J J Alibert

P Seppecher
email: seppecher@imath.fr

Closure of the Set of Diffusion Functionals -the One Dimensional Case

Keywords: Homogenization, Mosco-convergence, -convergence, Local functionals, Dirichlet forms Mathematics Subject Classifications (2000) 49J45, 49N10, 35J25

We characterize the closure with respect to Mosco or -convergence of the set of diffusion functionals in the one dimension case. As commonly accepted we find this closure is a set of local Dirichlet forms. The difficulty is to identify the right notion of locality. We compare different possible definitions. We give a representation theorem for the elements of the considered closure.

Introduction

We are interested in the characterization of the different limits which can be reached by a sequence of diffusion functionals, i.e. of functionals of the kind

F α (u) := α(x)|∇u(x)| 2 dx , ( 1.1)
where the positive diffusion coefficient α belongs, like its inverse, to L ∞ ( ). These functionals belong to the set of Dirichlet forms. As this set has been proved [START_REF] Mosco | Composite media and asymptotic Dirichlet forms[END_REF] to be closed for the -convergence, it is the natural framework of our study.

In dimension greater than two, well-known examples [START_REF] Hashin | A variational approach to the theory of the elastic behavior of multiphase materials[END_REF][START_REF] Milton | Composite materials with Poisson's ratios close to -1[END_REF][START_REF] Murat | Topics in the mathematical modelling of composite materials[END_REF][START_REF] Tartar | Estimations de coefficients homogénéisés[END_REF] given by homogenization theory show that the limit functional can be non isotropic: it is then described by a diffusion matrix A(x) and the limit functional takes the form: ∇u(x) • A(x) • ∇u(x) dx .

(1.

2)

It has been proved [START_REF] Murat | A strange term brought from somewhere else[END_REF] that the limit of functionals Eq. 1.1 takes the form Eq. 1.2 when the sequence of diffusion coefficients (α n ) and their inverses α -1 n are bounded by a fixed real M. This is still true under weaker assumptions [START_REF] Buttazzo | -limits of integral functionals[END_REF] but not in the general case.

In dimension greater than three, examples have been given (cf. [START_REF] Bellieud | Homogénéisation en présence de fibres de grandes conductivité[END_REF][START_REF] Bellieud | Homogenization of elliptic problems in a fiber reinforced structure[END_REF][START_REF] Buttazzo | -limits of integral functionals[END_REF][START_REF] Mosco | Composite media and asymptotic Dirichlet forms[END_REF]), [START_REF] Briane | Fibered microstructures for some nonlocal Dirichlet forms[END_REF][START_REF] Camar-Eddine | Non-local interactions resulting from the homogenization of a linear diffusive medium[END_REF] in which non local interactions arise at the limit. These interactions are represented by a non-negative measure γ on × and the limit functional F contains the nonlocal term or jumping term : × (u(x)u(y)) 2 γ (dx dy).

(

Other examples [START_REF] Murat | A strange term brought from somewhere else[END_REF] have been given in which the limit functional contains a socalled killing term of the form (u(x)) 2 ν(dx), (1.4) where ν is a non-negative measure on . Let us notice immediately that we only consider in this paper functionals which vanish on constant fields c, or in an equivalent way, which are invariant when adding a constant :

∀c ∈ R, F(u + c) = F(u). ( 1.5)
We call "objective" the functionals which satisfy this property. Any limit of a sequence of such functionals will inherit this property and we do not have to consider killing terms.

Recently, it was proved [START_REF] Camar-Eddine | Closure of the set of diffusion functionals and that of elasticity with respect to the Mosco-convergence[END_REF][START_REF] Camar-Eddine | Closure of the set of diffusion functionals with respect to the Mosco-convergence[END_REF] that, in dimension greater than three, the closure of the set of diffusion functionals coincides with the set of all objective Dirichlet forms.

In dimension two the characterization of the closure is still an open problem: very recently some fundamental differences between the two dimension case and greater dimension cases has been pointed out [START_REF] Briane | Non-local effects in two-dimensional conductivity[END_REF].

In the one dimension case it is a fact commonly accepted that any limit remains local. The goal of this paper is to establish this fact rigorously. In the literature there are many different ways for defining locality but we need to introduce a new definition adapted to our purpose. At the end of this paper we discuss the relations between the different notions of locality.

The paper is organized as follows: in Section 2 we describe the framework of objective Dirichlet forms and of Mosco or -convergence. We set precisely the closure problem. We also give a simple example which shows the necessity of dealing with non regular Dirichlet forms.

In Section 3 we propose a new definition of locality and we state our main results. First we give a representation theorem for any local Dirichlet form. Then our main result states that this locality property characterizes the closure of the set of diffusion functionals. The proofs of these two theorems are rather long. There are splitted in several parts. In Subsection 3.2 we establish some characterizing properties of local Dirichlet forms. Subsection 3.3 is devoted to the proof of the representation theorem : many steps are necessary, the longest one being the construction of the underlying measure. Subsection 3.4 is devoted to the proof of the closure theorem.

The major difficulty lies in the density result. The previous representation theorem plays here a crucial role.

In Section 4 we discuss the different notions of locality which can be found in the literature and we compare them by considering examples. These examples are based on the existence of "essential partitions" of the interval (0, 1), that is, partitions such that the intersection of any open interval with any piece of the partition has a non vanishing Lebesgue measure. We show that all the considered notions of locality are indeed different (and also different from the one we propose). We also prove that none of them are closed for the Mosco-convergence.

Notations

As this paper is concerned only by the one dimension case, denotes a bounded open interval := (0, 1) ⊂ R and L 2 ( ) is the usual space of square integrable (class of) functions with respect to the Lebesgue measure on . Note that in this whole paper, unless differently specified, the considered measure is the Lebesgue measure. We recall that the support of u ∈ L 2 ( ) is the smallest closed subset of such that u(x) = 0 for almost every x ∈ \ .

Let us first, following [START_REF] Fukushima | Dirichlet Forms and Markov Processes[END_REF] introduce the set of Dirichlet forms.

Definition 1

We call Dirichlet form any functional on L 2 ( ) satisfying the following property i) F is non negative: it takes values in [0, +∞].

ii) F is quadratic: its domain D(F) := {u ∈ L 2 ( ) : F(u) < +∞} is a linear subspace of L 2 ( ) and there exists a positive semi-definite bi-linear form B such that F(u) = B(u, u) for every u in D(F). iii) F is lower semi-continuous: it satisfies, for any u ∈ L 2 ( ) and any sequence (u n ) converging to u :

lim inf n→∞ F(u n ) ≥ F(u) . (2.1) iv) F is Markovian: it satisfies for any u ∈ L 2 ( ) F( u ) ≤ F(u), ( 2.2)
where u denotes the truncated function u := max(0, min(1, u)).

Remark 1 By proposition 11.9 of [START_REF] Dal Maso | An introduction to -convergence[END_REF], Property (ii) is equivalent to the fact that, for any u and v in L 2 ( ) and any t ≥ 0,

F(u) ≥ 0, F(tu) ≤ t 2 F(u), F(u + v) + F(u -v) ≤ 2F(u) + 2F(v). (2.3)
Remark 2 Note that Property (ii) ensures that the domain D(F) is non empty. In the present definition of Dirichlet forms the domain D(F) is not necessarily dense in L 2 ( ). We emphasize that, unlike many authors (e.g. [START_REF] Bouleau | Dirichlet forms and analysis on Wiener space[END_REF]), we do not assume this density property. Indeed, as we will see later (see Example 1), this property is not preserved when passing to the limit with respect to the considered convergence of functionals.

Definition 2

We call objective Dirichlet form any functional satisfying in addition to properties (i)-(iv), v) F is objective: it satisfies F(c) = 0 for any constant function c.

We denote Q the set of all objective Dirichlet forms.

Remark 3 Property (v) is equivalent to ∀u ∈ L 2 ( ), ∀c ∈ R, F(u + c) = F(u) . (2.4)
To check this equivalence, it is enough to remark that, when F(u) is finite, the quantity F(u + c) -F(u) -F(c) is linear in c and lower-bounded by -F(u).

Definition 3 We say that F ∈ Q is a diffusion form if there exists α ∈ L ∞ ( , R + * ) with α -1 ∈ L ∞ ( , R + * ), such that F(u) = α(x)(u (x)) 2 dx if u ∈ H 1 ( , R), +∞ otherwise. (2.5)
We denote D the set of all diffusion forms.

Definition 4 A sequence (F n ) in Q Mosco-converges to F if
and only if it satisfies the following two properties: i) Lower-bound inequality : For any sequence (u n ) converging weakly to some u in L 2 ( ), the following lower-bound inequality holds :

lim inf n→∞ F n (u n ) ≥ F(u) . (2.6)
ii) Upper-bound inequality : For every u in L 2 ( ), there exists an approximating sequence (u n ) converging to u strongly in L 2 ( ) such that

lim sup n→∞ F n (u n ) ≤ F(u).
(2.7)

Definition 5 Let U be a subset of Q, we call Mosco-closure of U and denote U the set of all possible Mosco-limits of all sequences in U.

Our goal is then to characterize D.

Remark 4 As the properties (i), (iii), (iv), (v) and also (ii) in the form Eq. 2.3 easily pass to the limit, the set Q is closed for the Mosco-convergence: Q = Q. The set Q is actually a good framework for our problem.

Remark 5 Mosco-convergence in the L 2 ( ) topology is clearly a stronger notion than -convergence for the strong topology of L 2 ( ) (refer to [START_REF] Dal Maso | An introduction to -convergence[END_REF] for definition and properties of -convergence). Then the Mosco-closure of a set is contained in its -closure, i.e. in the set of all -limits of all sequences in U. However, the previous closure result remains true even if one uses the -convergence in the strong topology of L 2 ( ) [START_REF] Mosco | Composite media and asymptotic Dirichlet forms[END_REF]. Therefore our results can be interpreted in terms of -convergence for the strong topology of L 2 ( ).

