Digital transcoding: an extension of translation? 1

The impact of theoretical models on the concept of extended translation

1. 1 The concept of translation as an "umbrella" model Søren Pold & all (2018) propose to extend the concept of (linguistic) translation in 4 directions: translinguistics, transcoding, transmedia and transcreation. In this perspective, updating, remediation, adaptation of a medium and classical linguistic translation would be based on a single model: translation. What would be the added value of such a broadening of the concept of translation, and what biases could it bring?

Grouping different concepts in the same class is not without danger. Indeed, bringing concepts together under the "umbrella" of another, considering them in fine as belonging to the same class defined by this "umbrella" concept changes the way of thinking about these concepts by projecting onto them a new understanding and constraints stemming from the philosophical and ideological characteristics of the generative concept of the class. This is not insignificant at all. Linguistic translation is based on two fundamental characteristics: the existence of a source text and the fact that the entire source text is translated into a single target text. The existence of a source text is a sine qua non of linguistic translation, and ideally it should be an original text. This means that, as part of an extended concept of translation, all the features of the original source digital work must be translated as a whole into a single target digital work in which, on the one hand, the target program is a translation of the original program and, on the other hand, the screen rendering of the target program while running, translates the original rendering. This is exactly what it says: "we could say that reinventeing sound ans meaning correlations […] or reinventing sound, meaning and visual correlations […] are theoretically equivalent to reinventing the correlations between thows verbal and visual layers, on the hand, and software layers, on the other, in electronic literature." (Pold & all, 2018: 3) We will argue that the concept of "extended translation" comes up against this concomitance of several "translations".

2 The underlying theoretical model of the digital work

Pold & all explicitly base their argument on cybertext theory [START_REF] Aarseth | Cybertext: Perspectives on Ergodic Literature[END_REF]:

"Translation of this textoniclayer (in Aarseth's terminology) can be described as the translation of processes rather than texts [START_REF] Cayley | The Translation of Process[END_REF]. Translation of electronic literature thus seems to require that our focus is placed on the programmed compositional processes as much as on the textual and media instantiations that constitute the more or less transient configuration of the scriptonic layers" (Pold & all, 2018: 2) Cybertext and its alternative procedural model

1 The cybertext model

The following schema, which applies to any operator (author as well as reader), defines the cybertext Figure 1: schema of the cybertext (Aarseth, 1997: 21) This model has many qualities but also several limitations that make it ineffective for analyzing certain properties of real digital works. Let's examine the implicit and limitations of cybertext in the case of the digital.

2

The work in the cybertext model a)

Cybertext is classically based on the "reader's point of view"

The theory of cybertext is part of the classical paradigm of the work, which can be formulated, for digital works, in the following way: "the work is what can be read on the screen". This point of view appears implicitly in AArseth's definition of the fundamental objects of his model:

"A text, then, is any object with the primary function to relay verbal information (…) It is useful to distinguish between strings as they appear to readers and strings as they exist in the text, since these may not always be the same. For want of better terms, I call the former scriptons and the latter textons (…) In addition to textons and scrip'tons, a text consists of what I call a traversal function-the mechanism by which scriptons are revealed or generated from textons and presented to the user of the text." (Aarseth, 1997: 62) An absent author: The author is the great absentee of cybertext. The model never talks about what the text machine is to the author. Although Aarseth refutes the theoretical relevance of the distinction between author and reader, in his examples he only mentions reading operations and only talks about "manipulation". One can only manipulate what already exists. Verbal signs have to be created for the text machine to work. Indeed, a "verbal" sign is a sign readable by man in his language, in other words a sign that emerges on reading. We deduce from this that, in a very classical way, writing is conceived as a reading anticipated by the reader; its sole purpose is the creation of the textual machine of reading.

Lack of semiotic autonomy of textons: Aarseth explicitly states that verbal information is relayed between textons and scriptons through the implementation of a traversal function. The traversal function relays all the information contained in the textons. This implies that, even if a given reader does not discover all the information allowed by the machine, the infinite set of readers will, globally, read all the information contained in the textons. There is therefore no semiotic autonomy of textons, i.e. under no circumstances can they carry information that would never appear on reading. This subordination of the created sign to the read sign is the direct consequence of the author's absence: since the author is only a reader by anticipation, the meaning of any sign he creates (texton) is necessarily readable in reading situations, even if the form of the sign that actualizes it is different (scripton). In semiotic terms, we can say that textons and scriptons point to the same meaning in the encyclopedia or, put another way, that scriptons are merely a re-mediatization of textons. The true nature of the traversal function then appears: it is a process of re-mediatization.

The traversal function is not a transformation of the code into a media but a transmission: The traversal function is not in fine interested, neither in the computing specificity of the code, nor in the media built during the execution of this code, but only in the transmission of the information of the textons to the scriptons. Cybertext therefore implicitly takes into account only part of the computer code and part of the media produced. This truncated consideration of physical reality is also characteristic of a classical conception according to which the text is an abstract, purely linguistic object, an "object in itself" disconnected from all materiality. The cybertext schema (Figure 1) explains this conception of text as an "object in itself". In the classical conception this induces the concept of support (the text is in material relation with its support) and the cybertext is only an extrapolation of this conception in which the support becomes a machine which is nothing but a set of manipulable supports: the screen is the support of scriptons and the code is the support of textons.

Cybertext falls into the category of "front-end dispositives": Finally, we can say that cybertext is part of a long western tradition that the development of printing has forged, then later cinema and recorded music, and which I call the "front-end dispositive". In this conception, writing only aims at constructing readability via a single reading modality. In other words, the text is "transparent" for this reading modality, it has no hidden parts. Everything happens as if the author and the reader were on the same side of the text, one writing what the other reads or, in the case of cybertext, one building the machine that the other manipulates. The communicational model underlying this dispositive is none other than that of Shannon and Weaver.

b)

Cybertext does not address the actual technical properties of the "text machine".

The cybertext model says nothing about the actual technical nature of the traversal function, nor about the actual physical dispositive in which it is embedded. In this model, the relationship between textons and scriptons is causal: the scripton is totally determined by the texton and the traversal function. However, the real digital dispositive has a fundamental property that cybertext does not deal with: lability. The lability consists in the dependence of the result of the execution of a program on the technological context of this execution. It is above all a technological fact. Clearly, the same program that is neither generative nor hypertextual, but merely describes a displayed process, for example an animation, actually produces different animations depending on the technological context in which it is executed. This is why this feature was only discovered in 1990 [START_REF] Bootz | Notes ». alire livret[END_REF] on animated poetry and not on text generators, the generative character masking it. This technical lability induces of course meaning effects and is therefore accompanied by semiotic lability; it impacts reading, sometimes in a very significant way [START_REF] Bootz | signs and apparatus in digital poetry: the example of Jean-Marie Dutey's Le mange-texte[END_REF].

Considering that the effective traversal function during an execution is an instance of a class of functions would make it possible to account for some of the observed effects. In this case, we should consider that Aarseth's analysis of traversal functions is relative, not to the actual traversal functions at work, but to the classes they instantiate. In other words, the real traversal function would be virtual, dependent on an unpredictable context of execution. This approach does not destroy the causal character of the traversal function. Unfortunately, the actual observation of lability phenomena shows that sometimes scriptons appear, signs in the read media, which have no cause in the code, which do not emanate from any texton. This phenomenon is logical because lability is not caused by the program but by the complete technological context at runtime. This phenomenon is also observed in cases where the technological context is modified by a hardware change that has no impact on the execution of the program. For example, Xavier Hautbois and I observed that the reading of a video file of a sequence from Hans Richter's Rythm 21 generated two different semiotic units depending on whether it was viewed from the same computer on a CRT or LCD screen. The cybertext model cannot account for this observation.

For all these reasons, I have proposed a more pragmatic alternative model of the digital work: the procedural model [START_REF] Bootz | Un modèle fonctionnel des textes procéduraux[END_REF][START_REF] Bootz | Formalisation d'un modèle fonctionnel de communication à l'aide des technologies numériques appliqué à la création poétique[END_REF].

3

The procedural model

a) Presentation of the model

It is a general model of communication, which has been mainly constructed from the analysis of digital literary works but that is not limited to this type of production. Its generic presentation also uses examples that are not digital [START_REF] Bootz | Une approche modélisée de la communication ; Application à la communication par des productions numériques[END_REF]. Let us present it very quickly: This model makes a strict difference between 3 dimensions: material (and technological), semiotic, and psychological in the broadest sense (influence of culture, waiting horizon, emotional state, spatio-temporal context of individuals...). It considers that the situation of communication results from a physical transformation of a material work material, the source, into another work material, the transitoire observable, via an apparatus. This transformation then defines a "main communicational axis" that carries the apparatus and two particular roles: Author and Reader who are at the two extremities of the transformation.

In digital productions, the source is the source program, a form that the author can manipulate in her authoring environment. The transitoire observable is the physical phenomenon appearing on the reader's screen. The apparatus consists of all the machines involved in the transformation between the source and the transitoire observable. In the case of a Web production, the apparatus includes part of the Internet. Source and transitoire observable are not signs but physical phenomena that an ad-hoc dispositive can capture, for example a printer for an encoded source and a video capture dispositive for a transitoire observable produced on a screen. These physical events constitute the stimulus, in the Klinkenberg (1996) sense 1 , of texts which are: the texte-auteur (on the source) and the texteà-voir (on the transitoire observable). It is quite obvious that, in a programming or writing operation, an author "writes" a texte-auteur, even if he acts strictly speaking on the source, and a reader, conversely, reads a texte-à-voir, even if he perceives a transitoire observable. Physically, he does not manipulate the transitoire observable but rather the apparatus.

Materiality is not described by a "machine" but by a "dispositive" that encompasses all the technological artifacts that participate in the transformation and all the human actors in contact, in one way or another, with the main communication axis. These individuals are divided, at a given time, into classes that define roles: Author, Reader, technical agents and meta-reader. These roles may be collective, just as, conversely, a given individual may alternate various roles during his or her participation in the dispositive. This dispositive is structural. In other words, the transformation can be synchronous (face-to-face conversation, for example) or asynchronous (reading a previously written book). The part of the dispositive that carries the transformation and thus defines the main communication axis is called the main dispositive (shaded part in figure 3).

The psychological constraint is represented in the model by an external influence on perception and cognition, called "profondeur de dispositive", which influences, for each actor, the construction of the texte-à-voir and the texte-auteur.

b) Autonomy of texte-à-voir and texte-auteur

The physical transformation is not a traversal function. Due to the existence of lability and of the profondeur de dispositif, the procedural model establishes a total independence between the texte-auteur and the texte-à-voir. The model is therefore symmetrical: texteauteur and texte-à-voir can be considered separately. No causal relation exists between them, only correlations, even if the intentionality of the writing aims at creating a causal relation.

c) Braids and snatches in texts

In this model, the concepts of texton and scripton are not relevant. This is shown in figure 4: a concrete poem is spotted on a portion of the texte-auteur of first screening (BpNichol, 1984), but the execution of this fragment of source code does not produce in the texte-à-voir any sign of the same level of articulation, but only the beginning of a sentence. In other words, in this example, the textons form a lexie [START_REF] Barthes | [END_REF] but not the associated scriptons. This is why the procedural model does not decompose texte-auteur and texte-à-voir into textons and scriptons, but into braids and snatches, respectively. Braids and snatches are lexies. They therefore depend directly on the individual who perceives and constructs the texte-auteur and/or the texte-à-voir. Depending on one's culture, a lexie for one person, for example a concrete poem, is not necessarily a lexie for another. Because of the autonomy of the texte-auteur and the texte--voir, no semiotic transformation function links the braids to the snatches. The physical transformation connecting the part of the source stimulus of a perceived braid to the part of a transitoire observable stimulus of a perceived snatch only exists, just as the physical transformation that has transformed part of the source into the part of the transitoire observable stimulus of a perceived snatch exists.

Concrete Texte-auteur concret in source screen production by this source Let us insist on one point: any individual participating in a role having access to the source builds on it a texte-auteur and any individual having access to the transitoire observable builds on it a texte-à-voir. The model indicates that a particular role, the metareader, has joint access to these two material components.

