Mohammed Bachir
email: mohammed.bachir@univ-paris1.fr

THE SPACE OF PROBABILISTIC 1-LIPSCHITZ MAPS

Keywords: Probabilistic metric space, Probabilistic 1-Lipschitz map, Probabilistic Banach-Stone type theorem and Isometries. msc: 46S50, 47S50, 54E70 Mathematics Subject Classification. 46S50, 54E70, 47S50

We introduce and study the natural notion of probabilistic 1-Lipschitz maps. We use the space of all probabilistic 1-Lipschitz maps to give a new method for the construction of probabilistic metric completion (respectively of probabilistic invariant metric group completion). Our construction is of independent interest. We prove that the space of all probabilistic 1-Lipschitz maps defined on a probabilistic invariant metric group can be endowed with a semigroup structure. Then, we caracterize the probabilistic invariant complete Menger groups by the space of all probabilistic 1-Lipschitz maps in the sprit of the classical Banach-Stone theorem.

Introduction

The general concept of probabilistic metric spaces was introduced by K. Menger, who dealt with probabilistic geometry [START_REF] Menger | Statistical metrics[END_REF], [START_REF] Menger | Probabilistic geometry[END_REF], [START_REF] Menger | Géométrie générale[END_REF]. The decisive influence on the development of the theory of probabilistic metric spaces is due to B. Schweizer and A. Sklar and their coworkers in several papers [START_REF] Schweizer | Espaces metriques aleatoires[END_REF], [START_REF] Schweizer | Sklar Statistical metric spaces[END_REF], [START_REF] Schweizer | Sklar Triangle Inequalities in a Class of Statistical Metric Spaces[END_REF], [START_REF] Schweizer | The metrization of statistical metric svaces[END_REF], see also [START_REF] Sherwood | On E-Spaces and their Relation to Other Classes of Probabilistic Metric Spaces[END_REF], [START_REF] Sherwood | Complete probabilistic metric spaces Z. Wahrscheinlichkeitstheorie und Verw[END_REF] and [START_REF] Hadžić | Pap Fixed Point Theory in Probabilistic Metric Space[END_REF]. For more informations about this theory we refeer to the excellent monograph [START_REF] Schweizer | Probabilistic metric spaces[END_REF].

The purpose of this paper is to make a new contributions to this theory. We introduce, in a natural way, the concept of probabilistic Lipschitz maps defined from a probabilistic metric space G into the set of all distribution functions that vanish at 0, classically denoted by ∆ + . We study the properties of the space of all probabilistic 1-Lipschitz maps denoted by Lip 1 ⋆ (G, ∆ + ) in the paper. As in the classical determinist analysis, the space of probabilistic 1-Lipschitz maps will play important role. To motivate the interest of this notion, we mention as example some results obtained about the completions of probabilistic metric spaces. Indeed, the well known result of H. Sherwood in [START_REF] Sherwood | Complete probabilistic metric spaces Z. Wahrscheinlichkeitstheorie und Verw[END_REF], shows that every probabilistic metric space with continuous triangle function, has a completion. To obtain this result, H. Sherwood used the method of the quotient space by Cauchy sequences. As consequence of our study, we give a new method of constructing of the completions of probabilistic metric spaces by using the concept of the probabilistic 1-Lipschitz maps instead of the quotient space by Cauchy sequences (see Theorem 3 and Corollary 1). Our method allows at the same time to obtain the completion of a probabilistic invariant metric group (Corollary 1), thus completing the result of H. Sherwood, just mentioned, in the group framework (in the deterministic case, the group completion for invariant metric groups is a known result of V.L. Klee [START_REF] Klee | Invariant metrics in groups (solution of a problem of Banach)[END_REF]). We mention here that our method is of independent interest and will be used for proving a probabilistic Banach-Stone type theorem.

We will start our study with relatively basic property by giving, for example, a probabilistic analogue of the classical result of E. Mac Shane [START_REF] Shane | Extension of range of functions[END_REF] on the extension of Lipschitz maps from a subset to the whole space (Theorem 1). We prove that the space Lip 1 ⋆ (G, ∆ + ) has also a probabilistic metric structure in which the probabilistic metric space G embeds isometrically . In the case where G is endowed with a group structure, we prove that we can also equip the space Lip 1 ⋆ (G, ∆ + ) with a natural "Sup-Convolution" operation denoted by ⊙ so that (Lip 1 ⋆ (G, ∆ + ), ⊙) become a probabilistic metric monoid. Then, we prove in a general setting our main results, which implies that the probabilistic invariant complete metric group structure of G is completely determined by the probabilistic metric monoid structure of (Lip 1 ⋆ (G, ∆ + ), ⊙). Our results apply in particular to probabilistic invariant complete Menger groups (Corollary 2). This gives a probabilistic version of the well known Banach-Stone theorem and extend some results obtained recently by the author for classical invariant metric groups in [START_REF] Bachir | The Inf-Convolution as a Law of Monoid. An analogue to the Banach-Stone theorem[END_REF], [START_REF] Bachir | A Banach-Stone type theorem for Invariant Metric Groups[END_REF] and [START_REF] Bachir | Representation of Isometric Isomorphisms between Monoids of Lipschitz Functions[END_REF]. Recall that the classical Banach-Stone theorem asserts that a compact Hausdorff space K is completely determined by the Banach structure of the space (C(K), . ∞ ) of all real-valued continuous functions equipped with the sup-norm. In other words, two Hausdorff compact spaces K and L are homeomorphic if and only if (C(K), . ∞ ) and (C(L), . ∞ ) are isometrically isomorphic as Banach spaces. Finally, note that several authors was interested on the extention of classical results obtained in metric spaces to the general framework of probabilistic metric spaces. For example, the theory of fixed points has several extensions in the setting of probabilistic metric spaces (see [START_REF] Hadžić | Pap Fixed Point Theory in Probabilistic Metric Space[END_REF] and [START_REF] Hadžić | On the Local Uniqueness of the Fixed Point of the Probabilistic q-Contraction in Fuzzy Metric Spaces Published by Faculty of Sciences and Mathematics[END_REF]). Also, the classical result of Mazur-Ulam on isometries in normed vector spaces has been extended to the probabilistic normed spaces (see for instance [START_REF] Chu | On the Mazur-Ulam problem in linear 2-normed spaces[END_REF], [START_REF] Cobzas | A Mazur-Ulam theorem for probabilistic normed spaces[END_REF]). This paper is organized as follows. In Section 2, we recall some classical and known notions about probabilistic metric spaces. In Section 3 we introduce the concept of probabilistic Lipschitz maps and probabilistic sup-convolution, we then deal with some basic properties. Section 4 and Section 5 contains the main results of the paper. In Section 4, we give our first main results Theorem 3, Theorem 4 and Corollary 1. On one hand, these results will permit to give a new construction of the completions of probabilistic metric spaces and probabilistic invariant metric groups. On the other hand, they will play an important role in Section 5. In Section 5, we give our second main results (Theorem 5, Theorem 6 and Corollary 2), which will permit to state a probabilistic Banach-Stone type theorem.

Classical Notions of Probabilistic Metric Spaces.

In this section, we recall somme general well know definitions and concepts about probabilistic metric spaces. All these concepts can be found in [START_REF] Hadžić | Pap Fixed Point Theory in Probabilistic Metric Space[END_REF] and [START_REF] Schweizer | Probabilistic metric spaces[END_REF].

A distribution function is a function F : [-∞, +∞] -→ [0, 1] which is nondecreasing and left-continuous with F (-∞) = 0; F (+∞) = 1. The set of all distribution functions such that F (0) = 0 will be denoted by ∆ + . For F, G ∈ ∆ + , the relation F ≤ G is meant by F (t) ≤ G(t), for all t ∈ R. For all a ∈ R, the distribution H a is defined as follow H a (t) = 0 for all t ≤ a 1 for all t > a For a = +∞,

H ∞ (t) = 0 for all t ∈ [-∞, +∞[
1 for all t = +∞ It is well known that (∆ + , ≤) is a complete lattice with a minimal element H ∞ and the maximal element H 0 . Thus, for any nonempty set I and any familly (F i ) i∈I of distributions in ∆ + , the function F = sup i∈I F i is also an element of ∆ + . Definition 1. (see [START_REF] Schweizer | Probabilistic metric spaces[END_REF], [START_REF] Hadžić | Pap Fixed Point Theory in Probabilistic Metric Space[END_REF]) A triangle function τ is a binary operation on ∆ + that is commutative, associative, and non-decreasing in each place, and has H 0 as identity. In other words, τ satisfies the axioms

(i)-(v): (i) τ (F, L) ∈ ∆ + for all F, L ∈ ∆ + . (ii) τ (F, L) = τ (L, F ) for all F, L ∈ ∆ + . (iii) τ (F, τ (L, K)) = τ (τ (F, L), K), for all F, L, K ∈ ∆ + . (iv) τ (F, H 0 ) = F for all F ∈ ∆ + . (v) F ≤ L =⇒ τ (F, K) ≤ τ (L, K) for all F, L, K ∈ ∆ + . Definition 2.
A tiangle function τ is said to be sup-continuous (see for instance [START_REF] Cobzas | Completeness with respect to the probabilistic Pompeiy-Hausdorff metric[END_REF]) if for all nonempty set I and all familly (F i ) i∈I of distributions in ∆ + and all L ∈ ∆ + , we have

sup i∈I τ (F i , L) = τ (sup i∈I (F i ), L).
For simplicity of notations, in what follows, the triangle function τ will be denoted by the binary operation ⋆ as follows:

τ (L, K) := L ⋆ K.
It follows from the axioms (i)-(v) that (∆ + , ⋆) is an abelian monoid having H 0 as identity element. Definition 3. Let ⋆ be a triangle function on ∆ + .

(

) A sequence (F n ) of distributions in ∆ + converges weakly to a function F in ∆ + if (F n (t)) converges to F (t) at each point t of continuity of F . In this case, we write indifferently F n w -→ F or lim n F n = F . (2) We say that the law ⋆ is continuous at (F, L) ∈ ∆ + × ∆ + if we have F n ⋆ L n w -→ F ⋆ L, whenever F n w -→ F and L n w -→ L. 1
Definition 4. (see [START_REF] Schweizer | Probabilistic metric spaces[END_REF], [START_REF] Hadžić | Pap Fixed Point Theory in Probabilistic Metric Space[END_REF]) A triangular norm (t-norm for short) is a binary operation on the unit interval

[0, 1], that is, a function T : [0, 1] × [0, 1] -→ [0, 1]
such that for all x, y, z ∈ [0, 1] the following four axioms are satisfied:

(T 1) T (x, y) = T (y, x) ( commutativity); (T 2) T (x, T (y, z)) = T (T (x, y), z) (associativity); (T 3) T (x, y) ≤ T (x, z) whenever y ≤ z (monotonicity ); (T 4) T (x, 1) = x (boundary condition).

Several basic examples of left-continuous t-norm are given in [START_REF] Hadžić | Pap Fixed Point Theory in Probabilistic Metric Space[END_REF] and [START_REF] Schweizer | Probabilistic metric spaces[END_REF]. We give below a classical continuous triangle functions.

Examples 1. (see [START_REF] Hadžić | Pap Fixed Point Theory in Probabilistic Metric Space[END_REF] and [START_REF] Schweizer | Probabilistic metric spaces[END_REF]) Let T : [0, 1] × [0, 1] -→ [0, 1] be a left-continuous t-norm, then the operation ⋆ T defined for all F, L ∈ ∆ + and for all t ∈ R by

(F ⋆ T L)(t) := sup s+u=t T (F (s), L(u)) (1)
satisfies the axioms (i)-(v) and is both continuous and sup-continuous.

Definition 5. (see Example 3 below) Let G be a set and let D : G×G -→ (∆ + , ⋆, ≤) be a map. We say that (G, D, ⋆) is a probabilistic metric space if the following axioms (i)-(iii) hold:

(i) D(p, q) = H 0 iff p = q.
(ii) D(p, q) = D(q, p) for all p, q ∈ G (iii) D(p, q) ⋆ D(q, r) ≤ D(p, r) for all p, q, r ∈ G If moreover the set (G, •) is a monoid, then we say that (G, •, D, ⋆) is a probabilistic metric monoide and if (G, •) is group and satisfies the following additional axiom, (iv) D(pr, qr) = D(rp, rq) = D(p, q) for all p, q, r ∈ G then, we say that (G, •, D, ⋆) is an invariant probabilistic metric group. Definition 6. Two probabilistic metric spaces (G, D, ⋆) and (G ′ , D ′ , ⋆) (resp. monoids, groups) are said to be isometric (resp. isometrically isomorphic), if there exists a bijective map (resp. isomorphism of monoids, isomorphism of groups) I : G -→ G ′ such that D ′ (I(x), I(y)) = D(x, y) for all x, y ∈ G.

