is input-to-state stable (ISS) (in semi-maximum formulation) if Σ i is forward complete and there are γ ij , γ j ∈ K ∪ {0} for all j ∈ I, and β i ∈ KL such that for all initial states x i ∈ X i , all internal inputs w =i = (w j ) j∈I\{i} ∈ PC b (R + , X =i ), all external inputs u ∈ U and t ≥ 0: Here we assume that the functions γ ij satisfy sup j∈I γ ij (r) < ∞ for every r ≥ 0 (implying that the sum on the right-hand side is finite) and γ ii = 0. ⊳ The functions γ ij and γ i in this definition are called (nonlinear) gains.
Assuming that all systems Σ i , i ∈ I, are ISS in semi-maximum formulation, we can define a nonlinear monotone operator Γ ⊗ : ℓ ∞ (I) + → ℓ ∞ (I) + from the gains γ ij by In general, Γ ⊗ is not well-defined. It is easy to see that the following assumption is equivalent to Γ ⊗ being well-defined.
Lemma 4.5 Assumption 4.4 is equivalent to the existence of ζ ∈ K ∞ and a ≥ 0 such that sup i,j∈I γ ij (r) ≤ a + ζ(r) for all r ≥ 0.
⊳ Finally, we provide a criterion for continuity of Γ ⊗ (this criterion with a slightly different statement can already be found in [14,Lem. 2.1], however, without proof).
Proof First, we show that Γ ⊗ is well-defined. Fixing some r > 0, the family {γ ij } (i,j)∈I 2 is uniformly equicontinuous on the compact interval [0, r], which follows by a compactness argument. Hence, we can find δ > 0 so that We can assume that δ is of the form r/n for an integer n. Then for all (i, j) ∈ I 2 . Hence, Γ ⊗ is well-defined.
Now we prove continuity. Choose any ε > 0, fix some s 0 ∈ ℓ ∞ (I) + and let s ∈ ℓ ∞ (I) + so that s − s 0 ℓ∞(I) ≤ δ for some δ > 0 to be determined. By the required equicontinuity, we can choose δ small enough so that |γ ij (s 0 In the last inequality, we use the estimate and the analogous estimate in the other direction.

⊓ ⊔
Another formulation of ISS for the systems Σ i is as follows. In this formulation, we need to assume that I is countable.
Definition 4.8 Assume that I is a nonempty countable set. Given the spaces (X j , · Xj ), j ∈ I, and the system Σ i for a fixed i ∈ I, we say that Σ i is input-tostate stable (ISS) (in summation formulation) if Σ i is forward complete and there are γ ij , γ j ∈ K ∪ {0} for all j ∈ I, and β i ∈ KL such that for all initial states Here we assume that the functions γ ij are such that j∈I γ ij (r) < ∞ for every r ≥ 0 (implying that the sum on the right-hand side is finite) and γ ii = 0. ⊳ Remark 4.9 If a network has finitely many components, ISS in summation formulation, and ISS in semi-maximum formulation are equivalent concepts. Nevertheless, even for finite networks the gains in semi-maximum formulation and the gains in summation formulation are distinct, and for some systems one formulation is better than the other one in the sense that it produces tighter (and thus smaller) gains. This motivates the interest in analyzing both formulations. We illustrate this by examples in Sections 6.3, 6.4. In fact, also more general formulations of input-tostate stability for networks are studied in the literature [16], using the formalism of monotone aggregation functions. ⊳ Assuming that all systems Σ i , i ∈ I, are ISS, we can define a nonlinear monotone operator Γ ⊞ : ℓ ∞ (I) + → ℓ ∞ (I) + from the gains γ ij as follows: Again, Γ ⊞ might not be well-defined, hence we need to make an appropriate assumption.

Remark 4.11
Assume that all the gains γ ij , (i, j) ∈ I 2 , are linear functions. Then the gain operator Γ ⊞ can be regarded as a linear operator on ℓ ∞ (I) and Assumption 4.10 is equivalent to Γ ⊞ being a bounded linear operator on ℓ ∞ (I). ⊳ Proposition 4.12 Assume that the operator Γ ⊞ is well-defined. A sufficient criterion for continuity of Γ ⊞ is that each γ ij is a C 1 -function and sup i∈I j∈I Using the assumption that each γ ij is a C 1 -function and writing s 0 max := s 0 ℓ∞(I) , we can estimate this by By assumption, the last supremum is finite, which implies that δ can be chosen small enough so that the whole expression is smaller than ε.

⊓ ⊔
We also need versions of UGS for the systems Σ i .