The meta-reader role is a determining element of the model. In particular, any person analyzing a work is in a meta-reading situation. We will argue below that this role is also a real position of reception of the work, complementary to the digital reading to which we are accustomed (called narrow reading in the model), in that it gives access to aesthetic and semiotic dimensions that never appear on the screen. Any meta-reading is based on the joint use of several modalities of access to the material components of the dispositive among the 7 identified (figure 3). These modalities are instrumented, using protocols and investigative tools.

4 Comparison between the two models

The Aarseth text machine appears to be the form in which the main dispositive is seen from the position of the Reader in the procedural model. Yet profound differences exist between these two models.

In both models, text is divided into two sets of autonomous signs: scriptons and textons in the cybertext, texte-auteur and texte-à-voir in the procedural model. The textons are in the texte-auteur and the scriptons are in the texte-à-voir. However, the two models do not really deal with the same types of signs. In cybertext, textons and scriptons are "verbal signs", i.e. linguistic entities. In the procedural model, on the other hand, textes-auteur and textes-àvoir can consist of any type of signs. On the other hand, textons and scriptons are not equivalent to braids and snatches.

In cybertext, all readers who perform the same manipulations access the same textons and scriptons when the program, if it is generative, makes the same decisions. This is not the case in the procedural model: textes-auteur and textes-à-voir may differ due to lability and differences in the individual profondeurs de dispositif.

We have pointed out that lability requires a redefinition of the traversal function. Moreover, the position of meta-reader is totally absent in cybertext and there is no way to add it. For the meta-reader, texte-auteur and texte-à-voir are two distinct parts of the same set of signs. She can even add a third set: the Reader's behavior. These three sets are therefore not linked by a traversal function type transformation, but by rhetorical relationships, some of which we present below. Rather than considering a function that transforms signs, the procedural model considers a material transformation that transforms entities in the material world, and which therefore remains relevant whatever are the semiotic processes in the dispositive.

In the end, the procedural model is not compatible with the frontal dispositive, especially since the three roles access different and disjointed parts of the dispositive.

5 Rhetoric of Dual Signs in Meta-Reading a)

The three classes of signs in the main dispositive

The sign space develops in the procedural model in 3 physical spaces disjointed from the main dispositive (source, transitoire observable and reader's reaction) that only the metareader can apprehend simultaneously. The model thus defines 3 classes of signs, different but not exclusive, with the same sign belonging to one or two classes. These classes are:

-The unary signs: These signs are perceived in a texte-auteur or a texte-à-voir read autonomously, without any relation with the dispositive, as if the dispositive only is the support. Most of the analyses are limited to the consideration of a single texte-à-voir and therefore deal only with unary signs. All roles have access to unary sign spaces.

-The dual signs. These signs develop on two distinct physical spaces. They are the most interesting signs in digital literature because they allow a rich set of rhetorical figures. Only the roles Reader and meta-reader can access dual signs. For the Reader, the textual space of dual signs consists of the texte-à-voir and its reading activity (the signs corresponding to the latter constitute "double reading" [START_REF] Bootz | der/die leser ; reader/readers[END_REF]), whereas for the metareader, the dual signs relate his text-author and his text-to-view.

-Ternary signs: these develop on the texte-à-voir, the texte-auteur and the reader's reaction. Only the meta-reader accesses this category of signs through the joint implementation of meta-reading modalities 2, 4 and 5. These signs mainly concern interactive works and the aesthetics of frustration.

Signifying spaces are added to these spaces of signs. They are physical spaces that do not allow the construction of signs, such as the inside of the apparatus, but to which one can give meaning. [START_REF] Baldwin | Ping Poetics[END_REF] gives a typical example of this.

b) Rhetorical figures on dual signs

Figures of rhetoric develop between the texte-auteur and the texte-à-voir that a metareader perceives. Let us recall some examples that I have previously analyzed in several papers and lectures. They are a sequence of passage [START_REF] Bootz | passage[END_REF] -top illustration in figure 5, and an obfuscated C work by Eric Marshall (1986) -bottom illustration. The rhetorical figures detected are different depending on whether they are based solely on meta-reading modalities 2 and 5 or if we add to them the consideration (3) of the apparatus. The latter then constitute "material metaphors" [START_REF] Hayles | Writing Machines[END_REF] since they are based on the physical functioning of the main dispositive. In the sequence of passage, the texte-auteur includes a photo of the largest ammonite slab in Europe. This photo never appears in the texte-à-voir, but it is used to create a very moving and rhythmic texture in which geometric figures that are absent from the texte-auteur appear. Photo and textures constitute respectively a graphic braid and a graphic snatch. Observation through meta-readings modalities 2 and 5 puts them in relation within an oxymoron type rhetorical figure: the ammonite slab being a static object with a connotation of death, whereas the texture is moving and dynamic with a connotation of life. The text scrolling on the screen does not refer to the slab in the unary sign of the texte-à-voir, whereas in the dual sign, the ammonite slab appears as the referent, which sheds new light on this text and changes its meaning.

The apparatus can be perceived as meaningful. It constitutes the screen that prevents the reader from accessing the photo. The apparatus then plays the role of the rock covering the fossil slab: it must be broken (and metaphorically switched to meta-reading and not narrow reading) to access the slab. The material metaphor here is of a metaphorical type whereas it is based on the same dual sign! In Marshall's work, the comparison (2 and 5) between the texte-auteur, a locomotive calligram, and the texte-à-voir, the choo choo written on the screen, creates a rhetorical figure of metonymic type: the locomotive and the "choo choo" are in semantic continuity. On the other hand, taking into account the apparatus gives a metaphorical dimension to the main dispositive: the apparatus must be supplied with energy so that the program writes "choo choo" just as the locomotive must be supplied with energy so that it emits a sound typical of its operation.

c)

The overall rhetorical figure

Generally, the perception that the texte-auteur gives of the work in its unary signs space does not cover the perception that emanates from the texte-à-voir in its one. This difference cannot be perceived by the roles Author and Reader since they do not access the same spaces. On the other hand, it is manifested in meta-reading. This difference is then perceived as a trope: it constitutes the overall rhetorical figure that emanates from metareading.

This figure necessarily exists in digital literature simply because of the fundamental difference in status between the source and the transitoire observable: the source, as a source program, is perennial, and even printable, whereas the transitoire observable is ephemeral, it exists only when the transformation is active in the main dispositive. This perennial/ephemeral opposition is projected onto the texte-auteur/texte-à-voir couple. By the concomitant presence in meta-reading of these opposed qualities, the overall figure thus has at least an oxymoronic dimension. Following the advice of the mu group, which recommends that the names of linguistic rhetorical figures should not be used in other semiotic codes because the process of the figure and its characteristics are different there, I propose to name "proxymoron" (for procedural oxymoron) this overall figure.

The oxymoronic dimension of the proxymoron can be reinforced by other oppositions or supplemented by other dimensions. For example, in recent works, the texte-auteur is often fragmented, split on several components of the source (several files in the case of a coded program, timeline (or graph) and scripts in other cases...) whereas the texte-à-voir constitutes a single homogeneous whole, especially in non-hypertextual works. This feature amplifies the oxymoron dimension of the proxymoron.

As an example, I have noted the oppositions between the texte-auteur and texte-à-voir that construct the proxymoron in my meta-reading of Dutey's work Le mange-texte (1989). In figure 6, the general oppositions present in all the early works are indicated under the more specific oppositions in this work. The source is programmed in a very particular way, which a computer scientist would consider profoundly naive because the same scripts are repeated when it would have sufficed to reuse them, and because they are sometimes made up of exactly the same instructions, only the values of the data change. It would therefore have been wiser to introduce an argument to pass to a single script the different values used in the different parts of the work. Tibor knew how to program perfectly well, and therefore a meta-reader's eye is not fooled by this first impression left by the functional, technical dimension of the source. In the texte-auteur, the source program is not reduced to its computer dimension alone. The challenge in constructing a texte-auteur is precisely to capture the non-computer dimensions whose source is also a stimulus in the sense of Klinkenberg. In Orion, these computer "naiveties" are precisely the clue to true programmed writing: the order of repetition of the scripts marks the true sequences of the work. We can see in particular that they are based on the visual and not on sound. The "naïve" scripts that do not use variables are all time delay scripts. It is therefore understandable that the temporality in the texte-auteur is not continuous as it is in the texte-à-voir, but built on 2 bricks of elementary duration of 5 and 3 seconds that the author arranges a bit like measures on a score. Moreover, the sequences are actually non-linear constructions in the texte-auteur. If we consider that the sequence is an element at the level of a "paragraph", each sequence is constructed from two types of "sentences": the main pattern, different from one sequence to another but unique for each sequence, and the "transversal processes" which are animations. The main pattern is an animation on a background image. Transversal processes are animations of elements without a background image. The processing on the timeline of these two components is not the same: the main pattern develops on the timeline according to a succession of frames on which timing scripts can be added to break its rhythm, whereas the transversal processes are totally programmed, each on one or two frames. They thus pause, as it were, one frame of the main pattern. Transversal processes are never repeated, whereas the main pattern can be repeated, often with sonic variations. Thus, the programming of the work obeys a perfectly defined and rigorous grammar: it is by no means a "description" of what appears on the screen, but a real statement; a texte-auteur. In meta-reading, the opposition between this non-linear grammar of the texte-auteur and the linear grammar of the animation in the texte-à-voir reinforces the oxymoronic dimension of the proxymoron. Main pattern and transversal processes are braids. By destructuring the program so that they play alone, the transitoire observable correlated to them can be produced and recorded. The corresponding textes-à-voir are snateches that are found, depending on the case, intact or unstructured in the real texte-à-voir2 .

b) A pseudo-random sequence

A specific use of the random function in the sequence that tells Orion's story reinforces the proxymoron. In general, Tibor Papp only uses randomness to establish rhythmic variations within the texte-à-voir. But he uses it in a more generative way in this sequence to create random bifurcations in the content. The texte-à-voir in this sequence then seems generative. The unstructured and erratic nature of this sequence amplifies this generative combinatorial dimension because the main drawback of combinatorial generation is that it produces statements that often flirt with nonsense or, at best, a surrealist image.

In Orion, this sequence is long enough that, on reading, any attempt to compare it with a previous reading proves impossible because it puts the reader in cognitive overload. There is nothing like this in meta-reading. It is enough to make video captures (modality 5) and analyses of the program to understand the real mechanism of this sequence and see that it is only very weakly random and not at all combinatorial.

The story is told and broken down into 9 moments (noted from 01 to 09 in figure 8). The correct chronological sequence defines the pivot sequence (noted 6). These 9 moments are sequentially rearranged in a non-random way in 8 variants. These variants also use a sound variant (denoted 0102...) of 5 of the 9 sound files in the pivot sequence. In the end, the choice of the played sequence among these 9 possible sequences (numbered from 1 to 9)3 is random, but the number of variants thus allowed is far from the total number of 11.612.160 sequences that a purely random choice would have allowed. Transcoding in both models

1 Problematics of transcoding

In the cybertext model, transcoding simply means programming the translation of the scriptons. We can then consider that transcoding translates the traversal function. This conception means that transcoding must encode the technical performance of the initial computer program. It is limited to a recoding that produces the linguistic and media translations.

The procedural model views transcoding as more than simply the technical programming of the translated snatches.

Lability poses a first generic problem for translation: what initial state of the text to be translated is appropriate? I have shown in many circumstances, notably in [START_REF] Bootz | signs and apparatus in digital poetry: the example of Jean-Marie Dutey's Le mange-texte[END_REF], that lability is neither noise nor deterioration, but a transformation that can transform the rhetorical relations and signs of the texte-à-voir. There is therefore no reference state that can constitute a single source text to be translated.

In the procedural model, transcoding does not only act on the Reader' texte-à-voir, but also on the semiotic relations between the meta-reader's texte-auteur and texte-à-voir, and on the Author's texte-auteur. The code being the stimulus for the latter, transcoding must ensure its translation. It is then necessary to transcode according to 3 dimensions that may conflict with each other:

-The performative dimension: the new code must produce at runtime the same processes of animation and media transformation in the translated texte-à-voir as in the original. It is a performative translation, with no equivalent in print or subtitling.

-The syntactic dimension of the instruction: the new code must reproduce the meaning of each original instruction. This is a "line-by-line" transcoding, equivalent to a literal linguistic transcoding because an instruction is similar to a sentence in a computer code.