Definition 7. Let (G, D, ⋆) be a probabilistic metric space and ⋆ = ⋆ T , where

(K ⋆ T L)(t) = sup u+v=t T (K(u), L(v))
for a t-norm T . Then (G, D, ⋆) is the so called Menger space, which will be denoted by (G, D, T ). If moreover G is a group, we say that (G, D, T ) is a Menger group.

Introduction to the Probabilistic Lipschitz Maps

We are going to introduce the concept of probabilistic Lipschitz maps defined from a probabilistic metric space into ∆ + . Definition 8. Let (G, D, ⋆) be a probabilistic metric space and let f : G -→ ∆ + be a map.

(1) We say that

f is continuous at z ∈ G if f (z n ) w -→ f (z), whenever D(z n , z) w -→ H 0 . We say that f is continuous if f is continuous at each point z ∈ G.
(2) We say that f is 1-Lipschitz map if:

∀x, y ∈ G, D(x, y) ⋆ f (y) ≤ f (x).
We can also define a k-Lipschitz map f for a nonegative real number k ≥ 0 as follows ∀x,

y ∈ G, D k (x, y) ⋆ f (y) ≤ f (x),
where, for all x, y ∈ G and all t ∈ R, D k (x, y)(t) = D(x, y)( t k ) if k > 0 and D 0 (x, y)(t) = H 0 (t) if k = 0. In this paper, we deal only with 1-Lipschitz maps.

Examples 2. Let (G, d) be a metric space. Assume that ⋆ is a triangle function on ∆ + satisfying H a ⋆ H b = H a+b for all a, b ∈ R + (for example if ⋆ = ⋆ T where T is a lef-continuous t-norm). Let (G, D, ⋆) be the probabilistic metric space defined with the probabilistic metric

D(p, q) = H d(p,q) .
Let L : (G, d) -→ R + be a real-valued map. Then, L is a nonnegative 1-Lipschitz map if and only if f : (G, D, ⋆) -→ ∆ + defined for all x ∈ G by

f (x) := H L(x)
is a probabilistic 1-Lipschitz map. This example shows that the framework of probabilistic 1-Lipschitz maps encompasses the classical determinist case.

By Lip 1 ⋆ (G, ∆ + ) we denote the space of all probabilistic 1-Lipschitz maps

Lip 1 ⋆ (G, ∆ + ) := {f : G -→ ∆ + /D(x, y) ⋆ f (y) ≤ f (x)
; ∀x, y ∈ G}. For all x ∈ G, by δ x we denote the map

δ x : G -→ ∆ + y → D(y, x).
We set G(G) := {δ x /x ∈ G} and by δ, we denote the operator

δ : G -→ G(G) ⊂ Lip 1 ⋆ (G, ∆ + ) x → δ x .
3.1. Basic Properties of 1-Lipschitz Maps. In this section, we give some basic and elementary but useful properties of probabilistic 1-Lipschitz maps.

Proposition 1. Let (G, D, ⋆) be a probabilistic metric space. Then, we have that δ a ∈ Lip 1 ⋆ (G, ∆ + ) for each a ∈ G and the map δ :

G -→ Lip 1 ⋆ (G, ∆ + ) is injective. Proof. The fact that δ a ∈ Lip 1
⋆ (G, ∆ + ) for each a ∈ G follows from the property: D(x, y) ⋆ D(y, a) ≤ D(x, a) for all a, x, y ∈ G. Now, let a, b ∈ G be such that δ a = δ b . It follows that δ a (x) = δ b (x) for all x ∈ G. In particular, for

x = b we have that D(a, b) = δ a (b) = δ b (b) = H 0 , which implies that a = b. Let f ∈ Lip 1 ⋆ (G, ∆ +
) and F ∈ ∆ + , by f, F : G -→ ∆ + , we denote the map defined by f, F (x) := f (x) ⋆ F for all x ∈ G. We easly obtain the following propostion.

Proposition 2. Let (G, D, ⋆) be a probabilistic metric space. Then, for all f ∈ Lip 1 ⋆ (G, ∆ + ) and all F ∈ ∆ + , we have that f, F ∈ Lip 1 ⋆ (G, ∆ + ). In a classical metric space (M, d), we know that the distance function to a nonempty subset A of M defined by x → d(x, A) := inf y∈A d(x, y) is a real-valued 1-Lipschitz map. We have the analogous of this property in the probabilistic metric case. The following classical proposition will be used very often:

Proposition 4. Let (F n ), (L n ), (K n ) ⊂ (∆ + , ⋆). Suppose that (a) the triangle function ⋆ is continuous, (b) F n w -→ F , L n w -→ L et K n w -→ K. (c) for all n ∈ N, F n ⋆ L n ≤ K n , then, F ⋆ L ≤ K.
Proof. It is well known that an increasing function from R into R is continuous outside a set which is at most countable. Clearly, from Definition 3 we have that

(F ⋆ L)(t) ≤ K(t) for all t ∈ R which is a point of continuity of both F ⋆ L and K. Since both F ⋆ L and K are let-continuous, we deduce that (F ⋆ L)(t) ≤ K(t) for all t ∈ R.
We also recall the following deep results due to D. Sibley: From [25, Theorem 1. and Theorem 2.] we know that the topological space (∆ + , w) (equipped with the weak convergence) is metrizable by the modified Lévy metric and from [START_REF] Sibley | A metric for weak convergence of distribution functions[END_REF]Theorem 3.] we know that (∆ + , w) is a compact metrizable space. This last result will be used very often in this paper. We have the following proposition. Proposition 5. Let (G, D, ⋆) be a probabilistic metric space such that ⋆ is continuous. Then, every probabilistic 1-Lipschitz map defined on G is continuous.

Proof. Let f be a probabilistic 1-Lipschitz map, x ∈ G and (x n ) n ⊂ G be a sequence such that lim n D(x n , x) = H 0 . Since (∆ + , w) is a compact metrizable space, it suffices to prove that every (weak) convergent subsequence (f (x ϕ(n) )), converges necessary to f (x). Suppose that a subsequence (f (x ϕ(n) )) converges to some distribution F and let us prove that F = f (x). Indeed, Since f is a probabilistic 1-Lipschitz map, then for all n ∈ N we have

D(x ϕ(n) , x) ⋆ f (x) ≤ f (x ϕ(n) ).
Using Proposition 4 and the fact that D(x ϕ(n) , x) w -→ H 0 , we get from the above inequality that f (x) ≤ F . Also we have

D(x ϕ(n) , x) ⋆ f (x ϕ(n) ) ≤ f (x).
Again, using Proposition 4, the fact that D(x ϕ(n) , x)

w -→ H 0 and f (x ϕ(n) ) w -→ F , we get that F ≤ f (x). Thus F = f (x) and so, f (x n ) w -→ f (x)
. This finish the proof. [START_REF] Shane | Extension of range of functions[END_REF]: if (X, d) is a metric space, A a nonempty subset of X and f : A -→ R is k-Lipschitz map, then there exist a k-Lipschitz map f : X -→ R such that f|A = f . We give bellow an analogue of this result for probabilistic 1-Lipschitz maps.

A Probabilistic Analogue of Mac Shane Result's. Recall the Lipschitz extention result of Mac Shane in

Theorem 1. Let (G, D, ⋆) be a probabilistic metric space such that ⋆ is supcontinuous and let A be a nonempty subset of G. Let f : A -→ ∆ + be a probabilistic 1-Lipschitz map. Then, there exists a probabilistic 1-Lipschitz map f :

G -→ ∆ + such that f|A = f . Proof. We define f : G -→ ∆ + as follows: for all x ∈ G, f (x) := sup a∈A D(a, x) ⋆ f (a).
We first prove that f (x) = f (x) for all x ∈ A. Indeed, let x ∈ A. On one hand we have f

(x) = H 0 ⋆ f (x) = D(x, x) ⋆ f (x) ≤ sup a∈A D(a, x) ⋆ f (a) = f (x). On the other hand, since f is probabilistic 1-Lipschitz on A and x ∈ A, then D(a, x)⋆f (a) ≤ f (x) for all a ∈ A. It follows that f (x) := sup a∈A D(a, x) ⋆ f (a) ≤ f (x). Thus, f (x) = f (x) for all x ∈ A. Now, we show that f is probabilistic 1-Lipschitz on G. Indeed, Let x, y ∈ G. For all a, x, y ∈ A we have that D(a, x) ⋆ D(x, y) ≤ D(a, y). So, D(a, x) ⋆ f (a) ⋆ D(x, y) ≤ D(a, y) ⋆ f (a)
. By taking the supremum over a ∈ A and using the sup-continuity of ⋆ we get f (x) ⋆ D(x, y) ≤ f (y), for all x, y ∈ G. Hence, f is probabilistic 1-Lipschitz map on G that coincides with f on A.

3.3.

Probabilistic Sup-Convolution. In this section, we assume that G is a group. The aim of this section is to prove that if (G, •, D, ⋆) is a probabilistic invariant metric group having e as identity element, then we can equip the set Lip 1 ⋆ (G, ∆ + ) of probabilistic 1-Lipschitz maps with a Sup-Convolution law denoted by ⊙, so that (Lip 1 ⋆ (G, ∆ + ), ⊙) becomes a monoid having δ e as identity element (Theorem 2). This will allow, subsequently, to embed G as group into Lip 1 ⋆ (G, ∆ + ). The probabilistic sup-convolution introduced below is an extension of the inf-convolution treated recently in the classical invariant metric groups framework in [START_REF] Bachir | The Inf-Convolution as a Law of Monoid. An analogue to the Banach-Stone theorem[END_REF], [START_REF] Bachir | A Banach-Stone type theorem for Invariant Metric Groups[END_REF] and [START_REF] Bachir | Representation of Isometric Isomorphisms between Monoids of Lipschitz Functions[END_REF].

Let G be a group and let f, g : G -→ ∆ + be two maps. The Sup-Convolution of f and g denoted f ⊙ g : G -→ ∆ + is a map defined for all x ∈ G as follows

(f ⊙ g)(x) := sup y,z∈G/yz=x f (y) ⋆ g(z) = sup y∈G f (y) ⋆ g(y -1 x).
In order to prove that (Lip 1 ⋆ (G, ∆ + ), ⊙) has a monoid structure, we need to establish the three following propositions. Recall that a monoid is a semigroup having an identity element. Now, we prove that that δ a ⊙ δ b = δ ab for all a, b ∈ G. Indeed, let x ∈ G and t ∈ R, using the change variable u = zb -1 and the invariance of D we have:

δ a ⊙ δ b (x) = sup z∈X δ a (xz -1 ) ⋆ δ b (z) = sup z∈G δ e (xz -1 a -1 ) ⋆ δ e (zb -1 ) = sup u∈G δ e (xb -1 t -1 a -1 ) ⋆ δ e (u) = sup u∈X δ e (a -1 xb -1 u -1 ) ⋆ δ e (u) = (δ e ⊙ δ e )(a -1 xb -1 ) = δ e (a -1 xb -1 ) = δ e (xb -1 a -1 ) = δ e (x(ab) -1 ) = δ ab (x). Thus, δ a ⊙ δ b = δ ab for all a, b ∈ G.
When G is a probabilistic invariant metric group, the space Lip 1 ⋆ (G, ∆ + ) turns out to be naturally linked to the operation of Sup-Convolution. It is characterized by being the maximal set of maps from G into ∆ + having δ e as identity element for the operation ⊙. Proposition 7. Let (G, •, D, ⋆) be an invariant probabilistic metric group. Then, we have

Lip 1 ⋆ (G, ∆ + ) = {f : G -→ ∆ + map /f ⊙ δ e = δ e ⊙ f = f }. Proof. Let f ∈ Lip 1 ⋆ (G, ∆ + ).
We first prove that δ e ⊙f (x) = f ⊙δ e (x) for all x ∈ G. Using the variable change u = xy -1 and the invariance of D, we have for all x ∈ G,

(f ⊙ δ e )(x) = sup y∈G f (xy -1 ) ⋆ δ e (y) = sup u∈G f (u) ⋆ δ e (u -1 x) = sup u∈G δ e (xu -1 ) ⋆ f (u) = (δ e ⊙ f )(x). Now, let f ∈ Lip 1 ⋆ (G, ∆ + ) then for all x, y ∈ G we have f (x) ≥ d(x, y) ⋆ f (y) = d(xy -1 , e) ⋆ f (y) = δ e (xy -1 ) ⋆ f (y).
Taking the supremum over y ∈ G, we get

f (x) ≥ (δ e ⊙ f )(x) = (f ⊙ δ e )(x) for all x ∈ G.
For the converse inequality we have

f ⊙ δ e (x) = sup y∈G f (xy -1 ) ⋆ δ e (y) ≥ f (x) ⋆ δ e (e) = f (x) ⋆ H 0 = f (x).
Thus f ⊙δ e = δ e ⊙f = f for all f ∈ Lip 1 ⋆ (G, ∆ + ) and so we have that

Lip 1 ⋆ (G, ∆ + ) ⊂ {f : G -→ ∆ + map /f ⊙ δ e = δ e ⊙ f = f }. Now, let f ∈ {f : G -→ ∆ + map /f ⊙ δ e = δ e ⊙ f = f }. It follows that for all x ∈ G f (x) = f ⊙ δ e (x) = sup yz=x f (y) ⋆ δ e (z) ≥ f (y) ⋆ δ e (y -1 x) = D(x, y) ⋆ f (y). It follows that f ∈ Lip 1 ⋆ (G, ∆ + ). Finally we have Lip 1 ⋆ (G, ∆ + ) = {f : G -→ ∆ + map /f ⊙ δ e = δ e ⊙ f = f }.
Proposition 8. Let (G, •, D, ⋆) be an invariant probabilistic metric group. Suppose that ⋆ is sup-continuous. Then, the set (Lip 1 ⋆ (G, ∆ + ), ⊙) has the associative property.

