A Small-Gain Theory for Abstract Systems on Topological Spaces

We develop a small-gain theory for systems described by set-valued maps between topological spaces. We introduce an abstract notion of stability unifying the continuity properties underlying different existing concepts, such as Lyapunov stability of equilibria, sets, or motions, (incremental) input–output stability, asymptotic gain properties, and continuity with respect to fast-switching inputs. Then, we prove that a feedback interconnection enjoying a given abstract small-gain property is stable. While, in general, the proposed small-gain property cannot be decomposed as the union of stability of the subsystems and a contractiveness condition, we show that it is implied by standard assumptions in the context of input-to-state stable systems. Finally, we provide application examples illustrating how the developed theory can be used for the analysis of interconnected systems and design of control systems.

The aforementioned small-gain results typically differ in terms of the class of systems considered and the stability requirements for the systems involved in the interconnection, but they all share a common paradigm from which a "small-gain principle" can be drawn: the interconnection of stable systems satisfying a certain "small-gain condition" is itself stable. In qualitative terms stability of the subsystems + small-gain property ⇒ stability of the interconnection (1) In this article, we develop a small-gain theory extending the small-gain principle (1) to set-valued maps between topological spaces. This leads to the following three main contributions.
1) It unifies different existing theorems developed for metric spaces of trajectories and gives new insights on the topological nature of the small-gain principle. 2) It enables the study of interconnections out of reach of existing small-gain theorems (e.g., formed by systems that do not satisfy ISS-like conditions). 3) It extends the small-gain principle to general maps between topological spaces not necessarily representing trajectories of a dynamical system (See Example 2). In this respect, we point out that the main methodological corpus of this work is composed of Items 1) and 3), which establish a common framework for the small-gain principle. Instead, item 2) is illustrated through examples (see Section VI). Indeed, the application of the presented results to specific cases requires the definition of suitable topological spaces and a preliminary analysis, both of which are problem-specific and, hence, not treated here systematically.
Going into the specifics of the framework and the main result of the article, we describe systems in terms of set-valued maps between arbitrary sets. These sets can be endowed with different topologies turning them into topological spaces. For each choice of such topologies, we define "stability" as a property similar to upper semicontinuity generalizing the continuity properties implied by the usual notions of Lyapunov stability of equilibria, sets, or motions, global or local (incremental) stability, and asymptotic gain. In particular, the continuity conditions underlying any of these properties can be obtained in terms of the proposed notion of stability for a specific choice of the involved topological spaces.
Given a feedback interconnection of two systems of this kind, we introduce an abstract small-gain property, and we prove a small-gain theorem stating that such property implies stability of the interconnection. The proposed small-gain property is an abstraction of the joint condition "stability of the subsystems + small-gain property" of (1) that, however, does not admit a similar decomposition but is a unique requirement. Nevertheless, we show that, in an ordinary ISS context, "stability of the subsystems + small-gain condition" implies the proposed small-gain property.
Finally, in this connection, we emphasize that the presented results only concern the continuity conditions implied by the global stability and asymptotic gain properties [11] and not directly ISS. While for finite-dimensional systems these properties imply (local) ISS, this is not generally true, for instance, for hybrid (even of finite dimension) [38,Remark 3.3] and infinite-dimensional [39], [40] systems. Therefore, in an ISS context, the conclusions that can be drawn on the feedback interconnection from the proposed theory are in general weaker than ISS. Nevertheless, we remark that this is not necessarily a shortcoming of the proposed theory, as it deals with spaces where ordinary notions of uniform convergence or boundedness may not make sense. Moreover, stronger properties, such as "uniform asymptotic gain" [11], may be obtained by suitably redefining or extending the input and output spaces and their topologies, as discussed in Section V-E.
The article is organized as follows. In Section II, we introduce the basic notions of systems and interconnections. In Section III, we define the notion of stability and connect it to the usual global stability and asymptotic gain properties in metric spaces. In Section IV, we define the small-gain property, we establish the main result, and we show that ISS implies the proposed small-gain property. In Section V, we discuss further connections between stability and other existing notions. Finally, in Section VI, we present three examples illustrating how the proposed theory can be used to handle interconnections of systems falling outside the scope of existing small-gain theorems.
Notations and Preliminaries. We denote by R and N the set of real and natural numbers, respectively (0 ∈ N). If ∼ is a relation on a set S and s ∈ S, we let S ∼s := {z ∈ S : z ∼ s}.
We denote by dom F the set of x ∈ X for which F (x) is nonempty, and by ran F : A topological space is a pair (X , τ) where X is a set and τ is a collection of subsets of X which contains ∅ and X itself and is closed under finite intersections and arbitrary unions. The elements of τ are called open sets. A neighborhood of a point x ∈ X is a subset of X containing an open set containing x. A neighborhood of a set X ⊆ X is a subset of X containing a neighborhood of every point of X. The set of all the neighborhoods of X ⊆ X is denoted by When τ is clear from the context we omit it and, for instance, we write X for (X , τ) and N (·) for N τ (·). If not otherwise specified, we shall assume every X ⊆ X to be endowed with the subset topology τ X := {O ∩ X : O ∈ τ } and, if (X 1 , τ X 1 ), . . . , (X n , τ X n ) are topological spaces, their product X 1 × · · · × X n will be assumed to be endowed with the product topology denoted by τ X 1 ⊗ · · · ⊗ τ X n . A net on a set X is a map x : I → X from a directed set I to X. We denote nets also by (x j ) j∈I .
For t > 0, we denote by C t (X) the set of continuous functions [0, t) → X, and we let C (0,∞] (X) := ∪ t∈(0,∞] C t (X). A continuous function k : R ≥0 → R ≥0 is of class-K if it is strictly increasing and k(0) = 0. We denote by k −1 : ran k → R ≥0 the inverse of k. Notice that, if k is of class-K, there always exists > 0 so that [0, ) ⊆ ran k = dom k −1 . Hence, k −1 (s) exists for all sufficiently small s > 0.