-The semantic dimension: transcoding must translate or preserve the braids of the texte-auteur, independently of any relation with the texte-à-voir, i.e. independently of any computer consideration. This dimension concerns the autonomy of the texte-auteur and is equivalent to a translation of meaning.

As in linguistic translation, it is better to privilege the semantic dimension over the syntactic dimension in case of conflict. The performative dimension must be ensured in any case.

2 Non-preservation of the texte-auteur in the cybertext model

When transcoding is carried out in accordance with the cybertext model, the translation of the texte-auteur is impossible because transcoding, in this model, only preserves the performative dimension of the source program. This can be seen by comparing the texteauteur of the first screening original program in basic, created on Apple IIe, and the transcoding into JavaScript by Marko Niemi (2006). This example is exemplary because Marko Niemi has tried, on the one hand, to preserve the temporality of the animation he saw on his screen when executing the program of the work, and on the other hand, to preserve as much as possible the literal dimension of the original program by transcoding the basic instructions by instructions that most resemble them: "Some parts of the javascript code are meant to be as directly equivalent to the original BASIC code as possible; for instance, the 'Print (vtab, htab, text)' javascript function works pretty much the same as 'VTAB Y: HTAB X: PRINT "<TEXT>"' lines in BASIC (VTAB and HTAB define the coordinates of the text in question; in javascript, the top-left position of the screen is at (0,0) whereas in BASIC it's at (1,1)). SPEED variables have their equivalents in 'pause()' arrays (SPEED=255 is the fastest tempo in BASIC and SPEED=0 probably the slowest, whereas in javascript the smaller the 'pause', the faster the tempo), and the 'Home()' function clears the screen the same way as the HOME command in BASIC. " (Niemi, 2006, mail answer to Jim Andrews about the JavaScript version) Marko Niemi also points out that he used an Apple IIe emulator to compare the transitoire observable produced by the original program and his program. As lability is linked to the technical compatibility of the devices, one can hope that the emulator used emulates the Apple IIe well enough to have a low lability. The question of the reference state is then resolved.

We can therefore affirm that Marko Niemi has achieved a good "computer" transcoding, i.e. one that respects the letter and spirit of the instructions, and thus the computer dimension of the program. The comparison of the two programs shows that, unfortunately, the other dimensions of the original texte-auteur are not preserved, especially the concrete snatches. The JavaScript program modifies the explicit nature of the data: from text to list, which gives a different semiotic vision of the program. It introduces non-linearity with an if...else structure as well as an explicit reference to time (with the setTimeOut method) that do not exist at all in the original program. By numbering the lines4 , the original program insists on linearity. If it would have been easy to preserve the snatches, the predominance of the concept of text over that of data and avoid the if...else structure, it would have been at the price of a literal and computer "naive" writing of the JavaScript program. It is understandable that a programmer refuses to do so. On the other hand, it is impossible to avoid calling a JavaScript callback 5 function to manage timers. It is therefore impossible to avoid the nonlinearity of the JavaScript code that results from the asynchronous nature of callbacks and the presence of functions. Whereas this transposition is not conceived as a translation (no modification of the natural language), we can see how difficult, if not impossible, it is to conceive of transcoding as a translation operation: literal translation and translation of meaning are simultaneously impossible, as in classical translation, but with consequences that can be assumed to be much more dramatic: a translation that is too far removed from literal translation would risk corrupting the transposition of the transitoire observable via the other programming language and thus distorting the translation of the latter. I have also observed, with regard to translations in German and English of a Reader's texte-à-voir from the rabot poète [START_REF] Bootz | Le rabot poète[END_REF], that the linguistic semantic translation could impose a modification of the animation script, which profoundly modifies the text-auteur built on the source program.

"Reading machines" as an alternative

Whatever it is (translation, transcoding, transcreation...), transferring a literary digital work poses a difficult challenge when one wants to transpose all the facets of the original into a single target work. One of the solutions is to overemphasize the importance of transcreation and to consider the result produced by this transfer, not as a new version of the work, but as a work in its own right, an adaptation, just as one adapts a novel to the cinema.

The "reading machines" constitute an alternative that makes it possible to stay as close as possible to the texte-auteur and texte-à-voir of the original production and to preserve the rhetoric of the dual signs. A reading machine, in the procedural model, is a second discourse emanating from meta-reading (figure 10). It is a digital or video production that reconstructs for screen reading the dimensions perceived in meta-reading that do not appear in narrow reading. It is therefore a component that allows the Reader to access all the perceptible dimensions of the work. These reading machines can quite possibly contain translations. The reading machine is based on a deconstruction in lexies of all the sign spaces of the work. There is no need to bring them all together in a single object. One can thus perform the linguistic translation(s) of the initial texte-à-voir videotapes when possible, or construct a target transitoire observable via programs that only "translate" the functional dimension of the original program. They do not even need, then, to constitute a literal translation of the original program. One can even envisage different translations to translate different snatches that may overlap. On the other hand, the proxymoron and the braids of the texte-auteur can be given to read from the original source program. The same applies to ternary signs.

The first reading machines should be installed at BnF on the Richelieu site in 2021.

Figure 2 :

 2 Figure 2: treatment of lability in cybertext by virtualization of the traversal function

Figure 3 :

 3 Figure 3: schema of the procedural model

Figure 4 :

 4 Figure 4: braid on the source of first screening and screeshot of its execution

Figure 5 :

 5 Figure 5: rhetorical figures in dual signs

Figure

  Figure 6: proxymoron in le mange-texte

Figure 7 :

 7 Figure 7: grammatical reinforcement of the proxymoron in an Orion sequence

Figure 8 :

 8 Figure 8: pseudo-random sequence in Orion

  3

Figure 9 :

 9 Figure 9: comparison of the 2 first screening programs

Figure 10 :

 10 Figure 10 : reading machines in the procedural model

You can find a video of the main pattern of the sequence analyzed in figure7recorded using this process in http://bootz.fr/orion/aaaOrionD8501-nuits-sur-carte-fond%20lin%c3%a9aire.mp4 and a video of 1 transversal process in http://bootz.fr/orion/aaOrionp02-sequence-nuits-sur-carte-processus-frame295.mp4. The total sequence is recorded in http://bootz.fr/orion/aaOrionp02-sequence-nuits-sur-carte.mp4

The sequence 1 is recorded in http://bootz.fr/orion/sequence-orion1.mp4 and sequence 9 in http://bootz.fr/orion/sequence-orion9.mp4

Irrespective of the fact that this is an inherent constraint of the basic and not of the author, even unintentionally, linearity is emphasized.


Samir Adly
email: samir.adly@unilim.fr

Hedy Attouch
email: hedy.attouch@umontpellier.fr

Van Nam Vo

Asymptotic behavior of Newton-like inertial dynamics involving the sum of potential and nonpotential terms

Keywords: Mathematics Subject Classifications proximal-gradient algorithms, inertial methods, Hessian driven damping, nonpotential term, cocoercive operators

In a Hilbert space H, we study a dynamic inertial Newton method which aims to solve additively structured monotone equations involving the sum of potential and nonpotential terms. Precisely, we are looking for the zeros of an operator A = ∇f + B where ∇f is the gradient of a continuously differentiable convex function f , and B is a nonpotential monotone and cocoercive operator. Besides a viscous friction term, the dynamic involves geometric damping terms which are controlled respectively by the Hessian of the potential f and by a Newton-type correction term attached to B. Based on a fixed point argument, we show the well-posedness of the Cauchy problem. Then we show the weak convergence as t → +∞ of the generated trajectories towards the zeros of ∇f + B. The convergence analysis is based on the appropriate setting of the viscous and geometric damping parameters. The introduction of these geometric dampings makes it possible to control and attenuate the known oscillations for the viscous damping of inertial methods. Rewriting the second-order evolution equation as a first-order dynamical system enables us to extend the convergence analysis to nonsmooth convex potentials. These results open the door to the design of new first-order accelerated algorithms in optimization taking into account the specific properties of potential and nonpotential terms. The proofs and techniques are original and differ from the classical ones due to the presence of the nonpotential term.

Introduction and preliminary results

Let H be a real Hilbert space endowed with the scalar product •, • and the associated norm • . Many situations coming from physics, biology, human sciences involve equations containing both potential and nonpotential terms. In human sciences, this comes from the presence of both cooperative and noncooperative aspects. To describe such situations we will focus on solving additively structured monotone equations of the type Find x ∈ H : ∇f (x) + B(x) = 0.

(1.1)

In the above equation, ∇f is the gradient of a convex continuously differentiable function f : H → R (that's the potential part), and B : H → H is a nonpotential operator 1 which is supposed to be monotone and cocoercive. To this end, we will consider continuous inertial dynamics whose solution trajectories converge as t → +∞ to solutions of (1.1). Our study is part of the active research stream that studies the close relationship between continuous dissipative dynamical systems and optimization algorithms which are obtained by their temporal discretization. To avoid lengthening the paper, we limit our study to the analysis of the continuous dynamic. The analysis of the algorithmic part and its link with first-order numerical optimization will be carried out in a second companion paper. From this perspective, damped inertial dynamics offers a natural way to accelerate these systems. As the main feature of our study, we will introduce into the dynamic geometric dampings which are respectively driven by the Hessian for the potential part, and by the corresponding Newton term for the nonpotential part. In addition to improving the convergence rate, this will considerably reduce the oscillatory behavior of the trajectories. We will pay particular attention to the minimal assumptions which guarantee convergence of the trajectories, and which highlight the asymmetric role played by the two operators involved in the dynamic. We will see that many results can be extended to the case where f : H → R ∪ {+∞} is a convex lower semicontinuous proper function, which makes it possible to broaden the field of applications.

Dynamical inertial Newton method for additively structured monotone problems

Let us introduce the following second-order differential equation which will form the basis of our analysis: ẍ(t) + γ ẋ(t) + ∇f (x(t)) + B(x(t)) + β f ∇ 2 f (x(t)) ẋ(t) + β b B (x(t)) ẋ(t) = 0, t ≥ t 0 . (DINAM)

We use (DINAM) as an abbreviation for Dynamical Inertial Newton method for Additively structured Monotone problems. We call t 0 ∈ R the origin of time. Since we are considering autonomous systems, we can take any arbitrary real number for t 0 . For simplicity, we set t 0 = 0. When considering the corresponding Cauchy problem, we add the initial conditions: x(0) = x 0 ∈ H and ẋ(0) = x 1 ∈ H.

The term B (x(t)) ẋ(t) is interpreted as d dt (B(x(t))) taken in the distribution sense. Likewise the term ∇ 2 f (x(t)) ẋ(t) is interpreted as d dt (∇f (x(t))) taken also in the distribution sense. Because of the assumptions made below these terms are indeed mesurable functions which are bounded on the bounded time intervals. So, we will consider strong solutions of the above equation (DINAM) . Throughout the paper we make the following standing assumptions:

         (A1) f : H → R is convex, of class C 1 ,
∇f is Lipschitz continuous on the bounded sets;

(A2) B : H → H is a λ-cocoercive operator for some λ > 0;

(A3) γ > 0, β f ≥ 0, β b ≥ 0 are given real damping parameters.

We emphasize the fact that we do not assume the gradient of f to be globally Lipschitz continuous. Developping our analysis without using any bound on the gradient of f is a key to further extend the theory to the nonsmooth case. As a specific property, the inertial system (DINAM) combines two different types of driving forces associated respectively with the potential operator ∇f and the nonpotential operator B. It also involves three different types of friction:

(a) The term γ ẋ(t) models viscous damping with a positive coefficient γ > 0.

(b) The term β f ∇ 2 f (x(t)) ẋ(t) is the so-called Hessian driven damping, which allows to attenuate the oscillations that naturally occur with the inertial gradient dynamics.

(c) The term β b B (x(t)) ẋ(t) is the nonpotential version of the Hessian driven damping. It can be interpreted as a Newton-type correction term.

Note that each driving force term enters (DINAM) with its temporal derivative. In fact, we have

∇ 2 f (x(t)) ẋ(t) = d dt (∇f (x(t))) and B (x(t)) ẋ(t) = d dt (B(x(t))) .
This is a crucial observation which makes (DINAM) equivalent to a first order system in time and space, and makes the corresponding Cauchy problem well posed. This will be proved later (see subsection 2.1 for more details). The cocoercivity assumption on the operator B plays an important role in the analysis of (DINAM), not only to ensure the existence of solutions, but also to analyze their asymptotic behavior as time t → +∞.

