Cinema Lyotard: An Introduction

Jean-Michel Durafour L et us dispel a persistent injustice. For reasons I will come back to later, it has taken several decades for the name of Jean-François Lyotard to be able to appear, without a feeling of arbitrary unfairness or of audacious anomaly, alongside those of Gilles Deleuze, André Bazin or Serge Daney in a dossier dedicated to French cinema theory (as much as Lyotard detested this word 'theory', which reeks of monotheism and accounting. . .). 1 Looking a little more closely at the facts, which are all equally as stubborn as the theoreticians, we fi nd it hard to understand how such an ostracism -there is no other word for it -has been able to impose itself in the discourse on cinema, despite the fact (we will see this later also) that numerous theoreticians, sometimes those very ones who keep obstinately quiet about it, have openly stolen Lyotard's whole box of methodological and operative tools (the fi gural), with more or less good fortune. (That is said in passing.)

Certainly, one will not fi nd in Lyotard anything comparable to the enterprise later conducted by Deleuze with his two Cinema volumes; and unlike his friend, the author of Discourse, Figure has not inspired a whole critical disciplinary trend, nor given, coram populi, a new face to cinema studies. But like Deleuze, Lyotard has in his own right extended a metamorphic gesture that could be traced back to Maurice Merleau-Ponty, and in particular to his celebrated 1945 conference on 'The Film and the New Psychology'. It was there that Merleau-Ponty posed the fundamentals, common to cinema and the philosophy of perception, of what he called 'a certain way of being, a certain view of the world which belongs to a generation ' (1964: 59), and thereby established a programme for more than sixty years of philosophy and refl ection on cinema. Mer leau-Ponty himself only offered a catch-phrase or slogan for this programme -and his predecessors had never done any more on this virgin territory -since he never supported his proposition about the cinematic (perceptive) process with even the slightest, most idiosyncratic analysis of fi lmic (aesthetic) facts. One can reasonably affi rm that the Lyotardian gesture signals itself fi rst, and perhaps foremost, by two specifi cities, as follows. At least for the second, our philosopher is the only one (among thinkers of calibre) of his generation.

1. The inscription of a refl ection on cinema in a broad and general aesthetic project which, far from obscuring it, gives cinema a central and select value. When in 1973 Lyotard wrote his fi rst important text on cinema, 'Acinema', this aesthetic had just been put in place in Discourse, Figure (1971), notably around an operator destined to become famous: the 'fi gural'. It is not my intention here to analyse in detail a work as dense and complex as Discourse, Figure . But we can grasp several points. The fi gural is distinguished from the fi gurative (even if the fi gurative is the fi gural 'cooled down'). While the fi gurative designates 'a property that applies to the plastic object's relation to what it represents' (DF 205), the fi gural names that which in the presentation of the plastic event is always singular and disruptive. The fi gural escapes from predictability (otherwise it would be pre-fi gured), from recognisability, identifi ability and referentiality; it escapes from codifi cation, from forms, and from isotopic and pre-established structures. In the fi gural, the event is welcomed for itself, in its sensible symptomatic expression. It thereby differs from the regimes of signifi cation and designation, as well as from the mimesis of the fi gurative tradition, in which the plastic event is taken as no more than an (abstract, separate) sign which is referred to an other (thing, model). The fi gural is the vacant space left by desire, that is to say by sense, in the visible, and in the sensible in general. Harrowing, turbulent, it is that which makes difference; that is to say, that by which the donation of the sensible, in its constitutive difference, is possible -while words crush all intensities on the homogeneous spatiality of language and the generality of concepts. 2 Words are incapable of safekee ping the sensible event we task them with representing: 'one can say that the tree is green, but saying so does not put color in the sentence' (DF 50).

It is consequently, stricto sensu, impossible to say the fi gural (and still less to defi ne it). And in fact Lyotard tells us that Discourse, Figure is a 'bad' book, a failed book, because it misses the singularity of its subject. But it also poses an ideal of the book on art that Lyotard sought all his life to write, or not to write; in any case a book which would be badly written, badly constructed, even badly thought (since the concept is not in its proper place in images). Such a book would be the opposite of the argumentative essay approved by the philosophical tradition, a book more to be seen than to be read. Needless to say, therein lies the major diffi culty for all verbal (written, spoken) aesthetics subject to the nonverbal and non-dianoetic thought of works of art. And Lyotard did not cease, under one form or another, to encounter this diffi culty. The whole problem is announced right away in Discourse, Figure : how can we express that which, in art and in particular in visual art (painting, literature), escapes the readable and the sayable ('[r]eading is understanding, not seeing' (DF 211))?

How can we express that which takes place behind language -since at the same time one cannot situate oneself in complete exteriority to articulated language? One sees because one speaks (did the Argentinian writer Jorge Luis Borges not ask himself in which language he would die?) since at the same time one cannot but resort to concepts in order to account for the otherness of aesthetic experience. Is not philosophy still needed in order to express what escapes philosophy? How can we express the symptom which dazzles, uncodable, uncharacterisable, on this side of language and fi gures, with words that the image leaves us with, and which replay in the present something of the prepredicative and originary experience of the world, of the time before the distinction between subject and object, before knowledge and refl ection? 3 But if it is true that, on the one hand, sense is 'muddled, inaudible, as if inexistent', 'until the military front of the words -so to speak -has contacted it' (WPh 75), the fact remains that, on the other hand, speech 'is already inarticulately present to what is not yet said' (WPh 82), just as the word, the muttum, always keeps a vestige -the sounds impish, treacherous -of the muttus which structures it; that there is a 'colloquy prior to all articulated dialogue' (WPh 93). Elsewhere: 'One needn't be immersed in language [langage] in order to be able to speak. . . . What speaks is something that must remain outside of language as system and must continue to remain there even when it speaks' (DF 8). How then to keep fi delity to the fact that 'one does not paint in order to speak, but in order to keep silent' (Lyotard 2015: 38)?

2. The extension of the theoretical gesture in a certain practice of fi lm. But this practice also precedes the theoretical gesture, and gives weight to it, in a creative process. Lyotard did in fact make several fi lms on celluloid or video, from an essentially experimental perspective. From the end of the 1960s, within a collective composed equally with Dominique Avron, Claudine Eizykman and Guy Fihman, he participated in the creation of the video L'Autre scène, a six-minute short fi lm on the analogy between the work of the dream and the work of fi lm, intended as the extension of a seminar on Freud. A fi rst, silent version dates from 1969, but the fi lm in its defi nitive form, with sound, was not completed until 1972. Two years later, Lyotard made on his own an experimental three-minute short fi lm on 16mm, Mao Gillette (1974). The 1970s drew to a close with Tribune sans tribun (1978), an appropriation by Lyotard of a report for French television, for the programme Tribune Libre. He chose to turn it into an experimental video whose title indicates its subject: the absence of legitimation of the speakers on the televised stage, who speak of anything and everything. Thus ended a decade of experimentation with images, certainly on one level modest, but which, for a philosopher, is an event suffi ciently rare for it to be appreciated for its proper value. A last fi lm, from 1982, À blanc, breaks with the experimental gesture in that it presents itself fi rst and foremost as an exegetical essay in images, on someone else's aesthetic expression (which is why I excluded it from my survey in the little book that I scribbled on the question of cinema in Lyotard (Durafour 2009)): it's a brief video dedicated to a series of canvases by the painter René Guiffrey.

Lyotard's entry into cinema theory is hence made by a double path: the aesthetic of the fi gural, and the practice of experimental cinema. The latt er appears to put the former to work in a fi lmic fi gural aesthetic: that is to say, a pulsional aesthetic, subject to desire, to difference, through which the cinema would escape from industry (capital, revenue, the same, mimesis), and clearly would be an art. We will come back to this shortly. First, however, we can conclude this second point by noting that the Lyotardian concept of acinema directly inspired for several years the work of Guy Fihman and Claudine Eizykman (V.W. Vitesses Women, Ultrarouge-Infraviolet). Moreover, the experimental fi lmic moment played an essential role for Lyotard in the passage from the aesthetic paradigm of modern art (Cézanne, Klee) -which for the most part, even if not uniquely, characterises Discourse, Figure, and which is primarily a legacy of the phenomenology of Merleau-Ponty -to the great dialogues with contemporaries (Jacques Monory, Valerio Adami, Sam Francis, Karel Appel) which began thereafter.

1973

Everything on the theoretical plane begins then with the initial text 'Acinema'. It fi rst appeared in 1973 in the special issue of Revue d'esthétique edited by Dominique Noguez, 'Cinéma: Théorie, Lectures', and was subsequently collected, the same year, in Des Dispositifs pulsionnels. In this text, Lyotard appears to apply the principles of fi gural aesthetics developed in Discourse, Figure to cinema.

Experimental cinema is a cinema of irreverence and 'irreference', as Dominique Noguez (1979: 42) said; it refuses the conventions and the analogical imprint indexed on dominant mainstream cinema. It is a cinema of apparition more than of appearances; appearing, in effect, to comply with all the characteristics of the fi gural: a cinema which refuses mimesis, representation, narration (which the recognition of fi gures implies: what do they go on to become?). Lyotard calls it acinema: being the negation (a-) of the majoritarian (industrial, commercial) cinema, that is to say of the cinema norm, where the movement (kinêma) and the image are neutralised in a middle range acceptable to the largest number of viewers. Acinema is experimental cinema. But not all experimental cinema is acinema. Or, more precisely, Lyotard restricted the concept and excluded all cinema which still, despite everything, rests on the narrative and the fi guration (surrealist cinema, for example) -that is to say all cinema in which the indexical analogism still plays the role of a signifying centre. Acinema is cinema which accepts 'what is fortuitous, dirty, confused, unsteady, unclear, poorly framed, overexposed' (Ac 33): intensities, timbres, nuances, colours, drips, bursts, breaks, scratches, cuts, openings. In a word, the energetic: the tenuous, the unstable and shifting, which always escapes the deterministic and reductive constructions of the well-formed.

In this text Lyotard proposes, on either side of the 'normal' movements of mainstream cinema, a distinction between a cinema of extreme mobility, or 'lyric abstraction' (Ac 40) (for example, Hans Richter, Viking Eggeling), and a cinema of extreme immobilisation, or 'tableau vivant' (Ac 40) (he doesn't give any examples, but such a gesture characterises, among others, a fi lmmaker like Andy Warhol. . .). In my book Jean-François Lyotard: questions au cinéma. Ce que le cinéma se fi gure, I spoke of 'exo-cinema ' and 'endo-cinema' (Durafour 2009: 35-8). Exo-cinema corresponds to the cinema of agitation and makes representational fi guration explode by attacking it from the exterior: it's the cinema of the avant-gardes (notice the original military sense of the expression). Endo-cinema corresponds to the cinema of immobility and makes the fi guration implode by drilling it from its interior: it's the cinema called underground, literally 'of the sub-soil' (even if historically, we know, the term responded to other imperatives).