Definition 4.13
Given the spaces (X j , · Xj ), j ∈ I, and the system Σ i for a fixed i ∈ I, we say that Σ i is uniformly globally stable (UGS) (in semi-maximum formulation) if Σ i is forward complete and there are γ ij , γ j ∈ K ∪ {0} for all j ∈ I, and σ i ∈ K ∞ such that for all initial states Here we assume that the functions γ ij are such that sup j∈I γ ij (r) < ∞ for every r ≥ 0 (implying that the sum on the right-hand side is finite) and γ ii = 0. ⊳ Definition 4.14 Let I be a countable index set. Given the spaces (X j , · Xj ), j ∈ I, and the system Σ i for a fixed i ∈ I, we say that Σ i is uniformly globally stable (UGS) (in summation formulation) if Σ i is forward complete and there are γ ij , γ j ∈ K ∪ {0} for all j ∈ I, and σ i ∈ K ∞ such that for all initial states Here we assume that the functions γ ij are such that j∈I γ ij (r) < ∞ for every r ≥ 0 (implying that the sum on the right-hand side is finite) and γ ii = 0. ⊳

Stability of discrete-time systems
In this section, we study stability properties of the system Here, (X, X + ) is an ordered Banach space, A : X + → X + is a nonlinear operator on the cone X + , and the input u is an element of ℓ ∞ (Z + , X + ), where the latter space is defined as A solution of the equation (5.1) is a mapping x : Z + → X + that satisfies (5.1). We call a mapping x : As we will see, for the small-gain analysis of infinite interconnections, the properties of the gain operator and the discrete-time system (5.1) induced by the gain operator, are essential. So we now relate the stability of the system (5.1) to the properties of the operator A.
Definition 5. 1 The system (5.1) has the monotone limit property (MLIM) if there is ξ ∈ K ∞ such that for every ε > 0, every constant input u(·) :≡ w ∈ X + and every decreasing solution x : Definition 5.2 Let (X, X + ) be an ordered Banach space and let A : X + → X + be a nonlinear operator. We say that id − A has the monotone bounded invertibility (MBI) property if there exists ξ ∈ K ∞ such that for all v, w ∈ X + the following implication holds:

Proposition 5.3
Let (X, X + ) be an ordered Banach space and let A : X + → X + be a nonlinear operator. If system (5.1) has the MLIM property, then the operator id − A has the MBI property.
Since this holds for every ε > 0, we obtain v X ≤ ξ( w X ), which completes the proof.

⊓ ⊔
Whether the MBI property is strictly weaker than the MLIM property, or whether they are equivalent, is an open problem. In the next proposition, though, we show that they are equivalent under certain assumptions on the operator A or the cone X + . Later, in Propositions 7.16 and 7.17, we show their equivalence for linear operators and for the gain operator Γ ⊗ with linear gains, defined on ℓ ∞ (I).
The cone X + is said to have the Levi property if every decreasing sequence in X + is norm-convergent [1, Def. 2.44 (2)]. Typical examples are the standard cones in L pspaces for p ∈ [1, ∞), and the standard cone in the space c 0 of real-valued sequences that converge to 0. We note in passing that if the cone X + has the Levi property, then it is normal [1, Thm. 2.45].
Proposition 5.4 Let (X, X + ) be an ordered Banach space with normal cone and let A : X + → X + be a nonlinear, continuous and monotone operator. If the cone X + has the Levi property or if the operator A is compact (i.e., it maps bounded sets onto precompact sets), then the following statements are equivalent: (i) System (5.1) satisfies the MLIM property.
(ii) The operator id − A satisfies the MBI property.
Proof In view of Proposition 5.3, it suffices to prove the implication "(ii) ⇒ (i)". Hence, consider a constant input u(·) :≡ w ∈ X + and a decreasing sequence x(·) in X + such that x(k + 1) ≤ A(x(k)) + w for all k ∈ Z + . As A is monotone, the operatorÃ(x) := A(x) + w,Ã : X + → X + , is monotone as well; if A is compact, then so isÃ. Moreover, Now consider the sequence y(k) :=Ã(x(k)), k ∈ Z + . As x is decreasing, so is y.
Next, we note that y converges in norm. Indeed, if the cone has the Levi property, this is clear. If the cone does not have the Levi property, then A, and thusÃ, is compact by assumption. So y has a convergent subsequence; since y is decreasing and the cone is normal, it thus follows that y converges itself.
Taking the limit for k → ∞ and using continuity of A results in y * ≤Ã(y * ) = A(y * ) + w. Since this can be written as (id − A)(y * ) ≤ w, the MBI property of id − A gives y * ≤ ξ( w ). As X + is a normal cone, there is δ > 0 such that for every ε > 0 there is k > 0 large enough, for which This completes the proof. ⊓ ⊔ Nonlinear small-gain theorems for ISS of infinite interconnections 13 6 Small-gain theorems

