Matteo Della
email: matteo.dellarossa@uclouvain.be.

Rossa Aneel Tanwani

Luca Zaccarian

Nonpathological ISS-Lyapunov Functions for Interconnected Differential Inclusions

This article concerns robustness analysis for interconnections of two dynamical systems (described by upper semicontinuous differential inclusions) using a generalized notion of derivatives associated with locally Lipschitz Lyapunov functions obtained from a finite family of differentiable functions. We first provide sufficient conditions for input-to-state stability (ISS) for differential inclusions, using a class of non-smooth (but locally Lipschitz) candidate Lyapunov functions and the concept of Lie generalized derivative. In general our conditions are less conservative than the more common Clarke derivative based conditions. We apply our result to state-dependent switched systems, and to the interconnection of two differential inclusions. As an example, we propose an observer-based controller for certain nonlinear two-mode state-dependent switched systems.

Introduction

For analyzing stability or performance of integrated or large-scale dynamical systems, it is natural to consider them as a collection of several subsystems of lower dimension/complexity. After a certain abstraction, the behavior of the overall system can be obtained either by switching among the constituent subsystems, or through certain interconnections of the underlying subsystems, or through a combination of these. This viewpoint of analyzing complex systems provides the motivation to consider stability and robustness analysis for interconnections of switched systems.

Given a family of vector fields {f 1 , . . . , f K } ⊂ C 1 (R n × R m , R n ) and a switching signal σ : R n → {1, . . . , K}, we consider the system ẋ = f σ(x) (x, u).

For studying generalized solutions of such discontinuous systems, we extend the map f σ(•) (•, u), considering the Filippov regularization of (1) (see [START_REF] Filippov | Differential Equations with Discontinuous Right-Hand Side[END_REF]). This leads to a differential inclusion of the form (see Section 3 for details) ẋ ∈ F (x, u),

where F satisfies some regularity assumptions. Our first objective is to study asymptotic stability and robustness with respect to u for system [START_REF] Bacciotti | Nonsmooth optimal regulation and discontinuous stabilization[END_REF]. We then apply our results to the analysis of interconnected systems of the form

® ẋ1 ∈ F 1 (x 1 , x 2 , u), ẋ2 ∈ F 2 (x 1 , x 2 , u). (3)
In the theory of nonlinear control systems, the concept of input-to-state stability (ISS), introduced in [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF], has been widely used to study the robustness of dynamical subsystems to external disturbances. Because 1 of its elegant characterization in terms of Lyapunov functions, ISS is now perceived as a textbook tool for analyzing the performance of nonlinear systems, [START_REF] Khalil | Nonlinear Systems[END_REF]. For example, the ISS notion has been useful in analyzing interconnections of two dynamical systems, either in cascade form [START_REF] Sontag | Changing supply functions in input/state stable systems[END_REF], or in feedback by using the small-gain condition [START_REF] Jiang | Small-gain theorem for ISS systems and applications[END_REF][START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF][START_REF] Ito | An intuitive modification of max-separable Lyapunov functions to cover non-ISS systems[END_REF]. Moving away from the framework of conventional nonlinear systems, the ISS notion has been generalized to systems with continuous and discrete dynamics. In this regard, we find sufficient conditions in terms of slow switching for ISS of time-dependent switched systems in [START_REF] Vu | Input-to-state stability of switched systems and switching adaptive control[END_REF], characterization of ISS for hybrid systems with jump dynamics in [START_REF] Cai | Characterizations of input-to-state stability for hybrid systems[END_REF], [START_REF] Cai | Robust input-to-state stability for hybrid systems[END_REF], or for a class of differential inclusions in [START_REF] Jayawardhana | Input-to-state stability of differential inclusions with applications to hysteretic and quantized feedback systems[END_REF]. More recently, we have seen ISS results for interconnections of hybrid systems [START_REF] Liberzon | Lyapunov-based small-gain theorems for hybrid systems[END_REF][START_REF] Sanfelice | Input-output-to-state stability tools for hybrid systems and their interconnections[END_REF], and time-dependent switched systems [START_REF] Yang | A Lyapunov-based small-gain theorem for interconnected switched systems[END_REF][START_REF] Zhang | ISS Lyapunov functions for cascade switched systems and sampled-data control[END_REF].

By and large, most of the aforementioned results in the literature deal with smooth Lyapunov functions. This is partially justified by the fact that the existence of a smooth Lyapunov function is not only sufficient but also necessary for asymptotic stability of the equilibrium [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF], [START_REF] Teel | A smooth Lyapunov function from a class-KL estimate involving two positive semidefinite functions[END_REF], [START_REF] Dayawansa | A converse Lyapunov theorem for a class of dynamical systems which undergo switching[END_REF], and for ISS with respect to external perturbations [START_REF] Lin | A smooth converse Lyapunov theorem for robust stability[END_REF]. Under some assumptions, this implications holds true also in the context of hybrid systems, as proved in [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF], [START_REF] Cai | Robust input-to-state stability for hybrid systems[END_REF]. The recent survey on converse Lyapunov theorems [START_REF] Kellett | Classical converse theorems in Lyapunov's second method[END_REF] provides an insightful background on such developments. However, in the context of switched and hybrid systems, the "composite" structure itself provides the motivation to work with multiple smooth Lyapunov functions, see for example [START_REF] Liberzon | Switching in systems and control[END_REF]Chapter 3] and [START_REF] Johansson | Computation of piecewise quadratic Lyapunov functions for hybrid systems[END_REF]. More specifically, when dealing with time-dependent switched systems, these multiple Lyapunov functions can still be combined to get a smooth (with respect to the state) composite Lyapunov function. Such constructions have been seen in analyzing ISS of switched system [START_REF] Vu | Input-to-state stability of switched systems and switching adaptive control[END_REF] and certain interconnections [START_REF] Yang | A Lyapunov-based small-gain theorem for interconnected switched systems[END_REF][START_REF] Zhang | ISS Lyapunov functions for cascade switched systems and sampled-data control[END_REF]. When dealing with state-dependent switched systems, the patching of the Lyapunov functions may make the resulting common Lyapunov function non-differentiable, but locally Lipschitz in most cases. This element is seen in the analysis of asymptotic stability using piecewise differentiable functions [START_REF] Johansson | Computation of piecewise quadratic Lyapunov functions for hybrid systems[END_REF][START_REF] Baier | Linear programming based Lyapunov function computation for differential inclusions[END_REF], and to some extent for establishing ISS in [START_REF] Heemels | Input-to-state stability of discontinuous dynamical systems with an observer-based control application[END_REF], [START_REF] Heemels | Input-to-state stability and interconnections of discontinuous dynamical systems[END_REF]. For trajectory-based conditions for ISS of state-dependent switched systems, see the recent paper [START_REF] Liu | Global stability and asymptotic gain imply input-to-state stability for statedependent switched systems[END_REF].

This paper is about developing sufficient conditions for ISS using locally Lipschitz Lyapunov functions for the class of differential inclusions in [START_REF] Bacciotti | Nonsmooth optimal regulation and discontinuous stabilization[END_REF]. The concept of set-valued derivatives for locally Lipschitz functions, introduced in [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF], [START_REF] Bacciotti | Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions[END_REF], is crucial to properly define the notion of derivatives along the system's trajectories. In particular, we focus on two different notions of set-valued derivatives that we call Clarke and Lie derivatives, each of them being a set-valued map from the state space R n to the real numbers, see [START_REF] Ceragioli | Discontinuous ordinary differential equations and stabilization[END_REF] and [START_REF] Cortes | Discontinuous dynamical systems[END_REF] for the formal definitions. For a large class of locally Lipschitz functions called non-pathological functions (as phrased in [START_REF] Valadier | Entrainement unilateral, lignes de descente, fonctions Lipschitziennes non pathologiques[END_REF]), the notion of Lie derivative leads to less conservative stability conditions, see [START_REF] Ceragioli | Discontinuous ordinary differential equations and stabilization[END_REF] and our recent papers [START_REF] Della Rossa | Smooth approximation of patchy Lyapunov functions for switching systems[END_REF] and [START_REF] Della Rossa | Max-min Lyapunov functions for switched systems and related differential inclusions[END_REF]. As another example, the Lie derivative concept has been recently used in [START_REF] Kamalapurkar | On reduction of differential inclusions and Lyapunov stability[END_REF] to identify and remove infeasible directions of a differential inclusion of the form [START_REF] Bacciotti | Nonsmooth optimal regulation and discontinuous stabilization[END_REF], and for stability analysis using an invariance principle for state-dependent switched systems [START_REF] Kamalapurkar | Invariance-like results for nonautonomous switched systems[END_REF], based on the ideas already introduced in [START_REF] Sanfelice | Invariance principles for hybrid systems with connections to detectability and asymptotic stability[END_REF].

The technical content of this paper starts with the use of Lie derivatives for establishing Lyapunov-based ISS results for differential inclusions (2) (but with particular attention to switched systems as in (1)), while considering non-pathological candidate Lyapunov functions. The following original contributions are then presented.

• We propose a novel ISS Lyapunov result for state-dependent switched systems (1) using piecewise C 1 Lyapunov functions, also providing a numerical example that illustrates its relevance.

• We study ISS of the interconnection (3), using non-smooth Lyapunov functions that satisfy a mild decrease condition based on the Lie derivative, thus generalizing the existing small gain results for state-dependent switched systems based on the Clarke derivative/gradient, as [START_REF] Liberzon | Lyapunov-based small-gain theorems for hybrid systems[END_REF], [START_REF] Heemels | Input-to-state stability and interconnections of discontinuous dynamical systems[END_REF].

• When (3) is in cascade form, we combine two non-pathological Lyapunov functions to provide ISS certificates. These state-dependent switching results differ significantly from the existing results for interconnected time-dependent switched systems [START_REF] Yang | A Lyapunov-based small-gain theorem for interconnected switched systems[END_REF][START_REF] Zhang | ISS Lyapunov functions for cascade switched systems and sampled-data control[END_REF].

• We finally illustrate the usefulness of our results by performing output feedback stabilization of a statedependent switched system using an observer-based controller. The arising conditions are shown to become computationally tractable in the switched linear case.

The rest of the article is organized as follows: In Section 2 we provide the basic definitions from nonsmooth analysis, with particular attention to the nonpathological class of locally Lipschitz functions, together with the main result on ISS of system (2) using locally Lipschitz Lyapunov functions. In Section 3 we apply our result to state-dependent switched systems. In Section 4, we study interconnected differential inclusions, proposing a Lie derivative-generalization of classical small-gain and cascade arguments. We study the application of our results for feedback stabilization of switched systems in Section 5. In the Appendix, we prove some technical results on piecewise C 1 functions used in Section 3.

Fundamental Tools and Results

Basic notions for differential inclusions

We introduce here the formalism of differential inclusions with inputs, and recall the basic concepts of solutions and stability of equilibrium. Throughout this manuscript, we consider set valued maps F : R n × R m ⇒ R n , satisfying Assumption 1.

Assumption 1. Considering a set valued map F : R n × R m ⇒ R n , we suppose that:

• F has nonempty, compact and convex values;

• F is locally bounded (see [START_REF] Rockafellar | Variational Analysis, volume 317 of Gundlehren der mathematischen Wissenchaften[END_REF]Definition 5.14]);

• For every u ∈ R m , F (•, u) : R n ⇒ R n is upper semi-continuous; • For every x ∈ R n , F (x, •) : R m ⇒ R n is continuous.
The interested reader is referred to [START_REF] Rockafellar | Variational Analysis, volume 317 of Gundlehren der mathematischen Wissenchaften[END_REF] for a thorough discussion about continuity concepts for set-valued maps. We suppose that F (0, 0) = {0}, and consider the differential inclusion

ẋ ∈ F (x, u), (4)
where input u : R + → R m belongs to the set of measurable and locally essentially bounded functions, i.e.