Lyotard paid dearly for his fondness for experimental cinema, which did no more at the time than develop the underground movement theoretically, 4 and which had to remain in purgatory for a long time. (And incidentally, when he became interested in a completely different cinema later, people didn't really take it into account.) Pascal Bonitzer -characteristic of the attitude that we recalled in opening, and in his typically virulent style -has not hesitated to qualify acinema purely and simply as 'puke ' (1976: 70). Various commentators continued to be frequently discomforted by it. Thus Dominique Chateau, in Cinéma et philosophie, after having recalled, almost as an excuse, that ' [Lyotard's] contribution to the philosophy of cinema is, if not contradictory, at least relatively disparate ' (2003: 126), situates the essence of this contribution (incidentally he is not the only one to do so) in the fi gural (the word fi rst encountered in a working note of the late Merleau-Ponty). Yet without doubt we owe the recourse to the terminology of the fi gural in the fi lmic regime less to Lyotard himself than to those who are -more or less directly -infl uenced by him. In Discourse, Figure, cinema is mentioned only in a footnote concerning Méliès (yet it is to this foundational text that most of the fi gural analyses of cinema refer). And in 'Acinema', there is not the slightest mention of the word 'fi gural', even though it was readily available (and was still brand new). . . All this is, to say the least, curious, and opens up some complex questions that it is not possible to deal with here (among which the principal one is surely this: is cinema a plastic art?). This last point, which accounts for the ambiguous status of underground cinema, which is lodged in fi gurative representation, then makes possible another reading: a more accurate reading of this text which is usually poorly understood, and certainly very poorly liked (but perhaps because poorly understood?) -but also liked, when it is, for the wrong reasons.

TAKING INTO ACCOUNT THE FIGURATIVE

Lyotard always displayed a great suspicion of the critical gesture, which consists in wanting to take hold of an exteriority, but which only perpetuates in another sense what it rejects. To this posture he always preferred clandestine infi ltration, undercover work. 5 I recalled this earlier here, apropos of language. One can say the same for representation: it is impossible to position oneself in complete exteriority to it. 6 With language, we must situate ourselves in language outside of language, in the eye of the cyclone (the pivotal zone where everything is very calm), as it exists as a 'well of discourse' (DF 7). We affi rm that there exists in the same way a wellspring of fi guration in which the fi gure is secreted from representation; that is to say, from what is always already fi gurative. And years later, Lyotard will end up somewhere else by detaching himself from the fi gural. In What to Paint? (1988), he did not hesitate to write that Discourse, Figure is 'a book which makes a screen to the anamnesis of the visible' (WP 96). 7 For Lyotard produced more than one text on cinema (even if we silently pass over the multiple examples taken from cinema in the framework of his philosophical writings). While these writings are 'disparate' (Chateau's word is appropriate), this does not however permit the conclusion that they lack coherence or unity, or that they remain minor or imperfect thoughts. Lyotard simply never felt the necessity or the desire to collect them or to develop them into a book. There is no need to ask why: it is a fact. We just have to live with it.

What are these texts? It is not my project to make a catalogue or some kind of taxonomy. If I leave aside -in addition to 'Acinema', which I have just spoken about -pages here or there where mention is made of such and such a fi lm, such as interviews mentioning the fi lms presented during the exhibition Les Immatériaux at Beaubourg in 1985 (which require a proper perspective), we can propose as the most important the following texts, generally put in the garbage of cinema theory:

1. An article in English, 'The Unconscious as Mise-en-Scène' (1977), which appeared in the collection edited by Michel Benamou and Charles Caramello from the University of Wisconsin Press, Performance in Postmodern Culture. It presents, in the entire fi nal section, an analysis of Michael Snow's La Région central, starting from the work of the unconscious.

2. 'Deux métamorphoses du séduisant au cinema' [Two Metamorphoses of the Seductive in Cinema] (1980), a contribution to the book edited by Maurice Olender and Jacques Sojcher, La Séduction, which appeared from the publisher Aubier-Montaigne. Here the principal focus is Francis Ford Coppola's Apocalypse Now (1979), and just at the end, very rapidly, some fi lms of Hans-Jürgen Syberberg (Hitler, ein Film aus Deustchland, Winifred Wagner und die Geschichte des Hauses Wahnfried von 1914Wahnfried von -1975)).

3. Finally, the posthumous publication of the manuscript of a conference paper delivered in Munich in 1995, 'Idée d'un fi lm souverain' [The Idea of a Sovereign Film], included by Dolorès Lyotard in Misère de la philosophie in 2000. The subject of this crucial paper is fi lmic invention in the so-called 'modern' cinema (in Yasujirō Ozu, Orson Welles, etc.).

If the fi rst of these texts still inscribes itself in the continuity of the acinematographic libidinal theses, it is not at all the same with the other two. These two texts do not break purely and simply with acinema, but oblige us to reconsider its nature. Starting by coming back to the seminal article of 1973 and proposing a slightly different reading, we are clearly enabled, in my opinion, to grasp an important stake of this text which has so far escaped us. We can understand what it was about Lyotard's writings that made the cinema theorists uncomfortable: their refusal to constitute a system (system is closure, capitalisation, hoarding, theorisation), to offer a fully delivered, established global theory; their 'drift' (a Lyotardian word), that one could take for disorder, or see as superfi ciality. The necessity which is ours, to which they oblige us, is that of needing to present and represent them in order to 'squeeze the juice out of them', so to speak. But in this lies what in Lyotard's eyes always counted the most (and this was also the basis of his interest in experimental cinema): a fondness for differences, a fi delity to singularities, a passion for dissensus.

'Two Metamorphoses of the Seductive in Cinema' and 'The Idea of a Sovereign Film' do not then fully satisfy the model of acinema. Let us move forward from this observation. It is diffi cult, in effect, to see Apocalypse Now or The Magnifi cent Ambersons as experimental fi lms. That being so, they include some acinematographic moments (the helicopter attack with all its smoke and pyrotechnics in the fi rst, for example). This is what Lyotard calls, in the second of these texts, 'fi lmic facts [faits]' (ISF 68; my emphasis). In this regard, if one reads a little quickly, one could see a conservative weakening of the orthodox acinematographic position, particularly in 'The Idea of a Sovereign Film'. This text was undertaken after the publication of the two volumes of Cinema by Deleuze, from whom Lyotard acknowledges, in the note sent to the journal Libération on the occasion of his death, having appropriated several ideas (MP 193). Had Deleuze not presented a philosophy of the event and of singularity able to think all cinema, including when it is fi gurative and narrative (which it is never either straightforwardly; if such were the case, as Eugène Green (2003: 35) said, there would be no difference between a cinematic fi lm and a tourist fi lm 8 ), and even if the fi gurative and narrative axis is not the one retained by the author of The Movement-Image? 9 In this conference paper, Lyotard in fact displaces the acinematographic marker taken as fi lmic totality towards the 'fi lmic fact', which alone guarantees the sovereignty of fi lm. What does this mean? It means that sovereignty excludes totality. 10 Do we witness a frightful rupture?

1. One can fi rst of all say that the 'fi lmic fact' makes a difference, visually or auditorily, in relation to a globally maintained 'narrative-representative form' (ISF 64,68).

In this sense, it maintains fi delity to the difference of the event because such a difference can only be perceived on a homo-audio-visual ground from which it is detached (Merleau-Ponty, in his Phenomenology of Perception, called it the 'phenomenal fi eld'). When there is only difference, when difference is all there is, when difference makes a totality, it cancels itself as difference. When everything is different, nothing is different. It is therefore appropriate to keep the fi gurative ground, at least in this capacity. But there's more. In addition, the fi lmic fact is a 'faded' reality; it consists in 'moments [which correspond to] outcrops of the visual or the vocal in the surface of the visible and the audible' (ISF 69).

2. Next, this gesture permits us at once to widen the scope to Deleuzian proportions, 11 to make a case for the application of the acinematographic criterion, without abandoning it, to a cinema more expanded and less restrictive than acinema strictly speaking. To be given henceforth by examples of acinematographic moments, of 'fi lmic facts', of 'intense instants, temporal spasms, which are only transcendents because they emanate from immanence, that is to say from a realist experience and existence -one says in fi lmography: neo-realist' (ISF 62; my emphasis). Moreover, was acinema really as restrictive as it appeared to be? One will remember what we recalled earlier: that despite superfi cial appearan ces, the profound originality of the acinematographic thesis of 'Acinema' bore principally on the fi gurative and representative image (which begins to explain the absence of the explicit terminology of the fi gural. . .). This was precisely the underground cinema: attacking representation from the interior of representation (the under-ground), and not with any putative and deceptive exteriority, like the eye of the cyclone (one will recall the image from Discourse, Figure ) where it is neutralised at its centre. Thus, in 'Acinema', it is clearly underground cinema which Lyotard prefers, for reasons, one could say, of a 'general philosophical' nature (total exteriority is an illusion). Now what does he say of this cinema? This: 'the paradox of immobilisation is seen to be clearly distributed along the representational axis' (Ac 40; my emphasis). This is the only sentence in the article where Lyotard addresses himself directly to his reader, which would suffi ce to underline its importance, or at least its difference. Moreover, in placing the accent on the representation of the perceptive reality in the image, such a formula appears to be a counter-current to the fi gural perspective. Without doubt that's why this exclusively disruptive perspective will never really have existed (as will be said differently in later texts).

FILMIC MATTER (PRESENT POTENTIAL)

In the philosophical and aesthetic canon, matter, in contrast to form (which informs), has generally been taken to be purely passive. The source is above all Platonic -the Timaeus (50c) describes matter as 'the stuff from which everything is moulded', the principal 'wax' (ekmageîon), 'modifi ed and moulded by the things that enter it, with the result that it appears different at different times' (Plato 2008: 42). Matter is only the 'that in which' of the thing, whose form is the 'that for which'. It is, before all information, unlimited, therefore imperfect, tending towards non-being, since it is form which has defi nition and completion (form is closed).

Nevertheless we can say right away that such an opposition -to which materialism only takes the opposite stance, which means it fails to put it in question -is problematic. Henri Focillon, in The Life of Forms in Art (1934), had already attracted attention to this apparently extravagant proposition: matter determines form, matter is the form (the determination) of form. And he proposed to speak of matters rather than of matter, in order to designate the plural and complex 'formal vocation' (Focillon 1989: 97) of matter: grains, waves, notches. Such matters -marble, metal, wood, graphite, charcoal, voice, etc. -are not anodyne for an artistic project, nor are they equivalent to each other; they differ according to their fl exibility or the effects aimed at. '[A] charcoal drawing copied in wash . . . at once assumes totally unexpected properties; it becomes, indeed, a new work' (1989: 100). In the same way, the tool in its materiality, and not in the gesture which manipulates it (which is form, the design of the artist), contributes to making the fi nal form of the work: 'Certain plates reek of the tool and conserve a metallic aspect [from it]' (1989: 114). Moreover the term 'form', in an expression such as 'matter imposes its form on the form', need not be heard univocally in both occurrences: the 'form' of the matter is not the 'form' of the form. The formative power of the matter has its proper principles, which must not be confused with those of form.