Though we will deal essentially with non regular Dirichlet forms, it is not useless to introduce the set Q r of regular Dirichlet forms: let C 0 ( ) (or C 1 0 ( )) denote the set of continuous (resp. continuously differentiable) functions with compact support in . The form F is said to be regular if there exists a subset of D(F) ∩ C 0 ( ) dense in C 0 ( ) for the uniform norm and in D(F) for the norm u 2 L 2 ( ) + F(u). The Deny-Beurling formula [START_REF] Beurling | Dirichlet spaces[END_REF] states that any regular and objective Dirichlet form admits on C 1 0 ( ) the following representation (in which η is a non negative Radon measure on while γ is a symmetric non-negative Radon measure on × which does not concentrate on the diagonal):

F(u) = (u (x)) 2 η(dx) + × (u(x) -u(y)) 2 γ (dx dy).
(2.8)

One refers usually to the first term of this representation as the "diffusion term" or "local term" and to the second one as the "jump term" or "non-local term". The set D of diffusion forms is clearly included in the set of regular forms. This is not the case for the researched closure D. To illustrate this fact let us consider the following simple example.

Example 1 Let us consider the sequence of diffusion functionals (F n ) in D defined by

F n (u) = α n (x)(u (x)) 2 dx if u ∈ H 1 ( , R), +∞ otherwise.
where

α n (x) = n -1 if x ∈ 1 2 , 1 2 + 1 n , n otherwise.
We let the reader check that this sequence Mosco-converges to the functional F ∈ Q defined by

F(u) = (a -b ) 2 if u = a ∈ R, a.e. in 0, 1 2 , u = b ∈ R, a.e. in 1 2 , 1 , +∞ otherwise.
In this example D(F) is a subspace of L 2 ( ) of dimension 2. It is not a dense subspace. The functional F which belongs to D is definitively not a regular Dirichlet form.

Main Results

Statement of the Main Results

Recall that x ∈ is called a Lebesgue point of u ∈ L 1 oc ( ) if there is a real number, let us call it ũ(x), such that

lim ε→0 1 2ε x+ε x-ε |u(t) -ũ(x)|dt = 0. (3.1)
The Lebesgue points of u are thus points where u does not oscillate too much, in an average sense. As well known, almost every x ∈ is a Lebesgue point of u.

Let us introduce a notion of locality which is suitable for characterizing the closure of the set of diffusion forms.

μ u (O) = μ u (O \ {x})
The set of all such forms is denoted L.

The following theorem gives an integral representation for such local forms.

Theorem 1 Let F ∈ L.
Then there exists a unique closed subset S of and a unique Radon measure μ on \ S such that, for every u in the domain of F, the derivative u of u in the sense of distributions in \ S can be written u = du dμ dμ with du dμ in L 2 μ and

F(u) = \S du dμ 2 dμ (3.4)
This representation theorem is a key point for the proof of our following main result.

Theorem 2 The closure D of the set of diffusion functionals coincides with the set of local Dirichlet forms L.

Characterization of Local Forms

The following proposition gives a criterion to decide whether a functional F belongs to L or not. For each Lebesgue point x of u ∈ L 1 oc ( ), define

u x := u1 (0,x) + ũ(x)1 (x,1) and u r x := ũ(x)1 (0,x) + u1 (x,1) . (3.5) Proposition 1 A Dirichlet form F ∈ Q belongs to L if and only if F(u) = F u x + F u r x , (3.6)
for every u ∈ L 2 ( ) and every Lebesgue point x of u.

Proof Let F ∈ L, u ∈ L 2 ( ) and x be a Lebesgue point of u. The functional v → μ v ((s, t)
) is quadratic and objective for every open interval (s, t) ⊂ which implies that, if v = c a.e. in (s, t) for some constant c then μ v ((s, t)) = 0. Since x is also a Lebesgue point for u x and u r x , we have

F(u) = μ u ( ) = μ u ( \ {x}) = μ u ((0, x)) + μ u ((x, 1)) = μ u x ((0, x)) + μ u r x ((x, 1)) = μ u x ( \ {x}) + μ u r x ( \ {x}) = μ u x ( ) + μ u r x ( ) = F u x + F u r x
Conversely, assume that F ∈ Q is such that Eq. 3.6 holds. Let u ∈ L 2 ( ) and x ∈ be a Lebesgue point of u. For each δ > 0, there are two Lebesgue points a δ ∈ (0, δ),

x δ ∈ (x -δ, x) of u such that | ũ(a δ )| ≤ 1 δ δ 0 |u(t)|dt and | ũ(x δ ) -ũ(x)| ≤ 1 δ x x-δ |u(t) -ũ(x)|dt.
This implies that there exist two sequences

(a n ), (x n ) of Lebesgue points of u such that (a n ) decreases to 0, (x n ) increases to x, lim n a n ũ(a n ) 2 = 0 and lim n | ũ(x n )-ũ(x)|=0.
Then

u xn -u an + ũ(a n ) = u a.e. in (a n , x n ) and u xn -u an + ũ(a n ) converges to u x with respect to the L 2 ( ) norm. Hence F(u, (0, x)) ≥ lim inf n F u xn -u an + ũ(a n ) ≥ F u x ≥ F (u, (0, x)) . ( 3.7)
In the same way, one can prove that F(u, ) = F(u) and F(u, (x, 1)) = F(u r x ). Then for every Lebesgue points x < y of u,

F(u, (0, x)) = F u x , F(u, (x, y)) = F u y -u x , F(u, (y, 1)) = F u r y . (3.8)
Define the subset S u of by

S u := {x ∈ : 0 ≤ s < x < t ≤ 1 =⇒ F(u, (s, t)) = +∞) } . ( 3.9)
Let I be any connected component of the open set \ S u and x 1 < ... < x n be n Lebesgue points of u in I. By Eq. 3.8, we have

F(u, (x 1 , x n )) = n-1 i=1 F(u, (x i , x i+1 )) (3.10)
Then choose a Lebesgue point x of u in I and consider the non decreasing function

f I from I to R defined by f I (t) := F(u, (x, t)) if t ≥ x, f I (t) := -F(u, (t, x)) if t < x. Thus f I (t) -f I (s) = F(u, (s, t))
for every Lebesgue points s < t of u.

Let now y ∈ I be any Lebesgue point of u and let us choose two sequences

(s n ) and (t n ) of Lebesgue points of u such that (s n ) increases to y, (t n ) decreases to y, lim n | ũ(s n ) -ũ(y)| = 0 and lim n | ũ(t n ) -ũ(y)| = 0. Using Eq. 3.10 we obtain F(u, (s 1 , y)) + F(u, (y, t 1 )) = F(u, (s 1 , s n )) + F(u, (s n , t n )) + F(u, (t n , t 1 )) < +∞. Since lim inf n F(u, (s 1 , s n )) ≥ F(u, (s 1 , y)), lim inf n F(u, (t n , t 1 )) ≥ F(u, (y, t 1 )), we get lim n F(u, (s n , t n )) = 0.
Then f I has no jump at any Lebesgue point of u. This implies that there exists a unique Radon measure μ I on I such that for every Lebesgue points

s < t ∈ I of u, μ I ((s, t)) = f I (t) -f I (s) = F(u, (s, t)). Then for every s < t (Lebesgue points or not) such that [s, t] ⊂ I we have μ I ((s, t)) = F(u, (s, t)).
(3.11)

Define the measure μ u on the σ -field of Borel sets B ⊂ by

μ u (B) := +∞ if B ∩ S u = ∅ I μ I (B ∩ I) if B ∩ S u = ∅ (3.12)
where the sum is taken over all the connected components of \ S u . Now let us check that μ u is suitable. If (s, t) ∩ S u = ∅ then F(u, (s, t)) = +∞ which implies μ u ((s, t)) = F(u, (s, t)). If (s, t) ∩ S u = ∅ then there exists a unique connected component I of \ S u such that (s, t) ⊂ I which implies (by Eq. 3.11) that μ u ((s, t)) = F(u, (s, t)). Then for every (s, t) ⊂ , we have

μ u ((s, t)) = F(u, (s, t)). Let x ∈ be a Lebesgue point of u. If x ∈ \ S u then μ u ({x}) = μ I ({x}) = 0 where I is the connected component of \ S u containing x. If x ∈ S u then μ u ({x}) = +∞ and μ u (O) = +∞ for every open set O ⊂ such that x ∈ O. Then F ∈ L.
The next proposition shows that minimization of any F in L can be achieved by monotone functions. We do not know if this property characterizes the set L.

Proposition 2 If F ∈ L, u ∈ L 2 ( ) and x < y are two Lebesgue points of u then there exists v ∈ L 2 ( ) satisfying (i) v = u a.e. in \ [x, y], (ii) v is monotone in (x, y), (iii) v ∈ [ ũ(x), ũ(y)] a.e. in (x, y), (iv) F(v) ≤ F(u).
Proof Without loss of generality, let us assume that ũ(x) ≤ ũ(y). By induction, we construct a sequence of subdivisions σ n :

t n 0 = x < t n 1 < • • • < t n 2 n = y with a step size s n := sup i |t n i+1 -t n i |
tending to zero and an associated sequence of functions

(u n ) satisfying (a) u n = u a.e. in \ [x, y], (b) each t n i is a Lebesgue point of u n , (c) ũ(t n i ) ≤ u n ≤ ũ t n i+1 in t n i , t n i+1 , (d) F(u n ) ≤ F(u).
For n = 0 we consider the trivial subdivision x < y and the associated function u 0 defined by u 0 =: min ũ(y), max u, ũ(x) in [x, y] and u 0 := u in \ [x, y]. Proposition 1 and Markov property ensure that F(u 0 ) ≤ F(u).