Proof. Let f, g, h ∈ Lip 1 ⋆ (G, ∆ + ). Using the variable change u := yz together with the sup-continuity of ⋆, we obtain for all x ∈ G,

(f ⊙ g) ⊙ h(x) = sup z∈G {(f ⊙ g)(xz -1 ) ⋆ h(z)} = sup z∈G {(f ⊙ g)(xz -1 ) ⋆ h(z)} = sup z∈G {sup y∈G {f (xz -1 y -1 ) ⋆ g(y)} ⋆ h(z)} = sup z∈G {sup y∈G {f (xz -1 y -1 ) ⋆ g(y)} ⋆ h(z)} = sup z∈G {sup u∈G {f (xu -1 ) ⋆ g(uz -1 )} ⋆ h(z)} = sup u∈G {f (xu -1 ) ⋆ sup z∈G {g(tz -1 ) ⋆ h(z)}} = sup u∈G {f (xu -1 ) ⋆ (g ⊙ h)(u)} = f ⊙ (g ⊙ h)(x). Theorem 2. Let (G, •, D, ⋆) be an invariant probabilistic metric group. Suppose that ⋆ is sup-continuous. Then, the set (Lip 1 ⋆ (G, ∆ + ), ⊙) is a monoid having δ e as identity element. Moreover, the map δ : (G, •) -→ (Lip 1 ⋆ (G, ∆ + ), ⊙) is an injective group homomorphism. Proof. First we prove that f ⊙ g ∈ Lip 1 ⋆ (G, ∆ + ), whenever f, g ∈ Lip 1 ⋆ (G, ∆ + ). Using Proposition 7,it suffices to shows that δ e ⊙ (f ⊙ g) = (f ⊙ g) ⊙ δ e = f ⊙ g. Indeed, since f ∈ Lip 1
⋆ (G, ∆ + ), then by Proposition 7 we have that δ e ⊙ f = f . Thus, (δ e ⊙ f )⊙ g = f ⊙ g. Using the associative property (Proposition 8) we obatin that δ e ⊙ (f ⊙ g) = f ⊙ g. In a similar way, since g ⊙ δ e = g, we obtain (using the associative property) that (f ⊙ g) ⊙ δ e = f ⊙ g. Finally, we have

δ e ⊙ (f ⊙ g) = (f ⊙ g) ⊙ δ e = f ⊙ g, which implies that f ⊙ g ∈ Lip 1
⋆ (G, ∆ + ). This property with the associative property shows that (Lip 1 ⋆ (G, ∆ + ), ⊙) is a semigroup. Using again Proposition 7, we obtain that (Lip 1 ⋆ (G, ∆ + ), ⊙) is a monoid having δ e as identity element. Using Proposition 6 we get that the map δ : (G, •) -→ (Lip 1 ⋆ (G, ∆ + ), ⊙) is a group homeomorphism.

The Probabilistic Metric 1-Lipschitz Space and Completions

In this section, we give a new method of constructing the completion of probabilistic metric space. This method is based on the use of the probabilistic 1-Lipschitz maps and is of independ interest. Indeed, it will be necessary to treat Section 5 especially to prove theorem 5. Recall that in the classical metric spaces, there are at least two methods for constructing the completion of a metric space M . The first one uses the quotient by the Cauchy sequences of M and the second one uses the isometric embedding of M into a complete metric space of functions. The proof, in the probabilistic case, given by H. Sherwood in [START_REF] Sherwood | Complete probabilistic metric spaces Z. Wahrscheinlichkeitstheorie und Verw[END_REF] uses the first approach. We will follow here the second approach, by embedding isometrically a given probabilistic metric space G into a probabilistic metric space of 1-Lipschitz maps. Note also that the completion of probabilistic normed space was given by B. Lafuerza Guillen, J. A. Rodriguez Lallena, and C. Sempi [START_REF] Lafuerza Guillen | Completion of probabilistic normed spaces[END_REF], using also the method of the quotient space by Cauchy sequences. In this section, we give in particular (Corollary 1) the new result on the group completion of any probabilistic invariant metric group, which is the probabilistic version of the group completion given by V.L. Klee [START_REF] Klee | Invariant metrics in groups (solution of a problem of Banach)[END_REF] in the classical metric case.

We recall below the definition of probabilistic complete metric space. Definition 9. Let (G, D, ⋆) be a probabilistic metric space. A sequence (z n ) ⊂ G is said to be a Cauchy sequence if for all t ∈ R,

lim n,p-→+∞ D(z n , z p )(t) = H 0 (t). (Equivalently, if D(z n , z p ) w -→ H 0 , when n, p -→ +∞). A probabilistic metric space (G, D, ⋆) is said to be complete if every Cauchy sequence (z n ) ⊂ G weakly converges to some z ∞ ∈ G, that is lim n→+∞ D(z n , z ∞ )(t) = H 0 (t) for all t ∈ R, we will briefly note lim n D(z n , z ∞ ) = H 0 .
Examples 3. We give below general frameworks of probabilistic complete metric spaces and probabilistic invariant complete metric groups.

(1) Every complete metric space induce a probabilistic complete metric space. Indeed, if d is a metric on G and ⋆ is a triangle function on ∆ + satisfying H a ⋆ H b = H a+b for all a, b ∈ R + (the triangle function ⋆ T with a left-continuous t-norm T , satisfies this condition, see for instance [START_REF] Hadžić | Pap Fixed Point Theory in Probabilistic Metric Space[END_REF] and [START_REF] Schweizer | Probabilistic metric spaces[END_REF]), then (G, D, ⋆) where,

D(p, q) = H d(p,q) , ∀p, q ∈ G,
is a probabilistic complete metric space. Thus, if (G, •, d) is invariant complete metric group (Several examples of invariant complete metric groups are given in [START_REF] Bachir | A Banach-Stone type theorem for Invariant Metric Groups[END_REF]) then (G, •, D, ⋆) is a probabilistic invariant complete metric group.

(2) Let (G, F, ⋆) be a probabilistic complete normed vector space with a probabilistic norm F (the definition of probabilistic complete normed space can be found in [START_REF] Alsina | On the definition of a probabilistic normed space[END_REF], see also [START_REF] Hadžić | Pap Fixed Point Theory in Probabilistic Metric Space[END_REF] p.66 for example). For all p, q ∈ G, let us set

D(p, q) = F p-q .
Then, (G, +, D, ⋆) is a probabilistic invariant complete metric group (this example also works by remplacing probabilistic normed vector space by probabilistic normed groups, the definition can be found in [START_REF] Shaabani | On the definition of a probabilistic normed group[END_REF], [START_REF] Nourouzi | Pourmoslemi Probabilistic Normed Groups[END_REF]). Definition 10. Let (G, D, ⋆) be a probabilistic metric space. We define the subset

Π(G) of Lip 1 ⋆ (G, ∆ + ) as follows: f ∈ Π(G) if and only if, f is a probabilistic 1- Lipschitz map such that, there exist a Cauchy sequence (a n ) ⊂ G satisfying: for all x ∈ G, δ an (x) := D(a n , x) w -→ f (x), when n → +∞.
Note that G(G) is always a subset of Π(G) and we have G(G) = Π(G) if and only if G is complete. The main result of this section is the following theorem. Its proof will be given after two lemmas. (2) (Π(G), D, ⋆) is a probabilistic complete metric space.

(

) Π(G) = G(G) D . In other words, δ(G) := G(G) is dense in (Π(G), D, ⋆). 3
Lemma 1. Let (G, D, ⋆) be a probabilistic metric space (not assumed to be complete), where ⋆ is a continuous triangle function. Let (a n ) ⊂ G be a Cauchy sequence. Then, there exists a probabilistic 1-Lipschitz map f : G -→ ∆ + such that, for all x ∈ G we have D(a n , x)

w -→ f (x), when n -→ +∞. In particular, f (a n ) w -→ H 0 , when n -→ +∞.
Proof. We know from [START_REF] Sibley | A metric for weak convergence of distribution functions[END_REF] that (∆ + , w) is a compact metrizable space. Thus, for each fixed point x ∈ G, the sequence (D(a n , x)) has a convergent subsequence that is, there exists f (x) ∈ ∆ + and a strictly increasing map ϕ x : N -→ N such that D(a ϕx(n) , x) w -→ f (x) when n -→ +∞. From the properties of D, we have for all y ∈ G,

D(a ϕy(n) , y) ⋆ D(a ϕx(n) , a ϕy(n) ) ≤ D(a ϕx(n) , y) (2)
Also, for each y ∈ G the sequence (D(a ϕx(n) , y)) has a convergent subsequence by the compactness of (∆ + , w) that is, there exists g x (y) ∈ ∆ + and there exists a strictly increasing map ψ x,y : N -→ N such that

D(a ϕx(ψx,y(n)) , y) w -→ g x (y). (3)
when n -→ +∞. Using (2), (3), the continuity of ⋆, the fact that (a n ) is Cauchy sequence and taking the limit (see Proposition 4), we get

f (y) = f (y) ⋆ H 0 ≤ g x (y). (4)
On the other hand, we have D(y, x) ⋆ D(a ϕx(ψx,y(n)) , y) ≤ D(a ϕx(ψx,y(n)) , x)

So by Proposition 4,

D(y, x) ⋆ g x (y) ≤ f (x) (5)
Thus, from (4) and ( 5) we have for all x, y ∈ G D(y, x) ⋆ f (y) ≤ f (x).

This shows that the well defined map f is a probabilistic 1-Lipschitz map. Now, we are going to prove that D(a n , x) w -→ f (x) for all x ∈ G, when n -→ +∞. Indeed, let us fixe an x ∈ G. In order to prove that D(a n , x) w -→ f (x), we need only to prove that the sequence (D(a n , x)) has a unique cluster point, since (∆ + , w) is a metrizable compact set. Let (D(a ϕ(n) , x)) be a convergent subsequence to some f (x). Let us prove that necessarily f (x) = f (x). Indeed, from the properties of D, we have that

D(a ϕx(n) , x) ⋆ D(a ϕx(n) , a ϕ(n) ) ≤ D(a ϕ(n) , x). Since D(a ϕx(n) , x) w -→ f (x) and (a n ) is a Cauchy sequence, then from Proposition 4 f (x) = f (x) ⋆ H 0 ≤ f (x).
In a similar way, we have

D(a ϕ(n) , x) ⋆ D(a ϕ(n) , a ϕx(n) ) ≤ D(a ϕx(n) , x).
Since D(a ϕ(n) , x) w -→ f (x) and (a n ) is a Cauchy sequence, then from Proposition 4

f (x) = f (x) ⋆ H 0 ≤ f (x).
Thus f (x) = f (x) and so, D(a n , x) w -→ f (x). Now, to see that f (a n ) w -→ H 0 (when n -→ +∞), it suffices to observe that lim n f (a n ) = lim n (lim p D(a p , a n )) = H 0 , since (a n ) is a Cauchy sequence.

sup x∈G f (x) ⋆ g(x) = lim n,m D(a n , b m ) = lim n D(a n , b n ) = lim m f (b m ) = lim n g(a n ).
Proof. Let (a n ), (b n ) ⊂ G be any Cauchy sequences such that D(a n , x) w -→ f (x), and D(b n , x) w -→ g(x) for all x ∈ G, when n -→ +∞. In particular we have that

f (a n ) w -→ H 0 and g(b n ) w -→ H 0 , when n -→ +∞ (see Lemma 1). Since f is a proba- bilistic 1-Lipschitz then, for all m ∈ N f (x) ⋆ D(b m , x) ≤ f (b m ).
We prove that lim m f (b m ) = sup x∈G f (x) ⋆ g(x). To do this, it suffices to show that every convergent subsequence of (f (b m )) converges to sup x∈G f (x) ⋆ g(x) since (∆ + , w) is a compact metrizable space. Indeed, let (f ϕ(m) ) be a convergent subsequence to some F ∈ ∆ + . By using the continuity of ⋆ and taking the limit over m ∈ N in the above inequality (using Proposition 4), we get that for all x ∈ G, f (x) ⋆ g(x) ≤ F . So, sup

x∈G f (x) ⋆ g(x) ≤ F.
On the other hand, we have f (b ϕ(m) ) ⋆ g(b ϕ(m) ) ≤ sup x∈G f (x) ⋆ g(x). Using Proposition 4, the continuity of ⋆ and the fact that g(b ϕ(m) )

w -→ H 0 (when m -→ +∞), we obtain that

F ≤ sup x∈G f (x) ⋆ g(x)
.

