Emmanuel Bernuau

Emmanuel Moulay

Patrick Coirault

Stability of discontinuous homogeneous nonlinear sampled-data systems

Keywords: Sampled-data systems, Differential inclusions, Homogeneity, Lyapunov stability, Lyapunov functions

come

Introduction

The problem of stability for systems with sampled-data inputs has been renewed with the development of networked control systems [START_REF] Zhang | Stability of networked control systems[END_REF]. Due, for instance, to wireless communications, the quality of the channel between the controller and the actuator can be insufficient to ensure a quick updating of the feedback control. This leads to the problem of knowing if the stability properties pertain through sampled-data inputs [START_REF] Walsh | Stability analysis of networked control systems[END_REF]. Lot of works have been done to solve this problem such as the input/output approach [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF] or the sum of squares approach [START_REF] Bauer | Stability analysis of networked control systems: A sum of squares approach[END_REF]. Moreover, several different modeling alternatives as hybrid systems [START_REF] Zhang | Stability of networked control systems[END_REF][START_REF] Donkers | Stability analysis of networked control systems using a switched linear systems approach[END_REF], discrete-time systems [START_REF] Nešić | A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models[END_REF] or time-delay systems [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF][START_REF] Liu | Networked control systems: A time-delay approach[END_REF][START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF][START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF] have also been used to tackle this problem. Most of these strategies [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF][START_REF] Bauer | Stability analysis of networked control systems: A sum of squares approach[END_REF][START_REF] Omran | On the stability of input-affine nonlinear systems with sampled-data control[END_REF] are developed under the emulation method where the controller is first designed in continuous time and then implemented as a sampleddata controller [START_REF] Bemporad | Networked control systems[END_REF][START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF].

The notion of homogeneity was introduced in [START_REF] Zubov | On ordinary differential equations with generalized homogeneous right-hand sides (in Russian)[END_REF][START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF] ⋆ This paper was not presented at any IFAC meeting. Corresponding author E. Bernuau. Tel. +33 (0)1 69 93 50 83; Fax +33 (0)1 69 93 50 20.

Email addresses: emmanuel.bernuau@agroparistech.fr (Emmanuel Bernuau), emmanuel.moulay@univ-poitiers.fr (Emmanuel Moulay), patrick.coirault@univ-poitiers.fr (Patrick Coirault).

and developed for instance in [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF][START_REF] Kawski | Geometric homogeneity and stabilization[END_REF][START_REF] Moulay | Stability and stabilization of homogeneous systems depending on a parameter[END_REF]. Moreover, stability theorems for perturbed homogeneous systems were first studied in [START_REF] Hermes | Nilpotent and high-order approximations of vector field systems[END_REF]. Recently, we have developed in [START_REF] Bernuau | Stability of homogeneous nonlinear systems with sampled-data inputs[END_REF] a new nonlinear emulation strategy based on homogeneity which uses the input-to-state stability (ISS) properties of homogeneous systems in order to preserve their stability properties when they are subject to aperiodic sampled-data inputs. This strategy has been developed in the framework of continuous controllers but we can wonder if it still works in the case of discontinuous controllers. Homogeneous properties of discontinuous systems have been developed for instance in [START_REF] Orlov | Discontinuous systems: Lyapunov analysis and robust synthesis under uncertainty conditions[END_REF]. Recently, the ISS properties of homogeneous systems have been extended to discontinuous systems in [START_REF] Bernuau | On an extension of homogeneity notion for differential inclusions[END_REF]. Some papers have also stated robustness properties of discontinuous homogeneous systems (see for instance [START_REF] Bernuau | Robustness of homogeneous and homogeneizable differential inclusions[END_REF]). However, no robustness condition allowed us to assess stability properties for emulated discontinuous homogeneous systems.

To highlight the contribution of this paper, we would like to point out that, although systems controlled by discontinuous control laws are well known to present robustness properties with respect to exogenous perturbations (like external forces acting on the system), it is also well known that, when subject to endogenous perturbations (like sampling), these systems may present bad behaviors, chattering being the most famous. To understand the difference, let us present a simple scalar ex-ample showing the difficulties appearing when the continuity is lost. Consider the system ẋ = -⌊x + δ⌉ α , with x ∈ R and α ≥ 0, where δ is an unknown disturbance and ⌊z⌉ α = sign(z)|z| α . A commonly used technique consists in writing ⌊x + δ⌉ α = ⌊x⌉ α + (⌊x + δ⌉ α -⌊x⌉ α ) which gives the effect of the nominal system -⌊x⌉ α , added to a perturbative term ∆ = ⌊x + δ⌉ α -⌊x⌉ α . We would like the perturbative part to be dominated by the nominal part in order to bring the system to 0. When α > 0 the system is continuous and we can show that |∆| ≤ 2|δ| α , which implies that ∆ → 0 when δ → 0 uniformly in x. When α = 0 we reach the discontinuous case and then ∆ = 0 when |δ| < |x| and ±2 elsewhere. We still have ∆ → 0 when δ → 0 but this convergence is not uniform anymore, leading to a cascade of technical problems to prove the stability. Obviously, in this very specific and very simple example, solutions can be found by hand; but in the general case we need specific techniques to address the discontinuous case.

In this paper, we prove that, under well chosen technical assumptions, discontinuous homogeneous systems behave nicely when subject to endogenous perturbations, in particular samplings. The difficulty was to find a condition, namely Assumption 15, that would then allow us to use the same techniques than in [START_REF] Bernuau | Stability of homogeneous nonlinear systems with sampled-data inputs[END_REF], while not being too constraining to apply to a large set of systems. Moreover, the results fall into the framework of sliding mode control which has been used for sampled-data systems in [START_REF] Han | Sampled-data sliding mode observer for robust fault reconstruction: A time-delay approach[END_REF][START_REF] Nguyen | Output feedback sliding mode control for sampled-data systems[END_REF][START_REF] Young | A control engineer's guide to sliding mode control[END_REF]. Homogeneous sliding mode controllers has been proposed recently in [START_REF] Cruz-Zavala | Homogeneous high order sliding mode design: A Lyapunov approach[END_REF].

The article is organized as follows. After some preliminaries stated in Section 2, we develop a new result about the robustness of homogeneous systems in Section 3. Then we use this result in the context of homogeneous sampled-data systems for addressing the main result of the article in Section 4. A numerical application to the double integrator is given Section 5. Concluding remarks appears in Section 6.

Preliminaries

Mathematical notation

Throughout the paper the following notation is used:

• R + = {x ∈ R : x ≥ 0},

Differential inclusions

In this subsection, we recall the basic concepts of differential inclusions and the Filippov procedure. For details and proofs, see [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides[END_REF].

Definition 1 A set-valued map F : R n ⇒ R n is upper semicontinuous (USC) if for any x ∈ R n and any neighborhood V of F (x), there exists a neighborhood U of x such that for any y ∈ U, F (y) ⊂ V.

Definition 2 Let F : R n ⇒ R n be a set-valued map. We say that F verifies the standard assumptions if F is USC and if for any x ∈ R n , F (x) is a nonempty compact convex set.

Definition 3 Let g : R n × R m → R n be locally essentially bounded. We denote F x [g](x, ∆) the set

ε>0 N ∈N conv{g(y, ∆) : y ∈ B(x, ε) \ N }, (1)
where N denotes the set of all Lebesgue-negligible subsets of R n . The set-valued map

F x [g] : (x, ∆) → F x [g](x, ∆)
is called the Filippov's regularization of g.

Proposition 4 Let g : R n × R m → R n be locally essentially bounded. For any ∆ ∈ R m , the set-valued map

G ∆ : R n ⇒ R n defined by G ∆ (x) = F x [g](x, ∆
) verifies the standard assumptions.

We refer to [START_REF] Filippov | Differential Equations with Discontinuous Righthand Sides[END_REF] for the definitions of a differential inclusion and the corresponding notions of solutions, convergence and stability.

Input-to-State Stability

Let us consider the following nonlinear system:

ẋ = f (x, d), (2)
where x ∈ R n is the state, d ∈ L ∞ loc is the external input and f : R n+m → R n is continuous. We will also denote

f d (x) = f (x, d) and hence f 0 (x) = f (x, 0).
Let us introduce the definition of input-to-state stability given for instance in [START_REF] Sontag | On characterizations of the inputto-state stability property[END_REF].

Definition 5

The system ( 2) is called input-to-state stable (ISS), if for any input d ∈ L ∞ loc and any x 0 ∈ R n there exist some functions β ∈ KL and γ ∈ K such that for any solution x(t) of the equation ( 2) we have:

x(t) ≤ β( x(0) , t) + ess sup [0,t] γ( d ) ∀t ≥ 0.
The function γ is called a nonlinear asymptotic gain.

Homogeneity

Different notions of homogeneity appear in the litterature. The most common one is the weighted homogeneity, based on a particular choice of the coordinates, while the most generic one is the geometric homogeneity, which is coordinate free. We shall more use in the sequel the framework of geometric homogeneity than the framework of weighted homogeneity, so we will introduce the latter as a particular case of the former.

Definition 6 A vector field ν on R n is called an Euler vector field if ν is of class C 1 , complete (i.e. the maximal integral curves are defined on R) and if the origin is a globally asymptotically stable equilibrium of -ν.

Example 7 If we consider a matrix A ∈ R n×n such that -A is Hurwitz, the vector field defined by ν(x) = Ax is an Euler vector field. In particular, if A = Diag(r 1 , . . . , r n ) with r 1 , . . . , r n > 0, the vector field ν(x) = Ax is Euler.