II. SYSTEMS
In this section, we introduce the basic notions we use to model systems and their interconnections.

A. Systems as Mappings
Throughout the article, systems are represented by set-valued maps between sets.
Definition 1 (Systems): A system is a triple (D, Y, Ψ) in which D and Y are sets and Ψ : D ⇒ Y is a set-valued map.
The set D is called the input space, and its elements the inputs of the system. The set Y is called the output space, and its elements the outputs of the system The notion of systems provided by Definition 1 resembles that of [2] and [6], with the difference that here D and Y are generic sets, and not necessarily normed spaces of signals. Moreover, Definition 1 also fits the behavioral framework of [41], as the set graph Ψ is a behavior on D × Y in the sense of [41,Def. 1.2.1]. In this connection, we observe that seeing Ψ as a map D ⇒ Y, instead of a map Ψ L : Y ⇒ D, is a matter of convention as graph Ψ and graph Ψ L are isomorphic.
Definition 1 is sufficiently general to include most of the usual definitions of interest in control theory, such as transfer functions, ordinary/partial differential equations or inclusions, and hybrid systems, as shown in the following example.
Example 1: Consider the hybrid inclusions [42] ẋ is a solution pair to (2) with x originating at x 0 . Then, (D, Y, Ψ) is a system in the sense of Definition 1.
In addition, Definition 1 extends beyond dynamical systems. It can be used to model algebraic maps, solution mapping of optimization problems, or other relations capturing only some specific aspects of dynamics. For instance, Example 2 hereafter deals with limit sets, used in control to characterize the steadystate trajectories of a system [43], [44].
Example 2: In the setting of Example 1, fix u = 0 and let X 0 ⊂ R n be compact. Let S(X 0 ) be the set of all complete solutions x of (2) originating in X 0 and corresponding to u = 0. Suppose that S(X 0 ) = ∅ and, for each τ ≥ 0, is a filter base [45, Sec. 1.6] whose (possibly empty) set of cluster points Ω(X 0 ) := ∩ τ ≥0 R τ (X 0 ) is the Ω-limit set of (2) from X 0 (and with u = 0). Let D be the set of all compact subsets X 0 of R n , Y = R n , and Ψ : D ⇒ Y the set-valued map Ψ(X 0 ) := Ω(X 0 ). Then, (D, Y, Ψ) is a system in the sense of Definition 1 representing the mapping between sets of initial conditions and the corresponding attractor.

B. Interconnections
As a first case, assume that D 1 = D 2 and let D : The parallel interconnection of Σ 1 and Σ 2 is defined as the sys- As a second case, assume that Y 1 ⊆ D 2 . The series interconnection of Σ 1 and Σ 2 is defined as the system (D, Y, Ψ) with Finally, assume that, for some sets V 1 , V 2 , D 1 , and Fig. 1). Then, a feedback interconnection of Σ 1 and Σ 2 is a system (3) Let F(Σ 1 , Σ 2 ) denote the set of all feedback interconnections of Σ 1 and Σ 2 . Let be the partial order on F(Σ 1 , Σ 2 ) such that, for every two (D, , then Zorn's Lemma guarantees the following. Lemma 1: F(Σ 1 , Σ 2 ) has a unique maximal element. We call such a unique maximal element the feedback interconnection of Σ 1 and Σ 2 .
The parallel interconnection of any two nontrivial systems is nontrivial when dom Ψ 1 ∩ dom Ψ 2 = ∅. The series interconnection is nontrivial if dom Ψ 2 ∩ ran Ψ 1 = ∅. Nontriviality of feedback interconnections is instead not straightforward, as illustrated by the following example.
The following lemma, proved in Appendix A, provides an alternative characterization of Υ 1 and Υ 2 that plays an important role in the main result given later in Section IV.
Remark 1: In view of (7) and (8), the elements of Ψ(d) are fixed points of the maps Ψ 1 (Ψ 2 (·, d 2 ), d 1 ) and Ψ 2 (Ψ 1 (·, d 1 ), d 2 ), respectively. This, however, is only a necessary condition since, in general, Namely, pairs of fixed points of the aforementioned maps need not be elements of Ψ(d), although they always belong to