Thus, lim m f (b m ) = F = sup x∈G f (x) ⋆ g(x)
. Finally, we have that

sup x∈G f (x) ⋆ g(x) = lim m f (b m ) = lim m (lim n D(a n , b m )).
Using the same arguments by replacing f by g we also get that

sup x∈G f (x) ⋆ g(x) = lim n g(a n ) = lim n (lim m D(a n , b m )).
Hence,

sup x∈G f (x) ⋆ g(x) = lim m f (b m ) = lim n g(a n ) = lim n,m D(a n , b m ).
Finally, it is not difficult to see that lim On the other hand,

D(δ a , δ b ) = sup x∈G δ a (x) ⋆ δ b (x) ≥ δ a (b) ⋆ δ b (b) = D(a, b) ⋆ H 0 = D(a, b). Thus, D(δ a , δ b ) = D(a, b), for all a, b ∈ G.
(2) This part is given by the following two steps.

Step 1: (Π(G), D, ⋆) is a probabilistic metric space. Indeed, (i) First, we prove that D(f, f ) = H 0 for all f ∈ Π(G). By Lemma 2 we have D(f, f ) = lim n,m D(a n , a m ) = H 0 , where (a n ) is a Cauchy sequence such that D(a n , x) w -→ f (x) for all x ∈ G. Conversely, let f, g ∈ Π(G) and suppose that D(f, g) = H 0 . There exists a Cauchy sequence (a n ) ⊂ G, such that D(a n , x) w -→ f (x) for all x ∈ G. Since g is a probabilistic 1-Lipschitz map, then D(a n , x) ⋆ g(x) ≤ g(a n ) ≤ H 0 . Let (g(a ϕ(n) )) be any (weak) convergent subsequence to some limit L ∈ ∆ + . Since, f (x) ⋆ g(x) = lim n (D(a n , x) ⋆ g(x)) = lim n (D(a ϕ(n) , x) ⋆ g(x)), we have (by Proposition 4)

f (x) ⋆ g(x) ≤ L ≤ H 0 .
Taking the supremum over x ∈ G, we get that

H 0 = D(f, g) = sup x∈G f (x) ⋆ g(x) ≤ L ≤ H 0 .
In other words, lim n g(a ϕ(n) ) = H 0 for any convergent subsequence. Since (∆ + , w) is a compact metrizable space, we get that lim n g(a n ) = H 0 . Now, since g is a probabilistic 1-Lipschitz map, then D(a n , x) ⋆ g(a n ) ≤ g(x) and so by taking the limit and using Proposition 4 we obtain that f (x) = f (x) ⋆ H 0 ≤ g(x). Using a similar arguments, we also have that g

(x) ≤ f (x). Hence, f = g. (ii) it is clear that D(f, g) = D(g, f ) for all f, g ∈ Π(G). (iii) Let f, g, h ∈ Π(G) and let us prove that D(f, g) ⋆ D(g, h) ≤ D(f, h). There exists a Cauchy sequences (a n ), (b n ), (c n ) ⊂ G such that D(a n , x) w -→ f (x), D(b n , x) w -→ g(x)
and D(c n , x) w -→ h(x) for all x ∈ G. Then, using the properties of D, the continuity of ⋆ and Lemma 2, we have

(sup x∈G f (x) ⋆ g(x)) ⋆ (sup x∈G g(x) ⋆ h(x)) = lim n,m D(a n , b m ) ⋆ lim m,p D(b m , c p ) = lim n,m,p (D(a n , b m ) ⋆ D(b m , c p )) ≤ lim n,p D(a n , c p ) = sup x∈G f (x) ⋆ h(x) Hence, D(f, g) ⋆ D(g, h) ≤ D(f, h).
Step 2: The probabilistic metric space (Π(G), D, ⋆) is complete. Let (f k ) ⊂ Π(G) be a Cauchy sequence. For each k ∈ N, there exists a Cauchy sequence (a

(n,k) ) n∈N ⊂ G such that D(a (n,k) , x) w -→ f k (x), when n -→ +∞ for all x ∈ G. This also implies that for all k ∈ N D(δ a (n,k) , f k ) = f k (a (n,k) ) w -→ H 0 , (6)
when n -→ +∞. Thus, for all k ∈ N, there exists

N k ∈ N such that for all n ≥ N k 1 - 1 k ≤ D(δ a (n,k) , f k )( 1 k ) = f k (a (n,k) )( 1 k ) ( 7
)
Since D is a probabilistic metric, we have that for all n, m, k, p

∈ N D(δ a (n,k) , f k ) ⋆ D(f k , f p ) ⋆ D(f p , δ a (m,p) ) ≤ D(δ a (n,k) , δ a (m,p) ) (8)
= D(a (n,k) , a (m,p) ).

Using Proposition 4, Lemma 2, the above inequality and (6), we get for all k, p ∈ N *

D(f k , f p ) ≤ lim n,m D(a (n,k) , a (m,p) ).
We can choose a map ϕ : N -→ N such that ϕ(k) ≥ N k for all k ∈ N * and such that for all k, p, s ∈ N * ,

D(f k , f p )( 1 s ) ≤ lim n,m (D(a (n,k) , a (m,p) )( 1 s )) ≤ D(a (ϕ(k),k) , a (ϕ(p),p) )( 1 s ) + ( 1 k + 1 p ).
Since, elements of ∆ + are increasing, for all t > 0, there exists an integer number s t ∈ N such that 1 st ≤ t and

D(f k , f p )( 1 s t ) -( 1 k + 1 p ) ≤ D(a (ϕ(k),k) , a (ϕ(p),p) )( 1 s t ) ≤ D(a (ϕ(k),k) , a (ϕ(p),p) )(t).
By sending k, p to +∞, using the fact that (f k ) is Cauchy we have for all t > 0

1 = lim k,p→+∞ D(f k , f p )( 1 s t ) = lim k,p→+∞ D(a (ϕ(k),k) , a (ϕ(p),p) )(t).
In other words, the sequence (a (ϕ(k),k) ) k∈N is Cauchy in G. Using Lemma 1, there exists a Probabilistic 1-Lipschitz map f such that, for all x ∈ G we have

D(a (ϕ(k),k) , x) w -→ f (x) and in particular f (a (ϕ(k),k) ) w -→ H 0 , when k -→ +∞. This implies that f ∈ Π(G) and D(δ a (ϕ(k),k) , f ) = f (a (ϕ(k),k) ) w -→ H 0 , when k -→ +∞. ( 9
)
Since D is a probabilistic metric on Π(G), we have

D(f k , δ a (ϕ(k),k) ) ⋆ D(δ a (ϕ(k),k) , f ) ≤ D(f k , f ) (10) ≤ H 0 .
We know that:

(a) on one hand, from (7) (since ϕ(k) ≥ N k for all k), we have that:

∀t > 0, ∀k ∈ N * such that 1 t ≤ k, 1 - 1 k ≤ D(δ a (ϕ(k),k) , f k )( 1 k ) = f k (a (ϕ(k),k) )( 1 k ) ≤ f k (a (ϕ(k),k) )(t).
It follows that lim k→+∞ f k (a (ϕ(k),k) )(t) = 1 for all t > 0. In other words,

D(f k , δ a (ϕ(k),k) ) = f k (a (ϕ(k),k) ) w -→ H 0 , when k → +∞.
(b) on the other hand, we know from (9) that D(δ

a (ϕ(k),k) , f ) w -→ H 0 , when k -→ +∞.
Hence, using (a), (b) and ( 10) (with Proposition 4), we get that D(f k , f ) w -→ H 0 , when k -→ +∞. This finish the proof of (2).

(3) Since (Π(G), D) is complete and G(G) ⊂ Π(G), clearly G(G) D ⊂ Π(G). For the converse, let f ∈ Π(G). Thus, there exist a Cauchy sequence (a n ) ⊂ G such that D(a n , x) w -→ f (x) for all x ∈ G, in particular f (a n )
w -→ H 0 , when n → +∞ (since (a n ) is a Cauchy sequence). Hence, since f is 1-Lipschitz, we have

D(δ an , f ) = sup x∈G D(a n , x) ⋆ f (x) = f (a n ), which shows that D(δ an , f ) w -→ H 0 . In other words, f ∈ G(G) D . Thus, Π(G) = G(G) D and so G(G) is dense in (Π(G), D, ⋆).
In the group framework, we have the following theorem.

Theorem 4. Let (G, •, D, ⋆) be a probabilistic invariant metric group having e as identity element, where ⋆ is a continuous triangle function. Then (Π(G), ⊙, D, ⋆) is a probabilistic invariant complete metric group having δ e as identity element.

Proof. The fact that (Π(G), D, ⋆) is a probabilistic complete metric space is given by Theorem 3. Let us prove that (Π(G), ⊙) is a group having δ e as identity element.

The associative property is given by Proposition 8. We prove that f ⊙ g ∈ Π(G), whenever f, g ∈ Π(G). Indeed, from Lemma 1, there exists Cauchy sequences

(z n ), (s n ) ⊂ G such that D(z n , x) w -→ f (x) and D(s n , x) w -→ g(x)
for all x ∈ G, when n -→ +∞. We prove that (z n s n ) is a Cauchy sequence and that D(z n s n , x) w -→ (f ⊙ g)(x) for all x ∈ G, when n -→ +∞. Indeed, using the invariance of D we have, for all n, p ∈ N,

D(s n , s p ) ⋆ D(z n , z p ) = D(z n s n , z n s p ) ⋆ D(z n s p , z p s p ) ≤ D(z n s n , z p s p ).
This implies that (z n s n ) is a Cauchy sequence. Let us set h x (y) := g(y -1 x) for all x, y ∈ G. The sequence (xs -1 n ) is also a Cauchy sequence and we have D(y, xs

-1 n ) = D(s n , y -1 x)
w -→ g(y -1 x) = h x (y) for all x, y ∈ G, when n -→ +∞. Using Lemma 2 with f and h x for each x ∈ G, we have that

f ⊙ g(x) = sup y∈G f (y) ⋆ h x (y) = lim n D(z n , xs -1 n ) = lim n D(z n s n , x). (11)
Hence, (z n s n ) is a Cauchy sequence and D(z n s n , x)

w -→ f ⊙ g(x)
for all x ∈ G, when n -→ +∞. This shows that f ⊙ g ∈ Π(G). Now let f ∈ Π(G) and let us prove that there exists g ∈ Π(G) such that f ⊙ g = g ⊙ f = δ e . From the definition of Π(G) there exists a Cauchy sequence

(z n ) ⊂ G such that D(z n , x) w -→ f (x) for all x ∈ G, in particular f (z n ) w -→ H 0 , when n -→ +∞. Since the probabilistic metric D is invariant, then (z -1
n ) ⊂ G is also a Cauchy sequence. By Lemma 1, there exists a probabilistic 1-Lipschitz map g such that

D(z -1 n , x) w -→ g(x)
for all x ∈ G, when n -→ +∞. In particular g ∈ Π(G). We show that f ⊙ g = δ e . Indeed, on one hand we have for all x, y ∈ G

D(z n , y) ⋆ D(z -1 n , y -1 x) = D(e, yz -1 n ) ⋆ D(yz -1 n , x) ≤ D(e, x) = δ e (x)
. Thus, by using Proposition 4 we have that for all x, y ∈ G, f (y) ⋆ g(y -1 x) ≤ δ e (x) and hence f ⊙ g(x) ≤ δ e (x). On the other hand, we know that

f (z n ) w -→ H 0 , when n -→ +∞. Also we have that g(z -1 n x) w -→ δ e (x), when n -→ +∞. Indeed, lim n g(z -1 n x) = lim n (lim p D(z -1 p , z -1 n x)) = lim n (lim p D(z n z -1 p , x)) = D(e, x) = δ e (x). Since f (z n ) ⋆ g(z -1 n x) ≤ f ⊙ g(x)
, then by using Proposition 4 we get that δ e (x) ≤ f ⊙ g(x) for all x ∈ G. Finally, we proved that f ⊙ g = δ e and in a similar way we have that g ⊙ f = δ e . Now, let us show that D is invariant. Indeed, let f, g, h ∈ Π(G). So, there exists (z n ), (s n ), (r n ) ⊂ G Cauchy sequences such that D(z n , x)

w -→ f (x), D(s n , x) w -→ g(x)
and D(r n , x) w -→ h(x) , for all x ∈ G, when n -→ +∞. We proved above that f ⊙ h, g ⊙ h ∈ Π(G) and D(z n r n , x) w -→ f ⊙ h(x) and D(s n r n , x) w -→ g ⊙ h(x) for all x ∈ G, when n -→ +∞. From Lemma 2 and the invariance of D, we have that

D(f ⊙ h, g ⊙ h) = lim n D(z n r n , s n r n ) = lim n D(z n , s n ) = D(f, g). (12)
In a similar way we prove that D(h ⊙ f, h ⊙ g) = D(f, g). This complete the proof.

