Integral-Input-to-State Stability of Switched Nonlinear Systems Under Slow Switching

In this article we study integral-input-to-state stability (iISS) of nonlinear switched systems with jumps. We demonstrate by examples that iISS is not always preserved under slow enough dwell time switching, and then we present sufficient conditions for iISS to be preserved under slow switching. These conditions involve, besides a sufficiently large dwell time, some additional properties of comparison functions characterizing iISS of the individual modes. When the sufficient conditions that guarantee iISS are only partially satisfied, we are then able to conclude weaker variants of iISS, also introduced in this work. As an illustration, we show that switched systems with bilinear zero-input-stable modes are always iISS under sufficiently large dwell time.


I. INTRODUCTION
A SWITCHED system is a dynamical system that consists of several subsystems and a logical rule, called a switching signal, which governs the switching between these subsystems. Due to their significance both in theory development and in practical applications (see, e.g., the book [1] or the survey [2] and the references therein), switched systems have received a great deal of attention in the last couple of decades. Nevertheless, the study of switched systems is still a challenging topic in the modern engineering literature because a switched system does not necessarily inherit the properties of its subsystems in general. For example, even if all the subsystems are stable, the switched system may still be unstable. Additional assumptions are needed to guarantee stability, either on the dynamics of all the subsystems-such as assuming the existence of a common Lyapunov function, which gives stability under arbitrary switching [1, Th. 2.1] (the converse is also true under some mild assumptions, see [3])-or on the switching signals, such as not allowing the switches to happen too often [4], [5], or a mix of the two.
When the subsystems are nonlinear and there are external inputs, input-to-state stability (ISS) [6] and integral-input-tostate stability (iISS) [7] can be used for the stability analysis. When all the subsystems are ISS with some mild assumptions, it has been shown using a Lyapunov function approach that the switched system is ISS as well under dwell time switching or average dwell time switching [5], [8], [9]; that is, the switching cannot happen too frequently.
However, regarding the iISS counterpart, it is not trivial to identify the assumptions that the subsystems need to meet (in addition to iISS) to ensure the iISS property for the switched system.
Indeed, we have observed in our recent work [10] that some switched systems with iISS subsystems are never iISS no matter how long the dwell time is. We showed that for such switched systems, bounds on the norm of the initial state and input energy have to be known prior to determining how slow the switching should be so that the switched system is iISS. We will also provide another example in this article to emphasize this problem. Unlike the one in [10], the best we can do in this example is to derive an iISS-like estimate on the solution with some offsets; in other words, under any dwell time switching and when there are no inputs, the solution of the switched system may not converge to the equilibrium but only to a neighborhood of it. Those observations suggest that we may only conclude some weaker versions of iISS when analyzing a switched system under slow switching, even if all its subsystems are iISS. On the other hand, there are still "good" switched nonlinear systems which inherit the iISS property from their subsystems; for example, switched bilinear systems which will also be discussed in this article. With that being said, this article aims to fill in the gap in the study of iISS of switched nonlinear systems with three main objectives as follows.
1) Define some weaker variants of iISS for switched nonlinear systems under slow switching. 2) Derive sufficient conditions so that the switched systems admit those weaker variants of iISS properties. 3) Show that when all the sufficient conditions hold, the switched system is iISS under slow switching.
We briefly mention some relevant research here. ISS for switched systems is also studied in [11] and [12] and the very recent work [13], just to name a few. ISS for hybrid systems is first studied in [14] and ISS for impulsive systems is studied in [15] and [16]; those studies can be generalized to switched systems. In terms of iISS results for switched systems, a converse theorem for iISS of switched systems is proposed in [17] while iISS is studied in the hybrid system framework in [18]. Statedependent switching is studied in [19] and [20] to guarantee iISS of switched systems. Some connections between ISS and iISS for switched systems are discussed in [21], and further characterizations of iISS for switched systems are presented by the same authors in [22]. A weaker variant of iISS, quasi-iISS which depends on the switching signal, is introduced in our recent work [10] and we have shown that under some mild assumptions, a switched system with all iISS subsystems is guaranteed to be quasi-iISS under slow switching. In this work we prove that those assumptions in fact guarantee uniform quasi-iISS with respect to all switching signals with the same dwell time condition. In addition, we identify other conditions which, when combined with the sufficient conditions for quasi-iISS, imply that the switched system is iISS.
This article is organized as follows: Basic notations are introduced and the switched systems as well as variants of iISS are defined in Section II. Section III provides two examples of switched systems where iISS cannot be achieved by any dwell time switching, which shows why we need some weaker variants of iISS. Our main results are then stated in Section IV and they are proven in Section V after some supporting lemmas are provided. With these results, the earlier examples are revisited in Section VI and a result on switched bilinear systems is derived. Section VII contains some remaining discussions and future work, followed by the conclusion in Section VIII.