Recall that the operator B : H → H is said to be λ-cocoercive for some λ > 0 if

By -Bx, y -x ≥ λ By -Bx 2 , ∀x, y ∈ H.
Note that B is λ-cocoercive is equivalent to B -1 is λstrongly monotone, i.e. cocoercivity is a dual notion of strong monotonicity. It is easy to check that B is λ-cocoercive implies that B is 1/λ-Lipschitz continuous. The reverse implication holds true in the case where the operator is the gradient of a convex and differentiable function. Indeed, according to Baillon-Haddad's Theorem [START_REF] Baillon | Quelques propriétés des opérateurs angles-bornés et n-cycliquement monotones[END_REF], ∇f is L-Lipschitz continuous implies that ∇f is a 1/L-cocoercive operator (we refer to [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert spaces[END_REF]Corollary 18.16] for more details).

Historical aspects of the inertial systems with Hessian-driven damping

The following inertial system with Hessian-driven damping

ẍ(t) + γ ẋ(t) + β∇ 2 f (x(t)) ẋ(t) + ∇f (x(t)) = 0,
was first considered by Alvarez-Attouch-Peypouquet-Redont in [START_REF] Alvarez | A second-order gradient-like dissipative dynamical system with Hessian-driven damping. Application to optimization and mechanics[END_REF]. Then, according to the continuous interpretation by Su-Boyd-Candès [START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method[END_REF] of the accelerated gradient method of Nesterov, Attouch-Peypouquet-Redont [START_REF] Attouch | Fast convex minimization via inertial dynamics with Hessian driven damping[END_REF] replaced the fixed viscous damping parameter γ by an asymptotic vanishing damping parameter α t , with α > 0. At first glance, the presence of the Hessian may seem to entail numerical difficulties.

However, this is not the case as the Hessian intervenes in the above ODE in the form ∇ 2 f (x(t)) ẋ(t), which is nothing but the derivative with respect to time of ∇f (x(t)). So, the temporal discretization of these dynamics provides first-order algorithms of the form

y k = x k + α k (x k -x k-1 ) -β k (∇f (x k ) -∇f (x k-1 )) x k+1 = y k -s∇f (y k ).
As a specific feature, and by comparison with the classical accelerated gradient methods, these algorithms contain a correction term which is equal to the difference of the gradients at two consecutive steps. While preserving the convergence properties of the accelerated gradient method, they provide fast convergence to zero of the gradients, and reduce the oscillatory aspects. Several recent studies have been devoted to this subject, see Attouch-Chbani-Fadili-Riahi [START_REF] Attouch | First-order algorithms via inertial systems with Hessian driven damping[END_REF], Bot ¸-Csetnek-László [START_REF] Bot | Tikhonov regularization of a second order dynamical system with Hessian damping[END_REF], Kim [START_REF] Kim | Accelerated Proximal Point Method for Maximally Monotone Operators[END_REF], Lin-Jordan [START_REF] Lin | A Control-Theoretic Perspective on Optimal High-Order Optimization[END_REF], Shi-Du-Jordan-Su [START_REF] Shi | Understanding the acceleration phenomenon via highresolution differential equations[END_REF], and Alesca-Lazlo-Pinta [START_REF] Alecsa | An extension of the second order dynamical system that models Nesterov's convex gradient method[END_REF] for an implicit version of the Hessian driven damping. Application to deep learning has been recently developed by Castera-Bolte-Févotte-Pauwels [START_REF] Castera | An Inertial Newton Algorithm for Deep Learning[END_REF]. In [START_REF] Adly | Finite convergence of proximal-gradient inertial algorithms combining dry friction with Hessian-driven damping[END_REF], Adly-Attouch studied the finite convergence of proximal-gradient inertial algorithms combining dry friction with Hessian-driven damping.

Inertial dynamics involving cocoercive operators

Let's come to the transposition of these techniques to the case of maximally monotone operators. Álvarez-Attouch [START_REF] Alvarez | An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping[END_REF] and Attouch-Maingé [START_REF] Attouch | Asymptotic behavior of second order dissipative evolution equations combining potential with nonpotential effects[END_REF] studied the equation

ẍ(t) + γ ẋ(t) + A(x(t)) = 0, (1.2)
when A : H → H is a cocoercive (and hence maximally monotone) operator, (see also [START_REF] Bot | Second order forward-backward dynamical systems for monotone inclusion problems[END_REF]). The cocoercivity assumption plays an important role in the study of (1.2), not only to ensure the existence of solutions, but also to analyze their long-term behavior. Assuming that the cocoercivity parameter λ and the damping coefficient γ satisfy the inequality λγ 2 > 1, Attouch-Maingé [START_REF] Attouch | Asymptotic behavior of second order dissipative evolution equations combining potential with nonpotential effects[END_REF] showed that each trajectory of (1.2) converges weakly to a zero of A, i.e. x(t) x ∞ ∈ A -1 (0) as t → +∞. Moreover, the condition λγ 2 > 1 is sharp. For general maximally monotone operators this property has been further exploited by Attouch-Peypouquet [START_REF] Attouch | Convergence of inertial dynamics and proximal algorithms governed by maximal monotone operators[END_REF], and by Attouch-Laszlo [START_REF] Attouch | Continuous Newton-like Inertial Dynamics for Monotone Inclusions, Set-Valued and Variational Analysis[END_REF][START_REF] Attouch | Newton-like inertial dynamics and proximal algorithms governed by maximally monotone operators[END_REF]. The key property is that for λ > 0, the Yosida approximation A λ of A is λ-cocoercive and A -1 λ (0) = A -1 (0). So the idea is to replace the operator A by its Yosida approximation, and adjust the Yosida regularization parameter. Another related work has been done by Attouch-Maingé [START_REF] Attouch | Asymptotic behavior of second order dissipative evolution equations combining potential with nonpotential effects[END_REF] who first consider the asymptotic behavior of the second order dissipative evolution equation with f : H → R convex and B : H → H cocoercive

ẍ(t) + γ ẋ(t) + ∇f (x(t)) + B(x(t)) = 0, (1.3)
combining potential with nonpotential effects. Our study will therefore consist initially in introducing the Hessian term and the Newton-type correcting term into this dynamic.

Link with Newton-like methods for solving monotone inclusions

Let us specify the link between our study and Newton's method for solving (1.1). To overcome the illposed character of the continuous Newton method for a general maximally monotone operator A, the following first order evolution system was studied by Attouch-Svaiter [START_REF] Attouch | A continuous dynamical Newton-Like approach to solving monotone inclusions[END_REF],

v(t) ∈ A(x(t)) γ(t) ẋ(t) + β v(t) + v(t) = 0.
This system can be considered as a continuous version of the Levenberg-Marquardt method, which acts as a regularization of the Newton method. Remarkably, under a fairly general assumption on the regularization parameter γ(t), this system is well posed and generates trajectories that converge weakly to equilibria (zeroes of A). Parallel results have been obtained for the associated proximal algorithms obtained by implicit temporal discretization, see [START_REF] Abbas | Newton-like dynamics and forward-backward methods for structured monotone inclusions in Hilbert spaces[END_REF], [START_REF] Attouch | A dynamic approach to a proximal-Newton method for monotone inclusions in Hilbert Spaces, with complexity O(1/n 2 )[END_REF], [START_REF] Attouch | Global convergence of a closed-loop regularized Newton method for solving monotone inclusions in Hilbert spaces[END_REF]. Formally, this system is written as

γ(t) ẋ(t) + β d dt (A(x(t))) + A(x(t)) = 0.
Thus (DINAM) can be considered as an inertial version of this dynamical system for structured monotone operator A = ∇f + B. Our study is also linked to the recent works by Attouch-Laszlo [START_REF] Attouch | Continuous Newton-like Inertial Dynamics for Monotone Inclusions, Set-Valued and Variational Analysis[END_REF][START_REF] Attouch | Newton-like inertial dynamics and proximal algorithms governed by maximally monotone operators[END_REF] who considered the general case of monotone equations. By contrast with [START_REF] Attouch | Continuous Newton-like Inertial Dynamics for Monotone Inclusions, Set-Valued and Variational Analysis[END_REF][START_REF] Attouch | Newton-like inertial dynamics and proximal algorithms governed by maximally monotone operators[END_REF], according to the cocoercivity of B, we don't use the Yosida regularization, and exhibit minimal assumptions involving only the nonpotential component.

Contents

The paper is organized as follows. Section 1 introduces (DINAM) with some historical perspective. In section 2, based on the first order equivalent formulation of (DINAM), we show that the Cauchy problem is well-posed (in the sense of existence and uniqueness of solutions). In section 3, we analyze the asymptotic convergence properties of the trajectories generated by (DINAM). Using appropriate Lyapunov functions, we show that any trajectory of (DINAM) converges weakly as t → +∞, and that its limit belongs to S = (∇f + B) -1 (0). The interplay between the damping parameters β f , β b , γ and the cocoercivity parameter λ will play an important role in our Lyapounov analysis. In Section 4, we perform numerical experiments showing that the well-known oscillations in the case of the heavy ball with friction, are damped with the introduction of the geometric (Hessian-like) damping terms. An application to the LASSO problem with a nonpotential operator as well as a coupled system in dynamical games are considered. Section 5 deals with the extension of the study to the nonsmooth and convex case. Section 6 contains some concluding remarks and perspectives.

Well posedness of the Cauchy-Lipschitz problem

We first show the existence and the uniqueness of the solution trajectory for the Cauchy problem associated with (DINAM) for any given initial condition data (x 0 , x 1 ) ∈ H × H.

First-order in time and space equivalent formulation

The following first-order equivalent formulation of (DINAM) was first considered by Alvarez-Attouch-Bolte-Redont [START_REF] Alvarez | A second-order gradient-like dissipative dynamical system with Hessian-driven damping. Application to optimization and mechanics[END_REF] and Attouch-Peypouquet-Redont [START_REF] Attouch | Fast convex minimization via inertial dynamics with Hessian driven damping[END_REF] in the framework of convex minimization. Specifically, in our context, we have the following equivalence, which follows from a simple differential and algebraic calculation.

Proposition 2.1 Suppose that β f > 0. Then, the following problems are equivalent:

(i) ⇐⇒ (ii) (i) ẍ(t) + γ ẋ(t) + ∇f (x(t)) + B(x(t)) + β f ∇ 2 f (x(t)) ẋ(t) + β b B (x(t)) ẋ(t) = 0. (ii)        ẋ(t) + β f ∇f (x(t)) + β b B(x(t)) + γ - 1 β f x(t) + y(t) = 0; ẏ(t) -1 - β b β f B(x(t)) + 1 β f γ - 1 β f x(t) + 1 β f y(t) = 0.
Proof. (i) =⇒ (ii). For t ≥ 0, set

y(t) := -ẋ(t) -β f ∇f (x(t)) -β b B(x(t)) -γ - 1 β f x(t), (2.1)
which gives the first equation of (ii). By differentiating y(•) and using (i), we get

ẏ(t) = -ẍ(t) -β f ∇ 2 f (x(t)) ẋ(t) -β b B (x(t)) ẋ(t) -γ - 1 β f ẋ(t) = γ ẋ(t) + ∇f (x(t)) + B(x(t)) -γ - 1 β f ẋ(t) = ∇f (x(t)) + B(x(t)) + 1 β f ẋ(t). (2.2)
By combining (2.1) and (2.2), we obtain

ẏ(t) + 1 β f y(t) = 1 - β b β f B(x(t)) - 1 β f γ - 1 β f x(t). (2.3)
This gives the second equation of (ii).

