Héctor Ríos

Laurentiu Hetel

Denis Efimov

Nonlinear Impulsive Systems: 2D Stability Analysis Approach

Keywords: Nonlinear impulsive systems, Stability, 2D systems

This paper contributes to the stability analysis for nonlinear impulsive dynamical systems based on a vector Lyapunov function and its divergence operator. The new method relies on a 2D time domain representation. Different types of stability notions for a class of nonlinear impulsive systems are studied using a vector Lyapunov function approach. The results are applied to analyze the stability of a class of Lipschitz nonlinear impulsive systems based on Linear Matrix Inequalities. Some numerical examples illustrate the feasibility of the proposed approach.

Introduction

An important class of hybrid systems [START_REF] Goebel | Hybrid dynamical systems[END_REF] is represented by impulsive systems [START_REF] Bainov | Systems with Impulse Effects: Stability, Theory and Applications[END_REF]. This class of systems contain discontinuities or jumps in the state trajectories of the system governed by discrete dynamics. The dynamics of a very large variety of phenomena may be described by abrupt changes in the system state at certain instants, e.g. sample-data systems, power electronics, some models in economics, bursting rhythm models in medicine, etc. In accordance with the manner of impulses to be triggered, several kind of impulsive systems can be established: time-dependent impulsive systems, state-dependent impulsive systems, usually called reset systems, and the combination of both of them (see, e.g.

H. Ríos gratefully acknowledges the financial support from CONACyT 270504. This work was supported in part by Conseil Region Nord-Pas de Calais (ESTIREZ), the Government of Russian Federation (Grant 074-U01), the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031), and ANR ROCC-SYS (ANR-14-CE27-0008).

Email addresses: hriosb@correo.itlalaguna.edu.mx (Héctor Ríos), laurentiu.hetel@ec-lille.fr (Laurentiu Hetel), denis.efimov@inria.fr (Denis Efimov).

[2], [START_REF] Goebel | Hybrid dynamical systems[END_REF] and [START_REF] Zavalishchin | Dynamic Impulse Systems. Theory and Applications[END_REF]). In this work impulsive systems with time-triggered impulses are studied.

In the stability analysis framework of impulsive systems an important effort has been made for linear dynamics. A functional-based approach is developed for stability analysis in [START_REF] Briat | A looped-functional approach for robust stability of linear impulsive systems[END_REF]. Some dwell-time results are established by means of this method which introduces looped functional leading to Linear Matrix Inequality (LMI) conditions. In [START_REF] Naghshtabrizi | Exponential stability of impulsive systems with application to uncertain sampled-data systems[END_REF], exponential stability is proposed based on Lyapunov functions with discontinuity at the impulses for nonlinear time-varying impulsive systems. These stability conditions are mainly applied to the linear case. In the sample-data system context, there exist several constructive works where the impulsive systems are used to describe the behavior of aperiodic sample-data systems (see, e.g. [START_REF] Hu | Constrained robust sampleddata control of nonlinear uncertain systems[END_REF] and [START_REF] Sivashankar | Characterization of the L 2 -induced norm for linear systems with jumps with applications to sampled-data systems[END_REF]). In [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF], an equivalent correspondence between continuous and discrete time domains is provided.

With respect to the nonlinear impulsive systems, in [START_REF] Xu | Exponential stability with L 2 -gain condition of nonlinear impulsive switched systems[END_REF] the exponential stability and stabilization problems are considered for nonlinear impulsive switched systems with time-varying disturbances. By means of switched Lyapunov function, sufficient conditions expressed as LMIs are obtained. Recently, in [START_REF] Lee | Finite time stability of nonlinear impulsive systems and its applications in sampled-data systems[END_REF] a finite-time stabil-ity criteria has been established for nonlinear impulsive systems based on average impulse intervals. Nevertheless, to the best of our knowledge, the works dedicated to the stability conditions for nonlinear impulsive systems are still few. In this paper, since hybrid systems are inherently related to 2D times due to the continuous and discrete variables [START_REF] Goebel | Hybrid dynamical systems[END_REF], based on a 2D time domain equivalence (see e.g. [START_REF] Rogers | Control Systems Theory and Applications for Linear Repetitive Processes[END_REF] and [START_REF] Yeganefar | Lyapunov theory for 2-D nonlinear roesser models: Application to asymptotic and exponential stability[END_REF]), different types of stability notions for a class of nonlinear impulsive systems are studied using a vector Lyapunov function approach [START_REF] Lakshmikantham | Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems[END_REF]. It is worth mentioning that there exist some works related to the stability for nonlinear 2D discrete systems based on Lyapunov function approach (see [START_REF] Zhu | Stability and absolute stability of a general 2-d non-linear fm second model[END_REF] and [START_REF] Zhu | Lyapunov-type theorem of general two-dimensional nonlinear parameter-varying fm second model[END_REF]), and for nonlinear 2D systems based on vector Lyapunov function approach (see [START_REF] Emelianov | Stability and stabilization of differential nonlinear repetitive processes with applications[END_REF] and [START_REF] Emelianova | Vector Lyapunov function based stability of a class of applications relevant 2d nonlinear systems[END_REF]). However, in the last two results the boundary condition is known and decreasing. In the context of impulsive systems this means that the impulsive effects introduce stability to the system a priori. In the current work, contrarily to [START_REF] Emelianov | Stability and stabilization of differential nonlinear repetitive processes with applications[END_REF] and [START_REF] Emelianova | Vector Lyapunov function based stability of a class of applications relevant 2d nonlinear systems[END_REF]; some conditions are provided to ensure stability and convergence for impulsive systems. Moreover, note that the results given by [START_REF] Emelianov | Stability and stabilization of differential nonlinear repetitive processes with applications[END_REF] and [START_REF] Emelianova | Vector Lyapunov function based stability of a class of applications relevant 2d nonlinear systems[END_REF] are not directly applicable for impulsive systems since only repetitive processes are considered there and the system trajectories do not present any type of jump or discontinuity. Recently, in a previous work [START_REF] Ríos | Vector Lyapunov function based stability for a class of impulsive systems[END_REF], the 2D time approach was introduced to analyze the exponential stability of linear impulsive systems based on LMIs. However, only quadratic vector Lyapunov functions were considered as well as linear impulsive systems.

The main result of this work contributes to the development of a new stability analysis method for nonlinear impulsive dynamical systems based on a divergence operator of a vector Lyapunov function in a 2D time domain. Different types of stability notions for a class of nonlinear impulsive systems are studied using a vector Lyapunov function approach. The results are applied to analyze the asymptotic stability of a class of Lipschitz nonlinear impulsive system based on LMIs. Some numerical results illustrate the feasibility of the proposed approach.

The structure of this paper is as follows. The problem statement and preliminary definitions of different types of stability are given in Section 2. The main results as well as the stability conditions are described in Section 3. An LMI Approach for certain class of Lipschitz nonlinear impulsive systems is given in Section 4. A couple of simulation examples are considered in Section 5. Finally, some concluding remarks are discussed in Section 6.