Small-gain theorems in semi-maximum formulation
In this subsection, we prove small-gain theorems for UGS and ISS, both in semimaximum formulation. We start with UGS.
Theorem 6.1 (UGS small-gain theorem in semi-maximum formulation) Let I be an arbitrary nonempty index set, (X i , · Xi ), i ∈ I, normed spaces and Furthermore, let the following assumptions be satisfied: pointwise for all i ∈ I. (iii) Assumption 4.4 is satisfied for the operator Γ ⊗ defined via the gains γ ij from (i) and id − Γ ⊗ has the MBI property.
Then Σ is forward complete and UGS. Proof Abbreviatingφ j (·) =φ j (·, x j , (φ =j , u)) and using assumption (i), we can estimate From the inequalities (using continuity of s → φ(s, x, u)) From Assumption (ii), it follows that also the vectors σ(x) := (σ i ( x i Xi )) i∈I and γ(u) := (γ i ( u U )) i∈I are contained in ℓ ∞ (I) + . Hence, we can write the inequalities (6.1) in vectorized form as By Assumption (iii), this yields for some ξ ∈ K ∞ , independent of x, u: and we conclude that which is a UGS estimate with σ(r) := ξ(2σ max (r)), γ(r) := ξ(2γ max (r)) for Σ for all (t, x, u) ∈ D φ . Since Σ has the BIC property by assumption, it follows that Σ is forward complete and UGS. Theorem 6.2 (Nonlinear ISS small-gain theorem in semi-maximum formulation) Let I be an arbitrary nonempty index set, (X i , · Xi ), i ∈ I, normed spaces and Furthermore, let the following assumptions be satisfied: (i) Each system Σ i is ISS in the sense of Definition 4.3 with β i ∈ KL and nonlinear gains γ ij , γ i ∈ K ∪ {0}. (ii) There are β max ∈ KL and γ max ∈ K so that β i ≤ β max and γ i ≤ γ max pointwise for all i ∈ I. (iii) Assumption 4.4 holds and the discrete-time system
Then Σ is ISS.
Proof We show that Σ is UGS and satisfies the bUAG property, which implies ISS by Lemma 3.7.
UGS. This follows from Theorem 6.1. Indeed, the assumptions (i) and (ii) of Theorem 6.1 are satisfied with σ i (r) := β i (r, 0) ∈ K and the gains γ ij , γ i from the ISS estimates for Σ i , i ∈ I. From Proposition 5.3 and Assumption (iii) of this theorem, it follows that Assumption (iii) of Theorem 6.1 is satisfied. Hence, Σ is forward complete and UGS.
bUAG. As Σ is the interconnection of the systems Σ i and since Σ is forward complete, we have φ i (t, x, u) =φ i (t, x i , (φ =i , u)) for all (t, x, u) ∈ R + × X × U and i ∈ I, with the notation from Definition 4.2.

⊓ ⊔
For finite networks, Theorem 6.2 was shown in [37]. However, in the proof of the infinite-dimensional case there are essential novelties, which are due to the fact that the trajectories of an infinite number of subsystems do not necessarily have a uniform speed of convergence. This resulted also in a strengthening of the employed small-gain condition.
In the special case when all interconnection gains γ ij are linear, the small-gain condition in our theorem can be formulated more directly in terms of the gains, as the following corollary shows. Corollary 6.3 (Linear ISS small-gain theorem in semi-maximum formulation) Given an interconnection (Σ, U, φ) of systems Σ i as in Theorem 6.2, additionally to the assumptions (i) and (ii) of this theorem, assume that all gains γ ij are linear functions (and hence can be identified with nonnegative real numbers), Γ ⊗ is well-defined and the following condition holds: Then Σ is ISS.
Proof We only need to show that Assumption (iii) of Theorem 6.2 is implied by (6.6). The linearity of the gains γ ij implies that the operator Γ ⊗ is homogeneous of degree one and subadditive, see Remark 4.6. Then Proposition A.1 and Remark B.2 together show that (6.6) implies that the system is eISS (according to Definition 7.15), which easily implies the MLIM property for this system. ⊓ ⊔

Small-gain theorems in summation formulation
Now we formulate the small-gain theorems for UGS and ISS in summation formulation.
Theorem 6.4 (UGS small-gain theorem in summation formulation) Let I be a countable index set, (X i , · Xi ), i ∈ I, be normed spaces and Σ i = (X i , PC b (R + , X =i ) × U,φ i ), i ∈ I be forward complete control systems. Assume that the interconnection Σ = (X, U, φ) of the systems Σ i is well-defined. Furthermore, let the following assumptions be satisfied: (i) Each system Σ i is UGS in the sense of Definition 4.14 (summation formulation) with σ i ∈ K and nonlinear gains γ ij , γ i ∈ K ∪ {0}. (ii) There exist σ max ∈ K ∞ and γ max ∈ K ∞ so that σ i ≤ σ max and γ i ≤ γ max , pointwise for all i ∈ I. (iii) Assumption 4.10 is satisfied for the operator Γ ⊞ defined via the gains γ ij from (i) and id − Γ ⊞ has the MBI property.
Then Σ is forward complete and UGS.
Proof The proof is exactly the same as for Theorem 6.1, with the operator Γ ⊞ in place of Γ ⊗ . ⊓ ⊔ Theorem 6.5 (Nonlinear ISS small-gain theorem in summation formulation) Let I be a countable index set, (X i , · Xi ), i ∈ I be normed spaces and Σ i = (X i , PC b (R + , X =i )×U,φ i ), i ∈ I be forward complete control systems. Assume that the interconnection Σ = (X, U, φ) of the systems Σ i is well-defined. Furthermore, let the following assumptions be satisfied: (i) Each system Σ i is ISS in the sense of Definition 4.8 with β i ∈ KL and nonlinear gains γ ij , γ i ∈ K ∪ {0}. (ii) There are β max ∈ KL and γ max ∈ K so that β i ≤ β max and γ i ≤ γ max , pointwise for all i ∈ I. (iii) Assumption (4.10) holds and the discrete-time system with w(·), v(·) taking values in ℓ ∞ (I) + , has the MLIM property.
Then Σ is ISS.
Proof The proof is almost completely the same as for Theorem 6.2. The only difference is that instead of interchanging the order of two suprema sup s≥t and sup j∈I , we now have to use the estimate sup s≥t j∈I . . . ≤ j∈I sup s≥t . . ., which is trivially satisfied.