In the following corollary, we give as consequence the completion of probabilistic metric space (a result proved by H. Sherwood in [START_REF] Sherwood | Complete probabilistic metric spaces Z. Wahrscheinlichkeitstheorie und Verw[END_REF]) and also give the new result concerning the completion of probabilistic invariant metric group.

Corollary 1. Every probabilistic metric space (resp. Every probabilistic invariant metric group) (G, D, ⋆), where ⋆ is a continuous triangle function, has a probabilistic metric completion (resp. has a probabilistic invariant metric group completion). More precisely, the space (Π(G), D, ⋆) (resp. the group (Π(G), ⊙, D, ⋆)) is the probabilistic completion (resp. the probabilistic group completion) of (G, D, ⋆).

Proof. We know from Theorem 3 (resp. from Theorem 4) that δ : (G, D, ⋆) -→ (Π(G), D, ⋆) is an isometric embedding (resp. an isometric homomorphism embedding). We also know that (Π(G), D, ⋆) is complete and that δ(G) is dense in (Π(G), D, ⋆). To finish the proof, it remains to prove the uniqueness of the completion which is a classical and known fact. Suppose that (X, Λ, ⋆) is a probabilistic complet metric space (resp. a probabilistic invariant complete metric group) and ϕ : G -→ X is an isometry (resp. an isometric group homomorphism) such that ϕ(G) is dense in X. Let f ∈ Π(G), then there exists a Cauchy sequence (a n ) ⊂ G such that D(δ an , f ) w -→ H 0 . Since (a n ) ⊂ G is a Cauchy sequence in G, then (ϕ(a n )) is a Cauchy sequence in (X, Λ, ⋆) which is complet. So there exists g ∈ X such that Λ(ϕ(a n ), g) w -→ H 0 . Now, we define the map T : Π(G) -→ X by T (f ) = g. It is a well known fact that this map is a well defined surjective isometry (resp. isometric isomorphism).

A Probabilistic Banach-Stone Type Theorem

In this section, we will show that Lip 1 ⋆ (G, ∆ + ) can be provided with a probabilistic metric D, such that (Lip 1 ⋆ (G, ∆ + ), ⊙, D, ⋆) will be a probabilistic complete metric monoid in which the probabilistic invariant metric group G embeds isimetricaly a group. Then, we caracterize the set of all invertible elements of (Lip 1 ⋆ (G, ∆ + ), ⊙, D, ⋆) (Theorem 5) and prove a Banach-Stone type theorem (Theorem 6).

The Probabilistic Complete Metric Space (Lip 1

⋆ (G, ∆ + ), D, ⋆). Given a probabilistic metric space (G, D, ⋆), we build a probabilistic metric D :

Lip 1 ⋆ (G, ∆ + )× Lip 1 ⋆ (G, ∆ + ) -→ ∆ + extending the probabilistic metric D defined on Π(G) × Π(G) as follows: D(f, g) =    D(f, g) if (f, g) ∈ Π(G) × Π(G) H 0 if f = g H ∞ if f = g; (f, g) ∈ [Π(G) × Π(G)] C
where, the set

[Π(G) × Π(G)] C := (Lip 1 ⋆ (G, ∆ + ) × Lip 1 ⋆ (G, ∆ + )) \ (Π(G) × Π(G)) denotes the complement of Π(G) × Π(G) in Lip 1 ⋆ (G, ∆ + ) × Lip 1 ⋆ (G, ∆ + ).
The interest of the probabilistic metric D is that this allows to keep the distances between the points of G after embedding it in the space Lip 1 ⋆ (G, ∆ + ) via the map δ. This will play an important role in Theorem 6. Proposition 9. Let (G, D, ⋆) be a probabilistic metric space such that ⋆ is continuous. Then, we have

(1) the space (Lip 1 ⋆ (G, ∆ + ), D, ⋆) is a probabilistic complete metric space in which, the probabilistic metric space (G, D, ⋆) embeds isometrically via the map

δ : (G, D, ⋆) -→ (Lip 1 ⋆ (G, ∆ + ), D, ⋆). (2) if moreover G is a probabilistic invariant metric group and ⋆ is sup-continuous, the map δ : (G, •) -→ (Lip 1 ⋆ (G, ∆ + ), ⊙) is a group homomorphism Proof. It suffices to prove that (Lip 1 ⋆ (G, ∆ + ), D, ⋆
) is a probabilistic complete metric space. The fact that δ : (G, D, ⋆) -→ (Lip 1 ⋆ (G, ∆ + ), D, ⋆) is an isometry (resp. an isometric homomorphism) is given by Theorem 3 (resp. by Theorem 4).

Step 1: we show that D is a probabilistic metric on Lip

1 ⋆ (G, ∆ + ). (i) Let f, g ∈ Lip 1 ⋆ (G, ∆ + ). Then, D(f, g) = H 0 if and only if f = g since D is a probabilistic metric on Π(G). (ii) it is clear that D(f, g) = D(g, f ) for all f, g ∈ Lip 1 ⋆ (G, ∆ + ). (iii) let f, g, h ∈ Lip 1
⋆ (G, ∆ + ) and let us prove that D(f, g) ⋆ D(g, h) ≤ D(f, h). This is clear in the case where (f, g), (g, h) ∈ Π(G) × Π(G) since D is a probabilistic metric. This is also clear in the case where (f, g), (g, h)

∈ [Π(G) × Π(G)] C since H ∞ ⋆ H ∞ ≤ H 0 ⋆ H ∞ = H ∞ . Suppose now that (f, g) ∈ Π(G) × Π(G) and (g, h) ∈ [Π(G) × Π(G)] C . In this case, also (f, h) ∈ [Π(G) × Π(G)] C , thus D(f, g) ⋆ D(g, h) = D(f, g) ⋆ H ∞ ≤ H 0 ⋆ H ∞ = H ∞ = D(f, h).
Step 2: We show that (Lip

1 ⋆ (G, ∆ + ), D, ⋆) is complete. Indeed, let (f k ) ⊂ Lip 1 ⋆ (G, ∆ + ) be a Cauchy sequence. Case 1: Threre exists N ∈ N such that for all k ≥ N , f k ∈ Lip 1 ⋆ (G, ∆ + ) \ Π(G).
In this case, the sequence (f k ) is necessarily a stationary sequence and so converges.

Case 2: There exists a subsequence (f

ϕ(k) ) ⊂ Π(G). In this case, since (Π(G), D, ⋆) is complete, there exists f ∈ Π(G) such that D(f ϕ(k) , f ) w -→ H 0 . So, using the fol- lowing inequality D(f k , f ϕ(k) ) ⋆ D(f ϕ(k) , f ) = D(f k , f ϕ(k) ) ⋆ D(f ϕ(k) , f ) ≤ D(f k , f ),
and taking the limit when k -→ +∞, we get that D(f k , f ) w -→ H 0 .

The Group of Units of the Monoid (Lip 1

⋆ (G, ∆ + ), ⊙). We are going to caracterize the set of all invertible elements of the monoid (Lip 1 ⋆ (G, ∆ + ), ⊙). The main result of this section Theorem 5 will be used to obtain a probabilistic Banach-Stone type theorem.

Let U(∆ + ) (resp. U(Lip 1 ⋆ (G, ∆ + ))) denotes the group of all invertible element of (∆ + , ⋆) (resp. of (Lip

1 ⋆ (G, ∆ + ), ⊙)). Recall that if f ∈ Lip 1 ⋆ (G, ∆ + ) and F ∈ ∆ + , then f, F (x) := f (x) ⋆ F for all x ∈ G. Recall also that G(G) := {δ a /a ∈ G}. By Π(G), U(∆ + ) , we denote the following subgroup of (Lip 1 ⋆ (G, ∆ + ), ⊙) Π(G), U(∆ + ) := { f, U /f ∈ Π(G) and U ∈ U(∆ + )}. Note that if U(∆ + ) is trivial, that is U(∆ + ) = {H 0 }, then Π(G), U(∆ + ) = Π(G).
The symbol " ∼ =" means "isometrically isomorphic". Remark 1. Let ⋆ be a triangle function on ∆ + such that U(∆ + ) = {H 0 } (see Lemma 4). We equip ∆ + with the following probabilistic discret metric:

D(K, L) = H 0 if K = L H ∞ otherwise
Then, (∆ + , ⋆, D) ∼ = (Lip 1 ⋆ ({H 0 }, ∆ + ), ⊙, D). Lemma 3. Let (G, D, ⋆) be a probabilistic metric space (not necessarily complete). Let (z n ) ⊂ G be a Cauchy sequence and f : G -→ ∆ + be a probabilistic 1-Lipschitz map. Then, there exists F ∈ ∆ + such that f (z n ) w -→ F , when n -→ +∞.

Proof. Since (∆ + , w) is a compact metrizable space, it suffices to show that every (weak) convergent subsequence of (f (z n )) converges to the same limit. Indeed, let (f (z ϕ(n) ) and (f (z ψ(n) ) be two convergent subsequence and let F and K the respective (weak) limits. Since f is a probabilistic 1-Lipschitz map, then

D(z ϕ(n) , z ψ(n) ) ⋆ f (z ϕ(n) ) ≤ f (z ψ(n) ).
Using Proposition 4, and the fact that D(z ϕ(n) , z ψ(n) ) w -→ H 0 we get that F ≤ L. In a similar way, we have that L ≤ F . Theorem 5. Let ⋆ be a continuous and sup-continuous triangle function. Let (G, •, D, ⋆) be an invariant probabilistic metric group (not necessarily complete). Then, we have,

U(Lip 1 ⋆ (G, ∆ + )) = Π(G), U(∆ + ) . Moreover, ( 1
) for all f ∈ Π(G) and U ∈ (U(∆ + ), ⋆), we have f, U -1 = f -1 , U -1 . (2) if U(∆ + ) = {H 0 }, then (U(Lip 1 ⋆ (G, ∆ + )), ⊙, D, ⋆) = (Π(G), ⊙, D, ⋆). (3) if U(∆ + ) = {H 0 } and G is complete, then (U(Lip 1 ⋆ (G, ∆ + )), ⊙, D, ⋆) = (G(G), ⊙, D, ⋆) ∼ = (G, •, D, ⋆).
Proof. We first prove that { f, U /f ∈ Π(G), U ∈ U(∆ + )} ⊂ U(Lip 1 ⋆ (G, ∆ + )). Indeed, using the definition of the operation ⊙ and the sup-continuity of ⋆ , it is easy to see that f, U ⊙ g, V = f ⊙ g, U ⋆ V for all f, g ∈ Π(G) and all U, V ∈ U(∆ + ). Since Π(G) is a group by Theorem 4, f, U ⊙ f -1 , U -1 = δ e , H 0 = δ e for all f ∈ Π(G) and all U ∈ U(∆ + ). Hence, f, U -1 = f -1 , U -1 . Now, we are going to prove the converse. Indeed, let f ∈ U(Lip 1 ⋆ (G, ∆ + )). There exists g

∈ U(Lip 1 ⋆ (G, ∆ + )) such that f ⊙ g = δ e (13)
In particular, we have f ⊙ g(e) = δ e (e) = H 0 . Thus, we obtain (f ⊙ g)(e)(t) = 0, for all t ≤ 0 (f ⊙ g)(e)(t) = 1, for all t > 0 So, on one hand, we have that for all z ∈ G and for all t ≤ 0 (f (z) ⋆ g(z -1 ))(t) = 0 [START_REF] Menger | Probabilistic geometry[END_REF] On the other hand, we have that for all t > 0 sup z∈G {(f (z) ⋆ g(z -1 ))(t)} = 1 [START_REF] Menger | Géométrie générale[END_REF] For all n ∈ N, by taking t = 1 n in [START_REF] Menger | Géométrie générale[END_REF], there exists (z n ) ⊂ G such that for all n ∈ N * 1 -

1 n < (f (z n ) ⋆ g(z -1 n ))( 1 n ) ≤ 1
Since f (z n ) ⋆ g(z -1 n ) ∈ ∆ + is increasing then, for all t > 0 there exists N t ∈ N such that for all n ≥ N t we have 1 -

1 n < (f (z n ) ⋆ g(z -1 n ))(t) ≤ 1 (16)
From ( 14) and ( 16) we have that for all t ∈ R lim n→+∞

(f (z n ) ⋆ g(z -1 n ))(t) = H 0 (t) (17)
From the definition of f ⊙ g, we get that for all x, y ∈ G f (x) ⋆ g(y -1 ) ≤ (f ⊙ g)(xy -1 ) (

Using [START_REF] Schweizer | Espaces metriques aleatoires[END_REF], we get that (f (z n ) ⋆ g(z -1 p )) ⋆ (f (z p ) ⋆ g(z -1 n )) ≤ (f ⊙ g)(z n z -1 p ) ⋆ (f ⊙ g)(z p z -1 n ) By the commutativity of the operation ⋆, the invariance of D and (13), we obtain (f (z n ) ⋆ g(z -1 n )) ⋆ (f (z p ) ⋆ g(z -1 p )) ≤ δ e (z n z -1 p ) ⋆ δ e (z p z -1 n ) = D(z n , z p ) ⋆ D(z n , z p ).

