Jean-Christophe Pain
email: jean-christophe.pain@cea.fr

Kamerlingh Onnes pour la découverte de la supraconductivité

de Niels Bohr sur la structure atomique et le prix Nobel de

LES TROIS ARTICLES DE NIELS BOHR PARUS EN 1913

Niels Henrik David Bohr (1885Bohr ( -1962) ) entre en 1903 à l'université de Copenhague où il obtient un doctorat en 1911 intitulé « Sur la théorie électronique des métaux », contenant ses premières idées sur la structure atomique. Il obtient une bourse de la fondation Carlsberg (créée par le fondateur de la célèbre brasserie) et souhaite dans un premier temps travailler à l'université de Cambridge avec le professeur Joseph John Thomson dont le modèle atomique, sphère de charge positive dans laquelle sont plongés les électrons, ne le satisfait pas totalement. Bohr rencontre alors Ernest Rutherford (qui obtiendra le prix Nobel de chimie en 1908 pour la découverte de la radioactivité accompagnant la désintégration des éléments chimiques) qu'il rejoint à Manchester. Il rentre au Danemark en 1912 et se marie peu de temps après. De cette union naîtront six garçons, le plus connu étant Aage Bohr, lauréat du prix Nobel de physique de 1975 avec Ben Roy Mottelson et Leo James Rainwater pour la découverte du lien entre les mouvements collectifs et particulaires dans les noyaux atomiques et le développement de la théorie de la structure du noyau qui en résulte.

C'est en 1913 qu'un ami étudiant, Hans Magnus Hansen, demande à Bohr, alors assistant à la chaire de physique de l'université de Copenhague, si son modèle pourrait apporter des informations sur les spectres atomiques ; Bohr lui ayant répondu par la négative, Hansen lui conseille de jeter un coup d'oeil à la formule de Balmer. « Dès que je vis la formule de Balmer, tout devint clair pour moi », dira Bohr bien des années plus tard [START_REF] Van Delft | [END_REF]. En s'appuyant sur les théories de Rutherford, Bohr publie en 1913 un modèle de la structure de l'atome et de la liaison chimique dans une série de trois articles dans la revue « Philosophical Magazine » [START_REF] Onnes | On the Cryogenic Laboratory at Leyden and on the Production of Very Low Temperature[END_REF]3,4] dans lesquels il postule l'existence d'états stationnaires pour un système atomique dont le comportement peut être décrit par la mécanique classique, alors que la transition d'un état stationnaire à un autre est un processus non classique accompagné de l'émission ou l'absorption d'un quantum de rayonnement dont l'énergie est égale à la fréquence multipliée par la constante de Planck. Cette théorie présente l'atome comme un noyau autour duquel gravitent des électrons. Les orbites les plus éloignées, comprenant le plus d'électrons, déterminent les propriétés chimiques de l'atome. Les électrons ont la possibilité de passer d'une orbite à une autre, émettant un quantum d'énergie, le photon. Cette théorie est à la base de la mécanique quantique et Albert Einstein s'y intéresse de très près dès sa publication.

Durant la Première Guerre mondiale, le Danemark est un État neutre et Bohr peut rester dans la recherche civile, même en Angleterre où il occupe un poste de professeur à l'université de Manchester depuis 1914. Il en profite pour peaufiner son modèle atomique en y introduisant des effets relativistes du mouvement des électrons, théorie qui sera reprise et complétée par Arnold Sommerfeld. En 1916, Bohr devient professeur à l'université de Copenhague dans la chaire de physique théorique fraîchement créée. Il fonde un institut de physique théorique, qui prendra le nom de « Niels Bohr Institut » en 1965. Pendant les années 1920, il complète sa théorie, parvenant à établir une relation étroite entre le tableau de Mendeleïev et la structure électronique des atomes. Il reçoit le prix Nobel de physique en 1922 « pour ses études de la structure des atomes et des radiations qui en proviennent », publiées dans la « trilogie » de 1913.

Bohr est aussi à l'origine du principe de complémentarité : la lumière est à la fois une onde et un faisceau de photons, mais ces deux aspects ne peuvent être perçus simultanément. Il travaille également sur le principe de complémentarité en biologie. En octobre 1927, il rencontre pour la première fois Albert Einstein au cours du cinquième congrès Solvay avec qui il aura de très fréquents entretiens jusqu'en 1935. Einstein dira de lui «… que ces fondements contradictoires et peu sûrs aient suffi à un homme doué de l'instinct et de la perspicacité uniques de Bohr pour découvrir la loi capitale des raies spectrales et des couches électroniques de l'atome en même temps que leur signification du point de vue chimique m'apparut comme un miracle et apparaît encore comme tel aujourd'hui. C'est la forme la plus élevée de la musicalité dans la sphère de la pensée » [5]. Cependant, la théorie quantique ne satisfait pas Einstein du point de vue épistémologique [6]. Bohr, au contraire, considère qu'il s'agit d'une théorie parfaitement achevée. Lors d'un débat devenu célèbre, alors que Niels Bohr se dispute avec Albert Einstein à propos de la réalité de la physique quantique, Einstein, excédé, lui lance : « Dieu ne joue pas aux dés ! », ce à quoi Bohr répond : « Qui êtes-vous Einstein, pour dire à Dieu ce qu'il doit faire ! » [START_REF] Bohr | Discussion avec Einstein sur les problèmes épistémologiques de la physique atomique[END_REF]. Ces réflexions et discussions sont à l'origine de ses « Écrits philosophiques », publiés en quatre volumes (dont deux à titre posthume en 1963 et en 1998), dans lesquels il aborde notamment la question du langage.

À

BIBLIOGRAPHIE ET NETOGRAPHIE

[1] Segrè E. « Les physiciens modernes et leurs découvertes, des rayons X aux quarks », Fayard, collection « Le temps des sciences », 1990.

Encadré

Johan Jakob Balmer (1825-1898), professeur de lycée suisse, passionné de théorie des nombres, avait remarqué une régularité frappante dans les fréquences des raies spectrales de l'hydrogène.

Encadré

Le bohrium est un élément chimique, de symbole Bh et de numéro atomique 107. C'est un élément synthétique dont la demi-vie de l'isotope le plus stable 262 Bh est inférieure à 102 millisecondes. Huit isotopes ont été identifiés, tous radioactifs, du 260 Bh au 267 Bh et l'Union Internationale de Chimie Pure et Appliquée (UICPA) propose l'appellation « bohrium » en 1994 ; une polémique s'ensuit alors par crainte que ce nom puisse entraîner des confusions avec le nom anglais « boron » du bore (« nielsbohrium » est alors envisagé), mais bohrium est finalement adopté en 1997.

HEIKE KAMERLINGH ONNES, PRIX NOBEL DE PHYSIQUE 1913

Le physicien hollandais Heike Kamerlingh Onnes (1853Onnes ( -1926) ) reçoit le prix Nobel en 1913 pour ses travaux sur les propriétés de la matière à basse température conduiront, entre autres, à la production de l'hélium liquide. Onnes débute ses études universitaires à Groningen aux Pays-Bas. Il étudie ensuite à l'Université de Heidelberg (Allemagne) de 1871 à 1873 sous la direction de Robert Wilhelm Bunsen et Gustav Kirchhoff. Il retourne ensuite à Groningen où il soutient en 1879 sa thèse de doctorat intitulée « Nouvelles preuves de la rotation de la terre » [START_REF] Van Delft | [END_REF]. Onnes est l'élève de Johannes Diderik van der Waals (prix Nobel de physique en 1910 pour ses travaux sur l'équation d'état des gaz et des liquides).

De 1882 à 1923, Onnes est professeur de physique expérimentale à l'Université de Leyde. En 1904, il fonde un grand laboratoire de cryogénie et y attire d'autres chercheurs, ce qui contribue à sa reconnaissance par la communauté scientifique. Il est le premier à réussir à liquéfier de l'hélium le 10 juillet 1908 à l'aide de cryostats. En utilisant l'effet Joule-Thomson, il parvient à faire diminuer la température jusqu'à moins de 1 degré au-dessus du zéro absolu ; il atteint 0,9 K qui est la température la plus basse jamais atteinte alors [START_REF] Onnes | On the Cryogenic Laboratory at Leyden and on the Production of Very Low Temperature[END_REF]3].

A cette époque, certains scientifiques, comme William Thomson (Lord Kelvin), pensent qu'au sein d'un conducteur les électrons devraient être immobiles au zéro absolu, ce qui devrait correspondre à une résistivité électrique infinie. D'autres, dont Onnes, pensent que cette résistivité doit décroître progressivement jusqu'à zéro. En effet, Augustus Matthiessen a montré vers 1860 que la résistivité augmente généralement avec la température dans les métaux. Fidèle à sa devise « Door meten tot weten » (« par l'expérience, la connaissance »), Onnes entreprend de mesurer la résistance électrique du mercure refroidi par de l'hélium liquide.