Problem statement

Consider the class of impulsive dynamical systems for which the impulse times are external to the system and time-dependent, i.e.

ẋ(t) = f 1 (x(t)) , ∀t ∈ R + \ I, (1) x(t) = f 2 x(t -) , ∀t ∈ I, x(0) = x 0 (2)
where x, x 0 ∈ R n are the state of the system and the initial condition, respectively. The set of impulse times I := {t i } i∈N is a countable subset of R + with t i+1 -t i > 0, for all i ∈ N and t 0 = 0 and lim i→∞ t i = +∞ in order to avoid any zeno phenomena. The state trajectory is assumed to be right continuous and to have left limits at all times. The notation x(t - i+1 ) = lim t↑ti+1 x(t) denotes the left limit of x(t) as t goes to t i+1 from the left. The distance between the impulses, i.e. the dwelltime, is defined as T i := t i+1 -t i , and it is assumed that any sequence of impulse instants {t i } i∈N belongs to an interval, i.e. T i ∈ [T min , T max ], for all i ∈ N, where T min and T max are the minimum and maximum dwell-time, respectively. The nonlinear functions f 1 and f 2 are such that f 1 (0) = 0 and f 2 (0) = 0. It is assumed that for all x 0 ∈ R n , f 1 is such that system (1) has a unique forward solution over any time interval [t i , t i+1 ) for all i ∈ N while f 2 is a locally bounded function. In this sense, a sufficient condition for the existence of a forward solution over [t i , t i+1 ) is that f 1 is Lipschitz over the same time interval.

The proposed Vector Lyapunov function based approach relies on the embedding of system (1)-( 2) into a 2D time domain. Indeed, the entire state trajectory x(t) can be viewed as a sequence of the diagonal dynamics1 of the following 2D system:

dx t k dt = f 1 x t k , ∀t ∈ [t i , t i+1 ) , t 0 = 0, (3)
x t k+1 = f 2 x t k , ∀t = t i+1 , ∀i = k ∈ N, x 0 0 = x 0 , (4)
where

x t k : [t i , t i+1 ) × N → R n is the current state vec- tor, x ti+1 k+1 : t i+1 × N → R n represents the reset vec- tor state, x ti+1 k : t i+1 × N → R n denotes the value of x just before the impulse k + 1, for all i = k ∈ N, i.e. x ti+1 k = lim t↑ti+1 x t k , for t < t i+1 .
It is worth saying that the solutions of system (3) are unique for the diagonal dynamics, i.e. for all i = k ∈ N, t ∈ R + ; and they are represented by a sequence of functions

x ti+τ k , τ ∈ (0, t i+1 ] i , for all i = k ∈ N. Note that the discrete time k depicts the number of impulses on the system.

The goal in this work is to find the conditions for the stability and asymptotic stability of the impulsive systems (1)-( 2) by means of the 2D time representation (3)-(4).

Preliminaries

A continuous function σ : R + → R + belongs to class K if it is strictly increasing and σ (0) = 0; it belongs to class K ∞ if it is also unbounded. A continuous function β : R + × R + → R + belongs to class KL if, for each fixed s, β(r, s) ∈ K with respect to r and, for each fixed r, β(r, s) is decreasing with respect to s and β(r, s) → 0 as s → ∞. Let |q| denote the Euclidean norm of a vector q. The symbols ∧ and ∨ denote the logic operations "and " and "or ", respectively.

Let Ω be an open subset of R n , such that 0 ∈ Ω. The following stability definitions are introduced: Definition 1 A 2D system described by ( 3)-( 4) is said to be Diagonal Stable (DS) in Ω, if there exist functions

σ 1 , σ 2 ∈ K ∞ and some constants κ 1 , κ 2 > 0 such that, for all i = k ∈ N |x ti+1 k+1 | ≤ σ 1 (|x 0 |) + κ 1 , ∀x 0 ∈ Ω, ( 5
)
|x t k | ≤ σ 2 (|x ti k |) + κ 2 , ∀t ∈ [t i , t i+1 ) . (6)
Definition 2 A 2D system described by ( 3)-( 4) is said to be Asymptotically Diagonal Stable (ADS) in Ω, if there exist functions β ∈ KL and σ ∈ K ∞ such that, for all

i = k ∈ N |x ti+1 k+1 | ≤ β(|x 0 |, k), ∀x 0 ∈ Ω, (7)
|x t k | ≤ σ(|x ti k |), ∀t ∈ [t i , t i+1 ) . (8)
Definition 3 A 2D system described by (3)-( 4) is said to be Strongly Asymptotically Diagonal Stable (SADS) in Ω, if there exist functions β 1 , β 2 ∈ KL such that, for

all i = k ∈ N |x ti+1 k+1 | ≤ β 1 (|x 0 |, k), ∀x 0 ∈ Ω, ( 9
)
|x t k | ≤ β 2 (|x ti k |, t -t i ), ∀t ∈ [t i , t i+1 ) . (10)
If Ω = R n , then the 2D system is said to be globally DS (GDS ), ADS (GADS ), or SADS (GSADS ), respectively. Note that the Definition 3, i.e. strong asymptotical diagonal stability; is a particular case of the Definition 2, i.e. asymptotical diagonal stability.

Stability analysis

In order to give the stability conditions a vector Lyapunov approach is used [START_REF] Ríos | Vector Lyapunov function based stability for a class of impulsive systems[END_REF]:

V (t, x t k , x t i+1 k+1 ) =   V1(t, x t k ) V2(x t i+1 k+1 )   , (11)
where V 1 (t, x) > 0, V 2 (x) > 0, for all t ≥ 0 and x = 0, and V 1 (t, 0) = 0, V 2 (0) = 0, for all t ≥ 0. Now, the divergence operator of the candidate function V along the trajectories of system (3)-( 4) is introduced for all t ∈ [t i , t i+1 ):

divV (•) = dV1(t, x t k ) dt + V2(x t i+1 k+1 ) -V2(x t i+1 k ), (12)
i.e. V 1 is differentiable with respect to the continuous time t while the difference in V 2 is calculated in the discrete time k.