III. STABILITY
In this section, we introduce a notion of stability for systems satisfying Definition 1. We then discuss its relationship with the properties of global stability and asymptotic gain. This enables us to make a direct connection with ISS systems. Instead, connections with other notions, such as Lyapunov stability and incremental stability, are discussed later in Section V.

A. Stability as a Topological Notion
As for continuity, stability is defined with reference to a topology τ D defined on the input space D and a topology τ Y defined on the output space Y. Different choices of these topologies lead to different notions of stability. When τ D and τ Y are clear from the context they are omitted, and we say that the system is stable at D. If D = {d} is a singleton, we say that the system is stable at d instead of at {d}. Stability at D ⊆ D is implied by upper semicontinuity of Ψ at each point of D. Indeed, every Y ∈ N (Ψ(D)) contains a set of the form Nevertheless, the converse does not hold. Namely, stability at D does not imply upper semicontinuity of Ψ at each point of D. Indeed, as a trivial counterexample, notice that every system is stable at D (since D is a neighborhood of itself) without any relation to upper semicontinuity of Ψ at any point of D.
Remark 2: We underline that the notion of stability given in Definition 2 is aimed at generalizing the continuity properties implied by the notion of Lyapunov stability of autonomous differential/difference equations and global stability à la [11] for systems with input, both of which are properties of the map Ψ, and not convergence or attractiveness, which are instead properties of the specific inputs and outputs of Ψ.
Remarkably, parallel interconnections are stable at a point if so are the interconnected systems, whereas series interconnections of stable systems are stable at both points and sets.
denote their parallel and series interconnection (see Section II-B) whenever they make sense. Let D 1 , D 2 , Y 1 , and Y 2 be endowed with some topologies, and Y p = Y 1 × Y 2 with the product topology (in the parallel case, D p = D 1 = D 2 ; hence D 1 and D 2 are given the same topology). Then, the following holds. 1 Proposition 1: Proposition 1 is proved in Appendix B. Stability of feedback interconnections between Σ 1 and Σ 2 is instead more delicate, and it is indeed the main object of this article. We conclude this section with the following technical lemma (proved in Appendix C), which is used by several forthcoming results.
Lemma 3: Let A and B be topological spaces, and let A ⊆

B. Connections With Global Stability
Let (X, | · |), (U, | · |), and (Y, | · |) be seminormed linear spaces (for ease of notation, we denote all seminorms by | · |). Let (T U , ≥) and (T Y , ≥) be (possibly different) directed sets, and let (U , | · |) and (Y, | · |) be seminormed linear spaces (under the pointwise operations) of functions T Y → Y and T U → U , respectively. The elements of X may represent initial/boundary conditions or parameters. The elements of U represent exogenous input signals, and those of Y the system's outputs. Let D := X × U, and suppose a system Σ = (D, Y, Ψ) is defined such that, for some class-K functions α and κ, When |y| := sup t∈T Y |y(t)| and |u| := sup t∈T U |u(t)|, Condition (9) is a global stability property implied by ISS. If, instead, T U = R ≥0 and |u| := ( ∞ 0 |u(t)| 2 dt) 1/2 , we obtain an "integral" variant of global stability, implied by integral ISS [46]. In general, different notions of stability can be obtained for different choices of the seminorms [46]. In any case, (9) implies stability, in the sense of Definition 2, of Σ at D : where τ Y is the topology induced on Y by its seminorm, and τ D is the product topology on D induced by the seminorms on X and U . Indeed, (9) implies Ψ(D ) ⊆ {y ∈ Y : |y| = 0}, and every neighborhood Y of Ψ(D ) contains a set of the form B := {y ∈ Y : |y| < } for some > 0 small enough so as ∈ ran α ∩ ran κ. As τ D is induced by the seminorm |(x, u)| := max{|x|, |u|}, then the set U :