Le 8 avril 1911, il mesure l'annulation de la résistivité électrique du mercure au-dessous d'une certaine température appelée température critique Tc, de l'ordre de 4,2 K. C'est la première observation d'un état supraconducteur [4] : Onnes écrit alors que « le mercure est passé dans un nouvel état, qui du fait de ses propriétés électriques extraordinaires pourrait être appelé état supraconducteur ». On raconte parfois qu'Onnes aurait demandé à son étudiant Gilles Holst de surveiller une expérience, et que ce dernier aurait laissé le dispositif expérimental s'emballer, conduisant à des températures extrêmement faibles. Le véritable découvreur de la supraconductivité serait donc Gilles Holst ! En réalité, le cahier d'expériences écrit de la main même de Kamerlingh Onnes et découvert récemment par le directeur du Boerhaave Museum de Leiden montre que Onnes était bel et bien aux commandes de l'expérience ce jour-là [5], Gilles Holst mesurant la résistance électrique avec un Pont de Wheatstone à 30 mètres de distance (la pièce où était refroidi le mercure subissait trop de vibrations à cause des pompes). La Société Française de Physique a célébré en 2011 les 100 ans de la découverte de la supraconductivité par une série d'articles [6].

En 1950, une théorie phénoménologique dite de Ginzburg-Landau est élaborée par Lev Landau et Vitali Ginzburg. Cette théorie permet d'expliquer les propriétés macroscopiques des supraconducteurs grâce à l'équation de Schrödinger. En particulier, Alexeï Abrikossov montre qu'avec cette théorie on peut prévoir l'existence de deux catégories de supraconducteurs. Abrikossov et Ginzburg se verront attribuer, avec Anthony Leggett, le prix Nobel 2003 pour leurs contributions pionnières à la théorie des supraconducteurs et des superfluides (Landau étant décédé en 1968).

[ 2 ]

 2 Bohr Niels. « On the Constitution of Atoms and Molecules, Part I ». Philosophical Magazine, 1913, vol. 26, p. 1-24.

  la fin des années 1930, les recherches de Bohr s'orientent vers le noyau atomique. En 1943, il s'échappe du Danemark occupé vers les États-Unis et travaille au Laboratoire national de Los Alamos dans le cadre du projet Manhattan. Après la guerre, il rentre à Copenhague et milite pour une utilisation pacifique de l'énergie nucléaire, en particulier avec la création du Laboratoire national Risø (équivalent du Commissariat à l'Energie Atomique français) en 1956, ce qui lui vaut d'être lauréat de l' « Atoms for Peace Award » en 1957 (son fils Aage recevra également ce prix en 1969). Niels Henrik David Bohr meurt à Copenhague le 18 novembre 1962. En son honneur, l'élément de numéro atomique 107est baptisé bohrium en 1997 et l'Union astronomique internationale nomme une vallée lunaire la « Vallis Bohr » en 1976.

En 1957, un chimiste Hollandais découvre le premier supraconducteur organique synthétique, le di-(tétraméthyl-tétrasélénofulvalène)-hexafluorophosphate. Une théorie complète de la supraconductivité sera proposée en 1957 par John Bardeen, Leon Cooper et John Schrieffer. Connue sous le nom de théorie BCS, elle explique la supraconductivité par la formation de paires d'électrons (paires de Cooper) se comportant comme des bosons interagissant avec des phonons (modes de vibration du solide). Pour leur travail, les auteurs recevront le prix Nobel de physique en 1972. Le nom de Kamerlingh Onnes a été donné à un cratère lunaire.

BIBLIOGRAPHIE ET NETOGRAPHIE


Yassine Ariba
email: yariba@laas.fr

Frédéric Gouaisbaut

Input-output framework for robust stability of time-varying delay systems

The paper is devoted to the stability analysis of linear time varying delay. We first model the time varying delay system as an interconnected system between a known linear transformation and some operators depending explicitly on the delay. Embedding operators related to the delay into an uncertain set, stability of such system is then performed by adopting the quadratic separation approach. Having recognized that the conservatism comes from the choice of the feedback modeling and the operators definition, these first results are afterwards enhanced by using some redundant equation and scaling filter. At last, numerical examples are given to illustrate the results.

INTRODUCTION

Stability of linear time-delay systems has been intensively studied since several decades (see [START_REF] Fridman | Input-output approach to stability and l 2 -gain analysis of systems with timevarying delays[END_REF], [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF], [START_REF] Gu | Stability of Time-Delay Systems[END_REF] and references therein). A such success can be explained by their applied aspect. Indeed, many processes include dead-time phenomena such as biology, chemistry, economics, as well as population dynamics [START_REF] Kolmanovskii | Introduction to the Theory and Applications of Functional Differential Equations[END_REF] [START_REF] Niculescu | Delay Effects on Stability. A Robust Control Approach[END_REF]. Processing time and propagation time in actuators and sensors generally induce also such delays, especially if some devices are physically distant. That is the challenge of networked controlled systems [START_REF] Bushnell | Networks and control[END_REF] as well as network control [START_REF] Tarbouriech | Advances in communication Control Networks[END_REF].

In the case of constant delay and unperturbed linear systems, efficient criteria based on roots location [START_REF] Olgac | An exact method for the stability analysis of time-delayed linear time-invariant (lti) systems[END_REF] allow to find the exact region of stability with respect to the value of the delay. Beside these direct methods, numerous works based either on Lyapunov functionals [START_REF] Gu | Stability of Time-Delay Systems[END_REF][21] [START_REF] Fridman | A descriptor system approach to H ∞ control of linear time-delay systems[END_REF] or robustness framework (small gain theory [START_REF] Gu | Stability of Time-Delay Systems[END_REF], IQC [START_REF] Kao | Robust stability analysis of linear systems with time-varying delays[END_REF] or quadratic separation [START_REF] Gouaisbaut | Robust stability of time-delay systems with interval delays[END_REF]) have established interesting results to tackle the robust stability of delay systems with practical tools (like LMI). All resulting stability conditions are based on convex optimization (linear matrix inequality framework) and allow to conclude on stability intervals with respect to the delay and/or the uncertainty.

Regarding the case of time-varying delay systems, some authors have extended the upper cited results to address the stability issue of such systems. Nevertheless, the time-varying nature of the delay should be carefully handled rather than roughly adapted from methods originally developed for the constant delay case. In the Lyapunov-Krasovskii approach few studies [START_REF] Wu | Delay-dependent criteria for robust stability of time-varying delay systems[END_REF][7] [START_REF] He | Further improvement of free-weighting matrices technique for systems with time-varying delay[END_REF] [START_REF] Ariba | Delay-dependent stability analysis of linear systems with time-varying delay[END_REF] have proposed customized functionals able to significantly improve classical results. In the input-output approach [START_REF] Gu | Stability of Time-Delay Systems[END_REF][8] [START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF] [START_REF] Ariba | Stability analysis of time-varying delay systems in quadratic separation framework[END_REF], some terms (or operators) related to the delayed dynamics are embedded into an uncertain matrix and the method consists in ensuring the robust stability of the nominal system with respect to the uncertain domain. Hence, in one hand, the key issue relies on the choice of the interconnection modeling the delay systems (and thus the uncertain set), and in other hand, on the L 2 -norm bounds which fit the uncertain set. Although the Lyapunov and the input-ouput approaches are closely related [START_REF] Zhang | Stability of time-delay systems: Equivalence between Lyapunov and scaled small-gain conditions[END_REF][9], the second one states clearly the reasons of the conservatism and how it may be reduced.

In this paper, the quadratic separation principle, belonging to the input-output framework, is considered to deal with the stability analysis of linear time-varying delay systems. First, such systems are modeled as the interconnection of a linear matrix equation with an uncertain matrix of operators. Secondly, based on previous results [START_REF] Ariba | Stability analysis of time-varying delay systems in quadratic separation framework[END_REF] and [START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF] which provide bounds on some operators, integral quadratic constraints are built. At last, a redundant equation is introduced to construct a new modeling of the delay systems. To this end, an augmented state is considered which is composed of the original state vector and its derivatives. Then defining relationship between augmented states ẋ, ẍ, the delay h and its derivative ḣ as a set of integral quadratic constraints allows to improve the stability criterion. Conditions are expressed in terms of linear matrix inequalities (LMI) which can be solved efficiently with semi-definite programming (SDP) solvers.

After the introduction, the paper carries on with the definition of some operators and preliminaries on quadratic separation useful to present the main result. In section 3, the prior result on robust stability is exploited to derive a stability condition for time-varying delay systems. The robust stability for the case of uncertain systems is also addressed. Numerical examples that show the effectiveness of the proposed critera is provided in section 4.

PRELIMINARIES

Notations and problem statement

Throughout the paper, the following notations are used. The set of L n 2 consists of all measurable functions f : R + → stands for the block diagonal matrix:

C n such that the following norm f L 2 = ∞ 0 ( f * (t) f (t))
diag(A, B,C) =       A 0 0 0 B 0 0 0 C       .
Let consider the following time-varying delay system:

     ẋ(t) = Ax(t) + A d x(t -h(t)) ∀t ≥ 0, x(t) = φ (t) ∀t ∈ [-h max , 0] (1)
where x(t) ∈ 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 time-varying and the following constraints are assumed

h(t) ∈ [0, h max ] and | ḣ(t)| ≤ d, (2)
where h max and d are given scalar constants and may be infinite if delay independent condition and fast-varying delay condition, respectively, are looked for.