Minimum dwell-time results

Let Ω be an open neighborhood of the origin. Then, based on the vector Lyapunov function and its divergence, the following theorem is established:2

Theorem 1 (Diagonal stability -Minimum dwelltime). Assume that there exist positive constants

ε ∈ (0, 1), a 1 , a 2 , b 1 , b 2 , c 1 , c 2 , c 3 , c 4
and c 5 such that the vector Lyapunov function V (•) and its divergence along the trajectories of the system (3)-( 4) satisfy the following inequalities for all i = k ∈ N,

x t k ∈ Ω and t ∈ [t i , t i+1 ) c1|x t k | a 1 ≤ V1(t, x t k ) ≤ c2|x t k | a 2 , ( 13
)
c3|x p k | b 1 ≤ V2(x p k ) ≤ c4|x p k | b 2 , ∀p = ti, ti+1, (14)
divV

≤ -c5(|x t k | a 2 + |x t i+1 k | b 2 ), ( 15
) c4 ≤ c5, (16) b1
> a2 ∧ Tmin = c2 c5 γ < Ti, (17)
where γ = -ln[c

a2/b1 3 (1 -ε) /(c 5 +c a2/b1 3 (1-ε))].
Then, the 2D system (3)-( 4) is DS in Ω for any sequence

T i ∈ [T min , ∞).
Proof. From the divergence definition and inequalities (13), ( 14) and [START_REF] Nešić | Stability and performance of siso control systems with first-order reset elements[END_REF], it follows that

dV1(t, x t k ) dt ≤ -c5(|x t k | a 2 + |x t i+1 k | b 2 ) -V2(x t i+1 k+1 ) + V2(x t i+1 k ), ≤ -βV1(t, x t k ) + λV2(x t i+1 k ) -V2(x t i+1 k+1 ), (18)
where λ = (c 4 -c 5 )/c 4 and β = c 5 /c 2 . By means of the comparison principle, with respect to the time t, from [START_REF] Seuret | A novel stability analysis of linear systems under asynchronous samplings[END_REF], for all t ∈ [t i , t i+1 ), it is obtained that

V1(t, x t k ) ≤ e -β(t-t i ) V1(ti, x t i k ) + t t i e -β(t-τ ) × λV2(x t i+1 k ) -V2(x t i+1 k+1 ) dτ , = e -β(t-t i ) V1(ti, x t i k ) + ρ i (t) λV2(x t i+1 k ) -V2(x t i+1 k+1 ) , (19)
where

ρ i (t) = (1 -e -β(t-ti) )/β ≥ 0 for all t ∈ [t i , t i+1 ].
Now, the proof is split into two parts: convergence in the impulse times, and boundedness between the impulses, i.e. eqs. ( 5) and ( 6) from Definition 1, respectively. a) Convergence in the impulse times. Evaluating [START_REF] Sivashankar | Characterization of the L 2 -induced norm for linear systems with jumps with applications to sampled-data systems[END_REF] in t = t i+1 , it gives

V1(ti+1, x t i+1 k ) ≤ e -βT i V1(ti, x t i k ) + ρ i (ti+1)λV2(x t i+1 k ) -ρ i (ti+1)V2(x t i+1 k+1 ). (20)
From the inequalities ( 13) and ( 14), it follows that for all

i = k ∈ N V α 1 2 (x p k ) c α 1 4 ≤ |x p k | a 1 ≤ V1(p, x p k ) c1 , ∀p = ti, ti+1, (21)
V

β 1 1 (p, x p k ) c β 1 2 ≤ |x p k | b 1 ≤ V2(x p k ) c3 , ∀p = ti, ti+1, (22)
where

α 1 = a 1 /b 2 and β 1 = b 1 /a 2 . From (21), it is given that c 1 c -α1 4 V α1 2 (x ti+1 k ) ≤ V 1 (t i+1 , x ti+1 k
), and therefore from [START_REF] Sturm | Using SEDUMI 1.02, a MATLAB toolbox for optimization over symmetric cones[END_REF], it is obtained that

ρ i (ti+1)V2(x t i+1 k+1 ) ≤ e -βT i V1(ti, x t i k ) + ρ i (ti+1)λV2(x t i+1 k ) -c1c -α 1 4 V α 1 2 (x t i+1 k
). ( 23)

Consider that c 5 ≥ c 4 , i.e. λ ≤ 0 and ( 16) holds; using [START_REF] Yeganefar | Lyapunov theory for 2-D nonlinear roesser models: Application to asymptotic and exponential stability[END_REF] and ( 23) it follows that

V2(x t i+1 k+1 ) ≤ ϕ i V 1/β 1 2 (x t i k ), (24)
where

ϕ i = c 2 e -βTi /c 1/β 1 3
ρ i (t i+1 ); and then, by induction, it is obtained that

V2(x t i+1 k+1 ) ≤ ϕ p(k) i V (1/β 1 ) k 2 (x0) , (25)
where p(k) = k j=0 (1/β 1 ) j . Assume that β 1 > 1, i.e. b 1 > a 2 , then p(k) is a geometric series; that there exists a ε ∈ (0, 1), and that the following condition holds

ϕ i = c2e -βT i c 1/β 1 3 ρ i (ti+1) < 1 -ε ⇔ Tmin > c2 c5 γ.
Note that the last inequality is the same that ( 17) with γ = -ln[c

1/β 1 3 (1 -ε) /(c 5 + c 1/β 1 3 (1 -ε))]. Then, tak- ing into account that ρ i (t i+1 ) = c 2 (1 -e -βTi )/c 5 , one obtains from (25) that V2(x t i+1 k+1 ) ≤ ϕ i (V2 (x0) + 1) ≤ c5 (V2 (x0) + 1) c 1/β 1 3 (1 -e -γ )
. Thus, using ( 21) and [START_REF] Yeganefar | Lyapunov theory for 2-D nonlinear roesser models: Application to asymptotic and exponential stability[END_REF], it follows that for all

i = k ∈ N |x t i+1 k+1 | ≤ σ1(|x0|) + κ1, ∀x0 ∈ Ω, ( 26
)
where

σ 1 (|x 0 |) = c 4 c 5 |x 0 | b 2 c (a 2 +b 1 )/b 1 3 (1 -e -γ ) 1 b 1 , (27)
κ 1 = c 5 c (a 2 +b 1 )/b 1 3 (1 -e -γ ) 1 b 1 . ( 28
)
Therefore, it is concluded that the trajectories of system (3)-( 4) are stable (Definition 1, eq. ( 5)), under constraints ( 16) and [START_REF] Rogers | Control Systems Theory and Applications for Linear Repetitive Processes[END_REF].

b) Boundedness between the impulses. From [START_REF] Sivashankar | Characterization of the L 2 -induced norm for linear systems with jumps with applications to sampled-data systems[END_REF] the negative term ρ i (t)V 2 (x ti+1 k+1 ) can be disregarded, then it is given that

V1(t, x t k ) ≤ e -β(t-t i ) V1(ti, x t i k ) + ρ i (ti+1)λV2(x t i+1 k ). ( 29
)
Consider that c 5 ≥ c 4 , i.e. λ ≤ 0 and ( 16) holds; from (29), it follows that V 1 (t,

x t k ) ≤ e -β(t-ti) V 1 (t i , x ti k ) and boundedness is given, i.e. |x t k | ≤ σ 2 (|x ti k |), ∀t ∈ [t i , t i+1 ), where σ 2 (|x ti k |) = (c 2 |x ti k | a2 /c 1 ) 1/a1
. Thus, during each interval between impulses the trajectories of the system are bounded (Definition 1, eq. ( 6)), and due to the stability property given by (5), the 2D system described by ( 3)-( 4) is DS in Ω.

Remark 1 Note that if (26) is satisfied for all x 0 ∈ Ω = R n , then one gets global diagonal stability.

