Nicolas Marchand
email: nicolas.marchand@gipsa-lab.inpg.fr

Sylvain Durand
email: sylvain.durand@inrialpes.fr

Jose Fermi

Guerrero Castellanos
email: fguerrero@ece.buap.mx

J F Guerrero Castellanos

A general formula for event-based stabilization of nonlinear systems

In this paper, a universal formula is proposed for event-based stabilization of general nonlinear systems affine in the control. The feedback is derived from the original one proposed by E. Sontag in the case of continuous time stabilization. Under the assumption of the existence of a smooth Control Lyapunov Function, it is proved that an event-based static feedback, smooth everywhere except at the origin, can be designed so to ensure the global asymptotic stability of the origin. Moreover, the intersampling time can be proved not to contract at the origin. More precisely, it is proved that for any initial condition within any given closed set the minimal inter-sampling time is proved to be below bounded avoiding the infinitely fast sampling phenomena. Moreover, under homogeneity assumptions the control can be proved to be smooth anywhere and the inter-sampling time bounded below for any initial condition. In that case, we retrieve a control approach previously published for continuous time stabilization of homogeneous systems.

I. INTRODUCTION

 and the references therein), even in the case of delays, sampling jitter and measurement loss [2]. In the case of nonlinear control systems, one way to address a discrete-time feedback is to implement a continuous time control algorithm with a sufficiently small sampling

period [START_REF] Hsu | The effect of discretized feedback in a closed loop system[END_REF]. However, this can be constrained by hardware and reducing the sampling period to a level that guarantees acceptable closed-loop performance may be impossible. Other way to tackle this problem is the application of sampled-data control algorithms based on approximate discretetime models [START_REF] Nešić | A framework for stabilization of nonlinear sampled-data systems based on their approximate discrete-time models[END_REF] which is not a trivial task. Another proposed approach consist to modify a continuous time stabilizing control using a general formula to obtain a redesigned control suitable for sampled-data implementation [START_REF] Nešić | Lyapunov-based continuous-time nonlinear controller redesign for sampled-data implementation[END_REF]. Beside these works, sampling was also used for control theory purpose in order to establish an equivalence between controllability and stabilizability for nonlinear systems [START_REF] Clarke | Asymptotic controllability implies feedback stabilization[END_REF], [START_REF] Marchand | NonLinear Control in the Year 2000[END_REF]. However, in these approaches, the control remains time varying in between sampling instants contrary to the case addressed in this paper.

To overcome the complexity of periodic sampling, event-based control has been recently proposed. In this control strategy the control task is executed after the occurrence of an external event (event-triggered) or, as focused in this paper, after the occurrence of an internal event generated by some event function (self-triggered). In this scheme, the inter-sampling time denotes the time interval between two consecutive events. The sampling instants are not necessarily equidistant in time. Let us first consider general nonlinear systems of the form:

ẋ = f (x, u) (1)
where x ∈ X ⊂ R n , u ∈ U ⊂ R p , and f a Lipschitz function vanishing at the origin. For sake of simplicity, we only consider in this paper null stabilization with initial time instant t 0 = 0.

Event-based feedback usually means a set of two functions:

• an event function e : X ×X → R that indicates if one needs (e ≤ 0) or not (e > 0) to update the control value. Event function e takes the current state x as input and a memory m of x last time e became negative. There is a priori no constraint on the regularity of e. Memoryless (that is e(x)), time-varying (e(x, m, t)) or simply time index depending (e(x, m, k)) is not the purpose of the contribution.

• a feedback function k. The terms static (depending upon x) and dynamic (depending upon

x and t or k) will then be used as in the classical frame.

Typical event mechanisms are functions of the state variation (or the output) of the system, like in [START_REF] Årzén | A simple event-based PID controller[END_REF], [START_REF] Sandee | Event-driven control as an opportunity in the multidisciplinary development of embedded controllers[END_REF], [START_REF] Durand | Further results on event-based PID controller[END_REF], [START_REF] Sánchez | Comparative study of event-based control strategies: An experimental approach on a simple tank[END_REF], [START_REF] Sánchez | On the application of different event-based sampling strategies to the control of a simple industrial process[END_REF]. Although the event-triggered control is well-motivated and allows to relax the periodicity of computations, only few works report theoretical results about the stability, convergence and performance. In [START_REF] Åström | Comparison of Riemann and Lebesque sampling for first order stochastic systems[END_REF] for instance, it is proved that such an approach reduces the number of sampling instants for the same final performance. Recent works deal with the problem of scheduling the control task for continuous-time linear systems [START_REF] Åström | Comparison of Riemann and Lebesque sampling for first order stochastic systems[END_REF], [START_REF] Heemels | Analysis of event-driven controllers for linear systems[END_REF], [START_REF] Lunze | A state-feedback approach to event-based control[END_REF], [START_REF] Donkers | Output-based event-triggered control with guaranteed L∞-gain and improved eventtriggering[END_REF] and discrete-time linear system [START_REF] Eqtami | Event-triggered control for discrete-time systems[END_REF] where stability and some robustness proprieties such as ISS and L ∞ -performance are exploited. Furthermore, in [START_REF] Eqtami | Event-triggered control for discrete-time systems[END_REF] a Model Predictive Control scheme is used and the event-triggered policies relax the computationally demanding algorithms. Some of the above contributions do not need memory of the last sampling.

An alternative approach consists in taking e related to the variation of a Lyapunov functionand consequently to the state too -between the current state and its value at the last sampling, like in [START_REF] Mazo | On self-triggered control for linear systems: Guarantees and complexity[END_REF], [START_REF] Velasco | On Lyapunov sampling for event-driven controllers[END_REF], or in taking e related to the time derivative of the Lyapunov function. An important contribution for convergence and stability in the nonlinear case is studied in [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF], [START_REF] Anta | Self-triggered stabilization of homogeneous control systems[END_REF], [START_REF]To sample or not to sample: Self-triggered control for nonlinear systems[END_REF]. Their main contribution is the existence of a minimal inter-sampling time for any bounded initial conditions. The update policy is based on the existence of a Lipchitz stabilizing control law and an ISS-CLF, that is a CLF such that ∂V ∂x f (x, k(x + ε)) ≤ -α( x ) + β( ε ) where α and β are two functions of class K ∞ and ε = m -x denotes the measurement error. The update is then computed as soon as β( ε ) ≤ σα( x ) ensuring that way the strict decrease of the CLF with 0 < σ < 1. Results are given for general non linear systems, homogeneous systems (quite in the spirit of the second part of this paper) and polynomial systems. However, in these works is assumed that the control is Lipschitz at the origin. Unfortunately it is known that general formulas as treated in this paper can not ensure this strong property. Moreover, the event policy is based on the knowledge of an upper bound of the CLF with respect to state and measurement error, which is not the case of this paper.

The solution of (1) with event-based feedback (e, k) starting in x 0 ∈ X at t = 0 is then defined as the solution (when it exists) of the differential system:

ẋ = f (x, k(m)) (2)    m = x if e(x, m) ≤ 0, x = 0 ṁ = 0 elsewhere (3) with: x(0) = x 0 and m(0) = x(0) (4)
If f is assumed to be Lipschitz, a unique solution in the Caratheodory sense always exists without any smoothness assumption on k similarly to [START_REF] Clarke | Asymptotic controllability implies feedback stabilization[END_REF] with punctual events. However, this solution may not exists for all t ≥ 0 as shown by example 2 of section II. Let t → x(t, x 0 ) denote this solution. Given an event function e, and a feedback k defined as above, for any initial condition

x(t = 0) = x 0 it fully defines a sampling set T e,k,x 0 := {t 0 , t 1 , t 2 , . . . } as the set of time instant Notations: In the following, B(d, x) will stand for the ball of radius d centred at x and B(d) for the ball of radius d centred at the origin. x(t; x 0 , t 0 , u) will denote the solution of a differential system starting in x 0 at t 0 with control u. For sake of simplicity, u will be omitted when trivial and x(t; x 0 ) will stand for x(t; x 0 , 0). • will stand for the classical L 2 euclidian norm. • p will stand for the L p -norm. For diagonal matrices, the norm will denote the norm of its diagonal taken as a vector.

t 0 = 0,

II. WHAT CAN HAPPEN WITH EVENT-BASED CONTROL ?

To illustrate different phenomena that can arise with event-based feedback systems, we consider the simple linear integrator ẋ = u. All event functions considered in these examples are assumed to be memoryless that is to be just functions of X . Between two sampling instants t i and t i+1 , u remains constant so that: Then T e,k,x 0 := {0}. Now, taking x 0 = 1, the solution is x i = exp(-i), the inter-sampling time is constant (equal to 1 -exp(-1)) and T e,k,x 0 := {j • (1 -exp(-1)), j ∈ N}. For both initial conditions trajectories are well-defined for all t ∈ [0, +∞[.

x i+1 = x i + (t i+1 -t i ) • u, x i
2) Take k(x) = -x 1
2 , e(x) = 0 when |x| = 1 κ , κ ∈ Z and initial condition x 0 = 1. In that case, the inter-sampling interval t i+1 -

t i = 1 i 1 2 (i+1)
. And when i tends to infinity, the sampling interval tends to zero and t i converges to some t ≈ 1.86. The trajectory is then well-defined only for all t ∈ [0, t[.

3) Take k(x) = -x, e(x) = 0 when |x| = 1 κ , κ ∈ Z and initial condition x 0 = 1.
In that case, one can calculate that t i = i -i j=1 j-1 j and t i+1 -t i = 1 i . Hence the inter-sampling time tends to zero although the trajectory is well defined on [0, ∞[.

4) Take k(x) = -x 3 , e(x) = 0 when |x| = exp(-κ), κ ∈ Z and initial condition x 0 = 1.
Then, the inter-sampling time t i+1 -

t i = x i+1 -x i -x 3 i = exp(2i) • [1 -exp(-1)
] and when i tends to infinity, t i+1 -t i also tends to infinity. The trajectory is well-defined for all t ∈ [0, +∞[ as in case 1.

Consider now the unstable system ẋ = (x + u) 3 . The control u being constant between each sampling instant, the solution is the one of a Bernoulli differential system whose solution is:

x i+1 = x i + u 1 -2(t i+1 -t i ) • (x i + u) 2 -u Then, taking 5) k(x) = -2x, e(x) = 0 when |x| = exp(-κ), κ ∈ Z and initial condition x 0 = 1.
Then the inter-sampling duration is:

t i+1 -t i = exp(2i) 2 • 1 - 1 (2 -exp(-1)) 2
and when i tends to infinity, t i+1 -t i also tends to infinity. However, the origin of the closed loop system can be proved to be asymptotically stable and the trajectories well-defined on [0, +∞[ for any initial condition.

In cases 1 to 5, the system can trivially be proved to be globally null-asymptotically stable.

Cases 1 show that the sampling set is initial condition dependent. Cases 1 to 4 show that for the same system and initial condition, the sampling can be periodic, contractile or expansile (with a finite or infinite limit) depending upon the event function or the feedback. Case 5 shows the inconsistency of Shannon's sampling theorem in the event-based paradigm since the intersampling duration can infinitely increase even with an open-loop unstable system.

