Dr Thibault De Swarte
email: thibault.deswarte@enst-bretagne.fr

Bernard Marquet
email: bernard.marquet@orange-ftgroup.com

Potential Space as a source of creativity and terrifying anxiety Teens, students and their domestic uses of multimedia: second self, potential space or division of the subject ?

Keywords: Potential space, personal digital world, Creativity, digital second self, structure of the subject

. On the basis of our empirical material, can we talk about an "online" second self or a more or less antagonistic "on-screen" second self with our "original" self? The second part of the text will look at the potential space hypothesis: what is it about the domestic multimedia spaces that we looked at that are also transitional spaces in which the physical identity of each person is played, replayed and rebuilt? These two sections are an implicit series of theoretical preoccupations: does the "subject" build himself in the virtual world or not? Is it necessary to deconstruct the subject? On the contrary, should we deconstruct sociological, psychological and psychoanalytical analysis categories in order to find out more about multimedia culture? Without having a successfully completed form and by performing psychoanalysis, these theoretical reflections are eager to contribute towards achieving a deeper understanding of how ICT's (Information and Communication Technologies) are used.

Introduction

Our initial research objective was empirical, ethnographical and monographical. How could we look at how young people use multimedia at home? How do they build their own domestic multimedia space? How do they build this space whilst building themselves as people?

Past studies have also made us aware of the psychological and psychoanalytical dimension behind how this technology is used. Of course, if one is to use a computer, it is obviously going to be for "practical reasons" such as buying a train ticket or to listen to a piece of music that has been downloaded from the internet. Moreover, using a computer can also be a question of "being practical", for example, talking to friends and family who are physically far away or participating in an international discussion forum on a specific subject. Furthermore, computers can also be used for "impractical reasons" such as finding a soul mate from millions of theoretical possibilities or to play a war game in which the user virtually kills thousands of enemies, or to lead another online life through the medium of an avatar on the "second life" website or MSN. However, loving, waging war or being someone else also provides an open door to subjectivity, which acts as the basis of each person's uniqueness. Whether the subject socializes through loving or playing war games is not a new subject, it is this subject which corresponds in our plan below to socialization, the digital world above all appearing as a possibility of a supplementary game to play (from Winnicott's point of view). Interactive socialization seems to be more original in the digital world as the possibilities are practically endless (on MSN for example), whereas socializing in groups of twos was limited by having to be geographically nearby. Above all, the issue of deconstruction concerns the threat posed by the ever increasing virtual input and emphasis on how young people communicate. From our point of view, these three possible relationships between the subject and his digital world: socialization, interactive socialization and deconstruction constitute a potential space which we aim to investigate in the following pages.

This presentation is made up of two sections. The first section looks at the hypothesis of the "second self" put forward by [START_REF] Turkle | The Second Self: Computers and the Human Spirit, 20 th Anniversary Edition[END_REF]. On the basis of our empirical material, can we talk about an "online" second self or a more or less antagonistic "on-screen" second self with our "original" self? The second part of the text will look at the potential space hypothesis: what is it about the domestic multimedia spaces that we looked at that are also transitional spaces in which the physical identity of each person is played, replayed and rebuilt? These two sections are an implicit series of theoretical preoccupations: does the "subject" build himself in the virtual world or not? Is it necessary to deconstruct the subject? On the contrary, should we deconstruct sociological, psychological and psychoanalytical analysis categories in order to find out more about multimedia culture? Without having a successfully completed form and by performing psychoanalysis, these theoretical reflections are eager to contribute towards achieving a deeper understanding of how ICT's (Information and Communication Technologies) are used.

I The Hypothesis of the "Second Self"

The second self hypothesis is at the very heart of debates concerning the impact of the digital and virtual world on the structure of a person's personality, especially amongst young people. There would be our "true" self and another; it's more or less malevolent double. Such an old question has already been posed in the Old Testament and then revisited by Freud, where the subconscious is the symbolic equivalent of the devil who we must learn to live with in order to better combat him. Today, the virtual world is apparently "diabolical" and has an antisocial influence on the socialization process. We will present the results obtained by the research group. Subsequently, we will examine the conditions in which the fantasy of "young people sinking into virtuality" was formulated. Finally, we will look at the question of a second self, by looking at studies by Turkle, Freud and Lacan.