A. Notations
We use the convention that N = {0, 1, · · · } is the set of all natural numbers including 0 and N + = N\{0}. We denote the maximum between a and b by a ∨ b, and the minimum between a and b by a ∧ b. For convenience, we also use n i=1 a i (resp. n i=1 a i ) to denote the maximum (resp. minimum) in the set {a 1 , . . . , a n }.

B. Switched Systems
Let P ⊂ N + be a set of either finite or infinite cardinality. For all p ∈ P, let the vector fields f p (x, u) : R n × R m → R n be locally Lipschitz, uniformly with respect to p ∈ P and assume they have the common equilibrium property that f p (0, 0) = 0.
The differential equationṡ where x(t) ∈ R n is the state variable of the system and u(t) ∈ U ⊆ R m is the input variable, are the dynamics of the subsystems or modes of the switched system. We consider the input u being measurable, locally essentially bounded and we denote the set of such input functions taking values in U by M U . For all pairs (p, q) ∈ P 2 , define the jump maps g p,q (x) : R n → R n which are continuous in x, uniformly with respect to (p, q) ∈ P 2 and assume that they have the common equilibrium property that g p,q (0) = 0. Let Σ be the set of all left-continuous mappings (which are also measurable) from [0, ∞) to P, called switching signals. For each switching signal σ ∈ Σ, define the ordered set is the collection of times when switches occur. Note that by this definition we have assumed that there is no switch at the initial time t 0 = 0. Define the class of switching signals with some dwell Note that Σ(τ D ) is a more regular set of switching signals compared to Σ as the positive dwell time τ D prevents accumulated switches and chattering. For a given σ ∈ Σ(τ D ), a switched system is defined bẏ We denote the solution of (3) with initial state x 0 , input u, and switching signal σ by x(·; x 0 , u, σ). When x 0 , u, σ are clear from the context, the solution is also abbreviated by x(·). Note that by definition of (3), x(·; x 0 , u, σ) obeys some differential equation when there is no switch, and it jumps when there is a switch. When there are no state jumps, i.e., all g p,q functions are identity functions, (3a) is sufficient to describe the dynamics of the switched system. In this case we say that the switched system is jump-free. In general the jump maps g p,q could be functions of states and inputs so when there are jumps, the magnitude of the jumps could also depend on the input, which is discussed in the framework of hybrid systems in [14] or impulsive systems in [15]. The presence of inputs in the jump map would require additional care in treating u(t i ) for t i ∈ T (σ) when defining iISS. Our work can be easily extended to that more general framework; nevertheless, for the clarity of presentation we focus on the framework of (3) and the simpler stability definitions which are introduced in the next subsection.