U :=    u : R + → R m u measurable, ess sup 0≤τ ≤T |u(τ )| < ∞, ∀T > 0    .
For the unperturbed differential inclusion ẋ ∈ F (x, 0), the hypotheses that F (•, 0) : R n ⇒ R n has closed, convex and non-empty values together with upper semicontinuity are sufficient for the existence of solutions, and are sometimes referred as basic assumptions in the literature, (see, for example, [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]). On the other hand, the hypothesis that F : R n × R m ⇒ R n is continuous in the second argument is introduced to handle a large class of inputs like U.

We introduce here the concepts of solutions: Given a vector x 0 ∈ R n and an input u ∈ U, x : [0, T ) → R n (for some T > 0 and possibly T = ∞) is a (Carathéodory) solution of system (4) starting at x 0 if x : [0, T ) → R n is locally absolutely continuous, x(0) = x 0 , and ẋ(t) ∈ F (x(t), u(t)), for almost every t ∈ [0, T ). Under the stated assumptions on the map F : R n × R m ⇒ R n , we may prove the following existence result.

Proposition 1 (Local existence). Let F : R n × R m ⇒ R n satisfy Assumption 1. Given any input u ∈ U, system (4) has solutions from any initial point x 0 ∈ R n , i.e. there exists (at least) a Carathéodory solution x : [0, T ) → R n of system (4), for some T > 0, with x(0) = x 0 .

Proof. Considering any input u ∈ U, we define F u : R + × R n ⇒ R n by F u (t, x) := F (x, u(t)) and we prove the existence of solutions of the non-autonomous differential inclusion ẋ(t) ∈ F u (t, x(t)).

By hypothesis, F u is upper semicontinuous with respect to the x-argument. By continuity of F with respect to the second argument, for every x ∈ R n , we can extract a continuous function f (x, •) : R m → R n such that f (x, u) ∈ F (x, u), for every u ∈ R m , see for example [START_REF] Rockafellar | Variational Analysis, volume 317 of Gundlehren der mathematischen Wissenchaften[END_REF]Example 5.57] or [START_REF] Deimling | Multivalued Differential Equations[END_REF]Lemma 2

.1]. Thus, f (x, u(•)) : R + → R n is a measurable function such that f (x, u(t)) ∈ F (x, u(t)) = F u (t, x). Since F : R n × R m ⇒ R n is locally bounded, F u : R + × R n ⇒ R n is
locally essentially bounded, and hence it is locally bounded by integrable functions. We can then apply [START_REF] Deimling | Multivalued Differential Equations[END_REF]Corollary 5.2] to conclude local existence of solutions.

Next, we recall the input-to-state stability (ISS) concept, firstly introduced in [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF]. 4) is input-to-state stable (ISS) with respect to u if there exist a class KL function β, and a class K function1 χ such that, for any x 0 ∈ R n and for any input u ∈ U, all the solutions starting at

Definition 1. System (
x 0 satisfy |x(t)| ≤ β(|x 0 |, t) + χ ess sup 0≤τ ≤t |u(τ )| , ∀t ≥ 0. ( 5
)
Recalling the definition of U, bound (5) ensures that the solutions x are uniformly bounded, and thus complete, i.e. dom(x(•)) = [0, +∞). It is clear that ISS of (4) implies global asymptotic stability (GAS) in the unperturbed case u ≡ 0.

Generalized derivatives

Our aim is to prove ISS of system (4) via non-smooth Lyapunov functions, and thus in the following we collect various notions of generalized derivatives and gradients. Given a locally Lipschitz function V : R n → R we have the following characterization of the Clarke generalized gradient [7, Theorem 2.5.1, page 63] which is taken here as a definition. Let V : R n → R be a locally Lipschitz function, Clarke generalized gradient of

V at x is ∂V (x) := co lim k→∞ ∇V (x k ) | x k → x, x k / ∈ N V , (6)
where N V ⊆ R n is the set where ∇V is not defined, which has zero Lebesgue measure by Rademacher's Theorem, and co(S) denotes the convex hull of a set S ⊆ R n . We now introduce two different notions of generalized directional derivatives for locally Lipschitz functions with respect to differential inclusion (4), which appeared firstly in [START_REF] Bacciotti | Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions[END_REF].

Definition 2 (Set-valued directional derivatives [START_REF] Bacciotti | Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions[END_REF][START_REF] Cortes | Discontinuous dynamical systems[END_REF]). Consider the unperturbed differential inclusion (4) with u ≡ 0; given a locally Lipschitz continuous function V : R n → R, the Clarke generalized derivative of V with respect to F , denoted VF (x), is defined as

VF (x) := { p, f | p ∈ ∂V (x), f ∈ F (x, 0) }.
Additionally, we define the Lie generalized derivative of V with respect to F , denoted V F , as

V F (x) := {a ∈ R | ∃f ∈ F (x, 0) : p, f = a, ∀p ∈ ∂V (x)}.
These concepts can be extended to the case of a perturbed differential inclusion with input (4) as follows:

VF (x, u) := { p, f | p ∈ ∂V (x), f ∈ F (x, u) }, V F (x, u) := {a ∈ R | ∃f ∈ F (x, u) : p, f = a, ∀p ∈ ∂V (x)}. (7)
For each (x, u) ∈ R n ×R m the sets VF (x, u) and V F (x, u) are closed and bounded intervals, with V F (x, u) possibly empty, see [START_REF] Ceragioli | Discontinuous ordinary differential equations and stabilization[END_REF]. In particular V F (x, u) ⊆ VF (x, u).

Moreover, if V is continuously differentiable at x, one has ∂V (x) = {∇V (x)} and thus

V F (x, u) = VF (x, u) = { ∇V (x), f | f ∈ F (x, u)}.

nonpathological Functions

We now introduce a class of locally Lipschitz functions (and not necessarily C 1 ), firstly introduced in [START_REF] Valadier | Entrainement unilateral, lignes de descente, fonctions Lipschitziennes non pathologiques[END_REF].

Definition 3. [49]

A locally Lipschitz function V : R n → R is said to be nonpathological if, given any absolutely continuous function ϕ ∈ AC(R + , R n ), we have that for almost every t ∈ R + there exists a t ∈ R such that v, φ(t) = a t , for all v ∈ ∂V (ϕ(t)).

In other words, ∂V (ϕ(t)) is a subset of an affine subspace orthogonal to φ(t), for almost every t ∈ R + .

The usefulness of nonpathological functions is mainly given by the following result.

Proposition 2.

[49] If V : R n → R is nonpathological and ϕ : R + → R n is an absolutely continuous function, then the set

{ p, φ(t) | p ∈ ∂V (ϕ(t))},
is equal to the singleton { d dt V (ϕ(t))} for almost every t ∈ R + . Remark 1. Given a nonpathological function V : R n → R, for any u ∈ U, any initial condition x 0 ∈ R n and any solution x : dom(x(•)) → R n of (4), we have that

d dt V (x(t)) ∈ V F (x(t), u(t)) (9)
for almost every t ∈ dom(x(•)). In fact, by Proposition 2, for almost every t ∈ dom(x(•)), we have that

d dt V (x(t)) = { p, ẋ(t) | p ∈ ∂V (x(t))} ⊆ {a ∈ R | ∃f ∈ F (x(t), u(t)) : p, f = a, ∀p ∈ ∂V (x(t))} = V F (x(t), u(t)).
nonpathological functions form a large class of functions which clearly includes C 1 (R n , R), we recall here some important properties of this family of functions, for the proofs we refer to [START_REF] Valadier | Entrainement unilateral, lignes de descente, fonctions Lipschitziennes non pathologiques[END_REF] and [START_REF] Bacciotti | Nonsmooth optimal regulation and discontinuous stabilization[END_REF]. Lemma 1. The set of nonpathological functions is closed under addition, multiplication by scalars and pointwise maximum operator. More precisely

, if V 1 , V 2 : R n → R are nonpathological then V 1 + V 2 , max{V 1 (x), V 2 (x)} and λV (λ ∈ R) are nonpathological. Moreover, if V : R n → R
is locally Lipschitz continuous and has at least one of the following properties:

• Continuously differentiable,

• Clarke-regular (see [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF]Definition 2.3.4.]),

• convex/concave,

• semiconvex/semiconcave, then V is nonpathological.

In our paper [START_REF] Della Rossa | Max-min Lyapunov functions for switched systems and related differential inclusions[END_REF] the class of nonpathological functions obtained from pointwise maximum and minimum combinations over a finite set of continuously differentiable functions has been studied, by proposing sufficient conditions for asymptotic stability of state-dependent switching systems. In particular, explicitly exploiting the properties of the max-min structure, we have shown in [START_REF] Della Rossa | Max-min Lyapunov functions for switched systems and related differential inclusions[END_REF] the advantages of using the Lie derivative concept (Definition 2), providing several examples where the Clarke derivative approach is too conservative. In this work instead, we study ISS of differential inclusions with inputs, considering the broader class of nonpathological candidate Lyapunov functions, and investigating how the nonpathological property could be exploited in the context of interconnected differential inclusions. In Section 3 we will define a family of locally Lipschitz functions and we will prove that it is a subset of the nonpathological functions. A similar class is considered also in our parallel work [START_REF] Della Rossa | Piecewise structure of Lyapunov functions and densely checked decrease conditions for hybrid systems[END_REF], in the context of hybrid systems composed by continuous differential inclusions over restricted domains.

ISS-Lyapunov Result

We now provide sufficient conditions for ISS of system [START_REF] Cai | Characterizations of input-to-state stability for hybrid systems[END_REF]. In what follows, due to the fact that the set V F (x, u) is possibly empty, we adopt the convention max ∅ = -∞. The novelty of the following ISS-Lyapunov result lies in the fact that we require Lie generalized derivative of the Lyapunov function to be negative definite. Recalling the inclusion [START_REF] Cortes | Discontinuous dynamical systems[END_REF], this statement can be seen as a generalization of the existing results on ISS of differential inclusions relying on the notion of Clarke derivative, in particular [START_REF] Liberzon | Lyapunov-based small-gain theorems for hybrid systems[END_REF].

Theorem 1. Let V : R n → R be a locally Lipschitz and nonpathological function such that there exist

α, α ∈ K ∞ , ρ ∈ PD and γ ∈ K such that α(|x|) ≤ V (x) ≤ α(|x|), (10)
V (x) > γ(|u|) ⇒ max V F (x, u) ≤ -ρ(|x|), (11)
then system (4) is ISS w.r.t. u, and V is called a nonpathological ISS-Lyapunov function for system (4).

Proof. Let us consider any initial condition x 0 ∈ R n and any input u ∈ U. Let x : dom(x(•)) → R n be a solution of system (4) starting at x 0 and with input u. Let us note that the function V • x : dom(x(•)) → R is absolutely continuous because it is the composition of a locally Lipschitz continuous function and an absolutely continuous function. Then d dt V (x(t)) exists almost everywhere. Since V is nonpathological, recalling [START_REF] Cortes | Discontinuous dynamical systems[END_REF] in Remark 1, we have that d dt V (x(t)) ∈ V F (x(t), u(t)) for almost every t ∈ dom(x(•)). From equation [START_REF] Deimling | Multivalued Differential Equations[END_REF], we have that

d dt V (x(t)) ≤ -ρ(|x(t)|), for almost every t ∈ dom(x(•)) such that V (x(t)) > γ(|u(t)|).
The proof is completed by following standard approaches as those in [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF], [START_REF] Khalil | Nonlinear Systems[END_REF]Theorem 4.18]. For the interested reader, the argument is fully developed in [12, Proof of Theorem 5.4].