Goals of the research program on the "second self"

A) Methodology of the fieldwork

The class of 2007 was the third group of students to work on young people's domestic multimedia space under the supervision of researchers. In all, the students accumulated a total of approximately 800 hours of study on this issue in three years, since 2004. The guidelines for 2007 were as follows: to carry out approximately 20 hours of semi directive interviews with college students (15-18 years old) so that they can talk freely about their relationship with multimedia at home. The investigation excluded blogs and those explicitly related to their sex life. Therefore, it consisted of a rather "soft" set of instructions with which the researchers hoped to improve the empirical insight into young people's domestic multimedia organization and multimedia practices, however in no way invading their privacy.

B) Main Results from the study conducted on the teens uses

There is no doubt that the study presented here provides an excellent basis for generating constructive thoughts about the relationship teenagers maintain with technology (See [START_REF] Bounat | L'espace multimédia des 15-20 ans, étude « marketing et psychologie[END_REF]. It is true that the quality of the results is questionable, particularly because of the low number of people interviewed. However, we took the liberty to infer some theories from the results, as some solid ideas emerged.

It is commonplace to say that young people are sinking into virtuality because of wasting hours playing games or hanging around in chat rooms, yet the results of our investigation undoubtedly show that on several occasions teenagers are only attracted by virtual games and activities to a certain extent. Even those who use this kind of virtuality are quite skeptical about what they see there, especially in terms of realism and credibility. Moreover, it is usually admitted that young people are fond of new technologies, and that they are in complete control of computing. There again, circumspection is advised and even we believed that this cliché was true. Just imagine our surprise when we found out that the people consulted were not really all that interested in new technologies and used only a few of the possibilities available on their equipment. The mobile phone is a striking example of this phenomenon: these devices are condensed technology, but teenagers only use them to phone, even though the device contains other functionalities such as MP3 players, cameras, etc. One of the interviewees summed it up quite well: "A phone; it's made for phoning". Furthermore, it is impossible to find a teenager capable of explaining how to use a cell phone. In fact, they do not read the directions, due to finding them too complicated and useless.

Therefore, if teenagers seem to like technology, it is mainly because they are confronted with it at a much earlier age than the elderly. Consequently, they manage to handle the devices better in a superficial way, which proves to be sufficient in most cases, except for when facing a serious malfunction. We point out in our study that they give up quickly when encountering a problem, which illustrates their indifference towards the technique perfectly.

To conclude, we have to admit that Sherry Turkle's publication formulates some ideas which can be considered debatable when confronted with the results of our study. Thereby, her analysis into schizophrenia resulting from new technologies, or the parallel established between chats and the ventilation theory are interesting outlines that we believe are arguable, and far from being generally applicable.

A theoretical analysis of the empirical results

A) "Are young people sinking into virtuality?"

The "generation crisis" issue is anything but new. Plato was already concerned by it. Many teachers and human sciences researchers harbor a great deal of worry which in some way represents their "principal mission". In a forewarning publication in France in the field of socio-psychoanalysis, Mendel (1969) pointed out, by looking at generation crises post May 1968, that (p125):

"The technological ideal tends to "take root" in the person, ruining all previously acquired culture, in a phenomenon which we could possibly call technical acculturation rooting from particular narcissistic frustrations".

Therefore, worries concerning technical isolation are nothing new. Psychologists'worries about antisocial technology don't appear to be any newer than those about the crisis of generations. Therefore, we try to stand back on the basis of methodically observing usages of multimedia technology on the one hand and on the other hand a theoretical framework limiting their place in human sciences and psychoanalysis, but not exclusively. We try to remain in this interface between the research fields and to avoid too structured views which close all original research in advance.

Today as yesterday, most of the literature produced on "young people" is actually written by people old enough to be their parents, who generally don't have a technical culture and belong to the teaching, human sciences or media worlds. Therefore, the generation conflict is also in the process of producing discourse itself, with its implicit presuppositions and unconscious mechanisms, not only with its results. From this point of view, it is therefore necessary to question this discourse in which "young people are sinking into virtuality".

When it comes to our material, this discursive reasoning appears to be a new avatar for the problems associated with parental controls. The family being credible, whilst "making the rules" or representing the Symbolic Law (Lacan) which concerns the private lives of 15-20 year olds, everything goes by as if the control of the shared family resource, that is the computer, becomes a crucial point in the conflict of generations.

Effectively, the adolescent must "set up" his own multimedia universe in the family home, threatened by the risk of parents intruding and establishing rules. Therefore, we must note that access to this virtual universe is in reality and placed under strict and omnipresent parental control. Above all, the issue of "young people sinking into virtuality" would then be a fantasy exercised by worried parents and teachers confronted with a "virtual world" that they don't understand and which opens up a whole new potential world of freedom for their children and which calls into question the subconscious relationship that they relate to their own adolescence.