Assume that (σ n , u n ) is suitable. We define σ n+1 by setting t n+1 2i := t n i and by choosing for

t n+1 2i+1 a Lebesgue point of u n in 2 3 t n i + 1 3 t n i+1 , 1 3 t n i + 2 3 t n i+1 . For such a couple (σ n , u n ) we denote u n+1 := u1 \[x,y] + 2 n+1 1 ũn t n+1 i 1 t n+1 i-1 ,t n+1 i u n+1 := u1 \[x,y] + 2 n+1 1 ũn t n+1 i-1 1 t n+1 i-1 ,t n+1 i
Then we define u n+1 := min u n+1 , max u, u n+1 . Proposition 1 and Markov property ensure that F(u n+1 ) ≤ F(u n ). On the other hand, u n and u n are non decreasing in (x, y), u n , u n belongs to [ ũ(x), ũ(y)] a.e. in (x, y). Moreover

u n ≤ u n+1 ≤ u n+1 ≤ u n+1 ≤ u n
a.e. in , and

u n+1 -u n+1 L 1 ( ) ≤ 2 3 u n -u n L 1 ( ) .
Then

u n converges to some v ∈ L 2 ( ) with v monotone in (x, z). Moreover v ∈ [ ũ(x), ũ(y)] a.e. in (x, y), v = u a.e. in \ [x, y] and F(v) ≤ lim inf n F(u n ) ≤ F(u).
The characterization of L given by proposition 1 enables us to state the following lower-semi-continuity result.

Proposition 3 Let F ∈ L and (u n ) be a sequence in L 2 ( ) strongly converging to some u. Then lim inf n F(u n , (s, t)) ≥ F(u, (s, t)) for every open interval (s, t) ⊂ .
Proof Without loss of generality, assume that u n converges to u a.e. in and lim inf n F(u n , (s, t)) = lim n F(u n , (s, t)) < +∞. Let s < t two points in (s, t). Choose x ∈ (s, s ) and y ∈ (t , t) such that x and y are Lebesgue points of u and of every u n and such that lim n ũn (x) = ũ(x) and lim n ũ(y) = ũ(y). We have

lim n (u n ) y -(u n ) x -u y -u x L 2 ( ) = 0. Hence lim inf n F(u n , (s, t)) ≥ lim inf n F (u n ) y -(u n ) x ≥ F u y -u x ≥ F(u, [s , t ]).
The proposition is proved as the previous inequality holds for any [s , t ] ⊂ (s, t).

Proof of the Representation Theorem

Construction of the Set S

Let us introduce the set S where, roughly speaking, the value of F(u) does not provide any control on the variation of u.

Proposition 4

To any F ∈ Q, we associate the set S := {x ∈ : F( 1 (x,1) ) = 0}.

(3.13)

If F ∈ L, u ∈ L 2 ( ),
F(α1 (0,x) + β1 (x,1) ) = (β -α) 2 F 1 (x,1) = 0. (3.15) Let x ∈ S ∩ (s, t). Let (s n ) and (t n ) be two sequences in (s, t) of Lebesgue points of u such that x -1 n < s n < x < t n < x + 1 n and | ũ(s n )| ≤ n x x-1 n |u(t)|dt and | ũ(t n )| ≤ n x+ 1 n x |u(t)|dt Jensen's inequality implies that lim n (x -s n )| ũ(s n )| 2 = 0 = lim n (t n -x)| ũ(t n )| 2 . Define u n := u1 (0,sn) + ũ(s n )1 (sn,x) + ũ(t n )1 (x,tn) + u1 (tn,b ) . Since lim n ||u -u n || L2( ) = 0 we deduce from Proposition 3 that μ u (s, t) \ {x} = lim n μ u ((s, s n )) + μ u ((t n , t)) = lim n F(u, (s, s n )) + F(u, (t n , t)) = lim n F(u n , (s, t)) ≥ F(u, (s, t)) ≥ μ u ((s, t)) Since μ u ((s, t)) < +∞ we conclude that ∀x ∈ S ∩ (s, t) μ u ({x}) = 0. (3.16)
Let S σ be the set of these x ∈ S such that S∩] x -ε, x + ε [= {x} for some ε > 0. Since S σ is at most countable, we deduce from Eq. 3.16 that μ u ((s, t) ∩ S σ ) = 0.

(3.17)

Let h > 0 and consider a subdivision s = t 0 < ... < t n+1 = t of (s, t) such that |t it i-1 | < h and t 1 , ..., t n are Lebesgue points of u. Denote I i :=] t i , t i+1 [ and let J be the set of these i ∈ {1, ..., n -1} such that

I i ∩ S \ S σ = ∅. If i ∈ J then I i contains at least two points s 1 i < s 2 i of S. Define v h ∈ L 2 ( ) by v h (x) := ⎧ ⎪ ⎨ ⎪ ⎩ ũ(t i ) if x ∈ t i , s 1 i , c i if x ∈ s 1 i , s 2 i , ũ(t i+1 ) if x ∈ s 2 i , t i+1 , (3.18) if x ∈ I i for some i ∈ J, v h (x) := u(x) otherwise. Here c i is a real tuned in such a way that Ii u(t)dt = Ii v h (t)dt.
As h tends to zero, the sequence (v h ) converges to u in L 2 ( ) then, by Proposition 3, lim inf h↓0 F(v h , (s, t)) ≥ F(u, (s, t)). On the other hand, using Eq. 3.15 we have, for any i ∈ J, F(v h , I i ) = 0. As, for any i / ∈ J, F(v h , I i ) = F(u, I i ), we have

μ u (s, t) ∩ S \ S σ ≤ F(u, I 0 ) + F(u, I n ) + i∈J F(u, I i ) ≤ F(u, (s, t)) -F(v h , (s, t)) + F(u, I 0 ) + F(u, I n ).
Passing to the limsup as h tends to zero, we get

μ u (s, t) ∩ S \ S σ ≤ 0. (3.19)
Proof of Proposition 4 is completed recalling Eq. 3.17. We need to state several properties for m(s, t) before proving Proposition 5.

Construction of the

Lemma 1 Let F ∈ L. Then i) S := {x ∈ : ∀s, t ∈ (s < x < t =⇒ m(s, t) = 0) }. ii) Let x ∈ , for every sequences (s n ) in (0, x) and (t n ) in (x, 1) converging to x, lim n m(s n , t n ) = m(x, x). iii) If [s, t] ⊂ \ S then m(s, t) > 0.
Proof Point (i) is obvious. For each integer n, owing to the properties of Dirichlet forms, there exists a minimizer u n ∈ A(s n , t n ) such that u n = 0 a.e. in (0, s n ), u n = 1 a.e. in (t n , 1). and

F(u n ) = m(s n , t n ). Since lim n u n -1 (x,1) L 2 ( ) = 0 we have m(x, x) ≥ lim sup n m(s n , t n ) ≥ lim inf n m(s n , t n ) ≥ lim inf n F(u n ) ≥ F(1 (x,1) ) ≥ m(x, x).
Assume that [s, t] ⊂ \ S and m(s, t) = 0. By Proposition 2 there is a non decreasing u ∈ A(s, t) such that F(u) = 0. Owing to (i), for each x ∈ [ s, t ] there exist Lebesgue points of u such that s

< x < t et m(s , t ) > 0. Since F(u) ≥ ũ(t ) -ũ(s ) 2 m(s , t ) , ũ(t ) = ũ(s ). Since u is non decreasing, u is constant in a neighborhood of [s, t] which contradicts u ∈ A(s, t). Lemma 2 Let F ∈ L and I be a connected component of \ S. If s ≤ s < t ≤ t ∈ I then 1 m(s, t) ≥ 1 m(s, s ) + 1 m(t , t) . ( 3

.21)

Proof There exist u ∈ A(s, s ) and v ∈ A(t, t ) satisfying : u = 0 a.e. in (0, s), v = 0 a.e. in (0, t ), u = 1 a.e. in (s , 1), v = 1 a.e. in (t, 1), F(u) = m(s, s ) and

F(v) = m(t , t). Any x ∈ (s , t ) is a Lebesgue point of the function w ∈ A(s, t) defined by w := F(v) F(u) + F(v) u+ F(u) F(u) + F(v) v.
Then m(s, t) ≤ F(w) and by Proposition 1,

F(w) = F w x + F w r x = F(u)F(v) F(u) + F(v) = m(s, s )m(t ,
1 m(s, t) = 1 m(s, x 1 ) + 1 m(x n , t) + n-1 i=1 1 m(x i , x i+1 ) (3.22)
for every x 1 < ... < x n belonging to I * (s,t) . ii) there exists a Lebesgue measurable set I * ⊂ I such that |I \ I * | = 0, m(x, x) = +∞ for every x ∈ I * and

1 m(x 1 , x n ) = n-1 i=1 1 m(x i , x i+1 ) (3.23)
for every x 1 < ... < x n belonging to I * .