Let us consider the particular case in which b 1 = a 2 , i.e. β 1 = 1. Then, the following result is obtained.

Corollary 2 (Strong asymptotic diagonal stability -Minimum dwell-time). Assume that there exist positive constants ε ∈ (0, 1), a 1 , a 2 , b 1 , b 2 , c 1 , c 2 , c 3 , c 4 and c 5 such that the vector Lyapunov function V (•) and its divergence along the trajectories of the system (3)-( 4) satisfy (13), ( 14), ( 15), [START_REF] Ríos | Vector Lyapunov function based stability for a class of impulsive systems[END_REF], and the following constraints, for all i = k ∈ N,

x t k ∈ Ω and t ∈ [t i , t i+1 ) b1 = a2 ∧ Tmin = c2 c5 γ < Ti, (30)
where

γ = -ln[c 3 (1 -ε) /(c 5 + c 3 (1 -ε))].
Then, the 2D system (3)-( 4) is SADS in Ω for any sequence

T i ∈ [T min , ∞).
Proof. Note that ( 19), ( 20), ( 21), ( 22) and ( 23) hold for any possible value of a 1 , a 2 , b 1 and b 2 . Then, recalling that c 5 ≥ c 4 , i.e. λ ≤ 0; from [START_REF] Zavalishchin | Dynamic Impulse Systems. Theory and Applications[END_REF] it follows that for

β 1 = 1, V 2 (x ti+1 k+1 ) ≤ ϕ i V 2 (x ti k )
, where ϕ i = c2e -βT i c3ρ i (ti+1) ; and then, by induction, it is given

V2(x t i+1 k+1 ) ≤ ϕ i k+1 V2 (x0) , (31)
which is decreasing if the condition

T min > c 2 γ/c 5 , with γ = -ln[c 3 (1 -ε)/(c 5 + c 3 (1 -ε))]; holds. Therefore, it is obtained that for all i = k ∈ N |x t i+1 k+1 | ≤ β 1 (|x0|, k), ∀x0 ∈ Ω, ( 32
)
with β 1 (|x 0 |, k) = c 1/b1 κ (k+1)/b1 1 |x 0 | b2/b1
, where c = c 4 /c 3 , and, from the fact that T min > c 2 γ/c 5 ; κ 1 = c 5 /c 3 (1 -e -γ ) < 1. Hence, the trajectories of system (3)-( 4) converge to zero in the impulse times under constraints T min > c 2 γ/c 5 and b 1 = a 2 , if [START_REF] Ríos | Vector Lyapunov function based stability for a class of impulsive systems[END_REF] holds.

Note that the boundedness between the impulses can be proven in the same way as in Theorem 1. In fact, from [START_REF] Sivashankar | Characterization of the L 2 -induced norm for linear systems with jumps with applications to sampled-data systems[END_REF], it follows that for c 5 ≥ c 4 , i.e. λ ≤ 0, both negative terms ρ i (t)λV 2 (x ti+1 k

) and

ρ i (t)V 2 (x ti+1 k+1 ) can be disregarded, then V 1 (t, x t k ) ≤ e -β(t-ti) V 1 (t i , x ti k ) holds and boundedness is provided, i.e. |x t k | ≤ β 2 (|x ti k |, t -t i ), ∀t ∈ [t i , t i+1 ), where β 2 (|x ti k |, t -t i ) = (c 2 e -β(t-ti) |x ti k | a2 /c 1 ) 1/a1
. Thus, during each interval between impulses the trajectories of the system are bounded (Definition 3, eq. ( 10)), and due to the convergence property given by ( 9), the 2D system described by ( 3)-( 4) is SADS in Ω. Hence, the corollary has been proven.

Remark 2 As in Theorem 1, note that the results become global if (32) holds for all x 0 ∈ Ω = R n .

Maximum or ranged dwell-time results

Now, the following results describe the diagonal stability under maximum or ranged dwell-time conditions which come from the case in which condition ( 16) is not satisfied, i.e. for the case c 4 > c 5 .

Theorem 3 (Diagonal stability -Maximum/Ranged dwell-time). Assume that there exist positive constants ε ∈ (0, 1), a 1 , a 2 , b 1 , b 2 , c 1 , c 2 , c 3 , c 4 and c 5 such that the vector Lyapunov function V (•) and its divergence along the trajectories of the system (3)-( 4) satisfy (13), ( 14), [START_REF] Nešić | Stability and performance of siso control systems with first-order reset elements[END_REF], and the following constraints, for all i = k ∈ N,

x t k ∈ Ω and t ∈ [t i , t i+1 ) c4 > c5 ∧ a1 > b2, (33)
c2c α 1 -1 4 (c4 -c5) ≤ c1c5 ∨ Ti < c2 c5 α = Tmax, ( 34
) b1 > a2 ∧ Tmin = c2 c5 γ < Ti, (35)
where γ = -ln[c

a2/b1 3 (1 -ε) /(c 5 + c a2/b1 3 (1 -ε))],
and

α = -ln c2c (a 1 -b 1 )/b 1 4 (c4 -c5) -c1c5 c2c (a 1 -b 1 )/b 1 4 (c4 -c5) , for all c 2 c (a1-b1)/b1 4 (c 4 -c 5 ) > c 1 c 5 . Then, the 2D sys- tem (3)-(4) is DS in Ω for any sequence T i ∈ [T min , T max ].
Proof. From (33) it follows that c 4 > c 5 , i.e. λ ∈ (0, 1); and

α 1 > 1, i.e. a 1 > b 2 . Then, it is satisfied that V2(x t i+1 k ) < V α 1 2 (x t i+1 k
), for V2(x

t i+1 k ) > 1, V2(x t i+1 k ) ≥ V α 1 2 (x t i+1 k
), for V2(x

t i+1 k ) ≤ 1.
It is worth mentioning that ( 19), ( 20), ( 21), ( 22) and ( 23) hold for any possible value of a 1 , a 2 , b 1 and b 2 . Therefore, similar arguments as in the proof of Theorem 1 may be used to prove this theorem.

a) Convergence in the impulse times. Let us consider two different cases for the convergence proof, i.e. V 2 (x ti+1 k

) > 1 and V 2 (x ti+1 k

) ≤ 1. Therefore, from [START_REF] Zavalishchin | Dynamic Impulse Systems. Theory and Applications[END_REF], it follows that when V 2 (x

ti+1 k ) > 1 ρ i (ti+1)V2(x t i+1 k+1 ) ≤ c2e -βT i c 1/β 1 3 V 1/β 1 2 (x t i k ) + ρ i (ti+1)λ -c1c -α 1 4 V2(x t i+1 k ). ( 36
)
Note that if the constraint ρ i (t i+1 )λ < c 1 c -α1 4 holds, then the term depended on V 2 (x ti+1 k

) can be disregarded. In this sense, in order to satisfy such a constraint, recalling that β = c 5 /c 2 , λ = (c 4 -c 5 )/c 4 and ρ i (t i+1 ) = c 2 (1 -e -βTi )/c 5 , the following condition is founded

c2c α 1 -1 4 1 -e -βT i (c4 -c5) < c1c5, 1 -e -βT i < c1c5 c2c α 1 -1 4 (c4 -c5)
.