Stability analysis via quadratic separation

Coming from robust control theory, the quadratic separation provides a fruitful framework to address the stability issue of non-linear and uncertain systems [START_REF] Iwasaki | Well-posedness of feedback systems: insights into exact robustnessanalysis and approximate computations[END_REF], [START_REF] Peaucelle | Quadratic separation for feedback connection of an uncertain matrix and an implicit linear transformation[END_REF]. Recent studies [START_REF] Gouaisbaut | Robust stability of time-delay systems with interval delays[END_REF] have shown that a such framework allows to reduce significantly the conservatism of the stability analysis of time-delay systems with constant delay. Then, in order to deal with the time-varying delay case, the quadratic separation method has been extended in [START_REF] Ariba | Stability analysis of time-varying delay systems in quadratic separation framework[END_REF] to handle not only the case of uncertain matrices but more generally uncertain operators. Indeed, based on the inner product and the L 2e space a suitable theorem is then proposed. This latter will be later used to derive stability conditions for time-varying delay systems.

Let consider the interconnection defined by Figure 1 where E and A are two, real valued, possibly non-square matrices and ∇ is a linear operator from L 2e to L 2e . For simplicity of notations, we assume in the present paper that E is full column rank. Assuming the well-posedness, we are interested in looking for conditions that ensure the stability of the interconnection.

Theorem 1 The interconnected system of Figure 1 is stable if there exists a Hermitian matrix Θ = Θ * satisfying both conditions

E -A ⊥ * Θ E -A ⊥ > 0 (3) ∀u ∈ L 2e ,    1 ∇    u, Θ    1 ∇    u ≤ 0 (4)
Proof 1 Inspired from [START_REF] Peaucelle | Quadratic separation for feedback connection of an uncertain matrix and an implicit linear transformation[END_REF], the proof is detailed in [START_REF] Ariba | Stability analysis of time-varying delay systems in quadratic separation framework[END_REF].

Basically, inequality (4) which forms an integral quadratic constraint, is built from definitions and informations on different operators which compose the matrix ∇. Then, the other one (3) provides the stability condition of the interconnection.

MAIN RESULTS

Defining operators

Toward modeling delay system as an interconnected system such as illustrated on Figure 1, it is required to define appropriate operators. Define the integral operator

I : L 2e → L 2e , x(t) → t 0 x(θ )dθ , (5)
and the delay operator (or shift operator)

D : L 2e → L 2e , x(t) → x(t -h), (6)
which constitute the fundamental elementary operators to describe a delay system. The related integral quadratic constraints are introduced in the following two lemmas. These latters will be helpful to construct inequality [START_REF] Bushnell | Networks and control[END_REF] and to derive then stability criteria for linear systems with time-varying delays in the next section.

Lemma 1 An integral quadratic constraint for the operator I is given by the following inequality ∀x ∈ L n 2e and for a positive definite matrix P,   

1 n I 1 n    x,    0 -P -P 0       1 n I 1 n    x < 0 Proof 2 Simple calculus shows that ∀T > 0, ∀x ∈ L n 2e , (x being truncated: x(t) = 0, ∀t > T )    1 n I 1 n    x,    0 -P -P 0       1 n I 1 n    x = -2 T 0 x(t) T P t 0 x(s)ds = -2 T 0 d dt (I x) T P(I x)dt = -( T 0 x(s)ds) T P( T 0 x(s)ds) < 0
The second step is to derive a parameterized IQC for the operator D :

Lemma 2 An integral quadratic constraint for the operator D is given by the following inequality ∀T > 0, ∀x ∈ L n 2e and for a positive matrix Q,

   1 n D1 n    x,    -Q 0 0 Q(1 -ḣ)       1 n D1 n    x < 0 (7)
Proof 3 We get that ∀T > 0, ∀x ∈ L n 2e , (x being truncated:

x(t) = 0, ∀t > T )    1 n D1 n    x,    -Q 0 0 Q(1 -ḣ)       1 n D1 n    x = -+∞ 0 x T (u)Qx(u)du + ∞ 0 x T d (t)Qx d (t)(1 -ḣ(t))dt = -+T 0 x T (t)Qx(t)dt + T -h(T ) -h(0) x T (u)Qx(u)du = -T T -h(T ) x(u) T Qx(u)du < 0 where x d (t) = x(t -h(t)).
In the next paragraph, applying the prior result exposed in Section 2.2 a rate and delay dependent stability condition for time-varying delay systems is provided.

Stability condition for time-varying delay systems

First, let us reformulate the dynamic of linear systems with time-varying delay as suggested on Figure 1 in order to apply the quadratic separation principle. System (1) can be described as the feedback

   x(t) x(t -h(t))    w(t) =    I 1 n D1 n    ∇    ẋ(t) x(t)    z(t) , (8)
over the feedforward equation

   1 0 0 1    E    ẋ(t) x(t)    z(t) =    A A d 1 0    A    x(t) x(t -h(t))    w(t)
.

This simplistic description of the system 1 gives rise, applying Theorem 1, to the well-known independent of delay (IOD) criterion [2][9]. Secondly, so as to develop delay dependent condition, an additional operator must be blended into ∇, enhancing then the time delay system description. Usually, the operator (1 -D) • I (or in Laplace domain for the constant delay case (1e -sh )s -1 ), bounded by h max , is added. This operator is applied to the signal ẋ(t) and the relationship

(1 -D) • I [ẋ i (t)] = (1 -D) • I [I -1 [x i (t)]] = x i (t) -x i (t -h(t)), i = {1, ..., n}
should be specified in the linear equation E z(t) = A w(t). In this paper, inspired from [START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF], instead of the integrator I , a scaling filter of dimension n φ of the form

     ẋφ (t) = A φ x φ (t) + B φ u(t), y(t) = C φ x φ (t) + D φ u(t), (9)
is considered. The key idea is now to apply the new operator (1 -D) • φ -1 to the filtered signal y(t, x i (t)) (each component of the state vector x(t) of system (1) is processed u(t) = x i (t), i = {1, ..., n}):

(1 -D) • φ -1 [y(t, x i (t))] = (1 -D) • φ -1 [φ [x i (t)]] = x i (t) -x i (t -h(t)), i = {1, ..., n}
The dynamical system (9) should be designed according to the following lemma.

Lemma 3 An integral quadratic constraint for the operator F = (1 -D)• φ -1 is given by the following inequality ∀x ∈ L n 2e and for a positive definite matrix R,

   1 n F 1 n    x,    -R 0 0 R       1 n F 1 n    x < 0,
where φ , defined in [START_REF] Gouaisbaut | Robust stability of time-delay systems with interval delays[END_REF], is a realization of any bounded rational transfert function Φ which satisfies

         |Φ( jω)| > 1 + 1 √ 1-d , if h max |ω| > 1 + 1 √ 1-d , |Φ( jω)| > h max |ω|, if h max |ω| ≤ 1 + 1 √ 1-d . (10)
Proof 4 In [START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF], it is shown that for all systems φ satisfying the above specifications

(1 -D) • φ -1 L 2 ≤ 1 holds.
It means that for any v, a L n 2 function,

F v L 2 ≤ v L 2 ∞ 0 (F v(t)) T F v(t) -v T (t)v(t)dt ≤ 0 is satisfied. Defining v(t) = R 1/2 x(t), R being a symmetric positive definite matrix, we have ∞ 0 (F x(t)) T RF x(t) -x T (t)Rx(t)dt ≤ 0.
Factorizing on both sides by [x T (t) (F x(t)) T ] and its transposed, the IQC of the lemma is recovered.

An example of Φ(s), proposed by [START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF], satisfying (10) is

Φ(s) = k h 2 max s 2 + ch max s h 2 max s 2 + ah max s + b ( 11
)
where

k = 8/(2 -d), a = √ 6.5 + 2b, b = √ 50 and c = √ 12.5.
The time-varying delay system (1) is now modeled as the interconnection of

         x(t) xφ (t) x(t -h(t)) v 1 (t)          w(t) = ∇          ẋ(t) ẋφ (t) x(t) φ [x]          z(t) (12)
with

∇ = diag I 1 n , I 1 n φ n , D1 n , (1 -D) • φ -1 1 n and    1 n(3+n φ ) 0    E z(t) =             A 0 A d 0 Bφ Ãφ 0 0 1 n 0 0 0 Dφ Cφ 0 0 1 n 0 -1 -1             A w(t) ( 13
)
where

v 1 (t) = x(t) -x(t -h(t)), Ãφ = 1 n ⊗ A φ , Bφ = 1 n ⊗ B φ , Cφ = 1 n ⊗ C φ , Dφ = 1 n ⊗ D φ , xφ =       x φ 1 . . . x φ n       , φ [x] =       φ [x 1 ] . . . φ [x n ]       , (14)
x i are the components of the state vector x. Note that there is one filter of dimension n φ associated to each x i . At this point, refering to the quadratic separation approach, Theorem 1 may be applied.