(ii) =⇒ (i). By differentiating the first equation of (ii), we obtain

ẍ(t) + β f ∇ 2 f (x(t)) ẋ(t) + β b B (x(t)) ẋ(t) + γ - 1 β f ẋ(t) + ẏ(t) = 0. (2.4)
Let us eliminate y from this equation to obtain an equation involving only x. For this, we successively use the second equation in (ii), then the first equation in (ii) to obtain

ẏ(t) = 1 - β b β f B(x(t)) - 1 β f γ - 1 β f x(t) - 1 β f y(t) = 1 - β b β f B(x(t)) - 1 β f γ - 1 β f x(t) + 1 β f ẋ(t) + ∇f (x(t)) + β b β f B(x(t)) + 1 β f γ - 1 β f x(t).
Therefore,

ẏ(t) = ∇f (x(t)) + B(x(t)) + 1 β f ẋ(t). (2.5)
From (2.4) and (2.5), we obtain (i).

Well-posedness of the evolution equation (DINAM)

The following theorem shows the well-posedness of the Cauchy problem for the evolution equation (DI-NAM).

Theorem 2.1 Suppose that β f > 0 and β b ≥ 0. Then, for any (x 0 , x 1 ) ∈ H × H, there exists a unique strong global solution x : [0, +∞[→ H of the continuous dynamic (DINAM) which satisfies the Cauchy data x(0) = x 0 , ẋ(0) = x 1 .

Proof. The system (ii) in Proposition 2.1 can be written equivalently as

Ż(t) + F (Z(t)) = 0, Z(0) = (x 0 , y 0 )
where

Z(t) = (x(t), y(t)) ∈ H × H and F (x, y) = β f (∇f (x), 0) + β b B(x) + γ - 1 β f x + y, -1 - β b β f B(x) + 1 β f γ - 1 β f x + 1 β f y , y 0 = -x 1 -β f ∇f (x 0 ) -β b B(x 0 ) -γ - 1 β f x 0 . Therefore, F = ∇Φ + G where Φ : H × H → R is the convex differentiable function Φ(x, y) := β f f (x)
and

G : H × H → H × H G(x, y) := β b B(x) + γ - 1 β f x + y, -1 - β b β f B(x) + 1 β f γ - 1 β f x + 1 β f y
is a Lipschitz continuous map. Indeed, the Lipschitz continuity of G is a direct consequence of the Lipschitz continuity of B. The existence of a classical solution to

Ż(t) + ∇Φ(Z(t)) + G(Z(t)) = 0, Z(0) = (x 0 , y 0 )
follows from Brézis [START_REF] Brézis | Opérateurs maximaux monotones dans les espaces de Hilbert et équations d'évolution[END_REF]Proposition 3.12]. In fact, the proof of this result relies on a fixed point argument.

It consists in finding a fixed point of the mapping

u ∈ C([0, T ], H) → K(u) ∈ C([0, T ], H) where K(u) = w is the solution of ẇ(t) + ∇Φ(w(t)) = -G(u(t)), w(0) = (x 0 , y 0 ).
It is proved that the sequence of iterates (w n ) generated by the corresponding Picard iteration

ẇn+1 (t) + ∇Φ(w n+1 (t)) = -G(w n (t)), w n+1 (0) = (x 0 , y 0 ),
converges uniformly on [0, T ] to a fixed point of K. When returning to (DINAM), that's equation (i) of Proposition 2.1, we recover a strong solution. Precisely, ẋ is Lipschitz continuous on the bounded time intervals, and ẍ taken in the distribution sense is locally essentially bounded.

Remark 2.1 Note that when ∇f is supposed to be globally Lipschitz continuous, the above proof can be notably simplified, by just applying the classical Cauchy-Lipschitz theorem.

3 Asymptotic convergence properties of (DINAM)

In this section, we study the asymptotic behavior of the solution trajectories of (DINAM). For each solution trajectory t → x(t) of (DINAM) we show that the weak limit, w-lim t→+∞ x(t) = x ∞ exists, and satisfies

x ∞ ∈ S, where

S := {p ∈ H : ∇f (p) + B(p) = 0}.
Before stating our main result, notice that B(p) is uniquely defined for p ∈ S.

Lemma 3.1 B(p) is uniquely defined for p ∈ S, i.e.

p 1 ∈ S, p 2 ∈ S =⇒ B(p 1 ) = B(p 2 ).
Proof. Since p 1 ∈ S, p 2 ∈ S we have

∇f (p 1 ) + B(p 1 ) = ∇f (p 2 ) + B(p 2 ) = 0.
By monotonicity of ∇f we have

∇f (p 2 ) -∇f (p 1 ), p 2 -p 1 ≥ 0.
Replacing ∇f (p 1 ) by -B(p 1 ) and ∇f (p 2 ) by -B(p 2 ), we get

B(p 2 ) -B(p 1 ), p 2 -p 1 ≤ 0,
which by cocoercivity of B gives λ B(p 2 ) -B(p 1 ) 2 ≤ 0, and hence B(p 2 ) = B(p 1 ).

General case

The general line of the demonstration is close to one given by Attouch-Laszlo in [START_REF] Attouch | Continuous Newton-like Inertial Dynamics for Monotone Inclusions, Set-Valued and Variational Analysis[END_REF][START_REF] Attouch | Newton-like inertial dynamics and proximal algorithms governed by maximally monotone operators[END_REF]. A first major difference with the approach developed in [START_REF] Attouch | Continuous Newton-like Inertial Dynamics for Monotone Inclusions, Set-Valued and Variational Analysis[END_REF][START_REF] Attouch | Newton-like inertial dynamics and proximal algorithms governed by maximally monotone operators[END_REF] is that in our context thanks to the hypothesis of cocoercivity on the nonpotential part, we do not need to go through the Yosida regularization of the operators. A second difference is that we treat the potential and nonpotential operators in a differentiated way. These points are crucial for applications to numerical algorithms, because the computation of the Yosida regularization of the sum of the two operators is often out of reach numerically.

The following Theorem states the asymptotic convergence properties of (DINAM).

Theorem 3.1 Let B : H → H be a λ-cocoercive operator and f : H → R be a C 1 convex function whose gradient is Lipschitz continuous on the bounded sets. Suppose that S = (∇f + B) -1 (0) = ∅. Consider the evolution equation (DINAM) where the involved parameters satisfy the following conditions:

β f > 0 and 4λγ > (β b -β f ) 2 β f + 2 β b + 1 γ + 2 β b + 1 γ 2 + (β b -β f ) 2 γβ f .
Then, for any solution trajectory x : [0, +∞[→ H of (DINAM) the following properties are satisfied:

(i) (convergence) x(t) converges weakly, as t → +∞, to an element of S.

(ii) (integral estimates) Set A := B + ∇f and p ∈ S. Then,

+∞ 0 ẋ(t) 2 dt < +∞, +∞ 0 ẍ(t) 2 dt < +∞, +∞ 0 B(x(t)) -B(p) 2 dt < +∞, +∞ 0 d dt B(x(t)) 2 dt < +∞, +∞ 0 A(x(t)) 2 dt < +∞, and +∞ 0 d dt A(x(t)) 2 dt < +∞.
(iii) (pointwise estimates)

lim t→+∞ ẋ(t) = 0, lim t→+∞ B(x(t)) -B(p) = 0, lim t→+∞ A(x(t)) = 0,
where B(p) is uniquely defined for p ∈ S.

Proof. Lyapunov analysis. Set A := B + ∇f and

A β := β b B + β f ∇f . Take p ∈ S. Consider the function t ∈ [0, +∞[ → V p (t) ∈ R + defined by V p (t) := 1 2 x(t) -p + c ẋ(t) + A β (x(t)) -A β (p) 2 + δ 2 x(t) -p 2 , (3.1)
where c and δ are coefficients to adjust. Using the derivation chain rule for absolutely continuous functions (see [START_REF] Brézis | Analyse fonctionnelle[END_REF]Corollary VIII.10]) and (DINAM), we get

Vp (t) = ẋ(t) -c(γ ẋ + A(x(t)), x(t) -p + c ẋ(t) + A β (x(t)) -A β (p) + δ ẋ(t), x(t) -p . (3.2)
Setting δ := cγ -1 > 0, from (3.2) we obtain

Vp (t) = -cA(x(t)), x(t) -p + c (1 -cγ) ẋ(t) -cA(x(t)), ẋ(t) + A β (x(t)) -A β (p) . (3.3)
We have

c (1 -cγ) ẋ(t) -cA(x(t)), ẋ(t) + A β (x(t)) -A β (p) = c(1 -cγ) ẋ(t) 2 + c(1 -cγ) ẋ(t), A β (x(t)) -A β (p) -c 2 A(x(t)), ẋ(t) -c 2 A(x(t)), A β (x(t)) -A β (p) , = c(1 -cγ) ẋ(t) 2 -c 2 β b B(x(t)) -B(p) 2 -c 2 β f ∇f (x(t)) -∇f (p) 2 +[c(1 -cγ)β b -c 2 ] ẋ(t), B(x(t)) -B(p) + [c(1 -cγ)β f -c 2 ] ẋ(t), ∇f (x(t)) -∇f (p) -c 2 (β b + β f ) B(x(t)) -B(p), ∇f (x(t)) -∇f (p) . (3.4)
Using the fact that p ∈ S, ∇f is monotone, and B is λ-cocoercive, we have

-c A(x(t)), x(t) -p = -c A(x(t)) -A(p), x(t) -p = -c ∇f (x(t)) -∇f (p), x(t) -p -c B(x(t)) -B(p), x(t) -p ≤ -cλ B(x(t)) -B(p) 2 . (3.5)
From (3.3)-(3.5), we deduce that

Vp (t) ≤ -cδ ẋ(t) 2 -[c 2 β b + cλ] B(x(t)) -B(p) 2 -c 2 β f ∇f (x(t)) -∇f (p) 2 -[cδβ b + c 2 ] ẋ(t), B(x(t)) -B(p) -[cδβ f + c 2 ] ẋ(t), ∇f (x(t)) -∇f (p) -c 2 (β b + β f ) B(x(t)) -B(p), ∇f (x(t)) -∇f (p) . (3.6)
Let Γ : [0, +∞[→ R be the function defined by

Γ(t) := f (x(t)) -f (p) -∇f (p), x(t) -p ,
and E p : [0, +∞[→ R be the energy function given by

E p (t) := V p (t) + [cδβ f + c 2 ]Γ(t).
Since f is convex, we have Γ(t) ≥ 0, for all t ≥ 0. This implies E p (t) ≥ 0 for all t ≥ 0 as well.

We have,

Γ(t) = ẋ(t), ∇f (x(t)) -∇f (p) , (3.7) Ėp (t) = Vp (t) + [cδβ f + c 2 ] Γ(t).
(3.8)

By using (3.7) and (3.8), the equation (3.6) can be rewritten as

Ėp (t) + cδ ẋ(t) 2 + [c 2 β b + cλ] B(x(t)) -B(p) 2 + c 2 β f ∇f (x(t)) -∇f (p) 2 (3.9) +[cδβ b + c 2 ] ẋ(t), B(x(t)) -B(p) + c 2 (β b + β f ) B(x(t)) -B(p), ∇f (x(t)) -∇f (p) ≤ 0.
Let us eliminate the term ∇f (x(t))-∇f (p) from this relation by using the elementary algebraic inequality

c 2 β f ∇f (x(t)) -∇f (p) 2 + c 2 (β b + β f ) B(x(t)) -B(p), ∇f (x(t)) -∇f (p) ≥ - c 2 (β b + β f ) 2 4β f B(x(t)) -B(p) 2 .
We obtain

Ėp (t) + cδ ẋ(t) 2 + [c 2 β b + cλ - c 2 (β b + β f ) 2 4β f ] B(x(t)) -B(p) 2 +[cδβ b + c 2 ] ẋ(t), B(x(t)) -B(p) ≤ 0.
Equivalently

Ėp (t) + cS(t) ≤ 0, (3.10)
where

S(t) := δ ẋ(t) 2 + [δβ b + c] ẋ(t), B(x(t)) -B(p) + [cβ b + λ - c(β b + β f ) 2 4β f ] B(x(t)) -B(p) 2 . Set X(t) = ẋ(t) and Y (t) = B(x(t)) -B(p). We have S(t) = q(X(t), Y (t)) where q : H × H → R is the quadratic form q(X, Y ) := a X 2 + b X, Y + g Y 2 with a = δ, b = δβ b + c, and g = cβ b + λ - c(β b + β f ) 2 4β f = λ - c(β b -β f ) 2 4β f .
According to Lemma 7.3, and since a = δ = cγ -1 > 0, we have that q is positive definite if and only if