III. PRELIMINARY DEFINITIONS FOR EVENT-BASED SYSTEMS

Usually, the set of event instants is of null measure, in the sense that the control is recomputed only at distinct t. However, taking e(x) = 0 for all x ∈ X would mean that one recomputes the control at each x and therefore that one applies a classical continuous-time feedback. On the sets of non null measure where e(x) = 0, the solution is understood in the classical sense (with all possible solution existence problems if the field is discontinuous). Elsewhere, the solution can be intended in the Caratheodory sense. To go further on that, we define: Definition 3.1 (Well-defined event-based control): An event-based control (k, e) will be said well-defined if and only if for any initial condition x 0 at t = 0, the solution t → x(t; x 0 ) exists for all t ≥ 0.

) := inf i∈N,t i ∈T (x 0 ) t i+1 -t i > 0.
The aim of this property is to avoid zero inter-sampling time leading to zeno phenomena at finite time as in example 2 or at infinity as in example 3 as explained in the last section. In case the MSI property is satisfied, the control is piecewise constant between each time sample and x i , i ∈ I ⊂ N with x 0 := x(t = 0) will denote the series of successive state values at which e is negative for a given initial condition x 0 . To this series corresponds naturally the time series t i , i ∈ I ⊂ N with t 0 := 0. It trivially follows (proof is omitted): Theorem 3.2: An MSI event-based control is well-defined. This minimal sampling period is useful for implementation purpose but also when the feedback k is discontinuous for robustness purpose [START_REF] Sontag | Clocks and insensitivity to small measurement errors[END_REF]. However, it would be more suitable to have such a bound less depending upon the initial condition: Property 3.3 (Semi-uniformly MSI event-based control): An event-based control (k, e) will be said semi-uniformly MSI iff for any δ > 0, τ (δ) := inf i∈N,t i ∈T (x 0 ),x 0 ∈B(δ) t i+1 -t i > 0 Property 3.4 (Uniformly MSI event-based control): An event-based control (k, e) will be said uniformly MSI iff τ := inf i∈N,t i ∈T (x 0 ),x 0 ∈X t i+1 -t i > 0.

Properties 3.1 to 3.4 can be specified adding the term "global" when X = R n in opposition to the term "local" that was omitted above for sake of simplicity. Now that the above notions for eventbased controlled systems are appropriately defined, notions like stability, asymptotic stability and stabilizability naturally follow since they rely on the resulting trajectory. The question that arises then is: does a universal formula for uniformly discrete event-based feedback stabilization exist similarly to the continuous time case ? This is the purpose of the next section.

IV. A UNIVERSAL FORMULA FOR EVENT-BASED STABILIZATION

In the sequel, the analysis is restricted to systems affine in the control:

ẋ = f (x) + g(x)u = f (x) + i g i (x)u i (5)
where f and g are smooth functions with f vanishing at the origin. We assume that a Control Lyapunov Function (CLF) exists for system [START_REF] Nešić | Lyapunov-based continuous-time nonlinear controller redesign for sampled-data implementation[END_REF], that is a smooth and positive definite function V : X → R so that for each x = 0 there is some u ∈ U such that:

∂V ∂x f (x) + ∂V ∂x g(x)u < 0 (6)
In addition, one may require that V fulfills the small control property [START_REF]Mathematical control theory, deterministic finite dimensional systems[END_REF], that is that for each ε > 0 there is some µ > 0 such that for any x in the ball B(µ)\ {0}, there is some u with u ≤ ε such that (6) holds. Then, it is known that it is possible to design a feedback control that asymptotically stabilizes the system [START_REF]Mathematical control theory, deterministic finite dimensional systems[END_REF]: Then k is such that for all non zero x, ∂V ∂x f (x) + ∂V ∂x g(x)k(x) < 0. Moreover, if the CLF satisfies the so called small control property, then taking q(b) := b, the control is continuous at the origin .

The main purpose of this paper is to establish that a universal formula also exists in the eventbased context up to a modification of the original formula proposed by Sontag: Theorem 4.2: If there exists a CLF for system (5), then the event-based feedback (e, k) defined below is semi-uniformly MSI, smooth on X \ {0}, and such that:

∂V ∂x f (x) + ∂V ∂x g(x)k(m) < 0, x ∈ X \ {0} (7)
where m is defined by (3) and:

k i (x) := -b i (x)δ i (x)γ(x) (8) e(x, m) := -a(x) -b(x)k(m) -σ a(x) 2 + θ(x)b(x)∆(x)b(x) T (9)
where

• a(x) and b(x) are as in Theorem 4.1,

• x → ∆(x) := diag(δ 1 (x), δ 2 (x), . . . , δ p (x)) is a smooth function of X \ {0} to R p×p , positive definite on S := {x ∈ X | b(x) = 0}
• x → θ(x) is a smooth positive function of X to R, such that θ(x) ∆(x) vanishes at the origin, and ensuring on S\ {0} the inequality a(x

) 2 + θ(x)b(x)∆(x)b(x) T > 0 • σ is a control parameter in [0, 1[, • γ : X → R is defined by: γ(x) :=    a(x)+ √ a(x) 2 +θ(x)b(x)∆(x)b(x) T b(x)∆(x)b(x) T if x ∈ S 0 if x / ∈ S (10)
As in Theorem 4.1, if the CLF satisfies the small control property, then the control is continuous at the origin. Moreover, if there exists some smooth function ω : X → R + such that on S\ {0},

ω(x)b(x)∆(x)b(x) T -a(x) > 0 (11)
then the control is smooth on X as soon as θ(x) ∆(x) vanishes at the origin with:

θ(x) := ω(x) 2 b(x)∆(x)b(x) T -2a(x)ω(x) (12)
Before giving the proof of Theorem 4.2, let us explain the ideas behind the construction of feedback [START_REF] Årzén | A simple event-based PID controller[END_REF]. In the event function, the term a(x)

+ b(x)k(m) is the time derivative of V where -a(x) 2 + θ(x)b(x)∆(x)b(x) T is the value of V if k(x) is applied instead of k(m).
Therefore, right after an event, e(x, m) necessarily takes a strictly positive value, more precisely, it equals (1 -σ) a(x) 2 + θ(x)b(x)∆(x)b(x) T and remains positive as long as

V ≤ -σ a(x) 2 + θ(x)b(x)∆(x)b(x) T .
Events will be more frequent with smaller σ. σ = 0 means updating the control when V = 0. The second crucial tuning parameter of the control law is the choice of the function ∆ that directly impacts the performance of the control as well as the frequency of events.

We next focus on homogeneous systems that gave rise to an important literature (see for instance [START_REF] Faubourg | Control Lyapunov functions for homogeneous Jurdjevic-Quinn systems[END_REF], [START_REF] Grüne | Homogeneous state feedback stabilization of homogeneous systems[END_REF] and the references therein) and more recently for event-based approaches (mainly in [START_REF]To sample or not to sample: Self-triggered control for nonlinear systems[END_REF], [START_REF] Mazo | On self-triggered control for linear systems: Guarantees and complexity[END_REF], [START_REF] Tabuada | Event-triggered real-time scheduling of stabilizing control tasks[END_REF]). We shortly recall few definitions:

Definition 4.1: Let r = (r 1 , . . . , r n ) ∈ R n ≥0
, and a set of coordinates x = (x 1 , . . . , x n ).

• The family of dilations (δ r λ ) λ>0 is defined by: δ r λ (x) := (λ r 1 x 1 , . . . , λ rn x n ), ∀λ > 0

• V : X → R is a δ r -homogeneous function of degree d if: ∀(x, λ) ∈ X ×R >0 , V (δ r λ (x)) = λ d V (x) • h : X → X is a δ r -homogeneous function of degree d if: ∀(x, λ) ∈ X ×R >0 , h(δ r λ (x)) = λ d δ r λ (h(x)) • A δ r -homogeneous norm x → x (r,π) is a map defined by x (r,π) = n i=1 |x i | π r i 1 π
To these classical definitions, and for the sake of brevity, we add: Definition 4.2: Let an event-based controlled δ r -homogeneous system of degree (d 1 , d 2 )

denote a system of form ( 1) with an event-based feedback (e, k) ,m). Property 4.3: Consider an event-based controlled δ r -homogeneous system of degree (d 1 , d 2 ), then the solution exists, is unique and such that x(t; δ r λ (x 0 )) = δ r λ (x(λ d 1 t; x 0 )). The proof trivially follows from the δ r -homogeneity of degree d 1 of f (x, k(x)) and using the relation e(Λ r x, Λ r m) = λ d 2 e(x, m).

such that f (δ r λ (x), k(δ r λ (m))) = λ d 1 f (x, k(m)) and e(δ r λ (x), δ r λ (m)) = λ d 2 e(x
For homogeneous systems, Theorem 4.2 becomes the following. It is actually the event-based formulation of the smooth homogeneous control proposed in [START_REF] Tsinias | Remarks on feedback stabilizability of homogeneous systems[END_REF] that appears as a particular case of Theorem 4.2 with an appropriate choice of functions ∆(x) and ω(x). Theorem 4.4: Assume that f , each g i and the CLF V are δ r -homogeneous respectively of degree d f , d g i , and d V , with ∀i ∈ {1, . . . , p}, d g i < d f . Then, event-based feedback (8-9) with θ(x) as in [START_REF] Sánchez | On the application of different event-based sampling strategies to the control of a simple industrial process[END_REF], ω(x) = ν x

d f -d V (r,π) , ν > 0 and δ i (x) = x -2dg i (r,π)
, is given by:

k i (x) := -νb i (x) x d f -2dg i -d V (r,π) (13)
For ν sufficiently large, this event-based feedback is such that:

1) the event-based controlled system is δ r -homogeneous of degree (d f , d V + d f )
2) the event-based control is smooth and uniformly MSI 3) the CLF is strictly decreasing for all x ∈ X \ {0} Proof of Theorem 4.2: We begin the proof establishing γ is smooth on X \ {0}. For this, consider the algebraic equation:

F (x, p) := b(x)∆(x)b(x) T p 2 -2a(x)p -θ(x) = 0 (14)
Note first that p = γ(x) is a solution of ( 14) for all x ∈ X . It is easy to prove that the partial derivative of F with respect to p is always strictly positive on X \ {0}:

∂F ∂p := 2b(x)∆(x)b(x) T p -2a(x)
Indeed, when b(x) = 0, equation ( 6) gives ∂F ∂p = -2a(x) > 0 and when b(x) = 0, equation [START_REF] Durand | Further results on event-based PID controller[END_REF] gives ∂F ∂p = 2 a(x) 2 + θ(x)b(x)∆(x)b(x) T > 0. Therefore ∂F ∂p never vanishes at each point of the form {(x, γ(x))|x ∈ X \ {0}}. Furthermore, F is smooth w.r.t. x and p since so are a, b, θ and ∆. Hence, using the implicit function theorem, γ is smooth on X \ {0}.