Lyotard will go a lot further than Focillon (who in the end remains rather conventional, especially from the point of view of 'well written' language) by inscribing himself ardently in this opening. From the Greeks -as we know -matter is opposed to logos. We cannot say matter, even with the scholastic (originally Aristotelean) title of the materia prima, pure potentiality, of perceptive experience, because it gives us nothing to grasp. Lyotard responds: no discourse, which is always putting-in-form, putting-in-signs, can signify matter. Matter can only be expressed.

Lyotard thinks matter in a fi rst period through the fi gural, and the privilege of the visual over the visi ble and the readable: the event of the image. (This period culminates in Discourse, Figure, still in some regards very much a conventional book in its format and organisation.) That being so, the pulsional energy of desire makes established fi gures burst like a saxifrage plant; it breaks the decorum of the monotone, of the predictable, of that which is set in stone. There is a transgression of codes and writing by the irruption of the primary processes (intensities, desires) in the secondary processes (language, actions) by virtue of what Freud called, apropos of the formation of dream images from unconscious thoughts, the Rücksicht auf Darstellbarkeit ('the taking into account of fi gurability'). It's this fi gural matter which, in refl ection on cinema (Philippe Dubois, Raymond Bellour) or art in general (see the work of Georges Didi-Huberman), has dominated and still largely dominates the fi eld of analysis.

But we have seen that the fi gural does not constitute the whole of the Lyotardian thought of the image, and that cinema breaks with it. And it's without doubt here that Lyotard's refl ections are more fascinating for both theoretical research and an actual aesthetic of cinema. In the last decade of his philosophical activity, Lyotard spoke of 'an energetics or . . . a general dynamics' (MT1 199; my emphasis). 12 This 'energetics' envisages matter, no longer from the fi gural point of view, but from the side of fi gurative invention. What is the difference between the two? In a word: the fi gural concerns the presentable, not the representable (that is to say it concerns the event, not the thing, the res). The 'general energetics' is the fact that there is, in the representation, the unpresentable. What is unpresentable is presence itself. The presence of sound is not sonorous, as the presence of blue is not blue, as the presence of a rock is not mineral. This amounts in the end to the grandeur of art, when it opens onto the originary opening of our rapport with the world: the presentation of unpresentable matter -which fi xes itself as objective (in the various social, professional, etc., uses that we make of matter, which present everything quite well) -is in fact always aporetic. The question of painting: how does the colour perceive itself in itself? The question of music: how does sound perceive itself in itself?

The presence of the sensible world that cannot be taught, that is anterior to any symbolism and accomplished anew in art through its initial chromatic stridency, has always persistently overtaken pale discourse and its wan digressions . . . Beyond the spirit that meditates and the eye that reads, vision is already in the process of seeing. (MT2 659) From this then comes the exigency of a new aesthetic orientation, which Lyotard here and there calls 'anaesthetic' (in opposition to the traditional aesthetic of forms). This aesthetic is concerned not with presented forms, but with matter; and a matter not only stolen, pilfered from between forms, but a matter which it is all the more possible to welcome in its unpresentability because it is itself unformed. Since the aesthetic subject, which here no longer recognises itself (recognition is form), is numb, groggy, precisely anaesthetised. One cannot see matter (one sees forms); one can only eye it up. In the sense not of objectifi cation (this is still form: constraint), but of the French verb mater, understood as eyeing up furtively as if one had not clearly seen. As if one could not get over having seen that one had not seen.

This matter fi nds a fertile ground for expression in the cinematic image. We must not let ourselves be misled by its photorealism. I have argued in my book already cited (it is impossible for me to repeat here the long demonstration) that, through a relationship with pictorial hyperrealism, which plays itself out in cinema, including in its more fi gurative and narrative aspects, there is an explosion of the criteriology of traditional, analogical mimesis (resemblance). In cinema, one has little business with an image which resembles, better than another image, the thing which it models. Rather, cinema is concerned with the fact that, for the eye habituated to old images, the thing resembles itself in a way never before seen (black-and-white changes nothing here: for decades, this has been the norm of the realism of the photographic image, and colour in cinema only started to increase the exoticism, the fantasy, the dream. . .). It's not a question of saying that cinema gives us the 'things themselves': how would we know what they are? But the thing, as Lyotard said apropos the hyperrealism that I am drawing out here for cinema, comes half-way to meet us, and stumbles; it becomes 'a bit too much before the eyes' (MT2 485; 13 my emphasis). It is not only abstraction which gives a skewed image of the world: 'showing reality' (MT2 471 14 ) can do this just as much. In this 'excess of presence' a certain conventional image of the world dematerialises itself, under the cover of its minute and scrupulous renewal, and materialises what can only be eyed up [être maté]: a donation of presence-matter contravening established forms so that the classic aesthetic subject (with its coded certitudes, its mastery) feels itself threatened through and through by being touched by nullity. The form is that by which the fi gure exists, the matter that by which it insists. And this insistence is, at the same time, my private concern, my business, but also a demand which intimidates me intimately.

I will give, in conclusion, a very short fi lmic example of an analysis according to anaesthesic matter. John Cassavetes' Faces (1968) is celebrated in the history of cinema for its big shots exacerbating the grain of the skin of the body; an operation intensifi ed in various ways by the Éclair NPR, Kodak 4x16mm fi lm, and powerful lighting by quartz lamps. One can do a fi gural reading of this fi lm by paying attention to the jump cuts, the broken speeches, the over-the-shoulder shots, the variations of focus, and so on and so forth; in short, by paying attention to the vicissitudes of the form. But the 'general energetic or dynamic' makes sense of it in another way: a way of insisting, and not of rerouting (the fi gural) with regard to images, including those more fi gurative, more coded. It's a question of a phobic matter, which places the spectator before the anxiety of his or her proper aesthetic condition: that of perception. Better: it performs here, in these various images, an affection -what I have called above a commanded and intimidating intimacy, coming from the unpresentable presence of matter (the unrepresentable in representation). This announces: that in virtue of which we are beings of sensation is not anything. In the face of these skins visible to the pores, in the face of these prominent holes, with all their doors and all their windows (Leibnizian language), it's not only the form of the skin that we see [voyons]; we see [matons] the anxiety of being penetrated ourselves because if we see, if we feel, it's because we are penetrated. Matter reveals sensation to us as anxiety by connecting us to the unpresentable condition of presentational perception. Like Klein bottles, we do not have a (full) body because we are only gaps, voids, atomic bonds. And this is because we have neither interior nor exterior (the fi lm then asks: where are feelings, emotions, held? Between bodies?. . .). Matter is 'the suffering of a body visually bewildered' (PF 231); in effect, unpresentable matter, inasmuch as Lyotard defi ned the artistic power of cinema in this way: 'I think that a fi lmmaker, if he or she is not a commercial trader of images, carries in him or herself the idea of a sovereign fi lm where from time to time the realist plot allows the presence of the ontological real to pass . . . to which [idea] no object, here no fi lm, can correspond in experience . . .' (ISF 69). and marginal place (gaseous perception), while the central paradigm for thinking cinematic perception is liquid (equivalence of all spatial points, a-centric, immanence. . .) in opposition to ordinary human perception of the terrestrial type (differentiation and hierarchisation of points, centric, transcendence). 9. Deleuze repeats it in several places: 'In art, and in painting as in music, it is not a matter of reproducing or of inventing forms, but of capturing forces. For this reason no art is fi gurative ' (2003: 56). 10. The argument is primarily political: totality aims at unity under a dominating 'grand narrative', and is therefore the enemy of singularities, feelings, events; of all others, all dissenters. In fact, where there is totality, the sovereignty of 'little narratives', of differences, is neutralised by homogenising and generalising reason. 11. The 'fi lmic facts' are described by Lyotard in terms which are manifestly very Deleuzian: for example, 'vacuoles, or blocks of time, in the realist-narrative progression' (ISF 64). 12. I develop this question of the place and role of matter in Lyotard's aesthetics in further detail in the epilogue to MT1. 13. TN: Translation modifi ed. 14. TN: Translation modifi ed. 15. TN: Thanks are due to Véronique Malcolm for generously taking the time to provide helpful comments on this translation.

Translated by Ashley Woodward

  ON THE COMMON SAYING: IT MAY BE TRUE IN THEORY, BUT IT DOESN'T APPLY IN PRACTICE . . .

  Every word immediately becomes a concept, inasmuch as it is not intended to serve as a reminder of the unique and wholly individualized original experience to which it owes its birth, but must at the same time fi t innumerable, more or less similar cases -which means, strictly speaking, never equal -in other words, a lot of unequal cases. Every concept originates through our equating what is unequal'(Nietzsche 1976: 46). 3. The topos of phenomenology since Husserl. On the notion of Lebenswelt, see in particular Husserl 1970: 103-13. 4. See, in addition to the academic work of Noguez from the end of the 1960s (on North American underground cinema), from the other side of the Atlantic: Parker Tyler's Underground Film (1969), the articles by Annette Michelson and Manny Farber in the journal Artforum, Jonas Mekas's Ciné-journal (1971), P. Adams Sitney's Visionary Film (1974), etc. 5. Once again, permit me to refer you to my book -Durafour 2009: 69. 6. The reading of Lyotard for cinema that I propose here distinguishes itself from this complementary work by Jean-Louis Déotte. Déotte is less interested, in Lyotard, in what is thought of cinema or what can aid us by fi rst beginning from refl ections on cinema, than to what, in his philosophy of language (The Differend), can serve the purpose of an understanding of the phenomenon of montage: namely, montage as the expression of a differend between frames thought of as analogues of linguistic phrases, as irreducible 'universes' (each phrase presents a type of universe), between which it is necessary, despite everything, to link (one will have recognised the axiomatisation of The Differend). See in particular Déotte 2004. 7. I do not have the liberty, in the context of the present introductory article, to develop the arguments advanced by Lyotard in this extremely rich book. We can just note a diffi culty attendant to the fi gural: at what time can we identify it? Because to identify is to recognise, and we can only recognise what is repeated, what reproduces itself a number of times. But is this then still the fi gural? 8. In Deleuze's two books, incidentally, experimental cinema occupies a restrained

	15
	NOTES
	1. This chapter was fi rst published in La Furia Umana, Paper #3 (2013), pp.
	121-36.
	2. '


Walid Djema
email: walid.djema@inria.fr

Frédéric Mazenc
email: frederic.mazenc@l2s.centralesupelec.fr

Catherine Bonnet
email: catherine.bonnet@inria.fr

Stability Analysis and Robustness Results for a Nonlinear System with Distributed Delays Describing Hematopoiesis. $

Keywords: Delay, Positive system, Lyapunov, Stability, Biological model

A nonlinear system with distributed delays describing cell dynamics in hematopoiesis is analyzed -in the time-domain-via a construction of suitable Lyapunov-Krasovskii functionals (LKFs). Two interesting biological situations lead us to re-investigate the stability properties of two meaningful steady states: the 0-equilibrium for unhealthy hematopoiesis and the positive equilibrium for the healthy case. Biologically, convergence to the 0-equilibrium means the extinction of all the generations of blood cells while the positive equilibrium reflects the normal process where blood cells survive. Their analyses are slightly different in the sense that we take advantage of positivity of the system to construct linear functionals to analyze the 0-equilibrium, while we use some quadratic functionals to investigate the stability properties of the positive equilibrium. For both equilibria, we establish the exponential stability of solutions and we provide an estimate of their rates of convergence. Moreover, a robustness analysis is performed when the model is subject to some nonvanishing perturbations. Numerical examples are provided.