Thus, it is clear that if c 2 c α1-1 4 (c 4 -c 5 ) ≤ c 1 c 5 holds then ρ i (t i+1 )λ < c 1 c -α1 4 is trivially satisfied. Otherwise e -βT i > c2c α 1 -1 4 (c4 -c5) -c1c5 c2c α 1 -1 4 (c4 -c5) ⇔ Tmax < c2 c5 α,
where

α = -ln c2c (a 1 -b 1 )/b 1 4 (c4 -c5) -c1c5 c2c (a 1 -b 1 )/b 1 4 (c4 -c5) > 0,
for all c 2 c α1-1

4

(c 4 -c 5 ) > c 1 c 5 . Note that these two possibilities, i.e. c 2 c α1-1 4 (c 4 -c 5 ) ≤ c 1 c 5 or T i < c 2 α/c 5 , are represented by (34) in Theorem 3. Therefore, if one of them is satisfied, from (36), one obtains

V 2 (x ti+1 k+1 ) ≤ ϕ i V 1/β 1 2
(x ti k ), and using similar arguments as in Theorem 1, if β 1 > 1, i.e. b 1 > a 2 , and

T i > c 2 γ/c 5 , with γ = -ln[c a2/b1 3 (1 -ε) /(c 5 + c a2/b1 3 (1 -ε))], it follows that for all i = k ∈ N, |x ti+1 k+1 | ≤ σ 1 (|x 0 |) + κ 1 , ∀x 0 ∈ Ω, for V 2 (x ti+1 k ) > 1 with σ 1 (|x0|) = c4c5|x0| b 2 c (a 2 +b 1 )/b 1 3 (1 -e -γ ) 1 b 1 , κ 1 = c5 c (a 2 +b 1 )/b 1 3 (1 -e -γ ) 1 b 1 .
Therefore, it is possible to conclude that when V 2 (x ti+1 k

) > 1 the trajectories of system (3)-( 4) are bounded.

To complete the convergence proof let us consider the second case, i.e. V 2 (x ti+1 k ) ≤ 1. Using [START_REF] Yeganefar | Lyapunov theory for 2-D nonlinear roesser models: Application to asymptotic and exponential stability[END_REF] and dividing by ρ i (t i+1 ) [START_REF] Zavalishchin | Dynamic Impulse Systems. Theory and Applications[END_REF], it follows that for V 2 (x

ti+1 k ) ≤ 1, V 2 (x ti+1 k+1 ) ≤ ϕ i V 1/β 1 2 (x ti k ) + λ. Then, by induction, it is given that 5 V2(x t i+1 k+1 ) ≤ ϕ p(k) i V (1/β 1 ) k 2 (x0) + k l=0 ϕ p(l)-(1/β 1 ) l i λ (1/β 1 ) l , where p(k) = k j=0 (1/β 1 ) j is a geometric series if β 1 > 1.
Then, taking into account that β 1 > 1, T min > c 2 γ/c 5 , ρ i (t i+1 ) = c 2 (1 -e -βTi )/c 5 and λ ∈ (0, 1); the previous inequality can be majorized by the following expression

V2(x t i+1 k+1 ) ≤ c5 (V2 (x0) + 2) c 1/β 1 3 (1 -e -γ ) 1 -λ 1/β 1 .
Thus, using ( 21) and ( 22), one gets that for all i = k ∈ N, |x

ti+1 k+1 | ≤ σ 1 (|x 0 |) + κ 1 , ∀x 0 ∈ Ω, for V 2 (x ti+1 k ) ≤ 1 with σ1(|x0|) = c -(a 2 +b 1 )/b 1 3 c4c5|x0| b 2 (1 -e -γ )(1 -λ a 2 /b 1 ) 1 b 1 , κ1 = 2c -(a 2 +b 1 )/b 1 3 c5 (1 -e -γ )(1 -λ a 2 /b 1 ) 1 b 1 .
Note that the above bound for x ti+1 k+1 is also valid when

V 2 (x ti+1 k
) > 1. Therefore, it is concluded that the trajectories of system (3)-( 4) are bounded, under constraints (33), (34), and (35), for all possible values of V 2 (x ti+1 k

). Thus, stability in the impulse times is proven. b) Boundedness between the impulses. From (20), disregarding the negative term ρ i (t i+1 )V 2 (x ti+1 k+1 ), and recalling that c 1 c -α1

4 V α1 2 (x ti+1 k ) ≤ V 1 (t i+1 , x ti+1 k
); one gets

c1c -α 1 4 V α 1 2 (x t i+1 k ) ≤ e -βT i V1(ti, x t i k ) + ρ i (ti+1)λV2(x t i+1 k ). (37)
Once again, let us consider two cases for the boundedness proof, i.e. V 2 (x

ti+1 k ) > 1 and V 2 (x ti+1 k
) ≤ 1. Then, from (37), it follows that for all V 2 (x

ti+1 k ) > 1 V2(x t i+1 k ) ≤ 1 c1c -α 1 4 -ρ i (ti+1)λ V1(ti, x t i k ). ( 38
) Note that ρ i (t i+1 )λ < c 1 c -α1 4
has to hold in order to satisfy inequality (38). However, as it was previously described, if

c 2 c α1-1 4 (c 4 -c 5 ) ≤ c 1 c 5 holds then ρ i (t i+1 )λ < c 1 c -α1 4 is trivially satisfied, otherwise T max < c 2 α/c 5 . Thus, applying (38) in (20) with the negative term ρ i (t)V 2 (x ti+1 k+1 ) disregarding; recalling that ρ i (t i+1 )λ ≤ c 2 /c 5 , it is given that V1(t, x t k ) ≤ e -β(t-t i ) + ρ i (ti+1)λ c1c -α 1 4 -ρ i (ti+1)λ V1(ti, x t i k ), ≤ c1c5 + c2c α 1 4 c5 (c1 -ρ i (ti+1)λc α 1 4 ) V1(ti, x t i k ). ( 39
)
Therefore, from (39), boundedness is obtained, i.e.

|x t k | ≤ σ 2 (|x ti k |), ∀t ∈ [t i , t i+1 ), for V 2 (x ti+1 k ) > 1 with σ 2 (|x t i k |) =   c2 c1c5 + c2c a 1 /b 2 4 c1c5 c1 -λc a 1 /b 2 4 |x t i k | a 2   1/a 1
, where = c 2 (1 -e -γ )/c 5 , for all i = k ∈ N. The boundedness proof is completed analyzing the case when

V 2 (x ti+1 k ) ≤ 1. From (37) it is obtained c1c -α 1 4 V α 1 2 x t i+1 k ≤ e -βT i V1(ti, x t i k ) + ρ i (ti+1)λ, V2(x t i+1 k ) ≤ c4 c 1/α 1 1 V 1/α 1 1 (ti, x t i k ) + c 1/α 1 2 c4 (c1c5) 1/α 1 . ( 40
)
Thus, applying (40) in ( 20) with the negative term