(

) 19
Using the fact that H 0 is the maximal element of ∆ + , we get that for all n, p ∈ N *

From ( 19), [START_REF] Schweizer | Sklar Triangle Inequalities in a Class of Statistical Metric Spaces[END_REF], we get (f (z n ) ⋆ g(z -1 n )) ⋆ (f (z p ) ⋆ g(z -1 p )) ≤ D(z n , z p ) ≤ H 0 .

Using the formulas ( 17) and ( 21) and the continuity of ⋆, we obtain lim n,p D(z n , z p ) = H 0 .

Hence, the sequence (z n ) ⊂ G is a Cauchy sequence and also (z -1 n ) ⊂ G is a Cauchy sequence since (G; D, ⋆) is a probabilistic invariant metric group. By Lemma 1 there exists l f , l g ∈ Π(G), 1-Lipschitz maps such that D(z n , x) w -→ l f (x) and D(z -1 n , x) w -→ l g (x) for all x ∈ G, when n -→ +∞. By Lemma 3, there exists L f , K g ∈ ∆ + such that f (z n ) w -→ L f and g(z -1 n )

w -→ K g . Using the continuity of the law ⋆ in the formula [START_REF] Schweizer | Probabilistic metric spaces[END_REF], we get that L f ⋆ K g = H 0 . So, L f , K g ∈ (U(∆ + ), ⋆) and K g = L -1 f . Using the definition of f ⊙ g and the fact that f ⊙ g = δ e , we have for all

x ∈ G f (x) ⋆ g(z -1 n ) ≤ f ⊙ g(xz -1 n ) = δ e (xz -1 n ) = D(x, z n ) Using Proposition 4, f (x) ⋆ K g ≤ l f (x) Thus, f (x) ≤ l f (x) ⋆ L f .
On the other hand, since f is 1-Lipschitz, then D(x, z n ) ⋆ f (z n ) ≤ f (x) and so using Proposition 4 we obtain that l f (x) ⋆ L f ≤ f (x). Hence, f (x) = l f (x) ⋆ L f =: l f , L f (x) for all x ∈ G. Thus, f = l f , L f with l f ∈ Π(G) and L f ∈ (U(∆ + ), ⋆). This finish the proof of the converse.

The parts (2) and ( 3) are easy consequences.

5.3. Banach-Stone Type Theorem. We obtain the following probabilistic Banach-Stone type theorem. Note that a general framework where the group of invertible elements of (∆ + , ⋆) is trivial (i.e. U(∆ + ) = {H 0 }) is given in Lemma 4 below. The following theorem shows that the structure of the probabilistic invariant complete metric group (G, •, D, ⋆) such that the group of invertible elements of (∆ + , ⋆) is trivial, is completely determined by the probabilistic metric monoid structure of Lip 1 ⋆ (G, ∆ + ). This is the case in particular for every invariant complete Menger groups (Corollary 2). Theorem 6. Let ⋆ be a continuous and sup-continuous triangle function such that the group of invertible elements of (∆ + , ⋆) is trivial. Let (G, •, D, ⋆) and (G ′ , •, D ′ , ⋆) be two probabilistic invariant complete metric groups. Then, the following assertions are equivalent.

(1) (G, •, D, ⋆) and (G ′ , •, D ′ , ⋆) are isometrically isomorphic as groups, (2) (Lip 1 ⋆ (G, ∆ + ), D, ⊙) and (Lip 1 ⋆ (G ′ , ∆ + ), D, ⊙) are isometrically isomorphic as monoids.

Proof. Let L, K ∈ ∆ + be such that L ⋆ T * K = H 0 . Then, for all t ∈ R, inf s+u=t T * (L(s), K(u)) = H 0 (t).

In other words, for all t ∈ R sup s+u=t T (1 -L(s), 1 -K(u)) = 1 -H 0 (t).

In particular, for each n ∈ N * , sup

s+u= 1 n T (1 -L(s), 1 -K(u)) = 0 (24)
Thus, from [START_REF] Sherwood | Complete probabilistic metric spaces Z. Wahrscheinlichkeitstheorie und Verw[END_REF], for all n ∈ N * ,

T (1 -L( 1 n ), 1 -K(0)) = 0.
Since K(0) = 0, using the fact that T (x, 1) = x for all x ∈ [0, 1], we obtain that L( 1 n ) = 1 for all n ∈ N * . Since L is nondecreasing, we get that for all t > 0, L(t) = 1. So that L = H 0 and so also K = H 0 .

Remark 2. Note that ⋆ T * is not sup-continuous and therefore not used in this paper.

Proposition 3 .

 3 Let (G, D, ⋆) be a probabilistic metric space. Suppose that ⋆ is sup-continuous. Let A be a nonempty subset of G. Then the map D(x, A) : x → sup y∈A D(x, y) is a probabilistic 1-Lipschitz map. Proof. Using the sup-continuity of ⋆, we have for all a, b ∈ G D(a, b) ⋆ D(a, A) := D(a, b) ⋆ sup y∈A D(a, y) = sup y∈A (D(a, b) ⋆ D(a, y) ≤ sup y∈A D(b, y) = D(b, A).

Proposition 6 .

 6 Let (G, •, D, ⋆) be an invariant probabilistic metric group. Then, we have that δ a ⊙ δ b = δ ab for all a, b ∈ G.Proof. We first prove that δ e ⊙δ e = δ e , where e is the identity element of G. Indeed, for all x ∈ G, (δ e ⊙ δ e )(x) := sup y,z∈G/yz=x D(y, e) ⋆ D(z, e) = sup y,z∈G/yz=x D(y, e) ⋆ D(e, z -1 ) by the invariance of D ≤ sup y,z∈G/yz=x D(y, z -1 ) by the property (iii) of D = sup y,z∈G/yz=x D(yz, e) by the invariance of D = D(x, e). = δ e (x) On the other hand, (δ e ⊙ δ e )(x) := sup y,z∈G/yz=x D(y, e) ⋆ D(z, e) ≥ D(x, e) ⋆ D(e, e) = D(x, e) ⋆ H 0 = D(x, e) = δ e (x)

Theorem 3 .

 3 Let (G, D, ⋆) be any probabilistic metric space, where ⋆ is a continuous triangle function. Let D : Π(G) × Π(G) -→ ∆ + be the map defined byD(f, g) = sup x∈G f (x) ⋆ g(x) Then,(1) for all a, b ∈ G, D(δ a , δ b ) = D(a, b).

Lemma 2 .

 2 Let (G, D, ⋆) be a probabilistic metric space such that ⋆ is continuous. Let f, g ∈ Π(G) and let (a n ), (b n ) ⊂ G be any Cauchy sequences such that D(a n , x) w -→ f (x), and D(b n , x) w -→ g(x) for all x ∈ G, when n -→ +∞. Then, lim n,m D(a n , b m ) exists and is independent from the choice of the Cauchy sequences (a n ) and (b n ). More precisely, we have that

  n,m D(a n , b m ) = lim n D(a n , b n ) by using the fact that (a n ), (b n ) ⊂ G are Cauchy sequences. This finish the proof. Now, we give the proof of Theorem 3. Proof of Theorem 3. (1) Let a, b ∈ G. We have D(δ a , δ b ) = sup x∈G δ a (x) ⋆ δ b (x)

D

  (z n , z p ) ⋆ D(z n , z p ) ≤ D(z n , z p ) ⋆ H 0 = D(z n , z p ) ≤ H 0 .

Proof. (1) =⇒[START_REF] Bachir | The Inf-Convolution as a Law of Monoid. An analogue to the Banach-Stone theorem[END_REF]. Suppose that I : G -→ G ′ is a group isomorphism such that, for all x, y ∈ G, D ′ (I(x), I(y)) = D(x, y). Then, it is easy to see that the map defined by Φ(f) := f • I -1 for all f ∈ Lip 1 ⋆ (G, ∆ + ) maps the space Lip 1 ⋆ (G, ∆ + ) into the space Lip 1 ⋆ (G ′ , ∆ + ) and satisfies Φ(f ⊙ g) = Φ(f ) ⊙ Φ(g) for all f, g ∈ Lip 1 ⋆ (G, ∆ + ). So Φ is a monoid isomorphism. Also we can easly see that D(Φ(f ), Φ(g)) = D(f, g) for all f, g ∈ Lip 1⋆ (G, ∆ + ), so Φ is an isometry. (2) =⇒[START_REF] Alsina | On the definition of a probabilistic normed space[END_REF]. Suppose that Φ :Lip 1 ⋆ (G, ∆ + ) -→ Lip 1 ⋆ (G ′ , ∆ +) is an isomorphism of monoids such that D(Φ(f ), Φ(g)) = D(f, g) for all f, g ∈ Lip 1

⋆ (G, ∆ + ). Since an isomorphism of monoids send the group of invertibles elements on the group of invertible elements, then by using Theorem 5, we obtain that the restriction Φ | G(G),U (∆ + ) : G(G), U(∆ + ) -→ G(G ′ ), U(∆ + ) of Φ is an isometric isomorphism. Since U(∆ In the following lemma, we prove that the set of all invertible element of (∆ + , ⋆) is trivial (i.e. U(∆ + ) = {H 0 }) in the case where the law ⋆ = ⋆ T is defined for all L, K ∈ ∆ + and for all t ∈ R by:

where, T :

The distribution H 0 is the only one invertible element of the monoid

In particular, for each n ∈ N * , sup

Thus, from [START_REF] Sherwood | On E-Spaces and their Relation to Other Classes of Probabilistic Metric Spaces[END_REF], there exists a sequence (s n ) of real numbers such that

Using the monotonocity of T and the fact that T (x, y) ≤ T (1, y) = y ≤ 1 for all x, y ∈ [0, 1], we obtain for all t > 0, there exists N t ∈ N such that for all n ≥ N t we have

Since T (0, x) = T (x, 0) = 0 for all x ∈ [0, 1], we deduce form the above inequality that necessarily 0 < s n < 1 n for all n ∈ N * (since, L(s n ) and K( 1 n -s n ) must be different from 0). Thus, s n ց 0 and for all t > 0, there exists

Since K is left-continuous, we get that K(t) = 1 for all t > 0. On the other hand, we know that K(t) = 0 for all t ≤ 0. It follows that K = H 0 and so also L = H 0 .

In consequence, we have U(∆ + ) = {H 0 }.

We obtain in the following corollary the result announced in the abstract. In other words, let T be a left-continuous t-norm then, every probabilistic invariant complete Menger group (G, •, D, T ) is completely determined by the probabilistic metric monoid structure of (Lip 1 ⋆ (G, ∆ + ), D, ⊙).

Corollary 2. Let T be a left-continuous t-norm and (G, •, D, T ), (G ′ , •, D ′ , T ) be two probabilistic invariant complete Menger groups. Then, the following assertions are equivalent.

(1) (G, •, D, T ) and (G ′ , •, D ′ , T ) are isometrically isomorphic as Menger groups (2) (Lip 1 ⋆ (G, ∆ + ), ⊙, D, ⋆) and (Lip 1 ⋆ (G ′ , ∆ + ), ⊙, D, ⋆) are isometrically isomorphic as probabilistic metric monoids.