Proof Let s < t in I and w ∈ A(s, t) be such that F(w) = m(s, t). Define

I * (s,t) := {x ∈ (s, t) : m(x, x) = +∞, x is a Lebesgue point of w}. By Lemma 3 |(s, t) \ I * (s,t) | = 0. Let x 1 , ..., x n ∈ I * (s,t) be such that x 1 < ... < x n . Denote x 0 = s, x n+1 := t, m i := m(x i-1 , x i ) if 1 ≤ i ≤ n + 1, α 1 := w(x 1 ), α n+1 := 1 -w(x n ) and α i := w(x i ) -w(x i-1 ), if 2 ≤ i ≤ n. Using Proposition 1, identity n+1
i=1 α i = 1 and Cauchy-Schwarz inequality, we obtain

F(w) = F w x1 + F w r xn + n i=2 F w xi -w xi-1 ≥ n+1 i=1 α i 2 m i ≥ n+1 i=1 1 m i -1
, which leads to

1 m(s, t) ≤ n+1 i=1 1 m(x i-1 , x i ) . (3.24)
For each i ∈ {1, ..., n}, there is

w i ∈ A(x i-1 , x i ) such that F(w i ) = m(x i-1 , x i ). Let x ( p) i
be a sequence of Lebesgue points of w i increasing to x i . By Proposition 1 we have

m(x i-1 , x i ) = F (w i ) x ( p) i + F (w i ) r x ( p) i ≥ wi x ( p) i 2 m x i-1 , x ( p) i + 1 -wi x ( p) i 2 m x ( p) i , x i ≥ m x i-1 , x ( p) i m x ( p) i , x i m x i-1 , x ( p) i + m x ( p) i , x i . Using Lemma 2 we obtain n+1 i=1 1 m(x i-1 , x i ) = 1 m(x n , x n+1 ) + n i=1 1 m(x i-1 , x i ) ≤ 1 m(x n , x n+1 ) + n i=1 1 m x i-1 , x ( p) i + n i=1 1 m x ( p) i , x i ≤ 1 m(s, t) + n i=1 1 m x ( p)
i , x i Using Lemma 1 and passing to the limit as p tends to ∞ in the above inequality, we obtain

1 m(s, t) ≥ n+1 i=1 1 m(x i-1 , x i ) (3.25)
which together with Eq. 3.24 leads to Eq. 3.22. Since we also have

1 m(s, x 1 ) + 1 m(x 1 , x n ) + 1 m(x n , t) = 1 m(s, t)
we obtain Eq. 3.23. The proof of Lemma 4 is completed by using sequences (s q ), (t q ) such that (s q ) decreases, (t q ) increases, I := q (s q , t q ) and defining I * := q I * (sq,tq) .

Proof of

f I (t) := ⎧ ⎪ ⎨ ⎪ ⎩ m(x, t) -1 if t > x, 0 if t = x, -m(t, x) -1 if t < x.
Clearly f I is non decreasing, then there exists a unique Radon measure μ I on I such that f I (t + )f I (s -) = μ I ([s, t]) for every s < t in I. By Lemma 1, f I has no jump at any point of

I * then Lemma 4 implies that μ I ([s, t]) = m(s, t) -1 if s, t in I * . Let x ∈ I and choose two sequences (s n ) and (t n ) in I * such that (s n ) increases to x, (t n ) decreases to x. By Lemma 1 μ I ({x}) = lim n μ I ([s n , t n ]) = lim n m(s n , t n ) -1 = m(x, x) -1 .
Let s < t in I and I * (s,t) be a Lebesgue measurable set as in part (i) of Lemma 4. Choose two sequences (s n ) and (t n ) in I * (s,t) ∩ I * such that (s n ) decreases to s, (t n ) increases to t and s n < t n . We have

m(s, t) -1 = lim n m(s, s n ) -1 + m(s n , t n ) -1 + m(t n , t) -1 = lim n m(s, s n ) -1 + μ I ([s n , t n ]) + m(t n , t) -1 = m(s, s) -1 + μ I ((s, t)) + m(t, t) -1 = μ I ({s}) + μ I ((s, t)) + μ I ({t}) = μ I ([s, t])
In order to complete the proof of Proposition 5, we define the measure μ on the σ -field of Borel sets B ⊂ \ S by

μ(B) := I μ I (B ∩ I).
where the sum is taken over all the connected components of \ S. and(s, t) be an open interval of . If F(u, (s, t)) < +∞ then u ∈ BV oc (s, t) \ S and there exists f ∈ L 2 μ ((s, t) \ S) such that

Proof of Theorem 1

Lemma 5 Let F ∈ L, u ∈ L 2 ( )
\S u(x)ϕ (x)dx = - \S ϕ f dμ for every ϕ ∈ C ∞ c ((s, t) \ S).
Proof Let I be a connected component of \ S and h > 0 be such that [xh, x + h] ⊂ (s, t) \ S for every x belonging to the compact support of ϕ. There are

x 0 < x 1 < ... < x n+1 in I ∩ (s, t) such that h = x i+1 -x i and (x 0 , x n+1 ) ⊃ I ∩ supportϕ. Denoting U i = h -1 xi xi-1 u(t)dt we have I ϕ (x)u(x)dx = ε(h) + n+1 i=1 ϕ(x i ) -ϕ(x i-1 ) U i = ε(h) - i even ϕ(x i )(U i+1 -U i - i odd ϕ(x i )(U i+1 -U i (3.26) For each even i ∈ [1, n] choose t i ∈ (x i-1 , x i ) and t i+1 ∈ (x i , x i+1 ) which are Lebesgue points of u, μ({t i }) = μ({t i+1 }) = 0 and ũ(t i+1 ) -ũ(t i ) 2 ≥ U i+1 -U i 2 . For each odd i ∈ [1, n] choose s i ∈ (x i-1 , x i ) and s i+1 ∈ (x i , x i+1 ) which are Lebesgue points of u,
Closure of the set of one dimensional diffusion functionals 349

μ({s i }) = μ({s i+1 }) = 0 and ũ(s i+1 ) -ũ(s i ) 2 ≥ U i+1 -U i 2 .
Using Cauchy-Schwarz inequality we obtain

i even ϕ(x i )(U i+1 -U i ) 2 ≤ i even ϕ(x i ) 2 μ ] t i , t i+1 [ × i even ũ(t i+1 ) -ũ(t i ) 2 m(t i , t i+1 ). then i even ϕ(x i )(U i+1 -U i ) 2 ≤ F(u, (s, t)) i even ϕ(x i ) 2 μ ] t i , t i+1 [ .
Similar estimate holds for odd i. Then passing to the limit in Eq. 3.26 as h ↓ 0 leads to

I ϕ (x)u(x)dx 2 ≤ 2F(u, (s, t)) ϕ 2 L 2 μ (I) ,
which is enough to complete the proof.

Lemma 6 Let F ∈ L, u ∈ L 2 ( ) and (s, t) be an open interval of . If F(u, (s, t)) < +∞ then F(u, (s, t)) ≥ (s,t)\S du dμ 2 dμ
where du dμ denotes the derivative of u in the sense of Radon-Nikodym in (s, t) \ S.

Proof By Lemma 5, u ∈ BV oc ((s, t) \ S). Let [x, y] ⊂ (s, t) \ S such that μ({x}) = μ({y}) = 0. This implies that x and y are Lebesgue points of u. Then we have

ũ(y) -ũ(x) = (x,y) du dμ dμ. Let I be a connected component of \ S, ϕ ∈ C ∞ c ((s, t) \ S) and x 0 < x 1 < ... < x n such that I ∩ support(ϕ) ⊂ [x 0 , x n ] ⊂ I ∩ (s, t), and μ({x i }) = 0. Denoting h = max i {x i -x i-1 } and choosing points s i ∈ (x i-1 , x i ) we have I ϕ du dμ dμ = n i=1 (xi-1,xi) ϕ du dμ dμ = ε(h) + n i=1 ϕ(s i ) ũ(x i ) -ũ(x i-1 ) .
Using Cauchy-Schwarz inequality we obtain

I ϕ du dμ dμ + ε(h) 2 ≤ n i=1 ϕ(s i ) 2 μ [ x i-1 , x i ] × n i=1 ũ(x i ) -ũ(x i-1 ) 2 m(x i-1 , x i ) ≤ F(u, (s, t)) n i=1 (ϕ(s i )) 2 μ [ x i-1 , x i ] .
Then passing to the limit as h tends to 0 leads to

I ϕ du dμ dμ 2 ≤ F(u, (s, t)) ϕ 2 L 2 μ (I) .
which is enough to complete the proof.

Lemma 7 Let F ∈ L, u ∈ L 2 ( ) and (s, t) be an open interval of . If F(u, (s, t)) < +∞ then F(u, (s, t)) ≤ (s,t)\S du dμ 2 dμ
where du dμ denotes the derivative of u in the sense of Radon-Nikodym in (s, t) \ S.

Proof Let I be a connected component of \ S and [x, y] ∈ I ∩ (s, t) with μ({x}) = μ({y}) = 0. Let x := x 0 < ... < x n := y in I ∩ (s, t) be such that μ({x i }) = 0 and denote h := max{x ix i-1 }. By Lemma 5, x i is a Lebesgue point of u. By Proposition 2, there exists u i such that (i) u i is monotone in (0, 1), (ii)

u i = ũ(x i-1 ) a.e. in (0, x i-1 ), (iii) u i = ũ(x i ) a.e. in (x i , 1) and (iv) F(u i ) = ũ(x i ) -ũ(x i-1 ) 2 m(x i-1 , x i ).
By Jensen inequality we have

F(u i ) ≤ (xi-1,xi) du dμ 2 dμ. ( 3.27)
Let u h ∈ L 2 ( ) be such that u h := u a.e. in \ [x 0 , x n ] and u h := u i a.e in (x i-1 , x i ). Since u ∈ BV[x, y] we have lim h↓0 uu h L 2 ( ) = 0. Then we deduce from Proposition 3 and Eq. 3.27 that

μ u ((x, y)) = F(u, (x, y)) ≤ lim inf h↓0 F(u h , (x, y)) ≤ lim inf h↓0 n i=1 F(u h , [x i-1 , x i ]) ≤ lim inf h↓0 n i=1 F(u i , [x i-1 , x i ]) ≤ lim inf h↓0 n i=1 F(u i ) ≤ (x,y) du dμ 2 dμ
Using Proposition 4, the above inequality and Lemma 6 we conclude that

F(u, (s, t)) ≤ μ u ((s, t)) ≤ μ u ((s, t) \ S) ≤ (s,t)\S du dμ 2 dμ ≤ F(u, (s, t)).
Proof of existence of (S, μ) in Theorem 1 is complete. Uniqueness is ensured by the fact that, when F is a functional represented as in Eq. 3.4, then S has to satisfy Eq. 3.13 and μ has to satisfy the condition of Proposition 5. Proof Let (F n ) be a sequence in L and F ∈ Q. Assume that, for any sequence (u n ) converging strongly in L 2 ( ) to some u, lim inf n F n (u n ) ≥ F(u). Assume also that for any u ∈ L 2 ( ) there exists (u n ) converging strongly to u and such that lim sup n F n (u n ) ≤ F(u).