Definition 8 Let ν be an Euler vector field on R n . We denote Φ s (x) the value of the flow of ν at time s with initial condition x.

• A function V : R n → R is ν-homogeneous of de- gree κ ∈ R if for all x ∈ R n and all s ∈ R we have V (Φ s (x)) = e κs V (x). • A vector field f on R n is ν-homogeneous of degree κ ∈ R if for all x ∈ R n and all s ∈ R we have f (Φ s (x)) = e κs d x Φ s f (x).
Remark 9 If we have ν(x) = Ax, with -A Hurwitz, the flow of ν verifies Φ s (x) = exp(As)x. In particular, if A = Diag(r 1 , . . . , r n ) with r 1 , . . . , r n > 0, we find Φ s (x) = Diag(e r1s , . . . , e rns )x. The homogeneity defined by such an Euler vector field has been the subject of a lot of works. It is usually referred to as weighted homogeneity, the coefficients r 1 , . . . , r n are called the weights and r = [r 1 , . . . , r n ] is called the generalized weight [START_REF] Bacciotti | Liapunov functions and stability in control theory[END_REF].

Let us finally mention that homogeneity with respect to an Euler vector field defined by a generalized weight r is usually simply referred to as r-homogeneity.

Equivalent conditions for homogeneity exist. Given that we will not need them below, we just refer to [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF][START_REF] Kawski | Geometric homogeneity and stabilization[END_REF] for a complete exposition of the homogeneity theory.

Let us now introduce a tool and two lemmas useful in the sequel.

Definition 10 Let ν be an Euler vector field on R n . A ν-homogeneous norm is a function N : R n → R such that:

(1) N is positive definite;

(2) N is ν-homogeneous of degree 1;

(3) N is continuous.

Lemma 11 Let V : R n → R be a continuous positive definite ν-homogeneous function of degree κ > 0. There exist σ -and σ + two functions of class

K ∞ such that for all x ∈ R n , we have σ -( x ) ≤ V (x) ≤ σ + ( x ).
Proof. Denote σ + (r) = sup x ≤r V (x). The function σ + is clearly continuous, increasing and verifies σ + (0) = 0. Let us show that σ + is strictly increasing. It is enough to prove that for any x 0 such that x 0 ≤ r and V (x 0 ) = σ + (r) verifies x 0 = r. Assume by contradiction that x 0 < r. By continuity of the flow Φ, there exists ε > 0 such that for all s ∈ [0, ε[, we have Φ s (x 0 ) < r and thus

V (Φ s (x 0 )) ≤ σ + (r) = V (x 0 ). But V (Φ s (x 0 )) = e κs V (x 0 ) > V (x 0 ) for s > 0, which is a contradiction. Hence we get that V (x) ≤ σ + ( x ). Now, given that, for x = 0, Φ s (x) → +∞ when s → +∞, we see that e κs V (x) = V (Φ s (x)) ≤ σ + ( Φ s (x)
) and finally by taking the limit in both sides we get lim r→+∞ σ

+ (r) = +∞, that is σ + ∈ K ∞ . The function σ -is defined by σ -1 -(r) = sup V (x)≤r x . We similarly prove that σ -1 -∈ K ∞ . Finally, V (x) ≤ sup y ≤ x V (y) = σ + ( x ) and x ≤ sup V (y)≤V (x) y = σ -1 -(V (x)), that is V (x) ≥ σ -( x ).
Lemma 12 [START_REF] Bernuau | Robustness of homogeneous and homogeneizable differential inclusions[END_REF] Let σ : R + → R be an increasing function such that lim x→0 + σ(x) = 0. Then there exists a class K function σ such that for all x ∈ R + , σ(x) ≤ σ(x).

Robustness of discontinuous homogeneous systems

We consider the following differential inclusion, depending on a parameter ∆:

ẋ ∈ F (x, ∆), x ∈ R n , ∆ ∈ R n . (3)
We make the following assumptions on the system (3).

Assumption 13

The origin is globally asymptotically stable for the system (3) with ∆ = 0.

Assumption 14 There exists κ < 0 and a matrix A ∈ R n×n such that -A is Hurwitz and such that for all x ∈ R n , all ∆ ∈ R n and all s ∈ R we have F (e As x, e As ∆) = e κs e As F (x, ∆).

Assumption 15 For all compact set X ⊂ R n , there exists σ X ∈ K such that for all x ∈ X and all

∆ ∈ R n F (x, ∆) ⊂ F (x + ∆, 0) + B(0, σ X ( ∆ )).
The following result is an extension to discontinuous systems of Theorem 11 in [START_REF] Bernuau | Robustness of homogeneous and locally homogeneous differential inclusions[END_REF].

Theorem 16 Under Assumptions 13, 14 and 15, the system (3) is ISS w.r.t. the input ∆. Moreover, if we denote ν(x) = Ax and if N is any ν-homogeneous norm, then there exists constants α > 0, b > 0 and C > 0 such that for any solution x(t) of (3):

N (x(t)) ≤ α max β N x(0) , t ; CN (∆) , (4)
where

β(r, t) = max{r -κ -bt, 0} -1 κ . (5)
Proof. Given Assumption 13, Theorem 3.22 in [START_REF] Bernuau | Robustness and stability of nonlinear systems: a homogeneous point of view[END_REF] gives us a Lyapunov pair (V, W ) of class C 1 , ν-homogeneous functions of degree µ > 0 and µ + κ > 0 respectively such that for all x = 0 we have V (x) > 0, W (x) > 0 and for all v ∈ F (z, 0) we have d x V v ≤ -W (x). Let x = 0 be fixed and, given ∆ ∈ R n , let us pick v ∈ F (x, ∆). We denote N = V 1/µ , S = {N = 1} and s = ln(N (x)), i.e. N (x) = e s . It yields that x = e -As x ∈ S. We define ∆ = e -As ∆ and v = e -κs e -As v. Assumption 14 gives us that v ∈ e -κs e -As F (x, ∆) = F (x, ∆). Hence we get d x V v = e µs d xV e -As e κs e As v = e (µ+κ)s d xV v. Following Assumption 15 with X = S, there exists w ∈ F (x + ∆, 0) and r ∈ B(0, σ S ( ∆ )) such that v = w + r. It yields

d x V v = e (µ+κ)s d x+ ∆V w + (d xV -d x+ ∆V )w + d xV r . (6) Now, defining K(ρ) = {̟ ∈ R n : ∃δ ∈ B(0, ρ), ∃z ∈ S, ̟ ∈ F (z+δ, 0)} and M (ρ) = sup{ ̟ : ̟ ∈ K(ρ)} we find that, since F (•, 0) is USC, M (
ρ) is finite for any ρ ≥ 0. Moreover, M is increasing. Upper bounding the three parts of the right-hand side of (6) yields

d x+ ∆V w ≤ -W (x + ∆) ≤ -inf K( ∆ ) W (d xV -d x+ ∆V )w ≤ M ( ∆ ) sup z∈S d z V -d z+ ∆V d xV r ≤ sup z∈S |||d z V |||σ S ( ∆ ))
For ∆ small enough, we have K( ∆ ) ⊂ R n \ {0} and hence inf K( ∆ ) W > 0. Moreover, ρ →inf K(ρ) W is increasing and therefore there exists a > 0 and

η 1 > 0 such that if ∆ < η 1 then -inf K( ∆ ) W < -a.
The continuity of dV shows that there exists

η 2 > 0 such that if ∆ < η 2 , then M ( ∆ ) sup z∈S d z V -d z+ ∆V < a 4 .
The continuity of σ S shows that there exists η 3 > 0 such that if ∆ < η 3 , then sup z∈S |||d z V |||σ S ( ∆ )) < a 4 . Finally we find that, denoting η = min{η 1 , η 2 , η 3 }, if ∆ < η then d x V v < -e (µ+κ)s a 2 < 0. Thus, following Lemma 11, there exists C > 0 such that if CN ( ∆) < 1 we have d x V v < 0. But ∆ = e -As ∆ and hence N ( ∆) = e -s N (∆) = N (∆) N (x) . We conclude that

d x V v < -a 2 V (x) µ+κ µ
when N (x) > CN (∆) which leads to (4) and ( 5) with b = -aκ 2µ . Finally, the constant α > 0 appears when taking another homogeneous norm.

Among the assumptions on which relies the Theorem 16, Assumption 13 is necessary when looking at ISS properties, Assumption 14 is a homogeneity-related assumption and Assumption 15 is a weak continuity assumption. Assumption 15 arises naturally when a continuous control system is controlled by a discontinuous feedback law as shown in the following lemma.

Lemma 17 Consider the following disturbed system ẋ = f (x, u(x + ∆)) [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF] with f : R n × R m → R n continuous, u : R n → R m a locally essentially bounded state feedback and ∆ ∈ L ∞ loc . The set-valued map F = F x [g], where g(x, ∆) = f (x, u(x + ∆)), verifies Assumption 15.