C. Connections With Asymptotic Gain
In this section, we show that the asymptotic gain property implies stability according to Definition 2 for a specific choice of the topology of the input and output spaces. In particular, let (Z, | · |) be a seminormed linear space, (T Z , ≥) be a directed set, and Z be a linear space (under the pointwise operations) of In the same setting of Section III-B, suppose that all u ∈ U and y ∈ Y are bounded and there exists a class-K function ρ such that Condition (10) is an asymptotic gain property (implied by ISS). Let τ U and τ Y denote the limsup topologies on U and Y, respectively, and let τ X be any topology on X. Then, for every S ⊆ X, (10) implies that Ψ is stable at D := S × {u ∈ U : lim sup |u| = 0} with respect to (τ X ⊗ τ U , τ Y ). Indeed, (10) implies that Ψ(D ) ⊆ {y ∈ Y : lim sup |y| = 0}. Hence, every neighborhood V of Ψ(D ) contains a set of the form Y ε := {y ∈ Y : lim sup |y| < ε} for a sufficiently small ε > 0 satisfying ε ∈ ran ρ. Therefore, with δ := ρ −1 (ε), the set U δ := {u ∈ U : lim sup |u| < δ} is such that X × U δ ∈ N (D ) and, in view of (10), y ∈ Y ε ⊆ V for all y ∈ Ψ(X × U δ ). Fig. 1). We associate with Σ the maps Γ 12 :

A. The Small-Gain Property
Moreover, for (i, j) = (1, 2), (2, 1) and n ∈ N ≥1 , we define the maps Γ n 12 and Γ n 21 according to the following recursion: for all y i ∈ Y i and all d ∈ D. The maps Γ 12 and Γ 21 satisfy the following property.
In view of (8), Suppose that, for some n ∈ N ≥1 , Then, (12) As (11) is (12) with n = 1, we conclude by induction that (12), and hence y i ∈ Γ n ij (y i , D), hold for every n ∈ N ≥1 . Since D and y i ∈ Υ i (D) were arbitrary, the claim follows.
We endow D with a topology τ D and Y with a topology τ Y . Then, the following definition formalizes the small-gain condition used in this article to establish stability, with respect to Remark 3: Condition (13) in Definition 3 is a "contraction" requirement for the maps Γ 12 and Γ 21 that plays in our setting the role of "stability of the subsystems + small-gain property" of typical ISS contexts [cf., (1)]. However, we stress that, unlike (1), Definition 3 is a single condition that, up to the authors' knowledge, cannot be expressed in terms of the composition of stability of Σ 1 and Σ 2 and a contraction property. Indeed, in general, the set D may not even be a product of the form D 1 × D 2 . Nevertheless, later in Section IV-C, we show that ISS implies Definition 3.
Remark 4: Computing Γ 12 and Γ 21 may be difficult, if not impossible, for nontrivial interconnections. Fortunately, their computation is generally not needed for checking whether the condition (13) holds. Indeed, (13) can be usually checked by using some known property of the systems involved. In the following, we provide several instances where this is the case, i.e., Propositions 2 and 3, and the examples in Section VI.

B. Main Result
In this section, we prove the main result of this article, establishing that a feedback interconnection of two systems satisfying the small-gain property of Definition 3 is stable. As in the previous section, we consider two systems Σ 1 = (D 1 , Y 1 , Ψ 1 ) and (3). Finally, we endow D with a topology τ D and Y with a topology τ Y .

C. Connections With Other Small-Gain Theorems
In this section, we establish a relationship between the smallgain property given in Definition 3 and some existing smallgain theorems applying to ISS systems. Specifically, we show that "stability of the subsystems + small-gain property" in (1) (with stability meaning ISS) implies the small-gain condition of Definition 3. For simplicity, we focus on stability of the origin. Nevertheless, the same arguments can be extended to stability of sets or motions, by resorting to the corresponding notions described later in Section V. With is a system in the sense of Definition 1. The set X i may represent initial/boundary conditions in an initial value problem, while U i contains exogenous inputs. We hold for all (y j , x i , u i , y i ) ∈ graph Ψ i . Condition (16a) relates to global stability and (16b) to asymptotic gain. Both are implied by ISS. Finally, we suppose that the following small-gain condition holds: First, we show that Conditions (16a) and (17) imply the small-gain property of Definition 3 with respect to the uniform seminorm topology. To this end, for i = 1, 2, we define on U i and Y i the seminorms |u i | := sup t∈T U i |u i (t)| and |y i | Likewise, the product topology τ Y on Y := Y 1 × Y 2 is generated by |(y 1 , y 2 )| := max{|y 1 |, |y 2 |}. Consider a feedback interconnection Σ = (D, Y, Ψ) of Σ 1 and Σ 2 , defined according to Section II-B, and let O D : Then, the following holds (the proof is given in Appendix D).
(18) Next, we show that, for the previously defined feedback interconnection Σ of Σ 1 and Σ 2 , the bounds (16b) and (17) also imply the small-gain condition of Definition 3 with respect to the limsup topology on U i and Y i (and any topology on X i ). To this end, we endow X i with an arbitrary topology τ X i , we give U i and Y i the respective limsup topologies, as described in Section III-C, and we let τ D and τ Y be the product topologies on D and Y, respectively. For i = 1, 2, define Then, the following result holds (the proof is in Appendix E).
In view of Proposition 3, we can use Theorem 1 to deduce from (16b) and (17) that the interconnection Σ is also (τ D , τ Y )-stable at L D . This implies that [cf., (18)] To summarize, Theorem 1 allows us to conclude from the global stability and asymptotic gain properties (16) (hence, from ISS), and the small-gain condition (17), that the feedback interconnection Σ of the two subsystems Σ 1 and Σ 2 satisfies the continuity conditions (18) and (19), which are the same continuity conditions implied by ISS of Σ.

