S Mariano

R Postoyan

L Zaccarian
email: zaccar-ian@laas.fr

† S Mariano

Finite-time stability properties of Lur'e systems with piecewise continuous nonlinearities *

We analyze the stability properties of Lur'e systems with piecewise continuous nonlinearities by exploiting the notion of set-valued Lie derivative for Lur'e-Postnikov Lyapunov functions. We first extend an existing result of the literature to establish the global asymptotic stability of the origin under a more general sector condition. We then present the main results of this work, namely additional conditions under which output and state finite-time stability properties also hold for the considered class of systems. We highlight the relevance of these results by certifying the stability properties of two engineering systems of known interest: mechanical systems affected by friction and cellular neural networks.

Introduction

Defining conditions to ensure stability properties of continuous-time linear systems subject to a cone-bounded nonlinear output feedback, namely, the socalled the Lur'e problem, has been widely investigated in the literature see, e.g., [START_REF] Khalil | Nonlinear Systems[END_REF][START_REF] Lur | On the theory of stability and control systems[END_REF][START_REF] Popov | On absolute stability of nonlinear automatic control systems[END_REF][START_REF] Yakubovich | Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities[END_REF]. This class of systems is ubiquitously used in various engineering domains, such as mechanical engineering to describe dynamical systems affected by friction and/or unilateral constraints [START_REF] De Bruin | Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities[END_REF], electrical and electronic engineering to capture the behavior of electrical circuits with switches or electronic devices [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics[END_REF][START_REF] Vasca | A new perspective for modeling power electronics converters: Complementarity framework[END_REF], or neural networks [START_REF] Soykens | Lur'e systems with multilayer perceptron and recurrent neural networks: absolute stability and dissipativity[END_REF]; see [START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF] for additional examples. However, to the authors' best knowledge, very few results are available on the finite-time stability properties of Lur'e systems, see [START_REF] Tang | Finite-time cluster synchronization of Lur'e networks: A nonsmooth approach[END_REF], which concentrates on cluster synchronization of networks of Lur'e systems. Finite-time stability properties are gaining increasing attention due to their relevance in many applications such as high-order sliding mode algorithms [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF], controllers for mechanical systems [START_REF] Bartolini | A survey of applications of second-order sliding mode control to mechanical systems[END_REF], spacecraft stabilization [START_REF] Vorotnikov | Partial stability, stabilization and control: some recent results[END_REF], observer design problems [START_REF] Andrieu | LMI sufficient conditions for contraction and synchronization[END_REF]; see [START_REF] Vorotnikov | Partial Stability and Control[END_REF] for additional examples. There is therefore a need for analytical tools to establish finite-time stability properties for this class of systems. In this context, we investigate the output and state finite-time stability properties of Lur'e system with piecewise continuous nonlinearities.

Historically, two different types of Lyapunov functions have been used to analyze the (absolute) stability of continuous-time Lur'e systems: quadratic functions of the state and the so-called Lur'e-Postnikov Lyapunov functions, which are the sum of a quadratic function of the state and a weighted sum of the integrals of the feedback nonlinearities [START_REF] Khalil | Nonlinear Systems[END_REF]. Lur'e-Postnikov Lyapunov functions are generally used to draw less conservative sufficient stability conditions [START_REF] Yakubovich | Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities[END_REF]. However, when the nonlinearities are piecewise continuous, as in, e.g., mechanical systems [START_REF] De Bruin | Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities[END_REF], neural networks [START_REF] Forti | Global convergence of neural networks with discontinuous neuron activations[END_REF], see also [START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF], the challenge is that Lur'e-Postnikov Lyapunov functions become only differentiable almost everywhere (being locally Lipschitz continuous) due to the discontinuity points of the nonlinearities. Indeed, when the system nonlinearities are piecewise continuous and a Lur'e-Postnikov Lyapunov function is considered, the standard tools used in the nonsmooth analysis, like Clarke's generalized directional derivatives, may lead to conservative algebraic Lyapunov conditions as we show in this paper; see also [START_REF] Rossa | Non-Smooth Lyapunov Functions for Stability Analysis of Hybrid Systems[END_REF]. This limitation is overcome in [START_REF] De Bruin | Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities[END_REF], where trajectory-based arguments are used to prove an input-to-state (ISS) stability property, but no finite-time stability property is provided.

In this work, we first extend one of the results in [START_REF] De Bruin | Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities[END_REF] to establish the global asymptotic stability of the origin for Lur'e systems with piecewise continuous nonlinearities under a more general sector condition. We resort for this purpose to a nonsmooth Lur'e-Postnikov Lyapunov function. We present algebraic Lyapunov decrease conditions by using the notion of set-valued Lie derivative [START_REF] Bacciotti | Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions[END_REF][START_REF] Valadier | Entraînement unilatéral, lignes de descente, fonctions Lipschitziennes non pathologiques[END_REF]. The set-valued Lie derivative is the key to overcoming the conservatism which the customarily used Clarke's generalized directional derivative may give, as we illustrate in a dedicated example. It has to be noted that in [START_REF] Tang | Finite-time cluster synchronization of Lur'e networks: A nonsmooth approach[END_REF] set-valued Lie derivatives are also used in the analysis of these interconnections, however, the Lyapunov function is quadratic (thus continuously differentiable), which, as mentioned above, leads to more conservative conditions and, more importantly, the problem setting is different. Our main results establish output and state finite-time stability properties for the considered Lur'e systems. To illustrate the usefulness of our results we focus on two engineering applications, considered respectively in [START_REF] De Bruin | Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities[END_REF][START_REF] Forti | Global convergence of neural networks with discontinuous neuron activations[END_REF] and that can be modeled as Lur'e systems. Indeed, we establish output finite-time and state-independent local asymptotic stability properties for mechanical systems subject to friction, which is a novelty compared to [START_REF] De Bruin | Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities[END_REF]. Furthermore, we certify that the cellular neural networks modeled as in [START_REF] Forti | Global convergence of neural networks with discontinuous neuron activations[END_REF] are state finite-time stable, thus retrieving the results in [START_REF] Forti | Global convergence of neural networks with discontinuous neuron activations[END_REF]Thm. 4] while coping with a more general class of Lur'e systems.

The rest of the paper is organized as follows. Notation and background material are given in Section II. The class of Lur'e systems under consideration is introduced in Section III. In Section IV, we address asymptotic stability characterizations with a novel algebraic Lyapunov proof. Finite-time stability results are given in Section V, while we discuss applications of these results in Section VI. In Section VII we give conclusions and some perspectives.