Proof. The function f is continuous, therefore for any compact set X ⊂ R n the function ϕ : R + → R + defined by

ϕ(ρ) = sup{ f (y + ∆, u(y + ∆)) -f (y, u(y + ∆)) , y ∈ X + B(0, 1), ∆ ≤ ρ} (8)
is increasing and ϕ(0) = 0. The continuous function f is uniformly continuous on the compact set (X + B(0, 1))× B(0, 1) and thus ϕ is continuous at 0. Lemma 12 yields that there exists a function σ X ∈ K such that for all ρ ≥ 0, ϕ(ρ) ≤ σ X (ρ). Hence, for all x ∈ X and all y ∈ B(x, 1):

f (y, u(y + ∆)) = f (y + ∆, u(y + ∆)) +f (y, u(y + ∆)) -f (y + ∆, u(y + ∆)) ∈ f (y + ∆, u(y + ∆)) + B(0, σ X ( ∆ ))
which leads to F (x, ∆) ⊂ F (x + ∆, 0) + B(0, σ X ( ∆ )).

Example 18 Consider the double integrator

ẋ1 = x 2 ẋ2 = u (9)
We define the control law

u(x) = -k 1 sign x 1 -k 2 sign x 2 (10)
with k 1 > k 2 > 0. It is well-known that the origin is a globally finite-time stable equilibrium of the closed-loop system (9)-( 10) [START_REF] Orlov | Finite time stability and robust control synthesis of uncertain switched systems[END_REF], which yields Assumption 13. Taking A = Diag(2, 1) we can easily check that Assumption 14 holds for the following disturbed system with κ = -1

ẋ1 = x 2 ẋ2 = -k 1 sign(x 1 + ∆ 1 ) -k 2 sign(x 2 + ∆ 2 ) (11)
Finally, from Lemma 17, Assumption 15 holds as well for the system (11) and finally Theorem 16 shows that there exists b < 0 and C > 0 such that, for any solution x(t) of (11) and any [2, 1]-homogeneous norm N :

N (x(t)) ≤ max β N x(0) , t ; CN (∆) ,
where β(r, t) = max{r + bt, 0}.

Sampled-data systems

Consider the following control system

ẋ = f (x, u) (12)
with x ∈ R n , u ∈ R m and f continuous. We suppose known a locally essentially bounded state feedback u(x) such that the origin is a globally asymptotically stable equilibrium for the closed-loop system

ẋ = f (x, u(x)). ( 13
)
We want to know how the system (13) reacts when subject to the sampling of the control law u(x), that is when

u(x(t)) is replaced by u(x(t k )) for a given sequence of instants (t k ) k∈N ẋ(t) = f (x(t), u(x(t k ))), t ∈ [t k , t k+1 ). ( 14
)
Denoting, for t ∈ [t k , t k+1 ), ∆(t) = x(t k )x(t), we find that the system ( 14) is equivalent to

ẋ = f (x, u(x + ∆)). ( 15
)
Given that u is discontinuous, the vector field defined in ( 15) is discontinuous and we shall replace it with the differential inclusion coming from the Filippov's procedure. Hence defining

F = F x [g], where g(x, ∆) = f (x, u(x + ∆)), we reformulate (15) by ẋ ∈ F (x, ∆). ( 16
)
Let us see now the consequences of Theorem 16 on the sampling effects.

Theorem 19 Consider a sequence of sampling times (t k ) k∈N for which there exists η, h ∈ R such that 0 < η ≤ t k+1t k ≤ h < +∞. Assume that the reformulated sampled differential inclusion ( 16) is such that Assumptions 13 and 14 hold. Consider ν(x) = Ax and N any ν-homogeneous norm. Then, there exists ξ > 0 such that the set N = {x ∈ R n : N (x) ≤ ξh -1/κ } is globally asymptotically stable w.r.t. the system [START_REF] Zubov | On ordinary differential equations with generalized homogeneous right-hand sides (in Russian)[END_REF].

The proof of Theorem 19 is largely inspired by the proof of Theorem 12 in [START_REF] Bernuau | Stability of homogeneous nonlinear systems with sampled-data inputs[END_REF]. A couple of technical details are however different, given the discontinuous nature of the system [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]. For the sake of clarity, a complete proof is thus provided below.

Proof. The proof of the theorem is divided into 4 steps.

I. We show that it is sufficient to prove the result for a given h when κ = 0. II. We show that we can apply Theorem 16. III. We prove a preliminary fact on the increase rate of N (x(t)). IV. We prove the theorem for κ < 0.

Step I. When κ = 0, assume that the results of Theorem 19 hold for a given maximal sampling period h 0 > 0. Let us prove that the results then hold for any h > 0. Indeed, consider s ∈ R such that h = e -κs h 0 . Consider a solution x(t) of ( 16) with (t k ) k∈N the sequence of sampling times. Then consider z(t) = exp(As)x(e κs t). If t k ≤ e κs t < t k+1 , we have ż(t) ∈ e κs exp(As)F (x(e κs t), x(t k )x(e κs t)).

From Assumption 14, it follows that ż(t) ∈ F (exp(As)x(e κs t), exp(As)(x(t k )x(e κs t)))

∈ F (z(t), z(e -κs t k )z(t)).

Hence the curve z(t) = exp(As)x(e κs t) is a solution of ( 16) with sampling times (e -κs t k ) k∈N . It follows that the mapping x(t) → exp(As)x(e κs t) is a bijection between the solutions of ( 16) with sampling times (t k ) k∈N and the solutions of ( 16) with sampling times (e -κs t k ) k∈N . Now, the local asymptotic stability for the maximal sampling period h 0 ensures the existence of β 0 ∈ KL such that

N (x(t)) ≤ β 0 (N (x 0 ), t) + ξh -1/κ 0 , ∀t ≥ 0.
By the point (2) of Definition 8 , we have

N (z(t)) = e s N (x(e κs t))
≤ e s β 0 (N (x 0 ), e κs t) + e s ξh -1/κ 0 ≤ e s β 0 (e -s N (z 0 ), e κs t) + ξ(e -κs h 0 ) -1/κ ≤ β s (N (z 0 ), t) + ξh -1/κ , with β s (r, t) = e s β 0 (e -s r, e κs t) ∈ KL.

Step II. The signal ∆ is piecewise continuous and verifies ∆ = -f (x, ∆) and ∆(t k ) = 0. It is straightforward to check that Assumptions 13 and 14 hold. Assumption 15 also holds from Lemma 17. The origin being a globally asymptotically stable equilibrium of the ν-homogeneous system ẋ ∈ F (x, 0) of degree κ, by virtue of Theorem 3.22 in [START_REF] Bernuau | Robustness and stability of nonlinear systems: a homogeneous point of view[END_REF], there exists a ν-homogeneous smooth Lyapunov function V of degree µ > 0 with µ + κ > 0. Let us define N (x) = V (x) 1/µ . We will prove the theorem for this particular homogeneous norm N , but Proposition 10 from [START_REF] Bernuau | Stability of homogeneous nonlinear systems with sampled-data inputs[END_REF] implies that the theorem then holds for all homogeneous norms.

Step III.

Step II and Theorem 16 lead to N (x(t)) ≤ max{β(N (x(0)), t); C ess sup τ ∈[0,t] N (∆(τ ))} ( 17) with C > 0 and β(r, t) = max{r -κbt, 0} -1/κ (see the proof of Theorem 16). We define S = {z ∈ R n : N (z) = 1} and for any ε > 0

α(ε) = sup{ dzV v : V (z) ≤ ε µ , y ∈ S and v ∈ F (y -z, z)} and H = sup 0<ε<1/C ε µ α(ε) ∈ (0, +∞],
where C comes from [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF]. Let us mention that for any finite ε > 0, α(ε) is finite, being the supremum of a USC multivalued function on a compact set. Fix h ∈ (0, H). There exists ε ∈ (0, 1/C) such that h ≤ ε µ α(ε) . With this couple (h, ε) we shall now prove the following fact. Fact. For any sampling time

t k such that N (x(t k )) -κ ≥ α(ε)h ε µ , we have N (x(t)) ≤ max {β(N (x(t k )), t -t k ); CεN (x(t k ))} ( 18
)
for all t ∈ [t k , t k+1 ].
Remark first that if x(t k ) = 0, then the asymptotic stability of the origin for the closed-loop system ẋ ∈ F (x, ∆) implies that x(t) = 0 for all t ∈ [t k , t k+1 ] and the fact is proved. Now, to prove the fact when x(t k ) = 0, let us study the variations of ∆(t). We denote s = ln(N (x(t k ))). Consider

t * = inf{t ≥ t k : N (∆(t)) > εN (x(t k ))} = inf{t ≥ t k : V (∆(t)) > ε µ V (x(t k ))}.
Thus on the interval [t k , t * ] we have V (∆(t)) ≤ ε µ V (x(t k )), which implies V (exp(-As)∆(t)) ≤ ε µ and exp(-As)x(t k ) ∈ S given that N (x(t k )) = e s . We have

V (∆(t * )) = t * t k d dt V (∆(t))dt ≤ (t * -t k ) sup t∈[t k ,t * ] sup v∈-F (x,∆) d ∆(t) V v ≤ (t * -t k ) sup{ d zV v : V (exp(-As)z) ≤ ε µ , exp(-As)ỹ ∈ S and v ∈ F (ỹ -z, z)}
and from Assumption 14, it follows that

V (∆(t * )) ≤ e (κ+µ)s (t * -t k ) sup{ d z V v : V (z) ≤ ε µ , y ∈ S and v ∈ F (y -z, z)} ≤ N (x(t k )) κ+µ (t * -t k )α(ε).
By continuity, we have V (∆(t * )) = ε µ V (x(t k )) = ε µ N (x(t k )) µ which leads to

ε µ N (x(t k )) µ ≤ N (x(t k )) κ+µ (t * -t k )α(ε) and then t * -t k ≥ ε µ N (x(t k )) -κ α(ε) ≥ h that is t * ≥ t k+1 .
Thus, for all t ∈ [t k , t k+1 ] we have N (∆(t)) ≤ εN (x(t k )). Finally, inserting this inequality into (17) yields [START_REF] Moulay | Stability and stabilization of homogeneous systems depending on a parameter[END_REF] and the fact is proved.