4ag -b 2 > 0. Equivalently 4δ λ - c(β b -β f ) 2 4β f -[δβ b + c] 2 > 0. (3.11)
Our aim is to find c such that cγ -1 > 0 and such that (3.11) is satisfied. Take δ := cγ -1 > 0 as a new variable. Equivalently, we must find δ > 0 such that

4δ λ - δ + 1 γ . (β b -β f ) 2 4β f -δβ b + δ + 1 γ 2 > 0.
After development and simplification we obtain

4λ > (β b -β f ) 2 γβ f + 2 γ β b + 1 γ + 1 γ 2 δ + β b + 1 γ 2 + (β b -β f ) 2 γβ f δ.
Therefore, we just need to assume that

4λ > (β b -β f ) 2 γβ f + 2 γ β b + 1 γ + inf δ>0 1 γ 2 δ + β b + 1 γ 2 + (β b -β f ) 2 γβ f δ .
Elementary optimization argument gives that

inf δ>0 C δ + Dδ = 2 √ CD.
Therefore we end up with the condition Note that cγ = 1 + δ and δ > 0 implies c > 0. Therefore, there exist positive real numbers c, µ such that

4λ > (β b -β f ) 2 γβ f + 2 γ β b + 1 γ + 2 γ β b + 1 γ 2 + (β b -β f ) 2 γβ f . Equivalently 4λγ > (β b -β f ) 2 β f + 2 β b + 1 γ + 2 β b + 1 γ 2 + (β b -β f ) 2 γβ f . ( 3
Ėp (t) + cµ ẋ(t) 2 + cµ B(x(t)) -B(p) 2 ≤ 0. (3.13)
Estimates. We have shown that there exist positive real numbers c, µ such that, for all t ≥ 0

Ėp (t) + cµ ẋ(t) 2 + cµ B(x(t)) -B(p) 2 ≤ 0. (3.14)
By integrating (3.14) on an interval [0, t], we obtain that for all t ≥ 0,

E p (t) + cµ t 0 ẋ(s) 2 ds + cµ t 0 B(x(s)) -B(p) 2 ds ≤ E p (0). (3.15)
From (3.15) and the definition of E p we immediately deduce

sup t≥0 x(t) -p < +∞, (3.16
) Let us return to (3.9). We recall that

sup t≥0 x(t) -p + c( ẋ(t) + A β (x(t)) -A β (p)) < +∞, (3.17
Ėp (t) + cδ ẋ(t) 2 + [c 2 β b + cλ] B(x(t)) -B(p) 2 + c 2 β f ∇f (x(t)) -∇f (p) 2 (3.20) +[cδβ b + c 2 ] ẋ(t), B(x(t)) -B(p) + c 2 (β b + β f ) B(x(t)) -B(p), ∇f (x(t)) -∇f (p) ≤ 0.
After integration on [0, t], and by using the integral estimates +∞ 0 ẋ(t) 2 dt < +∞, and +∞ 0 B(x(t))-B(p) 2 dt < +∞, we obtain the existence of a constant C > 0 such that

c 2 β f t 0 ∇f (x(s)) -∇f (p) 2 ds ≤ C + c 2 (β b + β f ) t 0 B(x(s)) -B(p) ∇f (x(s)) -∇f (p) ds.
Therefore, for any > 0, we have

c 2 β f t 0 ∇f (x(s)) -∇f (p) 2 ds ≤ C + c 2 (β b + β f ) t 0 1 4 B(x(s)) -B(p) 2 + ∇f (x(s)) -∇f (p) 2 ds.
By taking > 0 such that Moreover, we also have

β f > (β b + β f ) which is always possible since β f > 0, we conclude +∞ 0 ∇f (x(t)) -∇f (p) 2 dt < +∞.

Combining this with

+∞ 0 A β (x(t)) -A β (p) 2 dt = +∞ 0 β f (∇f (x(t)) -∇f (p)) + β b (B(x(t)) -B(p)) 2 dt ≤ (β 2 f + β 2 b ) +∞ 0 ∇f (x(t)) -∇f (p) 2 + B(x(t)) -B(p) 2 dt < +∞. ( 3
L := L R + 1 λ . Therefore d dt A(x(t)) ≤ d dt A(x(t)) ≤ L ẋ(t)
, for all t ≥ 0. Similarly, we also have

+∞ 0 d dt B(x(t)) 2 dt < +∞.
By using (DINAM) we have

ẍ(t) = -γ ẋ(t) -A(x(t)) - d dt A β (x(t)) = -γ ẋ(t) -A(x(t)) -β f d dt A(x(t)) -(β b -β f ) d dt B(x(t)).
Since the second member of the above equality belongs to L 2 (0, +∞; H), we finally get +∞ 0 ẍ(t) 2 dt < +∞.

The limit. To prove the existence of the weak limit of x(t), we use Opial's lemma (see [START_REF] Peypouquet | Evolution Equations for Maximal Monotone Operators: Asymptotic Analysis in Continuous and Discrete Time[END_REF] for more details). Given p ∈ S, let us consider the anchor function defined by, for every t ∈ [0, +∞[

q p (t) := 1 2 x(t) -p 2 .
From qp (t) = ẋ(t), x(t) -p and qp (t) = ẋ(t) 2 + ẍ(t), x(t) -p , we obtain

qp (t) + γ qp (t) = ẋ(t) 2 + ẍ(t) + γ ẋ(t), x(t) -p = ẋ(t) 2 -A(x(t)) + d dt A β (x(t)), x(t) -p ≤ ẋ(t) 2 - d dt A β (x(t)), x(t) -p .
Equivalently,

qp (t) + γ qp (t) + d dt A β (x(t)), x(t) -p ≤ ẋ(t) 2 . (3.26)
According to the derivation formula for a product, we can rewrite (3.26) as follows

qp (t) + γ qp (t) + d dt A β (x(t)) -A β (p), x(t) -p ≤ ẋ(t) 2 + A β (x(t)) -A β (p), ẋ(t) .
By Cauchy-Lipschitz inequality we get

qp (t) + γ qp (t) + d dt A β (x(t)) -A β (p), x(t) -p ≤ ẋ(t) 2 + A β (x(t)) -A β (p) ẋ(t) . (3.27)
Then note that the second member of (3.27)

g(t) := ẋ(t) 2 + A β (x(t)) -A β (p) ẋ(t)
is nonnegative and belongs to L 1 (0, +∞). Indeed, we have

+∞ 0 A β (x(t)) -A β (p) ẋ(t) dt ≤ 1 2 +∞ 0 A β (x(t)) -A β (p) 2 dt + 1 2 +∞ 0 ẋ(t) 2 dt.
Using (3.18) and (3.22), we deduce that +∞ 0 g(t)dt < +∞.

Note that the left member of (3.27) can be rewritten as a derivative of a function, precisely

qp (t) + γ qp (t) + d dt A β (x(t)) -A β (p), x(t) -p = ḣ(t) with h(t) = qp (t) + γq p (t) + A β (x(t)) -A β (p), x(t) -p . (3.28)
So we have ḣ(t) ≤ g(t), for every t ≥ 0.

Let us prove that the function h given in (3.28) is bounded below by some constant. Indeed, since the terms q p (t) and A β (x(t)) -A β (p), x(t) -p are nonnegative, we have

h(t) ≥ qp (t) ≥ -ẋ(t) x(t) -p .
Using (3.16) and the fact that ẋ(•) is bounded, we deduce that there exists some m ∈ R such that h(t) ≥ m for every t ≥ 0.

Let us introduce the real-valued function ϕ : R + → R, t → ϕ(t) defined by

ϕ(t) = h(t) - t 0 g(s)ds.
We have ϕ (t) = ḣ(t) -g(t) ≤ 0. Hence, the function ϕ is nonincreasing on [0, +∞[. This classically implies that the limit of ϕ exists as t → +∞. Since g ∈ L 1 (0, +∞), we deduce that lim t→+∞ h(t) exists.

Using the fact that A β (x(t)) -A β (p), x(t) -p tends to zero as t → +∞ (a consequence of (3.25) and x(•) bounded), we obtain qp (t) + γq p (t) = θ(t)

with limit of θ(t) exists as t → +∞. The existence of the limit of q p then follows from a classical general result concerning the convergence of evolution equations governed by strongly monotone operators (here γ Id, see Theorem 3.9 page 88 in [START_REF] Brézis | Opérateurs maximaux monotones dans les espaces de Hilbert et équations d'évolution[END_REF]). This means that for all p ∈ S lim t→+∞ x(t) -p exists.

To complete the proof via the Opial's lemma, we need to show that every weak sequential cluster point of x(t) belongs to S. Let t n → +∞ such that x(t n )

x * , n → +∞. We have A(x(t n )) → 0 strongly in H and x(t n )

x * weakly in H.

From the closedness property of the graph of the maximally monotone operator A in w -H × s -H, we deduce that A(x * ) = 0, that is x * ∈ S. Consequently, x(t) converges weakly to an element of S as t goes to +∞. The proof of Theorem 3.1 is thereby completed.

Case β b = β f

Let us specialize the previous results in the case β b = β f . We set β b = β f := β > 0, and A := ∇f + B.

We thus consider the evolution system

(DINAM) ẍ(t) + γ ẋ(t) + A(x(t)) + β d dt (A(x(t))) = 0, t ≥ 0.
The existence of strong global solutions to this system is guaranteed by Theorem 2.1. The convergence properties as t → +∞ of the solution trajectories generated by this system is a consequence of Theorem 3.1 and are given below. Remark 3.1 It is worth stating the result of Corollary 3.1 apart because this is an important case. This also makes it possible to highlight this result compared to the existing literature for second-order dissipative evolution systems involving cocoercive operators. Indeed, letting β go to zero in (3.29) gives the condition

λγ 2 > 1, (3.30)
introduced by Attouch-Maingé in [START_REF] Attouch | Asymptotic behavior of second order dissipative evolution equations combining potential with nonpotential effects[END_REF] to study the second order dynamic (1.3) without geometric damping. With respect to [START_REF] Attouch | Asymptotic behavior of second order dissipative evolution equations combining potential with nonpotential effects[END_REF], the introduction of the geometric damping, i.e., taking β > 0, provides some useful additional estimates.

Numerical illustrations

In this section, we give some numerical illustrations by using a temporal discretization of the dynamic (DINAM). Let us recall the condensed formulation of (DINAM)

ẍ(t) + γ ẋ(t) + A(x(t)) + d dt (A β (x(t))) = 0, (DINAM)
where A := ∇f + B and A β := β b B + β f ∇f . Take a fixed time step h > 0, and consider the following implicit finite-difference scheme for (DINAM):

1 h 2 (x k+1 -2x k + x k-1 ) + γ h (x k+1 -x k ) + 1 h (A β (x k+1 ) -A β (x k )) + A(x k+1 ) = 0. (4.1)
After expanding (4.1), we obtain

x k+1 + h 2 1 + γh A(x k+1 ) + h 1 + γh A β (x k+1 ) = x k + 1 1 + γh (x k -x k-1 ) + h 1 + γh A β (x k ). (4.2)
Set s := h 1 + γh and α := 1 1 + γh . So we have

x k+1 + sA h (x k+1 ) = y k , (4.3)
where

A h = (h + β b )B + (h + β f )∇f, (4.4) y k = x k + α(x k -x k-1 ) + sA β (x k ). (4.5)
From ( 4.3) we get

x k+1 = (Id +sA h ) -1 (y k ). (4.6)
Combining (4.3) and (4.6), we obtain the following algorithm (DINAAM):

Initialize:

x 0 ∈ H, x 1 ∈ H α = 1 1 + γh , s = h 1 + γh , y k = x k + α(x k -x k-1 ) + sA β (x k ) , x k+1 = (Id +sA h ) -1 (y k ). (4.7) Remark 4.1 (i)
The convergence analysis of the algorithm (DINAAM) will be postponed to an other research investigation. In the current version, we focus only on the continuous dynamic (DINAM) and its asymptotic convergence. The numerical experiments below are given for illustrative purposes.