Notation

Let R be the set of real numbers, R ≥0 := [0, ∞), R >0 := (0, ∞), Z ≥0 := {0, 1, . . . }, Z >0 := {1, 2, . . . } and C := {a + ib : a, b ∈ R} with i := √ -1. The notation R n stands for the n-dimensional Euclidean space with n ∈ Z >0 . The notation B n stands for the closed unit ball of R n centered at the origin and we write B when its dimension is clear from the context. We denote with ∅ the empty set. Given a vector x ∈ R n , we denote with x ℓ its ℓ-th element, ℓ ∈ {1, . . . , n}, and with |x| its Euclidean norm. The notation 0 n stands for the vector of R n , whose n ∈ Z >0 elements are all equal to 0. We use I n to denote the identity matrix of dimension n × n with n ∈ Z >0 while O n denotes the null matrix of dimension n × n with n ∈ Z >0 . Given two vectors

x 1 ∈ R n and x 2 ∈ R m with n, m ∈ Z >0 , we denote (x 1 , x 2 ) := [x ⊤ 1 x ⊤ 2 ]
⊤ for the sake of convenience. Given a matrix A ∈ R n×m with n, m ∈ Z >0 , A ℓ stands for its ℓ-th row where ℓ ∈ {1, . . . , n}, |A| is its spectral norm while ker(A) stands for its kernel. The notation diag(x 1 , . . . , x n ) stands for the diagonal matrix of R n×n whose n ∈ Z >0 diagonal elements are x 1 , . . . , x n ∈ R. We define a symmetric matrix P ∈ R n×n with n ∈ Z >0 to be positive (negative) definite, i.e., P > 0 (P < 0), if all its eigenvalues are real and positive (negative); we say that P is positive (negative) semidefinite, i.e., P ≥ 0 (P ≤ 0), if all its eigenvalues are real and non-negative (non-positive). Given a set

S ⊂ R n with n ∈ Z >0 , co S is its closed convex hull. Given a function f : X → Y , the domain of f is defined as dom f = {x ∈ X : f (x) ̸ = ∅}. A function f : X → R ≥0 with X ⊆ R n and n ∈ Z >0 is radially unbounded if f (x) → ∞ as |x| → ∞. Let f : R n → R and r ∈ R with n ∈ Z >0 , we denote by f (r) -1 the set {x ∈ R n : f (x) = r},
which may be empty. Let X and Y be two non-empty sets, T : X ⇒ Y denotes a set-valued map from X to Y . We will refer to class K, K ∞ and KL functions as defined in [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]Chap. 3]. Let f : R → R and s • ∈ R, then f ′ (s • ) := lim s→s• (f (s)-f (s • ))/(s-s • ), when it exists. A function f : R → R is piecewise continuous if for any given interval [a, b], with a < b ∈ R, there exist a finite number of points a ≤

x 0 < x 1 < x 2 < • • • < x k-1 < x k ≤ b with k ∈ Z ≥0
such that f is continuous on (x i-1 , x i ) for any i ∈ {1, . . . , k} and its one-sided limits exist as finite numbers. A function f : R → R is piecewise continuously differentiable if f is continuous and for any given interval [a, b], with a < b ∈ R, there exists a finite number of points a ≤

x 0 < x 1 < x 2 < • • • < x k-1 < x k ≤ b, with k ∈ Z ≥0 such
that f is continuously differentiable on (x i-1 , x i ) for any i ∈ {1, . . . , k} and the one-sided limits lim s→x + i-1 f ′ (s) and lim s→x - i f ′ (s) exists for any i ∈ {1, . . . , k}.

Problem statement

Consider the system of the form

ẋ = Ax + Bu y = Cx (1) u = -ψ(y),
where x ∈ R n is the state, u, y ∈ R p are respectively the input and the output and A, B and C are real matrices of appropriate dimensions. The function ψ : R p → R p is decentralized, namely for any y = (y 1 , . . . , y p ) ∈ R p , ψ(y) = (ψ 1 (y 1 ), . . . , ψ p (y p )). We suppose that ψ satisfies the next sector condition.

Assumption 1 For any i ∈ {1, . . . , p}, ψ i is piecewise continuous and there exists ζ i ∈ (0, +∞] such that

ψ i (y i )(ψ i (y i ) -ζ i y i ) ≤ 0, ∀y i ∈ R. (2)
□ The sector condition (2) is more general than the one considered in [START_REF] De Bruin | Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities[END_REF], that is recovered when ζ i = +∞ for all i ∈ {1, . . . , p}, in which case (2) reads

-ψ i (y i )y i ≤ 0, ∀y i ∈ R ∀i ∈ {1, . . . , p}. (3)
This generalization allows to derive less conservative stability conditions when the nonlinearities satisfy (2) with some finite ζ i . Assumption 1 characterizes a so-called Lur'e system [START_REF] Khalil | Nonlinear Systems[END_REF]Ch. 7], [START_REF] Yakubovich | Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities[END_REF].

In view of Assumption 1, system (1) may have a discontinuous right-hand side. Therefore, when we refer to the solutions to system (1), we consider its so-called (generalized) Krasovskii solutions, which coincide with the solutions obtained by the Krasovskii regularization [START_REF] Hájek | Discontinuous differential equations, I[END_REF] of [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics[END_REF], that is

ẋ ∈ F (x) := Ax -BΨ(Cx), y = Cx, (4)
where Ψ(y) = (Ψ 1 (y 1 ), . . . , Ψ p (y p )) is the Krasovskii regularization of ψ in (1), whose components are defined as We analyze the stability properties of system (4) in the sequel, thereby ensuring the same stability properties for the Krasovskii solutions of (1). As customary in the Lur'e systems literature and as shown in Fig. 1, we perform a loop transformation to interpret system (4) as the feedback interconnection of two passive systems.

Ψ i (y i ) := s>0 co ψ i (y i + sB), i ∈ {1, . . . ,
By following the steps in [18, Ch. 7.1.2] and [START_REF] De Bruin | Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities[END_REF] and by adopting the same mathematical notation found in [START_REF] Khalil | Nonlinear Systems[END_REF]Ch. 7], we define the dynamic multiplier with transfer function M(s)

:= I + Γs ∀s ∈ C, (5)
where Γ := diag(γ 1 , . . . , γ p ) and γ 1 , . . . , γ p > 0 are suitable parameters, as detailed in the sequel. We thus interpret system (4) as the feedback interconnection of the linear system Σ 1

Σ 1 : ẋ = Ax + Bu y = (C + ΓCA)x + (ΓCB + Z)u, (6)
where We assume that system Σ 1 is strictly passive from u to y with quadratic storage function U 1 defined as

Z := diag(ζ -1 1 , . . . , ζ -1 p ) with 1 ζ i ∈ (0, +∞] in Assumption 1, with the nonlinear system Σ 2 Σ 2 : ẏ = g(y, y) := -Γ -1 y + Γ -1 (y -Zu) u ∈ -Ψ(y). (7
U 1 (x) := 1 2 x ⊤ P x, ∀x ∈ R n , (8)
with P ∈ R n×n symmetric and positive definite, as formalized next.