Let u ∈ L 2 ( ) and x be a Lebesgue point of u. There exists a sequence

(u n ) in L 2 ( ) such that lim n u -u n L 2 ( ) = 0, lim n F n (u n ) = F(u)
and such that (u n ) converges to u almost everywhere in . Choose two sequences (s p ) and (t p ) in , such that (s p ) increases to x, (t p ) decreases to x, s p and t p are Lebesgue points of u and of all u n and such that lim n ũn (s p ) = ũ(s p ), lim n ũn (t p ) = ũ(t p ) and lim p ũ(s p ) = lim p ũ(t p ) = ũ(x). As F n belongs to L, we have by Proposition 1

F n (u n ) = F n (u n ) sp + F n (u n ) tp -(u n ) sp + F n (u n ) r tp .
The choice we made for the sequences (s p ) and (t p ) ensures that (u n ) sp converges strongly to (u) sp , as n tends to infinity and that (u) sp converges strongly to (u) x , as p tends to infinity. The same holds for (u n ) r tp . Then

F(u) ≥ lim inf p lim inf n F n (u n ) sp + lim inf n F n (u n ) r tp ≥ lim inf p F u sp + lim inf p F u r tp ≥ F u x + F u r x .
On the other hand, there exist

(v n ) and (w n ) such that lim n u x -v n L 2 ( ) = 0, lim n F n (v n ) = F u x and (v n ) converges to u x almost everywhere in , lim n u r x - w n L 2 ( ) = 0, lim n F n (w n ) = F(u r
x ) and (w n ) converges to u r x almost everywhere in . Choose two sequences (s p ) and (t p ) such that (s p ) increases to x, (t p ) decreases to x, all s p and t p are Lebesgue points of u, v n and w n , lim n ṽn (s p ) = ũ(s p ), lim n wn (t p ) = ũ(t p ) and lim p ũ(s p ) = lim p ũ(t p ) = ũ(x). We have

F u x + F u r x = lim n F n (v n ) + F n (w n ) ≥ lim inf p lim inf n F n (v n ) sp + F n (w n ) r tp ≥ lim inf p lim inf n F n (v n ) sp + (w n ) r tp ≥ lim inf p F u sp + u r tp ≥ F u x + u r x ≥ F(u).
Proposition 1 then implies that F ∈ L.

Remark 7 In fact we have proved a stronger result, namely: the set L is closed for the -convergence in the strong topology of L 2 ( ).

Proof of the Density Result

Let F be a diffusion form as defined in Eq. 2.5. It is easy to check that u ∈ H 1 ( ) if and only if u x ∈ H 1 ( ) and u r

x ∈ H 1 ( ). We then deduce from Proposition 1 that F ∈ L. In this particular case, the set S and the measure μ which, owing to Theorem 1, represent F are respectively the empty set and μ(dx) = α -1 (x) dx. Hence D is a subset of L.

Proposition 7 Let F ∈ L. Then there exists a sequence of diffusion forms (F n ) which Mosco-converges to F. Proof Let (S, μ) associated to F by Theorem 1. For any integer n and any i ∈ {1, . . . , n}, let us denote I n i the interval

I n i := ( i-1 n , i n ), s n i := 2i-1
2n its center, h n i the function defined on (0, 1) by

h n i (x) := ⎧ ⎪ ⎨ ⎪ ⎩ n(x -s n i-1 ) if s n i-1 < x ≤ s n i , n(s n i+1 -x) if s n i < x < s n i+1 , 0 otherwise,
and α n i the quantity defined by

1 α n i := n -1 + n \S h n i dμ if [ i-2 n , i+1 n ] ⊂ \ S, n 3
otherwise.

Denoting α n the piecewise constant function

α n := n i=1 α n i 1 I n i , we define F n ∈ D by F n (u) := u (x) 2 α n (x)dx if u ∈ H 1 ( ), +∞ otherwise.
Now let us prove that the sequence (F n ) Mosco converges to F. For any ϕ ∈ C ∞ c ( \ S) and for n large enough, we have ϕ

(x) α n (x) dx = 1 n \S ϕ(x)dx + i n I n i ϕ(t)dt h n i (x)μ(dx). Hence lim n ϕ(x) α n (x) dx = \S ϕ(x)μ(dx). (3.28)
Let (u n ) be a sequence converging to u with respect to the weak topology of L 2 ( ) and such that lim inf n F n (u n ) = M < +∞. By Cauchy-Schwarz inequality, we have,

for any ϕ ∈ C ∞ c ( \ S), u n (x)ϕ (x)dx 2 ≤ F n (u n ) ϕ(x) 2 α n (x) dx
Then, passing to the limit as n tends to infinity and using Eq. 3.28, we obtain

u(x)ϕ (x)dx 2 ≤ M \S ϕ(x) 2 μ(dx).
By Theorem 1, we deduce

F(u) = \S du dμ 2 dμ ≤ M,
which proves the lower-bound inequality. Let u ∈ L 2 ( ) be such that F(u) < +∞. For any i ∈ {1, . . . , n}, let us denote

m n i := n s n i+1 s n i u(t) dt. We have, when [ i-2 n , i+1 n ], m n i -m n i-1 = n s n i s n i-1 u t + 1 n -u(t) dt = n s n i s n i-1 t+ 1 n t du dμ dμ dt = s n i+1 s n i-1 h n i du dμ dμ = \S h n i du dμ dμ.
Using Cauchy-Schwarz inequality, we obtain

m n i -m n i-1 2 ≤ h n i dμ h n i du dμ 2 dμ . ( 3.29)
Let u n be the continuous function on [0, 1] which is affine on each I n i and satisfies

u n (0) = m n 1 , u n (1) = m n n-1 and u n i n = m n i for 1 ≤ i ≤ n -1.
We have lim n uu n L 2 ( ) = 0. Using Eq. 3.29, Theorem 1 and Jensen inequality, we obtain

F n (u n ) = n-1 i=2 n α n (s i ) m n i -m n i-1 2 ≤ n-1 i=2 \S h n i du dμ 2 dμ + 1 n 2 n-1 i=2 (m n i -m n i-1 ) 2 ≤ F(u) + 4 n u(x) 2 dx
The upper-bound inequality is obtained by passing to the limit.

Proof of Theorem 2 is concluded by collecting the closure result of Proposition 7 and the density result of Proposition 7.

About Locality

Owing to Deny-Beurling formula, it is quite clear what a local regular Dirichlet form is: in the representation formula Eq. 2.8 the measure γ should vanish.

In this section we discuss the extension of this notion of locality to non regular forms. Different criterions have been proposed in the literature which all coincide when applied to regular forms. We compare these different notions and we prove that none of them are preserved when passing to the limit with respect to Mosco or -convergence.

Definition 7

i) We denote V the set of all forms F ∈ Q which satisfy: for any ϕ and ψ in C ∞ o (R) with disjoint supports and any u ∈ D(F),

F(ϕ • u + ψ • u) = F(ϕ • u) + F(ψ • u). (4.1)
ii) We denote S the set of all forms F ∈ Q which satisfy: for any u and v in L 2 ( ) with disjoint supports,

F(u + v) = F(u) + F(v). (4.2)
iii) We denote M the set of all forms F ∈ Q which satisfy: there exists a mapping μ which associates to any u ∈ L 2 ( ) a Borel measure μ u such that Definition (i) can be found in [START_REF] Bouleau | Dirichlet forms and analysis on Wiener space[END_REF] while definition (ii) can be found in [START_REF] Applebaum | Levy Processes and Stochastic Calculus[END_REF] or [START_REF] Fukushima | Dirichlet Forms and Markov Processes[END_REF]. There are closely related : indeed any u defines a new quadratic and Markovian form ϕ → F(ϕ • u). Definition (i) means that these new forms satisfy definition (ii). Roughly speaking, definition (ii) forbids interactions between distant zones: indeed as the supports of u and v are closed, assuming that there are disjoint means that their distance is positive.

F(u) = μ u ( ) and μ u (B) = μ v (B) (4.
Definitions (iii) or (iv) like our Definition 6 associate to any function u ∈ L 2 ( ) an energy density (a measure) μ u . Note that this measure is not necessarily finite on compact sets. Definition (iii) simply asks that the measure μ u depends in a "local way" on u. The mapping μ corresponds to the functional introduced a priori in definition 15.21 of [START_REF] Dal Maso | An introduction to -convergence[END_REF] and the property, μ u (B) = μ v (B) whenever u and v coincide on an open set O ⊂ B, is precisely the characterization of locality given in [START_REF] Dal Maso | An introduction to -convergence[END_REF]. Definition (iv) is more precise: it asks μ u to coincide on every open interval I to the relaxed energy on I. In our definition 6 we ask, in addition, that μ u does not concentrate on the Lebesgue points of u.

The relations between these different notions of locality are made precise by the following proposition. ii) R, M, V and S are not closed with respect to the Mosco convergence.

All inclusions in point (i) are almost direct consequences of definitions. To check that these inclusions are strict and to prove (ii) it is enough to consider suitable examples.

In Example 1 we exhibited a functional F which is the limit of a sequence of functionals in D. It belongs to L = D. We let the reader check that it does not belong to V. This example proves that L is not included in V and that V is not closed.

