Alexandre Seuret

Frédéric Gouaisbaut

Emilia Fridman

Stability of discrete-time systems with time-varying delays via a novel summation inequality

Keywords: Summation inequalities, stability analysis, discrete-timedelay systems, time-varying delay

This paper is concerned with the stability analysis of discrete linear systems with time-varying delays. The novelty of the paper comes from the consideration of a new inequality which is less conservative than the celebrated Jensen inequality employed in the context of discretetime delay systems. This inequality is a discrete-time counterpart of the Wirtinger-based integral inequality that was recently employed for the improved analysis of continuous-tine systems with delays. However, differently from the continuous-time case, the proof of the new inequality is not based on the Wirtinger inequality. The method is also combined with an efficient representation of the improved reciprocally convex combination inequality in order to reduce the conservatism induced by the LMIs optimization setup. The effectiveness of the proposed result is illustrated by some classical examples from the literature.

I. INTRODUCTION

The phenomenon of delays arises in many practical situations like in biology, economy or mechanical engineering (see for instance [START_REF] Gu | Stability of time-delay systems[END_REF]) and references therein) and its introduction in the model can lead to stability problems. This is the reason why a huge number of results is devoted to the development of criteria dedicated to the stability analysis of linear time-delay systems. When the delay is time-varying, two different methodologies have been employed. In the first framework, we aim at transforming the original system into a closed loop between a nominal LTI system and a system depending on the delay. This last element is embedded into an uncertainty and the use of classical tools like Small Gain Theorem [START_REF] Mirkin | Some remarks on the use of time-varying delay to model sample-and-hold circuits[END_REF], IQCs [START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF], [START_REF] Kao | On stability of discrete-time LTI systems with varying time delays[END_REF] or Quadratic Separation [START_REF] Ariba | Robust stability of timevarying delay systems: Quadratic separation approach[END_REF] allows to conclude on stability.

Another technique is based on the construction of Lyapunov-Krasovskii functionals (see for instance [START_REF] Fridman | Delay dependent stability and H∞ control: constant and time-varying delays[END_REF], [START_REF] Fridman | Stability and guaranteed cost control of uncertain discrete delay systems[END_REF]). A first method for continuous-time delay systems is based on the so-called Complete functionals, [START_REF] Kharitonov | Lyapunov-Krasovskii approach to robust stability of time delay systems[END_REF], where the Lyapunov-Krasovskii functional derivative is well given and then the functional is constructed, provides necessary and sufficient stability conditions for linear systems with constant delays but hardly extendable to time-varying delays. A second popular approach relies on the a priori construction of a Lyapunov-Krasovskii functional. In this case, the design of such Lyapunov functionals is still an open problem and it is generally based on a number of steps, which somehow bring some conservatism. The first step is concerned with the choice of the Lyapunov-Krasovskii functionals. Some Lyapunov functionals are based on double or triple integral/summation terms, extended state functionals ( [START_REF] Ariba | Robust stability of timevarying delay systems: Quadratic separation approach[END_REF], [START_REF] Kim | Note on stability of linear systems with time-varying delay[END_REF]), or discretization of the state ( [START_REF] Gu | Stability of time-delay systems[END_REF], [START_REF] Peng | Improved delay-dependent stability criteria for discrete-time systems with a new finite sum inequality[END_REF]). Naturally, this selection entails an important conservatism. Once the increment of the Lyapunov functional calculated, the second step comes from the inequalities to be used to derive a tractable numerical optimization problem (see [START_REF] He | Further improvement of freeweighting matrices technique for systems with time-varying delay[END_REF], [START_REF] Park | Reciprocally convex approach to stability of systems with time-varying delays[END_REF], [START_REF] Shao | New delay-dependent stability criteria for systems with interval delay[END_REF], [START_REF] Sun | Improved delay-range-dependent stability criteria for linear systems with time-varying delays[END_REF]). In this spirit, and in the continuous-time case, all the cross terms of the derivative of V are bounded using some integral inequalities like the Jensen [START_REF] Gu | Stability of time-delay systems[END_REF], the Wirtinger [START_REF] Seuret | Wirtinger-based integral inequality: Application to time-delay systems[END_REF], [START_REF] Seuret | Stability of systems with fastvarying delay using improved Wirtinger's inequality[END_REF] or the Bessel inequalities [START_REF] Seuret | Complete quadratic Lyapunov functionals using Bessel-Legendre inequality[END_REF].

The objective of the paper is to derive simple but efficient stability conditions discrete-time time-varying delay systems. More specifically, unlike many papers in the literature, we aim at studying how to reduce the conservatism of the last stage of the procedure by the development of a less conservative summation inequality. To do so, this contribution extends [START_REF] Seuret | Wirtinger-based integral inequality: Application to time-delay systems[END_REF], where integral inequalities have been provided for the stability analysis of continuous-time systems with delays, in order to provide a new summation inequality and an associated stability analysis for discrete-time systems with interval time-varying delays. More precisely, by an extensive understanding of the integral inequality from [START_REF] Seuret | Wirtinger-based integral inequality: Application to time-delay systems[END_REF], we provide a new summation inequality, which is less conservative than the Jensen inequality. It is also worth mentioning that the resulting integral and summation inequalities have similar interpretations. Guided by this inequality, suitable Lyapunov-Krasovskii functionals are proposed. Finally the theoretical results are tested on an academic example and on a networked control system example.

The paper is organized as follows. Section II introduces the new summation inequality and a preliminary result on reciprocally convex inequality that will be employed. Section III presents the main results of the paper on stability of nominal and uncertain systems with timevarying delays. Section IV illustrates our results with some examples extracted from the literature.

Notations: Throughout the paper, Z denotes the set of integers, R n the n-dimensional Euclidean space with vector norm | • |, R n×m the set of all n × m real matrices. For any symmetric matrix P ∈ R n×n , the notation P 0 (or P ≺ 0) means that P is positive (or negative) definite. The set S + n refers to the set of symmetric positive definite matrices. For any matrices A, B in R n×n , the notation diag(A, B) denotes the block diagonal matrix [ A 0 0 B ]. For any square matrix, He(A) stands for A + A T .

II. A NOVEL SUMMATION INEQUALITIES FOR DISCRETE-TIME DELAY SYSTEMS

A. Problem formulation

Consider a linear discrete system with time-varying delay of the form:

x

(k + 1) = Ax(k) + A d x(k -h(k)), ∀k ≥ 0, x(k) = φ(k), ∀k ∈ [-h2, 0], (1)
where x(k) ∈ R n is the state vector, φ is the initial condition and A, A d ∈ R n×n are constant matrices. The delay h is a positive integer which is assumed to be time-varying but satisfies, for some integers h2 ≥ h1 ≥ 1,

h(k) ∈ [h1, h2] , ∀t ≥ 0. (2)
Diverse types of functionals have been provided to assess stability of such class of systems. It has been shown in [START_REF] Fridman | Stability and guaranteed cost control of uncertain discrete delay systems[END_REF] that one of the most relevant terms has the following form

VZ (x k ) = 0 j=-h+1 k i=k+j y T (i)Zy(i), ( 3
)
where h is a positive integer, Z is a positive definite matrix, k is an integer representing the time variable, x k is the sequence of the state defined by x k (i) = x(k -i), for all i = -h, -h + 1, . . . , 0 and where y(i) = x(i) -x(i -1), for all integer i. Computing the forward difference leads to

∆VZ (x k ) := VZ (x k+1 ) -VZ (x k ) = hy T (k + 1)Zy(k + 1) - k i=k-h+1
y T (i)Zy(i).

(4) The main problem related to the equation (4) arises when including this negative term in order to derive LMI conditions. Thus, the trick is to apply the Jensen inequality, which unavoidably introduces some conservatism. In this paper, we aim at presenting a new summation inequality, which encompasses the Jensen inequality, to derive less conservative stability conditions for discrete linear systems with time-varying delays. To this aim, we take advantage of the integral inequality provided in [START_REF] Seuret | Wirtinger-based integral inequality: Application to time-delay systems[END_REF] to derive a new summation inequality.

B. A new summation inequality

In this section, a novel summation is provided following the idea of [START_REF] Seuret | Wirtinger-based integral inequality: Application to time-delay systems[END_REF] where a new integral inequality has been developed thanks to an extensive use of one version of the Wirtinger inequalities. This lemma is recalled here.

Lemma 1: [START_REF] Seuret | Wirtinger-based integral inequality: Application to time-delay systems[END_REF] For a given symmetric positive definite matrix R ∈ S + n , any differentiable function x in [-h, 0] → R n , the following inequality holds:

0 -h ẋT (θ)R ẋ(θ)dθ ≥ 1 h Ω 0 Ω 1 T R 0 0 3R Ω 0 Ω 1 , (5)
where

Ω0 = x(0) -x(-h), Ω1 = x(0) + x(-h) -2 h 0 -h x(θ)dθ.
The adaptation of this integral inequality into the discrete-time framework does not seem to be an easy task. Therefore, we present an alternative and simpler proof, which helps us deriving a novel summation inequality. We will also show that this summation inequality encompasses the Jensen inequality. The new proof of Lemma 1 is detailed in the following.

Proof :

For any differentiable function x in [-h, 0] → R n , consider the function z given by z(θ) = ẋ(θ) - 1 h Ω0 -3 (h + 2θ) h 2 Ω1, ∀θ ∈ [-h, 0].
Reinjecting the definition of z in 0 -h z T (θ)Rz(θ)dθ and developing the integral yields

0 -h z T (θ)Rz(θ)dθ = 0 -h ẋT (θ)R ẋ(θ)dθ + 1 h 2 0 -h 1dθΩ T 0 RΩ0 + 9 h 4 0 -h (h + 2θ) 2 dθΩ T 1 RΩ1 -2 h 0 -h ẋT (θ)dθRΩ0 -6 h 2 0 -h (h + 2θ) ẋT (θ)dθRΩ1 -6 h 3 0 -h (h + 2θ)dθΩ T 0 RΩ1. (6)
Basic integral calculus and an integration by parts ensure that

0 -h 1dθ = h, 0 -h (h + 2θ)dθ = 0, 0 -h (h + 2θ) 2 dθ = 1 3 h 3 , 0 -h ẋ(θ)dθ = Ω0, 0 -h (h + 2θ) ẋT (θ)dθ = 2hΩ1. (7)
It thus follows that

0 -h z T (θ)Rz(θ)dθ = 0 -h ẋT (θ)R ẋ(θ)dθ -1 h Ω T 0 RΩ0 + 3Ω T 1 RΩ1 . (8)
Since R 0, the left hand side of ( 8) is positive definite, which concludes the proof. ♦

The proof of Lemma 1 relies on the construction of the function z, the vectors Ω0 and Ω1 and the two polynomials 1, (h + 2θ). In order to understand this construction, z can be interpreted as the approximation error function of the continuous function ẋ by a first order polynomial function in θ. This approximation is evaluated through the norm associated with the inner product 0 -h φ(θ)ψ(θ)dθ, for any continuous functions φ, ψ. In light of the consideration of the inner product, the three first equations in [START_REF] He | Output feedback stabilization for a discrete-time system with a time-varying delay[END_REF] can be interpreted as the orthogonality of the polynomials 1 and (h+2θ) and the evaluation of their norms. Finally, the two last equations of [START_REF] He | Output feedback stabilization for a discrete-time system with a time-varying delay[END_REF] show that Ω0 and 2hΩ1 are the projections of the vector ẋ onto these two polynomials.