Assumption 2 System Σ 1 in ( 6) is strictly passive from u to y with storage function U 1 in (8) [18, Def. 6.3], i.e., there exist matrices Γ > 0 diagonal, P = P ⊤ > 0 and a scalar η > 0 such that

M := P A + A ⊤ P + ηI n P B -(C + ΓCA) ⊤ B ⊤ P -(C + ΓCA) -2Z -ΓCB -(ΓCB) ⊤ ≤ 0. (9)

□

The linear matrix inequality [START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF] in Assumption 2 can be efficiently tested numerically. Several tools are also available in the literature to certify (9): the Kalman-Yakubovich-Popov lemma [18, Lemma 6.3] and the equivalent conditions given in [START_REF] Brogliato | Dissipative Systems Analysis and Control[END_REF]Ch. 3.1] for minimal realizations or the results surveyed in [START_REF] Brogliato | Dissipative Systems Analysis and Control[END_REF]Ch. 3.3] for nonminimal ones, to cite a few. On the other hand, it can be proven, as clarified later in Remark 1 in Section 4.2, that Σ 2 in ( 7) is passive from input y to output -u, by considering the piecewise continuously differentiable storage function U 2 defined as

U 2 (y) := p i=1 γ i yi 0 ψ i (σ)dσ, ∀y ∈ R p , (10)
where γ i > 0 for i ∈ {1, . . . , p} are the diagonal elements of Γ as defined after [START_REF] Bacciotti | Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions[END_REF].

We are now ready to proceed with the stability analysis of (4). First, we provide sufficient conditions to ensure global asymptotic stability of the origin for system (4) in Section 4. We then analyze its finite-time stability properties in Section 5.

Asymptotic stability 4.1 Nonsmooth Lur'e-Postnikov Lyapunov functions

Inspired by [START_REF] De Bruin | Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities[END_REF][START_REF] Khalil | Nonlinear Systems[END_REF] where Lur'e systems with continuous nonlinearities are considered, we characterize the stability of the origin for system (4) with a Lur'e-Postnikov Lyapunov function V given by

V (x) := U 1 (x) + U 2 (Cx) = 1 2 x ⊤ P x + p i=1 γ i Cix 0 ψ i (σ)dσ, ∀x ∈ R n , (11)
where P comes from Assumption 2. Function V is piecewise continuously differentiable, and thus locally Lipschitz, therefore there are points where its gradient is not defined. A standard tool to circumvent this is Clarke's generalized directional derivative, defined for each direction

f ∈ R n at each x ∈ R n as [11, page 11] V • (x; f ) := max{⟨v, f ⟩ : v ∈ ∂V (x)},
where ∂V (x) denotes Clarke's generalized gradient of V at x given by

∂V (x) := {v ∈ R n |v ∈ x ⊤ P + (Ψ(Cx)) ⊤ ΓC}. ( 12
)
However, the Lyapunov analysis of system (4) using Clarke's generalized directional derivative of V is often too conservative to establish asymptotic stability of the origin. Roughly speaking, for some x ∈ R n \ {0 n } there may exist a selection f bad ∈ F (x) that is never viable for any solution to (4) and such that V • (x, f bad ) > 0, thereby preventing to prove that the origin of the system is globally asymptotically stable, as illustrated in the next example.

Example 1 Consider system (4) with n = 2, p = 1 (SISO case),

A = -1 -1 1 -1 , B = 1 0 , C = 1 0 ,
and where Ψ is the Krasovskii regularization of ψ : R → [-1 4 , 1], defined as

ψ(s) = 1 if s > 0, ψ(s) = -1 4 if s < 0, and ψ(s) = 0 if s = 0; hence Ψ(0) = [-1/4, 1]
. This function ψ satisfies Assumption 1 with ζ 1 = +∞ (namely (3)). Consider V as in [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF] with P = I 2 and Γ = γ 1 = 1. The proposed selection of matrices A, B, C, P and Γ and the set-valued map Ψ are such that Assumptions 1 and 2 are satisfied. Function [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF] in this case is given by V

(x) = 1 2 (x 2 1 + x 2 2 ) + x1 0 ψ(σ)dσ for any x ∈ R 2 .
We have that V is positive definite and radially unbounded. Furthermore, V is not differentiable at {0} × R. By following [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]Ch. 4] as summarized in [START_REF] Rossa | Non-Smooth Lyapunov Functions for Stability Analysis of Hybrid Systems[END_REF]Ch. 2.4.2], to analyze the stability of the origin for the considered system, we study at any x ∈ R2 the maximum of V • (x; f ) over all allowable directions f ∈ F (x) with F as in [START_REF] Aubin | Differential Inclusions: Set-valued Maps and Viability Theory[END_REF]. In this regard, consider x = (0, 1 2 ), max

f ∈F (0, 1 2 ) V • ((0, 1 2 ); f ) = max ⟨v, f ⟩| v ∈ [-1 4 , 1] × 1 2 , f ∈ -3 2 , -1 4 × -1 2 = 1 8 . (13)
With this positive upper bound, in view of [23, Def. 2.16], we cannot establish asymptotic stability of the origin 2 [23, Thm. 2.18]. Nevertheless a direct inspection shows that V strictly decreases along all solutions outside the origin. The issue is overcome in the following by exploiting the notion of set-valued Lie derivative of V [START_REF] Valadier | Entraînement unilatéral, lignes de descente, fonctions Lipschitziennes non pathologiques[END_REF]. □

In [START_REF] De Bruin | Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities[END_REF], the authors overcame the limitations discussed in Example 1 by using trajectory-based Lyapunov arguments when Assumption 1 holds with ζ i = +∞ for any i ∈ {1, . . . , p}. In the next theorem, we establish global asymptotic stability of the origin for system (4). Compared to [START_REF] De Bruin | Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities[END_REF], the result relies on the more general sector condition in (2), and, importantly for the sequel, its proof uses algebraic Lyapunov arguments. Theorem 1 Consider system (4) and suppose that Assumptions 1 and 2 hold. Then the origin is GAS, i.e., there exists β ∈ KL such that all solutions x satisfy

|x(t)| ≤ β(|x(0)|, t), ∀t ∈ R ≥0 . (14)
□ The proof of Theorem 1 is given in Section 4.2, where we use the concept of set-valued Lie derivative that we now recall.

Set-valued Lie derivative and its properties

The set-valued Lie derivative of V with respect to F in (4) at x ∈ R n is defined as [START_REF] Bacciotti | Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions[END_REF] V

F (x) := {a ∈ R| ∃f ∈ F (x) : ⟨v, f ⟩ = a, ∀v ∈ ∂V (x)}, (15)
with ∂V (x) given in [START_REF] Clarke | Nonsmooth Analysis and Control Theory[END_REF]. Note that V F (x) is a subset of {⟨v, f ⟩|v ∈ ∂V (x), f ∈ F (x)} and that, by definition, at any x where V is differentiable, so that ∂V (x) is a singleton, this reduces to the set of all standard directional derivatives of V in any direction of f ∈ F (x). Notice that V F (x) may be the empty set as illustrated later in Example 2. In the next lemma, a useful and intuitive upper bound of the set-valued Lie derivative of V in [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF] along dynamics (4) is provided.