C. Stability Definitions
We start by defining iISS for switched systems. Definition 1: The switched system (3) is iISS under slow switching if there exist a dwell time τ D > 0 and functions β * ∈ KL and γ * , χ * ∈ K, such that The phrase "under slow switching" indicates that we only ask the iISS property to hold for sufficiently large τ D . Note that our definition of iISS is uniform with respect to the switching signal in the sense that β * , γ * are independent of σ. When there are no switches, the above definition reduces to the classical iISS notion for a nonswitched system [7], which can also be characterized by an iISS-Lyapunov function.
Lemma 2.1: The systeṁ is iISS if and only if there exist a C 1 function V : R n → [0, ∞) and functions α ∈ PD, α 1 , α 2 ∈ K ∞ , χ ∈ K satisfying and Note that (6) is different from the more standard iISS-Lyapunov function condition where α is evaluated on |x|, not V (x). Nevertheless, in the proof of [7,Th. 1] it is shown that the two conditions are equivalent, and hence we will always use (6) when defining iISS-Lyapunov functions.
Next, we introduce two weaker variants of iISS. Definition 2: The switched system (3) is quasi-iISS (qiISS) under slow switching if for all δ 1 , δ 2 > 0 there exist a dwell time τ D > 0 and functions β * ∈ KL and γ * , χ * ∈ K, such that the estimate (4) holds for all t ≥ 0, x 0 ∈ R n with |x 0 | ≤ δ 1 , u ∈ M U with ∞ 0 χ * (|u(τ )|)dτ ≤ δ 2 and all σ ∈ Σ(τ D ). Definition 3: The switched system (3) is integral-input-tostate practically stable (iISpS) under slow switching if for each δ 3 > 0 there exist a dwell time τ D > 0 and functions β * ∈ KL, γ * , and χ * ∈ K, such that The qiISS property is inspired by "quasi-disturbance-to-error stability" (qDES) in the work [24], and the prefix "quasi" means the nonlinear estimates are not global but depend on initial conditions and magnitude of inputs. Analogously, the prefix "quasi" is here intended in a similar way as the prefix "semiglobal" defined in the work [25] and [26], where the system is nonswitched. The iISpS property is inspired by "input-to-state practical stability" (ISpS) in the work [27], and the qualifier "practical" refers to the presence of an offset term. Note that we also use the phrase "under slow switching" in the definitions of qiISS and iISpS, which indicates that we only ask the estimates to hold for sufficiently large dwell time τ D .
The following proposition regarding iISS, qiISS, and iISpS can be stated.
Proposition II.2: The switched system (3) is iISS under slow switching if and only if it is both qiISS and iISpS under slow switching.
Proof: The implication from iISS under slow switching to qiISS and iISpS under slow switching is clear. To show the other direction, we start by assuming that the system (3) is both qiISS and iISpS under slow switching. Pick δ 1 , δ 2 , δ 3 > 0 with δ 3 < δ 1 . Note that iISpS implies the existence of β * 1 ∈ KL, γ * 1 , χ * 1 ∈ K and τ D 1 > 0, such that for all t ≥ 0, x 0 ∈ R n , u ∈ M U and all σ ∈ Σ(τ D 1 ) Consider the family of systems defined by parametrized Note that since switching signals with dwell time condition are uniformly incrementally bounded [28, Definition 2.2], strong global uniform asymptotic stability under zero input (0-GUAS) and strong iISS, defined by [28, Definition 2.1], are, respectively, equivalent to 0-GUAS and iISS for S as discussed in that work. 1 We next show that S is both uniformly bounded energy input bounded state (UBEBS) (which is also defined by [28, Definition 2.1]) and 0-GUAS. To this end, S being UBEBS can be directly concluded from (8), by noticing that β * Because all f i , g p,q are independent of t and time shift does not change the slow switching nature of σ, the combination of (9) and (10) implies that S is 0-GUAS. Now because f i 's satisfy the conditions listed in [22, Lemma 1], they satisfy [22,Assumption 1], which is the same as [28, Assumption 1]; on the other hand, g p,q 's can be verified to satisfy Assumption 2 by directly using the assumptions that they are continuous uniformly over (p, q) ∈ P 2 and g p,q (0) = 0. As a result, from [28, Proposition 2.3 and Th. 3.1] we conclude that S is iISS. Because our switched system (3) is a particular realization of the family of systems S, it is therefore iISS under slow switching. Remark 1: Recall that we have assumed the Lipschitzness of f i (x, u) to be uniform over i ∈ P and the continuity of g p,q (x) to be uniform over (p, q) ∈ P 2 . Those assumptions are critical for Proposition II.2, and they are automatically satisfied if P is a finite set. In fact it can be shown that those uniformity assumptions are not necessary for concluding all the other results in this work. Nevertheless, while those assumptions make the augments in this article slightly conservative, Proposition II.2 helps us streamline the proofs.
In the Section IV we point out the need for considering the two weaker variants of iISS through two examples of switched systems, in which, while all the subsystems are iISS, the switched systems are not iISS, no matter how large τ D is. These examples also illustrate the mechanisms of how iISS may be destroyed by switching.