Introduction

With the ultimate goal of determining a model describing cell dynamics in acute myeloid leukemia, which will be of use for the optimization of polychemotherapies, we start here with a model describing the process of fabrication of blood which was studied in [START_REF] Adimy | Discrete Maturity-Structured Model of Cells Differentiation with Applications to Acute Myelogenous Leukemia[END_REF] and revisited by input-output methods in [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF].

HSCs are immature unspecialized cells able to produce cells with the same maturity level and to differentiate into specialized cells. This is a simplified development scheme, which does not take into account other cell fates -increasingly highlighted in recent years-such as cell dedifferentiation [START_REF] Cai | Dedifferentiation: a new approach in stem cell research[END_REF]. In fact, the complex cascade of signals regulating hematopoiesis is not currently clearly identified. Therefore, the importance of this biological process has motivated many theoretical and experimental works that focus on the earliest generations of immature cells since they play a critical role in blood formation, and because they are the source of several hematological disorders. The long list of works devoted to blood cells dynamics includes [START_REF] Adimy | Discrete Maturity-Structured Model of Cells Differentiation with Applications to Acute Myelogenous Leukemia[END_REF], [START_REF] Adimy | Mathematical Model of Hematopoiesis Dynamics with Growth Factor-dependent Apoptosis and Proliferation regulations[END_REF], [START_REF] Adimy | Boundedness and Lyapunov function for a nonlinear system of hematopoietic stem cell dynamics[END_REF], [START_REF] Mahaffy | Hematopoietic Model with Moving Boundary Condition and State-delay: Applications in Erythropoiesis[END_REF], [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF], [START_REF] Özbay | Stability analysis of systems with distributed delays and application to hematopoietic cell maturation dynamics[END_REF], [START_REF] Mackey | Unified hypothesis of the origin of aplastic anemia and periodic hematopoiesis[END_REF], [START_REF] Pujo-Menjouet | Long period oscillations in a G 0 model of hematopoietic stem cells[END_REF], and [START_REF] Fridman | Stability of the cell dynamics in Acute Myeloid Leukemia[END_REF].

Acute Myelogenous Leukemia (AML) is a serious type of cancer, which is characterized by an overproduction of abnormal myeloblasts, simultaneously with an inability to develop further into mature white blood cells (a blockade in the maturation process). Due to their overproliferation, blasts invade the bone marrow and even -sometimes -the blood circulation (Figure 1-a), which prevents adequate production of mature healthy blood cells. Since we want to emphasize on AML, we consider that the model that we focus on describes the development hierarchy leading to white cell production in the myeloid lineage.

Relying on several essential contributions by Mackey and his colleagues ( [START_REF] Mackey | Unified hypothesis of the origin of aplastic anemia and periodic hematopoiesis[END_REF], [START_REF] Pujo-Menjouet | Long period oscillations in a G 0 model of hematopoietic stem cells[END_REF], [START_REF] Mahaffy | Hematopoietic Model with Moving Boundary Condition and State-delay: Applications in Erythropoiesis[END_REF], to name but a few), Adimy et al. introduced and analyzed in [START_REF] Adimy | Discrete Maturity-Structured Model of Cells Differentiation with Applications to Acute Myelogenous Leukemia[END_REF] a nonlinear system with distributed delays to model cell dynamics in several maturity stages. This is the model we study here, considering that it describes a cancer state when some of its biological parameters are abnormal (i.e. being different from healthy parameters, or becoming time-varying to model the effect of appropriate infused drugs) and it reflects a healthy situation when all its parameters are normal. Using a Lyapunov technique we improve some existing results in two different contexts: i) we provide theoretical conditions to eradicate cancer cells in what we assume to be a basic unhealthy situation, and, ii) we ensure the survival of healthy cells in normal hematopoiesis. A key point that we emphasize here is that the Lyapunov direct method offers strong tools to study exponential convergence of solutions, estimates on their decay rates (for both steady states), as well as estimating the basin of attraction of the positive equilibrium point and this, in our opinion, improves the way to study the phenomenon of hematopoiesis (see the concluding remarks in [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF]). On the other hand, the search for a suitable Lyapunov functional is generally quite difficult, since no systematic methods apply [START_REF] Malisoff | Constructions of Strict Lyapunov Functions, Serie : Communications and Control Engineering[END_REF][START_REF] Kolmanovskii | Introduction to the Theory and Applications of Functional Differential Equations[END_REF], and that is the challenging problem that we are dealing with in this contribution.

The paper is organized as follows. In Section 2 we briefly present the model of interest. Section 3 is devoted to the study of the 0-equilibrium of the system. We establish global exponential stability even when some parameters are timevarying, then we perform a robustness analysis. The strictly positive equilibrium X e of the nominal system is discussed in Section 4. An estimate of its basin of attraction is proposed via a construction of a novel Lyapunov functional, that also allows us to perform a robustness analysis of the perturbed system.

Description of the model and known results

We revisit from [START_REF] Adimy | Discrete Maturity-Structured Model of Cells Differentiation with Applications to Acute Myelogenous Leukemia[END_REF] the model described in Figure 1-b, where for all i ∈ I n = {1, . . . , n}, n ≥ 1, x i denotes the total density of resting cells of generation i. A resting cell is a cell that is not actively in the process of dividing. The reintroduction function from resting into proliferating subpopulation of the i-th generation is denoted β i (•). Proliferating cells can divide between the moment they enter the proliferating phase and a maximal age τ i > 0, while the apoptosis rate, γ i , represents the death rate of proliferating cells of the i-th generation. At each division, a proportion K i of dividing cells goes to the next resting stage of the development hierarchy of interest, while the other part (L i = 1-K i ) stays at the same level i (self-renewing process), with the convention that K 0 = 0. The constant δ i covers both the death rate of the resting cells of the i-th generation, together with their differentiation into lineages that we do not focus on. Finally, the dynamical system equation is in the form:

ẋi (t) = -δ i x i (t) -w i (x i (t)) + 2L i τi 0 g i (a)w i (x i (t -a))da + 2K i-1 τi-1 0 g i-1 (a)w i-1 (x i-1 (t -a))da + i (t), (1)
for each compartment i ∈ I n , and t ≥ 0, with

w i (x i ) = β i (x i )x i , g i (a) = e -γia f i (a)
, where the f i s are C 1 functions representing the cell division probability densities, such that f i (a) ≥ 0 for all a ∈ [0, τ i ], and τi 0 f i (a)da = 1, since it is assumed in [START_REF] Adimy | Discrete Maturity-Structured Model of Cells Differentiation with Applications to Acute Myelogenous Leukemia[END_REF] that the mitosis occurs before the age-limit τ i . Moreover, biological facts induce that the parameters δ i , L i , K i , τ i and γ i are positive for all i ∈ I n , with K 0 = 0 and K i ∈ (0, 1) for all i ∈ I n . The functions β i (•) are assumed to be differentiable and decreasing functions such that lim

a→+∞ β i (a) = 0.
For a later use, we introduce the following parameters:

C i = τi 0 g i (l)dl, and, α i = 2L i C i -1, (2)
where α i is assumed to be strictly positive, for all i ∈ I n (see [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF], Assumption 2). We will perform a robustness analysis of the model (1) under nonvanishing perturbation terms i (t) ∈ [0, i ], where i > 0, for all i ∈ I n and t ≥ 0. It is well-known that disturbances are in general due to the lack of accuracy when modeling the laws governing complex living organisms. More precisely, in the model that we study, uncertainty comes from the biological parameters and functions (e.g. the nonlinearity β i , introduced in [START_REF] Mackey | Unified hypothesis of the origin of aplastic anemia and periodic hematopoiesis[END_REF]), and from more complex phenomena which are difficult to model. In particular, the ability of differentiated cells to undergo lineage reversion (including dedifferentiation -the mechanism whereby differentiated cells regress to a less mature state [START_REF] Cai | Dedifferentiation: a new approach in stem cell research[END_REF] -and transdifferentiation from different types of cells and hierarchies) is not covered by the model illustrated in Figure 1-b. A basic representation of cells plasticity features is achieved by considering dedifferentiation and transdifferentiation as perturbation inputs. In fact, it can be proven that nonvanishing perturbations arise from cell plasticity, and uncertain re-introduction functions β i , leading to system (1) with i (t) ∈ (0, i ], for all t ≥ 0. Notation and definitions: Throughout the paper, we analyze the stability of the model described by [START_REF] Adimy | Discrete Maturity-Structured Model of Cells Differentiation with Applications to Acute Myelogenous Leukemia[END_REF], where for all i ∈ I n = {1, . . . , n}, x i (t) ∈ R n . The state of the system (1) at a time instant t is defined as the restriction of each component x i (t) of the solution x(t) = (x 1 (t), . . . , x n (t)), on the segment [t -τ i , t], for all i ∈ I n . We let x = (x 1 , . . . , x n ) and C in = C ([-τ i , 0], R) denote the space of all continuous R-valued functions defined on a given interval [-τ i , 0], for all i ∈ I n , and for all t ≥ 0, the function x it is defined by

x it (m) = x i (t + m) for all m ∈ [-τ i , 0].
Finally, we notice that negative steady states are excluded from this study, as well as equilibria belonging to the boundaries of the positive orthant, except the origin, because of their biological irrelevance. We focus on two meaningful steady states: the 0-equilibrium denoted by X 0 = (0, . . . , 0) and the strictly positive equilibrium point denoted by X e = (x e 1 , . . . , x e n ), x e i > 0, for all i ∈ I n .

We recall two simple but useful results (see [START_REF] Adimy | Discrete Maturity-Structured Model of Cells Differentiation with Applications to Acute Myelogenous Leukemia[END_REF] and [START_REF] Djema | Lyapunov Stability Analysis of a Model Describing Hematopoiesis[END_REF]).