It follows from this discussion, that inequality ( 5) is already optimized in the sense of that specific inner product. Moreover it also shows that, if the function ẋ is a polynomial of degree 1, then the inequality becomes an equality. More details on the construction of generalized functions z are fully described in [START_REF] Seuret | Complete quadratic Lyapunov functionals using Bessel-Legendre inequality[END_REF], [START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time-delay systems[END_REF].

Hence the idea is to translate the construction of the function z, the construction of the two vectors and of two polynomials but in the discrete-time framework. This summarizes the contribution of the following lemma.

Lemma 2: For a given symmetric positive definite matrix Z ∈ S + n , any sequence of discrete-time variable x in [-h, 0]∩Z → R n , where h ≥ 1, the following inequality holds:

0 i=-h+1 y T (i)Zy(i) ≥ 1 h Θ 0 Θ 1 T Z 0 0 3 h+1 h-1 Z Θ 0 Θ 1 , (9)
where y(i) = x(i) -x(i -1) and

Θ0 = x(0) -x(-h), Θ1 = x(0) + x(-h) -2 h+1 0 i=-h x(i).
Proof : The proof is split into two cases. If h = 1, it is easy to see that Θ1 = 0 so that equation ( 9) holds. Assume now that h > 1.

For any sequence x in [-h, 0] ∩ Z → R n , define the signal z given,

z(i) = y(i) - 1 h Θ0 -3 (h -1 + 2i) h(h -1) Θ1, i = -h + 1, . . . , 1, 0,
where Θ0 and Θ1 are defined in the statements of the lemma. The new vector z has been built following the ideas given in the proof of Lemma 1. Developing

0 i=-h+1 z T (i)Zz(i) using the definition of z, we get 0 i=-h+1 z T (i)Zz(i) = 0 i=-h+1 y T (i)Zy(i) + 1 h 2 0 i=-h+1 1 Θ T 0 ZΘ0 + 9 h 2 (h-1) 2 0 i=-h+1 (h -1 + 2i) 2 Θ T 1 ZΘ1 -2 h Θ T 0 Z 0 i=-h+1 y(i) -6 h(h-1) Θ T 1 Z 0 i=-h+1 (h -1 + 2i)y(i) + 6 h 2 (h-1) 0 i=-h+1 (h -1 + 2i) Θ T 0 ZΘ1. (10)
Then some elementary calculus ensure that

0 i=-h+1 1 = h, 0 i=-h+1 (h -1 + 2i) = 0, 0 i=-h+1 (h -1 + 2i) 2 = h(h 2 -1) 3 , 0 i=-h+1 y(i) = Θ0, 0 i=-h+1 (h -1 + 2i)y(i) = (h + 1)Θ1.
Re-injecting them into (10) yields

0 i=-h+1 z T (i)Zz(i) = 0 i=-h+1 y T (i)Zy(i) -1 h Θ T 0 ZΘ0 -3 h+1 h-1 Θ T 1 ZΘ1 . ( 11
)
The proof is concluded by noting that the left-hand side of the previous equality is positive definite. ♦

The proofs of the lemmas for continuous and discrete-time are based on the same idea and the two inequalities are closely related. The only difference relies on the term h+1 h-1 ≥ 1, which is consistent with the selection Θ1 = 0 when h = 1. Moreover, the previous proof shows that this inequality becomes an equality if h = 2.

In some application with time-varying delays, the factor h+1 h-1 might be difficult to handle. Thus the following corollary is provided in order to make disappear this factor from the inequality.

Corollary 3: For a given symmetric positive definite matrix Z ∈ S + n , any sequence of discrete-time variable x in [-h, 0] ∩ Z → R n , where h ≥ 1, the following inequality holds:

0 i=-h+1 y T (i)Zy(i) ≥ 1 h Θ0 Θ1 T Z 0 0 3Z Θ0 Θ1 , (12)
where y(i), Θ0, and Θ1 are defined in Lemma 2.

Proof : For any h > 1, the inequality h+1 h-1 ≥ 1 holds. Reinjecting this inequality into equation ( 9) leads to the result. This equation still holds when h = 1 because Θ1 = 0. Consequenlty, the previous equality holds for all h ≥ 1.

♦ Remark 1: The inequality provided in Corollary 3 (and also in Lemma 2) implies

0 i=-h+1 y T (i)Zy(i) ≥ 1 h Θ T 0 ZΘ0,
which is exactly the Jensen summation inequality. Therefore, Corollary 3 (and also in Lemma 2) is less conservative than the celebrated Jensen inequality since a positive quantity is added in the right-hand side of the inequalities. Remark 2: The right-hand sides of the inequalities provided in Lemma 1 for continuous-time and Corollary 3 for discrete-time are exactly of the same form. In both cases, the vectors Ω0 and Θ0 represent the average evolution of the variable x over the delay interval while Ω1 and Θ1 are proportional to the difference between the mean value of x and its average over the delay interval.

C. Reformulated reciprocally convex combination inequality

In order to derive efficient stability conditions, the previous results will be combined with the convex inequality from [START_REF] Park | Reciprocally convex approach to stability of systems with time-varying delays[END_REF] which is recalled and reformulated in the following lemma.

Lemma 4: [START_REF] Park | Reciprocally convex approach to stability of systems with time-varying delays[END_REF] Let n, m be two positive integers, and two matrices R1 in S + n and R2 in S + m . The improved reciprocally convex combination guarantees that, if there exists a matrix X in R n×m such that R 1 X X T R 2 0, then the following inequality holds for any scalar α in the interval (0, 1)

1 α R 1 0 0 1 1-α R 2 R 1 X X R 2 . ( 13
)
Proof : By Schur complement,

R 1 X X T R 2 0 is equivalent to R1 -XR -1 2 X T 0.
Multiplying by the positive scalar 1-α α and re-employing Schur complement, we obtain:

0 1-α α R 1 -X -X α 1-α R 2 = 1 α R 1 0 0 1 α-1 R 2 - R 1 X X R 2 ,
which is equivalent to [START_REF] Kwon | Stability and stabilization for discrete-time systems with time-varying delays via augmented Lyapunov-Krasovskii functional[END_REF]. ♦

In the remainder of the paper, stability conditions for discrete-time systems with a time-varying delay are derived based on the previous summation inequality lemma.

III. STABILITY OF DISCRETE-TIME SYSTEMS WITH TIME-VARYING DELAYS

A. Stability result

Based on the summation inequalities, the following stability theorem is provided.

Theorem 5: Assume that there exist matrices P in S 2n + , Q1, Q2, Z1, Z2 in S n + ,and a matrix

X in R 2n×2n such that Ψ 0, Φ(h1) ≺ 0, Φ(h2) ≺ 0, (14)
where

Φ(h) = F T 1   P +   h 2 1 Z1 + h 2 12 Z2 0 0 0 0 0 0 0 0     F1 -F T 2 P F2 + He(Γ T (h)P F12) + Q -Π T ΨΠ, Ψ =   Z1(h1) 0 0 0 Z2 X 0 X T Z2   , Q = diag(Q1, -Q1 + Q2, 0, -Q2, 0, 0, 0), Z1(h1) = diag(Z1, 3γ(h1)Z1), Z2 = diag(Z2, 3Z2), and
F1 =   A -I 0 A d 0 0 0 0 0 -I 0 0 I 0 0 0 0 -I -I 0 I I   , F2 =   0 0 0 0 0 0 0 -I 0 0 0 I 0 0 0 -I -I 0 0 I I   , Γ(h) =   I 0 0 0 0 0 0 0 0 0 0 h 1 I 0 0 0 0 0 0 0 (h -h 1 )I (h 2 -h)I   , Π =   M 0 2n×2n 0 2n×n M 0 2n×n 0 2n×2n M   , M = I -I 0 0 0 I I 0 0 -2I , (15)
and where h12 = h2 -h1, F12 = F1 -F2, γ(h1) = 1 if h1 = 1 and γ(h1) = (h1 + 1)/(h1 -1) if h1 > 1. Then system (1) is asymptotically stable for any time-varying delay h(k) ∈ [h1, h2].

Proof : Consider the Lyapunov-Krasovskii functional given by

V (x k ) = V1(x k ) + V2(x k ) + V3(x k ), (16)
where

V1(x k ) =   x(k) k-1 i=k-h 1 x(i) k-h 1 -1 i=k-h 2 x(i)   T P   x(k) k-1 i=k-h 1 x(i) k-h 1 -1 i=k-h 2 x(i)   , V2(x k ) = k-1 i=k-h 1 x T (i)Q1x(i) + k-h 1 -1 i=k-h 2 x T (i)Q2x(i), V3(x k ) = h1 0 i=-h 1 +1 k j=k+i y T (j)Z1y(j) +h12 h 1 i=-h 2 +1 k j=k+i y T (j)Z2y(j),
where h12 = h2 -h1 and y(i) = x(i) -x(i -1). Note that the proposed functionals is the exact transcription to the discrete-time case of the functional employed in [START_REF] Seuret | Stability of systems with fastvarying delay using improved Wirtinger's inequality[END_REF] for continuous-time delay systems. This functional is positive definite since P 0, Q1 0, Q2 0, Z1 0 and Z2 0. For simplicity, the delay h(k) will be denoted as h. In the next developments, we aim at expressing an upper bound of ∆V (x k ) thanks to the augmented vector

ζ(k) =           x(k) x(k -h1) x(k -h) x(k -h2) ν1(k) ν2(k) ν3(k)          
, where

ν1(k) = 1 h 1 +1 k i=k-h 1 x(i), ν2(k) = 1 h-h 1 +1 k-h 1 i=k-h x(i), ν3(k) = 1 h 2 -h+1 k-h i=k-h 2 x(i).
Therefore we aim at expressing ∆V1 using the augmented vector ζ(k) by noting that

      x(k) k-1 i=k-h 1 x(i) k-h 1 -1 i=k-h 2 x(i)       =   0 -x(k) + ν 1 (k) ν 2 (k) + ν 3 (k) -x(k -h 1 ) -x(k -h)   +   x(k) h 1 ν 1 (k) (h -h 1 )ν 2 + (h 2 -h)ν 3 (k)   = (F2 + Γ(h))ζ(k), and       x(k + 1) k i=k-h 1 +1 x(i) k-h 1 i=k-h 2 +1 x(i)       =   (A -I)x(k) + A d x(k -h) -x(k -h 1 ) + ν 1 (k) ν 2 (k) + ν 3 (k) -x(k -h) -x(k -h 2 )   +   x(k) h 1 ν 1 (k) (h -h 1 )ν 2 (k) + (h 2 -h)ν 3 (k)   = (F1 + Γ(h))ζ(k).
Note that the previous equalities still hold when h = h1 and h = h2. It holds