Step IV. The condition

N (x(t k )) -κ ≥ α(ε)h ε µ is equivalent to N (x(t k )) ≥ α(ε)h ε µ -1 κ = ρ. Hence N (x(t k )) ≥ ρ ensures that N (x(t k+1 )) ≤ N (x(t)) < N (x(t k )) for all t ∈ (t k , t k+1 ].
However, the set {x ∈ R n : N (x) < ρ} is not necessarily positively invariant. We therefore need to prove that, starting with N (x(t k )) < ρ, a trajectory cannot go "too far" from the origin. Let us define, for ω ≥ 0

θ(ω) = sup{ d z V v : V (z) ≤ 1 + ω, V (y) ≤ 1 , v ∈ F (z, y -z)}.
Denoting s = ln(1+ω) µ , we get

θ(ω) = sup{ d z V v) : V (exp(-As)z) ≤ 1, V (exp(-As)y) ≤ (1 + ω) -1 and v ∈ F (z, y -z)} = sup{ d exp(As)z V v : V (z) ≤ 1, V (ỹ) ≤ 1/(1 + ω)
and v ∈ F (exp(As)z, exp(As)(ỹz))}

and from Assumption 14 it follows that

θ(ω) = e (κ+µ)s sup{ d z V v : V (z) ≤ 1, V (ỹ) ≤ (1 + ω) -1 and v ∈ F (z, ỹ -z)} = (1 + ω) κ+µ µ sup{ d z V v : V (z) ≤ 1, V (ỹ) ≤ 1/(1 + ω) and v ∈ F (z, ỹ -z)}. Hence when ω → +∞ we have θ(ω) ∼ Ω ω κ+µ µ , with Ω = sup{ d zV v : V (z) ≤ 1, v ∈ F (z, -z)} and there- fore ω θ(ω) → +∞, given that κ+µ µ < 1 since κ < 0. Since the function ω → ω θ(ω) vanishes at ω = 0, there exists ω 0 > 0 such that ω0 θ(ω0) = ε µ α(ε) . Assume that N (x(t k )) < ρ, i.e. V (x(t k )) < ρ µ . Let us denote t * = inf {t ≥ t k : V (x(t)) ≥ (1 + ω 0 )ρ µ } > t k . Hence for all t ∈ [t k , t * ] we have V (x(t)) ≤ (1 + ω 0 )ρ µ . Therefore, we obtain V (x(t * )) = V (x(t k )) + t * t k d dt V (x(t))dt ≤ ρ µ + (t * -t k ) sup{ d x(t) V v : t ∈ [t k , t * ], v ∈ F (x(t), x(t k ) -x(t))} ≤ ρ µ + (t * -t k ) sup { d z V v : V (y) ≤ ρ µ , V (z) ≤ (1 + ω 0 )ρ µ and v ∈ F (z, y -z)} ≤ ρ µ + (t * -t k )ρ κ+µ θ(ω 0 ).
Also, by continuity, V (x(t * )) = (1 + ω 0 )ρ µ and therefore after simplification and using ρ

= α(ε)h ε µ -1 κ we get ω 0 ≤ (t * -t k ) α(ε)h ε µ -1 θ(ω 0 ) and given that ω0 θ(ω0) = ε µ α(ε) we find t * -t k ≥ h. Finally, t * ≥ t k+1 and then for all t ∈ [t k , t k+1 ] we have V (x(t)) ≤ (1 + ω 0 )ρ µ , which implies N (x(t)) ≤ (1 + ω 0 ) 1/µ ρ.
Let us summarize. We have three cases:

• if N (x(t k )) < ρ then we have N (x(t)) ≤ (1 + ω 0 ) 1/µ ρ for all t ∈ [t k , t k+1 ]; • if ρ ≤ N (x(t k )) ≤ (1 + ω 0 ) 1/µ ρ then we still have N (x(t)) ≤ (1 + ω 0 ) 1/µ ρ for all t ∈ [t k , t k+1 ]; • if N (x(t k )) ≥ (1 + ω 0 ) 1/µ ρ then N (x(t k )) -κ ≥ α(ε)h ε µ
and by the Fact of Step III we have

N (x(t)) ≤ max {β(N (x(t k )), t -t k ) ; CεN (x(t k ))} for all t ∈ [t k , t k+1 ].
We get from the first two points that the set {N (x) ≤

(1 + ω 0 ) 1/µ ρ} is positively invariant. Hence, if N (x 0 ) > (1 + ω 0 ) 1/µ ρ, denoting k * = max k≥0 {N (x(t k )) ≥ (1 + ω 0 ) 1/µ ρ}, for all 0 ≤ k ≤ k * , we have N (x(t k )) ≥ (1 + ω 0 ) 1/µ ρ.
Therefore the third point gives us that N (x(t k+1 )) ≤ N (x(t k )) for all 0 ≤ k ≤ k * and thus N (x(t k )) < N (x 0 ) for all 0 < k ≤ k * . Using again the shape of the function β, for 0 ≤ k < k * and t ∈ [t k , t k+1 ] we get

N (x(t)) ≤ max{β(N (x(t k )), η) ; CεN (x(t k ))} ≤ N (x(t k )) max{β(1, ηN (x(t k )) κ ) ; Cε} ≤ N (x(t k )) max{β(1, ηN (x 0 ) κ ) ; Cε} ≤ N (x(t k ))q
where q = max{β(1, ηN (x 0 ) κ ) ; Cε} < 1, given that β ∈ KL and β(r, 0) = r. Therefore N (x(t)) ≤ max{N (x 0 )q k ; (1 + ω 0 ) 1/µ ρ}, t ∈ [t k , t k+1 ] and we get the global asymptotical stability of the set {x ∈ R n : N (x) ≤ (1 + ω 0 ) 1/µ ρ} = {x ∈ R n : N (x) ≤ ξh -1/κ }.

Remark 20

The assumptions taken on the sampling sequence (t k ) imply that (t k ) is monotonously increasing and does not admit an accumulation point. Note that the result holds for any homogeneous norm.

Changing the homogeneous norm would only change the number ξ in the definition of the asymptotically set N .

Application to the double integrator

Consider the double integrator (9) and for t ∈ [t k , t k+1 ), we select the control law to be u(x(t k )) = -k 1 sign(x 1 (t k ))k 2 sign(x 2 (t k )) [START_REF] Hermes | Nilpotent and high-order approximations of vector field systems[END_REF] with k 1 > k 2 . For the simulations, we take k 1 = 5 and k 2 = 2.5. We choose a constant sampling period

t k = kT, T > 0, k ∈ N ( 20
)
in order to use well known results about linear systems having a constant sampling period. Indeed, we know that the closed-loop linear sampled-data system (9) under the linear controller u(x(t k )) = -k 1 x 1 (t k )-k 2 x 2 (t k ) with the samplings (20) is asymptotically stable if and only if the matrix Λ(T ) of its associated linear difference equation is Schur [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: an overview[END_REF][START_REF] Maalej | State-dependent sampling for linear time invariant systems: a discrete time analysis[END_REF]. For the double integrator with k 1 = 5 and k 2 = 2.5, Λ(T ) is Schur if and only if T < T Schur = 0.8s. However, Theorem 19 yields that, for any [2, 1]-homogeneous norm, there exists ξ > 0 such that the set {x ∈ R n : N (x) ≤ ξT } is globally asymptotically stable for the nonlinear closed-loop system ( 9)-( 19)- [START_REF] Bernuau | Stability of homogeneous nonlinear systems with sampled-data inputs[END_REF]. In Figure 1, the asymptotically stable sets are drawn for different constant samplings T larger than T Schur = 0.8s.

Conclusion

In this article, we have extended the results about stability of homogeneous sampled-data systems to discontinuous systems. In particular, we have designed a discontinuous sampled controller for the double integrator.

The robustness with respect to model uncertainties of discontinuous controllers compared with the continuous ones in the sampled-data context is an interesting issue for futures works.

Fig. 1 .

 1 Fig. 1. Asymptotically stable sets for different values of T

  we have d [a,b] < +∞, the function d is called locally essentially bounded. We denote L ∞ loc the set of locally essentially bounded functions d : R + → R n . • A continuous function α : R + → R + belongs to the class K if α(0) = 0 and the function is strictly increasing. A function α : R + → R + belongs to the class K ∞ if α ∈ K and it is increasing to infinity. • A continuous function β : R + × R + → R + belongs to the class KL if β(•, t) ∈ K ∞ for each fixed t ∈ R +

	a < b and if for each fixed s ∈ R + the function t → β(s, t)
	is decreasing to 0.
	• The notation d x V (resp. d x Φ) stands for the differen-
	tial of the function V (resp. the diffeomorphism Φ) at
	the point x.
	where R is the set of real
	numbers.
	• | • | denotes the absolute value in R, . denotes the Euclidean norm on R n . • B(y, ε) = {x ∈ R n : x -y < ε} and B(y, ε) is its
	closure.
	• If r 1 , . . . , r n ∈ R, we will denote Diag(r 1 , . . . , r n ) the diagonal matrix of dimension n × n with k th diagonal
	entry r k . • For a (Lebesgue) measurable function d : R + → R n ,
	we denote d [a,b] = ess sup t∈[a,b] d(t) . If for any 0 ≤


Emmanuel Bernuau
email: emmanuel.bernuau@agroparistech.fr

Emmanuel Moulay
email: emmanuel.moulay@univ-poitiers.fr

Patrick Coirault
email: patrick.coirault@univ-poitiers.fr

Stability of homogeneous nonlinear systems with sampled-data inputs ⋆

Keywords: Sampled-data systems, Nonlinear systems, Homogeneity, Lyapunov stability, Lyapunov functions

come

Stability of homogeneous nonlinear systems with sampled-data inputs

Introduction

Feedback control systems wherein the control loops are closed through a real-time network are called Networked Control Systems (NCSs) [START_REF] Zhang | Stability of networked control systems[END_REF]. In NCSs, the presence of packet-based communication, network delays, limited bandwidth and packet dropouts is inevitable. NCSs are widely studied in automatic control since several years [START_REF] Bemporad | Networked control systems[END_REF][START_REF] Wang | Networked control systems[END_REF].