We can then overlook the so-called young people's "technophilia" (their passion for technology). On the contrary, the adolescent's classic attitude of "withdrawal" from the real world seems to whisk them away: "60% of young people give up resolving a technical problem if they fail on their first attempt; if the first attempt fails, they prefer to abandon the issue" (Bounat et al. p.25). A secondary analysis of our results therefore shows the effects on intuitive youngsters which challenges the idea, which is often admitted by the youngsters themselves, of a generation of youngsters running away from the real world and sinking into a virtual world. We now look at the theory behind this.

B) The Issue of the Self

The question of the self is not an easy one. André Gide, in his unique and very famous novel "The Counterfeiters", which was written in the 1920's, has already deconstructed the author's self in a literary process whereby the author self observes himself writing. At the same time, the author is rewriting his "true" life by romancing it1 . Gide's pioneer work was then followed by the "Nouveau Roman" published in the fifties after the war and in the philosophical field by Derrida and his theory of deconstruction.

Thus, in many respects the "self" also seems to be in many respects a convenient fiction to which no one gives the same meaning, no more in psychoanalysis than elsewhere. In particular, the theory of the "self" is different on both sides of the Atlantic. The western side often states that it is necessary to reinforce and construct the self whereas the eastern side of the pond claims it is necessary to scrutinize the self. It is undeniable that Sherry Turkle is well aware of this issue, having written a document entitled "Psychoanalytic Politics: Jacques Lacan and Freud French Revolution". However, if this soft Lacanian approach constitutes her background theory, she also needs to include the American cultural context, or the self is despite all, more suited to be "at home" (Freud) than in the structure of signifiers (Lacan).

The "self" issue must be dealt with by taking into account what has changed since Turkle's book published in 1995. The development of the "Second Life" internet site which allows a second virtual identity to be created in a potential space is very recent. The explosion of MSN or the "second self" often takes the form of a pseudonym which is older, but has experienced exponential development for some years. Turkle has been one of the first researchers to work on developing a virtual personality on the Web. Turkle pointed out that cases of schizophrenia and radical dissociation amongst different personality levels have multiplied by 10 over a period of approximately 10 years. She believes that the "Freudian" self could increasingly go against the virtual self in the "Life on the screen". Developing a second onscreen life is apparently a cause, or at least a symptom, of increasing schizophrenic tendencies. Therefore, the results from our investigation say otherwise.

45% of the total number of pupils interviewed were fans of games such as "World of Warcraft" or "Second Life", with the majority of these pupils being male (M=85%, F=25%). Only 10% of them considered these games to be "very realistic", 20% as "realistic, but still fictional" and 30% as "possible". 40% of this group of pupils considered the games as "absurd" (Bounat et al p. 30).

At this stage we cannot say whether virtual online life has either a structuring or destructive impact on young people. Therefore, we cannot resolve the issue of knowing whether interactive socialization prevails over personality destruction no more than we can empirically come to a conclusion on the nature of interactive socialization or destruction. The game and potential space hypotheses appear to be more interesting than the hypothesis of the split self. Before looking at this in the following third section on our results, we will first return to the theory of the self and theoretical problems that exist in the domestic relationship between young people and multimedia.

C) Theoretical thoughts on the "online self": a) We return to Freud For Freud: "The ego2 is not master in its own home". Following the development of the digital adolescent, we could play on words and say that "the home is the place where the mastership of one's different egos gets built".

From this point of view, it appears that a certain number of psychoanalysts or psychologists are falling behind on societal evolution. For a long time sociologists have admitted that plural identities and social roles exist. On the contrary, one of the leaders in a long line of Lacanian tradition in France (Melman, 2002) is concerned about the rise in "social psychosis" linked to the fact that the person who "represents the significant master is henceforth the "communicator" (p. 114-116) rather than the "advisor or the priest", amongst others. Indeed, one of Lacan's greatest contributions was to take a new angle on Descartes and his implicit theory of the ego, whereby the subject exists because he thinks "cogito ergo sum". Likewise, the hypothesis of the fundamental division of the subject because he is "a speaking subject3 " opens vast perspectives in the "chat" and MSN explosion context. Therefore, we will now look at what Lacan tells us about the self.