The theory, established in the case of functionals G with dense domain [START_REF] Bouleau | Dirichlet forms and analysis on Wiener space[END_REF], states the existence of a measure σ u satisfying G(ϕ • u) = R ϕ (t)) 2 σ u (dt) for every ϕ ∈ C 1 c (R). Here it cannot be applied to F (the domain of F is not dense in L 2 ( )!). Moreover, one can directly check that F cannot be represented in this way.

Proving the other assertions needs more sophisticated examples. To construct these examples we need the following lemma For lack of space, we skip the proof of this lemma which is obtained by an induction argument. When n = 2 a construction analogous to the classical construction of Cantor set is used. In the following we refer to partitions satisfying Lemma 8 as essential partitions. We let the reader check that, when {A, \ A} is an essential partition, the functional G A belongs to R \ L.

Example 3 Let {A n , \ A n } be a sequence of essential partitions satisfying |A n | = (n + 1) -1 . Let us define

B n := 1 3 A 1 ∪ 1 3 + 1 3 A n ∪ 2 3 + 1 3 A 1 (4.5)
and

B ∞ := 1 3 A 1 ∪ 2 3 + 1 3 A 1 (4.6)
Note that {B n , \ B n } is still an essential partition while it is not the case for {B ∞ , \ B ∞ }. We let the reader check that i) the sequence of functionals G Bn Mosco converges to G B∞ ii) the limit functional G B∞ does not belong to S. This example proves that neither R, nor M, nor S are closed.

Definition 6

 6 For any F ∈ Q, any compact K ⊂ and any open interval I ⊂ we set F(u, K) := inf{F(v); v = u a.e. on some open set O ⊃ K}, (3.2) F(u, I) := sup{F(u, K); K compact, K ⊂ I}. (3.3) We say that a form F ∈ Q is local if and only if, for any u ∈ L 2 ( ), there exists a Borel measure μ u such that μ u (I) = F(u, I), for any open interval I ⊂ and such that for any open set O ⊂ and any Lebesgue point x of u,

Proposition 5

 5 Let I be any connected component of the open set \ S and I * be a Lebesgue measurable set as in point (ii) of Lemma 4. Choose x ∈ I * and define f I from I to R by

3. 4 6

 46 Proof of the Closure Result3.4.1 Locality is a Closed NotionProposition The set L is closed.

3 )

 3 whenever u and v coincide on an open set O containing the Borel set B. iv) We denote R the set of all forms F ∈ Q which satisfy: there exists a mapping μ which associates to any u ∈ L 2 ( ) a Borel measure μ u such that, for every open interval I ⊂ , μ u (I) = F(u, I) (4.4) where F(u, I) is defined by Eqs. 3.2-3.3.

Proposition 8 i)

 8 We have the following strict inclusions D L R M S and D V S.

Lemma 8

 8 Let n ∈ N and let {α i } 1≤i≤n be a family of positive reals with n i=1 α i = 1. Then there exists a partition {A i } 1≤i≤n of [0, 1] made by Borel sets A i satisfying |A i | = α i and |A i ∩ I| > 0 for every i and every non empty open interval I ⊂ [0, 1].

Example 2

 2 Let A ⊂ and G A be the functional defined byG A (u) = 0 if u = a1 A + b 1 \A for some a, b ∈ R, +∞ otherwise.

Example 4

 4 Let {A, \ A} be an essential partition. Define F byF(u) := (v (x)) 2 + (w (x)) 2 + (v(x) -w(x)) 2 dx, (4.7)

  let μ u be the measure associated to F and u byDefinition 6 and (s, t) an open interval of such that μ

u (s, t) < +∞. Then μ u ((s, t) ∩ S) = 0.

(3.14) Remark 6 Clearly \ S is an open set. Indeed, if (x n ) is a sequence in S converging to some x ∈ , we have lim n 1 (xn,1) -1 (x,1) L 2 ( ) = 0. The lower semi-continuity of F implies F 1 (x,1) ≤ lim inf n F 1 (xn,1) ≤ 0. Thus x belongs to S. Proposition 4 shows that energy never concentrates on S. Proof of Proposition 4 Let x ∈ S. For every α, β ∈ R we have

  If x 1 , .., x p are p elements of the set {x ∈ [s, t] : m(x, x) ≤ n} then Lemma 2 give us the estimate

						t) m(s, s ) + m(t , t)	,
	which leads to Eq. 3.21.
	p ≤	p i=1	n m(x i , x i )	≤	n m(s, t)	.

Lemma 3 If F ∈ L then the set {x ∈ \ S : m(x, x) < +∞} is at most countable.

Proof Let I be a connected component of \ S, n be a positive integer and s < t in I.

Thus {x ∈ [s, t] : m(x, x) ≤ n}

is a finite set. Lemma 4 Let F ∈ L and I be a connected component of \ S. Then i) for every s < t in I, there exists a Lebesgue measurable set I * (s,t) ⊂ (s, t) such that |(s, t) \ I * (s,t) | = 0, m(x, x) = +∞ for every x ∈ I * (s,t) and

if u = v1 A + w1 \A a.e. in , for some v, w in H 1 ( ) and F(u) := +∞ otherwise. We let the reader check that F belongs to M \ R.

Example 5 Let us consider an essential partition {A 1 , A 2 , A 3 } and the functional F defined by

otherwise.

We let the reader check that F belongs to S \ M.


Gilbert Habib

Jean-François Pétin

1e priorities within a Wireless Networked Discrete Control System

HAL is

I. INTRODUCTION

SING wireless communication within industrial control applications offers many advantages such as reducing the wiring cost, as well as facilitating the commissioning, the reconfigurability, and the mobility of the control devices. Moreover, plant operation by human operator is more efficient thank to wireless mobile hand devices performing data analysis. However, wireless communication suffers from many drawbacks such as information loss, stochastic delay, jitter phenomena or loss of messages. These problems are mainly induced by the non deterministic features of wireless communication that may have a negative impact on the stability, the real time performance and the dependability of the control system.

This paper focuses on the study of Discrete Control Systems (DCS) whose implementation is distributed on Programmable Logic Controllers (PLC) and/or remote Inputs/Outputs devices communicating through a IEEE 802.11 wireless network. It aims at reducing the communication negative impact on the performance of a Wireless Networked Discrete Control System (WNDCS) by promoting a co-design approach that tries to manage the network quality of service with regards to the control Manuscript received June 1, 2010. Authors are with the Centre de Recherche en Automatique de Nancy (CRAN), UMR 7039 Nancy Université -CNRS, Campus scientifique, BP 70239, 54506 Vandoeuvre-lés-Nancy, France e-mail: {gilbert.habib, jeanfrancois.petin, thierry.divoux}@cran.uhp-nancy.fr). requirements. To attend this target:

-IEEE 802.11e [1] has been chosen for its ability to work with priorities of the communication traffics that enable to increase the network quality of service for the higher priorities, -An algorithm that dynamically allocates the priorities on the communication traffics with regards to the control requirements and current states.

This paper is organized as follow. Section 2 introduces main features of the WNDCS and highlights the main issues that are induced by using wireless communication between control and plant devices. Section 3 presents the IEEE 802.11e standard with traffic priorities as an efficient alternative for improving the global performance of a WNDCS. Section 4 proposes a control-based algorithm to manage the IEEE 802.11e priorities by dynamically gathering communication traffics according to their criticality for the control current states. At last, section 5 illustrates the proposed algorithm using a case study and demonstrates its efficiency through OPNET simulations. Finally, conclusions and open issues for future research are discussed in section 6.

II. WIRELESS NETWORKED DISCRETE CONTROL SYSTEM: DESCRIPTION AND ISSUES

A. Description of a WNDCS

A networked control system is a distributed system where the control loops are closed through a real-time network. This paper addresses WNDCS whose control system is characterized by a discrete state space and evolutions at discrete points in time. These points correspond to occurrences of asynchronous generated discrete events (event driven). A WNDCS is a system composed of three main components: Controller, Wireless Network and Plant device.

Controller device implements the control rules and interact with plant by receiving/sending data from sensors/to actuators. PLC consists of processor(s), memory, I/O and communication cards. It periodically scans the input variables, executes the control rules and updates the output variables according to a cycle period, called Pplc.

Plant devices are the sensors and actuators that enable respectively to collect information about the plant status and to operate actions on the plant process. Considering WNDCS, sensors and actuators are supposed to exchange information with control devices using remote I/O communicating through wireless network.

Control-based algorithm for the management of IEEE 802.11e priorities within a Wireless Networked Discrete Control System

G. Habib, JF. Pétin, T. Divoux

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Network device ensures the exchange of information between the plant and control devices. In the WNDCS this paper is focusing on, wireless communication follows the IEEE 802.11 standard, also called Wi-Fi. All the devices of the WNDCS are equipped with communication card that allow sending information according to a cyclic period, called Pcard, and receiving information that are immediately processed.

In general, three layers are considered in WNDCS: Physical, MAC and Application layers. The Physical layer is responsible for transmitting and receiving signals. Many technologies can be chosen in wireless network example: 802.11a, b, and g. The MAC layer manages the access to the medium using CSMA/CA (Collision Avoidance) algorithm. A station that wants to send a message is listening for the medium during a Distributed Inter-Frame Space (DIFS) time. If the medium is busy, a random waiting time, called backoff time, is taken. This time is between 0 and a variable called Contention Window (CW), where CW is between CWmin and CWmax. The station begins to decrement backoff time as long as the medium is idle and it will be frozen when the medium becomes busy. When backoff time becomes equal to zero, the station sends immediately the packet. Transmission of a message may fail because of noise that can affect physical layer or because of collisions that can occur if two same backoff times are taken for two different stations by the MAC layer. The Application layer manages the information sending and receiving procedures: periodical encapsulation of information to be sent into packets according to predefined parameters (packet size, sampling period Pcard …) and transmission of packets to lower layers.