The decrease of the CLF is trivial to prove. Indeed, for each x i = 0, i ∈ N:

dV dt (x i ) = ∂V ∂x (x i )f (x i ) + ∂V ∂x (x i )g(x i )k(x i ) = -a(x i ) 2 + θ(x i )b(x i )∆(x i )b(x i ) T < 0
With the updated control, the event function becomes strictly positive:

e(x i , x i ) = (1 -σ) a(x i ) 2 + θ(x i )b(x i )∆(x i )b(x i ) T > 0
Therefore, by smoothness of V , f and g, it follows that dV dt (x(t; x i , t i )) < 0 for all t ∈ [t i , t i+1 [, that is until the next event occurs. t i+1 is necessarily bounded since, if not, V should converge to a constant value where dV dt = 0, which is impossible. The event function precisely prevents this phenomena detecting dV dt is close to vanish and updates the control if it happens. To prove that the event-based control is MSI, we have to prove that for any initial condition in a priori given set, the sampling intervals are bounded below. First of all, notice that events occur only when e becomes negative (with x = 0) and therefore from equation [START_REF] Sandee | Event-driven control as an opportunity in the multidisciplinary development of embedded controllers[END_REF], on

{x ∈ X | b(x) = 0} \ {0}: e(x, m) = -a(x) -σ |a(x)| = (1 -σ) |a(x)| > 0
It follows that there is no event on the set {x ∈ X | b(x) = 0} ∪ {0}. We hence restrict the study to the set {x ∈ X |β(x) = 0} \ {0} where θ and the δ i 's are strictly positive by assumption.

Let us rewrite the time derivative of the CLF along the trajectories:

dV dt (x) = a(x) + b(x)k(m) = -a(x) 2 + θ(x)b(x)∆(x)b(x) T + b(x)(k(m) -k(x)) (15)
Let us define for m ∈ S, the level ϑ m := V (m) and the set V ϑm := {x ∈ X |V (x) ≤ ϑ m }. Then in equation [START_REF] Lunze | A state-feedback approach to event-based control[END_REF], it follows from the choice of the event function that x belongs to V ϑx ⊂ V ϑm .

Note that if m belongs to S, this is not necessarily the case for x that can escape from this set.

First see that for t = t i (recall the t i 's denote the time instant when an event occurs), x = m and therefore, since i) θ(x) is such that a(x) 2 + θ(x)b(x)∆(x)b(x) T > 0 for all nonzero x in S, and ii) a(x) is necessarily nonzero on the frontier of S (except possibly at the origin):

dV dt (x) = -a(m) 2 + θ(m)b(m)∆(m)b(m) T ≤ - inf m∈S s.t. V (m)=ϑm a(m) 2 + θ(m)b(m)∆(m)b(m) T =: -χ(ϑ m ) < 0
Considering now the second time derivative of the CLF:

V (x) = ∂a ∂x (x) + k(m) T ∂b T ∂x (x) (f (x) + g(x)k(m))
By continuity of all the involved functions, both terms can be bounded for all x ∈ V ϑm by the following upper bounds 1 (ϑ m ) and 2 (ϑ m ):

1 (ϑ m ) := sup m∈S s.t. V (m)=ϑm x∈V ϑm ∂a ∂x (x) + k(m) T ∂b T ∂x (x) 2 (ϑ m ) := sup m∈S s.t. V (m)=ϑm x∈V ϑm f (x) + g(x)k(m)
Therefore, V is strictly negative at any event instant t i and can not vanish until a certain time τ (ϑ m ) is elapsed and this minimal sampling is only depending on the level ϑ m of the CLF in m (and hence of the radius of any ball inside):

τ (ϑ m ) ≥ χ(ϑ m ) 1 (ϑ m ) 2 (ϑ m ) > 0 (16)
which ends the proof, the event-based feedback (8-9) is semi-uniformly MSI.

To prove the continuity of k at the origin, we only need to consider the points in S since we already have k(x) = 0 if b(x) = 0. We have:

k(x) ≤ |a(x)| + a(x) 2 + θ(x)b(x)∆(x)b(x) T b(x)∆(x)b(x) T ∆(x)b(x) T ≤ 2 |a(x)| b(x)∆(x)b(x) T ∆(x)b(x) T + θ(x) b(x)∆(x)b(x) T ∆(x)b(x) T ≤ 2 |a(x)| b(x)∆(x)b(x) T ∆(x)b(x) T + θ(x) ∆(x)
With the small control property, for any ε > 0, there is µ > 0 such that for any x ∈ B(µ)\ {0}, there exists some u with u ≤ ε such that a(x) + b(x)u < 0 and therefore |a(x)| < b(x) ε.

It follows:

k(x) ≤ 2ε b(x) ∆(x)b(x) T b(x)∆(x)b(x) T + θ(x) ∆(x) Since the function (v 1 , v 2 ) → v 1 v 2 v T 1 v 2
is continuous w.r.t its two variables at the origin where it equals 1, since θ and ∆ are also continuous, since θ(x) ∆(x) vanishes at the origin, for any ε , there is some µ such that ∀x ∈ B(µ )\ {0}, k(x) ≤ ε which ends the proof of continuity.

Finally, with θ as in [START_REF] Sánchez | On the application of different event-based sampling strategies to the control of a simple industrial process[END_REF], the control becomes k i (x) = -b i (x)δ i (x)w(x) which is obviously smooth on X .

Proof of Theorem 4.4: Take ν such that:

ν > sup x∈{x∈X | x (r,π) =1,a(x)≥0} a(x) p i=1 b i (x) 2 x d f -2dg i -d V (r,π) (17)
As in [START_REF] Tsinias | Remarks on feedback stabilizability of homogeneous systems[END_REF], k is δ r -homogeneous of degree d f -d g and the system is therefore δ r -homogeneous of degree d f . In addition e clearly satisfies e(δ r λ (x), δ r λ (m)) = λ d V +d f e(x, m). Item 1 of Theorem 4.4 therefore holds.

On x ∈ X | x (r,π) = 1, a(x) ≥ 0 , equation [START_REF] Eqtami | Event-triggered control for discrete-time systems[END_REF] gives ω(x)b(x)∆(x)b(x) T -a(x) > 0.

Since the right-hand side of the inequality is δ r -homogeneous of degree d V + d f , the same inequality holds for all x = {0} such that a(x) ≥ 0. Using [START_REF] Clarke | Asymptotic controllability implies feedback stabilization[END_REF], it follows that this inequality is valid for x in S\ {0}. Therefore, equation [START_REF] Sánchez | Comparative study of event-based control strategies: An experimental approach on a simple tank[END_REF] holds with the above choice of ω. Since θ given by equation ( 12) is homogeneous of degree 2d f and using the property d g i < d f that holds for all i ∈ {1, . . . , p}, it follows that lim x→0 θ(x) ∆(x) = 0. Thanks to Theorem 4.2, item 3 holds and the control is smooth and semi-uniformly MSI. the control power of -0.08, that is higher than the -0.05 with the smooth control. Moreover, the decrease of the CLF is slower than with the smooth and non smooth control proposed here as illustrated in Figure 2. In this example, with the same system, the same CLF, the proposed control and control update policy seems more efficient (faster convergence, smaller control, fewer events).

1 )

 1 denoting the value of the state when the i th event occurs. The following examples illustrate what may happen. Take k(x) = -x, e(x) = 0 when |x| = exp(-κ), κ ∈ Z and initial condition x 0 = 0.

Property 3 . 1 (

 31 Minimal Sampling Interval -MSI): An event-based control (k, e) will be said to satisfy the Minimal inter-Sampling Interval property (MSI) iff for any initial condition x 0 at t = 0, there exists a non zero minimal sampling interval τ (x 0

Theorem 4 . 1 (a 2

 412 Sontag's universal formula): Assume that system (5) admits V as CLF. For any real analytic function q : R → R such that q(0) = 0 and bq(b) > 0 for b = 0, let φ : R 2 → R be defined by φ(a, b) := a+ √ +bq(b) b if b = 0 and zero when b = 0. Let the feedback k : X → U, smooth on X \ {0} be defined by k i (x) := -b i (x)φ(a(x), β(x)), i ∈ {1, . . . , p} with a(x) := ∂V ∂x f (x), b(x) := ∂V ∂x g(x), β(x) := b(x) 2 .

Fig. 1 .

 1 Fig. 1. System (18) under non smooth (left) and smooth (right) event-based control. The red dots represents the events.

Fig. 2 .

 2 Fig.2. Time evolution of the CLF using the non smooth and smooth version of the proposed control law compared to[START_REF] Anta | Self-triggered stabilization of homogeneous control systems[END_REF] and a continuous time implementation.

To finish the proof, remains to establish that the event-based feedback is uniformly MSI.

For this, we invoke the homogeneity of the Lyapunov level sets V ϑm together with the relation x(t; δ r λ (x 0 )) = δ r λ (x(λ d 1 t; x 0 )), that gives for all ϑ m > 0, τ (ϑ m ) = τ (1) that ends the proof of item 2.

V. EXAMPLES

A. Linear time invariant systems

Consider the linear time-invariant system ẋ = Ax + Bu. Take P , a positive definite matrix solution of the Riccati equation P A + A T P -4εP BB T P = -P . Then V (x) := x T P x is a CLF for the system since for all x, u = -2εB T P x renders V strictly negative for x = 0. Since a(x) = x T (P A + A T P )x, b(x) = 2x T P B and β(x) = 4x T P BB T P x, the Riccati takes the form of equation [START_REF] Sánchez | Comparative study of event-based control strategies: An experimental approach on a simple tank[END_REF] with ω(x) := ε and δ i := 1 for all i ∈ {1, . . . , p}. Therefore, taking θ(x) according to [START_REF] Sánchez | On the application of different event-based sampling strategies to the control of a simple industrial process[END_REF], the control is smooth everywhere and linear: k(x) = -εb(x) T .

B. Nonlinear system

We consider next the nonlinear system proposed in [START_REF] Anta | Self-triggered stabilization of homogeneous control systems[END_REF]:

We first take δ i := 1 for all i ∈ {1, . . . , p}, θ(x) = b(x) 2 and σ = 0.9. The time evolution of x, V (x), k(x) and the event function e is depicted in Figure 1 for x 0 = (0.1, 0.4).

it follows that taking θ(x) as in [START_REF] Sánchez | On the application of different event-based sampling strategies to the control of a simple industrial process[END_REF], the control is smooth everywhere. The resulting trajectory with σ = 0.9 is represented in Figure 1.

In both cases, event occurs twice in the 40 s of time horizon (the initial event is included).

With the discontinuous control, the convergence is faster but the control value is also almost ten times larger. When compared to [START_REF] Anta | Self-triggered stabilization of homogeneous control systems[END_REF] with the same initial condition and the parameter σ = 0.9

(the higher σ is, less frequent events are), 117 events are obtained with a maximum value of

VI. CONCLUSION

In this paper, we proposed an extension of the universal formula for feedback stabilization to event-based controlled systems. A modification of the original formula is necessary to ensure that there is a minimal sampling time between two consecutive events avoiding zeno phenomena like accumulation points. As in the original work, if the Control Lyapunov Function fulfils the small control property, then the control is continuous at the origin. With additional homogeneity assumptions, the control can be proved to be smooth everywhere and the minimal inter-sampling time bounded below for all initial conditions.