Lemma 1 Given function V in [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF] and F in (4),

sup V F (x) ≤ sup u∈-Ψ(Cx) (x ⊤ P -u ⊤ ΓC)(Ax + Bu) , ∀x ∈ R n , (16)
where we use the convention sup ∅ = -∞ fot the left-hand side when sup V F (x) = ∅. □ Proof: For each x ∈ R n and each element f = Ax + Bu ∈ F (x) with u ∈ -Ψ(Cx), as in (4), denote ρ(x, f ) := P x+C ⊤ Γu and note that ρ(x, f ) ∈ ∂V (x).

Notice that ρ and f are defined by selecting the same u ∈ -Ψ(Cx). In view of Lemma 8 in [START_REF] Mariano | Hybrid coupling rules for leaderless heterogeneous oscillators: uniform global asymptotic and finite-time synchronization[END_REF], exploiting this selection we have that sup

V F (x) ≤ sup u∈-Ψ(Cx) (ρ(x, f ) ⊤ f ), thus concluding the proof. ■
Exploiting [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF] and Lemma 1, we can establish the next algebraic Lyapunov conditions for system (4).

Proposition 1 Consider F in (4) and suppose that Assumptions 1 and 2 hold. Then there exist α 1 , α 2 , α 3 ∈ K ∞ such that function V in [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF] satisfies

α 1 (|x|) ≤ V (x) ≤ α 2 (|x|), ∀x ∈ R n , ( 17
) sup V F (x) ≤ -α 3 (V (x)), ∀x ∈ R n . (18)
□ Proof: From ( 2) and ( 11), V is positive definite, continuous on R n and radially unbounded. Therefore, [START_REF] Hershkowitz | Recent directions in matrix stability[END_REF] holds by [START_REF] Khalil | Nonlinear Systems[END_REF]Lemma 4.3]. Let x ∈ R n , we have from Lemma 1 that

sup V F (x) ≤ sup u∈-Ψ(Cx) (x ⊤ P -u ⊤ ΓC)(Ax + Bu) -u ⊤ (-Zu -Cx) - η 2 |x| 2 + u ⊤ (-Zu -Cx) + η 2 |x| 2 , (19)
with Z as in [START_REF] Bartolini | A survey of applications of second-order sliding mode control to mechanical systems[END_REF]. Therefore,

sup V F (x) ≤ sup u∈-Ψ(Cx) 1 2 x u ⊤ M x u + u ⊤ (Zu + Cx) - η 2 |x| 2 , (20)
with M as in [START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF]. In view of Assumption 1, it holds that u ⊤ (Zu + Cx) ≤ 0 for all u ∈ -Ψ(Cx) and any x ∈ R n , as Ψ(Cx) is convex. Moreover, M ≤ 0 in view of Assumption 2. Therefore, we have from ( 20)

sup V F (x) ≤ sup u∈-Ψ(Cx) (u ⊤ (Zu + Cx)) - η 2 |x| 2 ≤ - η 2 |x| 2 ≤ - η 2 (α -1 2 (V (x))) 2 =: -α 3 (V (x)), (21)
with α 3 ∈ K ∞ , which shows ( 18) and the proof is complete. ■ Based on Proposition 1 we provide an algebraic proof of Theorem 1. Proof of Theorem 1. Let x be a solution to [START_REF] Aubin | Differential Inclusions: Set-valued Maps and Viability Theory[END_REF]. In view of [START_REF] Mariano | Hybrid coupling rules for leaderless heterogeneous oscillators: uniform global asymptotic and finite-time synchronization[END_REF]Prop. 4] and [START_REF] Rossa | Non-Smooth Lyapunov Functions for Stability Analysis of Hybrid Systems[END_REF]Lemma 2.20], V is non-pathological, and thus [START_REF] Rossa | Non-Smooth Lyapunov Functions for Stability Analysis of Hybrid Systems[END_REF]Lemma 2.23] ensures that d dt V (x(t)) ∈ V F (x(t)) for almost all t ∈ dom x. Hence, in view of (18) in Proposition 1, we have that V (x(t)) ≤ -α 3 (V (x(t))), for almost all t ∈ dom x.

By following the steps of the proof of [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF]Lemma A.4], we have that dom x = R ≥0 and there exists β ∈ KL (independent of x) such that

V (x(t)) ≤ β(V (x(0)), t), ∀t ∈ R ≥0 . (23)
Equations ( 17) and [START_REF] Rossa | Non-Smooth Lyapunov Functions for Stability Analysis of Hybrid Systems[END_REF]

imply |x(t)| ≤ α -1 1 (V (x(t))) ≤ α -1 1 (β(α 2 (|x(0)|), t)) =: β(|x(0)|, t) for any t ∈ R ≥0 , with β ∈ KL, thus concluding the proof.
■ With the help of Theorem 1, we can now establish that the origin of the system in Example 1 is GAS.

Example 2

The system in Example 1 satisfies both Assumptions 1 and 2 with the given selections of Z, Γ and P . As a result x = 0 2 is GAS in view of Theorem 1. It is instructive to see how the notion of set-valued Lie derivative helps overcoming the issue highlighted in Example 1. In particular, the setvalued Lie derivative of V with respect to F at x = (0, 1 2 ) is the empty set. Indeed, for each f ∈ F (0, 1 2 ) and any two different directions v 1 , v 2 ∈ ∂V (0, 1 2 ) with v 1 ̸ = v 2 , we have ⟨f, v 1 ⟩ ̸ = ⟨f, v 2 ⟩, thus there exists no a ∈ R satisfying the condition in [START_REF] Goebel | Hybrid Dynamical Systems: Modeling, Stability, and Robustness[END_REF]. More specifically, given F (0, 1 2

) = -3 2 , -1 4 × -1 and ∂V (0, 1 2 ) = [-1 4 , 1] × 1 2 , by selecting v 1 , v 2 ∈ ∂V (0, 1 2 ) with v 1 ̸ = v 2 , and f ∈ F (0, 1 2 ) we have that ⟨f, v 1 ⟩ = f 1 v 1,1 -1 4 and ⟨f, v 2 ⟩ = f 1 v 2,1 -1 4 with f 1 ∈ -3 2 , -1 4 and v 1,1 ̸ = v 2,1 ∈ [-1 4 , 1]. Therefore, ⟨f, v 1 ⟩ = ⟨f, v 2 ⟩ if and only if f 1 (v 1,1 -v 2,1 )
= 0, which is impossible for the specified selection of f , v 1 and v 2 . Hence, there exists no a ∈ R and f ∈ F (0, 1 2 ) such that ⟨f, v⟩ = a for all v ∈ ∂V (0, 1 2 ), thus implying that V F (0, 1 2 ) = ∅. Besides this specific illustrative analysis, by exploiting Lemma 1 we may actually show that sup V F (x) < 0 for all x ∈ R 2 \ {0 2 }. Indeed, we have that sup

V F (x) ≤ sup u∈-Ψ(x1) (-x 2 1 + 2ux 1 - x 2 2 + ux 2 -u 2 ) = sup u∈-Ψ(x1) -x 2 1 + 2ux 1 -1 2 x 2 -u 2 -3 4 x 2 2 < 0, because
Assumption 1 implies ux 1 < 0. We, therefore, obtain that the supremum of the set-valued Lie derivative of V with respect to F (x) is strictly negative outside the origin, which was not possible to prove using the conservative upper bound [START_REF] Doris | Output-feedback design for non-smooth mechanical systems: Control synthesis and experiments[END_REF]. □

We explain in the next remark why Σ 2 in ( 7) is passive.