III. MOTIVATING EXAMPLES
Example 1: Consider the two-dimensional jump-free switched system with two modeṡ where This example is already mentioned in our earlier work [10], where it was shown that both subsystems are iISS (but not ISS). Meanwhile, it is already discussed in [1, Sec. 3.2] that the switched linear system with the modeṡ has no common Lyapunov function and for some particular switching signals the solution will diverge, as shown in Fig. 1(c). When u ≡ 0, it is observed that the vector fields of the subsystems of (11) are the same as the vector fields of (13) except that the magnitude is scaled by 1 1+|x| 2 ; hence the solution trajectories of (11) when u ≡ 0 will be the same as the trajectories of (13) up to a time reparameterization. We now claim that the switched system generated by (11) is not iISS under slow switching. Let us initially consider the unforced version of system (11). By a coordinate change of (11) into polar form we havė in either case we always have |θ| ≤ 2 1+|x| 2 . Note that in order to achieve a divergent solution trajectory as the one in Fig. 1(c), we need the switches to occur when the state is on either axis; in other words, a switch occurs every time when θ increases by π 2 . We also observe that for this divergent trajectory, there exists ρ ∈ K ∞ , such that |x(t)| ≥ ρ(|x(0)|) for all t ≥ 0. Now for any τ D > 0, we pick the initial condition, such that |x(0)| ≥ and it is on an axis. Suppose it requires Δt for θ to increase by π 2 ; in other words Thus we have Δt ≥ τ D , which means we can always find a switching signal σ ∈ Σ(τ D ) yet the solution is exactly as the one in Fig. 1(c) and it is divergent. Hence the switched system generated by (11) is not iISS under slow switching. Note that this fact can also be argued by picking an input with large but finite integral and only a very small support near t = 0 and showing that this input leads to an "initial" state with arbitrarily large magnitude; hence by a similar argument we can again find a divergent solution.
Example 2: Consider the two-dimensional jump-free switched system with two modeṡ where A p are the same as given in (12). Both its subsystems are iISS (in fact also ISS), which will be shown later in Section VI and hence not repeated here. Nevertheless, we still claim that the switched system generated by (14) is also not iISS under slow switching. To show this, we first observe that by the same argument as in the previous example, the solution trajectory of (14) when u ≡ 0 is the same as the trajectory of (13) up to a time reparameterization. Again by coordinate change into polar form we can show that |θ| ≤ 2|x|. For any τ D > 0, consider the threshold c = π 4τ D and let x(0) be on an axis and such that |x(0)| < c. Apply the switching signal which results in a locally divergent solution as in Fig. 1(c) until a time s at which |x(s)| = c. Keep the mode active at time s and let τ ≥ τ D be the first time, such that x(s + τ ) is on an axis and |x(s + τ )| < c (such a τ exists because of how the modes behave for u ≡ 0). We can then treat s + τ as the initial time and repeat the above process. Note that for |x| ≤ c, the time Δt needed for the state to travel from one axis to the other satisfies Thus again Δt ≥ τ D , and by construction our switching signal is in Σ(τ D ). On the other hand, the resulting solution satisfies |x(t)| = c at infinitely many times t, hence it cannot converge to the origin. Therefore, the switched system generated by (14) is also not iISS under slow switching.
In both examples we see that there is no uniform stabilizing dwell time with respect to all initial states or inputs, and this is the major reason why these switched systems are not iISS under slow switching. In the first example, the convergence of subsystem solutions is slow when the magnitude of initial state or the integral of input is too large; in the second example, the convergence of subsystem solutions is slow when the states are too close to the origin. Recall that our definitions of qiISS and iISpS exactly deal with the cases of either small initial states plus small integral of inputs, or states sufficiently far away from the origin. We will revisit these two examples in Section VI and it is not to the readers' surprise that by applying the criteria derived in this work, the switched system generated by (11) can be shown to be indeed qiISS under slow switching and the switched system generated by (14) can be shown to be indeed iISpS under slow switching.