Remark 2. As already observed in the literature, for example in [START_REF] Cai | Characterizations of input-to-state stability for hybrid systems[END_REF], under some assumptions the existence of ρ ∈ PD and γ ∈ K such that condition [START_REF] Deimling | Multivalued Differential Equations[END_REF] holds is equivalent to the existence of two functions

ρ ∈ K ∞ and γ ∈ K such that max V F (x, u) ≤ -ρ(|x|) + γ(|u|) ∀(x, u) ∈ R n × R m . (12)
Indeed, implication (12) ⇒ (11) holds by choosing ρ := 1 2 ρ and γ = ρ -1 • 2 γ. The converse implication (11) ⇒ (12) holds if F (0, 0) = {0} and ρ ∈ K ∞ . Indeed, it suffices to choose ρ := ρ and γ ∈ K such that γ(r) ≥ max{0, γ 0 (r)}, for all r ∈ R + , where

γ 0 (r) := max max V F (x, u) + ρ(|x|) | |x| ≤ γ(r), |u| ≤ r .
It is possible to show that γ 0 above is well defined and γ 0 ∈ K, using local boundedness of ∂V and regularity of F : R n × R m ⇒ R n . Moreover, using [START_REF] Dayawansa | A converse Lyapunov theorem for a class of dynamical systems which undergo switching[END_REF], another equivalent formulation of condition ( 12) corresponds to asking that there exist two functions ρ ∈ K ∞ and γ ∈ K such that max

V F (x, u) ≤ -ρ(V (x)) + γ(|u|), ∀(x, u) ∈ R n × R m . ( 13
)
The advantage of ( 13) is that in this formulation the function ρ represents the decay rate of V along the solutions.

ISS for State Dependent Switched Systems

In this section, we apply our ISS result to a specific differential inclusion with inputs arising from a suitable regularization of state-dependent switched systems. We introduce the concept of a "well-behaved" partition of the state space, of the associated switched system, and of a family of functions related to this partition, and finally we provide the specialization of Theorem 1 in this setting.

Definition 4 (Proper State-Space Partition). Given a finite set of indexes

I := {1, . . . K}, let us consider closed sets X 1 , . . . , X K ⊆ R n and open sets O 1 , . . . O K ⊆ R n such that a) K i=1 X i = R n , b) X i ⊆ O i , for all i ∈ I, c) int(X i ) = X i , for all i ∈ I, d) For every i ∈ I, bd(X i ) has zero Lebesgue measure, e) X i ∩ X j = bd(X i ) ∩ bd(X j ), for all i, j ∈ I, i = j,
In this situation, we say that X := {X i , O i } i∈I is a proper partition of R n . We define ∂X := ∪ i∈I bd(X i ), and we underline that ∂X has zero Lebesgue measure.

Given a proper partition X of R n , we can introduce an "index indicator map", that is a set valued map I X : R n ⇒ I defined as

I X (x) := {i ∈ I | x ∈ X i }. (14)
We underline that I X is almost everywhere single valued. In fact, by Definition 4, Item e), if x ∈ int(X ) for some ∈ I then

I X (x) = { }. Definition 5 (State-Dependent Switched System). Given X = {X i , O i } i∈I a proper partition of R n , consider f i ∈ C 1 (O i × R m , R n ), i ∈ I. A state-dependent switching signal associated to X is a function σ : R n → I such that σ(x) = i, if x ∈ int(X i ), (15)
and the (perturbed) state-dependent switched system associated to

{X i , O i , f i } i∈I is the differential equation ẋ = f σ(x) (x, u). (16)
We note here that, given a proper partition X = {X i , O i }, a state dependent switching signal associated to it is not uniquely defined: the value of σ remains unspecified on the null-measure set ∂X . We now clarify why this ambiguity does not affect the solution set of the corresponding state-dependent switched system. System ( 16) has a discontinuous right-hand side in the first argument thus it may not have any Carathéodory solutions at the discontinuity points of f σ(•) (•, u), see [START_REF] Cortes | Discontinuous dynamical systems[END_REF]. Many possible definitions of "generalized solutions" for discontinuous dynamical system are possible (see for example [START_REF] Ceragioli | Discontinuous ordinary differential equations and stabilization[END_REF] or [START_REF] Cortes | Discontinuous dynamical systems[END_REF]); we consider the concept of Filippov solutions, introduced firstly in [START_REF] Filippov | Differential Equations with Discontinuous Right-Hand Side[END_REF]. More formally, we define Fil, the Filippov regularization of the discontinuous map f σ , as

Fil(f σ )(x, u) := δ>0 µ(S)=0 co{f σ(y) (y, u) | y ∈ B(x, δ) \ S} =: F sw (x, u),
where µ denotes the Lebesgue measure. Under the hypotheses in Definitions 4 and 5, it can be proven that

F sw (x, u) = co{f i (x, u) | i ∈ I X (x)},
see for example [START_REF] Filippov | Differential Equations with Discontinuous Right-Hand Side[END_REF], [START_REF] Cortes | Discontinuous dynamical systems[END_REF] and [START_REF] Heemels | Input-to-state stability and interconnections of discontinuous dynamical systems[END_REF]. Summarizing, we consider the regularized differential inclusion

ẋ ∈ F sw (x, u) = co{f i (x, u) | i ∈ I X (x)} (17) considering again signals u ∈ U. Since it is easily verified that f σ(x) (x, u) ∈ F sw (x, u) for all (x, u) ∈ R n ×R m ,
we have that any solution of ( 16) is also a solution of [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]. Moreover, it can be seen that F sw : R n ×R m ⇒ R n satisfies Assumption 1, and thus by Proposition 1 we have existence of solutions of ( 17) from any initial condition and any input u ∈ U (which was not the case for [START_REF] Filippov | Differential Equations with Discontinuous Right-Hand Side[END_REF], see [START_REF] Cortes | Discontinuous dynamical systems[END_REF]). Then, characterizing desirable (stability/convergence) properties of the solutions of ( 17) indirectly also characterizes the solutions of ( 16).

In particular, a Filippov solution of system ( 16) is by definition a solution of the differential inclusion [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF], according to the definition given in Section 2.

We introduce here a family of locally Lipschitz functions that we propose as candidate Lyapunov functions for [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF].

Definition 6 (Piecewise C 1 Functions). Consider J = {1, . . . , N } and Y = {Y j , V j } j∈J , a proper partition of R n . A continuous function V : R n → R is called a piecewise C 1 function with respect to the proper partition Y (and we write V ∈ P(Y)) if there exist real-valued functions V 1 , . . . V N such that 1. V j ∈ C 1 (V j , R) for each j ∈ J , 2. V (x) = V j (x), if x ∈ Y j .
Piecewise C 1 functions with respect to a proper partition are a particular kind of "piecewise C 1 functions" defined in [START_REF] Scholtes | Introduction to Piecewise Differentiable Equations[END_REF]. Proposition 3. Consider V ∈ P(Y), with respect to a proper partition Y = {Y j , V j } j∈J , in the sense of Definition 6. Then the following hold:

1. V is locally Lipschitz, nonpathological and

∂V (x) = co {∇V j (x) | j ∈ I Y (x)} . ( 18
)
2. Given F sw : R n × R m ⇒ R n defined as in [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF], we have

V F sw (x, u) = ® a ∈ R ∃f ∈ F sw (x, u) : ∇V j (x), f = a, ∀j ∈ I Y (x)
´.

We postpone the proof of Proposition 3 to the Appendix. We can now specialize the results stated in previous sections in this setting. First, we consider a candidate Lyapunov function in the class of piecewise C 1 functions P(Y), where the proper partition Y does not necessarily coincide with X , the proper partition associated with the considered switched system. Then we present specifically the case X = Y, see the subsequent Remark 3 for further discussion.

Corollary 1 (ISS for state dependent switching). Consider a proper partition X = {X i , O i } i∈I and an associated switched system [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]. Let us consider another proper partition Y = {Y j , V j } j∈J and let V ∈ P(Y). Suppose that there exist α, α ∈ K ∞ , ρ ∈ PD and γ ∈ K such that:

A) for each j ∈ J , for each x ∈ Y j , α(|x|) ≤ V j (x) ≤ α(|x|); B) for each j ∈ J , for each x ∈ int(Y j ) \ ∂X and for each u ∈ R m , V (x) > γ(|u|) ⇒ ∇V j (x), f σ(x) (x, u) ≤ -ρ(|x|); C) for each (x, u) ∈ ∂X × R m , V (x) > γ(|u|) ⇒ max V F sw (x, u) ≤ -ρ(|x|);
then system (17) is ISS.

Proof. By the nonpathological property of V established in Proposition 3, we can apply Theorem 1. Getting inequality (10) from A) is straightforward. We check inequality [START_REF] Deimling | Multivalued Differential Equations[END_REF] decomposing R n as follows:

R n = j∈J int(Y j ) \ ∂X ∪ ∂X ∪ ∂Y \ ∂X . Consider first a point x ∈ int(Y j ) \ ∂X for some j ∈ J . Function V ∈ P(Y) is C 1 at x, and F sw (x, u) = {f σ(x) (x, u)} is single-valued. Thus, for any u ∈ R m , we have V F sw (x, u) = { ∇V j (x), f σ(x) (x, u) },
and by B), the implication in [START_REF] Deimling | Multivalued Differential Equations[END_REF] holds. If x ∈ ∂X, the assertion follows directly by C).

As the last step, consider a point x ∈ ∂Y \ ∂X, and thus x ∈ int(X i ) for some i ∈ I. In particular at x, we have F sw (x, u) = {f i (x, u)}. Consider u ∈ R m and suppose that V (x) > γ(|u|). Recalling [START_REF] Heemels | Input-to-state stability and interconnections of discontinuous dynamical systems[END_REF] in Lemma 3 and by Definition 4, for each j ∈ I Y (x), there exists a sequence x j k → x such that x j k ∈ int(Y j ) ∩ int(X i ) for all k ∈ N. By continuity of V and γ, we can suppose V (x j k ) > γ(|u|). At these points, from B), we get

∇V j (x j k ), f i (x j k , u) ≤ -ρ(|x k j |)
. By continuity of ∇V j , f i and ρ we have

∇V j (x), f i (x, u) ≤ -ρ(|x|),
for each j ∈ I Y (x).

We have proved [START_REF] Deimling | Multivalued Differential Equations[END_REF] for all x ∈ R n , concluding the proof.

We underline that we are not explicitly checking [START_REF] Deimling | Multivalued Differential Equations[END_REF] on the zero Lebesgue measure set ∂Y \ ∂X: this is possible thanks to the continuity of F sw when restricted to int(X i ) for some i ∈ I. As a special case of Corollary 1, we present the situation X = Y.

Corollary 2. Consider a proper partition X = {X i , O i } i∈I and the associated switched system [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]. Consider V ∈ P(X ) such that there exist α, α ∈ K ∞ , ρ ∈ PD and γ ∈ K satisfying

A)' for each i ∈ I, for each x ∈ X i α(|x|) ≤ V i (x) ≤ α(|x|); B)' for each i ∈ I, for each (x, u) ∈ int(X i ) × R m , V (x) > γ(|u|) ⇒ ∇V i (x), f i (x, u) ≤ -ρ(|x|); C)' for each (x, u) ∈ ∂X × R m , V (x) > γ(|u|) ⇒ max V F sw (x, u) ≤ -ρ(|x|);
then system (17) is ISS.

Remark 3 (Comparison between Cor. 1 and Cor. 2). To check the conditions of Corollary 2 for each i ∈ I, we must find a smooth Lyapunov function V i for the system f i on the set O i ⊃ X i . Then we must construct a continuous function V by "gluing together" the V i 's on ∂X and finally check the Lie-based condition C)' on the switching surface ∂X. On the other hand, in some situations, it can be difficult to construct a smooth Lyapunov function even for a single subsystem in its region of activation if the subsystem is unstable. For this reason, in Corollary 1 we allow the candidate Lyapunov functions to be possibly non-differentiable in the interior of the X i , and we do not need to check the conditions at the point of non-differentiability of V , since F (•, u) is continuous in a neighborhood of ∂Y \ ∂X.