Remark 1 System Σ 2 is passive from y to -u as discussed at the end of Section III. Let g as in ( 7), we have that, by Lemma 8 in [START_REF] Mariano | Hybrid coupling rules for leaderless heterogeneous oscillators: uniform global asymptotic and finite-time synchronization[END_REF] sup U 2,g (y) ≤ sup u∈-Ψ(y)

(u ⊤ (y + Zu) -u ⊤ y), ∀y ∈ R p . (24)
Let y be a solution of [START_REF] Bartolini | A survey of applications of second-order sliding mode control to mechanical systems[END_REF] -u ⊤ y(t), for almost all t ∈ dom y.

Hence, we obtain U 2 (y(t)) ≤ U 2 (y(0)) + t 0 -u(s) ⊤ y(s) ds for all t ≥ 0 with t ∈ dom y by integrating [START_REF] Sepulchre | Constructive nonlinear control[END_REF]. Thus Σ 2 is passive from y to -u as per Definitions 2.1 and 2.2 in [START_REF] Sepulchre | Constructive nonlinear control[END_REF]. Notice that, even if Definitions 2.1 and 2.2 are stated for singlevalued outputs, we can apply these same definitions without loss of generality to our case where u ∈ -Ψ(y). □

We conclude this section with a discussion about how the conditions of Theorem 1 can be extended to.

Extension under special properties of plant (1)

Since the conditions in Theorem 1 are only sufficient, we may prove that the origin is GAS for system (4) via Lyapunov analysis by exploiting additional structural properties of matrices A, B and C in (4). A set of alternative exploitable properties for system (4) is given next.

Property 1

The following holds for system (4).

(i) Assumption 1 is satisfied.

(ii) There exist matrices Γ > 0 diagonal, P = P ⊤ > 0 and a scalar η > 0 such that

M := P A + A ⊤ P + ηI n P B B ⊤ P -2Z -ΓCB -(ΓCB) ⊤ ≤ 0. ( 26
)
(iii) There exist H := diag(h 1 , . . . , h p ) such that ΓCA = HC and, for all i ∈ {1, . . . , p}, either h i ≤ -1 holds, or h i ≤ 0 and Z = O p holds, with Z in [START_REF] Bartolini | A survey of applications of second-order sliding mode control to mechanical systems[END_REF]. □

The conditions in items (ii) and (iii) in Property 1 impose extra properties of the matrices C and A (item (ii)) and a different matrix inequality compared to (9) (item (iii)), indeed as the off-diagonal terms of M differ from those in M in [START_REF] Brogliato | Dynamical systems coupled with monotone set-valued operators: Formalisms, applications, well-posedness, and stability[END_REF]. We show in the next lemma that Property 1 implies GAS of the origin for system (4). We will invoke this extension in Section 6 to analyze the stability properties of the neural networks studied in [START_REF] Forti | Global convergence of neural networks with discontinuous neuron activations[END_REF].

Lemma 2 Suppose that system (4) satisfies items (i)-(iii) of Property 1. Then the origin is GAS for system (4). □

Proof: Let x ∈ R n and consider V in [START_REF] Clarke | Optimization and Nonsmooth Analysis[END_REF]. We have from Lemma 1 that, for any

x ∈ R n , sup V F (x) ≤ sup u∈-Ψ(Cx) (x ⊤ P -u ⊤ ΓC)(Ax + Bu) -u ⊤ Zu - η 2 |x| 2 + u ⊤ Zu + η 2 |x| 2 . ( 27
)
with Z as in [START_REF] Bartolini | A survey of applications of second-order sliding mode control to mechanical systems[END_REF]. Therefore,

sup V F (x) ≤ sup u∈-Ψ(Cx) 1 2 x u ⊤ M x u + u ⊤ (Zu -HCx) - η 2 |x| 2 . ( 28
)
We note that, in view of Assumption 1, it holds that u ⊤ (Zu -HCx) ≤ 0 for all u ∈ -Ψ(Cx) and any x ∈ R. Indeed, because each entry of Ψ(y) = Ψ(Cx) is convex for any x ∈ R n , by (2) it holds that α i u i y i ≤ -αi ζi u 2 i , and u i y i + 1 ζi u 2 i +α i u i y i -α i u i y i ≤ 0, for any α i ≥ 0, u ∈ -Ψ(Cx) and i ∈ {1, . . . , p}. Therefore, we have

1 ζi u 2 i + (1 + α i )u i y i ≤ α i u i y i ≤ -αi ζi u 2 i ≤ 0. When h i ≤ -1, taking α i = -1-h i ≥ 0,
we deduce that, for any u ∈ -Ψ(Cx) and i ∈ {1, . . . , p}, 1 ζi u 2 i -h i u i y i ≤ 0 and thus u ⊤ (Zu -HCx) ≤ 0 for all u ∈ -Ψ(Cx). In the particular case where Z = O p , -u ⊤ HCx ≤ 0 is true for any negative semidefinite matrix diagonal H by ( 3), for all u ∈ -Ψ(Cx) and i ∈ {1, . . . , p}. Moreover, we assumed M ≤ 0 in item (ii) of Lemma 2. Therefore, similar to [START_REF] Polyakov | Finite-time and fixed-time stabilization: Implicit Lyapunov function approach[END_REF], from [START_REF] Soykens | Lur'e systems with multilayer perceptron and recurrent neural networks: absolute stability and dissipativity[END_REF] we have

sup V F (x) ≤ - η 2 |x| 2 ≤ - η 2 (α -1 2 (V (x))) 2 =: -α 3 (V (x)), (29)
with α 3 ∈ K ∞ . Then, as anticipated, by exploiting ( 17) and ( 29), and following similar steps of those in the proof of Theorem 1, we conclude that the origin is GAS for system (4). ■

5 Finite-time stability

Definitions and assumptions

In this section, we provide conditions to guarantee output and state finite-time stability properties for system [START_REF] Aubin | Differential Inclusions: Set-valued Maps and Viability Theory[END_REF]. In particular, we consider the next stability notions, see [START_REF] Sontag | Lyapunov characterizations of input to output stability[END_REF][START_REF] Zimenko | On necessary and sufficient conditions for output finite-time stability[END_REF]. Definition 1 Consider system (4). If its solutions are all forward complete3 , then we say that the system is:

(i) output globally asymptotically stable (oGAS) if there exists β ∈ KL such that for any solution x

|y(t)| ≤ β(|x(0)|, t), ∀t ∈ R ≥0 ;
(ii) state-independent output locally asymptotically stable (SIoLAS) if there exist r > 0 and β ∈ KL such that for all solution x,

|x(0)| < r ⇒ |y(t)| ≤ β(|y(0)|, t), ∀t ∈ R ≥0 ;
(iii) output finite-time stable (OFTS) if it is oGAS and for each solution x there exists 0 ≤ T < +∞ such that y(t) = 0 p for all t ≥ T ;

(iv) state finite-time stable (SFTS) if the origin is GAS and for each solution x there exists 0 ≤ T < +∞ such that x(t) = 0 n for all t ≥ T . □

To be able to prove the output stability properties in Definition 1, we make the next assumption. (ii) The origin is GAS for system (4).

(iii) Each ψ i , with i ∈ {1, . . . , p}, is discontinuous at the origin and both its left and right limits are non-zero, i.e., for any i ∈ {1, . . . , p} lim s→0 + ψ i (s) > 0 and lim

s→0 - ψ i (s) < 0. □ Item (i)
of Assumption 3 imposes extra conditions on the matrices C and B of system [START_REF] Acary | Numerical Methods for Nonsmooth Dynamical Systems: Applications in Mechanics and Electronics[END_REF]. Sufficient conditions to ensure item (ii) of Assumption 3 are provided in Theorem 1 and Lemma 2. Finally, item (iii) of Assumption 3 requires each ψ i , i ∈ {1, . . . , p}, to be non-zero at the origin and to have non-zero left and right limit at zero as well. Examples of engineering systems satisfying Assumption 3 (as well as Assumptions 1 and 2) are provided in Section 6.

Output and state finite-time stability

We are now ready to present the main result of this section, whose proof is given in Section 5.3. Theorem 2 Consider system (4) and suppose that Assumptions 1 and 3 hold, then system (4) is OFTS and SIoLAS. □ Theorem 2 establishes output finite-time stability properties for system (4). A natural question is then whether state finite-time stability properties can also be guaranteed. An answer to this question is given in the next theorem which establishes that, whenever Assumptions 1 and 3 are satisfied, system (4) is SFTS if and only if C is invertible. Theorem 3 Consider system (4) and suppose that Assumptions 1 and 3 are verified. Then the system is SFTS if and only if matrix C is inveritble. □ Proof: We start by proving that there exists ε > 0 such that, for any ξ ∈ ker(C)∩εB n , u = -(CB) -1 CAξ belongs to Ψ(0 p ) and CAξ +CBu = 0 p . First, note that CB is invertible as it is LDS by item (ii) of Assumption 3. Hence, for any ξ ∈ ker(C) ∩ εB n , u = -(CB) -1 CAξ is well-defined and CAξ + CBu = 0 p . Secondly, in view of item (iii) of Assumption 3 there exists

ψ • ∈ R >0 such that [-ψ • , ψ • ] p ⊆ Ψ(0 p ).
Therefore, there exists ε > 0 such that, for any ξ ∈ ker(C) ∩ εB n and any i ∈ {1, . . . , p}, |((CB)

-1 CA) i ξ| ≤ ψ • , thus implying u = -(CB) -1 CAξ ∈ [-ψ o , ψ o ] p ⊆ Ψ(0 p
), as to be proven. Now we are ready to prove the necessary and sufficient conditions of Theorem 3. The sufficient condition in Theorem 3 is a direct consequence of Theorem 2. We proceed by contradiction to prove the necessary condition in Theorem 3. We thus assume that C is not invertible and consider ε > 0 as at the beginning of this proof. Since for any x ∈ ker(C) ∩ εB n we can select u = -(CB) -1 CAx that belongs to Ψ(0 p ), we consider below solutions to ( 4)

satisfying ẋ = Ax -B(CB) -1 CAx, x ∈ ker(C) ∩ εB n , (30) which implies ẏ
= C ẋ = (CA -CB(CB) -1 CA)x = 0 p , x ∈ ker(C) ∩ εB n . (31)
We now exploit [START_REF] Vasca | A new perspective for modeling power electronics converters: Complementarity framework[END_REF] to attain a contradiction. By item (ii) of Assumption 3, there exists δ > 0 such that any solution starting in δB n does not leave εB n for all times. Let x p be a nonzero solution starting in ker(C)∩δB n , with output y p = Cx p , which evolves according to [START_REF] Valadier | Entraînement unilatéral, lignes de descente, fonctions Lipschitziennes non pathologiques[END_REF] and [START_REF] Vasca | A new perspective for modeling power electronics converters: Complementarity framework[END_REF]. Then y p (0) = Cx p (0) = 0 p and equation [START_REF] Vasca | A new perspective for modeling power electronics converters: Complementarity framework[END_REF] imply y p (t) = Cx p (t) = 0 p and ẋp (t) = (A-B(CB) -1 CA)x p (t) ̸ = 0 for all t ≥ 0. As a consequence, x p exponentially converges to the origin but does not converge in finite-time. Such a solution establishes a contradiction, thus completing the proof. ■ We can now analyze the finite-time stability property of the system in Example 1 in light of Theorems 2 and 3.

Example 3 Consider the system in Example 1. Assumption 3 holds with Γ = 1. As a result, the system is OFTS and SIoLAS. We also know from Theorem 2 that the system is not SFTS as C is not invertible. Another way to see it is to consider x(0) ∈ X := {0} × [-1 4 , 1 4 ]. A possible solution to (4) is x p (t) = (0, x 2 (0)e -t ), which belongs to the set X for all t ≥ 0. Moreover, we have that y p (t) = 0 and ẏp (t) = 0 for all t ≥ 0. Clearly, x p converges exponentially to the origin, but not in finite-time. □

Proof of Theorem 2

The proof of Theorem 2 relies on the next lemma and proposition. We also invoke the next proposition, which states algebraic properties of a piecewise continuously differentiable function, which is similar to the one in ( 10)

W (Cx) := 2 p i=1 γ i Cix 0 ψ i (σ)dσ, ∀x ∈ R n , (33)
where γ , . . . , γ p > 0 are positve parameters selected such that ΓCB+(CB) ⊤ Γ > 0, with Γ = diag(γ 1 , . . . , γ p ), which exist by item (i) of Assumption 3. Function W enjoys the following properties.