V. CONNECTIONS WITH OTHER STABILITY NOTIONS
In this section, we present some relevant cases, obtained for a specific choice of (D, τ D ), (Y, τ Y ), and Ψ, that connect the stability notion given by Definition 2 with more common notions of stability used in control and systems theory.
Let X 0 be given the Euclidean topology, and X be given the topology induced by ρ. Moreover, let Ψ be the solution map of (20), mapping initial conditions x 0 ∈ X 0 to maximal solutions x ∈ X satisfying Notice that, by definition of ρ, (21) implies that, for all initial conditions inside a sufficiently small neighborhood of x * 0 , the corresponding solutions are complete. We observe that, when Ψ is single-valued and every solution of (20) originating in X 0 is complete, condition (21) (which, we recall, is Definition 2 in this context) equals Lyapunov stability of the motion x * = Ψ(x * 0 ).

B. Lyapunov Stability of Sets
In the same setting of Section V-A, let A ⊆ X 0 be nonempty, closed, and forward-invariant (i.e., ran x ⊆ A for all x ∈ Ψ(A)). With c > 0 arbitrary, define where |x| A := inf a∈A |x − a| denotes the distance of x to A. Then, ρ A induces a topology on X as specified by the following lemma.
Proof: Both ∅ = O 0 and X = O c+1 belong to τ A . Moreover, since X = O for all > c, then, for every 1 A sequence (x n ) n∈N converges in τ A to a limit x if, for every > 0, there exists n * ( ) ∈ N, such that ρ A (x n ) < ρ A (x) + for all n ≥ n * ( ). Clearly, if (x n ) n∈N converges to x, it converges to every other z satisfying ρ A (z) = ρ A (x). In particular, if x n converges to an x satisfying ρ A (x) = 0 (namely, ρ A (x n ) → 0), we say that (x n ) n∈N converges to A and write x n → τ A A.
The topology τ A has the following property, establishing the equivalence between convergence of x to A in the usual sense and convergence of the "tails" of x to A in τ A .
Proposition 4: Let x ∈ X be such that dom x = [0, ∞). Then, lim t→∞ |x(t)| A = 0 if and only if there exists a sequence (t n ) n∈N in R ≥0 such that the sequence (x n ) n∈N in X with terms x n (·) := x(t n + ·) converges to A in τ A .
Proof: (Only If) If |x(t)| A → 0, for every sequence ( n ) n∈N with n ∈ (0, c) satisfying n → 0, there exists a sequence (t n ) n∈N in R ≥0 such that |x(s)| A < n for all s ≥ t n . This implies ρ A (x n ) = sup t≥0 |x n (t)| A = sup t≥t n |x(t)| A → n 0. Namely, x n → τ A A.
(If) Let (t n ) n∈N be a sequence in R ≥0 such that x n → τ A A. Then, for every ∈ (0, c), there exists n * ( ) ∈ N such that ρ A (x n ) < for all n ≥ n * ( ). This implies that |x(s)| A < for all s ≥ t n and all n ≥ n * ( ). Thus, lim t→∞ |x(t)| A = 0.
Let τ A be the topology on X 0 generated by the family {X 0 } ∪ {Q δ : δ > 0}, in which Q δ := {x 0 ∈ X 0 : |x 0 | A < δ}. Then, (X 0 , X , Ψ) is (τ A , τ A )-stable at A in the sense of Definition 2 if and only if for each > 0, there exists δ > 0, such that |x 0 | A < δ implies sup t∈dom x |x(t)| A < for all x ∈ Ψ(x 0 ). In turn, this coincides with the usual notion of Lyapunov stability of a closed forward invariant set A [47]. In the same setting of previous Section III-B, suppose now that instead of (9) we have ∀(x, u, y), (x , u , y ) ∈ graph Ψ,

C. Incremental Input-Output Stability
Condition (22) is the equivalent of (9) for incremental ISS systems [48]. . Endow δD and δY with the respective incremental topologies. Then, by the same arguments used in Section III-B, one can show that (22) implies that the system δΣ is stable at the diagonal set δD : = {((x, u), (x , u )) ∈ δD : |x − x | = 0, |u − u | = 0}. Remark 5: {O : > 0} is a special case of a diagonal uniformity [49,Sec. 3.15], generalizing in topology the notion of uniform continuity. Indeed, in this sense, input-output stability (9) relates to continuity in the same way as its incremental version (22) relates to uniform continuity.