Proposition 1. The solutions of the system (1) with positive initial conditions are positive.

Proof. Since the nonvanishing perturbations i (t), for all i ∈ I n , satisfy i (t) ≥ 0, for all t ≥ 0, the proof that the positive orthant is forward invariant is similar to the one proposed in [START_REF] Djema | Lyapunov Stability Analysis of a Model Describing Hematopoiesis[END_REF] for the nominal system.

Concerning steady states for the nominal system (1), we notice that X 0 always exists. Next, without proof (available in [START_REF] Adimy | Discrete Maturity-Structured Model of Cells Differentiation with Applications to Acute Myelogenous Leukemia[END_REF] and [START_REF] Djema | Lyapunov Stability Analysis of a Model Describing Hematopoiesis[END_REF]), we recall the necessary and sufficient condition of the existence of X e of the nominal system (1): Proposition 2. The nominal system (1) admits a positive equilibrium point X e = (x e 1 , . . . , x e n ) if and only if:

β 1 (0) > δ 1 α 1 . (3)
Throughout Section 4, we will assume that the condition (3) is satisfied and we will analyze the stability properties of X e using a new approach.

Stability analysis of the trivial steady state

Global exponential stability of the nominal system

We start by establishing global exponential stability of the origin X 0 . As a corollary of this result, we prove exponential stability of X 0 when some biological parameters are uncertain or time-varying.

Theorem 1. The nominal system (1) admits the origin, X 0 , as a globally exponentially stable equilibrium point if for all i ∈ I n , the inequalities

s i := δ i -(2C i L i -1) β i (0) > 0, (4)
are satisfied. If

s 1 := δ 1 -(2C 1 L 1 -1)β 1 (0) < 0, (5)
then no positive solution converges to X 0 .

Remark 1. i) We can readily check that if (4) is satisfied, then zero is the unique equilibrium of the nominal system (1). ii) Using a frequency domain approach, it was proved in [START_REF] Adimy | Discrete Maturity-Structured Model of Cells Differentiation with Applications to Acute Myelogenous Leukemia[END_REF] that if (5) is satisfied then the system is unstable, and that the conditions (4) guarantee local asymptotic stability of the origin. In [START_REF] Adimy | Boundedness and Lyapunov function for a nonlinear system of hematopoietic stem cell dynamics[END_REF], slightly more restrictive conditions than (4) (due to the fact that L i < 1, for all i ∈ I n ) were given to ensure global asymptotic stability of the origin.

In [START_REF] Djema | Lyapunov Stability Analysis of a Model Describing Hematopoiesis[END_REF], we proved global asymptotic stability of X 0 under conditions (4). In Theorem 1 of the present paper, we extend the result of [START_REF] Djema | Lyapunov Stability Analysis of a Model Describing Hematopoiesis[END_REF] to establish global exponential stability under conditions (4). iii) Even if the analytic expression of the Lyapunov functional that we will introduce in this section will be slightly different from the one used in [START_REF] Djema | Lyapunov Stability Analysis of a Model Describing Hematopoiesis[END_REF], they share the common feature of being unusual since they are approximated at the origin by linear functions.

Proof. First, let us pick a family of positive constants ρ i , to be selected later, and define for all i ∈ I n , the functionals

v i (x it ) = t t-τi t m e -ρi(t-m-τi) g i (m + τ i -a)w i (x i (a))dadm. (6)
For all i ∈ I n , the derivative of the functional (6) along the trajectories of the nominal system (1) satisfies

vi (t) = -ρ i v i (x it ) - t t-τi g i (t -a)w i (x i (a))da + w i (x i (t)) τi 0 e ρia g i (a)da ≤ -ρ i v i (x it ) - t t-τi g i (t -a)w i (x i (a))da + w i (x i (t))e ρiτi C i ,
where the last inequality is a consequence of (2). Let us introduce the following functional for the first compartment of hematopoietic stem cells:

V 1 (x 1t ) = x 1 (t) + 2L 1 v 1 (x 1t ). ( 7
)
Its derivative along the trajectories of the nominal system (1) satisfies

V1 (t) ≤ -δ 1 x 1 (t) -2L 1 ρ 1 v 1 (x 1t ) -[1 -2L 1 e ρ1τ1 C 1 ] w 1 (x 1 (t)). (8)
Since α 1 > 0, we conclude that for all ρ 1 > 0, the inequality 2L 1 e ρ1τ1 C 1 -1 > 0 is satisfied. Therefore, using w 1 (x 1 (t)) ≤ β 1 (0)x 1 (t), it follows from (8) that:

V1 (t) ≤ -[δ 1 -(2L 1 e ρ1τ1 C 1 -1) β 1 (0)] x 1 (t) -2L 1 ρ 1 v 1 (x 1t ). (9)
Now, if ( 4) is satisfied, we choose

ρ 1 = 1 2τ1 ln δ1+β1(0)+2L1C1β1(0) 4L1C1β1(0)
. Then we

obtain δ 1 -(2L 1 e ρ1τ1 C 1 -1) β 1 (0) ≥ s1 2 . It follows that the inequality (9) gives V1 (t) ≤ -s1 2 x 1 (t) -2L 1 ρ 1 v 1 (x 1t
), and from the definition of V 1 , we get

V1 (t) ≤ -s 1 V 1 (x 1t ) - s 1 4 x 1 (t), (10)
with s1 = min s1 4 , ρ 1 . Consequently, the origin of the subsystem i = 1 is globally exponentially stable. Next, in order to extend the result to the overall system, we introduce the following functional which takes into account the cells dynamics of the first and the second generations of immature cells:

V 2 (x 2t , x 1t ) =x 2 (t) + 2L 2 v 2 (x 2t ) + 2K 1 v 1 (x 1t ) + 8K 1 β 1 (0)e ρ1τ1 C 1 s 1 V 1 (x 1t ).
Using [START_REF] Kolmanovskii | Introduction to the Theory and Applications of Functional Differential Equations[END_REF], we prove that the derivative of V 2 along the trajectories of the nominal system (1) satisfies

V2 (t) ≤ -δ 2 x 2 (t) -(1 -2L 2 e ρ2τ2 C 2 ) w 2 (x 2 (t)) -2L 2 ρ 2 v 2 (x 2t ) -2K 1 ρ 1 v 1 (x 1t ) - 8K 1 β 1 (0)e ρ1τ1 C 1 s1 s 1 V 1 (x 1t ) -2K 1 e ρ1τ1 C 1 [β 1 (0) -β 1 (x 1 (t))] x 1 (t). ( 11
)
Using the assumption α 2 > 0, together with the fact that the function β 2 is strictly decreasing, it follows that,

V2 (t) ≤ -[δ 2 -(2L 2 e ρ2τ2 C 2 -1) β 2 (0)] x 2 (t) -2L 2 ρ 2 v 2 (x 2t ) -2K 1 ρ 1 v 1 (x 1t ) - 8K 1 β 1 (0)e ρ1τ1 C 1 s1 s 1 V 1 (x 1t ). ( 12
)
When the conditions (4) are satisfied, we select ρ 2 > 0 (similarly to ρ 1 ), such that the inequality

δ 2 -(2L 2 e ρ2τ2 C 2 -1) β 2 (0) ≥ s2
2 , is satisfied. It follows from ( 12) that there exists a strictly positive constant s2 , such that

V2 (t) ≤ -s2 V 2 (x 1t , x 2t ) - s 2 4 x 2 (t), (13)
is satisfied. Next, by induction, we easily check that there exist a positive constant sn and a family of strictly positive weighting constants λ i and λi , such that the derivative of the functional

V(x t ) = n i=1 λ i x i (t) + λi v i (x it )
, which is taking into account all the n generations of immature blood cells, along the trajectories of the nominal system (1), satisfies

V(t) ≤ -s n V(x t ). ( 14
)
From the inequality ( 14) and the properties of the functional V, we conclude that, if the conditions (4) are satisfied, the origin of the nominal model ( 1) is globally exponentially stable.

In order to complete the proof, we consider the case where the inequality ( 5) is satisfied and we show that no positive solution converges to X 0 . As in [START_REF] Djema | Lyapunov Stability Analysis of a Model Describing Hematopoiesis[END_REF], we prove this result by contradiction, i.e. we assume that a positive solution x(t) converges to X 0 . Now, we select ρ 1 = 0 and we observe that the derivative of the functional V 1 , introduced in (7), is given by

V1 (t) = [-δ 1 + α 1 β 1 (x 1 (t))] x 1 (t). ( 15
)
When ( 5) is verified, using the facts that the function β 1 is decreasing and x 1 (t) converges to zero, we deduce that there exists t r > 0 such that, for all t ≥ t r ,

-δ 1 + α 1 β 1 (x 1 (t)) ≥ -δ1+α1β1(0) 2 . It follows from (15) that, for all t ≥ t r , V1 (t) ≥ -δ 1 + α 1 β 1 (0) 2 x 1 (t). (16)
From [START_REF] Djema | Lyapunov Stability Analysis of a Model Describing Hematopoiesis[END_REF], and the positivity of the solutions, it follows that for all t ≥ t r , V1 (t) > 0. Consequently, we deduce that, for all

t ≥ t r , V 1 (x 1t ) ≥ V 1 (x 1tr ) > 0.
It follows that V 1 (x 1t ) does not converge to zero. On the other hand, V 1 (x 1t ) converges to zero because x 1 (t) converges to X 0 . This yields a contradiction.

Example 1. A possible selection of the cell division probability densities, which was considered in [START_REF] Özbay | Stability analysis of systems with distributed delays and application to hematopoietic cell maturation dynamics[END_REF] and [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF], is given by f i (a) = mi e m i τ i -1 e mia , with m i > 0, for all i ∈ I n . Let us consider the following biological functions and parameters:

β i (x i ) f i (a) δ i L i τ i γ i i = 1 1.22 1+x 2 1 5e 5a e 5τ 1 -1 0.9 0.85 1.2 0.22 i = 2 1.33 1+4x 4 2 7e 7a e 7τ 2 -1 0.96 0.8 1.3 0.33
The form given to β i [START_REF] Mackey | Unified hypothesis of the origin of aplastic anemia and periodic hematopoiesis[END_REF] normalizes the values taken by the total density x i . Simple calculations give:

(2L 1 C 1 -1) β 1 (0) = 0.4448, (2L 2 C 2 -1) β 2 (0) = 0.
4392. Therefore, according to Proposition 2, the positive equilibrium of system (1) does not exist. Moreover, according to Theorem 1, the origin X 0 = (0, 0) of system (1) is globally exponentially stable, as shown in Figure 2.

Global exponential stability under time-varying differentiation rates

Convergence to X 0 means the eradication of all the immature blood cells. This case may be suitable when the model is assumed to describe the dynamics of unhealthy cells. Recall that one of the characteristics of leukemia is the blockade in the differentiation process, which becomes also a target for the drugs used in treatments. From a theoretical point of view, it is interesting to consider the case where differentiation and self-renewing rates are uncertain or time varying.