⊓ ⊔
Again, we formulate a corollary for the case when all gains γ ij are linear.
Corollary 6.6 (Linear ISS small-gain theorem in summation formulation) Given an interconnection (Σ, U, φ) of systems Σ i as in Theorem 6.5, additionally to the assumptions (i) and (ii) of this theorem, assume that all gains γ ij are linear functions (and hence can be identified with nonnegative real numbers), the linear operator Γ ⊞ is well-defined (thus bounded) and satisfies the spectral radius condition r(Γ ⊞ ) < 1. Then Σ is ISS.

Example: a linear spatially invariant system
Let us analyze the stability of a spatially invariant infinite network where a, b > 0 and each Σ i is a scalar system with the state x i ∈ R, internal inputs x i−1 , x i+1 and an external input u, belonging to the input space U := L ∞ (R + , R).
Following the general approach in Section 4, we define the state space for the interconnection of (Σ i ) i∈Z as X := ℓ ∞ (Z). Similarly as for finite-dimensional ODEs, it is possible to introduce the concept of a mild (Carathéodory) solution for the equation (6.8), for which we refer, e.g., to [34]. As (6.8) is linear, it is easy to see that for each initial condition x 0 ∈ X and for each input u ∈ U the corresponding mild solution is unique and exists on R + . We denote it by φ(·, x 0 , u). One can easily check that the triple Σ := (X, U, φ) defines a well-posed and forward complete interconnection in the sense of this paper.
Having a well-posed control system Σ, we proceed to its stability analysis.
Proposition 6.7 The coupled system (6.8) is ISS if and only if a + b < 1.
Proof "⇒": For any a, b > 0, the function y : t → (e (a+b−1)t x * ) i∈Z is a solution of (6.8) subject to an initial condition (x * ) i∈Z and input u ≡ 0. This shows that a + b ≥ 1 implies that the system (6.8) is not ISS.
"⇐": By variation of constants, we see that for any i ∈ Z, treating x i−1 , x i+1 as external inputs from L ∞ (R + , R), we have the following ISS estimate for the x isubsystem: for any t ≥ 0, x i (0) ∈ R and all x i−1 , x i+1 , u ∈ L ∞ (R + , R).
This shows that the x i -subsystem is ISS in summation formulation and the corresponding gain operator is a linear operator Γ : ℓ + ∞ (Z) → ℓ + ∞ (Z), acting on s = (s i ) i∈Z as Γ (s) = (as i−1 + bs i+1 ) i∈Z . It is easy to see that and thus r(Γ ) < 1, and the network is ISS by Corollary 6.6. ⊓ ⊔

Example: a nonlinear spatially invariant system
Consider an infinite interconnection (in the sense of the previous sections) where a, b > 0. As in Section 6.3, each Σ i is a scalar system with the state x i ∈ R, internal inputs x i−1 , x i+1 and an external input u, belonging to the input space U := L ∞ (R + , R). Let the state space for the interconnection Σ be X := ℓ ∞ (Z).
First we analyze the well-posedness of the interconnection (6.9). Define for x = (x i ) i∈Z ∈ X and v ∈ R as well as . Furthermore, f is clearly continuous in the second argument. Let us show Lipschitz continuity of f on bounded balls with respect to the first argument. For any x = (x i ) i∈Z ∈ X, y = (y i ) i∈Z ∈ X and any v ∈ R we have shows Lipschitz continuity of f with respect to the first argument on the bounded balls in X, uniformly with respect to the second argument.
According to [3,Thm. 2.4], 3 this ensures that the Carathéodory solutions of (6.9) exist locally, are unique for any fixed initial condition x 0 ∈ X and external input u ∈ U. We denote the corresponding maximal solution by φ(·, x 0 , u). One can easily check that the triple Σ := (X, U, φ) defines a well-posed interconnection in the sense of this paper, and furthermore Σ has BIC property (cf. [9,Thm. 4.3.4]).
We proceed to the stability analysis: Proposition 6.8 The coupled system (6.9) is ISS if and only if max{a, b} < 1.
Proof "⇒": For any a, b > 0 consider the scalar equatioṅ subject to an initial condition z(0) = x * . The function t → (z(t)) i∈Z is a solution of (6.9) subject to an initial condition (x * ) i∈Z and input u ≡ 0. This shows that for max{a, b} ≥ 1 the system (6.9) is not ISS. "⇐": Consider x i−1 , x i+1 and u as inputs to the x i -subsystem of (6.9) and define q := max{ax 3 i−1 , bx 3 i+1 , u}. The derivative of |x i (·)| along the trajectory satisfies for almost all t the following inequality: Arguing as in the proof of direct Lyapunov theorems (x i → |x i | is an ISS Lyapunov function for the x i -subsystem), see, e.g., [46,Lem. 2.14], we obtain that there is a certain β ∈ KL such that for all t ≥ 0 it holds that This shows that the x i -subsystem is ISS in semi-maximum formulation with the corresponding homogeneous of degree one gain operator Γ : The previous computations are valid for all ε > 0. Now pick ε > 0 such that a 1 < 1 and b 1 < 1, which is possible as a ∈ (0, 1) and b ∈ (0, 1). The ISS of the network follows by Corollary 6.3. ⊓ ⊔