B. Impact of the network behaviour on the WNDCS

Introducing IEEE 802.11 network in a discrete control system may have a negative impact on the predictability of its global behaviour. Indeed, non determinism of backoff procedure of CSMA/CA leads to jitter phenomena due to a random end-to-end delay between source and destination. Moreover, packet loss due to noise or collision may aggravate this problem because it requires an unknown number (but limited to a maximum value) of packet retransmission that leads to more uncertainty about the endto-end delay or, in the worst case, information that is never received.

For continuous control system, this problem may disturb the stability of the control loop [START_REF] Georges | Confronting the performances of a switched Ethernet network with industrial constraints by using the network calculus[END_REF]. For DCS, it may lead to Response Time (ResT) that is not compliant with the system requirements in terms of temporal performance or safety constraints. Indeed, real-time discrete control applications often requires a limited time for the reaction to a stimulus input.

Response Time is defined by [START_REF] Jasperneite | Switched Ethernet for factory communication[END_REF] to reflect the performance of a NCS which represents the roundtrip time from client to server and back to client including a processing time in the server. Also it can be defined as the delay between the occurrence of an event in a sensor (i.e.: detecting the presence of an obstacle) and the reception by the actuators of the control reaction to this event (i.e.: order to stop the move of a device). This delay takes into account the time for this event to be received by the controller, the time required for control processing and the time for the control output to be sent to the actuators.

III. IMPROVING WNDCS PERFORMANCE

Many researchers have studied the influence of network parameters on the response time for non deterministic communication [START_REF] Denis | Measuring the impact of vertical integration on response times in Ethernet fieldbuses[END_REF] and more particularly for IEEE 802.11 wireless communication [5][6]. If these works provide influence parameters that impact the response time, they do not suggest ways to improve the situation. Three main approaches have been proposed.

Control oriented approaches take into account the impact of communication during the control design by including estimation or prediction of the network behaviour into the control models [7][8]; main drawback of these approaches is the rough estimate of the network.

Network oriented approaches aims at optimising network quality of service [START_REF] Kim | Real-time scheduling method fornetworked discrete control systems[END_REF] by tuning its controllable parameters (packet size, routings, topology, data rate …); main drawback of these approaches is that the control requirements are not really taken into account.

At last Control/Network oriented approaches propose codesign approaches that integrate both the control requirements and models and the network quality of service optimisation [START_REF] Xiangheng | Wireless Network Design for Distributed Control[END_REF] [START_REF] Branicky | Scheduling and Feedback Co-Design for Networked Control Systems[END_REF]; these approaches introduce efficient ways to adapt the network with regards to the application needs but suffer from a high level of complexity required for a detailed modelling of control and network behaviour.

Our work locates in the Control/Network oriented approach. It is based on communication priorities tuning provided by the standard IEEE 820.11e. Indeed, IEEE 802.11e extension of IEEE 802.11 enables giving priority to each traffic of the network. More precisely, 802.11e includes a mandatory EDCA (Enhanced Distributed Channel Access) MAC mechanism. It provides four levels of priority from 0 (lowest priority) to 3 (highest priority). Every level has its own buffer in the MAC layer managed by standalone CSMA/CA. Each buffer, i.e each priority, has its own backoff time and DIFS (called AIFS in IEEE802.11e) parameters. The more the priority of traffic is higher, less the backoff time and the AIFS are small. Thus, the traffic will have more possibility to access the medium if it has a higher priority. Note that another mechanism available in IEEE802.11e, called HCCA (HCF controlled channel access (HCCA)) provides deterministic management of the medium access without using CSMA/CA. Unfortunately, HCCA is few implemented by commercial devices due to its complexity. Consequently, EDCA has been selected for our study.

Theoretically, the priority between the traffics in 802.11e must give a better performance compared to 802.11. Some previous studies [START_REF] Grilo | Performance evaluation of ieee 802.11e[END_REF] [START_REF] Lu-Mlng | Performance evaluation on IEEE 802.11e considering emergency calls in congested situation[END_REF] do not show that when considering static and predefined priorities. Their simulations prove that the traffics with higher priorities will dominate the medium, thus this will penalize the traffic with less priorities. Therefore, the priority may play a negative effect on some traffic in the network. Classical recommendation to avoid this negative effect is to avoid using 0 priority. Our proposal is to promote a dynamic allocation of the priorities based on the estimated needs for the control application.

IV. PRIORITY MANAGEMENT ALGORITHM

In order to improve the WNDCS performances in terms of security and velocity, an algorithm for adapting the network with regards to control needs is proposed. This algorithm is based on a dynamic variation of the traffic priorities to ensure fairness between the different stations according to the control application needs. In other words, the controller will define dynamic classification of its input information with regards to its importance for its current state.

The control system is assumed to be modelled using Finite State Automata defined by (X, Σ, δ, x 0 , x f ) where X is a nonempty set of states, Σ is the alphabet, δ is the state-transition function: δ: X x Σ → X, x 0 is an initial state, an element of X and, x f is the set of final states (x f ⊆ X).

Three sets of devices are thus defined with regards to the information criticism they send.

The controller selects the devices that will send Critical information that the controller is waiting for. It corresponds to the critical information for current control decision making; therefore, this information must arrive on time.

The controller selects the devices that will send information which are not important to take current decision but that will be useful for next step decision making; so it must be monitored, we call it Normal information.

At least, controller selects the devices that send information that it should not used in a near future, it is called Unnecessary information.

More formally, the definition of those three sets is based on the current state x cur of the control model. A sub-set Σ plant of the control model alphabet Σ is defined as the set of events coming from the sensors of the plant (the events are considered as controller inputs):

Σ plant = s 1 , s 2 , … , s m ⊂ Σ
Note that each sensor i is assumed to be able to generate only one event s i . Consequently, defining a set of event s i is equivalent to define a set of sensor i , (S i ).

Definition 1.a: 𝐶 is defined as the set of events related to Critical information, i.e.: 𝐶 = s ∈ Σ plant ∶ ∃ w ∈ (Σ\Σ plant ) * , δ x cur , ws is defined where (Σ\Σ plant ) * represents all the words composed of events that belong to the set Σ but not to set Σ plant .

Definition 1.b: 𝑁 is defined as the set of events related to Normal information, i.e.: 𝑁 = s ′ ∈ Σ plant ∶ ∃ w, w ′ ∈ Σ\Σ plant * , s ∈ Σ plant , δ x cur , wsw ′ s ′ is defined Definition 1.c: 𝑈 is defined as the set of events related to Unnecessary information, i.e.: 𝑈 = s ′′ , s ′′ ∈ Σ plant \(𝐶 ∪ 𝑁)

The proposed algorithm behaves as following.

If the controller estimates that the network current quality of service may delay a Critical Information that may not arrive on time (and consequently generate a too late reaction), the priority of the device which will send this information must be increased. Evaluating the network current quality of service is done by monitoring the response time (ResT) of the devices and by comparing it to a theoretical value that is expected to ensure a normal behaviour of the control application. In other words, if the response time is upper than a threshold, the priority of the given device must be increased, otherwise the priority of the device is kept at its current level. Note that several methods for evaluating the response time are available: to keep the genericity of the proposed algorithm, the method we used will be further detailed in the case study section.

The priorities of the devices that send Normal information are kept in default priority value that has to be determined according to the use case (equal to 1 in most of the case).

The devices that send Unnecessary information are temporarily silenced. We consider that this action is equivalent to send a request for a priority equal to the value Null that will be interpreted by the device as a request for stopping transmissions.

To summarize, the proposed algorithm is implemented on the controller device within the control model section, that is run with a Pplc cycle (PLC sampling period) and involves: enumeration of 𝐶, 𝑁 and 𝑈 sets, estimation of the response time of sensors belonging to 𝐶 and management of the priorities according to 𝐶, 𝑁 and 𝑈 sets and response time. Note that this algorithm, and more specifically the definition of 𝐶, 𝑁 and 𝑈 sets, is based on the current state of the control model which is assumed to be described using finite state automaton. If the control model {A} is described in a modular way with several coordinated automata {A 1 , A 2 … A n }, the following lemmas for computing the algorithm can be applied.

Algorithm -Priorities allocation

Lemma 1 : If Σ Ai ∩ Σ Aj = ∅ , i.e. A i and A j alphabets does not share common events, 𝐶(𝐴 𝑖 𝐴 𝑗 ) = 𝐶(𝐴 𝑖 ) ∪ 𝐶(𝐴 𝑗 ), (resp. 𝑁 and 𝑈), i.e. the enumeration of 𝐶 (resp. 𝑁 and 𝑈) will be equivalent if being computed from the synchronous product Ai||Aj than if obtained by merging 𝐶 (resp. 𝑁 and 𝑈) enumerations given by each automata.

Lemma 2: If Σ Ai ∩ Σ plant = ∅ , i.e. A i alphabet does not contain sensor events, 𝐶 𝐴 𝑖 = 𝑁 𝐴 𝑖 = 𝑈 𝐴 𝑖 = ∅.

V. APPLICATION

A. Presentation of the case study

The proposed algorithm is experimented using a case study (Fig. 1) and through two simulation scenarios: with and without the proposed algorithm. The case study is composed of two cylinders (cylinder1 moves horizontally and cylinder2 vertically), their associated sensors (three sensors S1, S2, and S3 for cylinder1, two sensors S4 and S5 for cylinder2) and a PLC that control the system. We assume that each device (sensors, actuators and PLC) send and receive information using wireless I/O cards. This configuration is clearly unrealistic for this kind of application but is interesting for communication experiments and is representative of what could be industrial application in a near future when using wireless sensor networks. Temporal performance of the WNDCS is required for avoiding collision between the two cylinders that share a common area. The main communication is the end-to-end delay between occurrence of a sensor event and the receipt by an actuator of a reaction control event. Control is designed voluntarily to provoke a dangerous situation by asking the two cylinders to get out simultaneously. The control is then expected to stop cylinder 1 into intermediate position to avoid collision, to wait for cylinder 2 freeing the common zone and then to end the roundtrip. This cycle is continuously repeated until the end of simulation scenario.