b) Lacan and the ego

'An imaginary representation of the fundamental object of desire (the mother) thus the mirror image "self" will be formed in the original place where the child had situated the mother. As for the place where the infant was initially situated, he is consequently going to be driven to produce an imaginary representation of himself, thus his self "m", thus remembering the alienated status of "non-subject" that he was. (Lacan in Dor (B) p. 287):

Obviously, it is extremely difficult and simplistic to comment on this quote from Lacan in just a few lines. To begin with, we remind ourselves that for Lacan, the ego, "m" ("moi") is an imaginary representation. Therefore, according to him there is no opposition between the real ego and the virtual ego since the ego itself is an imaginary representation. Next comes the question on the origin of this imaginary ego, and it's relation to the subject's mother. We overlook the fact that Lacan missed out the construction of a girl's sexual identity and only talks about a boy's, from which it is effectively reasonable to admit that he desires his mother. By mirror 4 image, Lacan understands that this desire was built by a game of mirrors. Effectively, one's desire is built through a kind of optical process where the object of desire is reflected at each "mirror step". Therefore, since the mother is forbidden, this will only ever be a more or less distorted reflection of what his desire will be and the principle in the "self's" structure. Nevertheless, the issue for Lacan is what could stop the alienation induced by his dependence on his desire for his mother 5 .

D) Applying this to the case of multimedia domestic space

Unlike Turkle, Lacan is not embarrassed by the virtual relation's "imaginary" character since everything in us, including our self, is embedded in our imagination. From this point of view, virtual worlds are only a reflection or another image of our imagination projected onto the real world. Therefore, there would be no reason to grant the virtual worlds or "second life" with a particular power: they would be simple tools that the imagination could or could not make suitable to build/cobuild or destroy a given state of the self.

The issue of the relationship with the mother would only be worth studying for longer developments.

We simply note here that the information technology and multimedia world is a predominantly masculine world, as proven by the statistic that 80% of engineering students in France are male. Symbolically, it therefore seems plausible that this male world works as a place which breaks with feminine social values. We know that many psychoanalysts, especially Lacanians, are worried about paternal figures disappearing both in young people's social lives (feminization of the teaching world, single parent families….) as well as in symbolic systems (decline in paternal metaphors and paternal metonymies…). Let us remember here that in the Lacanian model, there is always a relation between the Real, the Imaginary and the Symbolic. Engineering schools would then be an island of classicism.

From the point of view of a symbolic relation, they would represent a floating island immersed within a sea of somewhat feminine societal values.

Conclusion of part I

Our sometimes empirical and sometimes speculative method of working has its limitations. Clinicians will criticize it for not being clinical in the material exploitation protocol. Psychoanalysts could find it too philosophical and excessively general. If we limit ourselves to our 3 explicitly envisaged hypotheses that we explained in the introduction and collected material analysis for:

1. The digital world has a role to play in the socialization process. The relation between the subject and his own digital word is valid, together with the idea that common sense must prevail: not a revolution, but several evolutions, with young people being careful and pragmatic. 2. The chosen method doesn't allow us to validate the co-construction of the self hypothesis, even if it is in this direction. 3. It also prevents us from empirically deciding upon the hypothesis on the destruction of the self.

To conclude, we can say that the issue of on-screen "self-multiplicity" remains open. Freud had built a binary system: the self and the ideal self. Lacan jumped into the theoretical plane by describing the entire self as an imaginary construction and therefore leaving the possibility of many selves coexisting 4 « Spéculaire » in French 5 The theoretical question of confrontation in the theses by Lacan and Winnicott on the role of potential space which establishes itself between the subject and his mother cannot be tackled within this paper's limited framework open. All things considered, Turkle adopts a classic position on Lacan's side by opposing the conscious self against another on-screen self. She insists on the risk of the self and the personality exploding, especially little or badly structured personalities.

Moreover, we know that authors such as Deleuze and Guattari have greatly criticized classic psychoanalysis by trying to elaborate an "anti-Oedipus" model where the self would be a "rhizome" a multiple budding driven by a machinery of desires. These authors are trying to build an analogy between on one hand the biological sense of multiplication of its flowers by a rhizome and on the other hand the budding of the "flews of desire" in our post Freudian societies.

In this hypothesis, the vast variety of possibilities of life on the screen would reappear, a minima a homology of structures between the kaleidoscopic nature of virtual life and the polymorphic structure of multiple selves. In other words, this would no longer mean opposing different and difficult to define visions of the self, but to set up a coexistence of many different identities amongst which lays the virtual identity.