Proof. The proof is a direct consequence of Theorem 6 and Lemma 4. Also, we have that U(∆ + ) = {H 0 } if ⋆ = ⋆ T * is defined for all L, K ∈ ∆ + and for all t ∈ R by:

where, T : [0, 1] × [0, 1] -→ [0, 1] is a t-norm and T * (x, y) := 1 -T (1 -x, 1 -y) for all x, y ∈ [0, 1] (T * is a dual function of T called t-conorm).

Proposition 10. The distribution H 0 is the only one invertible element of the monoid (∆ + , ⋆ T * ).


Sorin Olaru
email: sorin.olaru@supelec.fr

Didier Dumur
email: didier.dumur@supelec.fr

Simona Dobre
email: simona.dobre@cran.uhp-nancy.fr

EXPLICIT PREDICTIVE CONTROL LAWS On the geometry of feasible domains and the presence of nonlinearities

Keywords: Predictive control, parameterized polyhedra, explicit control laws

This paper proposes a geometrical analysis of the polyhedral feasible domains for the predictive control laws under constraints. The fact that the system dynamics influence the topology of such polyhedral domains is well known from the studies dedicated to the feasibility of the control laws. Formally the system state acts as a vector of parameters for the optimization problem to be solved on-line and its influence can be fully described by the use of parameterized polyhedra and their dual constraints/generators representation. Problems like the constraints redundancy or the construction of the associated explicit control laws at least for linear or quadratic cost functions can thus receive fully geometrical solutions. Convex nonlinear constraints can be approximated using a description based on the parameterized vertices. In the case of nonconvex regions the explicit solutions can be obtain by constructing Voronoi partitions based on a collection of points distributed over the borders of the feasible domain.

INTRODUCTION

The philosophy behind Model-based Predictive Control (MPC) is to exploit in a "receding horizon" manner the simplicity of the Euler-Lagrange approach for the optimal control. The control action u t for a given state x t is obtained from the control sequence k * u = [u T t , . . . , u T t+N-1 ] T as a result of the optimization problem:

min k u ϕ(x t+N ) + N-1 ∑ k=0 l(x t+k , u t+k ) subj. to : x t+1 = f (x t ) + g(x t )u t ; h(x t , k u ) ≤ 0 (1)
constructed for a finite prediction horizon N, cost per stage l(.), terminal weight ϕ(.), the system dynamics described by f (.), g(.) and the constraints written in a compact form using elementwise inequalities on functions linking the states and the control actions, h(.).

Unfortunately, the control sequence k * u is optimal only for a single initial condition -x t and produces an open-loop trajectory which contrasts with the need for a feedback control law. This drawback is over-come by solving the local optimization (1) for every encountered (measured) state, thus indirectly producing a state feedback law. The overall methodology is based on computationally tractable optimal control problems for the states found along the current trajectory. However, two important directions are to be studied in order to enlarge the class of systems which can take advantage of the MPC methodology. One is related to the fact that the measurements can be available faster than the optimal control sequence becomes available (as output of the optimization solver) and thus important information can be lost with irreversible consequences on the closeloop performances. Secondly, the lack of a closed form expression for the feedback law notifies about the difficulties that can be encountered when considering properties such as stability, typically established for regions in the state space.

For the optimization problem (1) within MPC, the current state serves as an initial condition and influences both the objective function and the feasible domain. Globally, from the optimization point of view, the system state can be interpreted as a vector of parameters, and the problems to be solved are part of the multiparametric optimization programming family. From the cost function point of view, the parametrization is somehow easier to deal with and eventually can be entirely translated towards the set of constraints to be satisfied (the MPC literature contains references to schemes based on suboptimality or even to algorithms restraining the demands to feasible solution of the receding horizon optimization [START_REF] Scokaert | Suboptimal model predictive control (feasibility implies stability)[END_REF]). Unfortunately, similar observation cannot be made about the feasible domain and its adjustment with respect to the parameters evolution.

The optimal solution is often influenced by the limitations, the process being forced to operate at the designed constraints for best performance. The distortion of the feasible domain during the parameters evolution will consequently affect the structure of the optimal solution. Starting from this observation the present paper focuses on the topological analysis of the domains described by the MPC constraints.

The structure of the feasible domain is depending on the model and the set of constraints taken into consideration in (1). If the model is a linear system, the presence of linear constraints on inputs and states can be easily expressed by a system of linear inequalities. In the case of nonlinear systems, these properties are lost and the domains are in general difficult to handle. However, there are several approaches to transform the dynamics to those of a linear system over the operating range as for example by piecewise linear approximation, feedback linearisation or the use of time-varying linear models.

As a consequence, specific attention for the linear constraints and the associated polyhedral feasible domains may be prolific. More than that, the use of polyhedral domains between the convex sets is not hazardous since they offer important advantages, like the closeness over the intersection or the fact that the polyhedral invariant sets (largely used for enforcing stability) are less conservative than the ellipsoidal ones for example. In the current paper, these polyhedral feasible domains will be analyzed with a focus on the parametrization leading to the concept of parameterized polyhedra (Olaru and Dumur, ):

min k u F(x t , k u ) subj. to :    A in k u ≤ b in + B in x t A eq k u = b eq + B eq x t h(x t , k u ) ≤ 0 (2)
where the objective function F(x t , k u ) is usually linear or quadratic. Secondly it will be shown that the optimization problem may take advantage during the real-time implementation either from the possible alleviation of the set of constraints for the on-line optimization routines either from the construction of the explicit solution on geometrical basis when possible. With these two aspects, one can consider that MPC awareness is improved both from the theoretical (insight on the global control law) and practical (computational aspects) point of view.

An important remark concerning the presence of nonlinearities in the constraints is that if the feasible domain remains convex, then an approximation in terms of parameterized polyhedra can lead to an approximate explicit solution in terms of piecewise linear control laws. But, if the feasible domain becomes non-convex due to the presence of nonlinearities, then in order to obtain the explicit solution some assumption have to be relaxed. A special role, in finding the explicit control laws in the nonlinear case, is played by the Voronoi partition.

In the following, Section 2 introduces the basic concepts related to the parameterized polyhedra and interprets the feasible domains of the receding horizon optimization problems [START_REF] Schrijver | Theory of Linear and Integer Programming[END_REF] in this context. Section 3 presents the use of the feasible domain analysis for the construction of the explicit solution for linear and quadratic objective functions. In Section 4 an extension to nonlinear type of constraints is addressed, simple examples illustrating the construction of explicit solutions.

PARAMETRIZATION OF POLYHEDRAL DOMAINS

Double representation

A mixed system of linear equalities and inequalities defines a polyhedron [START_REF] Motzkin | The Double Description Method[END_REF]. In the parameter free case, it is represented by the equivalent dual (Minkowski) formulation:

P = k u ∈ R p A eq k u = b eq ; A in k u ≤ b in ⇐⇒ P = conv.hullV + coneR + lin.spaceL generators (3)
where conv.hullV denotes the set of convex combinations of vertices V = {v 1 , . . . , v ϑ }, coneR denotes nonnegative combinations of unidirectional rays in R = r 1 , . . . , r ρ and lin.spaceL = {l 1 , . . . , l λ } represents a linear combination of bidirectional rays (with ϑ, ρ and λ the cardinals of the related sets). This dual representation [START_REF] Schrijver | Theory of Linear and Integer Programming[END_REF] in terms of genera-tors can be rewritten as:

P = k u ∈ R p | k u = ϑ ∑ i=1 α i v i + ρ ∑ i=1 β i r i + λ ∑ i=1 γ i l i ; 0 ≤ α i ≤ 1, ϑ ∑ i=1 α i = 1 , β i ≥ 0 , ∀γ i (4
) with α i , β i , γ i the coefficients describing the convex, non-negative and linear combinations in (3).

Numerical methods like the Chernikova algorithm [START_REF] Leverge | A note on chernikova's algorithm[END_REF] are implemented for constructing the double description, either starting from constraints (3) either from the generators (4) representation.

The parametrization

A parameterized polyhedron [START_REF] Loechner | Parameterized polyhedra and their vertices[END_REF] is defined in the implicit form by a finite number of inequalities and equalities with the note that the affine part depends linearly on a vector of parameters x ∈ R n for both equalities and inequalities:

P (x) = k u (x) ∈ R p A eq k u = B eq x + b eq ; A in k u ≤ B in x + b in } = k u (x)| k u (x) = ϑ ∑ i=1 α i (x)v i (x) + ρ ∑ i=1 β i r i + λ ∑ i=1 γ i l i 0 ≤ α i (x) ≤ 1, ϑ ∑ i=1 α i (x) = 1 , β i ≥ 0 , ∀γ i .
(5) This dual representation of the parameterized polyhedral domain reveals the fact that only the vertices are concerned by the parametrization (resulting the so-called parameterized vertices -v i (x)), whereas the rays and the lines do not change with the parameters' variation. In order to effectively use the generators representation in (5), several aspects have to be clarified regarding the parametrization of the vertices (see for exemple [START_REF] Loechner | Parameterized polyhedra and their vertices[END_REF] and the geometrical toolboxes like POLYLIB [START_REF] Wilde | A library for doing polyhedral operations[END_REF]). The basic idea is to identify the parameterized polyhedron with a non-parameterized one in an augmented space:

P = k u x ∈ R p+n | A eq -B eq k u x = b eq ; [ A in | -B in ] k u x ≤ b in (6)
The original polyhedron in (5) can be found for any particular value of the parameters vector x through P(x) = Proj k u P ∩ H(x) , for any given hy-

perplane H(x 0 ) = k u x ∈ R p+n | x = x 0 and
using Proj k u (.) as the projection from R p+n to the first p coordinates R p . Within the polyhedral domains P , the correspondent of the parameterized vertices in (5) can be found among the faces of dimension n. After enumerating these n-faces: F n 1 ( P ), . . . F n j ( P ), . . . , F n ς ( P ) , one can write: ∀i, ∃ j ∈ {1, . . . , ς} s.t. v i (x) T x T T ∈ F n j ( P ) or equivalently:

v i (x) = Proj k u F n j ( P ) ∩ H(x) (7)
From this relation it can be seen that not all the nfaces correspond to parameterized vertices. However it is still easy to identify those which can be ignored in the process of construction of parameterized vertices based on the relation: Proj x F n j ( P) < n with Proj x (.) the projection from R p+n to the last n coordinates R n (corresponding to the parameters' space). Indeed the projections are to be computed for all the n-faces, those which are degenerated are to be discarded and all the others are stored as validity domains -D v i ∈ R n , for the parameterized vertices that they are identifying:

D v i = Proj n F n j ( P) (8)
Once the parameterized vertices identified and their validity domain stored, the dependence on the parameters vector can be found using the supporting hyperplanes for each n-face:

v i (x) = A eq Āin j -1 B eq Bin j x + b eq bin j (9)
where Āin j , Bin j , bin j represent the subset of the inequalities, satisfied by saturation for F n j ( P). The inversion is well defined as long as the faces with degenerate projections are discarded.

The interpretation from the predictive control point of view

The double representation of the parameterized polyhedra offers a complete description of the feasible domain for the predictive control law as long as this is based on a multiparametric optimization with linear constraints.

Using the generators representation, with simple difference operations on convex sets one can compute the region of the parameters space where no parameterized vertex is defined:

ℵ = R n \ {∪D v i ; i = 1 . . . ϑ} (10)
representing from the MPC point of view, the set of infeasible states for which no control sequence can be designed due to the fact that the limitations are overly constraining. As a consequence the complete description of the infeasibility is obtained.

Remark: the presence of rays and lines in the set of generators doesn't imply that the infeasibility is avoided. The feasibility is strictly related with the existence of valid parameterized vertices for the given value of the parameter (state) vector.

The vertices of the feasible domain cannot be expressed as convex combinations of other distinct points and, due to the fact that from the MPC point of view, they represent sequences of control actions, one can interpret them in terms of extremal performances of the controlled system (for example in the tracking applications the maximal/minimal admissible setpoint [START_REF] Olaru | Compact explicit mpc with guarantee of feasibility for tracking[END_REF]).

TOWARDS EXPLICIT SOLUTIONS

In the case of sufficiently large memory resources, construction of the explicit solution for the multiparametric optimization problem can be an interesting alternative to the iterative optimization routines. In this direction recent results where presented at least for the case of linear and quadratic cost functions (see [START_REF] Seron | Characterisation of receding horizon control for constrained linear systems[END_REF], [START_REF] Bemporad | The Explicit Linear Quadratic Regulator for Constrained Systems[END_REF], [START_REF] Goodwin | Constrained Control and Estimation[END_REF], [START_REF] Borelli | Constrained Optimal Control of Linear and Hybrid Systems[END_REF], [START_REF] Tondel | Evaluation of piecewise affine control via binary search tree[END_REF]). In the following it will be shown that a geometrical approach based on the parameterized polyhedra can bring a useful insight as well.