Theorem 2 For given positive scalars h max and d, if there exists positive definite matrices P ∈ R n(1+n φ )×n(1+n φ )

and Q, R ∈ R n×n , then system (1) with a time varying delay constrained by ( 2) is asymptotically stable if the LMI condition (3) holds with Θ (setting ḣ = d), E and A defined as [START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF] and [START_REF] Iwasaki | Well-posedness of feedback systems: insights into exact robustnessanalysis and approximate computations[END_REF].

Proof 5 First, condition (4) must be satisfied for ∇ defined as [START_REF] He | Further improvement of free-weighting matrices technique for systems with time-varying delay[END_REF]. Invoking all Lemmas previously defined and combining all inequalities related to each operator, it is readily seen that the separator [START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF] fulfills the inequality [START_REF] Bushnell | Networks and control[END_REF], where

Θ =    Θ 11 Θ 12 Θ * 12 Θ 22    , Θ 11 = diag(0 n(1+n φ ) , -Q, -R), Θ 12 = diag(-P, 0 2n ), Θ 22 = diag(0 n(1+n φ ) , (1 -ḣ)Q, R), ( 15
)
and P ∈ R n(1+n φ )×n(1+n φ ) and Q, R ∈ R n×n are positive definite matrices. It is readily seen that inequality [START_REF] Fridman | An improved stabilization method for linear time-delay systems[END_REF] still holds for ḣ = d. Hence, regarding the separator, Θ 22 can be chosen as diag(0, (1d)Q, R). Then, the filter φ may be chosen as a realization of (11) defined by

             ẋφi (t) =    0 1 -b h 2 max -a h max    x φ i (t) +    0 1    u(t) y i (t) = -kb h 2 max kc-ka h max x φ i (t) + ku(t)
, where x φ i , i = {1, ..., n}, represents the different states of the same filter according to the different inputs u(t) = {x 1 (t), ..., x n (t)}. Considering Theorem 1 where the interconnection is given by ( 13)- [START_REF] He | Further improvement of free-weighting matrices technique for systems with time-varying delay[END_REF], and ( 4) being proved, the LMI (3) forms the stability criterion.

Model extension

Previous works [START_REF] Ebihara | Robust performance analysis of linear time-invariant uncertain systems by taking higher-order time-derivatives of the states[END_REF] and [START_REF] Bliman | Lyapunov equation for the stability of linear delay systems of retarded and neutral type[END_REF], [START_REF] Gouaisbaut | Robust stability of time-delay systems with interval delays[END_REF] have shown that redundant system modeling (for linear uncertain systems and constant delay systems, respectively) may increase the relevancy of the stability analysis. The rational behind this model extension is to provide some extra relations between the delay, its variations and the state. Using the derivative operator, an augmented state is constructed which is composed of the original state vector and its derivatives. Then defining relationship between augmented states ẋ, ẍ, the delay h and its derivative ḣ an enhanced stability condition is provided. Differentiating the system (1), we get:

ẍ(t) = A ẋ(t) + (1 -ḣ(t))A d ẋ(t -h(t)).
Consider the artificially augmented system

     ẋ(t) = Ax(t) + A d x(t -h(t)), ẍ(t) = A ẋ(t) + (1 -ḣ(t))A d ẋ(t -h(t)), (16)
with accordingly defined initial conditions. Introducing the augmented state

ζ (t) =    ẋ(t) x(t)    ,
and specifying the relationship between the two components of ζ (t) with the equality

[0 1] ζ (t) = [1 0]ζ (t), we
have the new descriptor augmented system

E ζ (t) = Āζ (t) + Ād ζ (t -h(t)), ( 17
)
where

E =       1 0 0 1 1 0       , Ā =       A 0 0 A 0 1       , Ād =       A d 0 0 (1 -ḣ)A d 0 0       .
Then, the new time-varying delay system (17) can be shaped as Figure 1 with

         ζ (t) xφ (t) ζ (t -h(t)) v 2 (t)          w(t) = ∇ ⊗ 1 2          ζ (t) ẋφ (t) ζ (t) φ [ζ ]          z(t) (18)
and

   diag E, 1 2n(2+n φ ) 0    E z(t) =             Ā 0 Ād 0 Bφ Ãφ 0 0 1 n 0 0 0 Dφ Cφ 0 0 1 n 0 -1 -1             A w(t) (19)
where

v 2 (t) = ζ (t) -ζ (t -h(t)
), ∇ defined as [START_REF] He | Further improvement of free-weighting matrices technique for systems with time-varying delay[END_REF] and redefining

Ãφ = 1 2n ⊗ A φ , Bφ = 1 2n ⊗ B φ , Cφ = 1 2n ⊗ C φ , Dφ = 1 2n ⊗ D φ , xφ =       x φ 1 . . . x φ 2n       , φ [ζ ] =              φ [ ẋ1 ] φ [x 1 ] . . . φ [ ẋn ] φ [x n ]              . (20)
Following the same line than in the previous section, we propose:

Theorem 3 For given positive scalars h max and d, if there exists positive definite matrices P ∈ R 2n(1+n φ )×2n(1+n φ ) , Q, R ∈ R 2n×2n and a matrix X ∈ R 2n(6+2n φ )×n(7+2n φ ) , then system (1) with a time varying delay constrained by [START_REF] Ariba | Stability analysis of time-varying delay systems in quadratic separation framework[END_REF] is asymptotically stable if the LMI condition

Θ [ j] + X[E -A [ j] ] + (X[E -A [ j] ]) T > 0 holds for j = 1, 2
, with E and A defined as [START_REF] Olgac | An exact method for the stability analysis of time-delayed linear time-invariant (lti) systems[END_REF]. Θ is of the form of ( 15) with appropriate dimension. Θ [ j] and A [ j] are the vertices of, respectively, Θ and A setting ḣ(t) = d j . The constraint [START_REF] Ariba | Stability analysis of time-varying delay systems in quadratic separation framework[END_REF] gives d 1 = -d and d 2 = d.

Proof 6 First, It follows the same idea that the one of Theorem 2. Hence, the stability of ( 17) will be proved if

E -A ( ḣ(t)) ⊥ * Θ( ḣ(t)) E -A ( ḣ(t)) ⊥ > 0
with E , A defined as [START_REF] Olgac | An exact method for the stability analysis of time-delayed linear time-invariant (lti) systems[END_REF] and Θ is of the form of ( 15) with appropriate dimension. Applying the Finsler's lemma, the above inequality is equivalent to

Θ( ḣ(t)) + X[E -A ( ḣ(t))] + (X[E -A ( ḣ(t))]) T > 0 ( 21
)
where X is a new decision variable of appropriate dimensions. Since, ḣ(t) appears linearly and is bounded, invoking argument of convexity, it is sufficient to test ( 21) on its vertices. The inequality proposed by Theorem 3 is thus recovered. System [START_REF] Kolmanovskii | Introduction to the Theory and Applications of Functional Differential Equations[END_REF] being stable, the whole state ζ (t) converges asymptotically to zero. Hence, its components x(t) converge as well. The original system ( 1) is thus also asymptotically stable.

Robustness issue

Coming from robust control, quadratic separation provides a suitable framework to address the stability analysis of uncertain delay systems:

ẋ(t) = A(∆)x(t) + A d (∆)x(t -h(t)) ( 22
)
where

A(∆) A d (∆) = A A d + B∆ C C d .
The second term of the right hand side of the above equation describes the uncertainty characterizing system [START_REF] Tarbouriech | Advances in communication Control Networks[END_REF].

The uncertain time-varying matrix ∆, belonging to Ω, satisfies

∆ T (t)∆(t) ≤ 1, ∀t ≥ 0, ∀∆ ∈ Ω, (23)
and models non-linear and neglected dynamics as well as parametric uncertainties. C, C d and B are constant matrices of appropriate dimensions which structure the uncertainty. Then, according to the set of admissible uncertainties and (23), we have to find a separator U such that

   1 ∆    x,    U 1 U 2 U * 2 U 3    U    1 ∆    x < 0, ∀∆ ∈ Ω. (24)
For instance, assume Ω is a set of diagonal real valued matrices with bounded uncertainties:

Ω = ∆ = diag(δ 1 , ..., δ N ) | |δ i | ≤ δi ,
Then, inequality [START_REF] Zhang | Stability of time-delay systems: Equivalence between Lyapunov and scaled small-gain conditions[END_REF] holds with

U = diag -δ 2 1 u 1 , . . . , -δ 2 N u N , u 1 , . . . , u N
where u i , i = {1, ..., N} are scalar decision variables. Eventually, we propose to analyze the robust stability of system [START_REF] Tarbouriech | Advances in communication Control Networks[END_REF] with the following Theorem.