In the following example, mainly inspired by [27, Example 1], we apply the result presented in Corollary 2 to a particular state-dependent switched system.

Example 1. Consider the proper state-space partition X = {X i , R 2 } i∈{1,2} of R 2 defined by

X i := {x ∈ R 2 | x Q i x ≥ 0}, with Q 1 = -1 0 0 1 , and Q 2 = -Q 1 . Define the switching system ẋ = ® A 1 x + Bu, if x ∈ X 1 , A 2 x + Bu, if x ∈ X 2 , with A 1 = -ε a1 -a2 -ε , A 2 = -ε a2 -a1 -ε with 0 < ε < a 1 ≤ a 2 and B ∈ R 2×m arbitrary.
We want to study the resulting Filippov regularization as defined in [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]; for a graphical representation of the unperturbed case u ≡ 0, for some specific selections of a 1 , a 2 , ε, see [14, Figure 1]. Consider the function V : R 2 → R defined by

V (x) = ® V 1 (x) = x P 1 x if x ∈ X 1 , V 2 (x) = x P 2 x if x ∈ X 2 ,
with P 1 = a2 0 0 a1 , P 2 = a1 0 0 a2 . We now show that V is an ISS Lyapunov function, in the sense of Corollary 2. Since P 2 -P 1 = (a 2a 1 )Q 2 , V is continuous (and thus V ∈ P(X )) and moreover it satisfies item A)' of Corollary 2 with α(s) = a 1 s 2 and α(s) = a 2 s 2 . Moreover, it can be seen that A i P i + P i A i + εI 0, for any i ∈ {1, 2}; thus, following the reasoning of [START_REF] Khalil | Nonlinear Systems[END_REF]Lemma 4.6], it can be proven that, for any 0 < ε < ε we have

|x| ≥ b|u| ⇒ x (A i P i + P i A i )x + 2x P i Bu ≤ -ε |x| 2 , ( 20
)
for some b > 0 depending on A 1 , A 2 , B, P 1 , P 2 and ε .

It is easy to check ∂X = {x ∈ R 2 | x Q 1 x = 0} = R 1 ∪ R 2 where R 1 = {[x 1 x 2 ] ∈ R 2 | x 1 = x 2 } and R 2 = {[x 1 x 2 ] ∈ R 2 | x 1 = -x 2 }. Consider z ∈ ∂X,
∂V (z) = 2µ co {v 1 , v 2 } and F (z, u) = µ co{f 1 , f 2 } + Bu, where v 1 = [a 2 a 1 ] , v 2 = [a 1 a 2 ] and f i = A i [1 1] for i ∈ {1, 2}. Recalling Definition 2, we have that V F sw (z, u) = ∅ if and only if there exists λ ∈ R, 0 ≤ λ ≤ 1, such that [µ(λf 1 + (1 -λ)f 2 ) + Bu] v 1 = [µ(λf 1 + (1 -λ)f 2 ) + Bu] v 2 .
Simplifying, we obtain (a 1 + a 2 )µ = [-1 1] Bu (note that the dependence on λ cancels out). Now, it is clear that if (a 1 + a 2 )µ ≥ √ 2 B |u|, this equation has no solution2 . Recalling the definition of µ we have proved that

z ∈ R 1 and |z| ≥ 2 a 1 + a 2 B |u| ⇒ V F sw (z, u) = ∅. (21)
Since A i = R A j R and P i = R P j R with R = 0 1 -1 0 for any i, j ∈ {1, 2}, i = j, it can be seen that ( 21) holds also for z ∈ R 2 . Fixing ε < ε in [START_REF] Hrbacek | Introduction to Set Theory[END_REF] and defining γ(s) := a 2 max{bs, 2 B a1+a2 s} and ρ(s) := ε s 2 , we have proved Items B)' and C)' of Corollary 2, establishing that V is a (nonpathological) ISS Lyapunov function, and thus the system is ISS.

Remark 4. The construction proposed in Example 1 can be generalized to broader settings. The main idea is the following: consider a proper partition X = {X i , O i } i∈I and the switched system [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF] with f i (x, u) = A i x + g i (u), with A i ∈ R n×n and g i : R m → R n satisfying a linear growth condition, i.e. ∃ L > 0 such that g i (u) ≤ L|u| for all u ∈ R m , for all i ∈ I. Suppose that a piecewise quadratic V ∈ P(X ) satisfies conditions A)' and B)' of Corollary 2. Then, if for the unperturbed system the Lie derivative on the discontinuity surface ∂X is empty, i.e.

V F sw (x, 0) = ∅, ∀x ∈ ∂X, it follows that condition C)' is satisfied. Intuitively speaking, this is due to the fact that the set V F sw (x, u) remains empty for all x ∈ R n and all u ∈ R m small enough. For further insight regarding the construction of piecewise C 1 (or piecewise quadratic) Lyapunov functions for (unperturbed) state-dependent switching systems, we refer the reader to [START_REF] Johansson | Computation of piecewise quadratic Lyapunov functions for hybrid systems[END_REF], [START_REF] Iervolino | Asymptotic stability of piecewise affine systems with Filippov solutions via discontinuous piecewise Lyapunov functions[END_REF], [START_REF] Della Rossa | Max-min Lyapunov functions for switched systems and related differential inclusions[END_REF] and references therein.

Interconnected Differential Inclusions

In this section, we use Theorem 1 to study stability of feedback and cascade interconnections of two systems modeled by differential inclusions.

Feedback Interconnection and Small Gain Theorem

For the system shown in Figure 1, we establish ISS of the interconnected system by constructing a Lyapunov function from two (nonsmooth) ISS-Lyapunov functions associated with the two subsystems.

Consider

F 1 : R n1 × R n2 × R m ⇒ R n1 and F 2 : R n1 × R n2 × R m ⇒ R n2
and suppose that they are locally bounded, have non empty, compact and convex values and are upper semicontinuous in the first two arguments and continuous in the third one. Consider the interconnection

ẋ1 ∈ F 1 (x 1 , x 2 , u), ( 22a
) ẋ2 ∈ F 2 (x 1 , x 2 , u). (22b)
We introduce the notation

x = (x 1 , x 2 ) := (x 1 , x 2 ) ∈ R n = R n1+n2 and the augmented differential inclusion ẋ ∈ F (x, u) := Å F 1 (x 1 , x 2 , u) F 2 (x 1 , x 2 , u) ã . ( 23
)
We start our construction by assuming the existence of ISS-Lyapunov functions for the two subsystems, in order to conclude ISS of the overall interconnection [START_REF] Ito | Robust stability of networks of iISS systems: Construction of sum-type Lyapunov functions[END_REF]. These types of assumptions characterize classical ISS approaches [START_REF] Jiang | Small-gain theorem for ISS systems and applications[END_REF][START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF], also used in the recent works [START_REF] Heemels | Input-to-state stability and interconnections of discontinuous dynamical systems[END_REF], [START_REF] Liberzon | Lyapunov-based small-gain theorems for hybrid systems[END_REF]. The novelty introduced here lies in the fact that we consider nonpathological ISS Lyapunov functions satisfying the "relaxed" conditions involving the Lie derivative presented in Theorem 1, as formalized in the following statement. Assumption 2. Suppose that there exist nonpathological functions

ẋ1 ∈ F 1 (x 1 , x 2 , u) u (x 2 , u) x 1 ẋ2 ∈ F 2 (x 1 , x 2 , u) u x 1 x 2 x 2 (x 1 , u)
V 1 : R n1 → R and V 2 : R n2 → R such that 1a) There exist α 1 , α 1 ∈ K ∞ satisfying α 1 (|x 1 |) ≤ V 1 (x 1 ) ≤ α 1 (|x 1 |), ∀ x 1 ∈ R n1 . 1b) There exist α 2 , α 2 ∈ K ∞ satisfying α 2 (|x 2 |) ≤ V 2 (x 2 ) ≤ α 2 (|x 2 |), ∀ x 2 ∈ R n2 .
2a) There exist ρ 1 ∈ PD, and

χ 1 , γ 1 ∈ K satisfying V 1 (x 1 ) > max{χ 1 (V 2 (x 2 )), γ 1 (|u|)} ⇓ max V 1,F1 (x 1 , x 2 ,u) ≤ -ρ 1 (V 1 (x 1 ))
2b) There exist ρ 2 ∈ PD, and χ 2 , γ 2 ∈ K satisfying

V 2 (x 2 ) > max{χ 2 (V 1 (x 1 )), γ 2 (|u|)} ⇓ max V 2,F2 (x 1 , x 2 ,u) ≤ -ρ 2 (V 2 (x 2 ))
Since we want to combine the functions V 1 and V 2 to obtain a nonpathological ISS function W : R n → R for the interconnected system (23), we need the following results from non-smooth analysis.

Fact 1. [7, Theorem 2.6.6] Consider a locally Lipschitz function V : R k → R and σ ∈ C 1 (R, R), and define

U := σ • V . We have ∂U (x) = σ (V (x))∂V (x), ∀ x ∈ R k ,
where σ (s) denotes the derivative of σ at s ∈ R.

Fact 2. [7, Proposition 2.3.12] Given two locally Lipschitz functions

V 1 : R k → R and V 2 : R k → R consider the function V (x) := max{V 1 (x), V 2 (x)}. Given any z ∈ R k such that V 1 (z) = V 2 (z), it holds that ∂V (z) ⊆ co{∂V 1 (z), ∂V 2 (z)}.
Moreover, we need this well-known comparison result, for the proof, see [25, Theorem 3.1].

Fact 3. Given χ 1 , χ 2 ∈ K such that χ 1 • χ 2 (r) < r, ∀r > 0, there exists a continuously differentiable σ ∈ K ∞ with σ (s) > 0 for all s ∈ [0, ∞), such that χ 2 (r) < σ(r), and χ 1 (σ(r)) < r, ∀ r > 0.

The geometrical intuition behind [START_REF] Jayawardhana | Input-to-state stability of differential inclusions with applications to hysteretic and quantized feedback systems[END_REF] is that the graph of the function σ lies between the graphs of χ 2 and χ -1 1 , see for example Fig. 1 in [START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF]. Finally, the following lemma will be used in the proof.