Small-gain conditions
Key assumptions in the ISS and UGS small-gain theorems are the monotone limit property and monotone bounded invertibility property, respectively. In this section, we thoroughly investigate these properties. More precisely, in Section 7.1 we characterize the MBI property in terms of the uniform small-gain condition, in Section 7.2, we relate the uniform small-gain condition to several types of non-uniform small-gain conditions which have already been exploited in the small-gain analysis of finite and infinite networks. In Section 7.3, we derive new relationships between small-gain conditions in the finite-dimensional case. Finally, in Section 7.4, we provide efficient criteria for the MLIM and the MBI property in case of linear operators and operators of the form Γ ⊗ induced by linear gains.

A uniform small-gain condition and the MBI property
As we have seen in Section 6, the monotone bounded invertibility is a crucial property for the small-gain analysis of finite and infinite networks. The next proposition yields small-gain type criteria for the MBI property. Although in the context of small-gain theorems in terms of trajectories, derived in this paper, we are interested primarily in the case of (X, X + ) = (ℓ ∞ (I), ℓ + ∞ (I)), we prove the results in a more general setting, which besides the mathematical appeal also has important applications to Lyapunov-based small-gain theorems for infinite networks, where other choices for X are useful, see, e.g., [34] where X = ℓ p for finite p ≥ 1.
Proposition 7.1 Let (X, X + ) be an ordered Banach space with a generating cone X + . For every nonlinear operator A : X + → X + , the following conditions are equivalent: (i) id − A satisfies the MBI property.
(ii) The uniform small-gain condition holds: There exists η ∈ K ∞ such that . Fix x ∈ X + and write a := (A − id)(x). Let ε > 0. We choose z ∈ X + such that a − z X ≤ dist(a, X + ) + ε and we set y := a − z. If the constant M > 0 is chosen as in (2.1), we can decompose y as y = u − v, where u, v ∈ X + and u X , v X ≤ M y X ≤ M dist(a, X + ) + M ε. Then we have so it follows from the MBI property of id − A that Consequently, Since ε was arbitrary, this implies (ii) with η : The uniform small-gain condition in Proposition 7.1(ii) is a uniform version of the well-known small-gain condition, sometimes also called no-jointincrease condition: It is important to point out that the distance to the positive cone which occurs in the uniform small-gain condition in Proposition 7.1 can be explicitly computed on many concrete spaces. Indeed, many important real-valued sequence or function spaces such as X = ℓ p or X = L p (Ω, µ) (for p ∈ [1, ∞] and a measure space (Ω, µ)) are not only ordered Banach spaces but so-called Banach lattices.
An ordered Banach space (X, X + ) is called a Banach lattice if, for all x ∈ X, the set {−x, x} has a smallest upper bound in X, which is usually called the modulus of x and denoted by |x|, and if x X ≤ y X whenever |x| ≤ |y|. In concrete sequence and function spaces, the modulus of a function is just the pointwise (respectively, almost everywhere) modulus. Now, assume that (X, X + ) is a Banach lattice and let x ∈ X. Then the vectors x + := |x|+x 2 ≥ 0 and x − := |x|−x 2 ≥ 0 are called the positive and negative part of x, respectively; clearly, they satisfy x + − x − = x and x + + x − = |x|. If X is a concrete sequence or function space, then x − is simply 0 at all points where x is positive, and equal to −x at all points where x is negative.
In a Banach lattice (X, X + ), we have the formula dist(x, X + ) = x − X for each x ∈ X, as can easily be verified. ⊳ If the cone of the ordered Banach space (X, X + ) has nonempty interior, the uniform small-gain condition from Proposition 7.1 can also be expressed by a condition that involves a fixed interior point of X + .
Proposition 7.4 Let (X, X + ) be an ordered Banach space, assume that the cone X + has nonempty interior and let z be an interior point of X + . For every nonlinear operator A : X + → X + , the following conditions are equivalent: (ii) The uniform small-gain condition from Proposition 7.1(ii) holds.
Proof (i) ⇒ (ii). Let (i) hold with some η ∈ K ∞ . By [20,Prop. 2.11], we can find a number c > 0 such that for every y ∈ X we have Assume towards a contradiction that (ii) does not hold. Then (7.1) fails, in particular, for the function cη. Thus, we can infer that there is Hence, there exists y ∈ X + such that Consequently, the vector (A−id)(x)−y has norm at most c, so it follows from (7.3) that (A − id)(x) − y ≥ −η( x X )z. Thus, which shows that (7.2) fails for the function η, a contradiction.
(ii) ⇒ (i). Let (ii) hold with a certain η ∈ K ∞ . We show that (7.2) holds for the function η 2 z X substituted for η. Assume towards a contradiction that (7.2) fails for the function η 2 z X . Then there is x ∈ X + \ {0} such that Hence, it follows that which shows that (7.1) fails for the function η.