Theorem 4 For given positive scalars h max and d, if there exists positive definite matrices P ∈ R n(1+n φ )×n(1+n φ ) , Q, R ∈ R n×n and matrices U k (k = 1, 2, 3) designed according to the uncertain set Ω (i.e. such that (24) holds), then system [START_REF] Tarbouriech | Advances in communication Control Networks[END_REF] with a time varying delay constrained by ( 2) is robustly asymptotically stable for any uncertainty 3) holds with Θ, E and A defined as ( 26) and ( 25).

∆ ∈ Ω if the LMI condition (
Proof 7 First, introducing the exogenous signals

w ∆ = ∆z ∆ , with z ∆ = Cx(t) + C d x(t -h(t)),
we rewritte system [START_REF] Tarbouriech | Advances in communication Control Networks[END_REF] as the interconnection of [START_REF] Kao | Robust stability analysis of linear systems with time-varying delays[END_REF]. Combining every IQC related to each operators defined by lemmas and the struture of the uncertainty leading to [START_REF] Zhang | Stability of time-delay systems: Equivalence between Lyapunov and scaled small-gain conditions[END_REF], a separator of the form

            x(t) xφ (t) x(t -h(t)) v 1 (t) w ∆ (t)             w(t) = ∇             ẋ(t) ẋφ (t) x(t) φ [x] z ∆ (t)             z(t) with ∇ = I 1 n , I 1 n φ n , D1 n , F 1 n , ∆ and    1 n(3+n φ )+n ∆ 0    E z(t) =                 A 0 A d 0 B Bφ Ãφ 0 0 0 1 n 0 0 0 0 Dφ Cφ 0 0 0 C 0 C d 0 0 1 n 0 -1 -1 0                 A w(t) (25)
Θ =    Θ 11 Θ 12 Θ * 12 Θ 22    , Θ 11 = diag(0 n(1+n φ ) , -Q, -R, U 1 ), Θ 12 = diag(-P, 0 2n , U 2 ), Θ 22 = diag(0 n(1+n φ ) , (1 -d)Q, R, U 3 ), (26)
fulfills the requirement [START_REF] Bushnell | Networks and control[END_REF]. Finally, condition (3) provides the robust (with respect to the uncertain set Ω) stability criterion.

NUMERICAL EXAMPLES

Example 1

Consider the following system,

ẋ(t) =    -2 0 0 -0.9    x(t) +    -1 0 -1 -1    x(t -h(t)). ( 27
)
For various d, the maximal allowable delay, h max , is computed. To demonstrate the effectiveness of our criterion, results are compared against those obtained in the literature (see Table 1). On this example, compared to Lyapunov technics, robust approaches [START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF], Theorem 2 and 3 reduce drastically the conservatism, especially when d is close to zero i.e. when the delay is slowly time varying. Using the same scaling filter for bounding operators Theorem 2 recovers the results of [START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF], whereas Theorem 3, taking into account the derivative equation, reduces the conservatism. Indeed, the stability analysis is further improved thanks to an appropriate modeling of time-varying delay systems which brings additional informations on the system.

Example 2

Consider now the following time-varying and uncertain system, extracted from [START_REF] Kim | Delay and its time-derivative dependent robust stability of time-delayed linear systems with uncertainty[END_REF],

ẋ(t) =    -2 + δ 1 cos(t) 0 0 -1 + δ 2 sin(t)    x(t) +    -1 + γ 1 cos(t) 0 -1 -1 + γ 2 sin(t)    x(t -h(t)). ( 28
)
The δ i and γ i are uncertain but bounded parameters: Results are summarized in Table 2. It shows that Theorem 4 enables to find higher maximal bounds on the delay h(t) than others results from the literature. The quadratic separation offers thus a suitable framework to address uncertainties. In that case, the conservatism is also related to the manner to handle uncertainties (as the design of ( 24)).

|δ 1 | ≤ 1.

CONCLUSIONS

In this paper, the problem of the delay dependent stability analysis of a time-varying delay systems has been studied by means of quadratic separation. The delay part is embedded into an uncertain matrix of operators. Inspired from [START_REF] Ariba | Stability analysis of time-varying delay systems in quadratic separation framework[END_REF] and [START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF], tight bounds of the L 2 induced norms of operators allow to reduce the conservatism of the approach.

Then using an augmented state, new modelling of time delay systems are introduced which emphasizes the relation

Figure 1 :

 1 Figure 1: Feedback system

6 ,B = 1 2 ,

 62 |δ 2 | ≤ 0.05, |γ 1 | ≤ 0.1, |γ 2 | ≤ 0.3.This example can be expressed as system[START_REF] Tarbouriech | Advances in communication Control Networks[END_REF] with C = diag(1.6, 0.05), C d = diag(0.1, 0.3).

Table 1 :

 1 The maximal allowable delays h m for system(27)

	d	0	0.1	0.2	0.5	0.8	1
	Fridman et al (2002) [7] 4.472 3.604 3.033 2.008 1.364 0.999
	Fridman et al (2006) [8] 1.632 1.632 1.632 1.632 1.632 1.632
	Wu et al (2004) [23]	4.472 3.604 3.033 2.008 1.364	-
	He et al (2007) [11]	4.472 3.605 3.039 2.043 1.492 1.345
	He et al (2007) [12]	4.472 3.605 3.039 2.043 1.492 1.345
	Ariba et al (2007) [1]	5.120 4.081 3.448 2.528 2.152 1.991
	Kao et al (2007) [15]	6.117 4.714 3.807 2.280 1.608 1.360
	Theorem 2	6.117 4.714 3.807 2.280 1.608 1.360
	Theorem 3	6.117 4.794 3.995 2.682 1.957 1.602
	with notations of

Table 2 :

 2 The maximal allowable delays h m for system (28)

	d	0	0.1	0.5	0.8
	[16]	0.241 0.234 0.188 0.110
	[23]	1.149 1.106 0.924 0.760
	[15]	1.416 1.302 0.974 0.829
	Theorem 4 1.515 1.422 1.105 0.910

between ḣ and signals ẋ and ẍ. The resulting criteria are then expressed in terms of a convex optimization problem with LMI constraints, allowing the use of efficient solvers. Finally, two numerical examples show that these methods reduced conservatism and improved the maximal allowable delay.


Sabrine Kobbi
email: sabrina.kobbi@yahoo.fr

Hajer Bougatef
email: hajer.bougatf93@gmail.com

Rafik Balti
email: rafik.balti@centralesupelec.fr

Remi Przybylski
email: remi.przybylski@live.fr

Assaad Sila
email: assaadsila@gmail.com

Pascal Dhulster
email: pascal.dhulster@univ-lille1.fr

Ali Bougatef
email: ali.bougatef@isbs.usf.tn

Naïma Nedjar
email: naima.nedjar@univ-lille.fr

Peptide from RuBisCO protein hydrolysate and its application in beef meat preservation

Keywords: Antioxidant, antimicrobial, bioactive peptide, beef meat

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Fresh products as minced meat contains adequate amount of water and abundance of proteins and essential nutrients with favorable pH for supporting microbial growth. Meat is highly perishable products and requires appropriate preservation (Domínguez et al. 2018).

Therefore, several synthetic additives have long been directly applied as food additives or used as processing aids with primary intention of prolonging shelf life and preserving quality of meat and meat products [START_REF] Munekata | Addition of plant extracts to meat and meat products to extend shelf-life and health-promoting attributes: An overview[END_REF]. The application of chemical preservatives such as butylated hydroxytoluene (BHT) and propyl gallate (PG) has been questioned because of the potential toxic and carcinogenic effects [START_REF] Hashemi | Efficiency of ohmic assisted hydrodistillation for the extraction of essential oil from oregano (Origanum vulgare subsp. viride) spices[END_REF]. Not only pressure of the consumer health concerns, but also a trend towards natural food additives so called "cleanlabeling" has driven exploring of natural antimicrobial compounds as an alternative to synthetic food additives (Domínguez et al. 2019;[START_REF] Lorenzo | Berries extracts as natural antioxidants in meat products: A review[END_REF]. Advancement in bioactive peptide research has showed that the peptides derived from food proteins could be natural potential candidates to replace the synthetic compounds (Chakrabarti et al., 2018;[START_REF] Wang | Identification and characterization of antioxidant peptides from enzymatic hydrolysates of duck meat[END_REF]. Indeed, antimicrobial peptides of industrial waste could be used as preservatives [START_REF] Lafarga | Bioactive peptides from meat muscle and byproducts: Generation, functionality and application as functional ingredients[END_REF]. To date, several animal peptides such as lysozyme, lactoferrin, and nisin are approved for application in meat and meat products [START_REF] Silveira | Antimicrobial peptides as a feed additive alternative to animal production, food safety and public health implications : An overview[END_REF].

Recently, the food industries have taken interest to peptides derived from agroindustrial waste. Green juice derived from bio-refinery of Alfalfa (Medicago sativa), is often considered as a source of pollution, but it is an excellent protein-rich source and has become an alternative for peptides generation with great interest for food formulation [START_REF] Hadidi | Optimisation of steam blanching on enzymatic activity, color and protein degradation of alfalfa (Medicago sativa) to improve some quality characteristics of its edible protein[END_REF]. Ribulose-1,5-bisphosphate carboxylase oxygenase, most commonly known by the shorter name RuBisCO. It's the most abundant plant protein and possesses unique characteristics, a favorable nutritional profile, beneficial bioactivities, and desirable physical attributes, making it a promising candidate for use in food applications (Grácio et al., 2023).