IV. MAIN RESULTS
Before we state the main result, we claim that a PD function can always be lower-bounded by the product of a nondecreasing function and a nonincreasing function. This claim will be used in our main theorem.
Lemma IV.1: Let α be a locally Lipschitz function and α ∈ PD. Then there exist a locally Lipschitz nondecreasing function ρ 1 ∈ PD and a locally Lipschitz nonincreasing function ρ 2 : Lemma IV.1 is adopted from [7, Lemma IV.1] and its proof is straightforward and hence omitted.
We need the following two assumptions in order to state our main results.
Assumption 1: There exist C 1 functions V p : R n → [0, ∞) for all modes p ∈ P and a locally Lipschitz function α ∈ PD and functions α 1 , α 2 ∈ K ∞ , χ ∈ K, independent of p, such that and Assumption 2: The functions V p in Assumption 1 satisfy is a continuous and nonincreasing function with the property that there exists δ > 0, such that for any We make some remarks regarding the two assumptions. Compared with Lemma II.1, Assumption 1 implies that the subsystems of the switched system are iISS. In general, the iISS estimation functions may vary from mode to mode; that is, instead of unique α and χ, we may have α p , χ p depending on p ∈ P. However, if the set P is finite, we can always pick α(s) := p∈P α p (s) and χ(s) := p∈P χ p (s) and the estimation (16) will hold uniformly. When P has infinite cardinality, our assumption is still valid as long as there exist a uniform lower bound on α p and a uniform upper bound on χ p and they belong to class PD and class K, respectively.
On the other hand, assuming a constant gain μ in the value of Lyapunov functions when a switch occurs is a common practice in the literature (see, e.g., [8], [9]). The idea of using nonconstant gain μ is borrowed from the work [13] and our Assumption 2 clearly includes the case of constant gain. We observe that when the Lyapunov functions used for subsystems are all quadratic or polynomial functions of same degree, there is no advantage in assuming nonconstant gain. However, in the case when lim |x|→∞ V p (x)/V q (x) = 1, nonconstant gain becomes critical for deriving the desired results.
We are now ready to state the first main result of the article. Theorem 1: Consider a switched system defined via (3) with a set of modes P and assume both Assumption 1 and Assumption 2 hold. Let β ∈ KL be such that β(s, ·) is the solution to the initial-value ordinary differential equation We have the following implications.
then (3) is iISpS under slow switching. 3) If both (19) and (20) hold, then (3) is iISS under slow switching. Remark 2: The condition (20) can also be equivalently restated as which has a similar form as (19). We also note that from monotonicity properties of the function h (which will also be used in the proof of Theorem 1) it can be shown that the infima appearing in (18) and (19) are actually limits as ε → 0 + and as t → ∞, respectively. We would like to provide some insight about the assumptions (19) and (20). Recall that the point of stabilizing a switched system via slow switching is to neutralize the possible destabilizing effect of switching by the stabilizing effect when the system dwells in some modes for sufficiently long time. The function h introduced by (18) is essentially the estimate of the combination of these two effects for a single switch. In order to have an overall stabilizing effect when the system switches slowly, we need the existence of τ D > 0, such that h(s, t, ε) < s for all t ≥ τ D . This property is easily seen to be guaranteed by the equivalent characterizations of the inequalities (19) and (20) given in Lemma V.2 later. On the other hand, because of the lim sup functions in (19) and (20), there is no need to compute the entire h function but only analyze its value when s approaches to the limits when applying Theorem 1. See Remark 4 for more discussion.
In the special case that when the switched system is jumpfree (recall that jump-free switched systems mean g p,q (x) = x ∀p, q ∈ P), we do not need the knowledge of β, μ at all. The next theorem provides a more direct qualitative statement on iISS-related properties for jump-free switched systems which is based only on α, α 1 , α 2 .
Theorem 2: Consider a jump-free switched system defined via (3a) with a set of modes P and assume that Assumption 1 holds. Further assume that α 1 , α 2 are C 1 and there exist M ≥ M > 0, such that We have the following implications.
3) If the assumptions in 1 and 2i, or assumptions in 1 and 2ii are satisfied, then (3a) is iISS under slow switching. We would like to make some remarks on the assumptions in Theorem 2. First, the assumption lim inf v→0 + α(v) v > 0 is sometimes called superlinearity at the origin, which means that there exists k, l > 0, such that α(v) ≥ kv for all v ∈ [0, l]. It also means zero-input local exponential stability (0-LES) of the subsystems, and hence the first implication in Theorem 2 implies the switched system is qiISS under slow switching if all its subsystems are iISS and 0-LES.
On the other hand, the assumption 2.i in Theorem 2 is superlinearity at infinity because it is equivalent to the existence of k, l > 0, such that α(v) ≥ kv for all v > l. When both assumptions 1 and 2i in Theorem 2 are true, we simply have which means that the subsystems are all ISS with exponential decay rate when they are unforced; hence the study in [8] in fact directly tells us that the switched system is ISS under slow switching. More discussion for this special case and how dwell time can be estimated will be given in Section VII. When α ∈ K, it might be upper bounded and the subsystems are only iISS but not ISS. In this case we are able to conclude iISpS of the switched system by 2ii of Theorem 2 when the other assumption in 2ii also holds, which requires that the gap between α 1 (v) and α 2 (v) be uniformly bounded for large v.
Lastly note that if α ∈ PD but lim v→∞ α(v) = 0, Theorem 2 does not allow us to prove iISpS under slow switching, nevertheless the first statement of Theorem 2 may still apply. In fact, it can be shown that in this case a sufficient condition for the switched system to also be iISpS under slow switching is that α 2 (v) − α 1 (v) converges to 0 as v → ∞ at a rate comparable to the rate at which α converges to 0. We do not elaborate on this result because these properties appear to be too restrictive; examples satisfying such properties will be extremely artificial.

A. Supporting Lemmas on Comparison Functions
In this subsection we list several lemmas which will be used in the proof of Theorem 1. Lemma V.1 is a direct consequence of Assumption 2. Lemma V.2 contains some straightforward analysis observations. Corollary V.3 is a direct combination of the two statements of Lemma V.2. Lemma V.4 is cited from [7, Lemma IV.2]. Lemma V.5 is a result on combining two class KL functions and Lemma V.6 is a simple fact regarding splitting class K functions. The proofs of Lemma V.1, V.2, and V.5 are provided in Appendix A.

2) The inequality (20) holds if and only if there exist
for all (s, t, ε) By combining Corollary V.3 with Theorem 1, it is seen that iISS switched system under slow switching is implied by the condition (25) holding for all (s, t) ∈ [0, ∞) × [τ D , ∞). An inequality similar to (23) also appears in [11], where it is used as the essential assumption to show ISS of the switched nonlinear system. Our assumption is weaker than theirs as we do not require (23) to hold for all s ≥ 0. On the other hand, the bound on h(s, t, ε) in (25) is strictly larger than the bound in (23). As discussed after Theorem 1, it is conjectured that an even weaker assumption that h(s, t, ε) < s for some ε > 0 and all (s, t) ∈ (0, ∞) × [τ D , ∞) is sufficient to show iISS. We will not discuss that assumption here.
Lemma V.4: Let α ∈ PD and consider two functions y : [0, ∞) → R being C 1 and z : [0, ∞) → [0, ∞) being continuous and nondecreasing, such thaṫ for almost all t ≥ 0. Then where β ∈ KL is the solution to the initial value ODE probleṁ is also a class KL function. Lemma V.6: Let α ∈ K and k 1 , k 2 , s 1 , s 2 ≥ 0. Then for any The inequality (29) is inspired by [29, Lemma 10] and proven similarly, therefore the proof is omitted. A coarser version of Lemma V.6 with fixed ε = 1 is used in our conference article version of this work [10]. However, in this article we can get tighter estimates on the dwell time by allowing arbitrarily small ε. See Section VII for more discussion on this.