D. Weak Stability and Switching Controls
Many applications of control, such as PWM-based regulation of electric machines [50] or sliding-mode control [51], employ control signals that switch between quantized values (see also the example in Section VI-C). As the switching frequency increases, the controlled system's trajectories get closer to those one would obtain with a control law given by the average of the switched signal. This is a fundamental phenomenon that, in the same setting of Section III-B, can be characterized in terms of stability according to Definition 2, in which the input space is endowed with the weak topology.
Specifically, consider the same setting of Section III-B with where f and g are continuously differentiable. For simplicity, suppose that Ψ(x, u) is single valued at each (x, u) ∈ dom Ψ. Let ((x i , u i )) i∈I be an equibounded net in dom Ψ. If, for some (x * , u * ) ∈ dom Ψ, x i → x * and u i → u * weakly, then Ψ(x i , u i ) → Ψ(x * , u * ) uniformly (this can be deduced by the same arguments used in [52, Th. 1]). Hence, with τ Y the topology of uniform convergence on Y, τ X the Euclidean topology on X, τ U the weak topology on U , and τ D the product topology In particular, let Q ⊆ U be a compact (possibly finite) set where "implementable" control inputs can take values (for instance, Q contains the upper and lower values of a PWM signal), and let co Q denote its convex hull. Let u * ∈ coQ be a nominal, but possibly "not implementable" (i.e., u * / ∈ Q), steady value for the control input on [0, t] and, with (T i ) i∈N a sequence of periods T i ∈ (0, t) satisfying T i → 0, let (u i ) i∈N be a sequence of periodic piecewise-continuous control inputs of period T i , with values in Q, and satisfying Then, by denoting by u * also the constant function equal to u * at every s ∈ [0, t], the following result holds. Lemma 7: u i → u * weakly. Lemma 7, proved in Appendix F, allows to conclude that we can substitute a given arbitrary reference control input u * , constant on subsequent intervals of the form [t k , t k+1 ), with a second control input u with quantized values possibly very different from u pointwise (provided that u is periodic on each [t k , t k+1 ) with mean value equal to u * (t k )). The resulting effect is a deviation of the corresponding solution y from the one produced by u * that can be made arbitrarily small on each compact interval by taking the period of u sufficiently small.

E. Uniform Asymptotic Gain
The asymptotic gain condition (10) does not say anything about the convergence rate of y. Typically, stronger conditions asking for uniformity of convergence may be useful, for instance, to establish uniform ISS. In the same setting of Section III-B and III-C, let (X, τ X ) be compact, and let U := {u ∈ U : |u| = 0}. By following [11], we say that the system (X × U, Y, Ψ) has the uniform asymptotic gain property if there exists a class-K function κ such that

F. Practical Stability
In this section, we consider the case in which, as in [9], a further bias term appears in properties of the kind of global stability (9) and asymptotic gain (10). We specifically develop the asymptotic gain case as a simple example to introduce a methodology that can be extended to other similar contexts. In particular, in the same setting of Section III-C, suppose that, instead of (10), the following property holds: ∀(x, u, y) ∈ graph Ψ, lim sup |y| ≤ ρ(lim sup |u|) + b, (25) in which b > 0 is a bias term. While the presence of b ruins stability when Y has the limsup topology, stability still applies if Y is given the topology

A. RMS Rejection of Unknown Disturbances
Consider the systemẋ in which x(t) ∈ R is the state variable, w : R ≥0 → R is a bounded unmeasured disturbance, u(t) ∈ R is a control input, and f is an uncertain Lipschitz function whose nominal value f has Lipschitz constant > 0. Define the asymptotic root mean square of x as Given an arbitrary but fixed ε > 0, we consider the problem of designing a controller ensuring that |x| aRMS ≤ ε at front of every bounded disturbance w unknown a priori and for all sufficiently small deviations of f from f . We assume to measure the state x of the plant subject to a small, bounded and smooth additive disturbance ν. In particular, we assume to measure y := x + ν with ν ∈ V, in which V denotes the set of bounded continuously differentiable functions R ≥0 → R with bounded derivative.
Similarly, with H 0 := R \ {0}, we model (26b) as a system and, for each (y, η 0 ) ∈ D 2 , Ψ 2 (y, η 0 ) is the solution of (26b) originating at η 0 and subject to y, i.e., which is defined for all t ≥ 0. We consider the feedback interconnection of Σ 1 and Σ 2 , resulting in the system Σ = (D, Y, Ψ) defined as specified in Section II- We give X 0 , F, H 0 , W, V, and Y 2 the trivial topology and Y 1 the topology τ Y 1 generated by the collection {Q μ : μ > ε} with Q μ := {y ∈ Y 1 : |y| aRMS < μ}. We then give D and Y the respective product topologies τ D and τ Y .