⊓ ⊔
A typical example of an ordered Banach space whose cone has nonempty interior is (X, X + ) = (ℓ ∞ (I), ℓ ∞ (I) + ) for some index set I. For instance, the vector 1 is an interior point of the positive cone in this space.

Non-uniform small-gain conditions
In Propositions 7.1 and 7.4, we characterized the MBI property in terms of the uniform small-gain condition. In this subsection, we recall several further smallgain conditions, which have been used in the literature for the small-gain analysis of finite and infinite networks [18,16,14], and relate them to the uniform small-gain condition.
In this subsection, we always suppose that (X, X + ) = (ℓ ∞ (I), ℓ + ∞ (I)) for some nonempty index set I (which is precisely the space in which gain operators act).

Definition 7.5 We say that a nonlinear operator
(ii) the strong small-gain condition if there exists ρ ∈ K ∞ and a corresponding operator D ρ : ℓ + ∞ (I) → ℓ + ∞ (I), defined for any x ∈ ℓ + ∞ (I) by (iii) the robust small-gain condition if there is ω ∈ K ∞ with ω < id such that for all i, j ∈ I the operator A i,j given by satisfies the small-gain condition (7.4); here, e i ∈ ℓ ∞ (I) denotes the i-th canonical unit vector. (iv) the robust strong small-gain condition if there are ω, ρ ∈ K ∞ with ω < id such that for all i, j ∈ I the operator A i,j defined by (7.6) satisfies the strong smallgain condition (7.5) with the same ρ for all i, j. ⊳ The strong small-gain condition was introduced in [18], where it was shown that if the gain operator satisfies the strong small-gain condition, then a finite network consisting of ISS systems (defined in a summation formulation) is ISS. The robust strong small-gain condition has been introduced in [14] in the context of the Lyapunov-based small-gain analysis of infinite networks.
Remark 7.6 For finite networks, also so-called cyclic small-gain conditions play an important role, as they help to effectively check the small-gain condition (7.4) in the case when A = Γ ⊗ , which is important for the small-gain theorems in the maximum formulation, see [37] for more discussions on this topic. For infinite networks, the cyclic condition for Γ ⊗ is implied by (7.4), see [14,Lem. 4.1], but is far too weak for the small-gain analysis. For max-linear systems, Remark B.2 and Corollary 6.3 are reminiscent of the cyclic small-gain conditions. ⊳ We say that a continuous function α : R + → R + is of class P if α(0) = 0 and α(r) > 0 for r > 0.
The following lemma is an extension of the considerations in [28, p. 130].
Lemma 7.7 The following statements hold: is in P, satisfies ρ(s) ≤ α(s) for all s ∈ R + , and is globally Lipschitz with Lipschitz constant L. (ii) If in (i) α ∈ K, then ρ given by (7.7) is a K-function.
Next, for any r 1 , r 2 ≥ 0 we have by the triangle inequality Similarly, using the triangle inequality for the second term, we obtain and thus ρ is globally Lipschitz with Lipschitz constant L, and is of class P.
If y 1 < r 2 , then (iii). Let α ∈ K ∞ . Assume to the contrary that ρ is bounded: ρ(r) ≤ M for all r. Then for every r there is r ′ with α(r ′ ) + L|r − r ′ | ≤ 2M . Looking at the second term, we see that r ′ → ∞ as r → ∞. But then α(r ′ ) → ∞, a contradiction.
⊓ ⊔ Now we give a criterion for the robust strong small-gain condition.
Proposition 7.9 A nonlinear operator A : ℓ + ∞ (I) → ℓ + ∞ (I) satisfies the robust strong small-gain condition if and only if there are ω, η ∈ K ∞ and an operator η : ℓ + ∞ (I) → ℓ + ∞ (I), defined by η(x) := (η(x i )) i∈I for all x ∈ ℓ + ∞ (I), (7.9) such that for all k ∈ I it holds that Proof "⇒": Let the robust strong small-gain condition hold with corresponding ρ, ω and D ρ . Then for any x = (x i ) i∈I ∈ ℓ + ∞ (I) \ {0} and any j, k ∈ I, it holds that ∃i ∈ I : As ρ ∈ K ∞ , there is η ∈ K ∞ such that id − η = (id + ρ) −1 ∈ K ∞ , which can be shown as in Lemma 7.8(ii). Thus, (7.11) is equivalent to As for each x ∈ ℓ + ∞ (I) there is j ∈ I such that x j ≥ 1 2 x ℓ∞(I) , the condition (7.12) with this particular j implies that which is up to the constant the same as (7.10).
"⇐": Let (7.10) hold with a certain η 1 ∈ K ∞ and a corresponding η 1 . By Lemma 7.8(i), one can choose η ∈ K ∞ , such that η ≤ η 1 and id − η ∈ K ∞ . Then (7.10) holds with this η and a corresponding η, i.e., for all k ∈ I we have As x ℓ∞(I) ≥ x j for any j ∈ I, this implies that for all j, k ∈ I it holds that and thus ∃i ∈ I : As η ∈ K ∞ satisfies id − η ∈ K ∞ , by Lemma 7.8(ii) there is ρ ∈ K ∞ such that (id − η) −1 = id + ρ, and thus for all j, k ∈ I property (7.11) holds, which shows that A satisfies the robust strong small-gain condition. ⊓ ⊔
Specialized to the strong small-gain condition, Proposition 7.9 reads as follows.
Corollary 7.10 A nonlinear operator A : ℓ + ∞ (I) → ℓ + ∞ (I) satisfies the strong small-gain condition if and only if there are η ∈ K ∞ and an operator η : ℓ + ∞ (I) → ℓ + ∞ (I), defined via (7.9) such that The next proposition shows that the uniform small-gain condition is at least not weaker than the robust strong small-gain condition.
Proposition 7.11 Let A : ℓ + ∞ (I) → ℓ + ∞ (I) be a nonlinear operator. If A satisfies the uniform small-gain condition, then A satisfies the robust strong small-gain condition.
Proof As A satisfies the uniform small-gain condition with η, from the proof of Proposition 7.4 with z := 1, we see that for all For any x ∈ ℓ + ∞ (I) and any k ∈ I, it holds that and by Proposition 7.9, A satisfies the robust strong small-gain condition. ⊓ ⊔