In this part, we extend the result of Theorem 1 to the nominal model that describes the immature cell dynamics under time-varying differentiation rates, K i (t) for all t ≥ 0, and i ∈ I n , and which is given by ẋi

(t) = 2K i-1 (t) τi-1 0 g i-1 (a)w i-1 (x i-1 (t -a))da + 2L i (t) τi 0 g i (a)w i (x i (t -a))da -δ i x i (t) -w i (x i (t)), (17)
where

K i (t) + L i (t) = 1 and L i (t) ∈ [L i min , L i max ] ⊂ (0, 1)
. We recall that, by convention, K 0 (t) = 0, for all t ≥ 0, and we assume that K i (•), L i (•) are of class C 0 , for all i ∈ I n . Based on Theorem 1, we prove the following result:

Corollary 1. The conditions

s i = δ i -(2L i max C i -1) β i (0) > 0, ∀i ∈ I n , (18)
ensure that the origin of the system (17) is globally exponentially stable.

Proof. We give some indications for the proof, which is slightly different from the one of Theorem 1. Here we consider L 1 max instead of L 1 in the definition of the functional V 1 (x 1t ) introduced in ( 7), and, similarly, we consider L 2 max , K 1 max = 1 -L 1 min and s 1 , instead of L 2 , K 1 , and s 1 , respectively, in the definition of the functional V 2 (x 2t , x 1t ).

Example 2. Let us consider n = 2 and for all t ≥ 0, L 1 (t) = 1 2 (1 + 0.97 cos(25t)) and L 2 (t) = 1 2 (1 + 0.97 sin(15t)). Sine function sounds reasonable to model the variation in differentiation rates since drugs are -usually -infused quasiperiodically. Nevertheless, many other time-varying functions may be used instead of sine ones. Let us assume that:

β i (x i ) f i (a) δ i τ i γ i i = 1 2.87 1+x 2 1
e a e τ 1 -1 0.973 0.8 0.9

i = 2 2.7 1+x 4 2
e a e τ 2 -1 0.965 0.7 0.97 Elementary calculations give: s 1 = 0.0592, and, s 2 = 0.0099. According to Corollary 1, X 0 = (0, 0), which is the unique equilibrium point of [START_REF] Özbay | Stability analysis of systems with distributed delays and application to hematopoietic cell maturation dynamics[END_REF], is globally exponentially stable. Figure 3 illustrates the trajectories x 1 and x 2 for the the parameters and biological functions of Example 2. Remark 2. At this juncture, we emphasize that Theorem 1 and Corollary 1 complement the results of [START_REF] Adimy | Discrete Maturity-Structured Model of Cells Differentiation with Applications to Acute Myelogenous Leukemia[END_REF], [START_REF] Adimy | Boundedness and Lyapunov function for a nonlinear system of hematopoietic stem cell dynamics[END_REF], and [START_REF] Djema | Lyapunov Stability Analysis of a Model Describing Hematopoiesis[END_REF], by establishing global exponential stability instead of asymptotic stability and by extending the result to cover the case of time-varying differentiating and self-renewing rates. Let us briefly comment these results in the AML case, in which we expect a blockade of differentiation, i.e. K i decreases in early maturity stages. Not surprisingly, the conditions [START_REF] Pujo-Menjouet | Long period oscillations in a G 0 model of hematopoietic stem cells[END_REF] suggest that therapeutic strategies to eradicate cells must be oriented towards increasing the death rates γ i (recall that increasing the apoptosis rate γ i decreases C i ), and δ i , and also towards decreasing L imax (i.e. increasing differentiation).

Although very partial results for particular cases of AML (with myelodysplastic syndrome, MDS), and on cell cultures only, have been obtained using tyrosine kinase inhibitors (TKIs, in particular dasatinb [START_REF] Lainey | Tyrosine kinase inhibitors for the treatment of acute myeloid leukemia: Delineation of anti-leukemic mechanisms of action[END_REF]) in stimulating differentiation, the only clinically efficient case of redifferentiation therapy known until recently was by using all-trans retinoic acid (ATRA) and arsenic tri-oxide in acute promyelocytic leukemia (APL). However, this therapeutic track has lately been relaunched by establishing that inhibition of Dihydroorotate Dehydrogenase (DHODH) is efficient in releasing cells from differentiation arrest [START_REF] Sykes | Inhibition of dihydroorotate dehydrogenase overcomes differentiation blockade in acute myeloid leukemia[END_REF]. Finally, increasing apoptosis may be achieved classically by using cytosine arabinoside.

Robustness analysis of the trivial steady state

In this section, we use the strict functionals V i , introduced in Theorem 1, to perform a robustness analysis in the case of nonvanishing perturbations i (t) ∈ (0, i ], for all i ∈ I n , t ≥ 0. Let us observe that the derivative of the functional V 1 , defined in [START_REF] Fridman | Introduction to time-delay systems: analysis and control[END_REF], along the trajectories of the perturbed system (1), satisfies,

V1 (t) ≤ -s 1 V 1 (x 1t ) - s 1 4 x 1 (t) + 1 . (19)
We consider any constant θ ∈ (0, 1) and we define the family of sets:

T i i = ϕ ∈ C [-τ i , 0], R + , V i (ϕ) ≤ i θs i . ( 20
)
Notice for later use that the sets T i i are the smallest possible for θ close to 1. Clearly, if

x 1t / ∈ T 1 1 , then (19) gives V1 (t) ≤ -(1 -θ)sV 1 (x 1t ) -s1 4
x 1 (t). Therefore, the state x 1t converges exponentially to the set T 1 1 . However, a refined result can be provided, in the sense that we can determine smaller positive invariant sets than the family T i i . For that, let us introduce the functional,

A 1 (x 1t ) = V 1 (x 1t ) -ψ † 1 x 1 (t). ( 21
)
It is worth mentioning that the functional A 1 is positive on the positive orthant for ψ † 1 = s1 8(δ1+β1(0)) < 1, where s 1 is the constant defined in (4). Using the expression of ψ † 1 , we can check that the derivative of A 1 , along the trajectories of the perturbed system (1), satisfies:

Ȧ1 (t) ≤ -s1 V 1 (x 1t ) - s 1 8 x 1 (t) -2ψ † 1 L 1 t t-τ1 g 1 (t -a)w 1 (x 1 (a))da + (1 -ψ † 1 ) 1 .
Now, if we define the family of sets:

Ti i = ϕ ∈ C [-τ i , 0], R + , V i (ϕ) + 2ψ † i L i si θ 0 -τi g i (a)w i (ϕ)da ≤ (1 -ψ † i ) i θs i ,
where 0 < ψ † i < 1, for all i ∈ I n . Observe that Ti i ⊂ T i i , for all ψ i > 0. Now, notice that for all x 1t / ∈ T1 1 , the derivative of the functional A 1 satisfies,

Ȧ1 (t) ≤ -s1 (1 -θ)V 1 (x 1t ) - s 1 8 x 1 (t) ≤ -s 1θ A 1 (x 1t ) - s 1 + ψ † 1 θ 8 x 1 (t), ( 22
)
where s1θ = min {s 1 (1 -θ), θ/8} > 0, for all θ ∈ (0, 1). Therefore, from the definition of the functional A 1 , we conclude that the state x 1t converges exponentially to T1 1 , and the decay rate of the trajectory x 1 (t) is smaller than, or equal to, s1θ . On the other hand, we readily check, by contradiction, that the sets Ti i are positively invariant (i.e. a trajectory in Ti i remains in Ti i for all the future time). Arguing as in the proof of Theorem 1, we generalize this result to the overall system. In other words, we have proved the following result:

Theorem 2. If the conditions s i > 0 are satisfied, for all i ∈ I n , then the states x it of the perturbed system (1), where i (t) ∈ (0, i ], for all t ≥ 0, converge exponentially to the sets Ti i , where 0 < ψ † i < 1, for all i ∈ I n .

Stability analysis of the strictly positive steady state

A strictly positive equilibrium X e reflects the surviving of all the generations of blood cells, which is the aim of a healthy hematopoiesis. When the condition (3) is satisfied, a unique X e exists. In this section, we answer one important open question about the problem of finding an estimate of the basin of attraction of X e . Let us start by looking to the reintroduction functions, β i 's, from the resting to the proliferating stages. Owing to some biological considerations, Hill functions were proposed by Mackey in [START_REF] Mackey | Unified hypothesis of the origin of aplastic anemia and periodic hematopoiesis[END_REF] to model β i . Therefore, we consider,

β i (x i ) = β i (0) 1 + b i x ni i , (23)
where β i (0) > 0, b i > 0 and n i ≥ 2. This typical choice was assumed in subsequent works ( [START_REF] Adimy | Discrete Maturity-Structured Model of Cells Differentiation with Applications to Acute Myelogenous Leukemia[END_REF] and [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF]). Actually, many other smoothly decreasing functions β i , with a finite maximum β i (0) and adjustable slope and inflection point, can be chosen to match the biological assumptions [START_REF] Mackey | Unified hypothesis of the origin of aplastic anemia and periodic hematopoiesis[END_REF]. Throughout this section, we consider the functions β i in the form (23) and we indicate later for which other forms our results remain valid. Since we are interested in the positive equilibrium X e , we perform the change of coordinates, xi = x i -x e i , for i ∈ I n . It follows from (1) that

ẋi (t) = -δ i [x i (t) + x e i ] -w i (x i (t) + x e i ) + 2L i t t-τi g i (t -a)w i (x i (a) + x e i ) da + 2K i-1 t t-τi-1 g i-1 (t -a)w i-1 xi-1 (a) + x e i-1 da. (24)
However, a better representation of (24) that eases the analysis of its origin, can be obtained. Indeed, observe that, with an abuse of notation,

w i (x i + x e i ) = w i (x e i ) + µ i xi + R i (x i )
, where,

µ i = β i (x e i ) + β i (x e i )
x e i , and, R i (x i ) =

x e i +xi

x e i [x i + x e i -l]w (2)
i (l)dl. ( 25) Moreover, we denote

β i * = δ i + µ i . It follows that (24) is equivalent to ẋi (t) = -β i * xi (t) + 2L i µ i t t-τi g i (t -a)x i (a)da -R i (x i (t)) + 2L i t t-τi g i (t -a)R i (x i (a))da + 2K i-1 µ i-1 t t-τi-1 g i-1 (t -a)x i-1 (a)da + 2K i-1 t t-τi-1 g i-1 (t -a)R i-1 (x i-1 (a))da. ( 26
)
Remark 3. Compared with Section 3, the stability analysis of the origin of (26) is more complicated due to the shifting. Indeed, linear functionals can no longer be used since the system (26) is not positive. A common approach to investigate the stability properties of such a class of systems is by using quadratic functions 205 or functionals, as illustrated in the sequel.