When studying NCSs, a crucial problem is to determine whether some stability properties pertain through sampled-data inputs [START_REF] Walsh | Stability analysis of networked control systems[END_REF]. Moreover, due to, for instance, data packet dropouts [START_REF] Liu | Networked control systems: A time-delay approach[END_REF], the inputs are not only sampled but also usually nonuniformly sampled with respect to time and this is called aperiodic sampled-data inputs [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: an overview[END_REF]. Several approaches have been developed to solve the problem of stability for systems with sampled-data inputs, such as the input/output approach [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF][START_REF] Omran | On the stability of input-affine nonlinear systems with sampled-data control[END_REF] or the sum of squares approach [START_REF] Bauer | Stability analysis of networked control systems: A sum of squares approach[END_REF]. Moreover, different dynamical models as hybrid systems [START_REF] Zhang | Stability of networked control systems[END_REF][START_REF] Donkers | Stability analysis of networked control systems using a switched linear systems approach[END_REF], discrete-time systems [START_REF] Nešić | A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models[END_REF] or time-delay systems [START_REF] Liu | Networked control systems: A time-delay approach[END_REF][START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF][START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF][START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF] have also been used to tackle this problem. Most of these strategies [START_REF] Nešić | Input-output stability properties of networked control systems[END_REF][START_REF] Omran | On the stability of input-affine nonlinear systems with sampled-data control[END_REF][START_REF] Bauer | Stability analysis of networked control systems: A sum of squares approach[END_REF] are developed under the emulation method where the controller is first designed in continuous time and then implemented as a sampled-data controller [START_REF] Bemporad | Networked control systems[END_REF][START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF]. The notion of homogeneity was introduced in [START_REF] Zubov | On ordinary differential equations with generalized homogeneous right-hand sides (in Russian)[END_REF] and [START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF] and developed by many authors, for instance in [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF][START_REF] Kawski | Geometric homogeneity and stabilization[END_REF][START_REF] Moulay | Stability and stabilization of homogeneous systems depending on a parameter[END_REF]. In [START_REF] Anta | To sample or not to sample: Selftriggered control for nonlinear systems[END_REF], a new nonlinear approach based on the input-to-state stability (ISS) properties of homogeneous systems has been introduced for studying self-triggered control for nonlinear systems. We know from [START_REF] Laila | Open-and closedloop dissipation inequalities under sampling and controller emulation[END_REF][START_REF] Nešić | Formulas relating KL stability estimates of discrete-time and sampled-data nonlinear systems[END_REF] that the emulation of a feedback law, which globally asymptotically stabilizes the origin of a nonlinear continuoustime system, leads in general to semi-global practical stability in the sampled-data context. In our article, we show that if the nonlinear system satisfies in addition a homogeneity property, then the global asymptotic stability, local asymptotic stability or global asymptotic set stability are preserved under emulation depending on the degree of homogeneity.

The main result shows that if it is possible to build a stabilizing static feedback control for a continuous system such that the closed-loop system with sampled-data inputs satisfies a homogeneity property, then it is possible to preserve different notions of stability that depend on the degree of homogeneity κ for the closed-loop system with aperiodic sampled-data inputs. When κ = 0, the closed-loop system is globally asymptotically stable if the maximum sampling period h is shorter than a constant H. Linear systems are special cases of homogeneous systems of degree 0 and our result hence shows that homogeneity explains this feature of linear systems. We refer to [START_REF] Zhang | Stability of networked control systems[END_REF][START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF][START_REF] Suh | Stability and stabilization of nonuniform sampling systems[END_REF] for different strategies used in the literature for estimating H in the linear case. When κ > 0, the closed-loop system achieves local asymptotic stability, while when κ < 0, it achieves global asymptotic set stability. In these two cases, the results remain true regardless of h < +∞, although the size of the domain of attraction and of the limit set do depend on h. With the use of an homogeneous observer, the results are then extended to continuous systems with an output. The results are finally applied to the case of the double integrator which is rather important in control theory despite its simplicity [START_REF] Bernuau | Robust finite-time output feedback stabilisation of the double integrator[END_REF][START_REF] Bhat | Continuous finitetime stabilization of the translational and rotational double integrators[END_REF].

The article is organized as follows. After some notations and definitions given in Section 2, we develop the main results of the article in Section 3. The example of the double integrator is treated in Section 4. Finally, a conclusion is addressed in Section 5.

Notations and definitions

Let us introduce the following notations:

• R + = {x ∈ R : x ≥ 0},
→ R + belongs to the class K ∞ if α ∈ K and it is unbounded. • A continuous function β : R + × R + → R + belongs to the class KL if β(•, t) ∈ K ∞ for each fixed t ∈ R + and if for each fixed s ∈ R + the function t → β(s, t) is decreasing to 0. • The notation d x V (resp. d x Φ
) stands for the differential of the function V (resp. the diffeomorphism Φ) at the point x. • ⌊x⌉ α = |x| α sign(x) where x ∈ R and α > 0. • Given that we will deal with a variety of suprema, we will use a compact notation in the computations. For instance, the notation sup{f (x, y) :

g(x) = 0, h(y) ≤ 1} stands for sup x∈E,y∈F f (x, y) where E = {x ∈ R n : g(x) = 0} and F = {y ∈ R n : h(y) ≤ 1}.
Consider the following system with f continuous

ẋ = f (x), x ∈ R n . ( 1
)
Let us recall the definitions of Lyapunov set stability given for instance in [START_REF] Bhatia | Stability theory of dynamical systems[END_REF].

Definition 1 A compact set K ⊂ R n is: • stable w.
= f (x, ∆) (2)
where x ∈ R n is the state, ∆ ∈ L ∞ loc is the external input and f : R n × R m → R n is continuous. Let us recall the definition of input-to-state stability given for instance in [START_REF] Sontag | On characterizations of the inputto-state stability property[END_REF].

Definition 2

The system (2) is called input-to-state stable (ISS) if there exist some functions β ∈ KL and γ ∈ K such that for any input ∆ ∈ L ∞ loc and any

x 0 ∈ R n it holds that ∥x(t)∥ ≤ β(∥x(0)∥, t) + ess sup τ ∈[0,t] γ(∥∆(τ )∥) ∀t ≥ 0
where x(t) is the solution of the system (2) satisfying x(0) = x 0 . The function γ is called a nonlinear asymptotic gain.

The most common notion of homogeneity is the weighted homogeneity, based on a particular choice of the coordinates, while the most generic one is the geometric homogeneity, which is coordinate free [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF][START_REF] Kawski | Geometric homogeneity and stabilization[END_REF]. We use in the sequel the framework of geometric homogeneity. Definition 3 A vector field ν on R n is called an Euler vector field if ν is of class C 1 , complete (i.e. the maximal solutions of ẋ = ν(x) are defined on R) and if the origin is a globally asymptotically equilibrium of -ν. Definition 4 Let ν be an Euler vector field on R n , Φ s (x) denotes the value of the flow 1 of ν at time s with initial condition

x. A function V : R n → R is ν-homogeneous of degree κ ∈ R if for all x ∈ R n and all s ∈ R we have V (Φ s (x)) = e κs V (x). A vector field f on R n is ν- homogeneous of degree κ ∈ R if for all x ∈ R n and all s ∈ R we have f (Φ s (x)) = e κs d x Φ s f (x).
A direct application of the chain rule then gives us the following result.

Proposition 5 Let ν be an Euler vector field on R n . If V : R n → R is differentiable and ν-homogeneous of degree κ, then for all x ∈ R n and all s ∈ R

d Φ s (x) V • d x Φ s = e κs d x V.
Remark 6 If we consider a matrix A ∈ R n×n such that -A is Hurwitz then the vector field defined by ν(x) = Ax is an Euler vector field and the flow of ν verifies Φ s (x) = exp(As)x. In particular, if A = Diag(r 1 , . . . , r n ) with r 1 , . . . , r n > 0, the vector field ν(x) = Ax is Euler and we find Φ s (x) = Diag(e r1s , . . . , e rns )x. The homogeneity defined by such an Euler vector field is usually referred to as weighted homogeneity, the coefficients r 1 , . . . , r n are called the weights and r = [r 1 , . . . , r n ] is called the generalized weight [START_REF] Bacciotti | Liapunov functions and stability in control theory[END_REF]. Let us finally mention that homogeneity with respect to an Euler vector field defined by a generalized weight r is usually simply referred to as r-homogeneity.