∆V1(x k ) = ζ T (k) (F1 + Γ(h)) T P (F1 + Γ(h)) -(F2 + Γ(h)) T P (F2 + Γ(h)) ζ(k) = ζ T (k) F T 1 P F1 -F T 2 P F2 +He(Γ T (h)P (F1 -F2)) ζ(k).
On the second hand, the computation of ∆V2(x k ) and ∆V3 yields

∆V2(x k ) = x T (k)Q1x(k) -x T (k -h1)(Q1 -Q2)x(k -h1) -x T (k -h2)Q2x(k -h2) = ζ T (k) Qζ(k). and ∆V3(x k ) = y T (k) T (h 2 1 Z1 + h 2 12 Z2)y(k) -h1 k i=k-h 1 +1 y T (i)Z1y(i) -h12 k-h 1 i=k-h 2 +1 y T (i)Z2y(i).
The last sum is split into two parts: a first one gathering the terms between k -h1 and k -h + 1 and the second between k -h and k -h2 + 1. We are now in the situation to apply Lemma 2 to the first term and Corollary 3 to the two last terms. Thanks to the definitions of the matrices F1 and Π, it yields

∆V3(x k ) ≤ ζ T (k)   F T 1   h 2 1 Z1 + h 2 12 Z2 0 0 0 0 0 0 0 0   F1 -Π T    Z1 0 0 0 h 12 h-h 1 Z2 0 0 0 h 12 h 2 -h Z2    Π    ζ(k).
This inequality is still valid when h is equal to h1 or h2 using a similar argument as in [START_REF] Liu | Note on stability of discrete-time time-varying delay systems[END_REF]. It consists in noting that h12/(h -h1)(x(k -h) -x(k -h1)) T Z2(x(k -h) -x(k -h1)) is zero when h = h1. A similar argument holds when h = h2. Finally the convexity argument of Lemma 4 ensures that, if there exists a matrix X ∈ R 2n×2n such that Ψ 0, then the following upper bound of ∆V d3 is obtained

∆V3(x k ) ≤ ζ T (k)   F T 1   h 2 1 Z1 + h 2 12 Z2 0 0 0 0 0 0 0 0   F1 -Π T ΨΠ ζ(k).
Re-injecting the expressions of ∆V1 and ∆V2 and the upper bound of ∆V3 into the expression of ∆V leads to

∆V (x k ) ≤ ζ T (k)Φ(h)ζ(k).
Since the matrix Φ(h) is affine with respect to the delay h, Φ(h) is definite negative if and only if Φ(h1) and Φ(h2) are negative definite, which concludes the proof. ♦ Remark 3: Theorem 5 is the discrete-time version of the result showed in [START_REF] Seuret | Stability of systems with fastvarying delay using improved Wirtinger's inequality[END_REF] for continuous-time systems with interval timevarying delays.

B. Robust stability of discrete-time systems with time-varying delays

Consider an uncertain linear discrete-time delay system of the form: [START_REF] Seuret | Wirtinger-based integral inequality: Application to time-delay systems[END_REF].

x(k + 1) = M i=0 λi(k) {Aix(k) + A di x(k -h(k))} , ∀k ≥ 0, x(k) = φ(k), ∀k ∈ [-h2, 0], (17)
where x(k) ∈ R n is the state vector, φ is the initial condition and Ai, A di ∈ R n×n are constant and given matrices. The time-varying coefficients λi belong to the unit simplex and satisfy ∀i = 0, . . . , M, λi(k) ≥ 0,

M i=0 λi(k) = 1. (18)
The delay function is assumed to verify the same constraints as in (2). The following theorem provides an extension of Theorem 5 to the case of uncertain systems Theorem 6: Assume that there exist matrices P in S 2n + , Q1, Q2, Z1, Z2 in S n + ,and a matrix X in R 2n×2n such that Ψ 0 and

∀i = 1 . . . M, Φi(h1) ≺ 0, Φi(h2) ≺ 0, (19)
where

Φi(h) = F T 1i   P +   h 2 1 Z1 + h 2 12 Z2 0 0 0 0 0 0 0 0     F1i -F T 2 P F2 + He(Γ T (h)P F12i) + Q -Π T ΨΠ,
where the matrices Ψ, Γ(h), Π, Q, Z1, Z2, F2 and the parameters h12, γ(h) are given in the statement of Theorem 5 and where

F1i =   A i -I 0 A di 0 0 0 0 0 -I 0 0 I 0 0 0 0 -I -I 0 I I   , F12i = F1i -F2.
Then system [START_REF] Park | Reciprocally convex approach to stability of systems with time-varying delays[END_REF] is robustly asymptotically stable for any timevarying delay h(k) ∈ [h1, h2].

Proof : The proof follows straightforwardly by use of Schur complement, which ensures the convexity of the LMI condition [START_REF] Liu | Note on stability of discrete-time time-varying delay systems[END_REF] with respect to the system matrices A and A d . ♦

IV. EXAMPLES

A. Stability example

Consider the following discrete-time linear time-delay system (1):

x(k + 1) = 0.8 0.0 0.05 0.9

x(k) + -0.1 0.0 -0.2 -0.1 x(k -h(k)). (20)
A detailed comparison to various theorems from the literature is provided in Table I. It is worth mentioning that the benefits of employing the new summation inequalities from Lemma 2 and Corollary 3 are twofold. It indeed requires less decision variables than in the existing conditions [START_REF] Kwon | Stability and stabilization for discrete-time systems with time-varying delays via augmented Lyapunov-Krasovskii functional[END_REF] while leading to the same result.

B. Robust stability : Application to networked control systems

Consider the linear sampled-data system borrowed from [START_REF] Zhang | Stability of networked control systems[END_REF] given, for all t in [t k , t k+1 ) by [START_REF] Seuret | Complete quadratic Lyapunov functionals using Bessel-Legendre inequality[END_REF] where {t k } k≥0 represents a strictly increasing sequence of time instants for which there exist positive scalars 0 < T1 < T2 such that

ẋ(t) = 0 1 0 -0.1 x(t)+ 0 0 -0.375 -1.15 x(t k-h(k) ),
T k := t k+1 -t k ∈ [T1, T2], ∀k ≥ 0.
This class of examples differs notably from the problem addressed in [START_REF] Liu | Wirtinger's inequality and Lyapunov-based sampled-data stabilization[END_REF], [START_REF] Cloosterman | Stability of networked control systems with uncertain timevarying delays[END_REF], where the control input is computed with x(t k -τ ), where τ is a positive scalar representing a continuous-time delay. Here, the sampling instants and the delayed samples employed to generate the sampled input coincide. This time-delay example has been already studied in [START_REF] Seuret | Stability analysis of asynchronous sampled-data systems with discrete-time constant input delay[END_REF] but only in the case of constant delay h. The discretization of the sampled-data system leads to

x(k + 1) = A(T k )x(k) + A d (T k )x(t k-h(k) ), ∀k ≥ 0, (22)
where II where a tradeoff appears between the allowable uncertainty in the sampling period and the allowable variations in the delay function.

A(T k ) := e 0 1 0 -0.1 T k = 1 10(1 -λ(T k )) 0 λ(T k ) , and
A d (T k ) = T k 0 e 0
When the delay is constant and h1 = h2 = 1 and h1 = h2 = 5, an eigenvalue analysis shows that the maximal acceptable constant sampling period Tm = TM = T for which the system remains asymptotically stable is T = 0.76 and T = 0.21, respectively. Note that as we have mentioned earlier in the paper, the summation inequality is not conservative when h1 = h2 = 1. Hence, the conservatism in this case comes from the approximation the uncertain system [START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time-delay systems[END_REF]. This conservatism may be reduced by employing more advanced approximation methods provided for instance in [START_REF] Heemels | Comparison of overapproximation methods for stability analysis of networked control systems[END_REF]. This illustrates that the stability analysis provided in this paper is not conservative.

V. CONCLUSIONS

In this paper, a new summation inequality is presented and is shown to be relevant for the stability analysis of discrete-time systems with interval time-varying delays. This inequality has been derived from an appropriate interpretation of the Wirtinger-based integral inequality and is proved to be less conservative than the usual Jensen inequality. Combined with the reciprocal convexity lemma, two stability theorems have been provided, which are less conservative or less computationally demanding than the usual Lyapunov-Krasovskii approaches together with the Jensen inequality.

  375 -1.15 ds = 32.5(1 -λ(T k )) -3.75T k 115(1 -λ(T k )) -11.5T k 3.75(λ(T k ) -1) 11.5(λ(T k ) -1), with the notation λ(T k ) = e -T k /10 . Using the convexity property of the exponential function, we show that(T k , λ(T k )) ∈ Co{(Tm, λ(Tm)), (TM , λ(TM )), (Tm, λ(TM )(1 + (TM -Tm)/10))},where the notation Co refers to the convex of hull the considered vertices. The vertex (Tm, λ(TM )(1 + (TM -Tm)/10)) has been computed thanks to the tangent line of the exponential function T → λ(T ) = e -T /10 at T = TM . Then, a polytopic modeling of the discrete-time system can be defined and the system (22) becomes suitable for the application of Theorem 6. The results are given in Table

ADMISSIBLE

  UPPER BOUND h 2 FOR VARIOUS T 2 WITH T 1 = 0.1 AND h 1 = 1 FOR SYSTEM DESCRIBED IN EXAMPLE[START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time-delay systems[END_REF]. FOR VARIOUS T 2 WITH T 1 = 0.1 AND h 1 = 5 FOR SYSTEM DESCRIBED IN EXAMPLE[START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time-delay systems[END_REF].

	T 1 = 0.1 T 2	0.11 0.15 0.2 0.3 0.5 0.64
	h 1 = 1	h 2	9	6	4	3	1	1
				TABLE II
	T 1 = 0.1	T 2	0.11 0.15 0.2 0.202
	h 1 = 5	h 2	9	6	5	5
				TABLE III
	ADMISSIBLE UPPER BOUND h 2


Nahla Ben Amor
email: nahla.benamor@gmx.fr

Fatma Essghaier
email: essghaier.fatma@gmail.com

Hélène Fargier
email: fargier@irit.fr

Solving multi-criteria decision problems under possibilistic uncertainty using optimistic and pessimistic utilities

This paper proposes a qualitative approach to solve multicriteria decision making problems under possibilistic uncertainty. Depending on the decision maker attitude with respect to uncertainty (i.e. optimistic or pessimistic) and on her attitude with respect to criteria (i.e. conjunctive or disjunctive), four ex-ante and four ex-post decision rules are defined and investigated. In particular, their coherence w.r.t. the principle of monotonicity, that allows Dynamic Programming is studied.