ρ i (t)V 2 (x ti+1
k+1 ) disregarding, it follows that

V1(t, x t k ) ≤ e -β(t-t i ) V1(ti, x t i k ) + c2c4 c 1/α 1 1 c5 V 1/α 1 1 (ti, x t i k ) + c α 1 +1/α 1 2 c4 c 1/α 1 1 c α 1 +1/α 1 5 . (41)
Therefore, from (41), it is given that for all i = k ∈ N,

|x t k | ≤ σ 2 (|x ti k |) + κ 2 , ∀t ∈ [t i , t i+1 ), for V 2 (x ti+1 k ) ≤ 1 with σ2(|x t i k |) = c2 c1 |x t i k | a 2 + c c1 |x t i k | a 2 b 2 /a 1 1/a 1 , c = c (a 1 +b 2 )/a 1 2 c4/c b 2 /a 1 1 c5, κ2 = c (a 1 +b 2 )/a 1 2 c4/(c1c5) (a 1 +b 2 )/a 1 1/a 1 .
Thus, one can take the worst case between (39) and (41), and conclude that during each interval between impulses the trajectories of the system are bounded (Definition 1, eq. ( 10)) for every possible value of V 2 (x ti+1 k

). Therefore, due to the stability property given by ( 9), the 2D system is DS in Ω.

= k ∈ N, |x t k | ≤ 1 and t ∈ [t i , t i+1 ).
Finally, let us consider the particular case in which a 1 = b 2 and b 1 = a 2 , i.e. α 1 = 1 and β 1 = 1, respectively. Then, the following corollary is proposed.

Corollary 4 (Asymptotic diagonal stability -Maximum/Ranged dwell-time). Assume that there exist positive constants ε ∈ (0, 1), a 1 , a 2 , b 1 , b 2 , c 1 , c 2 , c 3 , c 4 and c 5 such that the vector Lyapunov function V (•) and its divergence along the trajectories of the system (3)-( 4) satisfy (13), ( 14), [START_REF] Nešić | Stability and performance of siso control systems with first-order reset elements[END_REF], and the following constraints, for all i = k ∈ N,

x t k ∈ Ω and t ∈ [t i , t i+1 ) c4 > c5 ∧ a1 = b2, (42) c2 (c4 -c5) ≤ c1c5 ∨ Ti < c2 c5 α = Tmax, ( 43
) b1 = a2 ∧ Tmin = c2 c5 γ < Ti, ( 44
)
where γ = -ln[c 3 (1 -ε)/(c 5 + c 3 (1 -ε))],
and

α = -ln c2 (c4 -c5) -c1c5 c2 (c4 -c5) ,
for all c 2 (c 4 -c 5 ) > c 1 c 5 . Then, the 2D system (3)-( 4) is ADS in Ω for any sequence

T i ∈ [T min , T max ].
Proof. Recalling that c 4 > c 5 , i.e. λ ∈ (0, 1); using [START_REF] Yeganefar | Lyapunov theory for 2-D nonlinear roesser models: Application to asymptotic and exponential stability[END_REF] and ( 23) it follows that

ρ i (ti+1)V2(x t i+1 k+1 ) ≤ c2e -βT i c3 V2(x t i k ) + ρ i (ti+1)λ -c1c -1 4 V2(x t i+1 k ).
Then, as it was previously demonstrated, if

c 2 (c 4 -c 5 ) ≤ c 1 c 5 holds, then ρ i (t i+1 )λ < c 1 c -1
4 is trivially satisfied, otherwise T max < c 2 α/c 5 , with α defined as in Theorem 3 for α 1 = 1; and then one obtains

V 2 (x ti+1 k+1 ) ≤ ϕ i V 2 (x ti k )
, and by means of similar arguments as in Corollary 2, if

T min > c 2 γ/c 5 , with γ = -ln[c 3 (1 -ε)/(c 5 + c 3 (1 -ε))]; holds, then it is obtained that for all i = k ∈ N |x t i+1 k+1 | ≤ β(|x0|, k), ∀x0 ∈ Ω, (45)
with β(|x 0 |, k) = c 1/b1 κ (k+1)/b1 |x 0 | a1/b1 , where c = c 4 /c 3 , and κ = c 5 /c 3 (1 -e -γ ) < 1. Hence, the trajectories of system (3)-( 4) converge to zero in the impulse times under constraints

T min > c 2 γ/c 5 , a 1 = b 2 , b 1 = a 2 and ρ i (t i+1 )λ < c 1 c -1 4 , if c 4 > c 5 holds.
The boundedness between the impulses is proven as follows. From (37) it is given that

c1c -1 4 V2(x t i+1 k ) ≤ e -βT i V1(ti, x t i k ) + ρ i (ti+1)λV2(x t i+1 k ),
and then, it follows that V2(x

t i+1 k ) ≤ 1 c1c -1 4 -ρ i (ti+1)λ V1(ti, x t i k ). (46)
Thus, applying (46) in ( 20) with the negative term

ρ i (t)V 2 (x ti+1 k+1
) disregarding, and recalling that ρ i (t i+1 )λ ≤ c 2 /c 5 , it is given that

V1(t, x t k ) ≤ c1c5 + c2c4 c5 (c1 -ρ i (ti+1)λc4) V1(ti, x t i k ). (47)
Therefore, from (47), boundedness is obtained, i.e.

|x t k | ≤ σ(|x ti k |), ∀t ∈ [t i , t i+1 ), where σ(|x t i k |) = c2 (c1c5 + c2c4) c1c5 (c1 -λc4) 1/a 1 |x t i k | a 2 /a 1 , with = c 2 (1 -e -γ )/c 5 , for all i = k ∈ N.
Hence, the trajectories of the system are bounded (Definition 2, eq. ( 8)), and due to the convergence property given by ( 7), the 2D system described by ( 3)-( 4) is ADS in Ω. Thus, the corollary has been proven.

Remark 4

Since a 1 = b 2 and b 1 = a 2 , the results become global if (45) holds for all x 0 ∈ R n .

Note that, from the statements given in Theorem 3 and Corollary 4, the diagonal stability is straightforward for the case a 1 > b 2 and b 1 = a 2 .

Contrary to the results given by [START_REF] Goebel | Hybrid dynamical systems[END_REF] (Theorem 3.18, Proposition 3.24, Proposition 3.27) and [START_REF] Nešić | Stability and performance of siso control systems with first-order reset elements[END_REF], where asymptotic stability is obtained by means of a single Lyapunov function whose derivative is negative definite; the statements given in this work rely on a vector Lyapunov function. However, in our case the divergence operator, and not the derivative of a Lyapunov function, needs to satisfy condition (15), i.e. be negative definite.

Particularly, the proposed results for minimal dwelltime are consistent with the ones in Proposition 3.24 (Persistent Flowing); while for maximum and ranged dwell-time with the ones in Proposition 3.27 (Persistent Jumping) in [START_REF] Goebel | Hybrid dynamical systems[END_REF], respectively.