Introductory result

To get a first intuition, let us consider the subsystem (26) for i = 1. A linear approximation at its origin is obtained by neglecting the terms where R 1 is present. The following linear system is obtained:

ż1 (t) = -β 1 * z 1 (t) + 2L 1 µ 1 t t-τ1 g 1 (t -a)z 1 (a)da. ( 27
)
Let us consider the positive definite quadratic function

Q(a) = 1 2 a 2 . ( 28
)
We apply the Razumikhin's theorem (see, for instance, [START_REF] Fridman | Introduction to time-delay systems: analysis and control[END_REF]): Pick q > 1 and consider t ≥ 0 such that qQ(z 1 (t)) ≥ Q(z 1 (a)), ∀a ∈ (t -τ 1 , t) . Then the derivative of Q along the trajectories of the system (27) satisfies:

Q(t) ≤ -2β 1 * Q(z 1 (t)) + 4 Q(z 1 (t))L 1 |µ 1 | t t-τ1 g 1 (t -a) Q(z 1 (a))da ≤ -2 [β 1 * -2 √ qL 1 |µ 1 |C 1 ] Q(z 1 (t)). ( 29
)
We conclude from Razumikhin's theorem that the condition β 1 * -2L 1 |µ 1 |C 1 > 0 is sufficient for the asymptotic stability of the origin of the system (27). This leads us to introduce, for all i ∈ I n , the constants

ς i = β i * -2L i |µ i |C i = δ i + µ i -2L i |µ i |C i , (30)
that will be of use later in the stability analysis of the nonlinear system, in the analytic expression of the quadratic Lyapunov-Krasovskii functionals and in the size of the region of attraction that we will provide.

Estimate of the basin of attraction of the positive steady state

Contrary to Section 3, the nonpositivity of the system under study motivates the use of the positive definite function (28), as well as the following functionals:

Ω i (ϕ it ) = t t-τi t l g i (l -a + τ i )Q(ϕ i (a))dadl, (31)
Λ i (ϕ it ) = t t-τi e l-t t l g i (l -a + τ i )Q(ϕ i (a))dadl. ( 32
)
Notice that other types of functionals may be used instead of (31) and (32). However, for the sake of clarity, we use a weighted combination of them in order to compensate the distributed delayed terms and estimate the exponential decay rates. Moreover, we define for all i ∈ I n , the following functionals:

S i (x it ) = 1 2 Q(x i (t)) + L i |µ i |Ω i (x it ), (33)
N 1 (x 1t ) = S 1 (x 1t ) + ς 1 2C 1 Λ 1 (x 1t )
, and for all i ∈ {2, . . . , n}, (34)

N i (x it , xi-1t ) = S i (x it ) + ς i 2C i Λ i (x it ) + ψ i Λ i-1 (x i-1t ), (35)
with ς i the constants defined in Section 4.1, and ψ i are appropriate strictly positive constants to be selected later, for all i ∈ {2, . . . , n}.

Next, for a later use, we prove in Appendix A the following assertion:

Claim 1. There exist constants ŝi > 0, for all i ∈ I n , which depend on the biological parameters of the model and on the strictly positive equilibrium X e , such that, for all xi > -x e i , where x e i > 0, the following inequality holds true:

|R i (x i )| ≤ ŝi Q(x i ), where, ŝi > 0 are given in Appendix A. (36)
Remark 4. It is worth mentioning that the stability analysis which will be performed for the origin of the nonlinear system (26) is available for many other reintroduction functions β i , as long as they satisfy the sector conditions (36).

Furthermore, in order to ease the notation, we denote

I i (x it ) = t t-τi g i (t -a)Q(x i (a))da. (37)
Finally, we define the constants ki = ςi 8ŝi , ki = ςi 4CiLi ŝie τ i and N i = min k2 i , k2 i . Notice that for all i ∈ I n , ki and ki are only dependent on the constant biological parameters of the model. Now, we prove the following result: Theorem 3. Let the system (26) be such that

ς i > 0, (38)
for all i ∈ I n . Then all the solutions of (26) with initial conditions φi ∈ C([-τ i , 0], (-x e i , +∞)) satisfying

N i ( φi , φi-1 ) < N i , (39)
converge exponentially to the origin.

Remark 5. Generally, Lyapunov theory provides sufficient conditions for stability. Nevertheless, due to earlier published works we can comment conditions (38). In previous works (using frequency domain approaches), it was claimed in [START_REF] Adimy | Discrete Maturity-Structured Model of Cells Differentiation with Applications to Acute Myelogenous Leukemia[END_REF] that the origin is locally asymptotically stable if δ i + (2L i C i + 1) µ i > 0 is satisfied. However, in [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF], it was shown that the previous assertion holds true only when -δ i < µ i < 0. We notice that our stability conditions (38) are equivalent to those of [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF] on that interval. Next, when µ i > 0, our exponential stability conditions (38) (which are provided without specifying a particular form of f i ), correspond to the conditions for local stability provided in [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF] (and which have been slightly improved using Nyquist criterion for a typical selection of the functions f i in [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF]). It remains the case µ i < -δ i which is not covered by the Lyapunov approach proposed here, and which was addressed in [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF]. The region of attraction defined in (39) is rather difficult to interpret. In fact, based on some numerical simulations and the conjecture made in [START_REF] Özbay | Stability analysis of cell dynamics in leukemia[END_REF], we suggest that the region defined in (39) is conservative.

Proof. First, let us observe for later use that the derivatives of the functionals Ω i and Λ i , for all i ∈ I n , along the trajectories of (26) satisfy,

Ωi (t) = C i Q(x i (t)) - t t-τi g i (t -a)Q(x i (a)
)da, and,

Λi (t) ≤ -Λ i (x it ) -e -τi t t-τi g i (t -a)Q(x i (a))da + C i Q(x i (t)),
where the last inequality is a consequence of (2). For the sake of clarity, we will decompose now the proof of Theorem 3 in two parts: we prove the exponential stability of solutions of the first compartment (i = 1), and then we extend the result to any number of compartments (i ≥ 1).

240

i) LKF for the first compartment: We start with the first generation of hematopoietic stem cells. Using (36), one can prove that the derivative of the function Q(x 1 (t)), introduced in (28), along the trajectories of (26) satisfies

Q(t) ≤2 [-β 1 * + L 1 |µ 1 |C 1 ] Q(x 1 (t)) + ŝ1 |x 1 (t)|Q(x 1 (t)) + 2L 1 (|µ 1 | + ŝ1 |x 1 (t)|) I 1 (x 1t ). ( 40
)
It follows that the derivative of the functional N 1 , introduced in (34), satisfies

Ṅ1 (t) ≤ - ς 1 8 Q(x 1 (t)) + ς 1 2C 1 Λ 1 (x 1t ) + ŝ1 2 |x 1 (t)| - ς 1 4 Q(x 1 (t)) - ς 1 8 Q(x 1 (t)) + L 1 ŝ1 |x 1 (t)| - ς 1 e -τ1 2C 1 I 1 (x 1t ). (41)
On the other hand, from the definition of N 1 we observe that

N 1 (x 1t ) ≤ 1 2 Q(x 1 (t)) + ς 1 + 2L 1 C 1 |µ 1 |e τ1 2C 1 Λ 1 (x 1t ). (42)
From ( 41) and (42), we deduce that for all ς1 ∈ 0, min ς1 4 , ς1 ς1+2L1C1|µ1|e τ 1 , the derivative of the functional N 1 satisfies

Ṅ1 (t) ≤ -ς1 N 1 (x 1t ) + ŝ1 2 |x 1 (t)| - ς 1 4 Q 1 (x 1 (t)) + L 1 ŝ1 |x 1 (t)| - ς 1 e -τ1 2C 1 I 1 (x 1t ) - ς 1 8 Q(x 1 (t)).
From the definition of N 1 , which is given in (34), we notice that |x

1 (t)| ≤ 2 N 1 (x 1t ). A direct consequence is that Ṅ1 (t) ≤ -ς1 N 1 (x 1t ) + ŝ1 N 1 (x 1t ) - ς 1 4 Q 1 (x 1 (t)) + 2L 1 ŝ1 N 1 (x 1t ) - ς 1 e -τ1 2C 1 I 1 (x 1t ) - ς 1 8 Q(x 1 (t)).
Now, we conclude that if the condition (39) is satisfied, then

Ṅ1 (t) ≤ -ς1 N 1 (x 1t ) - ς 1 8 Q(x 1 (t)). ( 43
)
This allows us to conclude that the origin of the subsystem (26), for i = 1, is exponentially stable, with a decay rate smaller than ς1 .

ii) LKF for the overall system: Here we take into account all generations of immature blood cells. We use the inequality |x i (t

)x i-1 (a)| ≤ ξ i Q(x i (t)) + 1 ξi Q(x i-1 (a)
), with ξ i > 0 for all i > 1, and the inequality |x i (t)x i (a)| ≤ Q(x i (t))+Q(x i (a)), for all i ∈ I n , and we select

ψ i = Ki-1|µi-1|e τ i-1 ξi + ςie -τ i Ki-1 ŝi-1 2Li ŝiCi e τi-1 .
Then the derivatives of the functions Q(x i (t)), for all i > 1, along the trajectories of (26) satisfy

Q(t) ≤2 [-β i * + L i |µ i |C i ] Q(x i (t)) + ŝi |x i (t)|Q(x i (t)) + 2L i (|µ i | + ŝi |x i (t)|) I i (x it ) + 2K i-1 |µ i-1 |C i-1 ξ i Q(x i (t)) + 2K i-1 ŝi-1 |x i (t)| + |µ i-1 | ξ i I i-1 (x i-1t ). (44)
Moreover, we choose

ξ i = ςi 4Ki-1|µi-1|Ci-1 . It follows that Ṅi (t) ≤ -ςi N i (x i , xi-1 ) + ψ i C i-1 Q(x i-1 (t)) - ς i 8 - 1 2 ŝi |x i (t)| Q(x i (t)) - ς i 16 Q(x i (t)) + L i ŝi |x i (t)| - ς i e -τi 2L i ŝi C i I i (x it ) + K i-1 ŝi-1 |x i (t)| - ς i e -τi 2L i ŝi C i I i-1 (x i-1t ), (45)
with ςi > 0. Finally, we conclude that if the conditions (39) are satisfied, then

Ṅi (t) ≤ -ςi N i (x it , xi-1t ) - ς i 16 Q(x i (t)) + ψ i C i-1 Q(x i-1 (t)). (46)
As we had done in [START_REF] Djema | Stability of Immature Cell Dynamics in Healthy and Unhealthy Hematopoiesis[END_REF], we can prove that the derivative of the functional W ( Xt ) = n i=1 p i N i (x it , xi-1t ), with an abuse of notation for N 1 , and where

p i = 2 n-i n k=i+1 8ψ k C k-1
ς k-1 , p n = 1, and X = (x 1 , . . . , xn ), satisfies,

Ẇ (t) ≤ - n i=1 p i ςi N i (x it , xi-1t ) - ς 1 8 Q(x 1 (t)) - ς n 16 Q(x n (t)) - 1 2 n-1 i=1 p i ς i 8 Q(x i (t)).
Finally, we obtain Ẇ (t) ≤ -ςW ( Xt ), with ς = min {ς 1 , ..., ςn } > 0.