The finite-dimensional case
The case of a finite-dimensional X is particularly important as it is a key to the stability analysis of finite networks.
Proposition 7.12 Assume that (X, X + ) = (R n , R n + ) for some n ∈ N, where R n is equipped with the maximum norm · and R n + denotes the standard positive cone in R n . Further assume that the operator A is continuous and monotone. Then the following statements are equivalent: Additionally, if A is either Γ ⊞ or Γ ⊗ , then the above conditions are equivalent to (v) A satisfies the robust strong small-gain condition. (vi) A satisfies the strong small-gain condition.
(ii) ⇒ (i). This follows from Proposition 5.4 since the cone R n + has the Levi property. (iv) ⇒ (v). Follows by Proposition 7.11.
(v) ⇒ (vi). Clear.  [24] for couplings of two systems, and in [41,25] for any finite number of finitedimensional systems. The authors are not aware of such trajectory-based results for networks with infinite-dimensional components and/or infinite networks. ⊳

Systems with linear gains
Here we show that in the case of linear and sup-linear gain operators the MBI and MLIM properties are equivalent and can be characterized via the spectral condition.
Definition 7.15 Let (X, X + ) be an ordered Banach space. System (5.1) is exponentially input-to-state stable (eISS) if there are M ≥ 1, a ∈ (0, 1) and γ ∈ K ∞ such that for every u ∈ ℓ ∞ (Z + , X + ) and any solution x(·) = (x(k)) k∈Z+ of (5.1) it holds that x(k) X ≤ M x(0) X a k + γ( u ∞ ) for all k ∈ Z + . (7.13) For linear systems, we obtain the following result, that we use to formulate an efficient small-gain theorem in summation formulation, see Corollary 6.6.
Proposition 7.16 Let (X, X + ) be an ordered Banach space with a generating and normal cone X + . Let the operator A : X + → X + be the restriction to X + of a positive linear operator on X. Then the following statements are equivalent: Proof The implication "(i) ⇒ (ii)" is trivial. By Proposition 5.3, (ii) implies (iii).
(iii) ⇒ (iv). It is easy to check that if A is homogeneous of degree one and id − A satisfies the MBI property with a certain ξ ∈ K ∞ , then id − A satisfies the MBI property with r → ξ(1)r instead of ξ. For sup-linear systems, MBI is again equivalent to eISS, and the following holds: Proposition 7.17 Assume that the gains γ ij , (i, j) ∈ I 2 , are all linear and that the associated gain operator Γ ⊗ is well-defined. Then the following statements are equivalent: (i) The operator id − Γ ⊗ satisfies the MBI property.