(ii) A general method to generate monotone cocoercive operators which are not gradients of convex functions is to start from a linear skew symmetric operator A and then take its Yosida approximation A λ . As a model situation, take H = R 2 and start from A equal to the rotation of angle π 2 . We have

A = 0 -1 1 0 .
An elementary computation gives that, for any λ > 0

A λ = 1 1 + λ 2 λ -1 1 λ ,
which is therefore λ-cocoercive. As a consequence, for λ = 1 we obtain that the matrix

B = 1 -1 1 1 is 1 2 -cocoercive.
With these basic blocks, one can easily construct many other cocoercive operators which are not potential operators. For that, use Lemma 7.1 which gives that the sum of two cocoercive operators is still cocoercive, and therefore the set of cocoercive operators is a convex cone.

Example 4.1 Let us start this section by a simple illustrative example in R 2 . We take H = R 2 equipped with the usual Euclidean structure. Let us consider B as a linear operator whose matrix in the canonical basis of R 2 is defined by B = A λ for λ = 5. According to Remark 4.1, we can check that B is λcocoercive with λ = 5 and that B is a nonpotential operator. To observe the classical oscillations, in the heavy ball with friction, we take f : R 2 → R defined by

f (x 1 , x 2 ) = 50x 2 2 .
We set γ = 0.9. It is clear that f is convex but not strongly convex. We study 3 cases: (1)

β b = 1, β f = 0.5, ( 2
)
β b = 0.5, β f = 1 and (3) β b = β f = 0.5.
As a straight application of Theorem 3.1, we obtain that the trajectory x(t) generated by (DINAM) converges to x ∞ , where x ∞ ∈ S = (B + ∇f ) -1 (0) = {0}. The trajectory obtained by using Matlab is depicted in Figure 1, where we represent the components x 1 (t) and x 2 (t) in red and blue respectively. given by f

(x) = 1 2 M x -b 2 , where M ∈ R m×n and b ∈ R m . We have ∇f (x) = M (M x -b), ∇ 2 f (x) = M M.
Since M M is positive semidefinite for any matrix M , the quadratic function f is convex. Furthermore, if M has full column rank, i.e. rank(M) = n, then M M is positive definite. Therefore f is strongly convex. Take B defined as below

B =          1 -1 0 • • • 0 1 1 0 • • • . . . 0 0 1 • • • . . . . . . . . . . . . . . . 0 . . . . . . . . . 1          ∈ R n×n .
Then, B is cocoercive. Indeed, for any x, y ∈ R n , we have

Bx -By, x -y = x 1 -y 1 2 + x 2 -y 2 2 + • • • + x n -y n 2 ≥ 1 2 2( x 1 -y 1 2 + x 2 -y 2 2 ) + x 3 -y 3 2 + • • • + x n -y n 2 = 1 2 Bx -By 2 .
If the matrix M has not full column rank with M M + B nonsingular, then

B(x) + ∇f (x) = 0 if and only if x = (M M + B) -1 M b.
In our experiment, we pick M a random 10 × 100 matrix which has not full column rank. Set γ = 3, β b = 1, β f = 1 and the operator B as presented above. Thanks to Corollary 3.1, we conclude that the trajectory x(t) generated by the system (DINAM) converges to x ∞ = (M M + B) -1 M b. Implementing the algorithm (DINAAM) in Matlab, we obtain the plot of k versus the norm of B(x k ) + ∇f (x k ). Similarly, we study several cases by changing the parameters β b , β f . This is depicted in Fig 3. Before ending this part, we discuss an application of our model to dynamical games.

The following example is taken from Attouch-Maingé [START_REF] Attouch | Asymptotic behavior of second order dissipative evolution equations combining potential with nonpotential effects[END_REF] and adapted to our context.

Example 4.3

We make the following standing assumptions:

(i) H = X 1 × X 2 is the Cartesian product of two Hilbert spaces equipped with norms • X 1 and • X 2 respectively. In which, x = (x 1 , x 2 ), with x 1 ∈ X 1 and x 2 ∈ X 2 , stands for an element in H;

(ii) f : X 1 × X 2 → R is convex function whose gradient is Lipschitz continuous on bounded sets;

(iii) B = (∇ x 1 L, -∇ x 2 L
) is the maximally monotone operator which is attached to a smooth convexconcave function L : X 1 × X 2 → R. The operator B is assumed to be λ-cocoercive with λ > 0.

In our setting, with x(t) = (x 1 (t), x 2 (t)) the system (DINAM) is written

               ẍ1 (t) + γ ẋ1 (t) + ∇ x 1 f (x 1 (t), x 2 (t)) + ∇ x 1 L(x 1 (t), x 2 (t)) + β f d dt (∇ x 1 f (x 1 (t), x 2 (t))) + β b d dt (∇ x 1 L(x 1 (t), x 2 (t))) = 0 ẍ2 (t) + γ ẋ2 (t) + ∇ x 2 f (x 1 (t), x 2 (t)) -∇ x 2 L(x 1 (t), x 2 (t)) + β f d dt (∇ x 2 f (x 1 (t), x 2 (t))) -β b d dt (∇ x 2 L(x 1 (t), x 2 (t))) = 0. (4.8)
According to Theorem 3.1,

x(t) x ∞ = (x 1,∞ , x 2,∞ ) weakly in H, where (x 1,∞ , x 2,∞ ) is solution of ∇ x 1 f (x 1 (t), x 2 (t)) + ∇ x 1 L(x 1 (t), x 2 (t)) = 0 ∇ x 2 f (x 1 (t), x 2 (t)) -∇ x 2 L(x 1 (t), x 2 (t)) = 0. (4.9)
Structured systems such as (4.9) contain both potential and nonpotential terms which are often present in decision sciences and physics. In game theory, (4.9) describles Nash equilibria of the normal form game with two players 1, 2 whose static loss functions are respectively given by

F 1 : (x 1 , x 2 ) ∈ X 1 × X 2 → F 1 (x 1 , x 2 ) = f (x 1 , x 2 ) + L(x 1 , x 2 ) F 2 : (x 1 , x 2 ) ∈ X 1 × X 2 → F 2 (x 1 , x 2 ) = f (x 1 , x 2 ) -L(x 1 , x 2 ). (4.10)
f (•, •) is their joint convex payoff, and L is a convex-concave payoff with zero-sum rule. For more details, we refer the reader to [START_REF] Attouch | Asymptotic behavior of second order dissipative evolution equations combining potential with nonpotential effects[END_REF]. As an example, take X 1 = X 2 = R, and L : R 2 → R given by L

(x) = 1 2 (x 2 1 - 2x 1 x 2 -x 2 2 ). Then B = (∇ x 1 L, -∇ x 2 L) = 1 -1 1 1 . Pick f (x) = 1 2 (3x 2 1 -2x 1 x 2 +x 2 2 )-x 1 -2x 2 .
The Nash equilibria described in (4.9) can be solved by using (DINAAM). Take γ = 3, β b = 0.5, β f = 0.5 and x 0 = (1, -1), ẋ0 = (-10, 10) as initial conditions, then the numerical solution for (DINAM) converges to x ∞ = ( 3 4 , 1) which is the solution of (4.9) as well. The numerical trajectories and phase portrait of our model applied to dynamical games are depicted in Figure 4.

The nonsmooth case

The equivalence obtained in Proposition 2.1 between (DINAM) and a first-order evolution system in time and space allows a natural extension of both our theoretical and numerical results to the case of a convex, lower semicontinuous and proper function f : H → R ∪ {+∞}. It suffices to replace the gradient of f by the convex subdifferential ∂f . We recall that the subdifferential of f at x ∈ H is defined by

∂f (x) = {z ∈ H : z, ξ -x ≤ f (ξ) -f (x) for every ξ ∈ H},
and the domain of f is equal to domf = {x ∈ H : f (x) < +∞}. This leads to consider the system (g-DINAM)

       ẋ(t) + β f ∂f (x(t)) + β b B(x(t)) + γ - 1 β f x(t) + y(t) 0; ẏ(t) -1 - β b β f B(x(t)) + 1 β f γ - 1 β f x(t) + 1 β f y(t) = 0.
The prefix g in front of (DINAM) stands for generalized. Note that the first equation of (g-DINAM) is now a differential inclusion, because of the possiblity for ∂f (x(t)) to be multivalued. By taking f = f 0 + δ C , where δ C is the indicator function of a constraint set C, the system (g-DINAM) allows to model damped inelastic shocks in mechanics and decision sciences, see [START_REF] Attouch | A second-order differential system with Hessian-driven damping; Application to nonelastic shock laws[END_REF]. The original aspect comes from the fact that (g-DINAM) now involves both potential driven forces (attached to f 0 ) and nonpotential driven forces (attached to B). As we will see, taking into account shocks created by nonpotential driving forces is a source of difficulties.

Let us first establish the existence and uniqueness of the solution trajectory of the Cauchy problem. Proof. The proof is parallel to that of Theorem 2.1. The system (g-DINAM) can be equivalently written as

Ż(t) + ∂Φ(Z(t)) + G(Z(t)) 0, Z(0) = (x 0 , y 0 ), (5.1)
where the function Φ(Z) = Φ(x, y) = β f f (x) is now convex lower semicontinuous proper on H × H. The operator G is unchanged, and is globally Lipschitz continuous. The above equation falls under the setting of the Lipschitz perturbation of an evolution system governed by the subdifferential of a convex function. The existence and uniqueness of the strong solution to (5.1) follows from Brézis [23, Proposition 3.12], and the fact that (x 0 , y 0 ) ∈ domΦ.

The solution set S is defined by

S := {p ∈ H : ∂f (p) + B(p) 0}.
Before stating our main result, notice that B(p) is uniquely defined for p ∈ S. Proof. The proof is similar to that of Lemma 3.1. It is based on the monotonicity of the subdifferential of f and the cocoercivity of the operator B.

For sake of simplicity, we give a detailed proof in the case β f = β b = β > 0. The system (g-DINAM) takes the simpler form:

(g-DINAM)        ẋ(t) + β∂f (x(t)) + βB(x(t)) + γ - 1 β x(t) + y(t) 0; ẏ(t) + 1 β f γ - 1 β x(t) + 1 β y(t) = 0.
To formulate the convergence results and the corresponding estimates, we write the first equation of (g-DINAM) as follows

ẋ(t) + βξ(t) + βB(x(t)) + γ - 1 β x(t) + y(t) = 0, (5.2)
where ξ(t) ∈ ∂f (x(t)) and we set A(x(t)) = ξ(t) + B(x(t)). Proof. Let us adapt the Lyapunov analysis developed in the previous sections to the case where f is nonsmooth. We have to pay attention to the following points. First, we must invoke the (generalized) chain rule for derivatives over curves (see [START_REF] Brézis | Opérateurs maximaux monotones dans les espaces de Hilbert et équations d'évolution[END_REF]Lemme 3.3]), that is, for a.e t ≥ 0

d dt f (x(t)) = ξ(t), ẋ(t) .
The second ingredient is the validity of the subdifferential inequality for convex functions. So, let us consider the function t ∈ [0, +∞[ → E p (t) ∈ R + defined by

E p (t) := 1 2 x(t) -p + c ẋ(t) + A(x(t)) 2 + δ 2 x(t) -p 2 + [cδβ f + c 2 ]Γ(t). (5.4)
When derivating E p (t), we use the formulation (g-DINAM) which gives

ẋ(t) + βA(x(t)) = -γ - 1 β x(t) -y(t)
which allows to derivate ẋ(t) + βA(x(t)), and obtain similar formulas as in the smooth case. Then a close examination of the Lyapunov analysis shows that we can obtain the additional estimate ∞ 0 A(x(t)), x(t) -p dt < +∞.