LMI approach for a class of Lipschitz nonlinear impulsive systems

In this section an LMI procedure to apply the main results is given to deal with impulsive systems containing linear terms but also Lipschitz nonlinearities.

Consider the following nonlinear impulsive system, consistent with (3)-(4), i.e.

dx t k dt = Ax t k + φ 1 (x t k ), ∀t ∈ [ti, ti+1) , (48)
x t k+1 = Ex t k + φ 2 (x t k ), ∀t = ti+1, ∀i = k ∈ N, (49)
where A ∈ R n×n and E ∈ R n×n are constant matrices whilst φ 1 (x t k ) : Ω → R n and φ 2 (x ti+1 k

) : Ω → R n are Lipschitz nonlinearities for all i = k ∈ N and t ∈ [t i , t i+1 ), i.e.

|φ 1 (x t k ) -φ 1 (x t k )| ≤ ψ 1 |x t k -xt k |, |φ 2 (x t i+1 k ) -φ 2 (x t i+1 k )| ≤ ψ 2 |x t i+1 k - xt i+1 k |,
where ψ 1 and ψ 2 are known positive constants for all x t k , xt k ∈ Ω. Define the vector Lyapunov function as follows

V (t, x t k , x t i+1 k+1 ) =   x t k T P1(t)x t k x t i+1 k+1 T P2(k + 1)x t i+1 k+1   , (50) 7
where, inspired by [START_REF] Allerhand | Robust stability and stabilization of linear switched systems with dwell time[END_REF] and [START_REF] Briat | Convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled-data systems under dwell-time constraints[END_REF] Then, based on Corollary 2 (minimum dwell-time) or 4 (ranged dwell-time), the following result is stated.

Corollary 5 (LMI approach). Consider the vector Lyapunov function (50). If there exist matrices P 11 = P T 11 > 0, P 12 = P T 12 > 0, P 21 = P T 21 > 0 and P 22 = P T 22 > 0, and a constant c 5 > 0 such that

  -µ 1 I P11 + P12 P11 + P12 Θ1(θ)   ≤ 0, ( 51
)   -µ 1 I P11 P11 Θ2(θ)   ≤ 0,   -µ 2 I P21E E T P21 Υ(θ)   ≤ 0, (52)
with Proof. From divergence definition it is obtained that for all t ∈ [t i , t i+1 )

Θ1(θ) = (P11 + P12)A + A T (P11 + P12) -θ -1 P12 +(µ 1 ψ 2 1 + c5)I, Θ2(θ) = P11A + A T P11 -θ -1 P12 + (µ 1 ψ 2 1 + c5)I, Υ(θ) = E T P21E -(P21 + θP22)
divV ≤ x t k T P1(t)A + A T P1(t) + dP1(t) dt x t k + x t i+1 k T E T P2(k + 1)E -P2(k) x t i+1 k + 2x t k T P1(t)φ 1 + 2x t i+1 k T E T P2(k + 1)φ 2 + φ T 2 P2(k + 1)φ 2 .
Applying the well-known inequality 2X T Y ≤ µ -1 X T X+ µY T Y for any scalar µ > 0, and taking into account that φ

T 1 φ 1 ≤ ψ 2 1 |x t k | 2 and φ T 2 φ 2 ≤ ψ 2 2 |x ti+1 k
| 2 hold due to the Lipschitz condition; it is given that divV is bounded as follows

divV ≤ x t k T Ξ1(t)x t k + x t i+1 k T Ξ2(k)x t i+1 k
, where

Ξ1(t) = P1(t)A + A T P1(t) + dP1(t) dt + µ -1 1 P 2 1 (t) + µ 1 ψ 2 1 I, Ξ2(k) = E T P2(k + 1)E -P2(k) + µ -1 2 E T P 2 2 (k + 1)E +ψ 2 2 (µ 2 I + P2(k + 1)),
for any scalars µ 1 , µ 2 > 0. Then, by Schur's complement, it follows that divV satisfies [START_REF] Nešić | Stability and performance of siso control systems with first-order reset elements[END_REF] if the following matrix inequality, as well as the right one in (52), holds, i.e.

  -µ 1 I P11 + (T i -τ t k ) T i P12 P11 + (T i -τ t k ) T i P12 Θ(t)   ≤ 0, (53)
where

Θ(t) = P11A + A T P11 + (Ti -τ t k ) Ti (P12A+A T P12)- 1 Ti P12+(µ 1 ψ 2 1 +c5)I,
for all θ ∈ {T min , T max }. Since (53) is affine in τ t k , its negative definiteness is given by the negativeness over the finite set τ t k ∈ {0, T i }; hence, the LMI (51) and the left one in (52) are provided.

Hence, divV satisfies [START_REF] Nešić | Stability and performance of siso control systems with first-order reset elements[END_REF], for a 1 = b 2 = 2 and b 1 = a 2 = 2, if the LMIs (51) and (52) hold for the finite set θ ∈ {T min , T max }, with c 5 > 0.

Then, if all the conditions of Corollary 2 (minimum dwell-time) or 4 (ranged dwell-time) hold for c 1 = λ min (P 11 ), c 2 = λ max (P 11 + P 12 ), c 3 = λ min (P 21 ), and c 4 = λ max (P 21 + θP 22 ); the Lipschitz nonlinear impulsive system is SADS or ADS (GSADS or GADS if φ 1 and φ 2 are globally Lipschitz).

For the particular case of linear time-invariant systems, our method can be seen as a generalization of the result in [START_REF] Briat | Convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled-data systems under dwell-time constraints[END_REF]. One may remark that taking P 1 and P 2 in the same form for the statements given by Corollary 5, leads to the conditions given by Theorem 2.2 (ranged dwelltime) and 2.3 (minimum dwell-time) in [START_REF] Briat | Convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled-data systems under dwell-time constraints[END_REF]. However, note that P 1 and P 2 are not independent with each other since they are coupled by means of the constants c 1 , c 2 , c 3 , c 4 and c 5 , which determine the stability and the type of dwell-time condition.

Note that Corollary 5 provides an easy way to deal with stability of Lipschitz nonlinear impulsive systems by means of simple quadratic Lyapunov functions and the Λ-inequality. In this sense, more complex tools like sumof-squares [START_REF] Briat | Convex conditions for robust stability analysis and stabilization of linear aperiodic impulsive and sampled-data systems under dwell-time constraints[END_REF], convex characterizations [5], or loopedfunctional approach [START_REF] Briat | A looped-functional approach for robust stability of linear impulsive systems[END_REF], may be applied to improve the application of our method as well as alternative inequalities to Λ-inequality [START_REF] Briat | Linear Parameter-Varying and Time-Delay Systems -Analysis, Observation, Filtering[END_REF].