Several studies showed that RuBisCO have been reported to contain several bioactive peptides compared to commonly consumed food proteins, except milk proteins (Corrêa et al., 2023). In recent previous studies antibacterial and antioxidant peptides derived from the enzymatic hydrolysis of Alfalfa RuBisCO were isolated and identified [START_REF] Kobbi | Antibacterial activity of novel peptides isolated from protein hydrolysates of RuBisCO purified from green juice alfalfa[END_REF][START_REF] Kobbi | Purification and Recovery of RuBisCO Protein from Alfalfa Green Juice: Antioxidative Properties of Generated Protein Hydrolysate[END_REF]. The antibacterial peptide (Arg-Asp-Arg-Phe-Leu) was previously identified from protein hydrolysates of Alfalfa RuBisCO prepared with pepsin ® [START_REF] Kobbi | Antibacterial activity of novel peptides isolated from protein hydrolysates of RuBisCO purified from green juice alfalfa[END_REF]. Arg-Asp-Arg-Phe-Leu a small (705 Da) peptide with one positive charges at pH 7 belongs to the great subunit of the RuBisCO, which has a large antibacterial spectrum as against Gram positive and Gram-negative bacteria [START_REF] Kobbi | Antibacterial activity of novel peptides isolated from protein hydrolysates of RuBisCO purified from green juice alfalfa[END_REF]. This bioactive peptide has several attractive properties and attributes that make it a potential promising candidate for application as preservative in the storage and distribution of various meat products. The objetive of this work was to purify a small (705 Da) and hydrophilic antimicrobial peptide (Arg-Asp-Arg-Phe-Leu) from RuBisCO protein hydrolysate and to evaluate its effect on the microbiological and oxidative stability of minced beef meat during refrigeration.

Materials and methods

All common chemicals and solvents were of analytical grade from commercial sources: Sigma-Aldrich (Saint-Quentin Fallavier, France) or Flandres Chimie (Villeneuve d'Ascq, France). Synthetic peptide was obtained from GeneCust Europe (Biotechnology Laboratory of Luxembourg S.A). The beef meat was obtained from a local purveyor.

The purification of RuBisCO was realized by a simple method based on pH and solvent effect, according to the method described by [START_REF] Kobbi | Purification and Recovery of RuBisCO Protein from Alfalfa Green Juice: Antioxidative Properties of Generated Protein Hydrolysate[END_REF]. Primary purification of an interest protein from a crude medium is specifically to eliminate impurities depending on the solubility and chemical properties of RuBisCO, as its overall charge, solubility or pHi.

In order to isolate a soluble fraction, 100 g of green juice was solubilized in 1 L of distilled water or 1 L of ammonia (NH4OH) at 0.1 M and pH 11. Suspensions were centrifuged at 10,000 g for 20 min at 4°C to eliminate the green protein and all debris. The RuBisCO containing base 1 at pH 10 was repurified to retrieve the second supernatant 2. Base 2 was repurified again to recover supernatant 3. After three successive centrifugations at pH 10, all insoluble fractions were removed, and, hence, three supernatants were summed up to recover all RuBisCO. Proteins were then precipitated by the acidification of the medium at pH 3 using 10 mM of formic acid. This was followed by a last centrifugation step at 10,000 g at 4 °C for 20 min. The precipitate was washed by ultrapure water and RuBisCO was then lyophilized.

A 10% (w/v) aqueous solution of the lyophilized RuBisCO was adjusted to pH 10 and then centrifuged at 10,000 g for 5 min. The supernatant was adjusted to optimal pH (pH 3.0) and temperature (37°C) for pepsin ® . The hydrolysis reaction was started by the addition of the enzyme at a 1:100 (U/mg) enzyme/ protein ratio. The enzymatic reaction was conducted at 37 °C and pH 3 for 24 h. The hydrolysis reactions are stopped after 24 h by increasing the pH to 10 with 1 M NaOH and then stored at -20 °C for further use. The proportion of none hydrolyzed RuBisCO and then hydrolyzed RuBisCO was deduced from the peak areas by HPLC.

The degree of hydrolysis (DH) is defined as the ratio of the number of peptide bonds cleaved to the total number of peptide bonds of protein substrate. DH is determinate by the OPA (o-phthaldialdehyde) method using leucine as standard [START_REF] Nielsen | Improved method for determining food protein degree of hydrolysis[END_REF]. A fresh OPA solution was prepared extemporaneously with 25 mL of 100 mM sodium tetrahydroborate, 5 mL of 10% SDS solution (w/v), 80 mg of OPA (dissolved in 1 mL of methanol) and 0.01 mL of β-mercaptoethanol were adjusted to a final volume of 50 ml with distilled water. To assay proteolysis with RuBisCO protein as substrates, an aliquot (usually 0.01-0.05 ml containing 5-100 mg proteins) was added to 1.0 ml of OPA reagent in a 1.5 ml quartz cuvette. The solutions were mixed briefly and incubated for 2 min at room temperature, and then the absorbance at 340 nm was measured.

The RuBisCO hydrolysis was analyzed by RP-HPLC on a C4 column according to the process previously described with slight modifications (Trovalset et al., 2007). The liquid chromatographic system consisted of a Waters 600E automated gradient controller pump module, a Waters Wisp 717 automatic sampling device and a Waters 996 photodiode array detector. Spectral and chromatographic data were stored on a NEC Image 466 Image 466 computer. Millennium software was used to plot, acquire, and analyze chromatographic data.

Chromatographic processes were performed with a VydacC4 column (250 mm×10 mm). The mobile phase composed of water/trifluoroacetic acid (1000:1, v/v) was used as eluent A, while acetonitrile/trifluoroacetic acid (1000:1, v/v) was used as eluent B. The flow rate was 0,6 ml/min. Samples were filtered through 0.22mm filters and then injected. The gradient applied was 0-10 min, 100% A; 25-95min, 21-81% B; 96-110min, 100-0% B; and the HPLC system were equilibrated for 10 min with 100% A. On-line UV absorbance scans were performed between 215 and 325 nm at a rate of one spectrum per second with a resolution of 1.2 nm. Chromatographic analyzes were completed with Millennium software.

The RuBisCO hydrolysate was adjusted to pH 7.5 and NaCl was added to a fixed volume to reach a final concentration of 2.0 M of salt [START_REF] Adoui | A simple method to separate the antimicrobial peptides from complex peptic casein hydrolysate and 15 identification of novel antibacterial domains within the sequence of bovine αs-casein[END_REF]. The solution was stirred for 30 min at room temperature before being centrifuged at 10,000 g for 10 min at 4°C. The pellet was washed twice by centrifugation with phosphate buffer (5 mM, pH 7.5) to remove soluble proteins, and then resuspended in phosphate buffer (5 mM, pH 7.5, 100 mM NaCl) to 20 times the original hydrolysate volume. The final pH of the peptide hydrolysate was adjusted between 1.5 to 2. After one hour stirring at 7 °C, the pellet was centrifuged (10,000 g, 10 min), and the supernatant was recovered. The solution was stored at -20°C until performing the antibacterial assay and analysis by RP-HPLC.

The molecular mass and peptide sequencing were done in positive ion mode using Electrospray ionisation-mass spectrometry (ESI-MS) and the tandem mass spectrometry (MS/MS), respectively. ESI mass spectrometry was performed using a triple quadrupole instrument Applied Biosystems API 3000 (PE Sciex, Toronto, Canada) equipped with an electrospray ion source. The system is controlled by the Analyst Software 1.4, allowing the control of the spectrometer, the analysis and the processing data. Interpretations of spectra MS-MS were made with the Bioanalyst software. The freeze-dried samples from RP-HPLC were dissolved in acetonitrile/water (20/80; v/v) containing 0.1% formic acid for the positive mode. The solution was injected (nebulised) uninterrupted, by a pump (Model 22, Harward Apparatus, South Natick, USA) with a flow rate of 5 μl/min. The potential of ionisation was of 5000 V in positive mode. At the time of the recording of the spectrum, 30 scans on average were added (MCA mode) for each spectrum. The gases used (nitrogen and air) were pure (up to 99%) and produced by a compressor Jun-Air 4000-40M and a nitrogen generator Whatman The antimicrobial peptide (Arg-Asp-Arg-Phe-Leu) was chemically synthesized by the conventional method: Fmoc solid phase synthesis, according to the method described by Fields and Noble (1990). Arg-Asp-Arg-Phe-Leu peptide was synthesized on a Fmoc-Arg (Pbf)-WangTentagel® S resin (0.25 mmol/g from Iris Biotech, 50 µmol scale) using an Intavis peptide synthesizer (Proteigene) and standard Fmoc/tert-butyl. Activation of the amino acids (10 eq) was performed separately prior to the coupling step using a mixture of TBTU/HOBt/DIEA: 10 eq/10 eq/30 eq in DMF. A capping step was performed after each coupling with a mixture of Ac2O/NMM/DMF: 5/10/75 v/v/v. At the end of the synthesis, the Fmoc protecting group was removed using 20% piperidine in DMF. The resin was washed with DMF (4 × 2 min) and CH2Cl2 (4 × 2 min) and dried in vaccuo. Deprotection and cleavage of the peptide from the resin was performed with a mixture of TFA/TIS/H2O: at room temperature. The mixture was precipitated in cold diethyl ether. The peptide was collected by centrifugation, dissolved in deionized water and lyophilized. Arg-Asp-Arg-Phe-Leu was purified by RP-HPLC.