Introduction

A popular criterion to compare decisions under risk is the expected utility model (EU ) axiomatized by Von Neumann and Morgenstern [START_REF] Morgenstern | Theory of Games and Economic Behavior[END_REF]. This model relies on a probabilistic representation of uncertainty: an elementary decision is represented by a probabilistic lottery over the possible outcomes. The preferences of the decision maker are supposed to be captured by a utility function assigning a numerical value to each consequence. The evaluation of a lottery is then performed through the computation of its expected utility (the greater, the better). When several independent criteria are to be taken into account, the utility function is the result of the aggregation of several utility functions u i (one for each criterion). The expected utility of the additive aggregation can then be used to evaluate the lottery, and it is easy to show that it is equal to the additive aggregation of the mono-criterion expected utilities.

These approaches presuppose that both numerical probability and additive utilities are available. When the information about uncertainty cannot be quantified in a probabilistic way the topic of possibilistic decision theory is often a natural one to consider. Giving up the probabilistic quantification of uncertainty yields to give up the EU criterion as well. The development of possibilistic decision theory has lead to the proposition and the characterization of (monocriterion) possibilistic counterparts of expected utility: Dubois and Prade [START_REF] Dubois | Possibility theory as a basis for qualitative decision theory[END_REF] propose two criteria based on possibility theory, an optimistic and a pessimistic one, whose definitions only require a finite ordinal scale for evaluating both utility and plausibility. Likewise, qualitative approaches of multi-criteria decision making have been advocated, leading to the use of Sugeno Integrals (see e.g. [START_REF] Dubois | The use of the discrete sugeno integral in decision-making: A survey[END_REF][START_REF]An axiomatic approach of the discrete sugeno integral as a tool to aggregate interacting criteria in a qualitative framework[END_REF]) and especially weighted maximum and weighted minimum [START_REF] Dubois | Weighted minimum and maximum operations in fuzzy set theory[END_REF].

In this paper, we consider possibilistic decision problems in the presence of multiple criteria. The difficulty is here to make a double aggregation. Several attitudes are possible: shall we consider the pessimistic/optimistic utility value of a weighted min (or max)? or shall we rather aggregate with a weighted min (or max) the individual pessimistic (or optimistic) utilities provided by the criteria? In short, shall we proceed in an ex-ante or ex-post way?

The remainder of the paper is organized as follows: Section 2 presents a refresher on possibilistic decision making under uncertainty using Dubois and Prade's pessimistic and optimistic utilities, on one hand, and on the qualitative approaches of MCDM (mainly, weighted min and weighted max), on the other hand. Section 3 develops our proposition, defining four ex-ante and four ex-post aggregations, and shows that when the decision maker attitude is homogeneous, i.e. either fully min-oriented or fully max-oriented, the ex-ante and the ex-post possibilistic aggregations provide the same result. Section 4 studies the monotonicity of these decision rules, in order to determine the applicability of Dynamic Programming to sequential decision making problems. 12 Background on one-stage decision making in a possibilistic framework

2.1 Decision making under possibilistic uncertainty (U + and U -) Following Dubois and Prade's possibilistic approach of decision making under qualitative uncertainty, a one stage decision can be seen as a possibility distribution over a finite set of outcomes also called a (simple) possibilistic lottery [START_REF] Dubois | Possibility theory as a basis for qualitative decision theory[END_REF]. Since we consider a finite setting, we shall write L = λ 1 /x 1 , . . . , λ n /x n s.t.

λ i = π f (x i )
is the possibility that decision f leads to outcome x i ; this possibility degree can also be denoted by L[x i ]. We denote L the set of all simple possibilistic lotteries. In this framework, a decision problem is thus fully specified by a set ∆ of possibilistic lotteries on a set of consequences X and a utility function u : X → [0, 1]. Under the assumption that the utility scale and the possibility scale are commensurate and purely ordinal, Dubois and Prade have proposed the following qualitative degrees for evaluating any simple lottery L = λ 1 /x 1 , . . . , λ n /x n : Optimistic utility (U + ) [START_REF] Dubois | Possibility theory as a basis for qualitative decision theory[END_REF][START_REF] Yager | Possibilistic decision making[END_REF][START_REF] Zadeh | Fuzzy sets as a basis for a theory of possibility[END_REF]:

U + (L) = max xi∈X min(λ i , u(x i )) (1)
Pessimistic utility (U -) [START_REF] Dubois | Possibility theory as a basis for qualitative decision theory[END_REF][START_REF] Whalen | Decision making under uncertainty with various assumptions about available information[END_REF]:

U -(L) = min xi∈X max((1 -λ i ), u(x i ))
(2) The value U -(L) is high only if L gives good consequences in every "rather plausible" state. This criterion generalizes the Wald criterion, which estimates the utility of an act by its worst possible consequence. U -(L) is thus "pessimistic" or "cautious". On the other hand, U + (L) is a mild version of the maximax criterion which is "optimistic", or "adventurous": act L is good as soon as it is totally plausible that it gives a good consequence.

2.2 Multi-criteria decision making (MCDM) using Agg + and Agg -

The previous setting assumes a clear ranking of X by a single preference criterion, hence the use of a single utility function u. When several criteria, say a set C = {c 1 ...c p } of p criteria, have to be taken into account, u shall be replaced by a vector u = u 1 , . . . , u p of utility functions u j : X → [0, 1] and the global (qualitative) utility of each consequence x ∈ X can be evaluated either in a conjunctive, cautious, way according to the Wald aggregation (Agg -(x) = min j=1,p u j (x)), or in an disjunctive way according to its max-oriented counterpart (Agg + (x) = max j=1,p u j (x)). When the criteria are not equally important, a weight w j ∈ [0, 1] can be associated to each c j . Hence the following definitions relative to multi-criteria utilities [START_REF] Dubois | Weighted minimum and maximum operations in fuzzy set theory[END_REF]:

Agg + (x) = max j=1,p min(w j , u j (x)).
(3)

Agg -(x) = min j=1,p max((1 -w j ), u j (x)). (4)
These utilities are particular cases of the Sugeno integral [START_REF] Dubois | The use of the discrete sugeno integral in decision-making: A survey[END_REF][START_REF]An axiomatic approach of the discrete sugeno integral as a tool to aggregate interacting criteria in a qualitative framework[END_REF][START_REF] Sugeno | Theory of fuzzy integrals and its applications[END_REF]:

Agg γ,u (L) = max λ∈[0,1] min(λ, γ(F λ )) (5)
where F λ = {c j ∈ C, u j (x) ≥ λ}, γ is a monotonic set-function that reflects the importance of criteria's set. Agg + is recovered when γ is the possibility measure based on the weight distribution (γ(E) = max cj ∈E w j ), and Agg -is recovered when γ corresponds to necessity measure (γ(E) = min cj / ∈E

(1 -w j )).

Multi-criteria decision making under possibilistic uncertainty

Let us now study possibilistic decision making in a multi-criteria context. Given a set X of consequences, a set C of independent criteria we define a multi-criteria decision problem under possibilistic uncertainty as triplet ∆, w, u 2 where:

-∆ is a set of possibilistic lotteries;

2 Classical problems of decision under possibilistic uncertainty are recovered when |C| = 1 ; Classical MCDM problems are recovered when all the lotteries in ∆ associate possibility 1 to some xi and possibility 0 to all the other elements of X: ∆ is identified to X, i.e. is a set of "alternatives" for the MCDM decision problem.

w ∈ [0, 1] p is a weighting vector: w j denotes the weight of criterion c j ; -u = u 1 , . . . , u p is a vector of p utility functions on X: u j (x i ) ∈ [0, 1] is the utility of x i according to criterion c j ;

Our aim consists in comparing lotteries according to decision maker's preferences relative to their different consequences (captured by the utility functions) and the importance of the criteria (captured by the weighting vector). To do this, we can proceed in two different ways namely ex-ante or ex-post:

-The ex-ante aggregation consists in first determining the aggregated utilities (Agg + or Agg -) relative to each possible consequence x i of L and then combine them with the possibility degrees. -The ex-post aggregation consists in computing the (optimistic or pessimistic) utilities relative to each criterion c j , and then perform the aggregation (Agg + or Agg -) using the criteria's weights.

We borrow this terminology from economics and social welfare economics, were agents play the role played by criteria in the present context (see e.g. [START_REF] Harsanyi | Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility[END_REF][START_REF] Myerson | Utilitarianism, egalitarianism, and the timing effect in social choice problems[END_REF]). Setting the problem in a probabilistic context, these works have shown that the two approaches can lead to different results (the so-called "timing effect") coincide iff the collective utility is affine. As a matter of fact, it is easy to show that the expected utility of the weighted sum is the sum of the expected utilities.

Let us go back to possibilistic framework. The decision maker's attitude with respect to uncertainty can be either optimistic (U + ) or pessimistic (U -) and her attitude with respect to criteria can be either conjunctive (Agg + ) or disjunctive (Agg -), hence the definition of four approaches of MCDM under uncertainty, namely U ++ , U +-, U -+ and U --; the first (resp. the second) indices denoting the attitude of the decision maker w.r.t. uncertainty (resp. criteria).

Each of these utilities can be computed either ex-ante or ex-post. Hence the definition of eight utilities: Definition 1. Given a possibilistic lottery L on X, a set of criteria C defining a vector of utility functions u and weighting vector w, let:

U ++ ante (L) = max xi∈X min(L[x i ], max cj ∈C min(u j (x i ), w j )). (6)
U -- ante (L) = min xi∈X max((1 -L[x i ]), min cj ∈C max(u j (x i ), (1 -w j ))). ( 7
) U +- ante (L) = max xi∈X min(L[x i ], min cj ∈C max(u j (x i ), (1 -w j ))). (8)
U -+ ante (L) = min xi∈X max((1 -L[x i ]), max cj ∈C min(u j (x i ), w j )). ( 9
)
U ++ post (L) = max cj ∈C min(w j , max xi∈X min(u j (x i ), L[x i ])). ( 10
) U -- post (L) = min cj ∈C max((1 -w j ), min xi∈X max(u j (x i ), (1 -L[x i ]))). ( 11
) U +- post (L) = min cj ∈C max((1 -w j ), max xi∈X min(u j (x i ), L[x i ])). ( 12
)
U -+ post (L) = max cj ∈C min(w j , min xi∈X max(u j (x i ), (1 -L[x i ]))). ( 13
)
Interestingly, the optimistic aggregations are related to their pessimistic counterparts by duality as stated by the following proposition.