(5.5)

Set 0 ∈ ∂f (p) + B(p). To obtain (5.5), we return to (3.5), and consider the following minorization that we split into a sum with coefficients and 1 -(where > 0 will be taken small enough)

c A(x(t)), x(t) -p = c A(x(t)), x(t) -p + c(1 -) A(x(t)) -Ap, x(t) -p ≥ c A(x(t)), x(t) -p + c(1 -) B(x(t)) -B(p), x(t) -p ≥ c A(x(t)), x(t) -p + c(1 -)λ B(x(t)) -B(p) 2 .
(5.6)

Note that in the second above inequality we have used the monotonicity of ∂f . So the proof continues with λ replaced by (1 -)λ. This does not change the conditions on the parameters since in our assumptions, the inequality λγ > β + 1 γ is strict, it is still satisfied by (1 -)λ when is taken small enough. So, after integrating the resulting strict Lyapunov inequality, we obtain the supplementary property (5.5). Until (3.22) the proof is essentially the same as in the case of a smooth function f . We obtain the integral estimates

+∞ 0 ẋ(t) 2 dt < +∞, +∞ 0 B(x(t)) -B(p) 2 dt < +∞, +∞ 0 A(x(t)) 2 dt < +∞.
But then we can no longer invoke the Lipschitz continuity on the bounded sets of ∇f . To overcome this difficulty, we modify the end of the proof as follows. Recall that given p ∈ S, the anchor function is defined by, for every t ∈ [0, +∞[

q p (t) := 1 2 x(t) -p 2 ,
and that we need to prove that the limit of the anchor functions exists, as t → +∞. The idea is to play on the fact that we have in hand a whole collection of Lyapunov functions, parametrized by the coefficient c. Recall that we have obtained that the limit of E p (t) exists as t → +∞, and this is satisfied for a whole interval of values of c. So, for such c, the limit of W c (t) := 1 cδβ + c 2 E p (t) as t → +∞ exists, where

W c (t) = 1 2(cδβ + c 2 ) x(t) -p + c ẋ(t) + βA(x(t)) 2 + δ 2(cδβ + c 2 ) x(t) -p 2 + Γ(t).
Take two such values of c, let c 1 and c 2 , and make the difference (recall that δ = cγ -1). We obtain As a consequence of (5.5) and of the previous estimates, we have that the limit of the two above integrals exists as t → +∞. Therefore, according to the convergence of W (t) we obtain that lim t→+∞ γw(t) + d dt w(t) exists.

W c 1 (t) -W c 2 (t) = 1 (c 1 γ -1)β + c 1 - 1 (c 2 γ -1)β + c 2 W (
The existence of the limit of w follows from a classical general result concerning the convergence of evolution equations governed by strongly monotone operators (here γ Id, see Theorem 3.9 page 88 in [START_REF] Brézis | Opérateurs maximaux monotones dans les espaces de Hilbert et équations d'évolution[END_REF]). In turn, using the same argument as above, we obtain that for all p ∈ S lim t→+∞ x(t) -p exists.

As in the smooth case, the strong convergence of A(x(t)) 2 dt < +∞, which implies that A(x(t)) converges strongly to zero in an "essential" way. According to Opial's lemma, this allows to complete the convergence proof as in the smooth case. This is a seemingly difficult question to examine in the future.

(ii) A particular situation is the case γ = 1 β , in wich case the system (g-DINAM) can be written in an equivalent way u(t) + γu(t) = 0 where ẋ(t) + βA(x(t)) u(t).

The convergence of the trajectory t → x(t) is then a consequence of the convergence of the semigroup generated by the sum of a cocoercive operator with the subdifferential of a convex lower semicontinuous and proper function, see Abbas-Attouch [START_REF] Abbas | Dynamical systems and forward-backward algorithms associated with the sum of a convex subdifferential and a monotone cocoercive operator[END_REF]. Note that is this case the condition for the convergence of the trajectories generated by (g-DINAM) does not depend any more on the cocoercivity parameter λ.

Conclusion, perspectives

In this paper, in a general real Hilbert space setting, we investigated a dynamic inertial Newton method for solving additively structured monotone problems. The dynamic is driven by the sum of two monotone operators with distinct properties: the potential component is the gradient of a continuously differentiable convex function f , and the nonpotential is a monotone and cocoercive operator B. The geometric damping is controlled by the Hessian of the potential f and by a Newton-type correction term attached to B. The well-posedness of the Cauchy problem is shown as well as the asymptotic convergence properties of the trajectories generated by the continuous dynamic. The convergence analysis is carried out through the parameters β f and β b attached to the geometric dampings as well as the parameters γ and λ (the viscous damping and the coefficient of cocoercivity respectively). The introduction of geometric damping makes it possible to control and attenuate the oscillations known for viscous damping of inertial systems, giving rise to faster numerical methods. It would be interesting to extend the analysis for both the continuous dynamic and its discretization to the case of an asymptotic vanishing damping γ(t) = α t , with α > 0 as in [START_REF] Su | A differential equation for modeling Nesterov's accelerated gradient method[END_REF]. This is a decisive step towards proposing faster algorithms for solving structured monotone inclusions, which are connected to the accelerated gradient method of Nesterov. The study of the corresponding splitting methods is also an important topic which needs further investigations. In fact, replacing ∇f by a general maximally monotone operator A, whose resolvent can be computed easily, would be an interesting problem to study an inertial forward-backward algorithm with Hessian-driven damping for solving structured monotone inclusions of the form: Ax + Bx 0. This is beyond the scope of the current manuscript and will be the subject of a future separate work.

Appendix

Technical lemmas

Let us show that the sum of two cocercive operators is still cocoercive. For further properties concerning cocoercive operators see [START_REF] Bauschke | Convex Analysis and Monotone Operator Theory in Hilbert spaces[END_REF].

Lemma 7.1 Let T 1 , T 2 : H → H be two cocoercive operators with respective cocoercivity coefficients λ 1 , λ 2 > 0. Then, T := T 1 + T 2 : H → H is λ-cocoercive with λ = λ 1 λ 2 λ 1 +λ 2 . Proof. According to the cocoercivity assumptions of T 1 and T 2 , we have T 1 y -T 1 x, y -x ≥ λ 1 T 1 y -T 1 x 2 , ∀x, y ∈ H, T 2 y -T 2 x, y -x ≥ λ 2 T 2 y -T 2 x 2 , ∀x, y ∈ H.

Let us show that the sum T = T 1 + T 2 is still cocoercive. Using elementary computation in Hilbert spaces, for all x, y ∈ H, we have T y -T x 2 = T 1 y -T 1 x + T 2 y -T 2 x 2 = T 1 y -T 1 x 2 + T 2 y -T 2 x 2 + 2 T 1 y -T 1 x, T 2 y -T 2 x

≤ T 1 y -T 1 x 2 + T 2 y -T 2 x 2 + λ 1 λ 2 T 1 y -T 1 x 2 + λ 2 λ 1 T 2 y -T 2 x 2 = λ -1 1 + λ -1 2 λ 1 T 1 y -T 1 x 2 + λ 2 T 2 y -T 2 x 2 .
Since T 1 , T 2 are cocoercive, we deduce that

T y -T x 2 ≤ λ -1 1 + λ -1 2 ( T 1 y -T 1 x, y -x + T 2 y -T 2 x, y -x ) = λ -1 1 + λ -1 2
T y -T x, y -x .

Equivalently,

T y -T x, y -x ≥ λ 1 λ 2 λ 1 + λ 2 T y -T x 2 , ∀x, y ∈ H.

So, T is still λ-cocoercive with λ = λ 1 λ 2 λ 1 +λ 2 > 0. Let us show that this estimate is sharp. Take T 1 : H → H, x → λ -1 1 x and T 2 : H → H, x → λ -1 2 x. It is easy to check that T 1 , T 2 are two cocoercive operators with cocoercivity coefficients λ 1 , λ 2 respectively. Then their sum operator is equal to T x = λ -1 1 + λ -1 2

x = λ -1 x with λ = λ 1 λ 2 λ 1 +λ 2 , and hence is λ cocoercive. This shows that we cannot obtain a better estimate.

The next lemma is a classical result in integration theory.

. 12 )

 12 When β b = β f = β we recover the condition λγ > β + 1 γ .

) +∞ 0 ẋ

 0 (t) 2 dt < +∞, (3.18) +∞ 0 B(x(t)) -B(p) 2 dt < +∞. (3.19)

+∞ 0 B

 0 (x(t)) -B(p) 2 dt < +∞, it follows immediately +∞ 0 A(x(t)) -A(p) 2 dt < +∞. (3.21)

. 22 )

 22 According to(3.16) the trajectory x(•) is bounded. Set R := sup t≥0 x(t) . By assumption, ∇f is Lipschitz continuous on the bounded sets. Let L R < +∞ be the Lipschitz constant of ∇f on B(0, R). Since B is λ-cocoercive, it is 1 λ -Lipschitz continuous. Therefore A is L-Lipschitz continuous on the trajectory with

(3. 23 )

 23 Using(3.21) and(3.23) we deduce that u(t) := A(x(t)) satisfies the condition of Lemma 7.2 (with p = 2 and r = 2). Therefore, lim t→+∞ A(x(t)) = 0. (3.24) Likewise, according to (3.22) we have lim t→+∞ A β (x(t)) -A β (p) = 0. (3.25) Returning to (3.17)-(3.18), we deduce that ẋ(•) is bounded and lim t→+∞ ẋ(t) = 0. By using the same argument as in (3.23), we obtain that d dt A β (x(t)) is bounded. From (3.23) we also get that

Corollary 3 . 1

 31 Let B : H → H be a λ-cocoercive operator and f : H → R be a C 1 convex function whose gradient is Lipschitz continuous on the bounded sets. Suppose that the solution set S = (∇f + B) -1 (0) = ∅. Consider the evolution equation (DINAM) where A = ∇f + B, β b = β f := β > 0 and where the involved parameters satisfy the following conditions γ > 0, β > 0, and λγ > β + 1 γ . (3.29)Then, for any trajectory solution x : [0, +∞[→ H of (DINAM) the following properties are satisfied:(i) (convergence) The trajectory x(•) is bounded, and x(t) converges weakly, as t → +∞, to an element x * ∈ S.(ii) (integral estimate) ) = 0, and lim t→+∞ A(x(t)) = 0.

  Case β b = 1, β f = 0.5.

  Case β b = 0.5, β f = 1.

  Case β b = 0.5, β f = 0.5.

Figure 1 : 5 ,Figure 2 :Example 4 . 2

 15242 Figure 1: Trajectories of (DINAM) for different values of the parameters β b , β f .

βb = 1 ,Figure 3 :

 13 Figure 3: The behaviour of (DINAAM) for high dimension problem.

Figure 4 :

 4 Figure 4: An application of (DINAAM) to dynamical games: trajectories (a) and phase portrait (b).

Theorem 5 . 1

 51 Let f : H → R ∪ {+∞} be a convex, lower semicontinuous and proper function. Suppose that β f > 0 and β b ≥ 0. Then, for any (x 0 , x 1 ) ∈ domf × H, there exists a unique strong global solution x : [0, +∞[→ H of (g-DINAM) which satisfies the Cauchy data x(0) = x 0 , ẋ(0) = x 1 .

Lemma 5 . 1 B

 51 (p) is uniquely defined for p ∈ S, i.e.p 1 ∈ S, p 2 ∈ S =⇒ B(p 1 ) = B(p 2 ).

Theorem 5 . 2 3 )

 523 Let B : H → H be a λ-cocoercive operator. Let f : H → R ∪ {+∞} be a convex, lower semicontinuous, proper function. Suppose that S = {p ∈ H : 0 ∈ ∂f (p) + B(p)} = ∅. Consider the evolution equation (g-DINAM) where the parameters satisfy the conditions: β f = β b = β > 0 and γ > 0, β > 0 and λγ > β Then, for any solution trajectory x : [0, +∞[→ H of (g-DINAM) the following properties are satisfied: (i) (integral estimates) Set A(x(t)) := ξ(t) + B(x(t)), with ξ(t) ∈ ∂f (x(t)) as defined in (5.2) and p ∈ S. Then, t)) -B(p) 2 dt < +∞, +∞ 0 A(x(t)) 2 dt < +∞, ∞ 0 A(x(t)), x(t) -p dt < +∞. (ii) (convergence) For any p ∈ S, 1. lim t→+∞ x(t) -p exists. 2. lim t→+∞ B(x(t)) -B(p) = 0, where B(p) is uniquely defined for p ∈ S.

i.e.B is not supposed to be equal to the gradient of a given function.

for almost every t > 0. Then lim t→∞ u(t) = 0.

In the proof of Theorem 3.1, we use the following elementary result concerning positive quadratic forms.

Lemma 7.3 Let a, b, c be three real numbers. The quadratic form q :

where the positive real number µ := 1 2 a + c -(a -c) 2 + 4b 2 is the smallest eigenvalue of the positive symetric matrix associated with q.