The hemolytic activity of the Arg-Asp-Arg-Phe-Leu peptide was determined by methods of [START_REF] Dathe | Peptide helicity and membrane surface charge modulate the balance of electrostatic and hydrophobic interactions with lipid bilayers and biological membranes[END_REF], slightly modified. Five milliliters of bovine blood were centrifuged at 3500 rpm for 10 min to isolate erythrocytes, which were then washed three times with 10mM sodium phosphate, pH 7.5, containing NaCl 9 g/L (NaCl/Pi). The cell concentration stock suspension was adjusted to 109 cellules /mL. The cell suspension (12 mL) along with varying amounts of peptide stock solution and the buffer were pipetted into Eppendorf tubes to give a final volume of 50 mL. The Eppendorf tubes with 2.5 108 cells/mL were then incubated at 37 °C during 40 min. After centrifugation (5000 rpm, 5 min), 30 mL of supernatant was diluted in 500 mL water. The absorbance of the diluted solution was measured at 420 nm. The absorbance obtained after treating erythrocytes with only NaCl/Pi and SDS (0.2%) was taken as 0 and 100%, respectively. The test was carried out in triplicate.

The beef patties were composed with commercial beef containing 0.28% (w/w) of salt, 14% (w/w) of fat and about 18% of proteins. Salt (70% NaCl and 30% KCl) was added at a rate of 1.5% to the minced meat. The sample was subdivided into four treatments: a negative control, the meat without additive, a positive control, the meat with two different BHT concentrations (0.1 and 0.5% (w/w)) and the meat with two different Arg-Asp-Arg-Phe-Leu peptide concentrations (0.1 % and 0.5% (w/w)). The meat sample preparation was realized under strong aseptic conditions to avoid contaminations. Patties of 10 g were placed in plastic foam meat trays, wrapped with film polyethylene and kept at 4 °C for fourteen days. Four samples were taken from each treatment at fixed day in order to evaluate their oxidation and the potential microbial contamination. A strict sanitation procedure was followed during the preparation of meat samples to avoid microbial contamination.

Lipid oxidation was evaluated using the thiobarbituric acid-reactive substance (TBARS) as previously described by [START_REF] Witte | A new extraction method for determination 2thiobarbituric acid values of pork and beef during storage[END_REF]. For each prepared formula five grams of minced meat were added to 12.5 mL of solution containing trichloroacetic acid (20% w/v) and phosphoric acid (2 M). The mixture was then adjusted to 25 mL of ultrapure water, shaken and filtered using Buchner funnel through Whatman paper. After that, 5 mL of the filtrate was then transferred into a glass test tube and an equal volume of 2-thiobarbituric acid (5 mM) was added. The test tube was then well shaken and incubated at room temperature for 17 h.

The absorbance of the mixture was measured at 520 nm and the results were expressed as malondialdehyde equivalents per meat kilogram (mg MDA/kg meat).

Microbial count was carried out at days 0, 1, 3, 6, 9, and 11 of storage at 4°C according to the following procedure. 45 mL sterilized peptone solution (0.1% w/v) was added to each five grams of beef patty in a sterile stomacher bag, then homogenized. Decimal dilutions up to 105 prepared from the initial concentration (100 mg/mL) and 100 µL of the appropriate dilutions were spread out on different selective media. Total viable counts were determined using plate count agar (PCA), potato dextrose agar (PDA) for yeasts and molds counts, Man, Rogosa and Sharpe (MRS) agar for lactic acid bacteria and violet, red bile lactose (VRBL) agar for coliform counts were used after incubation at 37 °C for 24h. Microbiological count was expressed as the log10 of colony-forming units per gram of patty (log10 CFU/g).

All data were submitted to Analysis of Variance (ANOVA) and differences between means were evaluated by Duncan's Multiple Range Test. The data were analyzed using the Statistical Package for the Social Sciences (SPSS) version 10.0 (Chicago, Illinois, USA).

Differences were considered significant at p<0.05.

Results and Discussion

Figure 1 shows the RP-HPLC chromatogram from the peptic RuBisCO proteins purified from green juice of Alafalfa. Generated hydrolysate (DH = 18%) was mainly intermediate peptides (produced from the beginning of hydrolysis and disappearing gradually as hydrolysis advance) [START_REF] Abou-Diab | Impact of conductivity on the performances of electro-acidification and enzymatic hydrolysis phases of bovine hemoglobin by electrodialysis with bipolar membranes for the production of bioactive peptides[END_REF]. This standard condition following hydrolysis process led to a complex hydrolysate containing many peptides, intermediate or final, with various characteristics [START_REF] Kleekayai | Protein Hydrolysates and Peptides[END_REF]. The chromatographic profile (Figure 2A)

showed the peptide population with retention times between 10 and 50 min which corresponding to antimicrobial peptides. According to the mass spectrometry analyses (Figure 2B), one antimicrobial pentapeptide was identified: Arg-Asp-Arg-Phe-Leu. The MIC of the purified peptide varied between 2.17 ± 0.1 mM (for Listeria innocua and Bacillus subtilis) and 2.83 ± 0.2 mM (for Escherichia coli and Micrococcus luteus). Peptides showed an outstanding antibacterial activity against both Gram positive and Gram-negative bacteria by damaging the cell membrane integrity [START_REF] Bi | Antibacterial Activity and Potential Application in Food Packaging of Peptides Derived from Turbot Viscera Hydrolysate[END_REF][START_REF] Tu | Identification and characterization of a novel casein anticoagulant peptide derived from in vivo digestion[END_REF]Medeiros et al., 2018).

Arg-Asp-Arg-Phe-Leu peptide did not cause hemolysis. The hemolytic activity of the peptide against human erythrocytes was determined as an indication of its toxicity towards mammalian cells. The highest concentration tested corresponding to this concentration of peptide. No hemolysis was observed up to a peptide concentration corresponding to five times the MIC. The positively charged residues can significantly reduce the toxicity of the peptides towards the mammalian cells, so with the increasing positive charges, the hemolytic activity of the peptides reduced reasonably (Lohan et al., 2023;[START_REF] Sengkhui | A novel designed membrane-active peptide for the control of foodborne Salmonella enterica serovar Typhimurium[END_REF].

Significant changes in TBARS occurred over the sampling period when Arg-Asp-Arg-Phe-Leu peptide was added to chilled minced beef meat. Analysis of variance of the TBARS values was determined between the different samples. The result showed that the TBARS values increased throughout the eleven days experiment. This fact was clearly marked for the control sample (without additives) which highest TBARS values value than all tested conditions by the end of storage (It goes from value 0.56 ± 0.07 mg MDA for day 1 to a value of 2.31 ± 0.18 mg MDA for day 11). While the TBARS values of the meat supplemented with either 0.5% or 0.1% (w/w) Arg-Asp-Arg-Phe-Leu peptide, were significantly lower (p < 0.05) than the meat without additives, used as control. Moreover, the efficiency in ground meat preservation increased proportionally with peptide concentration. But this difference was clear from day 6 (Figure 3) because no significant effect between the four treated groups has been observed during the first three days of the experiment, but their TBARS values were significantly lower than the control (Figure 3). From day 6, the power antioxidant of Arg-Asp-Arg-Phe-Leu at 0.5% (w/w) was higher (0.83 ± 0.04 mg MDA/kg) than the other two treated groups (peptide at 0.1% and BHT at 0.1%) with an average value of 1.75± 0.31 mg MDA and 1.25 ± 0.13 mg MDA respectively. Thus, the peptide at 0.5% (w/v) can protect beef meat against lipid oxidation and extend their storage time. The TBARS values of the meat supplemented with either peptide 0.5% or BHT 0.5% (w/v), used as reference, have no significant difference and were similar after 9 days of storage at 4 °C of an average value TBARS of 0.91 ± 0.21 mg MDA that is two timeless the control (2.145± 0.36 mg MDA) (Figure 3). Consequently, the BHT at 0.5% (w/v) and Arg-Asp-Arg-Phe-Leu at 0.5% (w/w) reduced the lipid oxidation on meat of 60% (Figure 3) compared to the control. This tendency continued until 11 days. At the end of the experiment, the BHT at 0.5 (w/w) and Arg-Asp-Arg-Phe-Leu 0.5% (w/w) had no significant difference to protect against the phenomenon of lipid oxidation with a TBARS-value about of 0.96 mg MDA/kg meat. The results of the present study show that adding the Arg-Asp-Arg-Phe-Leu' peptide derived from protein hydrolysates of Alfalfa RuBisCO at 0.5% (w/w) protects beef patties against lipid oxidation.