Proposition 1. Let P = ∆, w, u be a qualitative decision problem, let P τ = ∆, w, u τ be the inverse problem, i.e. the problem such that for any

x i ∈ X, c j ∈ C, u τ j (x i ) = 1 -u j (x i ). Then, for any L ∈ ∆: U ++ ante (L) = 1 -U τ -- ante (L) U ++ post (L) = 1 -U τ -- post (L) U -- ante (L) = 1 -U τ ++ ante (L) U -- post (L) = 1 -U τ ++ post (L) U +- ante (L) = 1 -U τ -+ ante (L) U +- post (L) = 1 -U τ -+ post (L) U -+ ante (L) = 1 -U τ +- ante (L) U -+ post (L) = 1 -U τ +- post (L)
As previously said the ex-ante and the ex-post approaches coincide in the probabilistic case. Likewise, the following Proposition 2 shows that when the decision maker attitude is homogeneous, i.e. either fully min-oriented or fully maxoriented, the ex-ante and the ex-post possibilistic aggregations provide the same result.

Proposition 2. For any L ∈ L, U ++ ante (L) = U ++ post (L) and U -- ante (L) = U -- post (L).
Hence, for any multi-criteria decision problem under possibilistic uncertainty,

U ++ ante (resp. U -- ante ) is equal to U ++ post (resp. U -- post )
. Such an equivalence between the ex-ante and ex-post does not hold for U +-nor for U -+ , as shown by the following counter-example.

Counter-example 1 Consider a set C of two equally important criteria c 1 and c 2 , and a lottery L (cf. Figure 1) leading to two equi-possible consequences x 1 and x 2 such that x 1 is good for c 1 and bad for c 2 , and x 2 is bad for c 1 and good for c 2 ; i.e. L[

x 1 ] = L[x 2 ] = 1, w 1 = w 2 = 1, u 1 (x 1 ) = u 2 (x 2 ) = 1 and u 2 (x 1 ) = u 1 (x 2 ) = 0. 1 1 x 1 , u(x 1 ) = <1,0> L x 2 , u(x 2 ) = <1,0> 1 L'
x 3 , u(x 3 ) = <0.5,0.5> The ex-ante and ex-post approaches may lead to different rankings of lotteries. Consider for instance, a lottery L ′ leading to the consequence x 3 for sure, i.e.

L ′ [x 3 ] = 1 and L ′ [x i ] = 0, ∀i = x 3 (such a lottery is called a constant lottery), with u 1 (x 3 ) = u 2 (x 3 ) = 0.5 . It is easy to check that U +- ante (L ′ ) = U +- post (L ′ ) = 0.5 i.e. U +- ante (L) < U +- ante (L ′ ) while U +- post (L) > U +- post (L ′ ).
Using the same lotteries L and L ′ , we can show that:

U -+ ante (L) = 1 = U -+ post (L) = 0 and that U -+ ante (L ′ ) = U -+ post (L ′ ) = 0.5; then U -+ post (L ′ ) > U -+ post (L) while U -+ ante (L ′ ) < U -+
ante (L): like U +-, U -+ are subject to the timing effect.

In summary, U -+ and U +-suffer from the timing effect, contrary to U --and U ++ .

Multi-criteria sequential decision making under possibilistic uncertainty

Possibilistic sequential decision making relies on possibilistic compound lotteries [START_REF] Dubois | Possibility theory as a basis for qualitative decision theory[END_REF], that is a possibility distributions over (simple or compound) lotteries.

Compound lotteries indeed allow the representation of decision policies or "strategies", that associate a decision to each decision point: the execution of the decision may lead to several more or less possible situations, where new decisions are to be made, etc. For instance, in a two stages decision problem, a first decision is made and executed; then, depending on the observed situation, a new, one stage, decision is to be made, that lead to the final consequences. The decisions at the final stage are simple lotteries, and the decision made at the first stage branches on each of them. The global strategy thus defines to a compound lottery .

To evaluate a strategy by U + , U -or, in the case of MCDM under uncertainty by any of the eight aggregated utility proposed in Section 3, the idea is to "reduce" its compound lottery into an equivalent simple one. Consider a compound lottery L = λ 1 /L 1 , . . . , λ m /L m ; the possibility of getting consequence x i ∈ X from one of its sub lotteries

L k is π k,i = min(λ k , L k [x i ])
(for the shake of simplicity, suppose that L ′ k are simple lotteries; the principle trivially extends to the general case). Hence, the possibility of getting x i from L is the max, over all the L k 's, of π k,i . Thus, [START_REF] Dubois | Possibility theory as a basis for qualitative decision theory[END_REF] have proposed to reduce a compound lottery λ 1 /L 1 , . . . , λ m /L m into a simple lottery, denoted by Reduction( λ 1 /L 1 , . . . , λ m /L m ), that is considered as equivalent to the compound one: Reduction( λ 1 /L 1 , . . . , λ m /L k ) is the simple lottery that associate to each x i the possibility degree max k=1..m min(λ k , L k [x i ]) (with L[x i ] = 0 when none of the L k 's give a positive possibility degree to consequence x i ). See Figure 2 for an example. From a practical point of view, sequential decision problems are generally stated through the use of compact representation formalisms, such as possibilistic decision trees [START_REF] Fargier | On the complexity of decision making in possibilistic decision trees[END_REF], possibilistic influence diagrams [START_REF] Garcias | Possibilistic influence diagrams[END_REF][START_REF] Guezguez | Qualitative possibilistic influence diagrams based on qualitative possibilistic utilities[END_REF] or possibilistic Markov decision processes [START_REF] Sabbadin | Empirical comparison of probabilistic and possibilistic markov decision processes algorithms[END_REF][START_REF] Sabbadin | Possibilistic markov decision processes[END_REF]. The set of potential strategies to compare, ∆, is generally exponential w.r.t. the input size. So, an explicit evaluation of each strategy in ∆ is not realistic. Such problems can nevertheless be solved efficiently, without an explicit evaluation of the strategies, by Dynamic Programming algorithms as soon as the decision rule leads to transitive preferences and satisfies the principle of weak monotonicity. Formally, for any decision rule O (e.g. U + , U -or even any of the decision rules proposed in the previous Section) over possibilistic lotteries, ≥ O is said to be weakly monotonic iff whatever L, L ′ and L ′′ and whatever (α,β) such that max(α, β) = 1:

L O L ′ ⇒ α/L, β/L ′′ O α/L ′ , β/L ′′ . ( 14
)
Such property ensures that each sub-strategy of an optimal strategy is optimal in its sub-problem. This allows Dynamic Programming algorithms to build optimal strategies in an incremental way (e.g. in decision tree, from the last decision to the root of the tree). [START_REF] Garcias | Possibilistic influence diagrams[END_REF][START_REF] Fargier | On the complexity of decision making in possibilistic decision trees[END_REF] have shown that U + and U -are monotonic. Let us now study whether it is also the case for the ex-ante and ex-post rules proposed in the previous Section. The ex-ante approaches are the easiest to handle: once the vectors of utilities have been aggregated according to Agg -(resp. Agg + ), these approaches collapse with the classical U + and U -approaches. It is then easy to show that: Proposition 3. U ++ ante , U -- ante , U +- ante and U -+ ante satisfy the weak monotonicity. Concerning U ++ post and U -- post , recall that the full optimistic and full pessimistic expost utilities are equivalent to their ex-ante counterparts thanks to Proposition 2. This allows us to show that: Proposition 4. U -- post and U ++ post satisfy the weak monotonicity. It follows from Propositions 3 and 4 that when the decision is based either on an ex-ante approach, or on U ++ post or U -- post , the algorithms proposed by Sabbadin et al. [START_REF] Sabbadin | Possibilistic markov decision processes[END_REF][START_REF] Garcias | Possibilistic influence diagrams[END_REF] can be used on multi-criteria possibilistic decision trees and influence diagrams after their transformation into single-criterion problems: it is enough to aggregate the vectors of utilities leading to the consequences x into single utilities using Agg + (for U ++ ante , U -+ ante , U ++ post ) or Agg -(for U -- ante , U +- ante , U -- post ) to get an equivalent single criterion problem where the criterion to optimize is simply U + (for

U ++ ante , U +- ante , U ++ post ) or U -(for U -- ante , U -+ ante , U -- post )
. Such approach cannot be applied when optimizing U -+ post or U +- post . First because U +- post (L) = U +- ante (L) and U -+ post (L) = U +- ante (L), i.e. the reduction of the problem to the optimization w.r.t. U + (resp. U -) of a single criterion problem obtained by aggregating the utilities with Agg -(resp. Agg + ) can lead to a wrong result. Worst, it is not even possible to apply Dynamic Programming, since U -+ post and U +- post do not satisfy the weak monotonicity property, as shown by the following counter-example:

Counter-example 2 Let X = {x 1 , x 2 , x 3 } and consider two equally important criteria c 1 and c 2 (w 1 = w 2 = 1) with :

u 1 (x 1 ) = 1, u 1 (x 2 ) = 0.8, u 1 (x 3 ) = 0.5; u 2 (x 1 ) = 0.6, u 2 (x 2 ) = 0.8, u 2 (x 3 ) = 0.8. Consider the lotter- ies L = 1/x 1 , 0/x 2 , 0/x 3 , L ′ = 0/x 1 , 1/x 2 , 0/x 3 and L ′′ = 0/x 1 , 0/x 2 , 1/x 3 :
L gives consequence x 1 for sure, L ′ gives consequence x 2 for sure and L ′′ gives consequence x 3 for sure. It holds that:

U -+ post (L) = Agg -(x 1 ) = max(1, 0.6) = 1 U -+ post (L ′ ) = Agg -(x 2 )
= max(0.8, 0.8) = 0.8. Hence L > U -+ post L ′ with respect to the U -+ post rule. Consider now the compound lotteries L 1 = 1/L, 1/L ′′ and L 2 = 1/L ′ , 1/L ′′ . If the weak monotonicity principle were satisfied, we would get:

L 1 > U -+ post L 2 .Since: Reduction( 1/L, 1/L ′′ ) = 1/x 1 , 0/x 2 , 1/x 3 and Reduction( 1/L ′ , 1/L ′′ ) = 0/x 1 , 1/x 2 , 1/x 3 . We have: U -+ post (L 1 ) = U -+ post (Reduction( 1/L, 1/L ′′ )) = 0.6. U -+ post (L 2 ) = U -+ post (Reduction( 1/L ′ , 1/L ′′ )) = 0.8. Hence, L 1 < U -+ post L 2 while L > U -+ post L ′ , which contradicts weak monotonicity.
Using the fact that U +- post = 1 -U τ -+ post , this counter-example is modified to show that also U +- post does not satisfy the monotonicity principle. Consider two equally important criteria, c τ 1 and c τ 2 with w 1 = w 2 = 1 with:

u τ 1 (x 1 ) = 0, u τ 1 (x 2 ) = 0.2, u τ 1 (x 3 ) = 0.5; u τ 2 (x 1 ) = 0.4, u τ 2 (x 2 ) = 0.2, u τ 2 (x 3 ) = 0.2.
Consider now the same lotteries L, L ′ and L ′′ presented above. It holds that:

U +- post (L) = Agg -(x 1 ) = 0 < U +- post (L ′ ) = Agg -(x 2 ) = 0.2, while U +- post (Reduction( 1/L, 1/L ′′ )) = 0.4 > U +- post (Reduction( 1/L ′ , 1/L ′′ )) = 0.2.
The lack of monotonicity of U -+ post is not as dramatic as it may seem. When optimizing U -+ post (L), the decision maker is looking for a strategy that is good w.r.t. U -for at least one criterion. This means that if it is possible to get for each criterion c j a strategy that optimizes U -according to this criterion (and this can be done by Dynamic Programming, since U -do satisfy the principle of monotonicity), the one with the higher U -is optimal w.r.t. U -+ post (L). Formally:

Proposition 5. U -+ post (L) = max j=1,p min(w j , U - j (L)) where U - j (L) = min xi∈X max((1 -L[x i ]), u j (x i ))
is the pessimistic utility of L according to the sole criterion j.