To summarize, by virtue of the properties of the functionals N i , for all i ∈ I n , and since the original system (1) is a positive system, we conclude that the set

A = {ϕ i ∈ C([-τ i , 0], R + ) : N i (ϕ i -x e i , ϕ i-1 -x e i-1 ) < N i }, (47)
is a subset of the basin of attraction of the positive steady state of system (1).

Example 3. In this numerical example, we consider the system with the following biological functions and parameters for n = 3:

β i (x i ) f i (a) δ i τ i γ i K i i = 1 0.5 1+x 2 1 10e 10a e 10τ 1 -

Robustness analysis of the positive equilibrium point

Here we return to the perturbed system (1), that we rewrite in a form similar to (26) by performing the change of coordinate xi (t) = x i (t) -x e i . Based on the functionals constructed in Theorem 3, we prove the following result: Corollary 2. Let the system (26) be perturbed by nonvanishing additive disturbances i (t) ∈ (0, i ], i > 0, for all t > 0, and i ∈ I n . If the conditions

ς i > 0 ( 48
)
are satisfied for all i ∈ I n , then all the solutions of (26) with initial conditions

ϕ i ∈ C([-τ i , 0], R + ) satisfying 2 i θς i 2 ≤ N i (ϕ i -x e i , ϕ i-1 -x e i-1 ) < N i , (49)
with θ ∈ (0, 1), converge exponentially to the domain,

G i = ϕ i ∈ C([-τ i , 0], R + ), N i (ϕ i -x e i , ϕ i-1 -x e i-1 ) ≤ 2 i θς i 2 . (50)
Proof. Let us prove the previous result for i = 1. Arguing as we did in the proof of Theorem 3, one can generalize to the overall system. First, observe that the derivative of Q(x 1 (t)) along the trajectories of the perturbed system satisfies:

Q(t) ≤2 [-β 1 * + L 1 |µ 1 |C 1 ] Q(x 1 (t)) + ŝ1 |x 1 (t)|Q(x 1 (t)) + 2L 1 (|µ 1 | + ŝ1 |x 1 (t)|) I 1 (x 1t ) + |x 1 (t)| i . (51)
Consequently, the derivative of the functional N 1 , introduced in (34), along the trajectories of the perturbed system, verifies

Ṅ1 (t) ≤ - ς 1 8 Q(x 1 (t)) + ς 1 2C 1 Λ 1 (x 1t ) + ŝ1 2 |x 1 (t)| - ς 1 4 Q(x 1 (t)) - ς 1 8 Q(x 1 (t)) + L 1 ŝ1 |x 1 (t)| - ς 1 e -τ1 2C 1 I 1 (x 1t ) + |x 1 (t)| i . (52)
Using (42), and the fact that

|x 1 (t)| ≤ 2 N 1 (x 1t ), we obtain Ṅ1 (t) ≤ -ς1 N 1 (x 1t ) + ŝ1 N 1 (x 1t ) - ς 1 4 Q 1 (x 1 (t)) + 2L 1 ŝ1 N 1 (x 1t ) - ς 1 e -τ1 2C 1 I 1 (x 1t ) - ς 1 8 Q(x 1 (t)) + 2 i N 1 (x 1t ), where ς1 ∈ 0, min ς1 4 , ς1 ς1+2L1C1|µ1|e τ 1 . Therefore, when N 1 (ϕ 1 -x e 1 ) < N 1 is satisfied, we deduce that Ṅ1 (t) ≤ -ς1 N 1 (x 1t ) - ς 1 8 Q(x 1 (t)) + 2 i N 1 (x 1t ). (53)
Now, let us consider any θ ∈ (0, 1) and observe that for all initial conditions ϕ 1 satisfying N 1 (ϕ 1 -x e 1 ) < N 1 with ϕ 1 / ∈ G i , the inequality (53) gives

Ṅ1 (t) ≤ -(1 -θ)ς 1 N 1 (x 1t ). (54)
We conclude that the states x 1t satisfying (49) converge exponentially to the invariant set G 1 , defined in (50), such that the decay rate of the trajectory x 1 (t) is smaller than, or equal to, (1-θ)ς1 2 .

Conclusion

With the aim of constantly refining and improving the modeling and the analysis of hematopoietic mechanisms, we proposed explicit constructions of suitable strict LKFs for nonlinear hematopoietic systems with distributed delays. Our approach allowed us to solve some practical and technical issues, which complement already published results on the topic. In comparison with previous works, our results provided exponential stability with an estimate on the decay rate of the solutions, and are derived without any extra assumption on the mitosis functions. Moreover, a robustness analysis was performed under nonvanishing perturbations (that may represent dedifferentiation flux, together with model uncertainties), and we have also covered some practical situations such that time-varying differentiating rates (to model the action on the blockade in differentiation and re-differentiation). Particular emphasis was given to the positive steady state that represents healthy hematopoiesis, and for which we provided an explicit formulation of a subset of its region of attraction. Future work will enhance the role of dedifferentiation and transdifferentiation by considering hematopoietic models where cell plasticity is no more a marginal phenomenon, and cannot be considered as a perturbation, but has to be fully

Figure 1 :

 1 Figure 1: (a) Blast cells are not typically found in the circulating blood of healthy individuals. The picture is from the National Cancer Institute. (b) Schematic representation of the earliest stages in the myeloid lineage [1].

Figure 2 :Figure 3 :

 23 Figure 2: Trajectories of Example 1.

  210

Figure 4 :

 4 Figure 4: Trajectories x 1 , x 2 and x 3 for the parameters of Example 3.

- 1 e h 2 z 1 )-2 |z| 3 + 1 [ 1 +- 1 e |c 3 | 2 . 2 .

 1213111322 modeled.Denote h = 1 + bx n e and observe that, 1 p(z) = 1 h ρ(z) + 1 h . It follows that ρ(z) = -nbx n-1 e ρ(z) h + 1 h 2 z + C(z). Consequently, we get the intermediate result:ρ(z) = -nbx n+ C(z) -On the other hand, observe thatJ(z) = ρ(z) + 1 h z + x e ρ(z) = c 1 z + c 2 C(z) + c 3 ρ(z)z, (A.2)where the last equality is a direct consequence of (A.1), withc 1 = 1 h -nbx n e h 2 , c 2 = x e and c 3 = 1 -nbx n e h. Now, we readily check that|C(z)| ≤ nb(n -1) p(z) (|z| + x e ) n-2 z 2 2 . (A.3) It follows that |ρ(z)| ≤ nbx n-1 e p(z) |z| + |C(z)|. Using (A.3), we get |zρ(z)| ≤ nbx n.2), we deduce that,|J(z) -c 1 z| ≤ nb(n -1)|c 3 | 2p(z) (|z| + x e ) nnb(n -1)|c 2 | (|z| + x e ) 1+b(z+xe) n ]h . Therefore, when z ≥ 0, we have 1 p(z) = 1 [1 + b(|z| + x e ) n ] h ,and when z ≤ 0 i.e. z ∈ (-x e , 0] b(2x e ) n [1 + b(|z| + x e ) n ] h .Consequently, for all z > -x e , we have1 p(z) ≤ 1 + b(2x e ) n [1 + b(|z| + x e ) n ] h .20We deduce that|J(z) -c 1 z| ≤ nb(n -1)|c 3 |(1 + b(2x e ) n ) 2h (|z| + x e ) n-2 |z| 1 + b(|z| + x e ) n + nb(n -1)|c 2 | (|z| + x e ) n-2 (1 + b(2x e ) n ) 2 [1 + b(|z| + x e ) n ] h + nbx n(1 + b(2x e ) n ) [1 + b(|z| + x e ) n ] h zBy distinguishing between the two cases |z| + x e ≥ 1 and |z| + x e ≤ 1, one can prove that the following inequalities are satisfied for all z > -x e , (|z| + x e )n-2 |z|1 + b(|z| + x e ) n ≤ (|z| + x e ) n-1 1 + b(|z| + x e ) n ≤ max b, b -1 .It follows that |J(z) -c 1 z| ≤ c 4 z 2 , with the positive constantc 4 = nb(n -1)(1 + b(2x e ) n ) (x e + |c 3 |) max b, b -1 2h + nbx n-1 e (1 + b(2x e ) n ) |c 3 | hOn the other hand, we easily check that µ = β(0)c 1 , with µ defined in (25). It follows that |R(z)| ≤ β(0)c 4 z 2 . Since Q(z) = 1 2 z 2 , we conclude that |R(z)| ≤ ŝQ(z), where, ŝ = 2c 4 β(0).

  We select constant initial conditions: ϕ 1 = 0.6850, ϕ 2 = 0.782 and ϕ 3 = 0.979. Therefore, we get,N 1 (ϕ 1 -x e 1 ) = 7.16 × 10 -5 < N 1 , N 2 (ϕ 2 -x e 2 , ϕ 1x e1 ) = 6.65 × 10 -7 < N 2 , and, N 3 (ϕ 3 -x e 3 , ϕ 2 -x e 2 ) = 1.94 × 10 -4 < N 3 . According to Theorem 3, the positive steady state (x e 1 , x e 2 , x e 3 ) is exponentially stable (Figure4).

		From the selected parameters, it follows that
			x e i			α i			ς i			ŝi	N i
		i = 1 0.70036 0.40422 0.08924 0.65070 2.5935 × 10 -4
	250	i = 2 0.78225 0.19888 0.02329 3.00487 9.3935 × 10 -7
		i = 3	1.0050	0.20422 0.33938 2.98491	2.02 × 10 -4
	255
				1.2
												x 1
												x 2
			cells/kg)	1								x 3
			Total density of quienscent cells (×1.62 × 10 8	0.2 0.4 0.6 0.8
				0
				0	20	40	60	80	100	120	140	160	180	200
									Time (days)
							1	0.1356 1.109402	0.3	0.05
		i = 2 i = 3	1 1+x 4 2 3 1+x 2 3	10e 10a e 10τ 2 -1 2e 2a e 2τ 2 -1	0.1669 0.3559			1.2 1.36	0.4 0.45 0.085 0.07

Appendix A. Proof of Claim 1

For notational convenience, we drop the subscript "i" and we use x e instead of x e i to denote the positive equilibrium. Using the expression of β given in (23), we observe that for all x e > 0 and z > -x e , R(z) = β(0)J(z) -µz, where

First, let us study the function