B. Proof of Theorem 1
To show that the switched system is iISS (resp. qiISS or iISpS) under the assumptions given in Theorem 1, we need to show that the estimate (4) holds for all solutions [resp. the estimate (4) holds for solutions with bounded initial state and bounded integral of input, or the estimate (7) holds for all solutions]. In our previous work [10], a similar but nonuniform qiISS is defined and proven for switched systems satisfying similar assumptions. In that work, for any chosen switching signal σ, qiISS is shown by providing an estimation of solutions. In order to make the iISS-related properties uniform with respect to the switching signal σ ∈ Σ(τ D ), the estimation function needs be recursively defined as the supremum of some nested class KL functions and this new proof is presented in this section.
Throughout this section we will use the notation This is a nonincreasing function on [0, ∞) as μ is nonincreasing, and h(s, t, ε) = β(φ ε (s)s, t).
for any t i ∈ T (σ) and any t ∈ (t i , t i+1 ], where β ∈ KL is constructed from α as in Lemma V.4. Because y(t) =ỹ(t) or y(t) = 0, we further conclude that In addition, (32b) implies thatỹ(t Combining (33) and (34), picking ε ∈ (0, ε 0 ), where ε 0 comes from Lemma V.2 and applying Lemma V.6, for all t ∈ (t i , t i+1 ] we have where β y (s, t) := β(φ ε (s)s, t) = h(s, t, ε) and β z (s, t) := β(ε −1 φ ε (0)s, t). We need to invoke the following Lemma at this point. Lemma V.7: Let β y , β z ∈ KL, σ ∈ Σ(τ D ) for some τ D > 0 and let T (σ) = {t 1 , t 2 , . . . } be the set of switching times. Let z : [0, ∞) → [0, ∞) be a nonincreasing function and y : [0, ∞) → [0, ∞) with the properties that for any t i ∈ T (σ) and any t ∈ [t i , t i+1 ) For i ∈ N + , recursively define two families of functions h i : and Then h i ∈ KL and l i ∈ K ∞ for all i ∈ N + . In addition, for any i ∈ N, t j ∈ T (σ) and t ∈ (t j+i−1 , t j+i ] The proof of Lemma V.7 is given in Appendix B. Applying Lemma V.7, with j = 1, we conclude that y(t) ≤ h i (y(t 1 ), t − t 1 ) ∨ l i (z(t)) for any t ∈ (t i , t i+1 ], where h i , l i are defined in (36) and (37). In addition (33) also implies that y(t 1 ) ≤ β(y(0), t 1 ) ∨ z(t 1 ). Combining these two inequalities and pretending that the exact value of t 1 is unknown, we find an upper bound for y(t) by taking the supremum over t 1 Notice that the estimation of y(t) given in (39) is only true when there are i switches up to time t. In order to obtain an estimation of y(t) for arbitrary number of switches up to any time t ≥ 0, define We claim that whenever H ∞ (s, t) is finite for all t ≥ 0. Clearly by comparing (41) with (39), in order to show the claim it suffices to show that l i (s) ≤ H ∞ (ε −1 φ ε (0)s ∨ s, 0) for all i ≥ 2 and all s ≥ 0. This is indeed guaranteed by the recursive definition of l i in (37), and the fact that while either (19) or (20) hold, Lemma V.2 always implies that The next Lemma tells that the function H ∞ defined in (40) has some KL-like properties under some assumptions.
Lemma V.8: Let h be defined as in (18) . The proof of Lemma V.8 is also given in Appendix B. With this lemma, we can finally finish the proof of Theorem 1.
To show qiISS when (19) holds, let δ 1 , δ 2 > 0 be arbitrary. By the first statement in Lemma V.2, (19) implies the existence of λ ∈ (0, 1), such that for there exists τ D > 0, such that (23) holds for all (s, t) ∞). Thus by the first conclusion in Lemma V.8 there exists which is class KL by Lemma V.5. Also denote which is class K ∞ . Conditions (41), (43), and (44) imply that for all v(0), z(t), such that the first arguments in the two H ∞ functions in (41) are in the domain [0, b]. In addition, from (15) and (31) we have where we have also used the fact that α( n i=1 s i ) ≤ n i=1 α(ns i ) for any α ∈ K. Notice that the previously mentioned constraints on the first arguments in the two H ∞ functions in (41) are also satisfied since we have assumed that |x 0 | ≤ δ 1 and for all t ≥ 0.  (43), (44), and using (41), we conclude that h(b, 0, ε).
Since we have already proven that the system (3) is qiISS and iISpS when both (19) and (20) hold, the last statement in Theorem 1 is a direct consequence of Proposition II.2 and this completes the proof.