Definition 7 Let ν be an Euler vector field on R

n . A ν- homogeneous norm is a function N : R n → R such that: (1) N is positive definite; (2) N is ν-homogeneous of degree 1; (3) N is continuous. Example 8 Let r = [r 1 , . . . , r n ] be a generalized weight. For any ρ > 0, a r-homogeneous norm is defined by N ρ (x) = ( ∑ n i=1 |x i | ρ r i ) 1 ρ . Furthermore, N ∞ (x) = sup i |x i | 1
r i also defines a r-homogeneous norm.

Remark 9 A homogeneous norm is always radially unbounded, following [18, Lemma 4.1]

The following proposition is a direct consequence of [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF]Lemma 4.2].

Proposition 10

Let N 1 and N 2 be two ν-homogeneous norms. Then there exist a, b > 0 such that, for all x ∈ R n :

aN 1 (x) ≤ N 2 (x) ≤ bN 1 (x).
1 See [START_REF] Teschl | Ordinary Differential Equations and Dynamical Systems[END_REF], Chapter 6 for details.

Main results

Let us consider the following nonlinear system

ẋ = F (x, u) (3)
defined by a continuous function F : R n × R m → R n and assume known a continuous static feedback u : R n → R m such that the origin is a globally asymptotically stable equilibrium of the closed-loop system ẋ = F (x, u(x)). Remark that, in this formulation, the control u may depend on the state x through a static output feedback u(x) = v(y), where y = g(x) is an output. Since in networked communication, the state information is only updated at discrete time instants, we attempt at giving conditions based on homogeneity under which some stability properties pertain for the system (3). More precisely, we consider a maximum sampling period 0 < h < +∞, a minimum sampling period η > 0 and a sequence of sampling times (t k ) k∈N such that t 0 = 0 and

η ≤ t k+1 -t k ≤ h. ( 4
)
Due to the sampling, the control is now u(t) = u(x(t k )) for all t ∈ [t k , t k+1 ) (sample and hold). The system can therefore be rewritten under the following form

ẋ(t) = F (x(t), u(x(t k ))), t ∈ [t k , t k+1 ). (5)
We consider the following assumption on system (5).

Assumption 11 There exists a matrix A ∈ R n×n such that -A is Hurwitz and a degree κ ∈ R such that F (exp(As)x, u(exp(As)z)) = e κs exp(As)F (x, u(z))

for all x, z ∈ R n and all s ∈ R.

Assumption 11 is a homogeneity property of system [START_REF] Liu | Networked control systems: A time-delay approach[END_REF]. Indeed, if we consider the vector field F defined on R n × R n by F (x, z) = (F (x, u(z)), 0) T , then Assumption 11 is equivalent to the ν-homogeneity of F with ν(x, z) = (Ax, Az) T . In particular, this assumption implies that F (x, u(x)) is homogeneous. A class of systems that satisfy Assumption 11 is, for example, the chain of integrators and its nonlinear extension studied in [START_REF] Hong | Finite-time stabilization and stabilizability of a class of controllable systems[END_REF].

Let us now give the main result of the article.

Theorem 12 Assume that the sampled system (5) is such that the sampling times satisfy [START_REF] Walsh | Stability analysis of networked control systems[END_REF] and that Assumption 11 holds. Consider ν(x) = Ax and N any νhomogeneous norm.

(1) If κ > 0 then the origin is a locally asymptotically stable equilibrium of the system (5) and there exists ξ > 0 such that the set {x ∈ R n : N (x) ≤ ξh -1/κ } is a subset of the domain of attraction.

(2) If κ = 0 then there exists H > 0 such that if h < H then the origin is a globally asymptotically stable equilibrium of the system (5). (3) If κ < 0 then there exists ξ > 0 such that the set {x ∈ R n : N (x) ≤ ξh -1/κ } is globally asymptotically stable w.r.t. the system (5).

We will need the following result to prove Theorem 12.

Lemma 13 [START_REF] Bernuau | On the robustness of homogeneous systems and a homogeneous small gain theorem[END_REF]Theorem 11] Assume that f : R n × R m → R n is continuous and that there exist ν an Euler vector field on R n with flow denoted Φ s (x), ν an Euler vector field on R m with flow denoted Φs (d) and κ ∈ R such that for all x ∈ R n and all z ∈ R m ,

f (Φ s (x), Φs (z)) = e κs d x Φ s f (x, z). ( 6
)
Assume also that the origin is a globally asymptotically stable equilibrium of the system ẋ = f (x, 0). Then the system (2) is ISS w.r.t. an input ∆. Furthermore, for any ν-homogeneous norm N and any ν-homogeneous norm Ñ , there exist C > 0 such that

N (x(t)) ≤ max { β(N (x(0)), t) ; C ess sup τ ∈[0,t] Ñ (∆(τ )) } (7) with β ∈ KL defined by β(r, t) =      re -a 2 t κ = 0 max { [ r -κ/µ + aκ 2µ t ] -µ/κ , 0 } κ ̸ = 0 ( 8
)
where a > 0 and µ > max(-κ, 0) are positive constants.

Proof of Theorem 12. The proof of the theorem is divided into 6 steps.

I. We show that it is sufficient to prove the result for a given h when κ ̸ = 0. II. We rewrite the system and introduce some notation. III. We prove a preliminary fact on the increase rate of N (x(t)). IV. We prove the theorem for κ > 0. V. We prove the theorem for κ = 0. VI. We prove the theorem for κ < 0.

Step I. When κ ̸ = 0, assume that the results of Theorem 12 hold for a given maximal sampling period h 0 > 0. Let us prove that the results then hold for any h > 0. Indeed, consider s ∈ R such that h = e -κs h 0 . Consider a solution x(t) of ( 5) with (t k ) k∈N the sequence of sampling times. Then consider z(t) = exp(As)x(e κs t). If t k ≤ e κs t < t k+1 , we have ż(t) = e κs exp(As)F (x(e κs t), u(x(t k ))).

From Assumption 11, it follows that ż(t) = F (exp(As)x(e κs t), u(exp(As)x(e κs e -κs t k )))

= F (z(t), u(z(e -κs t k ))).

Hence the curve z(t) = exp(As)x(e κs t) is a solution of (5) with sampling times (e -κs t k ) k∈N . It follows that the mapping x(t) → exp(As)x(e κs t) is a bijection between the solutions of ( 5) with sampling times (t k ) k∈N and the solutions of ( 5) with sampling times (e -κs t k ) k∈N . Now, in the case κ > 0 the local asymptotic stability for the maximal sampling period h 0 ensures the existence of

β 0 ∈ KL such that N (x 0 ) ≤ ξh -1/κ 0 ⇒ N (x(t)) ≤ β 0 (N (x 0 ), t), ∀t ≥ 0.
By the point (2) of Definition 7 , N (x 0 ) ≤ ξh

-1/κ 0 is equivalent to N (z 0 ) ≤ e s ξh -1/κ 0 = ξ(e -κs h 0 ) -1/κ = ξh -1/κ , we get N (z(t)) = e s N (x(e κs t))
≤ e s β 0 (N (x 0 ), e κs t) ≤ e s β 0 (e -s N (z 0 ), e κs t)

≤ β s (N (z 0 ), t),
with β s (r, t) = e s β 0 (e -s r, e κs t) ∈ KL, and finally

N (z 0 ) ≤ ξh -1/κ ⇒ N (z(t)) ≤ β s (N (z 0 ), t)
for all t ≥ 0. In the case κ < 0, a similar method can be used, based on the following recast of the announced result:

N (x(t)) ≤ β 0 (N (x 0 ), t) + ξh -1/κ , ∀t ≥ 0 with β 0 ∈ KL.
Step II. We shall now base the sequel of this proof on a rewriting of the system [START_REF] Liu | Networked control systems: A time-delay approach[END_REF]. Noting that u(x

(t k )) = u (x(t) + [x(t k ) -x(t)]), we define f (x, z) = F (x, u(x + z)
) and hence we see that the system (5) can be rewritten under the equivalent form

{ ẋ = f (x, ∆) ∆(t) = x(t k ) -x(t), t ∈ [t k , t k+1 ) (9)
The signal ∆ is piecewise continuous and verifies ∆ = -f (x, ∆) and ∆(t k ) = 0. It is straightforward to check that, under Assumption 11, the hypotheses of Lemma 13 hold. Since here x(t) and ∆(t) belong to the same space, namely R n , and ν(x) = ν(x) = Ax we can always select the homogeneous norm Ñ of Equation ( 7) to be equal to N . The origin being a globally asymptotically stable equilibrium of the ν-homogeneous system ẋ = f (x, 0) of degree κ, by virtue of [START_REF] Rosier | Homogeneous lyapunov function for homogeneous continuous vector field[END_REF]Theorem 2] there exists a ν-homogeneous smooth Lyapunov function V of degree µ > 0 with µ+κ > 0. Let us define N (x) = V (x) 1/µ . We will prove the theorem for this particular homogeneous norm N , but Proposition 10 implies that the theorem then holds for all homogeneous norms.

Step III.