Linear cost function

The linear cost functions are extensively used in connection with model based predictive control and especially for robust case ( [START_REF] Bemporad | Robust Model Predictive Control: Piecewise Linear Explicit Solution[END_REF], [START_REF] Kerrigan | Feedback minmax model predictive control using a single linear program: Robust stability and the explicit solution[END_REF]). In a compact form, the multiparametric optimization problem is:

k u * (x t ) = min k u f T k u subject to A in k u ≤ B in x t + b in (11)
The problem deals with a polyhedral feasible domain which can be described as previously in a double representation. Further the explicit solution can be constructed based on the relation between the parameterized vertices and the linear cost function (as in [START_REF] Leverge | A note on chernikova's algorithm[END_REF]). The next result resumes this idea.

Proposition: The solution for a multiparametric linear problem is characterized as follows:

a) For the subdomain ℵ ∈ R n where the associated parameterized polyhedron has no valid parameterized vertex the problem is infeasible; b) If there exists a bidirectional ray l such that f T l = 0 or a unidirectional ray r such that f T r ≤ 0, then the minimum is unbounded; c) If all bidirectional rays l are such that f T l = 0 and all unidirectional rays r are such that f T r ≥ 0 then there exists a cutting of the parameters in zones where the parameterized polyhedron has a regular shape j=1...ρ R j = R n -ℵ. For each region R j the minimum is computed with respect to the given linear cost function and for all the valid parameterized vertices:

m(x) = min f T v i (x)|v i (x) vertex of P (x) (12)
The minimum m(x) is attained by constant subsets of parameterized vertices of P (x) over a finite number of polyhedral zones in the parameters space R i j (∪R i j = R j ). The complete optimal solution of the multiparametric optimization is given for each R i j by:

S R i j (x) = conv.hull {v * 1 (x), . . . , v * s (x)} + + cone {r * 1 , . . . , r * r } + lin.spaceP(p) (13)
where v * i are the vertices corresponding to the minimum m(x) over R i j and r * i are such that f T r * i = 0 This result provides the entire family of solutions for the linear multiparametric optimization, even for the cases where this family is not finite (for example there are several vertices attaining the minimum).

Remark: For the regions of the parameters space characterized by the case (a), the set of constraints cannot be fulfilled and the feasible domain is empty.

Remark: If the solution of the optimization problem is characterized by the case (b), then the control law based on such an optimization is not well-posed as the optimal control action needs an infinite energy in order to be effectively applied.

Remark: Due to the fact that the parameterized vertices have a linear dependence on the parameter vector, the explicit solution will be piecewise linear. However, the solution is not unique as it can be seen from the case (c) and equation ( 13) and thus for the practical control purposes a continuous piecewise candidate is preferred, eventually by minimizing the number of partitions in the parameters space.

Quadratic cost function

The case of a quadratic const function is one of the most popular at least for the linear MPC. The explicit solution based on the exploration of the parameters space ( [START_REF] Bemporad | The Explicit Linear Quadratic Regulator for Constrained Systems[END_REF], [START_REF] Borelli | Constrained Optimal Control of Linear and Hybrid Systems[END_REF], [START_REF] Tondel | Evaluation of piecewise affine control via binary search tree[END_REF]) is extensively studied lately. Alternative methods based on geometrical arguments or dynamical programming ( [START_REF] Goodwin | Constrained Control and Estimation[END_REF], [START_REF] Seron | Characterisation of receding horizon control for constrained linear systems[END_REF]) improved also the awareness of the explicit MPC formulations. The parameterized polyhedra can serve as a base in the construction of such explicit solution (Olaru and Dumur, ), for a quadratic multiparametric problem:

k u * (x t ) = arg min k u k T u Hk u + 2k u T Fx t subject to A in k u ≤ B in x t + b in (14)
In this case the main idea is to consider the unconstrained optimum:

k sc u (x t ) = H -1
Fx t and its position with respect to the feasible domain given by a parameterized polyhedron as in ( 5).

If a simple transformation is performed:

ku = H 1/2 k u
then the isocost curves of the quadratic function are transformed from ellipsoid into circles centered in ksc u (x t ) = H -1/2 Fx t . Further one can use the Euclidean projection in order to retrieve the multiparametric quadratic explicit solution.

Indeed if the unconstrained optimum ksc u (x t ) is contained in the feasible domain P (x t ) then it is also the solution of the constrained case, otherwise existence and uniqueness are assured as follows:

Proposition: For any exterior point ku (x t ) / ∈ P (x t ), there exists an unique point characterized by a minimal distance with respect to ksc u (x t ). This point satisfies:

( ksc u (x t ) -k * u (x t )) T ( ku -k * u (x t )) 0, ∀ ku ∈ P (x t )
The construction mechanism uses the parameterized vertices in order to split the regions neighboring the feasible domain in zones characterized by the same type of projection.

Remark: The use of these geometrical arguments makes the construction of explicit solution to deal in a natural manner with the so-called degeneracy [START_REF] Bemporad | The Explicit Linear Quadratic Regulator for Constrained Systems[END_REF]. This phenomenon is identified by the parameters' values where the feasible domain changes its shape (the set of parameterized vertices is modified).

GENERALIZATION FOR NONLINEAR PROGRAMS

If the feasible domain is described by a mixed linear/nonlinear set of constraints then the convexity properties are lost and a procedure for the construction of exact explicit solutions do not exist for the general case.

Nonlinear constraints handling

As already mentioned, in dealing with the presence of nonlinearities constraints, a special case is when the associated feasible domain is convex. In this case, the main ideas in finding the explicit control laws are the following:

-considering the augmented space (formed by the extended arguments and parameters space ((x, k u )), find a set of points situated on the borders of the feasible domain (points that will correspond to the parameterized vertices in the associated linear feasible domain); -using this set of extremal points, construct the dual representation in terms of parameterized polyhedra (as a fact, in the presence of linear constraints, this set of points could represent the input to the linear algorithm, as well as a set of linear constraints);

-build the corresponding explicit solution by reporting the unconstrained optimum to these parameterized vertices and their validity domains.

Remark: The solution will be a continuous piecewise affine function in the state vector, due to the nature (convexity) of the feasible domain, and it is obtained by projecting the unconstrained optimum on the linear subset of constraints (associated with the nonlinear ones). In this nonlinear convex case, the precision of the solution is directly dependent of the linearization of the nonlinear constraints using a finite set of points on the frontier of the feasible domain (knowing that the rest of the algorithm retains the qualities of the linear algorithm).

Simple example of a multiparametric nonlinear program

Consider the discrete-time linear system:

x t+1 = 0.9 1 0 1

x t + 1 -1 u t ( 15
)
and a predictive control law with a prediction horizon of three sampling times and a control horizon of two steps. A nonlinear set of constraints will be also considered:

   ∑ 2 k=0 u 2 t+k ≤ 1 ∑ 2 k=0 u 2 t+k ≤ ln( 0 1 x t + 1) 0 1 x t+k ≥ 0; k = 0, 1, 2 (16)
It is obvious that the topology of the feasible domain is changing with the system dynamics, which means that the state vector represents in fact a parameter. More precisely, in our case only the second component of the state, x t is influencing the shape of the feasible domain and thus one can draw this dependence on the parameter as in figure 2a. Further this parameterized convex shape can be approximated with a set of parameterized linear inequalities and obtain a double description of a parameterized polyhedron as in figure 2b. A precutting in zones with regular shape (figure 2c) can help in the development of explicit solution due to the important degree of redundancy. Finally the nonlinear MPC law for the system (15) and the constraints ( 16) can be approximated by the explicit solution found in terms of a piecewise linear control law as in figure 3.

Nonconvex feasible domains

In the case when the convexity is lost, one can still construct the convex hull (and an explicit solution) associated with the non-convex domain. From the resulting piecewise affine function one should retain only those regions where the control law is feasible w.r.t. the nonlinear constraints. For the violated constraints a distribution of points will be obtained. Finally the missing regions in the explicit solution will be completed with the Voronoi partition corresponding to these points on the concave border of the feasible domain.

Algorithm:

1. Obtain a set of points (V ) on the frontier of the feasible domain D (based on the nonlinear constraints).

2. Considering this set of points, construct the convex C V (which will include the non-convex domain D and some other infeasible domains from the MPC point of view). The following notations where used:

Split the set

V as V L ∪ V NL ∪ V ∪ V • V ∈ Int(C V ) • V ∈ F(C V ) and C V = C V \ V • V L ∈ F(C V ), V L ∩ V = / 0
S = v ∈ V |∀x ∈ C V s.t. Sat( H , x) = K ; B(R NL , x) = Sat(R NL , v
F(X)
The frontier of a compact set X

Int(X)

The interior of a compact set X R L (D)

The set of linear constraints in the definition of the feasible domain D R NL (D)

The set of nonlinear constraints in the definition of the feasible domain D

Sat(R * , x)
The subset of constraints in R * (either R L either R NL ) saturated by the vector x

B(R * , x)
The subset of constraints in R * . violated by the vector x

Numerical example

Consider the MPC problem implemented using the first control action of the optimal sequence:

k * u = arg min k u N y -1 ∑ i=0
x

                                        x t+k+1|t = A 1 1 0 1 x t+k|t + B 1 0 2 1 u t+k|t k 0 -2 -2 u t+k|t 2 2 0 k N y -1 (u 1 t+k|t ) 2 + u 2 t+k|t -2 2 √ 3 0 k N y -1 (u 1 t+k|t ) 2 + u 2 t+k|t + 2 2 √ 3 0 k N y -1
u t+k|t = 0.59 0.76 -0.42 -0.16

K LQR x t+k|t N u k N y -1
One can observe the presence of both linear and nonlinear constraints. By following the previous algorithm, in the first stage, the partition of the state space is performed by considering only the linear constraints (figure 3).

Each such region correspond with a specific projection law. By simply verifying the regions where this projection law obey the nonlinear constraints, the exact part of the explicit solution is obtained (fig. 4).

Further, a distribution of points on the nonlinear frontier of the feasible domain has to be obtained and based on this distribution of points the associated Voronoi partition constructed. By superposing it to the regions not covered (white zones in the figure 4) at the previous step, one obtain a complete covering of the arguments space. Figure 5 depicts such a complete partition for distribution of 10 points for each nonlinear constraint.

By correspondence, the figure 6 describes the partition of the state space for the explicit solution.

Finally the complete explicit solution is depicted in figure 7. The discontinuities are observable in the regions generated upon the Voronoi partition. In order to give an image of the complexity it can mentioned that the explicit solutions contains 31 regions the computational effort was less than 2s mainly spent in the construction of the Voronoi partition.

CONCLUSION

The parameterized polyhedra offer a transparent characterization of the MPC degrees of freedom. Once the complete description of the feasible domain as a parameterized polyhedron is obtained explicit MPC laws can be constructed using the projection of the unconstrained optimum. The topology of the feasible domain can lead to explicit solution even if nonlinear constraints are taken into consideration. The price to be paid is found in the degree of suboptimality.

  Figure 1: (a) The nonlinear dependence of the feasible domain on the parameters (b) The approximation by a parameterized polyhedron; (c) Regions in the parameters' space corresponding to redundancy-free constraints sets.

Figure 2 :

 2 Figure 2: Explicit solution as a piecewise linear function.

  and V L saturate at least one linear constraint in the MPC constraints; • V NL ∈ F(C V ) and V NL saturate no linear constraint 4. C V is described in the dual representation by the intersection of halfspaces H (which represent the faces of the convex hull C V ). Split this set in H ∪ H • H ⊂ H such that ∃x ∈ C V with Sat( H , x) = / 0 and B(R NL , x) = / 0 • H = H \ H 5. Compute the unconstrained optimum k * u 6. Project the unconstrained optimum on C V : k * u ← Pro j C V {-c} 7. If k * u saturates a subset of constraints K ⊂ H (a) Retain the set of points:

  )} (b) Construct the Voronoi partition for the colection of points in S (c) Position k * u w.r.t. this partition and map the suboptimal solution k * u ← v where v is the vertex corresponding to the active region 8. If the quality of the solution is not satisfactory, improve the distribution of the points V by augmenting the resolution around k * u and restart from the step 2.

Figure 3 :

 3 Figure 3: Partition of the arguments space (linear constraints only).

Figure 4 :

 4 Figure 4: Retention of the regions with feasible linear projections.

Figure 5 :

 5 Figure 5: Partition of the arguments space (nonlinear case) -10 points per nonlinear constraint.

Figure 6 :

 6 Figure 6: Partition of the state space -10 points per nonlinear constraint.