Proposition 6. Let L be the set of lotteries that can be built on X and let:

-

∆ * = {L * 1 , . . . , L * p } s.t. ∀L ∈ L, j ∈ 1 . . . p, U - j (L * j ) ≥ U - j (L); -L * ∈ ∆ * s.t. ∀L * j ∈ ∆ * :max j=1,p min(w j , U - j (L * )) ≥ max j=1,p min(w j , U - j (L * i )).
It holds that, for any L ∈ L,

U -+ post (L * ) ≥ U -+ post (L).
Hence, it is enough to optimize w.r.t. each criterion separately and to compare the results to get an optimal strategy w.r.t. U -+ post (L). Let us finally study U +- post ; an analog of Proposition 6 exists:

Proposition 7. U +- post (L) = min j=1,p max((1 -w j ), U + j (L)) where U + j (L) = max xi∈X min(L[x i ], u j (x i ))
is the optimistic utility of L according to the sole criterion j.

But this proposition is helpless, since the lottery L maximizing this quantity is not necessarily among those maximizing the U + j 's: one lottery optimal for U + 1 w.r.t. criterion c 1 can be very bad for U + 2 and thus bad for U +- post .

Conclusion

This paper has provided a first decision theoretical approach for evaluating multicriteria decision problems under possibilistic uncertainty. The combination of the multi-criteria dimension, namely the conjunctive aggregation with a weighted min (Agg -) or the disjunctive aggregation with a weighted max (Agg -) and the decision maker's attitude with respect to uncertainty (i.e. optimistic utility U + or pessimistic utility U -) leads to four approaches of MCDM under possibilistic uncertainty. Considering that each of these utilities can be computed either ex-ante or ex-post, we have proposed the definition of eight aggregations, that eventually reduce to six: U ++ ante (resp. U -- ante ) has been shown to coincide with U ++ post (resp. U -- post ); such a coincidence does not happen for U +-and U -+ , that suffer from timing effect.

Then, in order to use these decision rules in sequential decision problems, we have proven that all ex-ante utilities (i.e. U ++ ante , U -- ante , U +- ante , U -+ ante ) satisfy the weak monotonicity while for the ex-post utilities, only U ++ post and U -- post satisfy this property. This result means that Dynamic Programming algorithms can be used to compute strategies that are optimal w.r.t. the rules . We have also shown that the optimization of U +- post can be handled thanks to a call of a series of optimization of pessimistic utilities (one for each criterion). The question of the optimization of U -+

post still remains open. This preliminary work call for several developments. First of all we intend, as future work, to propose and test suitable algorithms to solve sequential qualitative multi-criteria decision problems, e.g. influence diagrams and decision trees. From a more theoretical point of view we have to propose a general axiomatic characterization of our six decision rules. Finally, considering that the possibilistic aggregations used here are basically specializations of the Sugeno integral, we aim at generalizing the study of MCDM decision making under uncertainty through the development of double Sugeno Integrals.

Appendix: Proofs

Proof (Proposition 1). The proofs relative to these results are quite similar, so we show that U +- post (L) = 1 -U τ -+ post (L), (i.e. 1 -U +- post (L) = U τ -+ post (L)) and we can apply the same principle to prove other utilities. Proof (Propositon 3). To prove that U ante utilities satisfy the weak monotonicity property, we proceed by proving that U ++ ante (resp. U -- ante ) and U +- ante (resp. U -+ ante ) are equivalent to their equivalent in context of mono-criterion decision making i.e. U + (resp.U -) that are known as monotonic. While the principle is always the same, we will limit to the proof relative to the reduction of U ++ ante to U + .

U ++ ( α\L1, β\L2, . . . γLm ). = U ++ (red( α\L1, β\L2, . . . γLm )). = U + (Agg + (red( α\L1, β\L2, . . . γLm ))).

Fig. 1 .

 1 Fig. 1. Lotteries L and L ′ relative to counter-example 1

Fig. 2 .

 2 Fig. 2. A compound lottery L1 (a) and its reduction L2 (b)

1 - 1 -

 11 U +- post (L) = 1 -[ min cj ∈C max((1 -w j ), max xi∈X min(u j (x i ), L[x i ]))] = max cj ∈C [max((1 -w j ), max xi∈X min(u j (x i ), L[x i ]))] = max cj ∈C min 1 -[((1 -w j ), max xi∈X min(u j (x i ), L[x i ]))] = max cj ∈C min(w j , 1 -[max xi∈X min(u j (x i ), L[x i ])]) = max cj ∈C min(w j , min xi∈X 1 -[min(u j (x i ), L[x i ])]) = max cj ∈C min(w j , min xi∈X max 1 -[(u j (x i ), L[x i ])]) = max cj ∈C min(w j , min xi∈X max(1-u j (x i ), (1-L[x i ]) = max cj ∈C min(w j , min xi∈X max(u τ j (x i ), (1-L[x i ]) = U τ -+ post (L). ⊓ ⊔ Proof (Propositon 2).In the following, we illustrate the proof relative to equivalence between U ++ ante (L) and U ++ post (L). The proof relative to U --is similar by replacing max by min, L[x i ] by (1 -L[x i ]) and w j by (1 -w j ). U ++ ante (L) = max xi∈X min(L[x i ], max cj ∈C min(u j (x i ), w j )) = max xi∈X min(max(min(u 1 (x i ), w 1 ), min(u 2 (x i ), w 2 ), . . . , min(u p (x i ), w p )), L[x i ]) = max xi∈X max(min(min(u 1 (x i ), L[x i ]), w 1 ), min(min(u 2 (x i ), L[x i ]), w 2 ), . . . , min(min(u p (x i ), L[x i ]), w p )) = max cj ∈C max xi∈X (min(min(L[x i ], u j (x i ))), w j ) = max cj ∈C max(min(min(L[x 1 ], u j (x 1 )), w j ), min(min(L[x 2 ], u j (x 2 )), w j ), . . . , min(min(L[x n ], u j (x n )), w j )) = max cj ∈C min(max(min(L[x 1 ], u j (x 1 )), min(L[x 2 ], u j (x 2 )), . . . , min(L[x n ], u j (x n ))), w j ) = max cj ∈C min(max xi∈X (min(L[x i ], u j x i )), w j ) = U ++ post (L). ⊓ ⊔

Proofs relative to this paper are omitted for lack of space; they are available on ftp://ftp.irit.fr/IRIT/ADRIA/PapersFargier/ipmu14.pdf

Proof (Proposition 4). We have proven that U ++ ante (resp. U -- ante ) equals to U ++ post (resp. U -- post ), also we have shown that U ++ ante and U -- ante are monotonic. So, we can conclude that U ++ post and U ++ post satisfy the weak monotonicity property.

Proof (Proposition 5). we have defined

. So, U -+ post (L) can be expressed as follows:

Proof (Proposition 6). We suppose that

Proof (Proposition 7). We have defined U +- post (L) = min


Dorra Attiaoui
email: attiaoui.dorra@gmail.com

Pierre-Emmanuel Doré
email: pierre-emmanuel.dore@univ-rennes1.fr

Arnaud Martin
email: arnaud.martin@univ-rennes1.fr

Boutheina Ben
email: boutheina.yaghlane@ihec.rnu.tn

Inclusion within Continuous Belief Functions

Defining and modeling the relation of inclusion between continuous belief function may be considered as an important operation in order to study their behaviors. Within this paper we will propose and present two forms of inclusion: The strict and the partial one. In order to develop this relation, we will study the case of consonant belief function. To do so, we will simulate normal distributions allowing us to model and analyze these relations. Based on that, we will determine the parameters influencing and characterizing the two forms of inclusion.

I. INTRODUCTION

The theory of belief functions (also called Evidential Theory or The Dempster-Shafer Theory) is a well known framework for reasoning under uncertainty [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF]. It is widely used to model imperfect information. We can distinguish the imprecision (lack of accuracy), the uncertainty (lack of compliance comparing to the real world, due to the source of information) and even the inconstancy where is characterized by a high level of conflict.

The discrete case of belief functions, knowning a large succes, have been applied to several research fields, medical, military, risk management, information fusion [START_REF] Martin | Conflict measure for the discounting operation on belief functions[END_REF], [START_REF] Martin | Toward a combination rule to deal with partial conflict and specificity in belief functions theory[END_REF], [START_REF] Donoeux | Conjunctive and disjunctive combination for belief functions induced by nondistinct bodies of evidence[END_REF], etc.

Otherwise, Smets in [START_REF] Smets | Belief functions on real numbers[END_REF], defined the basic notions of continuous belief functions to describe them in the extended set of reals. Recently Attiaoui et al. [START_REF] Attiaoui | A distance between continuous belief functions[END_REF] proposed a similarity measure for continuous belief functions based on Smets' formalism using the distance of Jousselme [START_REF] Jousselme | A new distance between two bodies of evidence[END_REF].

Our work in this paper will consider the notion of inclusion and how two continuous belief functions can be included in each other. This operation will help us later to take into account this specific characteristic during the information fusion and the measurement of the conflit. Thus, we will define two forms: the strict inclusion and the partial one. To do so, we will present a measure called a degree of inclusion of an interval (the focal element of a continuous belief function) in the second one. This work presents a new way to determine the relation of inclusion by considering a new vision for the continuous case within the theory of belief functions.

II. THEORY OF BELIEF FUNCTIONS: AN OVERVIEW

This section recalls the necessary background related to the evidential theory. It has been developed by Dempster in his work on upper and lower probabilities [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF]. Based on that, he was able to represent more precisely the observed data. Later, in his book "A mathematical Theory of evidence" [START_REF] Shafer | A Mathematical Theory of Evidence[END_REF], Shafer, presented that, any information defined by an expert characterized by basic belief assignments has two functions: a credibility and a plausibility function corresponding respectively to the lower and upper probabilities of Dempster. The theory was further developed by Smets in [START_REF] Smets | The transferable belief model[END_REF], [START_REF] Smets | Constructing the pignistic probability function in a context of uncertainty[END_REF] who proposed the Transferable Belief Model (TBM). This model presents a pignistic probability induced by a belief function which is built by defining a uniform probability from each positive mass. Moreover, in terms of upper and lower probabilities, it can be considered as the center of gravity of the set of probabilities dominating the belief functions. He also introduced new tools for information fusion and decision making according to [START_REF] Dubois | New semantics for quantitative possibility theory[END_REF]. The objective of the evidential theory is to represent the information which is transmitted by a source concerning an event. A belief function must take in consideration all the possible events on which a source can describe a belief. Based on that, we can define the frame of discernment.