Step II and Lemma 13 lead to

N (x(t)) ≤ max { β(N (x(0)), t) ; C ess sup τ ∈[0,t] N (∆(τ )) } (10
) with C > 0 and β ∈ KL of Lemma 13. We define S = {z ∈ R n : N (z) = 1} and for any ε > 0

α(ε) = sup{∥d z V F (y-z, u(y))∥ : V (z) ≤ ε µ and y ∈ S} and H = sup 0<ε<1/C ε µ α(ε) ∈ (0, +∞].
Let us mention that for any finite ε > 0, α(ε) is finite, being the supremum of a continuous function on a compact set. Fix h ∈ (0, H). There exists ε ∈ (0, 1/C) such that h ≤ ε µ α(ε) . With this couple (h, ε) we shall now prove the following fact.

Fact. Consider k such that N (x(t k )) -κ ≥ α(ε)h ε µ , then we have N (x(t)) ≤ max {β(N (x(t k )), t -t k ) ; CεN (x(t k ))} (11) for all t ∈ [t k , t k+1 ].
Remark first that if x(t k ) = 0, then the asymptotic stability of the origin for the closed-loop system ẋ = F (x, u(x)) implies that x(t) = 0 for all t ∈ [t k , t k+1 ] and the fact is proved. Now, to prove the fact when x(t k ) ̸ = 0, let us study the variations of ∆(t). We denote s = ln(N (x(t k ))). Consider

t * = inf{t ≥ t k : N (∆(t)) > εN (x(t k ))} = inf{t ≥ t k : V (∆(t)) > ε µ V (x(t k ))}.
Thus on the interval [t k , t * ] we have V (∆(t)) ≤ ε µ V (x(t k )), which implies V (exp(-As)∆(t)) ≤ ε µ and exp(-As)x(t k ) ∈ S given that N (x(t k )) = e s . We have

V (∆(t * )) = ∫ t * t k d ∆(t) V f (x(t), u(x(t k )))dt ≤ (t * -t k ) sup t∈[t k ,t * ] ∥d ∆(t) V F (x(t), u(x(t k )))∥ ≤ (t * -t k ) sup{∥d z V F (ỹ -z, u(ỹ))∥ : V (exp(-As)z) ≤ ε µ and exp(-As)ỹ ∈ S}
and from Assumption 11 and Proposition 5, it follows that

V (∆(t * )) ≤ e (κ+µ)s (t * -t k ) sup{∥d z V F (y -z, u(y))∥ : V (z) ≤ ε µ and y ∈ S} ≤ N (x(t k )) κ+µ (t * -t k )α(ε). By continuity, we have V (∆(t * )) = ε µ V (x(t k )) = ε µ N (x(t k )) µ which leads to ε µ N (x(t k )) µ ≤ N (x(t k )) κ+µ (t * -t k )α(ε) and then t * -t k ≥ ε µ N (x(t k )) -κ α(ε) ≥ h that is t * ≥ t k+1 .
Thus, for all t ∈ [t k , t k+1 ] we have N (∆(t)) ≤ εN (x(t k )). Finally, inserting this inequality into (10) yields ( 11) and the fact is proved.

We shall now split the proof into three cases, whether κ > 0 or κ = 0 or κ < 0.

Step IV:

κ > 0 The condition N (x(t k )) -κ ≥ α(ε)h ε µ is equivalent to N (x(t k )) ≤ ( ε µ α(ε)h ) 1/κ
. Let us denote

B(h) = { z ∈ R n : N (z) ≤ ( ε µ α(ε)h ) 1/κ } . The set B(h) is a compact neighborhood of the origin. If x 0 ∈ B(h), (11) leads to N (x(t)) ≤ N (x(t k )) for all k ∈
N and all t ∈ (t k , t k+1 ]. Therefore the system is stable and N (x(t)) converges to a limit ℓ ≥ 0 when t → +∞. Now, by using [START_REF] Nešić | A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models[END_REF], we obtain for all t ∈ [t k , t k+1 ] and

k ∈ N N (x(t k+1 )) ≤ max{β(N (x(t k )), t k+1 -t k ); CεN (x(t k ))} ≤ max {β(N (x(t k )), η) ; CεN (x(t k ))} .
Following (8), we have β(r, t) = rβ(1, tr κ/µ ) therefore the limit ℓ satisfies ℓ ≤ max {ℓβ(1, ηℓ κ ) ; Cεℓ} ≤ ℓ. Thus ℓ = max { ℓβ(1, ηℓ κ/µ ) ; Cεℓ } = ℓβ(1, ηℓ κ/µ ) since Cε < 1, κ > 0 and β(r, 0) = r. Hence ℓ = 0 or β(1, ηℓ κ/µ ) = 1, but this second condition is also equivalent to ℓ = 0. The local asymptotic stability follows with B(h) a subset of the domain of attraction.

Step

V: κ = 0 The condition N (x(t k )) -κ ≥ α(ε)h ε µ is equivalent to h ≤ ε µ α(ε)
and therefore always holds given the choice we have made for ε in Step III. By similar arguments to the case κ > 0, we obtain that, if h < H, the origin is a globally asymptotically stable equilibrium of the system (5).

Step VI:

κ < 0 The condition N (x(t k )) -κ ≥ α(ε)h ε µ is equivalent to N (x(t k )) ≥ ( α(ε)h ε µ ) -1 κ = ρ. Hence N (x(t k )) ≥ ρ en- sures that N (x(t k+1 )) ≤ N (x(t)) < N (x(t k )) for all t ∈ (t k , t k+1 ].
However, the set {x ∈ R n : N (x) < ρ} is not necessarily positively invariant. We will therefore need another analysis to prove that, starting with N (x(t k )) < ρ, a trajectory cannot go "too far" from the origin. Let us define, for ω ≥ 0

θ(ω) = sup{∥d z V F (z, u(y))∥ : V (z) ≤ 1+ω , V (y) ≤ 1}.
Denoting s = ln(1+ω) µ , we get

θ(ω) = sup{∥d z V F (z, u(y))∥ : V (exp(-As)z) ≤ 1 and V (exp(-As)y) ≤ 1/(1 + ω)} = sup{∥d exp(As)z V F (exp(As)z, u(exp(As)ỹ))∥ : V (z) ≤ 1 and V (ỹ) ≤ 1/(1 + ω)}
and from Assumption 11 and Proposition 5, it follows that

θ(ω) = e (κ+µ)s sup{∥d z V F (z, u(ỹ))∥ : V (z) ≤ 1 and V (ỹ) ≤ 1/(1 + ω)} = (1 + ω) κ+µ µ sup{∥d z V F (z, u(ỹ))∥ : V (z) ≤ 1 and V (ỹ) ≤ 1/(1 + ω)}. Hence when ω → +∞ we have θ(ω) ∼ Ω ω κ+µ µ , with Ω = sup{∥d z V F (z, u(0))∥ : V (z) ≤ 1} and there- fore ω θ(ω) → +∞, given that κ+µ µ < 1 since κ < 0. Since the function ω → ω θ(ω) vanishes at ω = 0, there exists ω 0 > 0 such that ω0 θ(ω0) = ε µ α(ε) . Assume that N (x(t k )) < ρ, i.e. V (x(t k )) < ρ µ . Let us denote t * = inf {t ≥ t k : V (x(t)) ≥ (1 + ω 0 )ρ µ } > t k . Hence for all t ∈ [t k , t * ] we have V (x(t)) ≤ (1 + ω 0 )ρ µ . Therefore, we obtain V (x(t * )) = V (x(t k )) + ∫ t * t k d dt V (x(t))dt ≤ ρ µ + (t * -t k ) sup{∥d x(t) V F (x(t), u(x(t k )))∥ : t ∈ [t k , t * ]} ≤ ρ µ + (t * -t k ) sup {∥d z V F (z, u(y))∥ : V (z) ≤ (1 + ω 0 )ρ µ and V (y) ≤ ρ µ } ≤ ρ µ + (t * -t k )ρ κ+µ θ(ω 0 ).
Also, by continuity, V (x(t * )) = (1 + ω 0 )ρ µ and therefore after simplification and using ρ =

( α(ε)h ε µ ) -1 κ we get ω 0 ≤ (t * -t k ) ( α(ε)h ε µ ) -1 θ(ω 0 ) and given that ω0 θ(ω0) = ε µ α(ε) we find t * -t k ≥ h. Finally, t * ≥ t k+1 and then for all t ∈ [t k , t k+1 ] we have V (x(t)) ≤ (1 + ω 0 )ρ µ , which implies N (x(t)) ≤ (1 + ω 0 ) 1/µ ρ.
Let us summarize. We have three cases:

• if N (x(t k )) < ρ then we have N (x(t)) ≤ (1 + ω 0 ) 1/µ ρ for all t ∈ [t k , t k+1 ]; • if ρ ≤ N (x(t k )) ≤ (1 + ω 0 ) 1/µ ρ then we still have N (x(t)) ≤ (1 + ω 0 ) 1/µ ρ for all t ∈ [t k , t k+1 ]; • if N (x(t k )) ≥ (1 + ω 0 ) 1/µ ρ then N (x(t k )) -κ ≥ α(ε)h ε µ
and by the Fact of Step III we have

N (x(t)) ≤ max {β(N (x(t k )), t -t k ) ; CεN (x(t k ))} for all t ∈ [t k , t k+1 ].
We get from the first two points that the set