The work confirms the results obtained by [START_REF] Kobbi | Purification and Recovery of RuBisCO Protein from Alfalfa Green Juice: Antioxidative Properties of Generated Protein Hydrolysate[END_REF] which revealed that the Alfalfa RuBisCO hydrolysate (DH = 18.8 %) exhibited strong antioxidant abilities, including the prevention of linoleic acid oxidation, reduction of ferric ion, and stability of ABTS*.

Therefore, green juice derived from bio-refinery of Alfalfa offers a promising source of antioxidant peptides of interest for various functional food applications. The antioxidant activity of this peptide was probably due to its low molecular weight (705 Da) and the presence of aromatic residue Phenylalanine (Phe) which can quench free radicals by direct electron transfer to act as an antioxidant [START_REF] Geng | Encapsulating vitamins C and E using food-grade soy protein isolate and pectin particles as carrier: Insights on the vitamin additive antioxidant effects[END_REF][START_REF] Jeampakdee | The apoptotic and free radical-scavenging abilities of the protein hydrolysate obtained from chicken feather meal[END_REF][START_REF] Chi | Isolation and characterization of three antioxidant peptides from protein hydrolysate of bluefin leatherjacket (Navodon septentrionalis) heads[END_REF]. Peptides with highly potent inhibitory activity have hydrophobic amino acids as leucine (Leu), at the N-terminal position in the sequences [START_REF] Shi | A Novel Insight into Screening for Antioxidant Peptides from Hazelnut Protein: Based on the Properties of Amino Acid Residues[END_REF][START_REF] Nwachukwu | Structural and functional properties of food proteinderived antioxidant peptides[END_REF]. The microbial count increased proportionally with the storage time and reached the highest values at the end of chill period. Among the experimental groups, the negative control group showed the most rapid increase in the number of total microorganisms (from 4.29 ± 0.04 to 8.14 ± 0.31 log CFU/g), followed by samples treated with Arg-Asp-Arg-Phe-Leu peptide and BHT (Figure 4a). Arg-Asp-Arg-Phe-Leu peptide efficiency on bacterial growth inhibition was tested using 0.1%, and 0.5 % concentrations. In general, the microbial growth decreased significantly with the addition of Arg-Asp-Arg-Phe-Leu peptide and with the highest concentrations of this compound during the storage period (Figure 4a). Nevertheless, the group with a 0.5% (w/w) of Arg-Asp-Arg-Phe-Leu peptide showed the slowest increase in the number of total viable colonies until the eleventh day (7.12 ± 0.12 log CFU/g). The coliform count on a VRBL medium shows that no significant difference (p > 0.05) has been observed between the four treated groups with additives during the first three days (Figure 4b). But beyond this day, the group with Arg-Asp-Arg-Phe-Leu peptide at 0.5% (w/w) showed efficiency more important and comparable to that of BHT 0.5% (w/w) with the coliform counts of 4.73± 0.04 log CFU/g, at day 11. Therefore, this peptide can reduce by 32% the number of total coliforms in meat compared to the control. For the yeasts and molds, the most effective additives were the Arg-Asp-Arg-Phe-Leu at 0.5% (w/w) and BHT 0.5% (w/w), with the yeasts and molds counts of 5.64 ± 0.25 and 5.63± 0.06 log CFU/g, respectively, at day 11 (Figure 4c). These effects were similar to BHT 0.5% (w/w) (no significant difference). On the other hand, the groups treated with BHT 0.1% (w/w) and Arg-Asp-Arg-Phe-Leu at 0.1% (w/w) are less effective and they were presented no significant difference with the yeasts and molds counts 6.28 ± 0.25 and 6.19 ± 0.25 log CFU/g respectively, at day 11, compared to group without additive with 7.11 ± 0.62 log CFU/g. At last, for the growth of lactic acid bacteria (Figure 4d), during the first three days no bacterial proliferation has been observed for the four treated groups with additives (from 6.01 ± 0.42 to 6.35 ± 0.12 log CFU/g), except for the untreated group, the number of lactic acid bacteria was increased from 2.23 ± 0.81 log CFU/g at day 0 to 3.53 ± 0. 12 log CFU/g at day 3. After 6 days, three treated groups (with peptide 0.5%, BHT 0.1% and BHT 0.5%) showed a significant difference with the negative control group (Figure 4d). Once again, the most effective group were Arg-Asp-Arg-Phe-Leu at 0.5% (w/w) and BHT 0.5% (w/w). At this time, the microbial counts were of 4.67 ± 0.83 log CFU/g for the untreated group and 3.24 ± 0.18 log CFU/g for the group treated with the Arg-Asp-Arg-Phe-Leu peptide at 0.5% (w/w).

Many studies have reported the impact of antimicrobial peptides on meat and its products [START_REF] Zagorec | Meat microbiology and spoilage[END_REF]. Antimicrobial peptide can potentially be used to extend the quality of highly perishable foods during storage [START_REF] Gogliettino | Extending the Shelf-Life of Meat and Dairy Products via PET-Modified Packaging Activated With the Antimicrobial Peptide MTP1[END_REF].

Antimicrobial peptides have unique structural characteristics. The length of amino acids and the composition of antimicrobial peptides that have already been studied are extremely diverse. Their tendency to adapt to various structural changes when they come into contact with membranes is exclusive [START_REF] Torres | Peptide design principles for antimicrobial applications[END_REF]. The cationic peptides exert antibacterial activity by interacting with negatively charged bacterial membrane to increase membrane permeability and lead to cell membrane lysis and cell content release. Upon approaching the cytoplasmic membrane through electrostatic interaction with the microbial membrane, antimicrobial peptides bind to the microbial membrane and interact with the anionic components of the plasma membrane [START_REF] Rajagopal | Envelope structures of gram-positive bacteria[END_REF][START_REF] Baek | Lipopolysaccharide-bound structure of the antimicrobial peptide cecropin P1 determined by nuclear magnetic resonance spectroscopy[END_REF].

Conclusion 1.

A small antibacterial peptide from RuBisCO was isolated and purified.

2. Improvement of oxidative stability of minced beef meat during refrigeration.

3.

The peptide reduced the lipid oxidation about 60% to delay meat rancidity.

4.

Inhibition of the microbial growths under refrigeration during 14 days.

5.

The peptide can be effectively utilized as food ingredients in meat products.

antioxidants to be used in meat industry: A review. Food Research International, v.113, p.93-101, 2018

Control

Arg-Asp-Arg-Phe-Leu 0.1% BHT 0.1% Arg-Asp-Arg-Phe-Leu 0.5% BHT 0.5%

  model 75-72 (Whatman Inc, Haverhill, MA the USA). The polypropylene glycol (PPG) was used for the calibration and the optimisation of the machine. The peptide sequence was determined from the CID spectrum of the protonated analyse [M+H] + by MS/MS experiments. Peptide sequences were done using the bioanalyst software (Applied Biosystems, USA).

Figure 1 .

 1 Figure 1. Chromatographic profiles of RuBisCO without hydrolysis (DH=0%) and RuBisCO hydrolysis at 215 nm by RP-HPLC with a C4 column at 18.8% hydrolysis degree for 24 h.

Figure 2 .

 2 Figure 2. Reverse-phase HPLC profile of Arg-Asp-Arg-Phe-Leu peptide prepared by precipitation from RuBisCO hydrolysate with 2 M of NaCl and at pH 7.5 (A), ESI-MS spectrum of the Arg-Asp-Arg-Phe-Leu peptide, MS measurements were performed in positive ion mode using electrospray ionization (B).

Figure 3 .

 3 Figure 3. Lipid oxidation evaluation of the meat treated with BHT and Arg-Asp-Arg-Phe-Leu peptide at concentrations of 0.1 and 0.5% (w/w) during storage under refrigeration.

Figure 4 .

 4 Figure 4. Microbial changes in meat treated with BHT and Arg-Asp-Arg-Phe-Leu peptide at concentrations of 0.1 and 0.5% (w/w) during storage under refrigeration. (a) Total viable counts, (b) Coliform counts, (c) Yeasts and molds counts (d) Lactic acid bacteria counts.

International Journal of Peptide and

  . DOI: https://doi.org/10.1016/j.foodres.2018.06.073.

	FIELDS,	G.B.;	NOBLE,	R.L.	Solid	phase	peptide	synthesis	utilizing
	9fluorenylmethoxycarbonyl amino acids.

Acknowledgements

This work was funded by the Tunisian Ministry of Higher Education and Scientific Research. The authors would like to thank the Communauté urbaine du Grand Reims, Département de la Marne, Région Grand Est and the European Union (FEDER Champagne Ardenne 2014-2020) for their financial support of the Chair of Biotechnology of CentraleSupélec and the Centre Européen de Biotechnologie et de Bioéconomie (CEBB).