B. Robust Global Attractiveness Without ISS
Consider the forced Susceptible-Infected systeṁ with β > γ > 0, S(0), I(0) ≥ 0, and v ∈ V, where V is the set of bounded continuous functions R ≥0 → R ≥0 . It is well known that, when v = 0, the set A := {(S, I) ∈ R 2 ≥0 : I = 0} is globally attractive. Yet A is not Lyapunov stable. Moreover, the I subsystem is not ISS with respect to the input (S, v), nor the S subsystem is ISS with respect to I. Hence, attractiveness of A cannot be concluded by means of canonical small-gain arguments for ISS systems. Instead, as detailed in the rest of this section, it can be proved by using Theorem 1. In particular, we prove following stronger "robust attractiveness" property: where |(S, I)| A denotes the distance of (S, I) to A. Let Y 1 be the set of bounded nonincreasing continuous functions R ≥0 → R ≥0 , and Y 2 the set of continuous functions I : R ≥0 → R ≥0 that either are Lipschitz or satisfy I(t) → ∞. Define the system Σ 1 := (D 1 , Y 1 , Ψ 1 ), in which D 1 = Y 2 × R ≥0 and Ψ 1 is the solution map ofṠ = −βSI mapping pairs (I, S(0)) ∈ D 1 to complete solutions S ∈ Y 1 . Define the system Σ 2 := (D 2 , Y 2 , Ψ 2 ), in which D 2 = Y 1 × R ≥0 × V and Ψ 2 the solution map ofİ = βSI − γI + v mapping triples (S, I(0), v) ∈ D 2 to complete solutions I ∈ Y 2 . Then, system (29) can be seen as a feedback interconnection Σ = (D, Y, Ψ) of the previous two systems (see Section II-B), with D = R 2 ≥0 × V, Y = Y 1 × Y 2 , and Ψ mapping triples (S(0), I(0), v) to complete solutions (S, I) of (29). We endow R 2 ≥0 and Y 1 with the respective trivial topologies, and we give Y 2 and V the respective limsup topologies (see Section III-C).
Therefore, by using Theorem 1, we finally conclude that Σ is stable at D . As the set R 2 ≥0 of initial conditions has the trivial topology, and |(S, I)| A = |I|, this implies (30).

C. Automatic Frequency Regulation in PWM Control
Consider an electrical motor described by the linear systeṁ with x(t) ∈ R n , n ∈ N, u(t), y(t) ∈ R, and A Hurwitz. The output y represents the rotor's angular velocity, and u is the input voltage. We consider a control system that can only generate quantized switching voltages taking the value V or −V at a given time instant, with V > 0. This is typical of controllers implemented by power converters [50]. Given an ideal input profile u : R ≥0 → [−V, V ] (which can take any value in-between −V and V ), the controller approximates u by means of a function where The sequence T = (T k ) k∈N represents a time-varying switching period, and it is a degree of freedom. Moreover, due to the presence of unavoidable uncertainty in the controller implementation, we suppose that the actual generated control is u T :=û T + ν, with ν : R ≥0 → R a bounded additive perturbation. The controller measures the error e T := y − y T between the ideal output y that would be produced by (32) with u and x(0) = 0, and the actual output y T produced by u T for some x(0) ∈ R n . The control goal is to tune the sequence T k online to eventually reduce the error below a prespecified threshold ε > 0. 3 The error e T is bounded, and by means of arguments similar to those used in proving Lemma 7, it can be shown that, when ν = 0, for every ε > 0, there existsσ =σ(ε) > 0 such that, for every y T obtained with a sequence T satisfying T k ≤σ(ε) for all but at most finite k, lim sup |e T | ≤ ε. We assume that T min < σ(ε ), so as the set of sequences T for which lim sup |e T | ≤ ε when ν = 0 is nonempty.
We focus on decision strategies (continuous-or discrete-time) adapting T k iteratively in such a way that  initial conditions T 0 ∈ [T min , ∞) to sequences T ∈ T , and Σ 2 = (D 2 , Y 2 , Ψ 2 ) (D 2 = T × R n × V and Y 2 = E), mapping sequences T ∈ T , initial conditions x(0) ∈ R n , and perturbations ν ∈ V to error signals e T ∈ E. According to Section II-B, D = [T min , ∞) × R n × V, Y = T × E, and Ψ is given by (3).
Therefore, in view of Theorem 1, we claim that Σ is stable at D . As [T min , ∞) and R n have the trivial topology, this means that, regardless of the actual value of the initial conditions x(0) and T 0 , the controller produces an error whose limsup is larger than ε by an amount that continuously increases with the size of the disturbance ν (in particular, if ν = 0, lim sup |e| ≤ ε ).
We underline that we made no strong stability assumption, such as ISS, to achieve the above-mentioned result. Moreover, how to characterize Σ 1 in terms of ISS is also unclear. Indeed, a map Ψ 1 satisfying P1 and P2 may not be continuous in general. For example, take Ψ 1 as the function mapping every (e, T 0 ) ∈ D 1 to a sequence T with initial condition T 0 and satisfying Clearly, the sequence generated in this way fulfills P1 and P2.
Consider the metric |T − T | := sup k∈N |T k − T k | on T , and Thus, Ψ is not continuous. Fig. 3 shows a simulation obtained with the following decision policy for T (m, ρ, and σ are auxiliary variables): I1 start with m(t 0 ) = p(t 0 ) = 0 and σ(t 0 ) = T 0 ; I2 for every k ∈ N: r Define t k+1 := t k + σ(t k ). r Integrate the following equations over [t k , t k+1 ]: r If t k+1 − m(t k+1 ) ≥ 3, update the variables as The kth term T k of the sequence T generated in this way is given by T k = t k+1 − t k = σ(t k ). The error signal e T is obtained as e T = Cx − y withẋ = Ax + Bu andx(0) = 0.
Specifically, Fig. 3 shows three simulations obtained withû T defined as a PWM signal with variable period (according to T ) and, within each period, a duty cycle chosen in such a way that (33) holds, T min = 0.001, ε = 0.05, V = 2, and ν given by the interpolation of a pseudorandom signal uniformly sampled from [0, A ν ], where A ν equals, respectively, 0.01, 0.5, and 1 in the three simulations. As shown in the figure, the asymptotic amplitude of the error gradually increases with |ν| ∞ . This is consistent with our results that claim continuity of such increase. The sequence T , instead suffers from an evident discontinuity (in the previously defined uniform norm) when the amplitude of ν provokes spikes of e(t) above ε . In such case, indeed, T k decreases to T min despite the actual value of ν, whereas in the other two cases with A ν = 0.01 and A ν = 0.5, only a small variation of T k is observed. This behavior is caused by the fact that, as for (35), also in this case Ψ 1 is discontinuous. Indeed, while discontinuity of Ψ 1 does not invalidate our results, it nevertheless implies that we have no guarantee that "small" changes of ν reflect on "small" changes of T (with respect to the previously defined metric).