Proposition 2 Suppose that Assumption 1 and items (i) and (iii) of Assumption 3 hold. Given function W in [START_REF] Vorotnikov | Partial stability, stabilization and control: some recent results[END_REF], there exist µ ∈ (0, ν], with ν as in Lemma 3, and α 4 , α 5 ∈ K ∞ such that

α 4 (|Cx|) ≤ W (Cx) ≤ α 5 (|Cx|), ∀x ∈ µB n , (34)
sup Ẇ F (Cx) ≤ -cω, ∀x ∈ µB n \ ker(C), [START_REF] Zimenko | On necessary and sufficient conditions for output finite-time stability[END_REF] with c as in Lemma 3, ω := λ 1 (c -2µ λ2 λ1 ) > 0, λ 1 is the smallest eigenvalue of ΓCB + (CB) ⊤ Γ, and λ 2 := |ΓCA|. □ Proof: From (33) and Lemma 3, for any x ∈ µB n \ ker(C), W (Cx) > 0 while W (Cx) = 0 for any x ∈ ker(C) ∩ µB n . Moreover, we have that W is continuous on µB n . Therefore, [START_REF] Yakubovich | Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities[END_REF] holds in view of [START_REF] Khalil | Nonlinear Systems[END_REF]Lemma 4.3]. Let x ∈ µB n \ ker(C), from Lemma 1, by imposing P = 0 and Γ = Γ in ( 16), we have

sup Ẇ F (Cx) ≤ sup u∈-Ψ(Cx) (-2u ⊤ ΓC(Ax + Bu)). (36)
Using the Cauchy-Schwarz inequality, we obtain

sup Ẇ F (Cx) ≤ sup u∈-Ψ(Cx) (-u ⊤ (ΓCB + (CB) ⊤ Γ)u + 2|ΓCA||x||u|). (37)
Thus, in view of item (i) of Assumption 3, we have that We are now ready to prove Theorem 2. To prove the OFTS property of system (4), we proceed by steps. We first show that, for solutions to (4) initialized in a neighborhood of the origin, the corresponding output converges to the origin in finite-time and then, leveraging the GAS property of the origin for (4), we prove OFTS of (4). Proof of Theorem 2. We start by proving that solutions initialized sufficiently close to the origin converge to ker(C) in finite time by integrating [START_REF] Zimenko | On necessary and sufficient conditions for output finite-time stability[END_REF]. To do so, we recall that, by the GAS property of the origin, there exists κ > 0 such that solutions starting in κB will not leave µB, with µ as in Proposition 2 and we note that the set µB n ∩ ker(C) is forward invariant for any solution starting κB n ∩ ker(C). Indeed, suppose that there exists a solution x bad to (4) such that x bad (0) ∈ κB n ∩ ker(C) and x bad (t * ) / ∈ µB n ∩ ker(C) for some t * > 0 with t * ∈ dom x bad . Since x bad is continuous with respect to the time, we can choose t * > 0 such that x bad (t) ∈ κB n ∩ ker(C) for all t ∈ [0, t * ) and x bad (t * ) ∈ µB n \ ker(C). Hence, from [START_REF] Vorotnikov | Partial stability, stabilization and control: some recent results[END_REF] and [START_REF] Zimenko | On necessary and sufficient conditions for output finite-time stability[END_REF], and from the fact that W is positive definite on µB n and non-pathological, we have 0 = W (Cx bad (t)) < W (Cx bad (t * )), for all t ∈ [0, t * ), which establishes a contradiction by the continuity property of W . Consequently, solutions cannot leave µB n ∩ ker(C) after reaching the set κB n ∩ ker(C). Therefore, by combining [START_REF] Zimenko | On necessary and sufficient conditions for output finite-time stability[END_REF] with the fact that W is nonpathological, and the forward invariance of µB ∩ ker(C) for solutions starting in κB ∩ ker(C), for any solution x initialized so that x(0) ∈ κB n \ ker(C), we obtain by integration for any t ∈ dom x such that x(t) ∈ µB n \ ker(C)

sup Ẇ F (x) ≤ sup u∈-Ψ(Cx) (-λ 1 |u| 2 + 2|ΓCA||x||u|), = sup u∈-Ψ(Cx) (-(λ 1 |u| -2|ΓCA||x|)|u|), ≤ sup u∈-Ψ(Cx) -λ 1 |u| -2 λ 2 λ 1 |x| |u| . ( 38
W (Cx(t)) ≤ -cωt + W (Cx(0)), (39)
and thus

W (Cx(t)) ≤ max(-cωt + W (Cx(0)), 0), ∀x(0) ∈ κB n , ∀t ∈ R ≥0 . (40)
Thus, in view of (40) and by the GAS property of the origin, we conclude that, for any solutions starting in κB, there exists a T y , depending on κ, such that x(t) ∈ κB ∩ ker(C) for any t ≥ T y . We now leverage the GAS property of the origin to prove that (4) is OFTS. We recall that, for any solution x to (4), by the GAS property of the origin there exists a time T κ ≥ 0 such that x(t) ∈ κB for all t ≥ T κ . Therefore, we conclude that y(t) = 0 p for all t ≥ T := T κ + T y .

(iv) For any i ∈ {1, . . . , n}, function ψ i is nondecreasing, i.e., for any a > b ∈ dom ψ i it holds that ψ i (a) ≥ ψ i (b), is piecewise continuous and satisfies Assumption 1 with ζ i = +∞ and item (iii) of Assumption 3. □ Property 2 trivially implies Assumption 1 and items (i) and (iii) of Assumption 3. We show below that it also implies item (ii) of Assumption 3 so that we can invoke Theorems 1 and 2 to prove GAS of the origin for system (4) and that system (4) is SFTS, thus providing alternative proofs of the stability results given in [START_REF] Forti | Global convergence of neural networks with discontinuous neuron activations[END_REF]Thm. 3 and 4]. Indeed, we recall that, by proving stability properties for system (4), we ensure the same stability properties for the Krasovskii solutions of (1).

Lemma 4 Suppose that system (1) satisfies Property 2. Then the origin is GAS for system (4), and system (4) is SFTS. □ Proof: We prove below that there exist matrices Γ > 0 diagonal, P = P ⊤ > 0 and a scalar η > 0 satisfying [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF]. Since B is LDS, there exists a Γ > 0 diagonal such that ΓB + (ΓB) ⊤ =: Σ > 0 and such that ΓA ≤ -I n . With this selection, we can rewrite matrix M in (26) as,

M = P A + A ⊤ P P B B ⊤ P -Σ + diag(ηI n , 0 n ), (44)
noting that Z is the null matrix due to item (iv) of Property 2. Define where S = P -1 . Since A is Hurwitz by item (i) of Property 2, there exists S • = S ⊤ • > 0 such that S • A ⊤ + AS • = Π < 0. Therefore, by selecting S = αS • with α > 0, to be chosen, we have that

N = αΠ B B ⊤ -Σ < 0 ∀α > α ⋆ , (45)
where α ⋆ > 0 satisfies -α ⋆ λ Π > |BΣB ⊤ |, with λ Π > 0 denoting the smallest eigenvalue of Π. Hence, with the given selection of α and P , matrix N and thus M are negative definite. Therefore, by selecting 0 < η < -| M | we have that M ≤ 0 thus proving [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF] and item (ii) of Property 1. Consider now item (iii) of Property 1 and note that matrix H = ΓA < 0 is diagonal negative definite and satisfies ΓCA = ΓA = H = HC. Since Z = O p due to item (iv) of Property 2, then item (iii) of Property 1 holds and we can invoke Lemma 2 to certify that the origin is GAS for system (4) and render Property 2. Furthermore, since Assumptions 1 and 3 hold, then, by Theorem 3, system (4) is also SFTS because C is invertible. ■ We envision applying our results to a broader class of neural networks with piecewise continuous activation functions. Due to the short length of a technical note submission, we do not pursue such generalizations here and we regard them as future work.