Lemma 2. Suppose V 1 : R n1 → R and V 2 : R n2 → R are two nonpathological functions satisfying Assumption 2. Consider σ ∈ C 1 (R + , R + ) such that σ (s) > 0 for all s > 0 and the composite function

U 1 := σ • V 1 . Let W (x 1 , x 2 ) := max{U 1 (x 1 ), V 2 (x 2 )}, and consider a point z = (z 1 , z 2 ) = (0, 0), z 1 ∈ R n1 , z 2 ∈ R n2 such that U 1 (z 1 ) = V 2 (z 2 ). It holds that ∂W (z 1 , z 2 ) = co ¶ ∂ " U 1 (z 1 ), ∂ " V 2 (z 2 ) © ( 25
)
where

∂ " U 1 (z 1 ) := Å ∂U 1 (z 1 ) 0 ã = Å σ (V 1 (z 1 ))∂V 1 (z 1 ) 0 ã and ∂ " V 2 (z 2 ) := Å 0 ∂V 2 (z 2 ) ã . Proof. Consider a point z = (z 1 , z 2 ) = (0, 0) such that U 1 (z 1 ) = V 2 (z 2 ), the inclusion ∂W (z) ⊆ co{∂ " U 1 (z 1 ), ∂ " V 2 (z 2 )} is obtained by Fact 2.
For the converse inclusion, due to convexity of ∂W (z 1 , z 2 ), it suffices to show that ∂ "

U 1 (z 1 ) ⊆ ∂W (z 1 , z 2 ) and ∂ " V 2 (z 2 ) ⊆ ∂W (z 1 , z 2 )
. We only prove the first inclusion, as the other one can be proved with a similar reasoning. To prove ∂ " U 1 (z 1 ) ⊆ ∂W (z 1 , z 2 ), we note that, recalling the definition of Clarke generalized gradient [START_REF] Ceragioli | Discontinuous ordinary differential equations and stabilization[END_REF] and by convexity of ∂W (z 1 , z 2 ), it suffices to show that, for each sequence x k 1 ∈ R n1 where U 1 is differentiable, with x k 1 → z 1 and with v 1 := lim k→∞ ∇U 1 (x k 1 ), we have v := (v 1 , 0) ∈ ∂W (z 1 , z 2 ). From 1b) and 2b) of Assumption 2, the function V 2 has no local minima other than 0 because V 2 is a Lyapunov function for the unperturbed system ẋ2 ∈ F 2 (0, x 2 , 0). Thus, considering any point x 2 = 0, V 2 is decreasing along the solutions starting at (0, x 2 ) with zero input. By local existence of solutions from any initial point, we have that x 2 cannot be a local minimum of V 2 . Thus z 2 is not a local minima for V 2 , and we can consider a sequence x 2 → z 2 such that V 2 (z 2 ) > V 2 (x 2 ), for all ∈ N. Now, by continuity of U 1 and V 2 , for each ∈ N, there exists k ∈ N such that

U 1 (x k 1 ) > V 2 (x 2 ). ( 26
)
Consider the sequence x :=

Å x k 1 x 2 ã ∈ R n . We have x → z = (z 1 , z 2 )
, and from equation ( 26)

W (x ) = max{U 1 (x k 1 ), V 2 (x 2 )} = U 1 (x k 1 ), ∀ ∈ N. Thus, W is differentiable at all x ∈ R n and lim →∞ ∇W (x ) = Å lim →∞ ∇U 1 (x k 1 ) 0 ã = Å v 1 0 ã = v.
By definition of v and the generalized gradient, it follows that v ∈ ∂W (z 1 , z 2 ) and hence ∂ " U 1 (z 1 ) ⊆ ∂W (z 1 , z 2 ). Similarly, one can prove that ∂ " V 2 (z 2 ) ⊆ ∂W (z 1 , z 2 ), and thus the equality (25) holds.

We have now all the necessary tools to present a small gain theorem involving nonpathological ISS functions, adapting the idea firstly proposed in [START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF].

Theorem 2 (Generalized Small Gain Theorem). Consider the nonpathological functions V 1 , V 2 satisfying Assumption 2 and suppose that

χ 1 • χ 2 (r) < r, ∀r > 0. ( 27
)
Considering a function σ ∈ K ∞ ∩ C 1 (R + , R + ) satisfying property [START_REF] Jayawardhana | Input-to-state stability of differential inclusions with applications to hysteretic and quantized feedback systems[END_REF] in Fact 3, define W : R n → R as

W (x 1 , x 2 ) := max{σ(V 1 (x 1 )), V 2 (x 2 )}. ( 28
)
Then W is a nonpathological ISS function and thus system (23) is ISS w.r.t. u.

Proof. We want to show that W : R n → R satisfies all the conditions of Theorem 1. To this end, it is enough to show that

A) σ • V 1 : R n1 → R is nonpathological, and W : R n → R is nonpathological.
B) There exist ρ ∈ PD and γ ∈ K such that

W (x) > γ(|u|) ⇒ max Ẇ F (x, u) ≤ -ρ(|x|). ( 29
)
Proof of A): We recall that V 1 : R n1 → R is nonpathological and σ ∈ C 1 (R, R) and σ (r) > 0 for all r > 0. Defining U 1 := σ • V 1 , by Fact 1 we have ∂U 1 (x) = σ (V 1 (x))∂V 1 (x)
for all x ∈ R n1 . Moreover for any absolutely continuous function ϕ : [0, T ) → R n1 , by Definition 3 we have that ∂V 1 (ϕ(t)) is a subset of an affine subspace orthogonal to φ(t), for almost every t ∈ [0, T ), and the same holds for ∂U 1 (ϕ(t)) = σ (V 1 (ϕ(t))∂V 1 (ϕ(t)). Thus U 1 : R n1 → R is nonpathological. The non-pathology of W : R n → R follows from the fact that pointwise maximum of nonpathological functions is nonpathological, as stated in Lemma 1.

Proof of B):

We proceed by considering three cases. Let us define the sets

O 1 :={(x 1 , x 2 ) ∈ R n | V 2 (x 2 ) < σ(V 1 (x 1 ))}, O 2 :={(x 1 , x 2 ) ∈ R n | V 2 (x 2 ) > σ(V 1 (x 1 ))}, Γ :={(x 1 , x 2 ) ∈ R n | V 2 (x 2 ) = σ(V 1 (x 1 ))}.
For z = (z 1 , z 2 ) ∈ O 1 , by continuity there exists a neighborhood U of z where W (x) = σ(V 1 (x 1 )), for all x = (x 1 , x 2 ) ∈ U. By Fact 1, we have that

∂W (z) = σ (V (z 1 ))∂V 1 (z 1 ) × {0}. Thus f = (f 1 , f 2 ) ∈ F (z, u) is such that p, f = a, ∀p ∈ ∂W (z) if and only if f 1 satisfies p 1 , f 1 = a, ∀p 1 ∈ σ (V (z 1 ))∂V 1 (z 1 ). From (19), we thus get Ẇ F (z, u) = σ (V (z 1 )) V 1,F1 (z 1 , z 2 , u). ( 30
)
Recalling that z ∈ O 1 and equation ( 24), we have

χ 1 (V 2 (z 2 )) < χ 1 (σ(V 1 (z 1 ))) < V 1 (z 1 )
. Thus, by condition 2a) of Assumption 2, we have from [START_REF] Kellett | A compendium of comparison function results[END_REF] that

W (z) > γ 1 (|u|) ⇒ max Ẇ F (z, u) ≤ -ρ 1 (W (z)), ∀z ∈ O 1 , (31)
where

ρ 1 (s) := σ (σ -1 (s)) ρ 1 (σ -1 (s)) is a positive definite function and γ 1 (s) := σ(γ 1 (s)) is class K. For z = (z 1 , z 2 ) ∈ O 2
, following the same reasoning (but without the complications introduced by σ), one has that

W (z) > γ 2 (|u|) ⇒ max Ẇ F (z, u) ≤ -ρ 2 (W (z)), ∀z ∈ O 2 . ( 32
)
Before addressing z = (z 1 , z 2 ) ∈ Γ, using an idea proposed in [START_REF] Kamalapurkar | On reduction of differential inclusions and Lyapunov stability[END_REF], we introduce the following notation motivated by definition [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF]: Given F : R n × R m ⇒ R n and a locally Lipschitz function V : R n → R we define

F V (z, u) := {f ∈ F (z, u)|∃a ∈ R : v, f = a, ∀v ∈ ∂V (z)} .
By Definition 2, it is clear that

V F (z, u) = v, f | v ∈ ∂V (z), f ∈ F V (z, u) . ( 33
)
We continue by using the following set inclusion whose proof is postponed a few lines to avoid breaking the flow of the exposition:

F W (z, u) ⊆ F V1 1 (z 1 , z 2 , u) × F V2 2 (z 1 , z 2 , u). ( 34
)
Consider z = (z 1 , z 2 ) ∈ Γ and take any w ∈ ∂W (z), by Lemma 2, there exist 33), we may proceed as in [START_REF] Kellett | Classical converse theorems in Lyapunov's second method[END_REF] and [START_REF] Khalil | Nonlinear Systems[END_REF] and use continuity of W to get, for all z ∈ Γ = bd(O 1 ) ∩ bd(O 2 )

v 1 ∈ ∂V 1 (z 1 ), v 2 ∈ ∂V 2 (z 2 ) and λ ∈ [0, 1] such that w = Å λσ (V 1 (z 1 ))v 1 (1 -λ)v 2 ã . Consider f = Å f 1 f 2 ã ∈ F W (z, u), so that, from (34), f 1 ∈ F V1 1 (z 1 , z 2 , u) and f 2 ∈ F V2 2 (z 1 , z 2 , u). Using (
W (z) > γ 1 (|u|) ⇒ max f1∈F V 1 1 (z,u) v1∈∂V1(z1) σ (V 1 (z 1 )) v 1 , f 1 ≤ -ρ 1 (W (z)) W (z) > γ 2 (|u|) ⇒ max f2∈F V 2 2 (z,u) v2∈∂V2(z2) v 2 , f 2 ≤ -ρ 2 (W (z)). ( 35
)
Using [START_REF] Lin | A smooth converse Lyapunov theorem for robust stability[END_REF] we finally get that

W (z) > max{ γ 1 (|u|), γ 2 (|u|)} implies w, f = λσ (V 1 (z 1 )) v 1 , f 1 + (1 -λ) v 2 , f 2 ≤ -λ ρ 1 (W (z)) -(1 -λ)ρ 2 (W (z)) ≤ -min{ ρ 1 (W (z)), ρ 2 (W (z))}.
Thus, letting γ(s) := max{ γ 1 (s), γ 2 (s)} and ρ(s) := min{ ρ 1 (s), ρ 2 (s)}, we have

W (z) > γ(|u|) ⇒ max Ẇ F (z, u) ≤ -ρ(W (z)), ∀z ∈ Γ. (36)
Collecting ( 31), ( 32) and ( 36) we can conclude [START_REF] Kamalapurkar | Invariance-like results for nonautonomous switched systems[END_REF], and prove item B). We complete the proof by proving [START_REF] Liberzon | Lyapunov-based small-gain theorems for hybrid systems[END_REF]. To this, take any f ∈ F W (z, u). By definition of F in ( 23), we

have that f = Å f 1 f 2 ã for some f 1 ∈ F 1 (z 1 , z 2 , u) and f 2 ∈ F 2 (z 1 , z 2 , u)
. By Lemma 2 and Fact 1, for any

v 1 ∈ ∂V 1 (z 1 ), the vector w = Å σ (V 1 (z 1 ))v 1 0 ã ∈ ∂W (z) and thus w, f = σ (V 1 (z 1 )) v 1 , f 1 .
By varying v 1 in ∂V 1 (z 1 ) and recalling that f ∈ F W (z, u) (and thus w, f is constant for all w ∈ ∂W (z)), we obtain that

f 1 ∈ F V1 1 (z 1 , z 2 , u), that is v 1 , f 1 constant w.r.t. v 1 ∈ ∂V 1 (z 1 ). The same reasoning applies to f 2 , considering a vector w = Å 0 v 2 ã ∈ ∂W (z), with v 2 ∈ ∂V 2 (z 2 )
, concluding the proof of the claim.

Remark 5. The idea of analyzing the derivative of the composite function W in the three sets O 1 , O 2 , Γ, appeared firstly in [START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF], and is the common idea of many results on small-gain theorems for interconnected systems, see for example [START_REF] Liberzon | Lyapunov-based small-gain theorems for hybrid systems[END_REF] or [START_REF] Ito | An intuitive modification of max-separable Lyapunov functions to cover non-ISS systems[END_REF]. The analysis in O 1 and O 2 was straightforward, but because of nondifferentiability of V 1 and V 2 , the analysis in the set Γ is different from [START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF]. In particular the additional tools from nonsmooth analysis have been used to study the Lie-derivative of W along F on the set Γ.

Cascade System

We now apply Theorem 1 to cascade interconnections as in Figure 2. More precisely, given two maps

F 1 : R n1 × R n2 × R m ⇒ R n1
, and F 2 : R n2 × R m ⇒ R n2 we consider the cascade system defined as follow:

ẋ1 ∈ F 1 (x 1 , x 2 , u), ( 37a
) ẋ2 ∈ F 2 (x 2 , u). ( 37b
) ẋ2 ∈ F 2 (x 2 , u) ẋ1 ∈ F 1 (x 1 , x 2 , u) u u x 2 x 1 x 2
Figure 2: The cascade system in [START_REF] Rockafellar | Variational Analysis, volume 317 of Gundlehren der mathematischen Wissenchaften[END_REF].