The frame of discernment is a finite set of disjoint elements noted Ω where Ω = {ω 1 , ..., ω n }. This theory allows us to affect a mass on a set of hypotheses not only a singleton like in the probabilistic theory. Thus, we are able to represent ignorance, imprecision and uncertainty.

m : 2 Ω → [0, 1]. ( 1
) X⊆Ω m(X) = 1. (2)
The principle functions in the belief functions theory are:

The credibility function (bel): this function measures the strength of the evidence in favor of a set of propositions for all X ∈ 2 Ω : bel

(X) = Y ⊆X,Y =∅ m(Y ). ( 3
)
The credibility is interpreted as a degree of justified support given to proposition X by the available evidence.

The plausibility function (pl): expresses the maximum amount of specific support that could be given to a proposition X ∈ 2 Ω . pl(X) is then obtained by summing the bba ′ s given to the subsets Y such that Y ∩ X = ∅ :

pl(X) = Y ∈2 Ω ,Y ∩X =∅ m(Y ). (4)
It measures the degree of belief committed to the propositions compatible with X.

The commonality function (q): this function measures the set of bbas affected to the focal elements included in the studied set, for all X ∈ 2 Ω :

q(X) = Y ⊇X m(Y ).
(5)

A. Inclusion as a conflict measure for discrete belief functions

Recently Martin in [START_REF] Martin | About conflict in the theory of belief functions[END_REF] defined a degree of inclusion as involved in the measurement made in order to determine the conflict during the combination of two discrete belief functions. The author presented an index of inclusion having binary values where Inc(X 1 , Y 2 ) = 1 if X 1 ⊆ Y 2 and 0 otherwise with X 1 , Y 2 being respectively the focal elements of m 1 and m 2 . This index is then used to measure the degree of inclusion of the two mass functions and defined as:

d inc = 1 |F 1 ||F 2 | X1∈F1 Y2∈F1 Inc(X 1 , Y 2 ) (6)
σ inc (m 1 , m 2 ) = max(d inc (m 1 , m 2 ), d inc (m 2 , m 1 )) (7)
Where d inc is the degree of inclusion of m 1 in m 2 and inversely. This inclusion is used as a conflict measure for two mass functions, using it like presented:

Conf (m 1 , m 2 ) = 1 -σ inc (m 1 , m 2 )d(m 1 , m 2 ) (8)
where d(m 1 , m 2 ), is the distance of Jousselme:

d(m 1 , m 2 ) = 1 2 (m 1 -m 2 ) T D(m 1 -m 2 ); (9)
where D is a matrix based on the dissimilarity of Jaccard expressed by

D(A, B) = 1 if A = B = ∅ otherwise, D(A, B) = |A∩B| |A∪B| if ∀A, B ∈ 2 Ω III. CONTINUOUS BELIEF FUNCTIONS
In the previous sections, we have presented different specifications of discrete belief functions. Unfortunately, these functions do not allow us to manipulate continuous data that can be provided by sensors in different areas like: search and rescue problems [START_REF] Doré | Belief functions induced by multimodal probability density functions, an application to the search and rescue problem[END_REF], classification issues, information fusion, etc. Some researches were interested in representing belief functions in continuous frame of discernment like Strat in [START_REF] Strat | Continuous belief function for evidential reasoning[END_REF], and Smets in [START_REF] Smets | Belief functions on real numbers[END_REF].

In following sections, we will present the several proposals that allows us to describe continuous belief functions. First, we explain how to extend these functions on the real numbers. To do so, we will focus on Smets' approach to represent continuous belief functions by using probability densities. Later, we will remember the other approaches for the continuous case of the theory of belief functions.

A. Continuous belief functions on IR

Smets based on the TBM's background, used the same representation than Strat, and proposed the belief functions in the extended set of reals noted IR = IR ∪ {-∞, +∞}. However, using the belief function framework to model information in a continuous frame is not an easy task mainly to the complex nature of the focal elements. Comparing to the discrete domain, on real numbers, in [START_REF] Smets | Belief functions on real numbers[END_REF]) bba becomes basic belief densities (bbd) defined on an interval [a, b] of IR.

1) Basic belief densities:

A generalization of the classical bba into a basic belief density (bbd) denoted m I on the interval I. He defined the bbd where all focal elements are closed intervals or ∅. Given a normalized bbd m I , Smets defined another function f on IR 2 , where:

f (a, b) = m I ([a, b]), a ≤ b, f (a, b) = 0, a > b. (10)
f is called a probability density function (pdf ) on IR 2 .

B. Continuous belief functions induced by probability density functions

Let's consider several belief functions characterizing a unique source of information (the source is subjective and evidential). Smets proposed a pignistic transformation of the belief functions (representing the knowledge of the source) in order to obtain probabilities. The probabilities are used to ease the decision making. Pignistic probabilities are noted BetP having densities also noted betf . For each probability density, we have a set of belief functions with which they are compatible. This set is called an Isopignitic. The main issue is to choose one belief function from this set. To do so, we consider the principle of least commitment proposed in [START_REF] Dubois | The principle of minimum specificity as a basis for evidential reasoning[END_REF], [START_REF] Hsia | Characterizing belief with minimum commitment[END_REF]. This principle supports the idea of choosing the belief function that involves the least an expert. It can be considered as a natural approach to select the less committed bba from a subset. A particular type of belief functions describes the best this principle which are the consonant belief functions where focal elements are nested [START_REF] Dubois | Consonant approximations of belief functions[END_REF].

C. Continuous belief functions: other representations

Some other approaches have been proposed in order to describe continuous belief functions. [START_REF] Nguyen | On random sets and belief functions[END_REF] introduced in the notions of a source constituted by a probability space and a multivalued mapping which is able to define the lower probability defined by a Γ function.

This function can hold at the same time two notions: on one hand, it defines both of the lower and upper probability, on the other hand, it considers random sets. We can say that Γ as a multivalued mapping is measurable with respect to the spaces that it characterizes.

Moreover, he supposes that Γ is a measurable mapping, then it is a random set by specifying its probability distribution. Thus the probability distribution of a random set Γ is precisely the basic probability assignment.

We say that there is a correspondence established between belief functions on a source S and the probability distribution of random sets. This relation can be expressed by its density on P (S). [START_REF] Doré | Continuous belief functions to qualify sensors performances[END_REF], proposed a similar approach founded on an index function that can be assumed as Γ. This function can describe the set of focal elements of a continuous belief function. In this case, every index has its own probability measure where there is an allocated weight to a set of focal elements using a credal measure. Every set is described according to its index and its probability density.

Doré et al. in

The formalism of Smets takes into consideration only to closed intervals, in [START_REF] Doré | Belief functions induced by multimodal probability density functions, an application to the search and rescue problem[END_REF], the author extended classical continuous belief functions by proposing belief functions where focal elements are not represented by intervals. He uses αcuts to measure to area of the portions of multimodal distributions.

D. Similarity measure within continuous belief functions

Attiaoui et al. in [START_REF] Attiaoui | A distance between continuous belief functions[END_REF] proposed a similarity measure based on the distance of Jousselme using Smets' formalism. This distance uses a scalar product as a scalar product is defined on IR by:

f, g = +∞ x=-∞ +∞ y=-∞ f ([x, y])g([x, y])dxdy (11)
Here, the authors presented a new method to measure the similarity founded on the properties of belief functions on real numbers, they were able to define a distance between two densities in an interval I.

f 1 , f 2 = +∞ -∞ +∞ yi=xi +∞ -∞ yj =+∞ yj =xj (12)
f 1 (x i , y i )f 2 (x j , y j )δ(x i , x j , y i , y j )dy j dx j dy i dx i

The scalar product of the two basic belief densities is noted: f 1 , f 2 with a function δ defined as δ : IR -→ [0, 1] δ(x i , x j , y i , y j ) = λ( max(x i , x j ), min(y i , y j ) ) λ( max(y i , y j ), min(x i , x j ) )

where λ represents the Lebesgue measure used for the interval's length and δ(x i , x j , y i , y j ) is an extension of the dissimilarity of Jaccard applied for the intervals in the case of continuous belief functions.

a, b = ∅, if a > b [a,b], otherwise. (14)
Therefore, the distance between two basic belief densities is defined by the following equation:

d(f 1 , f 2 ) = 1 2 ( f 1 2 + f 2 2 -2 f 1 , f 2 ) (15)
We noticed that the standard deviation is influencing this measure. As long as the difference between the distributions grows, the more the distance is rising. This representation proposed a natural approach that allows us to manipulate and also study the behavior of continuous belief functions induced by normal and exponential distributions.

IV. INCLUSION WITHIN CONTINUOUS BELIEF FUNCTIONS

Within this section, we will consider two forms of inclusion: a strict and a partial inclusion. We will present their mathematical expressions, and explain how do we build them. But first, we enumerate several properties that must be satisfied by the relation of inclusion.

A. Properties of the inclusion

The inclusion defined between two intervals [x i , y i ] and [x j , y j ] in a set I satisfies the following requirements:

Property 1: Non-negativity

Inc(f i , f j ) ≥ 0.
Namely, the inclusion of the first interval in the second one must never be negative.

Property 2: asymmetry

Inc(f i , f j ) = Inc(f j , f i ), ∀f i = f j .
No need for the relation of inclusion to be symmetric.

Property 3: Upper bound

Inc(f i , f j ) = 1.
This property implies a total inclusion of the first interval in the second one other.

Property 4: Lower bound

Inc(f i , f j ) = 0.
This property implies the absence of any intersection or inclusion of the first interval in the second one.

B. Strict inclusion

Here, we will define the strict inclusion between two continuous belief functions represented by two basic belief densities bbd. First, we use these distributions to deduce a degree of inclusion between the bbds and then we can be able to measure the value inclusion between our continuous belief functions. Let's consider two continuous pdf s: f 1 and f 2 . If one distributions is included in the second one, then the strict inclusion is expressed by the following equation:

IncStr(f 1 , f 2 ) = +∞ -∞ +∞ yi=xi xi=+∞ xj =-∞ yj =+∞ yj =xj (16)
δ IncStr (x i , y i , x j , y j )f 1 (x i , y i )f 2 (x j , y j )dy j dx j dy i dx i

Where [x i , y i ], [x j , y j ] are the considered intervals and δ IncStr (x i , y i , x j , y j ) is the degree of strict inclusion that will allow us to measure the value related to the inclusion of the first interval in the second one.