{N (x) ≤ (1 + ω 0 ) 1/µ ρ} is positively invariant. Hence, if N (x 0 ) > (1 + ω 0 ) 1/µ ρ, denoting k * = max k≥0 {N (x(t k )) ≥ (1 + ω 0 ) 1/µ ρ}, for all 0 ≤ k ≤ k * , we have N (x(t k )) ≥ (1 + ω 0 ) 1/µ ρ. Therefore the third point gives us that N (x(t k+1 )) ≤ N (x(t k )) for all 0 ≤ k ≤ k * and thus N (x(t k )) < N (x 0 ) for all 0 < k ≤ k * . Using again the shape of the function β, for 0 ≤ k < k * and t ∈ [t k , t k+1 ] we get N (x(t)) ≤ max{β(N (x(t k )), η) ; CεN (x(t k ))} ≤ N (x(t k )) max{β(1, ηN (x(t k )) κ/µ ) ; Cε} ≤ N (x(t k )) max{β(1, ηN (x 0 ) κ/µ ) ; Cε} ≤ N (x(t k ))q
where q = max{β(1, ηN (x 0 ) κ/µ ) ; Cε} < 1, given that β ∈ KL and β(r, 0) = r. Therefore

N (x(t)) ≤ max{N (x 0 )q k ; (1 + ω 0 ) 1/µ ρ}, t ∈ [t k , t k+1 ]
and we get the global asymptotical stability of the set

{x ∈ R n : N (x) ≤ (1 + ω 0 ) 1/µ ρ} = {x ∈ R n : N (x) ≤ ξh -1/κ }.
Remark 14 With the notations of Theorem 12, in the case κ < 0, the conclusion is equivalent to

N (x(t)) ≤ β 0 (N (x(0)), t) + ξh -1/κ , ∀ 0 < h < H
with β 0 a class KL function, that is, the system (5) is ISS w.r.t. the input h with asymptotic gain γ(h) = ξh -1/κ . Consequently, the origin is globally practically stable, see [START_REF] Teel | Semi-global practical asymptotic stability and averaging[END_REF] for details. Let us also remark that the linear case is included in the case κ = 0.

The statement of Theorem 12 does not provide an explicit method for estimating the constants H and ξ. From this perspective, it could be seen as a qualitative result on sampled-data systems. However, up to a constant, the Theorem 12 gives a precise estimation of the dependency on h of the domain of attraction (when κ > 0) and of the stable set (when κ < 0), which is a quantitative data. Moreover, we would like to point out that, in most particular cases, the constants H and ξ can be estimated mathematically (if a Lyapunov function is known) or by computer simulations (see Section 4).

Even though Theorem 12 is written to emphasize the effects of the communication network on a system asymptotically stabilized by a state feedback, the result also applies to a system asymptotically stabilized by an observer-based controller, provided that this observer is homogeneous as well. We keep considering the system (3) verifying Assumption 11. We endow the system with an output y = j(x). As beforehand, we assume known a state feedback u(x) such that the origin is a globally asymptotically stable equilibrium for the closed-loop system. We assume that an observer is added to the system ẋ = G(y, x)

with G continuous, in such a way that the origin of R n × R n is a globally asymptotically stable equilibrium of the following system

{ ẋ = F (x, u(x)) ẋ = G(j(x), x) (12)
Now we want to study the impact of a sampling on the stability of this system. Indeed, the value of x will be updated only at the sampling instants t k , leading to

{ ẋ(t) = F (x(t), u(x(t k ))), t ∈ [t k , t k+1 ) x(t k+1 ) = x(t k ) + (t k+1 -t k )G(j(x(t k )), x(t k )) (13)
Proposition 15 Assume that the system [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF], defined on R n × R n , is ν-homogeneous of degree κ with ν(x, x) = (Ax, Ax). Assume also that the sampled system (13) is such that the sampling times satisfy ( 4).

(1) If κ > 0 then the origin is a locally asymptotically stable equilibrium of the system (13) and there exists ξ > 0 such that the set {(x, x) ∈ R n × R n : N (x) + N (x) ≤ ξh -1/κ } is a subset of the domain of attraction. (2) If κ = 0 then there exists H > 0 such that if h < H then the origin is a globally asymptotically stable equilibrium of the system (13).

(3) If κ < 0 then there exists ξ > 0 such that the set {(x, x) ∈ R n × R n : N (x) + N (x) ≤ ξh -1/κ } is globally asymptotically stable w.r.t. the system [START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF].

Proof. Let us denote F (X, U ) = ( F (X 1 , U 1 ) U 2 ) for X = (X 1 , X 2 ) ∈ R n × R n and U = (U 1 , U 2 ) ∈ R n × R n and consider Ẋ = F (X, U (X)). ( 14
)
If we apply to [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF] the state feedback given by {

U 1 (X) = u(X 2 ) U 2 (X) = G(j(X 1 ), X 2 ) (15) we find { Ẋ1 = F (X 1 , u(X 2 )) Ẋ2 = G(j(X 1 ), X 2 )
which is [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF], up to a renaming of the variables. Now, [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF] with sampled input [START_REF] Nešić | Explicit computation of the sampling period in emulation of controllers for nonlinear sampled-data systems[END_REF] becomes

{ Ẋ1 (t) = F (X 1 (t), u(X 2 (t k ))) Ẋ2 (t) = G(j(X 1 (t k )), X 2 (t k ))
and the right hand side of the second line being constant, the second line can be integrated. It yields

{ Ẋ1 (t) = F (X 1 (t), u(X 2 (t k ))) X 2 (t k+1 ) = X 2 (t k ) + (t k+1 -t k )G(j(X 1 (t k )), X 2 (t k ))
which is [START_REF] Mazenc | Robustness of nonlinear systems with respect to delay and sampling of the controls[END_REF], up to a renaming of the variables. Moreover, the Assumption 11 holds for F . Therefore, we apply Theorem 12 to (14) which concludes the proof.

The double integrator

Let us consider the double integrator with a sampleddata input

ẋ(t) = [ 0 1 0 0 ] x(t) + [ 0 1 ] u(t k ), t ∈ [t k , t k+1 ) (16)
which has been studied in [START_REF] Bernuau | Robust finite-time output feedback stabilisation of the double integrator[END_REF] as a typical example and the control law [START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF] with α > 0 and k 1 , k 2 > 0. By using

u(x) = -k 1 ⌊x 1 ⌉ α -k 2 ⌊x 2 ⌉ 2α 1+α
V (x) = k1 1+α |x 1 | 1+α + x 2 2 2 , we get V (x) = -k 2 |x 2 | 1+3α 1+α
≤ 0. A straightforward application of the LaSalle's principle shows that the origin of the continuous closed-loop system ( 16)-( 17) without the sampled-data input is a globally asymptotically stable equilibrium. In this example, we consider ν = Ax with A = Diag(1, 1+α

2 ), we see that the closed-loop system ( 16)-( 17) satisfies Assumption 11 with degree κ = α-1 2 . So, we can apply Theorem 12. Moreover, we choose a constant sampling period

t k = kT, T > 0, k ∈ N ( 18
)
in order to use well known results about linear systems having a constant sampling period. For the simulations, we take k 1 = k 2 = 1.

• If we select α = 1 then the closed-loop system ( 16)-( 17)-( 18) is linear. In this case, we know that the linear closed-loop sampled-data system ( 16)-( 17)-( 18) is asymptotically stable if and only if the matrix Λ(T ) of the linear difference equation associated with ( 16)- [START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF] and defined in [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: an overview[END_REF][START_REF] Maalej | State-dependent sampling for linear time invariant systems: a discrete time analysis[END_REF] is Schur. For the double integrator, Λ(T ) is Schur if and only if T < T Schur = 2s.

• If we select α > 1 then the closed-loop system ( 16)-( 17)-( 18) is homogeneous of positive degree. Due to Theorem 12, the local asymptotic stability pertains even if we have T > T Schur = 2s. We plot in simulation the shape of the domains of attraction for α = 8 and different values of T on Figure 1 and the shape of the domains of attraction for T = 6s and different values of α on Figure 2. • If we select α < 1 then the closed-loop system ( 16)-( 17)-( 18) is homogeneous of negative degree. Due to Theorem 12, the global asymptotic set stability pertains even if T > T Schur = 2s. We plot in simulation the asymptotically stable sets for α = 0.1 and T = 4s, T = 5s and T = 6s on Figure 3.

The case of the n-integrator can be treated in the same way by using for instance the controllers given in [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF][START_REF] Hong | Finite-time stabilization and stabilizability of a class of controllable systems[END_REF].

Conclusion

In this article, an emulation approach based on inputto-state stability properties of systems satisfying an homogeneous condition is developed for the stability of nonlinear systems with sampled-data inputs. The main result shows that if it is possible to build a stabilizing feedback control for a continuous system such that the closed-loop system with sampled-data inputs satisfies the homogeneity property, then it is possible to preserve different notions of stability that depend on the degree κ for the closed-loop system with aperiodic sampled-data inputs. Its usefulness is highlighted with the case of the double integrator.

  where R is the set of real numbers. • | • | denotes the absolute value in R, ∥ • ∥ denotes the Euclidean norm on R n . • For r 1 , r 2 , . . . , r n , Diag(r 1 , . . . , r n ) denotes the diagonal matrix of dimension n × n with k th diagonal entry r k . • A measurable function d : R + → R m is locally essentially bounded if for any 0 ≤ a < b, the function d is essentially bounded on the segment [a, b]. L ∞ loc denotes the set of locally essentially bounded functions d : R + → R m . • A continuous function α : R + → R + belongs to the class K if α(0) = 0 and the function is strictly increasing. A function α : R +

Fig. 1 .Fig. 2 .

 12 Fig. 1. Domains of attraction for α = 8

Fig. 3 .

 3 Fig. 3. Asymptotically stable sets for several T