C. Proof of Theorem 2
Let We first show that (17) holds for all x ∈ R n , p, q ∈ P. For the nontrivial case when x = 0, recall g p,q (x) = x for a jump-free switched system, so (15) implies that for any p, q ∈ P We then show that μ defined via (46) satisfies the conditions in Assumption 2. Denoting w = α −1 where we have used the inequality nonincreasing on (0, ∞). Thus the lim sup in (46) is in fact lim and Therefore μ is continuous and nonincreasing on [0, ∞). In addition, (21) also implies that Hence μ satisfies the conditions in Assumption 2.
To show the first implication in Theorem 2, it suffices to show that (19) holds if we construct a function β ∈ KL from α. From Lemma IV.1 we construct ρ 1 (v) = min w∈ [v,1] 1 2 ]. Define k := min s∈[0, 1 2 ] ζ(s) > 0. Recall that in Theorem 1, β(s, ·) is the solution to the initial value ODE problemv = −ρ 1 where recall φ ε is defined in (30). The inequality (19) can be shown subsequently and the first implication in Theorem 2 is proven by applying the first result in Theorem 1.
Next we show that 2ii implies iISpS. Because α ∈ K ∞ can always be replaced by α ∈ K while preserving (16), we can always assume lim s→∞ α(s) =: M 1 < ∞. Since α ∈ K, it follows that M 1 > 0. Now by definition (46) and the assumption on lim sup in this case As a result, for any δ > 0, there exists v, such that as long Therefore, for all s ≥ 2v, φ ε (s)s ≥ s ≥ 2v. Thus the initial condition is large enough and consequently

VI. EXAMPLES REVISITED
In this section we apply our main results to the two examples considered in Section III to draw some conclusions about their iISS properties.
We use the Lyapunov function V p = x M p x + 1 − 1 to show that the subsystems (11) are iISS, where M p is the solution to the Lyapunov equation . It can be computed a . On the other hand, by computing the derivative of α 1 , α 2 it is easy to see that Furthermore, it is not difficult to verify that Thus the assumption (21) is satisfied. In addition Hence we have It is easy to compute that α (0) = 1 a > 0 so by the first implication in Theorem 2, the jump-free switched system generated by (11) is qiISS under slow switching. On the other hand, neither assumption in the second implication in Theorem 2 applies here because α is not superlinear at infinity (in fact it converges to 0 at infinity) and α 2 (s) − α 1 (s) diverges at infinity. Indeed, we have already shown in Section III that this switched system is not iISS under slow switching.
Regarding the jump-free switched system with the modes (14) in Example 2, we pick the Lyapunov function V p = |x|x M p x, where M p again solves (47). Again the assumption (15) holds with α 1 (v) = av 3 , α 2 (v) = av 3 , where a, a are defined in (48). Thus = a a and the assumption (21) in Theorem 2 is satisfied. In addition for all x = 0 Hence we conclude α(v) = 3 2 ( v a ) 4 3 , χ(v) = 3a 2 2 v 2 , and each mode (14) is iISS (in fact also ISS). Clearly this α satisfies the assumption in 2i of Theorem 2; hence the switched system (14) is iISpS under slow switching. On the other hand, the assumption in the first implication in Theorem 2 does not apply because α (0) = 0. Indeed, we have already shown in Section III that this switched system is not iISS under slow switching. Similarly to what has been done for the qiISS example, let us provide a numerical estimation of the dwell time for this iISpS system. For this example, we have α(v) = 3 2 ( v 14.98 ) 4 3 , μ(v) = a a = 1.99; we also pick Δ := 0.001 and δ 3 := 2 which, it follows from (45) that for ε = 0.001, b = 10.02. With such a choice of parameters, it is numerically computed from the condition (24) that τ D = 7.12.
At last, we draw some conclusions on the switched bilinear systems with inputs, generated from finitely many modeṡ where p ∈ P = {1, 2, . . . , P } are the modes, x ∈ R n , u = u 1 · · · u m ∈ R m and A p ∈ R n×n , B p,j ∈ R n×n and C p ∈ R n×m . Proposition VI.1: The jump-free switched bilinear system generated from finitely many modes (49), where A p are all Hurwitz, is iISS under slow switching.