VII. CONCLUSIONS
In this article, we proposed a small-gain theory for interconnections of abstract systems described by set-valued maps between topological spaces. For systems of this kind, stability is defined as a continuity property generalizing and unifying the continuity conditions underlying commonly used stability notions, including Lyapunov stability of motions or sets, (incremental) input-output stability, and asymptotic gain properties. Given a feedback interconnection of two subsystems, the main result of the paper (Theorem 1) establishes the following implication: small-gain property ⇒ stability of the feedback interconnection where the "small-gain property" is formally defined in Definition 3 and represents an abstraction, in the context of topological spaces, of the joint condition "stability of the subsystems + small-gain condition" of ordinary small-gain theories for inputoutput operators or ISS systems. While the proposed small-gain property does not admit, in general, a similar decomposition, we proved in Section III that it is always implied by ISS.
The main contribution of the article is methodological, as the presented results provide a common framework for small-gain theories and extend the "small-gain principle" beyond interconnections of systems defined between metric spaces of trajectories. Yet, the application of Theorem 1 to practical problems might not be straightforward. The examples of Section VI do suggest that the developed small-gain theory can provide a useful tool to study complex interconnections uncovered by other existing paradigms. However, its application does require the definition of suitable topological spaces and a preliminary analysis that are problem-specific and may not be easy. In this respect, further research is required.
Moreover, the developed theory focuses on continuity at a set or point, which is a local property. How to cover properties, such as uniform convergence or Lagrange stability, within the same setting of this article is still an open problem deserving further research.
Hence, y i ∈ Ψ i (Υ j (d), d i ). Since y i was arbitrary, we conclude that Then, there exists y j ∈ Y j such that y i ∈ Ψ i (y j , d i ) and y j ∈ Ψ j (y i , d j ). This, in turn, implies that y i ∈ Ψ i (Ψ j (y i , d j ), d i ). Thus, y i ∈ S i (d). Hence, we conclude that Υ i (d) ⊆ S i (d).
Conversely, pick y i ∈ S i (d). Then, there exists y j ∈ Ψ j (y i , d j ) such that y i ∈ Ψ i (y j , d i ). By the definition (3), this implies that (y 1 , y 2 ) ∈ Ψ(d). Hence, y i ∈ Υ i (d), which proves S i (d) ⊆ Υ i (d).
Thus, in view of (17), proceeding as in the proof of Proposition 2, we can find n y ∈ N ≥1 such that Γ As T i → 0, we have ω i (b − a)T i → b − a. Hence, the second term of the right-hand side of (37) vanishes as i → ∞. Since u i is T i -periodic, then a+(k+1)T i a+kT i u i (s)ds = a+T i a u i (s)ds. Hence, in view of (23), the first term of (37) satisfies