Bernard Teissier
email: teissier@math.jussieu.fr

Some resonances of Lojasiewicz inequalities

published or not. The documents may come

Lojasiewicz exponents and Newton polygons

We consider the set of local Newton polygons in the first quadrant of R 2 . It is the set of boundaries of the convex hulls of finite unions of subsets of the form (a, b) + R 2 + where (a, b) ∈ R 2 + . There is a natural commutative operation, the sum, on Newton polygons. It is obtained by taking the boundary of the Minkowski sum of the infinite convex regions which two polygons bound. It is commutative and associative. Its unit element is the Newton polygon consisting of the union of the two coordinate half-lines.

The following notation, introduced in [T1], will be convenient for us: given two positive numbers, which for us will be integers, denote by ℓ h the elementary Newton polygon which is the boundary of the convex hull of the subset ((0, h) + R 2 + ) ((ℓ, 0) + R 2 + ) of R 2 . If we add to these finite elementary Newton polygons the two elementary polygons corresponding to the case where h or ℓ (but not both) may be taken to be infinite, we obtain a set of generators for the monoid of all Newton polygons. Every Newton polygon may be written as S 1991 Classification: 32S15 (14B05 14P15)

N = s i=1 ℓ i h i , 1 A.M.
1 and if we require that the inclinations ℓi hi be all different, this decomposition is unique. The unit element can be written as 0 0 . See [T1] or [L-T], Complément 2, for details.

Z Z Z Z Z h ℓ = ℓ h ℓ ′ = ℓ ′ ∞ ℓ h + ℓ ′ ∞ = h Z Z Z Z Z ℓ + ℓ ′ ℓ ′
Let now (X, O X ) be a reduced equidimensional complex analytic space, and I a sheaf of ideals defining a nowhere dense analytic subspace Y ⊂ X. The normalized blowing up π : N B I X → X of I is a proper and bimeromorphic analytic map, where the space N B I X is normal and the ideal IO N BI X is invertible.

Given a point x ∈ X let us consider the finitely many irreducible components (D i ) 1≤i≤t of the exceptional divisor of π which meet the compact fiber π -1 (x). For each component we may consider the general fiber of the map π : |D i | → |π(D i )| and the underlying reduced general fiber. Its degree with respect to the π-ample sheaf IO N BI X /I 2 O N BI X will be denoted by deg π |D i |. Now to any germ of analytic function g ∈ O X,x we can associate the order v Di (g •π) of vanishing along |D i | of the composition with π of a representative of g at a general point of |D i |. We may replace g by an ideal J if we define the order of vanishing v Di (J • π) as the infimum of the orders of vanishing of the (compositions with π of) elements of J, which is attained by some generators of the ideal. We can now define the Newton polygon of g with respect to I at x as

N I,x (g) = t i=1 deg π |D i | v Di (I) v Di (g) ,
provided that we have g ∈ √ I x , which ensures that none of the v Di (g) is zero, and the Newton polygon of J with respect to I as

N I,x (J) = t i=1 deg π |D i | v Di (I) v Di (J) , provided that we have J ⊂ √ I x .
Remarks 1.1. 1) Each element g ∈ √ I x determines a pre-ordering on the set of D i , by reading from left to right the compact faces of N I,x (g). It would be interesting to determine, for a given I and x, which pre-orderings are realizable in this manner. For example, if the germ I x is a primary ideal for the maximal ideal m x , the trivial pre-ordering, where all elements are equal, is realizable by a superficial element g for I x (see [L-T], Complément 2).

2) The strict transforms of the components D i in a log-resolution of the pair (X, Y ) may be called the dicritical components at x of the exceptional divisor of the resolution. In the special case where I is a primary ideal for the maximal ideal of a point on a normal surface X, the normalized blowing up of I is equal to the normalized blowing-up of the ideal generated by two general elements (f, g) of I and the strict transforms of the D i are exactly the components of the resolution where the quotient f g of the strict transforms of f and g is not constant. The computation of normalized blowing-ups is difficult in dimension > 2, although there exist algorithms to compute the normalization. A method of computation of the Lojasewicz exponent in the case of an ideal primary for the maximal ideal of O X,x , through an explicit log resolution is explained in [BA-E].

3) In the case where the ideal I x is primary for the maximal ideal of O X,x , so that all the components D i map to x, the valuations v Di are called the Rees valuations associated to that ideal (see [R1]).

What we call "the" Lojasiewicz exponent is of course the smallest real number for which the inequality holds in some neighborhood of a given point.

Proposition 1.1. (see [L-T], Théorème 4.6). Given a set of generators f 1 , . . . , f s for the germ I x ∈ O X,x and g ∈ O X,x , the smallest real number θ such that there exist a neighborhood U of x in X and a constant C > 0 such that the inequality

|g(z)| θ ≤ Csup s k=1 |f k (z)| holds for all z ∈ U , is equal to θ = max t i=1 v Di (I) v Di (g)
.

It is therefore a rational number. If g / ∈ √ I x , set θ = +∞.

For the rationality of the Lojasiewivz exponent in the real-analytic case see Risler's appendix in [L-T], [B-R], and [Fe].

Given g ∈ √ I x , denote by ν Ix (g) the largest integer n such that g ∈ I n

x . In [L-T], we prove that the limit ν Ix (g) = lim k→∞ νI x (g k ) k

exists and is equal to min t i=1 vD i (g) vD i (I) . Thus, the best Lojasiewicz exponent, which is the largest inclination of the compact edges of N I,x (g), can be understood as the inverse of this limit. We also give some indications on the history of these results.

Again, one can extend the definition of ν to an ideal, by defining ν Ix (J) to be the largest integer such that J ⊂ I n

x and proceeding as above. The fact that the Lojasiewicz exponents appears as (the inverse of) the inclination of a compact edge of a Newton polygon 2 is interesting in particular because other features of the same polygon have, at least in special cases, algebro-geometric interpretations:

• If the ideal I x is primary for the maximal ideal, the length of the horizontal projection of N Ix (g) is equal to the Samuel multiplicity e(I x ) of the primary ideal

I x in O X,x .
• The length of the vertical projection is equal to the degree function of g with respect to I x , studied by David Rees and which is the Samuel multiplicity of the 2 There are at least two other such constructions for ν I (g): one appears in [L-T], appendice au §4, and the other in [START_REF] Hickel | Fonction asymptotique de Samuel des sections hyperplanes et multiplicité[END_REF].

image of I x in the quotient O X,x /gO X,x . In particular if g is a general element of another primary ideal J, it is the mixed multiplicity e(I [d-1] x , J [1] ).

• As an immediate consequence of the convexity of the Newton polygon, setting d = dimX, we have the inequality ν Ix (J) ≤ e(I [d-1] x ,J [1] ) e(Ix) . 1.1. Lojasiewicz exponent and plane sections. In the paper [START_REF] Hickel | Fonction asymptotique de Samuel des sections hyperplanes et multiplicité[END_REF], Hickel has shown that given a primary ideal I in a regular local ring (O, m) of dimension d and equal characteristic zero, if we study the ν I (m), the ν IO H (i) (mO H (i) ) is well defined for a quotient O H (i) of O by the ideal generated by di "general" linear forms. This suggests the following:

Problem: Let O be a Cohen-Macaulay excellent equicharacteristic normal local ring of dimension d and let I and J be two primary ideals. Given a family g = (g 1 , . . . , g d-i ) of di elements of J, let us denote by J g the ideal of O which they generate. Is it true that the Newton polygon

N I O Jg (J O
Jg ) is independent of the family g provided it is "sufficiently general"? Such a result is true at least in the case where I ⊂ C{z 1 , . . . , z d } is the jacobian ideal of a hypersurface f (z 1 , . . . , z d ) = 0 with an isolated singularity at the origin i.e., the ideal generated by the partial derivatives, and J is the maximal ideal m. The Newton polygon N j(f ) (m) then takes the name of jacobian Newton polygon. There are four ingredients:

• One proves that the jacobian Newton polygon of the hypersurfaces of an analytic Whitney equisingular family is constant (see [T2]).

• The general hyperplane sections of an analytic space form a Whitney equisingular family (see [T2]).

• The image in the local ring of a hyperplane of the jacobian ideal of f and the jacobian ideal of the restriction of f to the hyperplane have the same integral closure if the hyperplane is "general" (a special case of the idealistic Bertini Theorem; see [T1], 2.8).

• The Newton polygon depends only on the integral closures of the ideals (see [L-T]). Here a hyperplane is "general" if it is not a limit position of tangent hyperplanes to the hypersurface along sequences of nonsingular points tending to the singular point.

The way to extend this argument to the general case is not clear.

1.2. The Lojasiewicz exponent at infinity. The Lojasiewicz inequalities of the type |F (z)| ≥ C|z| θ for families of polynomials defining a zero dimensional subspace of C d+1 were studied very early by P loski (see [P1]). In the paper [START_REF] Hickel | Solution d'une conjecture de C. Berenstein-A. Yger et invariants de contact à l'infini[END_REF] Hickel uses methods of normalized blowing up and interpretation of ν I (J) as the inverse of a Lojasiewicz exponent to prove:

Theorem 1.1. (Hickel) Let k be an algebraically closed field with a non trivial absolute value | |.

Let I = (p 1 , . . . , p m ) ⊂ k[X 1 , . . . , X n ] and V = {z ∈ k n /p 1 (z) = . . . = p m (z) = 0}. Set d i = degp i .
There exist a constant C > 0 and exponents θ 1 , θ 2 such that for all z ∈ k n the following inequality holds:

m i=1 |p i (z)| (1 + |z|) di ≥ C d(z, V ) θ1 (1 + |z|) θ2 ,
where the best exponents θ 1 , θ 2 are rational numbers, which can be computed from invariants which appear in the normalized blowing up of the sheaf of ideals on P n (k) corresponding to the homogeneizations of the polynomials p i .

Here d(z, V ) is the distance from z to V . Of course Hickel's statement is much more precise, and the techniques of proof use more global techniques than just local intersection theory, in particular the refined Bézout theorem. The close connexion between such inequalities and the effective Nullstellensatz has been known since the paper [B] of Brownawell, and the paper of Hickel makes important headway towards optimal bounds for s and the degrees of the q k f k in terms of n and the degrees of the polynomials f 1 , . . . , f t and g ∈ (f 1 , . . . , f t )k[x 1 , . . . , x n ] in equalities such as g s = k q k f k , in all characteristics. See also [L-T], Complément 5, and the article [C-K-T].

1.3. Lojasiewicz exponent and log canonical threshold. The gradient Lojasiewicz exponents of the restrictions to plane sections of all dimensions appear in an estimate of the log canonical threshold associated to the hypersurface singularity f (z 1 , . . . , z d+1 ) = 0 at the origin. I refer to the notes [Mu] of Must ¸atȃ and the book [La], Vol. II, of Lazarsfeld for the definition. It is of the nature of a Lojasiewicz exponent for differential forms.

The jacobian Newton polygon of a hypersurface with isolated singularity, say at the origin, can be written in the form:

N j(f ) (m) = t q=1 e q m q ,
where the q's index the branches of a general polar curve, which means the critical locus of a map F :

C d+1 → C 2 , z → (ℓ(z), f (z)
) with ℓ a general linear form.