VII. DISCUSSION AND FUTURE WORK
First of all, since the major focus of this article is to qualitatively determine whether a switched nonlinear system is iISS when it switches sufficiently slowly, the quantitative value of dwell time is not emphasized and it can be investigated in future research. Nevertheless, we point out here that the τ D in Corollary V.3 is a dwell time for the switched systems to be iISS. In the special case when the assumptions in the cases 1 and 2i in Theorem 2 both hold or, equivalently, when (22) holds for the subsystems, we have β(s, t) = se −kt , given by Lemma V.4. Thus, taking μ to be a constant and ignoring Δ, the inequality (25) gives Taking ε → 0 and λ → 1, we see that the infimum of dwell time is ln μ k , same as the lower bound on dwell time found in [8] which is based on ISS analysis. This suggests that our approach may also be promising for quantitatively analyzing general switched nonlinear systems in the sense that it can give tight results on the minimum dwell time.
Another possible future work direction is of course to extend dwell time to average dwell time. It is observed in the literature that linear decay rates in Lyapunov functions near the origin (such as the assumption in 2.i in Theorem 2) are essential when deriving the average dwell time. When this does not hold in general, nonlinear decay rates can be transformed into linear ones as by [32,Lemma 11], and the work [13] has formulated average dwell time conditions for ISS of hybrid systems based on that transformation. Using similar techniques, some promising results are presented in [33]. The very recent article [16] also discusses ISS properties of impulsive systems whose decay rates are nonlinear and the techniques used in that article may also be extended to study iISS-related properties of switched system under slow switching.
We can also study sufficient conditions for iISS of switched systems when there are unstable subsystems. Inspired by [9], the divergence of solutions due to the unstable subsystems can be either compensated by the convergence when the switched system is dwelling in a stable subsystem, or the stable jumps during switches. Reverse dwell time conditions may be concluded in this case to guarantee overall stability of the system. From our observations on the examples mentioned in Section III, the major reason that the systems (11) and (14) fail to be iISS under slow switching is due to the fact that the unforced systems are not 0-GUAS under slow switching. We do not really show how the interaction between input and dissipation may affect the stability of these switched systems. In fact, it can be inferred by [28] that a switched system is iISS under slow switching if and only if it is both 0-GUAS and UBEBS under slow switching, and this equivalence is also utilized in the proof of Proposition II.2 in this article. Nevertheless, while the above equivalence gives us some hints on how the input may affect the stability of the switched system in addition to the dissipation of the unforced system, "0-GUAS under slow switching" or "UBEBS under slow switching" are not stability properties that can be directly verified/designed for the switched system. In our framework we want to conclude some stability results of the switched system based on the sole knowledge of the stability properties of the subsystems and the features of the switching signals. Since iISS is equivalent to 0-GUAS plus UBEBS for a single-mode system as stated in [26], our next step can certainly be finding the sufficient conditions under which the UBEBS property can be passed from its subsystems to the switched system under slow switching. Meanwhile, inspired by the literature, it is also interesting to ask whether strong iISS, defined in [31], can be passed from its subsystems to the switched system under slow switching.
We also realize one drawback of our criteria that they rely on the Lyapunov functions of subsystems, which suggests that potentially the sufficient conditions based on α, α 1 , α 2 may not be invariant with respect to the choice of Lyapunov functions. Hence better criteria to test whether a switched system is iISS under slow switching will be directly relying on the stability properties of the subsystems. Certainly as discussed in this work, we need properties stronger than iISS for all the subsystems to hold. By a comparison between our "bad" systems (11) and (14) and the "good" bilinear system (49), an interesting question to ask is whether it is true that if the subsystems of a switched system are all globally exponentially stable under zero input and UBEBS, then the switched system is iISS. More research can be done in this direction.

VIII. CONCLUSION
In this article we have defined iISS, qiISS, and iISpS under slow switching for switched nonlinear systems. We then provided two sets of sufficient conditions, such that the switched system will have one of the aforementioned stability properties when either set of the proposed conditions is satisfied. In addition, if a switched system satisfies both sets of the proposed conditions, then it is iISS under slow switching. As a direct consequence from our result, we have shown that switched systems whose subsystems are 0-input stable bilinear ones are iISS under slow switching.

B. Proofs of Lemmas in Section V-B
Proof of Lemma V.7: The claim that h i ∈ KL is a direct result of Lemma V.5 and the claim that l i ∈ K ∞ follows from the definition (37). We now use induction on j to show (38). The base cases when i ∈ N + , j = 1 are trivially given by (35). Suppose the estimate (38) is true for all i ∈ N and all j ≤ N . We now want to find an upper bound on y(t) for t ∈ (t i+N , t i+N +1 ]. 2 We thank an anonymous reviewer for suggesting this shorter alternative proof of Lemma V.5.
Notice that there are N switches from time t i+1 to time t so by induction hypothesis we have where the second inequality comes from bounding y(t i+1 ) using (35) and it is split into two terms in the third inequality. In Thus by the monotonicity of z, β z and h N , we have Therefore and Lemma V.7 is proven by induction.
Proof of Lemma V. 8: Define Hence all h i (s, ·) converge to 0 uniformly and consequently lim t→∞ H ∞ (s, t) = 0. We conclude that there existsβ ∈ KL, such that H ∞ (s, t) ≤β(s, t) for all (s, t) ∈ Thus we have proven the claim and we conclude that