Defining again n := n 1 + n 2 we will write F : R n × R m ⇒ R n defined by

F (x 1 , x 2 , u) := Å F 1 (x 1 , x 2 , u) F 2 (x 2 , u)
ã .

The cascade system (37) can be seen as a system of the form ( 23) where F 2 does not depend on x 1 , see also Figure 2. Therefore Theorem 2 can be applied with χ 1 ≡ 0 and condition [START_REF] Johansson | Computation of piecewise quadratic Lyapunov functions for hybrid systems[END_REF] holds for any χ 2 ∈ K. On the other hand, the cascade structure allows us to construct a different ISS-Lyapunov function, based on two non-smooth ISS-Lyapunov functions associated with each subsystem. The function that we construct is in the so-called sum-separable form, that has some clear advantages with respect to the max-separable form (28) in Theorem 2, see [START_REF] Ito | Robust stability of networks of iISS systems: Construction of sum-type Lyapunov functions[END_REF] and references therein for a thorough discussion. In particular, the sum-separable architecture preserves regularity, and in our setting also leads to a more direct proof of ISS of the cascade interconnection.

In the following we adapt, in the framework of differential inclusions and nonpathological functions, the proof technique proposed firstly in [START_REF] Sontag | Changing supply functions in input/state stable systems[END_REF]. More specifically, we assume that both subsystems admit an ISS Lyapunov function, using the formulation (12) in Remark 2. Similar constructions can be found in [START_REF] Tanwani | On using norm estimators for event-triggered control with dynamic output feedback[END_REF] and [START_REF] Zhang | ISS Lyapunov functions for cascade switched systems and sampled-data control[END_REF]. Assumption 3. The following conditions hold for system (37):

(A.1) ISS of (37b). There exist a nonpathological function V 2 : R n2 → R and α 2 α 2 , ρ 2 ∈ K ∞ and γ 2 ∈ K satisfying

α 2 (|x 2 |) ≤ V 2 (x 2 ) ≤ α 2 (|x 2 |), ∀x 2 ∈ R n2 , max V 2,F2 (x 2 , u) ≤ -ρ 2 (V 2 (x 2 )) + γ 2 (|u|),
for all x 2 ∈ R n2 and for all u ∈ R m .

(A.2) ISS of (37a). There exist a nonpathological function

V 1 : R n1 → R and α 1 α 1 , ρ 1 , γ 1 ∈ K ∞ satisfying α 1 (|x 1 |) ≤ V 1 (x 1 ) ≤ α 1 (|x 1 |), ∀x 1 ∈ R n1 , max V 1,F1 (x 1 , x 2 , u) ≤ -ρ 1 (V 1 (x 1 )) + γ 1 (V 2 (x 2 )) + γ 2 (|u|), for all x 1 ∈ R n1 , x 2 ∈ R n2 and all u ∈ R m .
(A.3) Defining ν(s) := γ 1 (s)/ρ 2 (s), there exists a scalar M > 0 such that

lim s→0 + ν(s) = lim s→0 + γ 1 (s) ρ 2 (s) ≤ M.
In other words, γ 1 (s) ∈ O(ρ 2 (s)) as s → 0 + .

Remark 6 (Tightness of Assumption 3). Condition (A.3), which is used in the construction of W in the proof of Proposition 4, is not restrictive: if it does not hold it is possible to modify the function V 1 in such a way that it holds, following the same idea proposed in [START_REF] Sontag | Changing supply functions in input/state stable systems[END_REF]. Due to this fact, Proposition 4 establishes that when system ( 22) is in the cascade-form presented in equation [START_REF] Rockafellar | Variational Analysis, volume 317 of Gundlehren der mathematischen Wissenchaften[END_REF], it suffices to have ISS-Lyapunov functions (satisfying the Lie-derivative conditions presented in (A.1) and (A.2) ) for each subsystem, to conclude ISS of the interconnected system. In this context, the small gain condition required in the general construction of Theorem 2 is somehow trivially satisfied.

Using Assumption 3, we can construct a nonpathological Lyapunov function for the cascade system (37), by adapting a Lyapunov design developed in [START_REF] Sontag | Changing supply functions in input/state stable systems[END_REF] and [START_REF] Zhang | ISS Lyapunov functions for cascade switched systems and sampled-data control[END_REF]. Proposition 4. Consider the cascade system [START_REF] Rockafellar | Variational Analysis, volume 317 of Gundlehren der mathematischen Wissenchaften[END_REF], and suppose that Assumption 3 holds. There exists a continuous and nondecreasing function ν : R + → R + satisfying ν(s) ≥ 4ν(s), for all s ∈ R + . Moreover, the function

W (x 1 , x 2 ) := V2(x2) 0 ν(s)ds + V 1 (x) (38)
is a nonpathological ISS functions for system [START_REF] Rockafellar | Variational Analysis, volume 317 of Gundlehren der mathematischen Wissenchaften[END_REF]; that is there exist α, α ∈ K ∞ such that

α(|(x 1 , x 2 )|) ≤ W (x 1 , x 2 ) ≤ α(|(x 1 , x 2 )|) (39)
for all (x 1 , x 2 ) ∈ R n1 × R n2 , and there exist ρ ∈ K ∞ and γ ∈ K such that Since ν(s) ≥ 4ν(s) > 0, ∀s > 0, then is a class

max Ẇ F (x 1 , x 2 , u) ≤ -ρ(W (x 1 , x 2 )) + γ(|u|), (40)
for all (x 1 , x 2 ) ∈ R n1 × R n2
K ∞ function. Moreover ∈ C 1 (R + , R + ). We can thus rewrite W (x 1 , x 2 ) = • V 2 (x 2 ) + V 1 (x 1 ).
Non-pathology of W follows from Proposition 3 and Fact 1 since is C 1 by construction. Moreover, the functions α and α of equation ( 39) are easily constructed as α(s) := α 2 (s) 0 ν(r)dr + α 1 (s) and α(s) := α2(s) 0 ν(r)dr + α 1 (s).

Let us now define U 2 := • V 2 ; noting that (s) = ν(s) and using Fact 1, we have that U

2,F2 (x 2 , u) = ν(V 2 (x 2 )) V 2,F2 (x 2 , u). Recalling (A.1), we can write max U 2,F2 (x 2 , u) ≤ -ν(V 2 (x 2 ))ρ 2 (V 2 (x 2 )) + ν(V 2 (x 2 ))γ 2 (|u|), (41)
∀ x 2 ∈ R n2 , ∀ u ∈ R m . Defining θ(s) := ρ -1 2 (2γ 2 (s)), we prove the following inequality max U 2,F2 (x 2 , u) ≤ - 1 2 ν(V 2 (x 2 ))ρ 2 (V 2 (x 2 )) + ν(θ(|u|))γ 2 (|u|). (42)
Indeed, by (41 [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF] trivially holds. Otherwise, we see that

), if γ 2 (|u|) ≤ 1 2 ρ 2 (V 2 (x 2 )),
γ 2 (|u|) ≥ 1 2 ρ 2 (V 2 (x 2 )) ⇔ V 2 (x 2 ) ≤ ρ -1 2 (2γ 2 (|u|)) = θ(|u|),
and by the nondecreasing property of ν, inequality (42) holds. Before proceeding to proving [START_REF] Sanfelice | Invariance principles for hybrid systems with connections to detectability and asymptotic stability[END_REF] we observe the following equality

Ẇ F (x 1 , x 2 , u) = U 2,F2 (x 2 , u) + V 1,F1 (x 1 , x 2 ). (43)
To show [START_REF] Sontag | Changing supply functions in input/state stable systems[END_REF], we recall that any locally Lipschitz function

G : R n1 × R n2 → R defined by G(x 1 , x 2 ) = G 1 (x 1 ) + G 2 (x 2 ), satisfies ∂G(x 1 , x 2 ) = ßÅ v 1 v 2 ã | v 1 ∈ ∂G 1 (x 1 ), v 2 ∈ ∂G 2 (x 2 ) ™ (44)
and thus, using definition [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF], we obtain [START_REF] Sontag | Changing supply functions in input/state stable systems[END_REF]. From (A.2), ( 42) and ( 43), we have

max Ẇ F (x 1 , x 2 , u) ≤ -ρ 1 (V 1 (x 1 )) + γ 1 (V 2 (x 2 )) + γ 2 (|u|) - 1 2 ν(V 2 (x 2 ))ρ 2 (V 2 (x 2 ))+ν(θ(|u|))γ 2 (|u|).
From the assumption ν(s) ≥ 4 γ1(s) ρ2(s) for all s ∈ R + , and following [30, Lemma 10], we finally have

max Ẇ F (x 1 , x 2 ) ≤ -γ 1 (V 2 (x 2 )) -ρ 1 (V 1 (x 1 )) + ν(θ(|u|))γ 2 (|u|) + γ 2 (|u|) ≤ -ρ(W (x 1 , x 2 )) + γ(|u|),
where we have defined γ(s) := (ν(θ(s)) + 1) γ

:= min ß ρ 1 ( 1 2 s), γ 1 ( 1 2 -1 (s)) ™ . 2 (s), ρ(s)

Feedback Stabilization of a 2-mode State-dependent Switched System

We now use the tools developed in the previous section to solve an output feedback stabilization problem for a class of switched systems with two modes. In particular, we consider the state dependent switched system defined as

S :      ẋ = ® f 1 (x) + g(x)u if x ∈ X 1 := {q(x) ≥ 0}, f 2 (x) + g(x)u if x ∈ X 2 := {q(x) ≤ 0}, y = h(x), (45)
where x ∈ R n and u ∈ R m . The basic assumptions we impose on the system (45) are the following:

Assumption 4. The data in (45) is such that a) f 1 , f 2 ∈ C 1 (R n , R n ); b) g ∈ C 1 (R n , R n×m ); c) h ∈ C 1 (R n , R p ); d) q ∈ C 1 (R n , R) and X = {X i , R n } i∈{1,2}
form a proper partition of R n (recall Definition 4); e) q(0) ≥ 0 ⇒ f 1 (0) = 0 and q(0) ≤ 0 ⇒ f 2 (0) = 0; f) There exists β g ∈ C(R, R + ), β g (s) ≥ 0 for all s ≥ 0, such that g(x) ≤ β g (|x|), for all x ∈ R n , g) There exists

β f ∈ K ∞ such that |f 1 (x) -f 2 (x)| ≤ β f (|x|), for all x ∈ R n .
Example 2 (Regular Values and Partitions). Consider q ∈ C 1 (R n , R) such that 0 is a regular value of q, i.e. ∇q(x) = 0 for all x satisfying q(x) = 0; then condition d) of Assumption 4 is satisfied. Indeed, by the Implicit Function Theorem, X 0 := {x ∈ R n | q(x) = 0} is a (n -1)-dimensional C 1 manifold, and hence X 0 has Lebesgue measure 0. Let us prove that {X i , R n } i∈{1,2} is a proper partition, with X 1 and X 2 defined as in [START_REF] Tanwani | On using norm estimators for event-triggered control with dynamic output feedback[END_REF]. First of all, X 1 ∪ X 2 = R n and bd(X i ) ⊆ X 0 , for any i ∈ {1, 2}: in fact, if q(x) > 0 (resp. q(x) < 0), by continuity of q it holds that x ∈ int(X 1 ) (resp. x ∈ int(X 2 )). It remains to prove that int(X i ) = X i , for any i ∈ I. Consider w.l.o.g. i = 1; the inclusion int(X i ) ⊆ X i is trivial. Let us consider x ∈ X 1 , i.e. q(x) ≥ 0. If q(x) > 0, then x ∈ int(X 1 ) ⊂ int(X 1 ). If q(x) = 0, by assumption ∇q(x) = 0, therefore x is neither a maximum nor a minimum. Thus there exists a sequence x k → x such that q(x k ) > 0, ∀ k ∈ N, and hence, x ∈ int(X 1 ).