The integers e q , m q are then computed from a parametrization of each branch: m q is the multiplicity of that branch, and e q + m q is its intersection multiplicity with f = 0. See [T2], where it is proved that the best possible exponents for the gradient Lojasiewicz inequalities to hold in a neighborhood of the singular point,

|f (z)| θ1 ≤ C 1 |gradf (z)| and |z| θ2 ≤ C 2 |gradf (z)|,
are θ 1 = η 1+η and θ 2 = η where η = sup q eq mq . This shows not only that the Lojasiewicz exponents are rational numbers and that θ 1 < 1, a well known and very useful result of Lojasiewicz in the real case (see [L-T], Complément 1), but also that they are invariants of Whitney equisingularity, since the jacobian Newton polygon is.

Since the jacobian Newton polygon of a general hyperplane section is well defined, we may for each 1 ≤ i ≤ d + 1 define η (i) to be the gradient Lojasiewicz exponent and τ (i) to be the inf q e (i) q m (i) q for the restriction of f to a general i-dimensional linear space through the origin. They are respectively the largest and smallest inclinations of the edges of the jacobian Newton polygon of that restriction. We agree that the restriction of f to a general line, which is equivalent to z m 1 , gives θ (1) = τ (1) = m-1 where m is the order of f at x, the multiplicity at x of the hypersurface f = 0. Then we have: Theorem 1.2. (Loeser, see [START_REF] Loeser | Exposant d'Arnold et sections planes[END_REF], [START_REF] Loeser | Une estimation asymptotique du nombre de solutions approchées d'une équation p-adique[END_REF]) Assume that the isolated singularity of f (z) = 0 at x is not a rational singularity (this is equivalent to lct x (f ) ≤ 1 by a result of Morihiko Saito). The log canonical threshold lct x (f ) of f at x satisfies the inequalities

d+1 i=1 1 1 + ⌈η (i) ⌉ ≤ lct x (f ) ≤ d+1 i=1 1 1 + ⌊τ (i) ⌋ .
In the case where

f = z a1 1 + • • • + z a d+1
d+1 with a 1 ≤ a 2 ≤ • • • ≤ a d+1 , one computes (see [T2]) that τ (i) = η (i) = a i -1 and recovers the well known formula lct 0 (f ) = d+1 i=1 1 ai when this sum is ≤ 1.

In [START_REF] Loeser | Une estimation asymptotique du nombre de solutions approchées d'une équation p-adique[END_REF], one finds an application to arithmetical algebraic geometry. I had in [T3] conjectured the first inequality, without the ⌈ ⌉, for reasons related to the rate of vanishing as λ tends to zero of the volumes of the vanishing cycles on a Milnor fiber (f (z) = λ) ∩ B(x, ǫ).

It is shown in [T2] that: • The length of the projection to the horizontal (resp. vertical) axis of the jacobian Newton polygon N j(f |H (i) ) (m) of the restriction of f to a general vector subspace of dimension i + 1 is equal to the Milnor number µ (i+1) (f ) (resp. µ (i) (f )) of the restriction of f to H (i+1) (resp H (i) ). This jacobian Newton polygon is therefore a refinement of the elementary polygon µ (i+1) (f )

µ (i) (f ) .
• The double of the area bounded by the jacobian Newton polygon is the Milnor number of a general hyperplane section of the Thom-Sebastiani "double" f (z 1 , . . . , z d+1 )+ f (w 1 , . . . , w d+1 ) of the function f (z 1 , . . . , z d+1 ).

Remark 1.1. If f is a homogeneous polynomial of degree m with isolated singularity, one finds that all eq mq are equal to m -1, so that the optimal inequality is the expected "homogeneous" one:

|f (z)| m-1 ≤ C|gradf (z)| m .
But this is no longer true if the cone defined by the vanishing of the homogeneous polynomial f has a non isolated singularity at the origin; here is a counterexample:

f (z 1 , z 2 , z 3 ) = z 3 (z 1 z 3 3 + z 4 2 ) 2 + z 9 1 .
The plane projective curve determined by this equation has at the point z 1 = z 2 = 0 a singularity isomorphic to w 2z 36 = 0; it corresponds to a number η = sup q eq mq equal to 35, which must manifest itself in any neighborhood of the vertex of the cone defined by the equation, and prevents the exponent there from being 8 9 , since it must be at least equal to 35 36 . (This is an example I produced in the late 1980's in answer to a question of Amnon Neeman; the question of finding a sharp bound, in this homogeneous case, for θ 1 in terms of m is still open as far as I know).

The gradient Lojasiewicz exponent in dimension 2

In [GB-P] and [GB-K-P] some unexpected results, which find their roots in an earlier work of A. P loski, were proved, concerning the rational numbers which can appear as θ 2 , in the notations of the previous subsection, in the case d = 1 i.e., for plane curve singularities.

Theorem 2.1. (P loski, see [P2]) For d = 2 the rational number θ 2 belongs to the set

{N + b a , N, a, b ∈ Z, 0 ≤ b < a < N }.
Theorem 2.2. (García Barroso-P loski, see [GB-P]) The elements of the preceding set for which a = N -1, b > 1, gcd(a, b) = 1 are not exponents θ 2 for a plane curve.

There are more results of this kind in [GB-K-P].

Remark 2.1. The problem of determining from intrinsic invariants whether a given ideal is the jacobian ideal of a hypersurface is part of the folklore, and nobody knew how to begin. The previous theorem provides a partial answer in dimension 2: if a primary ideal I in C{z 1 , z 2 } is such that (ν I (m)) -1 belongs to the set determined in [GB-P], then it cannot be a jacobian ideal.

A theorem of M. Merle in [Me] implies that the jacobian Newton polygon of a plane branch is a total invariant of its equisingularity type: it determines and is determined by the Puiseux exponents, or the semigroup of values, or the topological type of its embedding in C 2 locally at x. This is no longer true for reduced plane curves, as was shown by Eggers (see [E]), and the correct generalization was achieved by E. García Barroso in [GB].

If one thinks that the jacobian Newton polygon is a natural extension of the gradient Lojasiewicz exponent it is natural to ask whether all Newton polygons meeting both axes (or convenient ) can be the jacobian Newton polygon of a hypersurface with isolated singularity, for a given dimension d. For d = 1 there is a very nice answer, for branches, due to E. García Barroso and J. Gwoździewicz in [GB-G]: there is an arithmetic characterization of the Newton polygons that are the jacobian Newton polygon of a plane branch. Moreover, they prove that if the jacobian Newton polygon of a plane curve is that of a branch, then the curve is a branch, giving rise to new criteria of irreducibility.

Lojasiewicz exponents in infinite dimensions

Let us briefly go back to a Lojasiewicz inequality in the real case, relating the distance to the zero set of a function to the value of the function:

Theorem 3.1. ( Lojasiewicz, [ L]) Let Ω be an open subset of R n , let f be an analytic function on Ω and V the locus of zeroes of f . For every compact K ⊂ Ω there exist C > 0, α ≥ 0 such that:

∀x ∈ K, dist(x, V ) ≤ C|f (x)| α ,
where d(x, V ) is the distance from x to V in R n . If V = ∅, we take α = 0 and agree that d(x, ∅) 0 = 1.

The key to understanding why this Lojasiewicz inequality is true is that the distance function to an analytic set behaves much like an analytic function: it is subanalytic. Note that in comparison with the previous inequalities, we have α = θ -1 . Let us reformulate the statement of the theorem as follows: for each sufficiently small ǫ, if |f (x)| ≤ ǫ, there exists x 0 ∈ V such that f (x 0 ) = 0 and |xx 0 | ≤ Cǫ α .

Let us now make a translation which is perhaps "classical" (see the review of [Sc]) and certainly well known to Hickel (see [START_REF] Hickel | Fonction de Artin et germes de courbes tracées sur un espace analytique[END_REF], 2.2): consider a space of analytic or formal functions in variables x 1 , . . . , x n , for example the maximal ideal M of the ring O = C{x 1 , . . . , x n } of convergent power series at the origin. Consider a finite set of algebraic (or analytic) equations Φ k (x 1 , . . . , x n , Y 1 , . . . , Y p ) = 0, with Φ k ∈ O[Y 1 , . . . , Y p ] (or in O{Y 1 , . . . , Y p }) and 0 ≤ k ≤ q, which we shall forget to write from now on; they define a subset V of the infinite dimensional space M p , which we shall call an algebraic subset (resp. an analytic subset) of this infinitedimensional space.

On M there is a metric given by the norm g = e -νM(g) , where ν M (g) is the M-adic order. We have g + g ′ ≤ max( g , g ′ ).

We extend this norm to M p by taking the maximum of the norms of the coordinates. In this context, Lojasiewicz's inequality becomes, after a small translation as above: For any sufficiently large integer N , if we have series without constant term y (N ) (x) = (y

= α ∈ R + , b ∈ R: ν M (y (N ) (x) -y(x)) ≥ aN + b,
which means that each coordinate is in M aN +b , so that we can reinterpret the Lojasiewicz inequality:

Proposition 3.1. Let V ⊂ M p be an algebraic subset defined by the polynomials

Φ k ∈ O[Y 1 , . . . , Y p ], 1 ≤ k ≤ q.
The following statements are equivalent: a) The subset V and its defining equations satisfy Lojasiewicz's inequality with respect to the norm f . b) There exist constants a ∈ R + , b ∈ R depending only on the system of equations Φ k such that for each sufficiently large integer N , if there exits a system of series without constant terms y (N ) (x) ∈ M p such that for all k we have ν M (Φ k (x, y (N ) (x))) ≥ N , then there exists a system y(x) ∈ M p such that Φ k (x, y(x)) = 0 and ν M (y (N ) (x)y(x)) ≥ aN + b.

It is natural to ask whether the Lojasiewicz inequality of Proposition 3.1 holds for algebraic and analytic subsets of M p .

Strong supporting evidence comes from M. Artin's theorem:

Theorem 3.2. (M. Artin, [A]) Given a system of equations such as Φ k (x, y), there exists an application β : N → N such that for any integer i, if there exists a system of series without constant term ỹ(x) such that

ν M (Φ k (x, ỹ(x))) ≥ β(i),
then there exists a system y 0 (x) such that Φ k (x, y 0 (x)) = 0 and ν M (ỹ(x)y(x)) ≥ i.

The only difference is that the function β(i) = i-b a obtained by inverting the map N → aN + b of statement 3.1 is linear, which increases the interest of finding linear upper bounds for Artin's beta function. The following result due to Hickel may help to throw some light and make one optimistic about the linearity:

Let our indeterminates consist of one indeterminate t and let the equation Φ be f (Y 1 , . . . , Y n ) = 0, where f (z 1 , . . . , z n ) is an analytic function with an isolated critical point at the origin. Solving the equation f (y 1 (t), . . . , y n (t)) = 0 with series without constant terms amounts to finding analytic arcs through the origin on the hypersurface defined by f (z 1 , . . . , z n ) = 0. Keeping the notation η = sup q eq mq introduced above, we have: Theorem 3.3. (Hickel,[START_REF] Hickel | Fonction de Artin et germes de courbes tracées sur un espace analytique[END_REF]) The function β of Artin satisfies in this case β(i) ≤ ⌊ηi⌋ + i.