Conclusion

We have analyzed the stability of the origin for Lur'e systems with piecewise continuous nonlinearities. We have first established the global asymptotic stability of the origin under a milder sector condition compared to [START_REF] De Bruin | Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities[END_REF] and by relying on a different, algebraic Lyapunov proof based on the concept of set-valued Lie derivative. We have then presented conditions under which finite-time stability properties can or cannot be established for the considered class of systems. These results have been applied to two engineering systems of interest: mechanical systems with friction and cellular neural networks.

Future research directions may include: systems affected by exogenous disturbances; weak stability analysis for the considered class of systems in the sense that only some solutions exhibit the desired stability properties; as well as the synchronization of interconnected Lur'e systems with piecewise continuous nonlinearities following the path of paved by [START_REF] Brogliato | Observer design for Lur'e systems with multivalued mappings: A passivity approach[END_REF][START_REF] Tang | Finite-time cluster synchronization of Lur'e networks: A nonsmooth approach[END_REF].

) 1 1 iFigure 1 :

 111 Figure 1: System (4) as the feedback interconnection of systems Σ 1 and Σ 2 .

Assumption 3

 3 The following holds. (i) Matrix CB is Lyapunov diagonally stable (LDS) [17, Def. 5.3], i.e., there exists a diagonal matrix Γ > 0 of appropriate dimensions such that ΓCB + (CB) ⊤ Γ > 0.

Lemma 3

 3 Under Assumption 1 and item (iii) of Assumption 3, there exist ν > 0 and c > 0, such that |u| ≥ c, ∀u ∈ -Ψ(y), ∀y ∈ νB p \ {0 p }. (32) □ Proof: In view of item (iii) of Assumption 3, there exist positive parameters ν • and c such that, for each i ∈ {1, . . . , p}, ψ i is continuous in the intervals [-ν • , 0) and (0, ν • ], and min(| lim s→0 + ψ i (s)|, | lim s→0 - ψ i (s)|) ≥ 2c. Hence, there exists ν ∈ (0, ν • ] such that, for any i ∈ {1, . . . , p} and s ∈ [-ν, 0) ∪ (0, ν], |ψ i (s)| ≥ c. Therefore, we have that for any y ∈ νB p \ {0 p } there exists i ∈ {1, . . . , p} such that |u| ≥ |u i | ≥ c for all u ∈ -Ψ(y) thus concluding the proof. ■

  ) Hence, in view of Lemma 3, by selecting µ ∈ (0, ν] we have that sup Ẇ F (Cx) ≤ sup u∈-Ψ(Cx) (-ω|u|) ≤ -cω, where ω = λ 1 c -2 λ2µ λ1 > 0, thus concluding the proof. ■

The Ryan's invariance principle[START_REF] Ryan | An integral invariance principle for differential inclusions with applications in adaptive control[END_REF] is also not applicable to guarantee asymptotic stability of the origin for this example.

A solution is forward complete if its domain is unbounded[START_REF] Angeli | Forward completeness, unboundedness observability, and their Lyapunov characterizations[END_REF].

* Work supported by the ANR under grant HANDY ANR-18-CE40-0010.

We have proved that, for any solution x, there exists T ≥ 0 such that y(t) = 0 p , for all t ≥ T . Moreover, system (4) is oGAS because it is GAS from item (i) of Assumption 3 and because |y| ≤ |C||x|. Therefore, system (4) is OFTS.

Finally, we prove that system (4) is also SIoLAS. Indeed, combining (34) and (40) yields, for any solution x with x(0) ∈ κB,

with β • ∈ KL, thus ending the proof. ■

Applications

In this section, we present two applications of the results of Sections 4 and 5.

6.1 Mechanical system affected by friction [START_REF] De Bruin | Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities[END_REF] Consider the rotor dynamic system with friction system given in [10, Sec. 5], i.e.,

with x = (α, ω u , ω ℓ ) ∈ R 3 , where α is the angular mismatch between two rotating discs connected by an angular spring and an angular dumper, and ω u and ω ℓ are the angular velocities of these two discs. Scalars J u , J ℓ , k u , k θ and b are positive system parameters whose values are reported in Table 6.1. The control input v ∈ R is used for state-feedback stabilization, while the set-valued maps T f u and T f ℓ in (42) are defined as

otherwise,

for suitable positive scalars f u,• , f ℓ,• , ∆f u , ∆f ℓ , q 1 , q 2 , q 3 and q 4 we give in Table 6.1 and with function sign : R → [-1, 1] defined as sign(s) = 1 if s > 0, sign(s) = -1 if s < 0, and sign(s) = 0 if s = 0, and for which we have that ( 2) is satisfied with

Like in [START_REF] Doris | Output-feedback design for non-smooth mechanical systems: Control synthesis and experiments[END_REF]Ch. 6], by considering the selection v = v p + v lin in (42), where

[kg m 2 /rad s] -0.0084 q 3

[s/rad] 0.05 q 4

[kg m 2 /rad s] 0.26

Table 1: Parameters identifying the system given in [START_REF] De Bruin | Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities[END_REF]Sec. 5].

which can be written in the Lur'e form (4), with n = 3 and p = 1, and

Assumption Since Assumption 1 is also satisfied, Theorem 1 implies that the origin is GAS for (4), thus retrieving the result originally presented in [START_REF] De Bruin | Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities[END_REF]. In addition, because Assumption 3 holds for the considered system, we establish here, from Theorem 2, that system (4) is OFTS and SIoLAS, which is a novelty compared to [START_REF] De Bruin | Control of mechanical motion systems with non-collocation of actuation and friction: A Popov criterion approach for input-to-state stability and set-valued nonlinearities[END_REF].

Cellular neural networks from [14]

In [START_REF] Forti | Global convergence of neural networks with discontinuous neuron activations[END_REF], cellular neural networks are modeled by system (1) (see [14, eq. (N1)-(N2)]), where the system data satisfies the next property according to [START_REF] Forti | Global convergence of neural networks with discontinuous neuron activations[END_REF]Prop. 3 and 4].

Property 2

The following holds for system (1).

(i) A is a diagonal, negative definite matrix.

(ii) B is LDS (as per Assumption 3).

(iii) C = I n .