We will consider that δ IncStr (x i , y i , x j , y j ) is having binary values where:

δ IncStr (x i , y i , x j , y j ) = 1, if [x i , y i ] ⊆ [x j , y j ] 0, otherwise. (17)
If we are in presence of two distributions that do touch each other, there is an intersection between them. The δ IncStr (x i , y i , x j , y j ) will have the value 1, and the strict inclusion will be weighted by the masses of our continuous belief functions. Otherwise δ IncStr (x i , y i , x j , y j ) will be null.

C. Partial inclusion

Considering two bbds represented by two intervals [x i , y i ] and [x j , y j ]: We say that [x i , y i ] is partially included in [x j , y j ] or inversely, if and only if their intersection is different of ∅. To represent the partial inclusion we define: δ IncP ar (x i , y i , x j , y j )f 1 (x i , y i )f 2 (x j , y j )dy j dx j dy i dx i with δ IncP ar (x i , y i , x j , y j ) is the degree of partial inclusion: δ IncP ar (x i , y i ,x j , y j )= max(0,min(y j , y i )-max(x i , x j ))

y i -x i (19)
The degree δ IncP ar (x i , y i , x j , y j ) represents the length of the intersection of the two probability density functions f 1 and f 2 on the length of f 1 if we are measuring IncP ar(f 1 , f 2 ) and the length of f 2 if we have to calculate the partial inclusion of f 2 in f 1 : IncP ar(f 2 , f 1 )

V. ASYMMETRY WITHIN THE INCLUSION

Let us consider four probability density functions defined by their means µ, and their standard deviations σ, like presented in table I and described in figure 1.

Once we apply the mathematical formula proposed in equation [START_REF] Dubois | The principle of minimum specificity as a basis for evidential reasoning[END_REF], we obtain the following table II strict inclusion. Otherwise, applying the partial inclusion expressed in equation [START_REF] Dubois | Consonant approximations of belief functions[END_REF], we obtain table III.

The property of asymmetry between two continuous belief functions can be confirmed when we observe the measures of inclusion in the table II and table III .

We witness for strict and partial inclusion that IncStr(pdf 2 , pdf 1 ) = IncStr(pdf 3 , pdf 4 ) = 0.8586 and IncP ar(pdf 2 , pdf 1 ) = IncP ar(pdf 3 , pdf 4 ) = 0.9595 do always have the highest values on both cases. These primary results confirm the distributions presented in Figure 1, Meanwhile, for the case of inclusion of pdf 1 in pdf 4 and pdf 4 in pdf 1 , where the difference between the means is very important we are dealing with very small values of inclusion. Knowing that σ 1 = σ 3 and σ 2 = σ 4 , when computing these partial inclusions, the difference between the standard deviations is maximal, generates a considerable value. For the strict inclusion we have some values where there is no intersection between the distributions and then we naturally obtain a null value, like for IncStr(pdf 1 , pdf 4 ).

According to the data in table II and table III, the primary results obtained using both of the strict and the partial inclusion, our proposed relation responds to all the properties previously announced (non negativity, asymmetry, upper-bound < 1).

A. Average of inclusion

Let us consider n distributions, and αf a set of bbds.We can measure the average of inclusion of a bbd f i in αf .

To do so, we present the following equations the first one related to the strict inclusion and the second one to the partial.

IncS(f i , αf ) = 1 n -1 n j=1,i =j IncStr(f i , f j ) (20) IncP (f i , αf ) = 1 n -1 n j=1,i =j IncP ar(f i , f j ) (21)
To model the measurement of the average of inclusion, we will apply the equations ( 19) and (20) to obtain respectively table IV and table V

VI. ILLUSTRATION OF THE STRICT AND PARTIAL

INCLUSIONS

In this section we will present our experimental phase to illustrate both of the strict and partial inclusion between two continuous belief functions. To illustrate both of the strict and partial inclusion between two continuous belief functions induced using a normal distribution. We decided to fix the value of the first distribution pdf 1 , where, it is characterized by its mean µ 1 = 0 and its standard deviation σ 1 = 1. For the second distribution pdf 2 , we will vary the values of µ 2 in [0, 10] and σ 2 in [0, 5].

Here, our purpose is to see the behavior of the inclusion when we modify one of the pdf s, and the parameters that infer in the obtained results.

A. Strict inclusion between belief densities induced by normal distributions

This strict inclusion is a natural approach, that allows us to perceive if there exists any intersection between two distributions, how they behave and the parameters that interfere during this process. In this case, when a degree of inclusion has binary values of 0 and 1, we will study this property between two continuous belief functions.

The more the value of the second standard deviation grows, the more difference between σ 1 and σ 2 develops. Here, the

Fig. 1 .

 1 Fig. 1. Modeling probalility distribution functions.

Fig. 2 .

 2 Fig. 2. Strict inclusion of pdf 1 in pdf 2.

TABLE I PROBABILITY

 I DENSITY DISTRIBUTIONS

strict inclusion between the two distributions, which generates a growth in the behavior of the obtained curve. We are then, in the presence of a growth of the strict inclusion of pdf 1 in pdf 2 due to the variation of the second distribution. Otherwise, we notice in Figure 2, relative to the strict inclusion, that it is not the gap between the means µ 1 and µ 2 that generates this growth. It is the difference between the two standard deviations, that is in the origin of this growing phenomenon of inclusion. As shown when σ 1 > σ 2 and µ 2 having its maximal value with µ 2 = 5, the strict inclusion is null. This value is due to the fact that the degree of strict inclusion δ IncStr (x i , y i , x j , y j ) = 0, which can be explained by the non-existence of any intersection between pdf 1 and pdf 2 . The part of the curve where the difference between the two standard deviations is low and at the same time the difference between the means is high. We have a small growing strict inclusion for example when µ 2 = 3 and σ 2 = 1.5, the strict inclusion IncStr(f 1 , f 2 ) = 0.1. At the meantime, even when µ 1 = µ 2 = 0 σ 1 = σ 2 = 1, there is an intersection between the two distributions and IncStr(f 1 , f 2 ) > 0. Moreover, when the gap between σ 1 and σ 2 gets higher, the curve increases, generating a bigger strict inclusion between pdf 1 and pdf 2 , until rising its maximal value where IncStr(f 1 , f 2 ) = 1, with δ IncStr (x i , y i , x j , y j ) = 1. In this specific case, we are in presence of a total inclusion of the first distribution pdf 1 in the second one pdf 2 .

We also witness a phenomenonin which µ 2 gets a high value, the strict inclusion drops considerably. Here we can state that the mean also has an impact on the generated inclusion.

B. Partial inclusion between belief densities induced by normal distributions

The partial inclusion is defined in order to give us the proportion of the intersection between two pdf s.

1) Partial inclusion of pdf 1 in pdf 2: During this experimentation, we keep the same values used for the strict inclusion and we obtain Figure 3 We notice that, when we are dealing with similar distribution where µ 1 = µ 2 = 0 and σ 1 = σ 2 = 1, the value of the partial inclusion is greater than zero. It is possible to state that when we are in presence of two distributions having the same values, there is not necessarily any total inclusion between them. We also take note, that as the difference between σ 1 and σ 2 rises, due to the of pdf 2 , the figure obtained grows faster, and reaches its maximal value IncP ar(f 1 , f 2 ) = 1, generating a curve more arched than the strict inclusion.

When the difference between µ 1 and µ 2 increases because of the variation of pdf 2 , the partial inclusion reaches a value of IncP ar(f 1 , f 2 ) = 0.85, which becomes lower when the gap between two standard deviations is the highest (σ 2 = 3), we obtain the maximal value for the partial inclusion: IncP ar(f 1 , f 2 ) = 1 like presented in Figure 3. In this specific case, we witness a full and total inclusion of pdf 1 in pdf 2 . This is similar to what we have presented regarding the strict inclusion. Here we have non-negative inclusions, that respect the lower and upper bounds where the values are [0, 1]. Besides, the property of asymmetry is also respected because, the inclusion (what ever is strict or partial) of a distribution in the other does not necessarily involve the inverse case with the same value.

When IncP ar = 1, we have a total inclusion of the first distribution pdf 1 in the second one pdf 2 , this situation is considered as a strict inclusion where, pdf 1 is fully included in pdf 2 .

2) Partial inclusion of pdf 2 in pdf 1: For this case, we have chosen to measure the partial inclusion of pdf 2 in pdf 1 , saving the same values for both distributions. Otherwise, the equation 18 becomes: δ IncP ar (x i , y i ,x j , y j )= max(0,min(y i , y j )max(x j , x i ))

We obtain the Figure 4. In this case, we notice a different phenomenon comparing to Figure 3. Here, we witness than when the two distributions are totally similar, the value of the partial inclusion equals 0.8183. This value is the maximal one that is expressed in this, and as stated before, two similar distributions can not be fully included in each other. We also observe in this figure, that as long as tha values of the second distribution µ 2 , and σ 2 rise, the value of the partial inclusion of pdf 2 in pdf 1 drops. More the two distributions are getting different from each other, more the partial inclusion decreases. Both parameters; the mean and the standard deviation have considerable impacts in this measure. This is proved by the behavior of the partial inclusion.

The difference between the Figure 3 and Figure 4 is very obvious considering the behaviors of the two curves. This is due to the nature of the focal elements of continuous belief functions which are expressed by intervals. Specially with the case of the inclusion where each time we measure the inclusion of the intervals of a normal distribution with those of a second normal distribution.

VII. STRICT INCLUSION VS PARTIAL INCLUSION

Comparing the results obtained in Figure 2 Figure 3, where we have the same values for the two distributions, we notice that the partial inclusion reaches the maximal value faster that the strict one. Besides, its area is bigger and larger. Thus, we can state that the partial inclusion is dominating. In Figure 2 and Figure 3, it seems that we have the same phenomena between the two types of inclusion. This can be explained by the fact that we are working with consonant belief functions (where focal intervals are nested). We can imagine that in presence in other nature of belief functions (categorical, or Bayesian belief functions) we can obtain different behaviors between the strict and the partial inclusion.

We notice the same phenomena that we have seen for the strict, when the values of parameters characterizing the second distribution grow, generating a big difference between the means, and the standard deviations, the partial inclusion decreases significantly and comparing when we are dealing with smaller values of µ 2 .

Thus, we can say that, both of the mean and the standard deviation do have a real impact on the measurement of the partial inclusion, having the same situation as the strict inclusion.

In the case of two distributions getting more and more different and especially when considering only the standard deviation, the phenomena of inclusion gets bigger. However, if the mean of one distribution is having non similar value than the other we can state than the inclusion has smaller values.

VIII. CONCLUSION

In this paper, we have emphasized the evaluation of the relation of inclusion between continuous belief functions induced by a normal distribution. We have defined two forms of inclusions: the strict and the partial one. Before that, we have detailed all preliminary background that will allow us to experiment this kind of relation. We have also provided the approach on which the evaluation of the inclusions will be based.

We have presented a relation of inclusion having normalized values that takes into account the nature of continuous belief functions described using intervals as focal elements. These two forms of inclusion respond to all the properties that must be satisfied. We also have shown that both of the mean and the standard deviation have different impacts in this phenomena each one with its specific output.