However, the existence in general of a linear upper bound for Artin's beta function, conjectured by Spivakovsky in [S], was shown to be false by G. Rond in [START_REF] Rond | Sur la linéarité de la fonction de Artin[END_REF]. So the nature of Lojasiewicz inequalities changes when one passes to infinite dimensional spaces, if we insist on taking the metric g = e -νM(g) .

A recent result of Moret-Bailly shows that if we seek roots of our equations Φ k (x 1 , . . . , x n , Y 1 , . . . , Y p ) = 0 not only in M but in the maximal ideal M ν of a henselian valuation ring R ν containing O and such that M ν ∩ O = M, and use an adapted metric, we recover a Lojasiewicz inequality. The following statement is the special case for an affine R ν -scheme, of Moret-Bailly's theorem: Let R ν be a henselian valuation ring such that the field of fractions K of its completion is a separable extension of its field of fractions K. Let Γ be the group of values of the valuation ν, which is totally ordered and, for γ ∈ Γ + , consider the valuation ideal P γ (R ν ) = {x ∈ R ν |ν(x) ≥ γ}.

  ), . . . , y (N ) p (x)) ∈ M p such that for all k the inequalities ν M (Φ k (x, y (N ) (x))) ≥ N hold, there exists a point y(x) ∈ V with y (N ) (x)y(x) ≤ Ce -N α . If we write the positive constant C in the form C = e -b this becomes, with a

Theorem 3.4. (Moret-Bailly, [MB]) Consider a finite set of algebraic equations

There exist a positive integer B and an element δ ∈ Γ + with the following properties: for each γ ∈ Γ + and y (γ) = (y

p ) ∈ P Bγ+δ (R ν ) for all k, there exists y ∈ R p ν such that Φ k (y) = 0 for all k and y (γ)y ∈ (P γ (R ν )) p . So, in this situation, the Artin function associated to the valuation ν (and not to ν M ) is indeed bounded by the linear function γ → Bγ + δ. If we restrict ourselves to valuations of rank one, for which the value group is an ordered subgroup of R, and take the metric x = e -ν(x) we are exactly in the same situation as at the beginning of this section. Indeed, this suggests that we must seek solutions in henselian valuation rings, and that the right "distance to the zero set" is measured by the valuation. Moret-Bailly's theorem gives us a version of the Lojasiewicz distance inequality in the space R p ν .


Yaël Kouzmine
email: yael.kouzmine@toulouse.inra.fr

European Union, Mediterranean region and energy issue

Keywords: Energy issue, Energy policies, Sustainable development, Europe, Mediterranean region, Barcelona Process: Union for the Mediterranean

The energy issue is an essential factor of economic development. Moreover, since the World War II, it is one of the bases of European Union political genesis. Nowadays, the debate about the cooperation process between European Union and Mediterranean countries is partly focused on the energy issue. On the one hand, European energy dependency is raising, on the other hand, most of the Southern and Eastern Mediterranean countries are economically dependent on their hydrocarbon exports. In this context, a new European energy policy emerged in the 90's structured by three main principles:  energy dependency requires to diversify the geographic origins of the energy imports;

 renewable energies have to be developed due to the environmental issue and to the increasing prices of fossil fuels;

 create an European energy market, which could become an Euro-Mediterranean one.

These elements lead to the emergence of a new cooperation between Europe and the other

Mediterranean countries. The Euro-Mediterranean Partnership project launched in 2007 could be a political framework able to facilitate this new form of governance. Energy issue is a pertinent thematic to analyse territorial coherence and stakes of sustainable development in the Euro-Mediterranean region.

In the first part, we will demonstrate the historical role of energy issue in European Union emergence. In the second part, we will present European initiatives in energy area which tend to include the other Mediterranean countries. Finally, we will discuss about energy stakes in the region and territorial differences between northern and southern coasts

Introduction

Energy represents one of the bases of the European Union construction since the end of the World War II, and is today in the middle of the Euro-Mediterranean cooperation. In a context of fossil resources depletion and energy consumptions increasing, the energy interdependence between Mediterranean countries is more and more obvious. The hydrocarbon reserves of the North Sea become exhausted quickly, while the economies of the Southern and Eastern Mediterranean countries (SEMCs) depend mainly on the receipts oil and gas exports.

Structural deficit of energy offer and European dependence on external suppliers, Mediterranean or not, caused partly a political reactivation of the process of cooperation within the Euro-Mediterranean region, in order to diversify the energy provisioning and to manage in the long term the energy market safety. The Euro-Mediterranean political framework, outlined since 1995, saw its energy dimension reinforced because of the new European policies since the end of the 1990s. If the concerns are directed fossil energies, renewable energies, solar one in particular, have to play in the future a more important part in the construction of a Euro-Mediterranean energy partnership.

The Mediterranean Solar Plan was announced on 13 July 2008 at the Paris Summit for the Mediterranean region in order to develop renewable energy production, in particular solar technologies, on the southern side of the Mediterranean. Main objectives are, on the one hand to encourage the local production in order to satisfy demand at local level, and on the other hand, to increase export to Europe and consequently to diversify the European energy consumptions. These prospects would require a technology transfer to the South.

The EU having any interest to support the development of this type of energy: to achieve its ambitious goals of reduction of greenhouse gas emissions, in the one hand, and to increase proportion of renewable energies in its consumption, on the other hand.

Between environmental urgency and energy supply acuity, construction of a Euro-Mediterranean energy area seems a possible and relevant way of regional dialogue. The energy interdependence of the Mediterranean countries could then evolve to a cooperation articulating Community concerns and individual ambitions of the States.

In the first part, we will demonstrate the historical role of the energy issue in the European Union emergence. In the second part, we will present the European initiatives in energy field, which tend to include the other Mediterranean countries. Finally, we will discuss about the energy stakes in the region (consumptions, productions, supplies and dependences) and territorial differences between northern and southern coasts.

European construction and energy

The beginnings

Energy field always constituted a discussion thread to all the treaties and process of European political construction. As [START_REF] Keppler | L'Union européenne et sa politique énergétique[END_REF] affirms: "The founding fathers of modern Europe had understood that the Union would be a union of energy or would not be". With a civil and peaceful vocation, it was thought like the tool which can make possible the reduction of the European energy dependence in a context of oil consumption increase and of geopolitical instability to the Middle East (Crisis of Suez of 1956).

The new energy policies in Europe

The permanent tensions on the energy market -prices hike, anticipation of "peak oil", energy supply problems, geopolitical situation in the Middle East, Central Asia and Latin Americacumulated with increasing environmental concerns -Summits of the Earth in Rio de Janeiro in 1992 and Johannesburg in 2002, Protocol of Kyoto in 1997 [START_REF] Lamy | D'un G8 à l'autre : sécurité énergétique et changement climatique[END_REF] -considerably reinforce the energy question within the Community policies since the 2000.

Facing this reality, the European Commission Green Papers on energy in 2000 and 2006 begin debate and ring the alarm bell about the European energy dependence and its impact on the economic growth. Green Papers propose priority areas in order to overcome the structural deficit of the energy production in EU; all the more deficit will be increasing because of European Union expansion, which will integrate countries more consumers than energy producers. Energy dependence on imported energy was evaluated in 2000 to approximately 50% of the European requirements -figure which could reach 70% by 2030. In 1999, "it costs the Union some EUR 240 billion, or 6% of total imports and 1.2% of the GNP" (Commission of the European Communities, 2000).

The European strategy to reduce the dependence, or all at least to maintain it on a reasonable level, rests on two objectives: on the one hand, an action on the offer by diversifying the various sources of supply (by product and by geographical region), and developing endogenous energies, of which renewable energies and, on the other hand, policies to control consumption (change of the behaviors). In addition, the establishment of a common and concerted policy is now in question: to harmonize practices in EU, to reinforce Union's influence on the international scene in negotiation about energy field, in particular in event of crisis or increased tensions on the markets. The structural weakness of economic and political pressure at the international scale, as well as the absence of common instruments, increase EU dependence and vulnerability in the strategy for security of supplies, more especially as "the risk of rupture of the international supply, [in particular] oil ones, certainly increased these ten last years" [START_REF] Noël | La doctrine Bush et la sécurité pétrolière[END_REF].

Whereas the Green Paper adopted in 2000 underlines risks correlated with energy dependence (economic, social and environmental), Green Paper released in 2006 priories action areas. The main objective is the construction of a European energy market, in order to guarantee the security of supply and to "ensures solidarity between Member States"; to diversify the energy mix by the development of renewable energy for "tackling climate change"; to encourage innovation; and to put in place a "coherent external energy policy" (Commission of the European Communities, 2006). Recognizing the decisive role of energy dimension in the process of European construction and in the strict continuity of the principles emerged during the preceding phase of consultation, the elements for a new "energy policy for Europe" were presented in 2007 (Rosseti di Valdalbero, 2008). Beyond the main measure previously exposed, the objectives posted by the European Commission are particularly ambitious. At environmental level, "the European Parliament gave its backing to the EU's climate change package [in December 2008] to ensure that the EU will achieve its climate targets by 2020": 20% reduction in greenhouse gas emissions, 20% improvement in energy efficiency and 20% share for renewables in the EU energy mix1 (European Parliament, 2008). One priority of the French EU Presidency was to manage an agreement between all Member States on these whole measures, before end-2008 [START_REF] Delcour | L'énergie, enjeu clé pour la présidence française de l'Union[END_REF] In foreign policy, EU has also large ambitions: creation or reaffirmation of the relations with bordering countries (Turkey, Ukraine); strengthening bonds with principal energy suppliers (Russia, OPEC, GCC2 , Central Asia countries or Brazil for biofuel); establishment of an African-European partnership on the energy field; and, finally, improvement of interactions with others world's largest consumer (USA, China, India). All these European provisions will have however to take into account past or in progress initiatives in direction of the Mediterranean countries, which take part of reorganization of the political and energy stakes on the Euro-Mediterranean partnership.

The construction of a Euro-Mediterranean energy partnership

The political framework: Barcelona Process

The Barcelona Process aims to establish a new relational framework between the European Union and Mediterranean countries. It constitutes the heritage of the political orientations followed by Council of the European Union, as well as the European Commission proposals in the beginning of the 1990s. Ratified in 1995 in Barcelona, the process then included the fifteen States of the EU, including three new members at 1 st January 1995 (Austria, Sweden and Finland), and twelve countries of the Mediterranean region (fig. 1).

Thus, this partnership concerned very heterogeneous States from the geographical point of view, with economic, social and political specificities.

Final Declaration of the Barcelona Euro-Mediterranean Ministerial Conference of 27 and 28

November 1995 establishes "a multilateral framework bringing together economic and security aspects and also comprises a social, human and cultural dimension". Three keys sector composed the Euromed partnership. First focused on the emergence of "a common area of peace and stability" founded on the respect of human rights. The second insists on the creation of a "an area of shared prosperity" in the Mediterranean Basin, passing by the introduction of a free trade area, the development of an economic cooperation and "a substantial increase in the European Union's financial assistance to its partners". Finally, the third sector concerns cultural, social and human exchange, with the objective to "bring peoples closer together, promote understanding between them and improve their perception of each other" (Council of the European Union, 1995).