II From the Second self hypothesis to the Potential Space hypothesis

In this section, we will firstly present our psychoanalysis orientated methodology. We will then investigate the transfer processes in action in this research context. With these research elements having been posed, we will be able to analyze the gathered material, especially the photos of the domestic multimedia spaces. In this framework, the PC has the status of a transitional object in according to Winnicott. This object is linked with cyberspace by the bias of a digital tie which analogically reproduces the structure of the mother-child relationship described in particular by Melanie Klein. We will therefore endeavor to present the relevance of the potential space concept in order to attempt to better understand the workings of young peoples' (aged 20-25) domestic multimedia space from a psychoanalytical perspective.

A Methodology Orientated by Psychoanalysis

Our objective and our methodology have been doubled. On the one hand we hope to allow the technologists to have the monopoly on the analysis on emerging technological usages and restore their dignity in a human sciences and methodological observation approach. On the other hand, our intuition has been to get out of the methodological psychoanalysis ghetto which the methodological psychoanalysis enjoys. The subconscious isn't included within the framework originally defined by Freud and the research protocols must take this into account.

However, we were conscious of the risk stated by [START_REF] Moscovici | La psychanalyse, son image, son public[END_REF] concerning the image and social representations of the psychoanalysis. To be constantly thinking about subconscious processes and to talk about it as least as possible so that we don't give rise to resistances to the psychoanalysis is all the more stronger given that the media world is saturated with vulgate psychoanalytic and that society appears to sometimes immerse itself in psychoanalysm [START_REF] Castel R | Le Psychanalysme[END_REF]. Therefore, we have chosen to abstain from all explicit references to psychoanalysis which would have biased the research with a pathological and sexually connoted image. We have limited the explicit theoretical references to "psychology" only6 . This is what Gabriel (1999) calls a situation whereby the researcher is "informed", and in our case, "orientated" by the psychoanalyst.

A prior stage to the research study with the students consisted of identifying the multimedia space's cartography and physical organization. Effectively, before looking at the potential space, it is necessary to start off by describing the real space. This method draws a great deal of inspiration from ethnologists and particularly from an ethnopyschoanalyst called G. Devereux. In the first step, we gathered together photos, technical descriptions and had discussions about how the students organized their domestic space. The second step consisted of creating suitable conditions for the students to benefit from transference and be able to self analyze themselves and their relation with multimedia.

The transfer is a basic condition in any pedagogical relations. The psychoanalysis is original in the sense that the analyst is in some way, its own teacher: the transfer via the analyst serves the discovery of its psyche by the subject himself.

Devereux has laid great stress upon the role of counter transference, knowing the risk that the analyst could project onto the patient with his pre-established analysis grid. Therefore, his proposition on the one hand consisted of the psychoanalysis favoring a counter transference self analysis. On the other hand, he suggests that the transfer and counter transference Dialectics are as such inserted into the research plan at a prior stage. The psychoanalysis is then a dynamic communication process between two subconscious', the subconscious of the researcher and the subconscious of the patient. In such a communication process the affect circulates whilst giving its contents to the transference process.

We must bear in mind this methodological framework and work on this pedagogic transfer based on a model of co-construction amongst the researchers who have performed interviews and the students who analyzed themselves whilst we endeavored to understand the psychological meaning of domestic multimedia organization. The research framework was not strictly psychoanalytical, and even less clinical, but orientated by psychoanalysis.

Our ability to listen to the Unconscious dimension

When we ask young scientists to analyze a vague humane situation, their first reaction is often to draw up a closed questionnaire. The fantasy of understanding social or psychological reality whilst gathering "objective" data is a profound fantasy. An assembly of psycho-analysts will not be surprised by such a point of view.

A psychoanalytically orientated study on transference must therefore be orientated to create a research plan where the subject of the observation accepts to be himself and for him himself to be an object of observation. There is an epistemological rupture and like all ruptures of that nature, this is a kind of break which can allow the subconscious contents to be expressed. Françoise Dolto was a Lacanian and therefore had very similar ideas to Lacan. Therefore, we could fear an excess of theories, but she also had a sharp clinical sense. This is why she affirmed that sometimes, a porter could play the role of a psychoanalyst. The key to being successful is the quality of their listening skills and that the transfer allows the subconscious to be expressed.

In a slightly surprising framework of marketing studies where young engineers are supposed to calculate and gather objective data, qualify and if possible segment the markets, they are going to be asked, on a voluntary basis, to analyze their own subjectivity from observing objective data in order to find know more about domestic multimedia organization and the meaning that this usage of domestic multimedia organization can have. This is what Jacques Girin, in the perspective that he is more ethnological than psychoanalytical, calls "objectifying subjective data".