Remark 7.18
The special form of the operator Γ ⊗ is used in Proposition 7.17 only for the proof of the implication (i) ⇒ (ii). The remaining implications are valid for considerably more general types of operators. Note that if s 0 is as in item (ii), then ts 0 also satisfies all conditions in item (ii), for any t > 0 and thus we can construct a path of strict decay t → ts 0 for the gain operator Γ ⊗ , which is an important ingredient for the proof of the Lyapunov-based ISS small-gain theorem, see [17]. ⊳ Proposition A.1 Let (X, X + ) be an ordered Banach space with a generating and normal cone X + . Consider system (5.1) and assume that the operator A : X + → X + is monotone and satisfies the following properties: (i) A is homogeneous of degree one, i.e., A(rx) = rA(x) for all x ∈ X + and r ≥ 0.
Then A is globally Lipschitz continuous and the following statements are equivalent: There is a globally Lipschitz V : X + → R + and L 1 , L 2 , ψ > 0, η > 1, such that and for any u ∈ ℓ ∞ (Z + , X + ), and any solution of (5.1) it holds that Proof A is Lipschitz continuous. Pick any x, y ∈ X + . As X + is generating, there are M > 0 (which does not depend on x, y) and a, b ∈ X + such that x − y = a − b and a X ≤ M x − y X , b X ≤ M x − y X . Hence, for all x, y ∈ X + we have Analogously, we obtain for all x, y ∈ X + that A(x) − A(y) ≥ −A(b). As X + is normal, due to [1,Thm. 2.38], there is c > 0, depending only on (X, X + ), such that A(x)−A(y) X ≤ c max{ A(a) X , A(b) X } ≤c max{ a X A(a/ a X ) X , b X A(b/ b X ) X } ≤ cCM x − y X .
"(a) ⇒ (b)": If (5.1) is eISS, then for u ≡ 0, any x ∈ X + and for the solution x(k + 1) = A(x(k)) of (5.1), the inequality (7.13) implies that A n (x) X ≤ M a n x X for all n ∈ Z + . Hence, sup x∈X + , x X =1 A n (x) 1/n X ≤ M 1/n a → a as n → ∞ and thus r(A) ≤ a < 1.
"(b) ⇒ (c)": From the assumptions (i) and (iii) together it follows that Consider the sequence a n := sup This sequence is submultiplicative, as for all m, n ∈ Z + it holds that a n+m = sup A m (A n (x)) X = sup A m (x) X = a n · a m .
By a submultiplicative version of the Fekete's subadditive lemma, lim n→∞ a 1 n n = inf n→∞ a 1 n n ≤ a 1 < ∞, and thus the limit in (A.1) exists. We fix η > 1 such that ηr(A) < 1 and define a function V : X + → R + by V (x) := sup n∈Z+ η n A n (x) X for all x ∈ X + .
Setting n := 0 in the supremum, we see that x X ≤ V (x) for all x ∈ X + . Since r(A) < η −1 , there exists N ∈ N so that sup x∈X + , x X =1 A n (x) X ≤ η −n for all n ≥ N.
By homogeneity of degree one of A, this implies η n A n (x) X = x X η n A n ( x x X ) X ≤ x X for all n ≥ N, x ∈ X + \ {0}. By (A.4), we have A(x) X ≤ C x X for all x ∈ X + . Due to homogeneity of A for all x ∈ X + , and by induction A n (x) X ≤ C n x X for all x ∈ X + .
Since η 0 A 0 (x) X = x X , with ψ := max 0≤n<N (ηC) n we have Also observe that As A is monotone and subadditive, it holds by induction for all n ∈ N that A n (x + y) = A n−1 (A(x + y)) ≤ A n−1 (A(x) + A(y)) ≤ A n (x) + A n (y), that is, A n are subadditive as well.
We can assume without loss of generality that the norm · X is monotone, i.e., 0 ≤ x ≤ y implies x X ≤ y X for any x, y ∈ X + . Otherwise, we choose an equivalent norm with this property, and note that eISS in one norm implies eISS in any other equivalent norm, and that the spectral radius does not depend on the choice of an equivalent norm.
As V is homogeneous of degree one, subadditive and monotone, V is Lipschitz continuous using the argumentation in the beginning of the proof.

B Systems, governed by a max-form gain operator
Here, we study the properties of the operator Γ ⊗ and its strong transitive closure. These results strengthen the corresponding results in [14,Sec. 4], and are motivated by them. We use these results to characterize the MBI and MLIM properties for the operator Γ ⊗ with linear gains in Proposition 7.17. However, the developments of this section are also useful for the construction of paths of strict decay for the nonlinear operator Γ ⊗ , which is essential for nonlinear Lyapunov-based small-gain theorems.

⊳
The following lemma can be found in [14,Lem. 4.1]. We include a rather simple proof for the sake of completeness.
Lemma B.3 Assume that Γ ⊗ satisfies the small-gain condition. Then all cycles built from the gains γ ij are contractions. That is, γ i1i2 • · · · • γ i k−1 i k (r) < r for all r > 0 if i 1 , . . . , i k is an arbitrary path with i 1 = i k .