Fig. 1 Evolution of the Euro-Mediterranean Partnership

The Barcelona process allowed the constitution of an area for political dialogue between countries of the Mediterranean Basin. Periodic meetings of the Ministers for Foreign Affairs are organized, in spite of the very complex and tended geopolitical context in the region.

Construction of this area of dialogue differs from the regional integration led in the Maghreb since 1989 (Arab Maghreb Union), which has never been really efficient [START_REF] Troin | Les perspectives du Plan Bleu sur le développement durable en Méditerranée[END_REF].

According to a recent report, assets of Barcelona Process concern three fields (Mediterranean Institute, 2007). The first is the interest of non-EU Mediterranean Countries, of which Germany and of new Member States, for the political process, but also for the economic potential. Second point is the evolution of institutions "which seems irreversible" in SEMCs.

Finally, on the security aspect, fight against traffic and terrorism, cooperation was reinforced significantly within Euro-Mediterranean partnership.

Nevertheless, beyond the construction of a political exchange framework and in spite of number assets, the results are thin regarded as ambitions posted in 1995. Indeed, the North-South economic divide is still relevant today. Disparities, as regards of richness creation, remain significant. The average GDP per capita in Europe were established in 2005 to EUR 28 300, and was only of EUR 8 733 for the whole SEMCs, and even lower if Israel is exclude. Thus, in spite of the integration of SEMCs into the global economy, the economic growth differences and social change between countries and within the Mediterranean countries are still present. During the period 2000-2006, the average economic growth at 4% in the EuroMed zone "was not sufficient to reduce the structural deficit" (Bauchard, 2008).

Moreover, relative weakness of direct investments abroad, which move mainly towards the "traditional income sectors" (real, oil and tourism), did not allowed a total economic convergence of Euro-Mediterranean space [START_REF] Beckouche | Méditerranée : d'un Euromed en panne à une région industrielle Nord-Sud[END_REF]. Finally, the relative share of the European investments in Mediterranean space decrease then, at the contrary, the United States and the Gulf States tend to intensify their investments.

The economic divide also results in unemployment variations between North and South of the Mediterranean. The rate within the UE-27 reached 7.1% of the population in 2007, whereas it can exceed 10-15% in certain SEMCs. In addition Alain Le [START_REF] Roy | L'Union pour la Méditerranée, une opportunité pour renforcer la coopération entre les rives de la Méditerranée. Entretien[END_REF], quoting OECD, estimates that "considering demographic pressure in the southern shores of the Mediterranean, it would be appropriate to create approximately 40 million jobs in the fifteen next years, to maintain the same employment rate". Taking into account demographic situation and economic growth, economic situation will probably not improve radically in the long term.

Concerning cultural and human exchanges, access to Schengen space becomes more and more difficult. Mobility between Mediterranean shores, in particular South-North influx, is not so easy, in spite of historical regional interactions. This aspect confirms the functional non-integration between Europe and "its South".

On the political level, in spite of significant evolutions and institutional reforms, the great majority of the SEMCs remains controlled by authoritative regimes, not very inclined to a real democratic opening. On the other hand, the area and its peripheries directly undergo the multiple consequences of 11 September 2001: Iraq conflict, bilateral American interventionism for prevention of terrorism, etc. [START_REF] Noël | La doctrine Bush et la sécurité pétrolière[END_REF]. Detractors of the Barcelona Process moreover criticized the heaviness of the new institutions created and the lack of articulation with the European neighbourhood policy (Mediterranean Institute, 2007). Thus, after ten years, the "multilateral framework based on a spirit of partnership" created to stimulate the emergence of a "common area of peace, stability and prosperity through the reinforcement of political dialogue and security, an economic and financial partnership and a social, cultural and human partnership" (European Union, 1995) in the Mediterranan Basin, did not bear its fruits. European voluntarism to cooperate with Mediterranean partners continues in 2000 with the adoption of the EU Common Strategy for the Mediterranean. This strategy "pursues cooperation between the EU, the Mediterranean region and Libya in a vast range of areas" (European Union, 2000a). If it does not propose new elements in the European action in direction of Mediterranean partners, this strategy confirms nevertheless the increase of political cooperation and economic exchange.

The relations between EU and Mediterranean partner countries on the energy filed thus began formalized with the installation of the Barcelona Process in 1995, liked with the wish to found a free trade area in the Mediterranean Basin. The regular organization of Euro-Mediterranean ministerial conference on energy, as the creation of a euro-Mediterranean forum on this field marks the installation of a regional political dialogue on energy.

Union for the Mediterranean: new perspectives?

In the continuity of Barcelona Process, the project of Union for the Mediterranean (UFM), initiated and supported by the President of the French Republic Mr. Sarkozy and his special advisor Mr. Guaino, clearly aims to restructure cooperation in the Mediterranean. This project is based on the relaunch of regional political dynamics and a restoration of the strategic objectives with partner countries. Evoked by Nicolas Sarkozy in February 2007, the emergence of the UFM project at the end of the year 2007 was marked by the French President speech in Tangier on 23 October 2007. President Sarkozy then did not hesitate to reaffirm that the future of Europe is played in South, and that "in turning the back on the Mediterranean, Europe would be cut not only of its intellectual, morals, spiritual sources, but also of its future. Because it is in the Mediterranean Basin that Europe will gain its prosperity, its safety, that it will find the Founding Fathers' impulse". Fifty years after the Treaties of Rome signature, and referring to the Europe Founding Fathers, the French President launches the concept of Union in order to build a new political experiment in the Mediterranean area.

The reception of this proposal by the European countries was at the very least mitigated:

Germany fearing the emergence of major divisions within the EU and, more prosaically, the exclusion of its companies of the process; Spain fearing Barcelona Process disappearance to which it particularly attached. As for the SEMCs, they "in their majority, expressed an interest for the project" (Bauchard, 2008). However, some reserves were expressed by Turkey which saw an attempt to impose a substitute on its request for EU adhesion [START_REF] Schmid | La Turquie et l'Union pour la Méditerranée : un partenariat calculé[END_REF] and by Libya which, in spite of recent changes in the geopolitical plan [START_REF] John | Redefining the Libyan revolution: the changing ideology of Muammar al-Qaddafi[END_REF], perceived the project like a new form of imperialism and refusing the idea of an union with Israel.

These political criticisms led to a new formalization of the project in the Rome Call launched jointly by Spain, France and Italy on 20 December 2007. The Call stipulates that Barcelona Process and European neighbourhood policy "will remain central in the partnership between the European Union as a whole and its Mediterranean partners", the UFM will have to "give them an extra boost seeking to complement and work in cooperation with all the existing institutions" (Présidence de la République Française, 2007b). Turkey and Croatia were assured that this new process should not interfere in the adhesion negotiations: the future Union for the Mediterranean is founded on a principle of cooperation and not of integration.

The project receives the German approval after modification and a Franco-German agreement was signed on 3 March 2008, making possible integration of the whole European States in the project. This agreement opened the way with acceptance of the project by the Council of the EU now called: "Process of Barcelona: Union for the Mediterranean" (fig. 1). Following the European Council of 19-20 June 2008, negotiations were engaged with all the countries called to be members of the UPM, "to preparing a joint declaration to be adopted at the Paris Summit for the Mediterranean on 13 July 2008" (European Union, 2008a) which gathered the whole candidates States. The objective is to find a "new shared governace" and "to increasing solidarities through concrete projects of regional size" (Présidence de la République Française, 2008a).

Towards an institutionalization of the European energy partnership?

Since 2000, elements show the reinforcement of the Euro-Mediterranean interactions around the energy area. On 29 May 2006, the Council of the European Union approved the Energy Community Treaty. It aims to create an integrated market of energy, natural gas and electricity, between the EU, six Balkan countries, Bulgaria and Romania (then candidate countries for accession to the EU). Moreover, today, five States have the Observer status (figure 2). This treaty explicitly aims "to promote High-Levels of gas and electricity provision to all citizens based on public service obligations, and to achieve economic and social progress" in a context where the 1990s conflicts dismantled the distribution networks in Balkans. It also facilitates the energy opening-up of Greece, encouraging the "connections to Caspian, North Africa and Middle East gas reserves", and includes a "the coordination of mutual assistance in case of serious disturbance to the energy networks or external disruptions" (European Union, 2006). Beyond, we can consider the intrinsic geopolitical aspect of the pipelines which can represent "the vectors of a mystic unit and peace" [START_REF] Simonet | Les pipelines internationaux, vecteurs de prospérité, de puissance et de rivalités. Oléoducs et gazoducs dans la géopolitique et les relations[END_REF] in this area.

In the energy field, this first cooperation extended to Mediterranean partners is registered in parallel of a "new Euro-Mediterranean energy partnership" inaugurated at the 5 th Euro-Mediterranean Energy Ministerial Conference on 17 December 2007 in Limassol (Cyprus).

The conference gathered 35 countries, including 24 Europeans, 10 Member States of the future Union for Mediterranean, as well as Croatia with Observer status; it should be noted the absence of Libya to the negotiating table, however major actor of gas and oil production in the area.

European Energy Commissioner Andris Piebalgs, with Ministers from the EU Member States and from the Mediterranean partner countries, endorsed the Priority Action Plan for Euro-Mediterranean cooperation. Three main areas was defined, "harmonization of energy markets and legislations", promotion of "sustainable development in the energy sector" and finally development of "initiatives of common interest in key areas, such as infrastructure extension, 2007). The project would make possible, on the one hand, preservation of the fossil reserves, and consequently perpetuation of export earnings for producer countries (Algeria, Egypt…) and, on the other hand, reduction of the energy dependence in the Mediterranean Basin, by supporting regional interactions. Moreover, development of a low-carbon energy in the Mediterranean Partner Countries exported to the North could allow the EU, not to achieve, but to approach the European goals of a 20% share of energy from renewable sources by 2020, as well as the Kyoto objectives. The real solar potential, although marginally exploited at the present time, is nevertheless recognized [START_REF] Harouadi | Les potentialités d'exploitation d'hydrogène solaire en Algérie dans un cadre euro -maghrébin Partie І: Phase d'étude d'opportunité et de faisabilité[END_REF][START_REF] Alnaser | Data bank, First solar radiation atlas for the Arab world[END_REF] and could in the long term constitute a local and convenient resource to tackle energy demand increase in SEMCs.

However, if these elements propose an interesting prospective vision, the current energy system in the Mediterranean Basin remains founded on the massive use of fossil fuels.

Interstate relations are largely based on this field. An analysis of dependences and interactions around energy within the Mediterranean Basin can partly explain the construction of a Euro-Mediterranean energy area.