Evaluating the temporal performance of the WNDCS is done using two criteria:

-number of collision that have not been avoided due to communication delay (safety property),

-number of roundtrips done by the two cylinders that characterise the control rate (vivacity property).

B. Modelling of the case study

To evaluate the algorithm, simulation scenarios using OPNET (http://www.opnet.com) are done. This tool is a network simulator, providing libraries for different protocol network and an editor for describing the behaviour of devices using EFSM (Extended Finite State Machine). This formalism has been shown as compliant with automata used in our models [START_REF] Habib | Evaluation de l'influence d'un réseau de communication sans fil sur la commande d'un SED[END_REF]. Our OPNET models are divided into three sections.

The plant model is an emulation model that represents the behaviour of the plant devices using EFSM and generates the sensors signals. Note that the actuators are modelled in a modular way as proposed by [START_REF] Marangé | Synthèse et filtrage robuste de la commande pour des systèmes manufacturiers sûrs de fonctionnement[END_REF]. For our study, two main attributes of plant models are considered: Sensor size and Spacing size sensors which are respectively translated into time needed by the cylinder to cross the sensor and time needed for reaching a sensor from another sensor.

The communication model represents the behaviour of the network:

-Physical and MAC layers are compliant with IEEE 802.11e standard and implemented using OPNET library.

-For Application layer, a packet created by PLC encapsulates all the actuator orders (A, B, E, D in Fig. 2) and is broadcasted. The broadcasting method minimizes the number of packet send by the PLC station. The packet created by each sensor is only composed by the efficient information and unicasted to PLC. This model also includes the definition of the sampling period Pcard. The PLC model describes the control cycle (as presented in 5.1.) and is represented by modular automata (See Fig. 3):

-M1 and M2 estimate the state of the two cylinders according to sensor inputs and generate output variables P_A_v1 and P_A_v2 according to this computed position, -M3 manage the control cycle and collision avoidance according to cylinder1 and cylinder2 states as given by P_A_v1 and P_A_v2.

Fig. 3. PLC model

This basic control is enriched by the algorithm implementation that has to define priorities for the communication traffics. PLC station encapsulates in the broadcasted packet five additional fields (field 3 till 7), which represent the priorities recommended for each sensor (S1 till S5), see Fig. 4. These fields can take a value from 1 up to 3 that denotes the priority of the station or 0 if the station must be silenced. As explained before, this algorithm is applied to the control model. It consists of selecting three sets: 𝐶, 𝑁 and 𝑈 which represent Critical, Normal and Unnecessary events respectively from Σ plant . In this case study, Σ plant = s 1 , s 2 , s 3 , s 4 , s 5 .

Crossing from one state to another in M3 depends on the P_A_v1 and P_A_v2 variables, which are not included in Σ plant . With regards to lemma 2, M3 generates empty sets for 𝐶3, 𝑁3 and 𝑈3. The models M1 and M2 can be considered as local for each cylinder and its associated sensors. More precisely:

Σ plant = Σ plant -M1 ∪ Σ plant -M2 where Σ plant -M1 = s 1 , s 2 , s 3 and Σ plant -M1 = s 4 , s 5 .

Consequently, alphabets of M1 and M2 have an empty intersection. According to lemma 1, the sets 𝐶, 𝑁 and 𝑈 can be respectively computed by merging respectively 𝐶1, 𝑁1 and 𝑈1 sets from M1 with respectively 𝐶2, 𝑁2 and 𝑈2 sets from M2.

Applying algorithm to control model is equivalent to apply separately the algorithm within M1 and M2 models.

For M1 and M2, parameters of the algorithm have been chosen as following:

-The threshold that is involved in the comparison with ResT has been chosen as Sensor size. This is justified by to provide the necessary time for stopping the cylinder before it leaves the sensor.

-Default priority has been chosen as equal to 1.

-Response Time (ResT) evaluation has been done according to the following procedure. The PLC encapsulates a flag as a random number (field 8 in Fig. 4) in the broadcasted packet. After recognising the flag, each sensor sends it back to the PLC within its information packet. The PLC station can then calculate the end-to-end communication time between sending the packet and receiving the response. Adding Pplc to this time give rise to response time (ResT).

At last, all this models are integrated into OPNET application using the following structure (Fig. 5): plant and control models are implemented into additional OPNET layers we introduced, application layer has been parameterised according to packet creation rules and sampling period we defined, and finally physical and MAC layers are implemented using OPNET libraries.

C. Algorithm evaluation using simulation scenarios

After modelling the whole system in OPNET, we study the influence of the network attributes onto the system in two cases:

-without algorithm, where all traffics have the same priority equal to 1, -with algorithm, where the priorities are computed in real time by taking into account communication quality of service and control state.

The static and dynamic parameters that are used by the simulation are given by Table 1. A Monte Carlo approach from 50 to 100 runs was taken for each simulation, depending on the gap between the current and mean values. Each simulation was launched for 200 seconds. As explained before, results of the simulation can be evaluated by considering the number of collisions between the two cylinders that have not been avoided due to communication delay (safety property) and, the number of roundtrips done by the two cylinders that characterise the control rate Cross points of Fig. 6 and Fig. 7 represent simulations done without using the proposed algorithm: Fig. 6 represents the number of collision occurrences with respect to Pcard and Fig. 7 shows the numbers of roundtrips executed by the two cylinders. These curves can be decomposed into three sections:

-Simulations where 1 ≤ Pcard ≤ 4ms give rise to some collisions. This result can be explained by a lot of packet sent to the medium and the consecutive network over load meaning important delay and packet losses. Consequences on control application are bad estimation of the plant state, too late or inappropriate reaction from the control.

-Simulations where 5 ≤ Pcard ≤ 10ms does not present collisions. Indeed, these values, the influence of the network on the global behaviour is minimized: the network induces small delay and few packet losses.

-Simulations where 11 ≤ Pcard ≤ 25ms give rise to many collisions. This result can be explained by the fact that Pcard is not compliant with real-time parameters of the plant. In this case, influence of network is very limited but some events to/from the plant may be not taken into account.

First section is important for WNDCS if we consider that some real-time control applications may require such time constraints. The resulting heavy load of the network may also be encountered for greater values of Pcard but with much more sending stations (in our case, only 6 stations are considered).

Fig. 6 and Fig. 7 (circle points) simulations are done using the proposed algorithm. For the simulations where 5 ≤ Pcard ≤ 25ms, the results are quite the same with and without algorithm, due to the light influence of the network. For the simulations where 1 ≤ Pcard ≤ 4ms, the algorithm improves significantly the results: no collisions are observed and a maximum number of roundtrips are obtained during the simulation. Nevertheless, one exception has to be mentioned. When Pcard is equal to 1ms, some collision happens. This is explained by the fact that Pcard (1ms) is near to the transmission time of a packet (approximately 0.8ms if we consider 802.11b-physical layer and ACK waiting time). In this case, the MAC layer buffers will fill rapidly and begins to drop packets coming from upper layer.

VI. CONCLUSION

In this paper, the impact of a wireless communication network on the WNDCS performance in terms of safety and velocity is studied. To avoid degradations of control performance, due to collisions and packet losses in case of network heavy load, an algorithm for 802.11e priorities management has been proposed. The originality of this algorithm relies on a dynamic allocation of traffic priorities by taking into account the control requirements and current states of execution. Simulations of a cased study performed using the OPNET software tool have highlighted the efficiency of the proposed algorithm. These first simulation results should be reinforced by an implementation of the algorithm on laboratory-scale and industrial-scale platforms to make the proposed approach effective in practice.

Fig. 1 .

 1 Fig. 1. Architecture of the WDNCS in the case study

Fig. 2 .

 2 Fig. 2. Broadcasted packet format without proposed algorithm

Fig. 4 .

 4 Fig. 4. Broadcasted packet format with proposed algorithm

Fig. 5 .

 5 Fig. 5. OPNET model

Fig. 6 .Fig. 7 .

 67 Fig. 6. Number of collisions occurrences without algorithm (cross points) and with algorithm (circle points)

REFERENCES [ 1 ]

 1 IEEE Std 802.11e, "IEEE Standard for Information technology Telecommunications and information exchange between systems Local and metropolitan area networks Specific requirements", Part 11: Wireless LAN Medium Access Control( MAC) and Physical Layer (PHY) specifications Amendment 8: Medium Access Control (MAC) Quality of Service Enhancements.

For each Pplc sampling period do Build C, N and U sets using traces from x cur state For each

	For each s i ∈ 𝑁 do
	P S i ⟵ default_priority
	End_for
	For each s i ∈ 𝑈 do
	P S i ⟵ Null /*Request to stop sending */
	End_for
	Return P S i
	End_for
	/* initialization */
	P S i ⟵ default_priority
	Threshold ⟵ default_TH
	s i ∈ 𝐶 do
	Evaluate ResT(S i ) (where S i is the sensor emitting s i
	and ResT(S i ) is the response time of the sensor S i );
	If ResT(S i ) > Threshold and P S i ≤ 3
	Then P S i ⟵ P S i + 1
	Else P S i ⟵ P S i
	End_if
	End_for

Table 1 .

 1 Simulation parameters

			Controller model layer		Plant model layer
	Physical	Mac		Application layer			Application layer
	header	header
			Buffer	Buffer	Buffer	Buffer	Buffer	Buffer	Buffer	Buffer
			with	with	with	with	with	with	with	with
			priority	priority	priority	priority	priority	priority	priority	priority
		8	0	1	2	3	0	1	2	3
		Field
				MAC layer			MAC layer
				Physical layer			Physical layer
						Medium (Air)