Example 3 (Switched Linear Case). As a simple paradigm, one can think of a state-dependent switched linear system, such as

f i (x) = A i x, g(x) ≡ B, h(x) = Cx,
where A i ∈ R n×n for i ∈ {1, 2}, B ∈ R n×m and C ∈ R p×n . Regarding the function q ∈ C 1 (R n , R), the simplest non-trivial cases are the halfspace partitions or the symmetric conic partitions, described respectively by the functions q v (x) := v, x or q Q (x) := x Qx, Under Assumption 4, we design next an observer-based controller for system [START_REF] Tanwani | On using norm estimators for event-triggered control with dynamic output feedback[END_REF] of the form

for some v ∈ R n , or Q ∈ Sym(R n×n ) := {S ∈ R n×n | S = S}, Q is neither
C :      ż = ® f 1 (z) + g(z)u + 1 (y -h(z)) if z ∈ X 1 , f 2 (z) + g(z)u + 2 (y -h(z)) if z ∈ X 2 , u = k(z), (46)
where 1 , 2 ∈ C 1 (R p , R n ), and the globally Lipschitz map k : R n → R m are design parameters. 3 The design of globally Lipschitz feedback laws is a rather common occurrence in stabilization problems for various kinds of nonlinear systems and in particular the design methods in [START_REF] Khalil | Nonlinear Systems[END_REF]Chapters 13,[START_REF] Della Rossa | Smooth approximation of patchy Lyapunov functions for switching systems[END_REF] can be adapted to meet this requirement. Moreover, when restricting the attention to initial states in a compact set, it is possible to develop semiglobal results to allow for locally Lipschitz feedbacks, as explained in [START_REF] Teel | Global stabilizability and observability imply semi-global stabilizability by output feedback[END_REF]. We consider the interconnected system (45)- [START_REF] Teel | Global stabilizability and observability imply semi-global stabilizability by output feedback[END_REF], and in particular its Filippov regularization, which can be written as follows

ẋ ∈ co {f i (x) | i ∈ I X (x)} + g(x)k(z) =: ‹ F x (x, z), (47a)
ż ∈ co {f i (z) + i (h(x) -h(z)) | i ∈ I X (z)} + g(z)k(z) =: ‹ F z (x, z), (47b)
where the function I X is defined as in [START_REF] Della Rossa | Smooth approximation of patchy Lyapunov functions for switching systems[END_REF]. The maps ‹ F x , ‹ F z : R n × R n ⇒ R n satisfy the conditions of Assumption 1: they have non-empty, compact and convex values, they are locally bounded, upper semicontinuous with respect to the states (x and z respectively) and continuous with respect to the inputs (z and x respectively). We can thus conclude local existence of solutions for the systems (47a), (47b) using Proposition 1.

To design [START_REF] Teel | Global stabilizability and observability imply semi-global stabilizability by output feedback[END_REF], we first characterize stability of the interconnection [START_REF] Teel | A smooth Lyapunov function from a class-KL estimate involving two positive semidefinite functions[END_REF]. To this end, we perform the change of coordinates (x, z) → (x, e) := (x, xz) and we construct the Filippov regularization of the corresponding dynamics, resulting in

ẋ ∈ F x (x, e) := ‹ F x (x, x -e) (48a) ė ∈ F e (x, e), (48b)
where the map F e : R n × R n ⇒ R n is defined as the Filippov regularization of the discontinuous map f e (x, e) :=

f1(x) -f1(z) -1(h(x) -h(z)) + g(x, z) if q(x) ≥ 0, q(z) ≥ 0, f2(x) -f1(z) -1(h(x) -h(z)) + g(x, z) if q(x) ≤ 0, q(z) ≥ 0, f1(x) -f2(z) -2(h(x) -h(z)) + g(x, z) if q(x) ≥ 0, q(z) ≤ 0, f2(x) -f2(z) -2(h(x) -h(z)) + g(x, z) if q(x) ≤ 0, q(z) ≤ 0, (49)
with g(x, z) := (g(x)g(z))k(z).

In our construction, we first use Theorem 2 to ensure ISS of (48a) based on two functions V 1 , V 2 , each of them associated to a mode. andψ x , ψ x , ρ x , α x ∈ K ∞ , such that, for each x ∈ R n , (-1) i-1 q(x) > 0 implies

Property 1. There exist V 1 , V 2 ∈ C 1 (R n , R),
ψ x (|x|) ≤ V i (x) ≤ ψ x (|x|), ( 50a
)
|∇V i (x)| ≤ ρ x (|x|), ( 50b
) ∇V i (x), f i (x) + g(x)k(x) ≤ -α x (|x|). (50c)
Moreover, there exists a function θ x ∈ K ∞ such that

θ x (s) ≤ α x (s) β g (s)ρ x (s) , ∀ s ∈ R + . (51)
Finally, defining

V x (x) := V i (x), if x ∈ X i i = 1, 2, (52)
we suppose that V x is continuous, that is,

(q(x) = 0) ⇒ V 1 (x) = V 2 (x), (53)
and there exist functions γ q x ∈ K and α q x ∈ PD such that for all x ∈ R n satisfying q(x) = 0, it holds that

(|x| ≥ γ q x (|e|) ) ⇒ max V x,Fx (x, e) ≤ -α q x (|x|). (54)
Based on Property 1, we may prove the next result.

Proposition 5. Under Property 1, there exists α x ∈ PD and γ x ∈ K such that

(|x| ≥ γ x (|e|) ) ⇒ max V x,Fx (x, e) ≤ -α x (|x|)
and thus system (48a) is ISS w.r.t. e.

Proof. First of all we rewrite system (48a) as

ẋ ∈ co {f i (x) | i ∈ I X (x)} + g(x)k(x) -g(x) [k(x) -k(z)] .
Let us note that equation (50a) assures continuity of the function V x , and thus V x is a piecewise C 1 function with respect to X , in the sense of Definition 6. Consider first a point x ∈ int(X i ) for some i ∈ {1, 2}, that is an x ∈ R n such that (-1) i-1 q(x) > 0. We have

F x (x, e) = {f i (x) + g(x)k(x) -g(x)[k(x) -k(z)]} =: { f i },
and thus by equation (50c) it follows that, for x ∈ X i ,

∇V x (x), f i = ∇V i (x), f i ≤ -α x (|x|) + |∇V i (x)| g(x) |k(x) -k(z)| ≤ -α x (|x|) + ρ x (|x|)β g (|x|)L k |x -z|,
where β g comes from Assumption 4, L k > 0 is the Lipschitz constant of the map k : R n → R and ρ x is given by equation (50b). Choosing 0 < ε < 1, we have

∇V x (x), f i ≤ -(1 -ε)α x (|x|) -εα x (|x|) + ρ x (|x|)β g (|x|)L k |x -z|
and thus, for i ∈ {1, 2}, and each x ∈ X i ,

∇V x (x), f i ≤ -(1 -ε)α x (|x|), if |e| ≤ εα x (|x|) L k ρ x (|x|)β g (|x|)
.

Thanks to (51), the function θ x (s) := εθx(s) γ c x := θ -1 x , by arbitrariness of i ∈ {1, 2}, the previous inequality implies that, for any x ∈ int(X 1 ) ∪ int(X 2 ),

L k ≤ εαx(s) L k ρx(s)βg(s) is of class K ∞ . Defining α c x := (1 -ε)α x and
(|x| ≥ γ c x (|e|) ) ⇒ max V x,Fx (x, e) ≤ -α c x (|x|). ( 55
)
Consider now a point x ∈ ∂X 1 ∪ ∂X 2 . By definition of the proper partition {X 1 , X 2 }, we have q(x) = 0, and thus implication (54) holds. Collecting (54) and ( 55) we obtain that, for all x, e ∈ R n ,

(|x| ≥ γ x (|e|) ) ⇒ max V x,Fx (x, e) ≤ -α x (|x|),
where γ x (s) := max{γ c (s), γ q x (s)} and α x (s) := min{α c x (s), α q x (s)}, concluding the proof.

Remark 7 (Clarke derivative based condition). It is possible to obtain a corollary of Proposition 5, based on the Clarke derivative, as in Definition 2. To this end, it is sufficient to replace implication (54) in Property 1, with the following:

(CL.1) For all x ∈ R n such that q(x) = 0, for each i ∈ {1, 2},

∇V i (x), f 3-i (x) + g(x)k(x) ≤ -α x (|x|),
where α x ∈ K ∞ satisfies also (50c).

The proof carries over straightforwardly, recalling the inclusion [START_REF] Clarke | Asymptotic stability and smooth Lyapunov functions[END_REF].

Example 3 (Continued). In the switched linear case of Example 3, Property 1 can be guaranteed with quadratic functions V i (x) := x P i x, i ∈ {1, 2}. Indeed, since the partitions given by q v (or q Q ) are conic, i.e. X i is a cone for each i ∈ {1, 2}, we can look for Lyapunov functions homogeneous of degree 2, see [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF] and the extension [START_REF] Tuna | Homogeneous hybrid systems and a converse Lyapunov theorem[END_REF]. From now on we focus on the case q Q (x) = x Qx. The half-space partition case (i.e. considering q v (x) = v, x ) can be developed analogously to [START_REF] Heemels | Input-to-state stability of discontinuous dynamical systems with an observer-based control application[END_REF]. Considering q Q (x), it suffices to find K ∈ R m×n , positive definite matrices

P 1 , P 2 ∈ R n×n , µ 12 , µ 21 , µ Q ∈ R, a x > 0 and µ 1 , µ 2 ≥ 0 such that P1 -P2 = µQQ; (56a) µ1Q + P1(A1 + BK) + (A1 + BK) P1 + axI ≺ 0; (56b) -µ2Q + P2(A2 + BK) + (A2 + BK) P2 + axI ≺ 0; (56c) µ12Q + P1(A2 + BK) + (A2 + BK) P1 + axI ≺ 0; (56d) µ21Q + P2(A1 + BK) + (A1 + BK) P2 + axI ≺ 0. ( 56e
)
and thus we can rewrite f e : R n × R n → R n as

f e (x, z) :=                f 1 (x) -f 1 (z) -1 (h(x) -h(z)) + g(x, z) + I -(q(x)) f (x) if q(z) ≥ 0, f 2 (x) -f 2 (z) -2 (h(x) -h(z)) + g(x, z) -I + (q(x)) f (x) if q(z) ≤ 0.
where we defined f (x) := f 2 (x)f 1 (x) and I + , I -are the indicator functions of the positive and negative real numbers respectively. We note that, by Assumption 4,

max{|I -(q(x)) f (x)|, |I + (q(x)) f (x)|} ≤ β f (|x|)
for all x ∈ R n . Thus, we can now apply the same reasoning used in proof of Proposition where α e := (1ε)α e and γ e := (εθ e ) -1 • β f , for some 0 < ε < 1. Note that condition (59) ensures that γ e ∈ K. The ISS property follows again from Theorem 1.

To clarify our construction, the idea behind Property 2 and Proposition 6 is to search for a common C 1 Lyapunov function for the two vector fields f i (x)f i (z)i (h(x)h(z)) + g(x, z), i ∈ {1, 2}. If x and the estimated state z are not in the same region X i , then the (xz)-dynamics is perturbed by a factor ±(f 1 (x)f 2 (x)), which is treated as an external disturbance. The injection gains i induce ISS with respect to these disturbances.

, (57b) and (58) are satisfied, and the function θ e ∈ K ∞ in (59) is defined as θ e (s) = ae 2λe s. We are finally ready to state our stability conditions, based on Theorem 2, for the interconnection in [START_REF] Teel | A smooth Lyapunov function from a class-KL estimate involving two positive semidefinite functions[END_REF].