Energy stakes in Euro-Med: North-South complementarities

Fossil fuels

Energy is one of the "key sectors" of the Euro-Mediterranean cooperation, because there is an "obvious complementarity between two sides of the Mediterranean" (Chevallier, 2008), in particular for fossil energies but also for renewable energies. Indeed, all the European countries are strongly or completely dependent on the hydrocarbon imports, except United Kingdom, Denmark and the Netherlands which are producer countries. On the other hand, North African countries are mainly exporters (fig. 3). This situation is all the more problematic as "even in the most optimistic conditions, Europe will have to buy at least 10 million barrels of oil and approximately a billion cubic meters of gas per day on the international market" [START_REF] Keppler | L'Union européenne et sa politique énergétique[END_REF].

Although projections envisage for 2025 an energy growth rate "four times higher in the Southern and Easter Mediterranean countries than in the North, linked with economic development and increasing population needs" (UNEP, 2006), principal poles of consumption are today mainly located in Europe. In 2007, EU countries accounted almost 95% of total UFM oil consumption and 90% of gas consumption; but produced 32,2% of total UFM oil productions and in 56,1% of gas [START_REF] Bp | Statistical review full report worldbook[END_REF]. Thus, even if the majority of the European countries do not have fossil resources, energy dependence of the EU is more related to the high level of consumption than to the low level of production.

Fig. 3 Energy interdependence in the Mediterranean Basin

Indeed, sum of natural gas productions in the European Union is higher than the production in countries of Southern side of the UFM (191.9 against 129.5 billion cubic meters per day), while oil productions are almost equivalent (2 394 thousand of barrels per day for the EU, 2 808 thousand of barrels per day for the Mediterranean partners). But oil and gas European reserves, mostly in North Sea, are much less important than those of the South: 0,5% of world oil proved reserves and 1,6% of world gas proved reserves in 2007 are in UE, whereas it respectively accounts for 1,3% and 3,7% in the African part of the UFM. Moreover, we assist at the progressive exhaustion of European reserves. Exploitable gas reserves of United Kingdom, the most important EU hydrocarbon producer, passed 640 billion cubic meters in 1987 to 770 billion in 1997 (after new discoveries) and then fall to 410 billion in 2006. At the current rate, it remains 5.7 years after a stop of exploitation. EU gas reserves follow the same tendency (3 850 billion cubic meters in 1997 against 2 940 in 2007) and, at the current exploitation rate, fifteen years of production are estimated [START_REF] Bp | Statistical review full report worldbook[END_REF].

Thus, dependence on South should increase, whereas question of durability and balance between consumption and production, which already arises today, will grow-up in the long term. Indeed, in the totality of the UFM countries, gas consumption accounts for 18.4% of world gas consumption, for 11.6% of production and for 6.4% of proved reserves. This observation is even more alarming for oil: 18.9% of world consumption; 8.8% of production and 5.2% of world proved reserves.

We only considered in table n°1 UFM countries, but if Libya is integrated, balance changes considerably. Indeed, gas production is not very important (15.5 billion cubic meters), but more than half is exported to EU (Eurostat, 2008). Moreover, Libya is an important oil producer (2.2% of the world oil production). Sum of the Algerian, Egyptian and Libyan oil productions, represents twice the European production and cumulated reserves of these three countries, oil and gas, constitute almost twice the whole African reserves. On the whole, these three countries have approximately 5% of the world hydrocarbon reserves [START_REF] Bp | Statistical review full report worldbook[END_REF].

OIL

Tab. 1 Oil and gas in the Euro-Mediterranean region

Considering natural gas more specifically, EU depends mostly on three supplier countries:

Russia, Norway and Algeria (fig. 4). Indeed, gas transportation by pipeline requires a relative geographical proximity and a geopolitical stability to allow the realization and the security of infrastructures. This European dependence concerns and weakens importing countries as much as producer-exporter ones, which are competing and must be ensured the outlets for their production. For example, Algeria exports more than 60% of its gas production to Europe [START_REF] Bp | Statistical review full report worldbook[END_REF][START_REF] Remme | Future European gas supply in the resource triangle of the former Soviet Union, The Middle East and Northern Africa[END_REF] and 95% of his currency resources come from oil and gas.

On another side, importing countries want to diversify supplier countries in order to safe their supply, "central factor for energy safety being the degree of diversity of the world offer" [START_REF] Copinschi | L'Afrique dans la géopolitique mondiale du pétrole[END_REF]. Fig. 5 The Euro-Mediterranean network of gas pipelines

Renewable energies

Even if fossil fuels constitute in the present day the fundamental base of the energy consumption in the Euro-Mediterranean countries, UFM intends to durably support renewable energies development in the Mediterranean Basin. This determination, theoretical for the moment, answers to rarefaction and price increase of fossil energies. All the more that prices of these non-renewable resources will inevitably increase, whatever operating costs. Thus, renewable energies development represents more than an ethical choice, but a real alternative.

In 2006, consumption of renewable energies (hydraulic, geothermic, solar and wind) in the Mediterranean Basin were estimated at approximately 3% of the primary energy supply, with respectively 3.1% for the northern countries and 2.6% for the SEMCs (UNEP, 2006). The Blue Plan (a UNEP Program) recently worked out two prospective scenarios on the energy evolution in Mediterranean Basin. The first scenario considers continuity in States and companies strategies working in the region, and bases the analysis on a perspective of offer increase. The second, with voluntarism orientation, imagines a common political will for a rational energy use, in order to "increase the energy system effectiveness" and develop renewable energies. Development of these energies (solar, wind, geothermic and small hydraulic) and "use of technologies already available" on the market, could lead to a 20-25% decrease in energy demand, and more in the SEMCs. Moreover, "renewable energies share in energy balance could approach 14%" as against 3% in the present day (UNEP et al., 2006).

In addition, the Blue Plan noted that "the international cooperation [in the Mediterranean] to bring energy consumption under control and to develop renewable energies production" was characterized by "a multitude of isolated projects without a strong structuring capacity in the long term" (UNEP et al., 2006). Diversity of national situations among SEMCs, as well as political basic divergences, are important obstacles for the realization of large scale projects, such as Mediterranean Solar Plan envisaged by the UFM. Nevertheless, bilateral North-South projects are developing, sometimes to the detriment of South-South projects. But attitude of SEMCs concerning development of renewable energies is very variable and largely depends on their energy resources. For example, Morocco which 90% depends on the imports for its energy supply already installed, at the beginning of the 2000s, a wind park between Tétouan and Tangier. This park constitutes the largest installation of Africa and supply 400 000 people for the electric consumption.

On the contrary, Algeria and other producer countries do not feel the urgency to develop this type of installations. Nevertheless, cooperation with Germany led the emergence of two significant projects. The first aims to build a gas-solar hybrid power station in the area of Hassi R'Mel in the north of the Algerian Sahara. The second, the Adrar-Aachen project, proposes the export of solar energy from the Algerian Sahara (area of Adrar) to Europe via Sardinia via high-voltage direct-current lines (HVDC). This project could generate a 10-15% reduction in transport losses. This action for solar energy is in the line of DESERTEC project, which aims to connect the whole of the Euro-Mediterranean countries to a supergrid for the transmissions of solar power. However, at the present time, solar energy is mainly used for rural electrification in the SEMCs. Costs for building of big solar power station remain still high, although solar energy can become competitive by 2020.

Perhaps Union for the Mediterranean, as well as the other mechanisms of cooperation on the energy question, will help the necessary development of the renewable energies use.

Moreover, and in spite of the weak productive and industrial integration of Mediterranean Basin [START_REF] Beckouche | Méditerranée : d'un Euromed en panne à une région industrielle Nord-Sud[END_REF], renewable energies can constitute, in the long term, a dynamic market, involving European companies and southern operators, for the benefit of the populations as regards employment and energy costs. But, getting these projects underway requires more than statement of principles.

Conclusion

The progressive construction of a Euro-Mediterranean energy area is in line with a changing context of regional political stakes in the Mediterranean Basin. Gulf which is equivalent to a "negative authority": aiming to prevent certain actions, more than to determine positively the future of the area" (Noël, 2008). Energy sector, beyond inevitable economic interdependences, can constitute a "common interest in the long term" (Bauchard, 2008). It can make possible to find a Euro-Mediterranean consensus, in particular, on the renewable energies field in a context of rarefaction of the European hydrocarbon reserves. Thus, renewable energies can, on the one hand represent a local response to a consumption increasing on the southern shore, and on the other hand take part in the diversification of the European energy supply. Moreover, the market of renewable technologies should constitute a considerable economic income for European industries, and a technology transfer to the South. This aspect could, in the long term, redefine energy relations in the Euro-Mediterranean region, and redraw the "energy dependence" map with the emergence of renewable energy producer countries.

  Thus, the first step of the European Union was the signature of the Treaty of Paris on 18 April 1951 which established the European Coal and Steel Community (ECSC) "based on a common market, common objectives and common institutions" (article 1). One of the objectives of this treaty was to create a convergence of interests between ex-belligerents through raw materials (coal and iron ores) necessary to the rebuilding and to the economic dynamism after the World War II. The second European Treaty directly relating with energy field created the European Atomic Energy Community (Euratom). It was established on 25 March 1957 along with the European Economic Community (EEC) by the Treaty of Rome. Euratom directly integrated economic and political dimensions of the energy question in the construction of a supranational entity.

  and the United Nations Climate Change Conference in Poznań in 2008 and then in Copenhagen in 2009.

  investment financing and research and development"(European Union, 2007). One of the main objectives of Limassol Conference is the development of infrastructures and networks for North-South and South-South supply. Thus, first partnerships aim a diversification of fossil fuel supplies and electricity supplies for the EU, and facilitating political dialogue and investments in the Mediterranean Basin.The UFM project in the energy field completes Energy community Treaty and Limassol Conference.

Fig. 2

 2 Fig. 2 The energy issue and the EuroMediterranean Partnership

Fig. 4

 4 Fig. 4 The European gas supplying

  Economic and political divide remain real and persistent between northern and southern shores. Strong European political integration contrasts with the failure of the concrete realization of the Arab Maghreb Union and with the political fragmentation of the Middle East. Moreover, the GDP per capita differentials, far from reabsorbing, accentuates cleavages between Mediterranean States. The various bridge-building attempts since the 1990s variable succeed, but they provided foundations for a political dialogue, in the absence of an efficient regional cooperation. Tensions remain multiple and geopolitical balances evolved, with the assertion of the foreign actors' role such as United States or Persian Gulf States, which invests financially or strategically this area. In the concert of geopolitical changes, Euro-Mediterranean dialogue was reaffirmed recently whereas energy field shows well reciprocal stakes and interdependences between States of the Mediterranean Basin. Moreover, with the fossil resources rarefaction and the increase of consumption, energy field was reinforced into the Community policies and takes an active part in the Euro-Mediterranean dialogue. Barcelona Process, Community of energy and Union for the Mediterranean concretely integrate this dimension and could help EU to play a structuring role in the region, contrary to traditional action "of the United States in the Persian

Year of reference for reductions remains to be solved (it will be a priori 1990, as for the Kyoto Protocol).

The Gulf Cooperation Council (GCC) is an economic and political policy-coordinating forum for the six member tates (Bahrain, Kuwait, Oman, Qatar, Saudi Arabia, and the United Arab Emirates).