Simulation examples

Example 1: Globally Lipschitz nonlinearities

Consider the nonlinear impulsive system (48)-(49) with

A =   1 2 1 3   , E =   -0.58 0.3 0.4 -0.68   , φ 1 (x t k ) =   sin(x1 t k ) 0   , φ 2 (x t k ) = 0.3   sin(x1 t k ) sin(x2 t k )   .
and initial conditions t 0 = 0, x 0 = (2 -2) T . Note that matrix A is not Hurwitz whilst matrix E is Schur, and the nonlinearities are globally Lipschitz.

Then, applying the statements given by Corollary 5 with µ 1 = µ 2 = 0.01, ψ 1 = 1 and ψ 2 = 0.3, and using SeDuMi solver among YALMIP in MATLAB [START_REF] Sturm | Using SEDUMI 1.02, a MATLAB toolbox for optimization over symmetric cones[END_REF][START_REF] Löfberg | Yalmip: A toolbox for modeling and optimization in MATLAB[END_REF], a solution for the corresponding LMIs is founded, i.e. and 5, the nonlinear impulsive system is GADS for all 0.1913 > T i > 0. It is easy to check that c 4 > c 5 and c 2 (c 4 -c 5 ) > c 1 c 5 hold, then the case for asymptotic diagonal stability is obtained. Some simulation results show that the system is unstable for T i ≥ 0.2140.

Example 2: A more general case

In order to depict the statements given by Theorem 3, let us consider the scalar nonlinear impulsive system given by ( 3)-(4) with

f1(x t k ) = t k 3 , ∀t ∈ [ti, ti+1) , t0 = 0 f2(x t k+1 ) = 0.8x t k 2 , ∀t = ti+1, ∀i, k ∈ N, x0 = 0.5.
Note that in this case the nonlinear impulsive system does not have a particular structure as in Example 1. However, following similar arguments, it is still possible to provided some LMIs to prove the stability of the system.

Define the vector Lyapunov function as follows

V (t, x t k , x t i+1 k+1 ) =   1 2 x t k 2 + (T i -τ t k )p 1 2T i x t k 2 x t i+1 k+1 1 2   ,
where p 1 > 0 is a positive constant and dτ t k /dt = 1, for all t ∈ [t i , t i+1 ), τ ti+1 k+1 = 0. Note that x ti+1 k 1 2 is welldefined for all x 0 ≥ 0. Let us consider the local case in which the trajectory satisfies |x t k | ≤ 1. Then, it is easy to check that for all |x t k | ≤ 1, the following inequalities hold: From divergence definition it is obtained that for all t ∈ [t i , t i+1 )

c1|x t k | 4 ≤ V1(t, x t k ) ≤ c2|x t k | 1 2 , c3|x p k | ≤ V2(x p k ) ≤ c4|x p k | 1
divV ≤ Ti + (Ti -τ t k )p1 Ti - p1 2Ti x t k 2 -0.1056x t i+1 k 3 .
Therefore, divV will satisfy [START_REF] Nešić | Stability and performance of siso control systems with first-order reset elements[END_REF], for the case a 1 > b 2 , b 1 > a 2 with a 1 = 4, a 2 = 1/2, b 1 = 1 and b 2 = 1/2 (Theorem 3); if the following LMI holds

  T i +(T i -τ t k )p 1 T i -p 1 2T i + c5 0 0 -0.1056 + c5   ≤ 0.
Hence, since previous inequality is affine in τ t k , the following LMIs are provided where θ ∈ {T min , T max }. Thus, it is concluded that divV satisfies [START_REF] Nešić | Stability and performance of siso control systems with first-order reset elements[END_REF], for the conditions of Theorem 3, if the previous LMIs hold for the finite set θ ∈ {T min , T max }, with c 5 > 0.

Then, using SeDuMi among YALMIP, the following solution for the LMIs is founded: p 1 = 0.4247 and c 5 = 0.0528. Thus, according to Theorem 3, the nonlinear impulsive system is locally DS for all 1.0384 > T i > 0. It is easy to check that c 4 > c 5 and c 2 c 3 4 (c 4 -c 5 ) > c 1 c 5 hold, thus the case for local diagonal stability is obtained. Some simulation results show that the system is unstable for T i ≥ 1.2001 which gives an idea about the conservatism of our results.

According to these results one may remark for both examples that, although only sufficient conditions are provided, the estimate of the maximum reset interval is not that far from the instability bound. From this point of view, the method provides an interesting trade off between accuracy of the analysis and simplicity.

Conclusions

In this paper the stability notion for a class of nonlinear impulsive systems is studied using a 2D vector Lyapunov function approach. The given methodology contributes to the development of a new stability analysis for nonlinear impulsive dynamical systems based on a vector Lyapunov function and its divergence operator in a 2D time domain. Different types of stability notions for a class of nonlinear impulsive systems are studied using this vector Lyapunov function approach. The results are applied to analyze the stability of a class of Lipschitz nonlinear impulsive systems based on LMIs, and some numerical results illustrate the feasibility of the proposed approach.

Remark 3

 3 These results are local since a 1 > b 2 and b 1 > a 2 , i.e. (33) and (35), respectively; only hold in a neighborhood of the origin, i.e. for all i

+ψ 2 2 (µ 2 I

 22 + P21 + θP22) + c5I, hold for all θ ∈ {T min , T max }, as well as all the conditions of Corollary 2 (minimum dwell-time) or 4 (ranged dwell-time), for c 1 = λ min (P 11 ), c 2 = λ max (P 11 + P 12 ), c 3 = λ min (P 21 ), and c 4 = λ max (P 21 + θP 22 ); then, the Lipschitz nonlinear impulsive system (48)-(49) is SADS (Corollary 2) or ADS (Corollary 4). If φ 1 and φ 2 are globally Lipschitz, i.e. Ω = R n ; then, (48)-(49) is GSADS or GADS.

with c 1

 1 = 0.1080, c 2 = 0.1723, c 3 = 0.1064, c 4 = 0.5845 and c 5 = 0.0150. Therefore, according to Corollary 4

2 ,

 2 ∀p = ti, ti+1, with c 1 = 1/2, c 2 = (p 1 + 1)/2, c 3 = 1 and c 4 = 1.

P

  , P 1 (t) = P 11 + 12 is a continuously differentiable with respect to t, symmetric, bounded, and positive definite matrix, with dτ t k /dt = 1, for all t ∈ [t i , t i+1 ), τ ti+1 k+1 = 0, P 11 = P T 11 > 0 and P 12 = P T 12 > 0; while P 2 (p) = P 21 + τ ti+1 p P 22 for p = k, k + 1, are symmetric and positive definite matrices for all i = k ∈ N, with P 21 = P T 21 > 0 and P 22 = P T 22 > 0.

	(Ti-τ t k )
	Ti

The diagonal dynamics make reference only to those dynamics given by (3)-(4) corresponding to i = k, for all i, k ∈ N and for all t ∈ R+.

It is worth mentioning that the results provided in this work generalize those given in[START_REF] Ríos | Vector Lyapunov function based stability for a class of impulsive systems[END_REF], for a larger class of systems.