Corollary 3. Assume that Properties 1 and 2 hold, and define η

1 := ψ x • γ e • ψ -1 e and η 2 := ψ e • γ x • ψ -1 x . If η 1 • η 2 (s) < s, ∀s > 0, (61)
then system (47) is globally asymptotically stable.

Proof. Since by Propositions 5 and 6 we can construct nonpathological ISS-Lyapunov functions V x and V e as in Assumption 2, it remains to check that condition (61) implies the small gain condition [START_REF] Johansson | Computation of piecewise quadratic Lyapunov functions for hybrid systems[END_REF] in Theorem 2.

First we note that, by (50a) and (57a),

V x (x) ≥ η 1 (V e (e)) = ψ x • γ e • ψ e -1 (V e (e))

⇔ ψ x -1 (V (x)) ≥ γ e • ψ e -1 (V e (e))

⇒ |x| ≥ γ e (|e|), since V x (x) ≤ ψ x (|x|), and V e (e) ≥ ψ e (|e|). By Proposition 5, this implies that (V x (x) ≥ η 1 (V e (e)) ) ⇒ max V x,Fx (x, e) ≤α x (|x|).

Following the same path for η 1 , we obtain the implication (V e (e) ≥ η 2 (V x (x)) ) ⇒ max V e,Fe (x, e) ≤α e (|e|), proving that (61) implies [START_REF] Johansson | Computation of piecewise quadratic Lyapunov functions for hybrid systems[END_REF].

Conclusions

We focused on ISS of differential inclusions using locally Lipschitz Lyapunov functions. We provided sufficient conditions based on the notion of Lie derivative of the candidate Lyapunov function, which generalize previous results relying on the study of the Clarke derivative. We applied our results to state-dependent switched systems and proposed a new formulation of the well-known small gain theorem in the context of interconnected differential inclusions. We finally studied the design of an observer-based output feedback controller for a bimodal switched system. As possible further research, we may investigate convex LMI-based algorithms, based on using Lipschitz non quadratic functions and Lie derivative.

A Properties of piecewise C 1 functions

In this Appendix we prove the two items of Proposition 3, first characterizing Clarke generalized gradient, and then showing that piecewise C 1 functions are nonpathological. We prove below that S 1 (x) = S 2 (x); then (62) follows from [START_REF] Ceragioli | Discontinuous ordinary differential equations and stabilization[END_REF]. If x ∈ int(X ) for some ∈ I, we have already noted that ∂V (x) = {∇V (x)} and thus S 1 (x) = S 2 (x) trivially holds. Let us suppose x / ∈ M =1 int(X ) which implies x ∈ ∂X. S 1 (x) ⊆ S 2 (x): Consider any v = ∇V (x) for some ∈ I X (x). By Definition 4, there exists a sequence x k ∈ int(X ) such that x k → x and thus ∇V (x k ) = ∇V (x k ). By continuity of ∇V we have lim k→∞ ∇V (x k ) = lim k→∞ ∇V (x k ) = ∇V (x), therefore v ∈ S 2 (x).

S 2 (x) ⊆ S 1 (x): Consider any v ∈ R n such that there exists a sequence x k → x of points where V is differentiable, such that the sequence ∇V (x k ) converges to v. By Definition 4, there exists a neighborhood O of x such that O ⊆ ∈I X (x) X , and thus for each k (large enough) there exists an index k ∈ I X (x) satisfying x k ∈ X k . By finiteness of I X (x) we can extract a subsequence of x k (without relabeling) and an index ∈ I X (x) such that x k ∈ X for each k ∈ N and ∇V (x k ) → v. Now for each k ∈ N we can take a sequence x k,l → x k satisfying x k,l ∈ int(X ). Thus ∇V (x k ) = lim l→∞ ∇V (x k,l ) = lim l→∞ ∇V (x k,l ) = ∇V (x k ). Summarizing, we have v = lim k→∞ ∇V (x k ) = lim k→∞ ∇V (x k ) = ∇V (x).

Since we proved S 1 (x) = S 2 (x), then (62) follows from [START_REF] Ceragioli | Discontinuous ordinary differential equations and stabilization[END_REF].

Having shown (62), equation [START_REF] Heemels | Input-to-state stability of discontinuous dynamical systems with an observer-based control application[END_REF] in Proposition 3 directly follows. The next statement completes the proof of Proposition 3. Lemma 4. Consider X = {X i , O i } i∈I , a proper partition of R n . If V ∈ P(X ), then V is nonpathological.

Proof. Recalling Definition 3 we must show that, given any ϕ ∈ AC(R + , R n ), ∂V (ϕ(t)) is a subset of an affine subspace orthogonal to φ(t), for almost all t ∈ R + , namely that ∃a t ∈ R such that v, φ(t) = a t , ∀v ∈ ∂V (ϕ(t)).

(63)

Since ϕ : R + → R n is absolutely continuous and V : R n → R is locally Lipschitz we have that ϕ and V (ϕ(t)) are differentiable almost everywhere, i.e. there exists a set of measure zero N ⊆ R + such that φ(t) and d dt V (ϕ(•))(t) both exist for every t ∈ R + \ N . Using (62) in Lemma 3, to ensure (63) it is enough to show that, for almost all t ∈ R + \ N , there exists a t ∈ R such that ∇V (ϕ(t)), φ(t) = a t , ∀ ∈ I X (ϕ(t)).

Fix any t ∈ R + \ N . Either (64) holds for that t, or there exist 1 , 2 ∈ I X (ϕ(t)) such that 1 = 2 and ∇V 1 (ϕ(t)), φ(t) = ∇V 2 (ϕ(t)), φ(t) .

In this second case we have

d dt (V 1 (ϕ(t)) -V 2 (ϕ(t)))
= ∇V 1 (ϕ(t)), φ(t) -∇V 2 (ϕ(t)), φ(t) = 0.

Thus, by continuity, there exists ε > 0 small enough such that V 1 (ϕ( t)) = V 2 (ϕ( t)) , for all t ∈ (tε, t + ε) \ {t}, which implies that either 1 / ∈ I X (ϕ( t)) or 2 / ∈ I X (ϕ( t)) (or both), for all such t, since, by Definition 6,

1 , 2 ∈ I X (x) ⇒ V (x) = V 1 (x) = V 2 (x),
for any x ∈ R n . Iterating the argument, this shows that for any point t where two or more scalar products ∇V j (ϕ(t)), φ(t) "disagree" in (64), t is isolated. We conclude by recalling that a set of isolated point is countable [START_REF] Hrbacek | Introduction to Set Theory[END_REF]Page 180] and thus has measure zero, as to be proven.

2 ;

 2 and firstly, let us suppose that z ∈ R 1 , and define µ := |z| √ we can write z = µ[1 1] (w.l.o.g. supposing that z lies in the first quadrant). By Proposition 3 and (17), we have that

Figure 1 :

 1 Figure 1: The interconnected system in (23).

  and for all u ∈ R m . Proof. The existence of function ν : R + → R + under (A.3) of Assumption 3 is established in [43, Lemmas 1 and 2]. Introduce the function : R + → R + defined by (s) := s 0 ν(r)dr, ∀ s ∈ R + .

  negative, nor positive semi-definite. These cases satisfy Assumption 4, by selecting β g (s) := B , and β f (s) := A 1 -A 2 s.

Example 3 (

 3 Continued). For the switched linear case presented in Example 3, Property 2 can be ensured using a quadratic function V e (e) := e P e e, with P e 0, by finding matrices L 1 , L 2 ∈ R n×p and a e > 0 such that P e (A i -L i C) + (A i -L i C) P e + a e I ≺ 0, (60) for each i ∈ {1, 2}. Indeed, defining λ e := λ min (P e ), λ e := λ max (P e ), ψ e (s) := λ e s 2 , ψ e (s) := λ e s 2 , ρ e (s) := 2λ e s, α e (s) := a e s 2 , equations

Remark 8 .λ x 3 λ e 3 λ x λ e a 2 x a 2 e < 1 .

 8321 Considering again the switched system presented in Example 3, we can check the small-gain condition (61) as follows. Recalling the definitions of γ x and γ e in the proofs of Propositions 5 and 6 we can writeγ x (s) = 2 B K λ x εa x s and γ e (s) = 2 A 1 -A 2 λ e εa e s.Thus, by arbitrariness of 0 < ε < 1, condition (61) holds if16 B 2 K 2 A 1 -A 2 2

Lemma 3 .

 3 Consider X = {X i , O i } i∈I , a proper partition of R n . If V ∈ P(X ) then V is locally Lipschitz and ∂V (x) = co {∇V (x) | ∈ I X (x)} . (62)Proof. For the proof that V is locally Lipschitz we refer to[START_REF] Scholtes | Introduction to Piecewise Differentiable Equations[END_REF] Proposition 4.1.2]. Given any x ∈ R n , define the setsS 1 (x) := {∇V (x) | ∈ I X (x)}, S 2 (x) := lim k→∞ ∇V (x k ) x k → x, x k / ∈ N V .

A function α : R ≥0 → R is positive definite (α ∈ PD) if it is continuous, α(0) = 0, and α(s) > 0 if s = 0. A function α : R ≥0 → R ≥0 is of class K (α ∈ K) if it is continuous, α(0) = 0, and strictly increasing; it is of class K∞ if, in addition, it is unbounded. A continuous function β : R + × R + → R + is of class KL if β(•,s) is of class K for all s, and β(r, •) is decreasing and β(r, s) → 0 as s → ∞, for all r.

Here, and in what follows, we consider the 2-induced matrix norm, i.e., given M ∈ R n×m we define M := sup |x|=1 |M x|

The globally Lipschitz assumption on k : R n → R m can be relaxed by asking that ∃α k ∈ K∞ such that |k(x) -k(y)| ≤ α k (|x -y|), for all x, y ∈ R n .

This work was partially supported by the ANR project ConVan with grant number ANR-17-CE40-0019-01.

Then all the conditions of Property 1 hold with V x (x) := x P i x, if x ∈ X i . Indeed, first we note that (56a) implies (53) of Property 1. Moreover, we can define

where λ max (P ), λ min (P ) represent respectively the largest and the smallest eigenvalues of a positive definite matrix P 0. The bound functions in (50a) and (50b) of Property 1 are thus obtained by defining ψ x (s) := λ x s 2 , ψ x (s) := λ x s 2 , ρ x (s) := 2λ x s.

Via the S-Procedure, equation (56b) implies

We have thus proved (50c) of Property 1 with α x (s) := a x s 2 . Similarly, using Finsler's Lemma, equations (56d) and (56e) imply item (CL.1) in Remark 7, again with α x (s) := a x s 2 . The function θ x ∈ K ∞ in (51) can be defined as θ x (s) := ax 2 B λx s. Let us now consider the error dynamics (48b) and characterize ISS from x, using a C 1 -Lyapunov function, satisfying the next property.

Property 2. Suppose that there exist V e ∈ C 1 (R n , R), and ψ e , ψ e , α e , ρ e ∈ K ∞ such that

∂V e ∂e (e) ≤ ρ e (|e|), ∀ e ∈ R n . (57b)

Moreover, for all x ∈ R n , for all z ∈ R n and for each i ∈ {1, 2},

with e = xz. Finally there exists θ e ∈ K ∞ such that

Based on Property 2 we can prove the next result.

Proposition 6. Under Property 2, there exist γ e ∈ K and α e ∈ K ∞ such that

for all f e ∈ F e (x, e), and thus the system (48b) is ISS w.r.t. x.

Proof. It is easy to see that the second and third expression in (49) can be rewritten respectively as

f 2 (x)f 2 (z) -2 (h(x)h(z)) + g(x, z) + (f 1 (x)f 2 (x))