Our approach has therefore been the following: to know a market, it is necessary to know the client first. To know the client is to take into account their radical otherness, which doesn't make him me and I am not going to look to plant, to "stick" as the psychoanalysts say, his reality to mine nor mix up the different levels of his reality. As expected, he wouldn't know about the issue of "psychoanalyzing" a customer. However it is still possible to self analyze in order to distinguish within a market, to be specific the multimedia It's in the course of a phrase, through an hesitation, a Freudian slip, sometimes a contradictory speech, or an approximation during the conversation that the unconscious could suddenly appear in the young engineers conversations. The situation is not a psychoanalytical one but the researchers are trying to listen on a psychoanalytical mode and a minima on a very open one. Robert Castel (1973) has criticized Psychoanalysm of being a method reducing social problems to individual problems by focusing only on the unconscious aspects while neglecting their social and political dimension. So we have tried to use psychoanalytical culture but not psychoanalytical vulgate (Mum, Dad, early childhood, sex, …) to understand a relation to the multimedia scene where psychoanalysis is an approach among other approaches.

From a methodological point of view in psychoanalysis, our research follows Freudian tradition in that the slips of the tongue in a conversation are when the subconscious is being expressed. From a more classically psychological point of view, it is necessary to thank the students for their intelligence and depth when carrying out this study. They accepted with great enthusiasm a research method which certainly didn't intend on exposing their subconscious, but which demanded them to study themselves and breaking with all that they were able to do until that point, especially in the case of the private French Scientific Universities.

The « self analysis » Research Group: from the domestic space to the potential space

In 2006, the aim of studying and researching was to construct cartography, a record and scenarios for the evolution of the domestic multimedia space. We have used a quantitative methodology. When presenting the project, the researchers insisted upon the creativity that was demanded: self analysis of their relationship with multimedia and compiling monographs, projection into the future (10-15 years).

Writing a science fiction on the subject had been proposed. The photos of the domestic multimedia space were demanded in order to better overcome the difficult Lacanian language barrier linked to the young engineers' lack of knowledge of psychological language. It was also necessary to be able to have iconographic material at our disposal.

A) Configurations of young peoples' domestic multimedia spaces

The domestic multimedia space of a teen

The domestic multimedia spaces of young adults (23 yo) having a job

Above: leaving at his parents.

Below : leaving in couple.

The domestic multimedia space of a nomadic 23 y.o. student (ENST Brittany)

B) Young engineers and the transitional object: domestic space vs potential space

From comparing these three photos, which conveys the concept of a potential space? We have generally noticed that there is much more disorder at the ENST Brittany College amongst students who are in a nomadic/temporary living situation than amongst those students who are "settled". For the nomadic student, all alone in his dwelling, technical possibilities of accessing a potential space on the Internet are nearly endless. Likewise, the usages are also practically unlimited because he has his own room and an individual internet installation. This "interior freedom" and this freedom of usages can be translated as the disorderliness at the heart of the physical space.

On the other hand, the family domestic multimedia space is constricted by the logic of sharing the resource and the rest of the family being able to see what happens in this space. A common space is more likely to be organized, as shown by the photos. A personal space doesn't necessarily have to be organized.

The Computer: a transitional object at the heart of the potential space?

Based on these observations, the location of computers at the heart of the ENST students' domestic multimedia spaces is entirely comparable to the location of transitional objects as defined by Winnicott (p. 8):

1. The nature of the object (which has to be valued affectively).

2. The student's ability to recognize the object as "non-self".

3. The location of the object inside or outside 4. The student's ability to create, imagine, invent and design in order to produce a multimedia system, to be specific.

The institution of an affectionate relationship

The observation over a long period of time of a population of young engineers, their self-observation work and their self-analysis to our contact and the comparison with more restricted domestic spaces therefore reinforces the hypothesis of a potential domestic multimedia space linked to the presence of a transitional object.

From the digital umbilical cord to the potential space

The idea to use Winicott for this research came to us when we « heard » -in the psychoanalytical sense described in our methodological approach-the students saying that for them the multimedia scene was a process of digital socialization and/or that they were highly dependent upon the « network ». It is obvious for a researcher familiar with group dynamics and psychoanalytical processes that such words are neither neutral nor innocent. The reference to the socialization process means that one has to deal with the most archaic link, that is to say the link which has been built with the mother. Melanie Klein has written extensively on this subject.

Effectively, the hypothesis of the link between the subject and the digital world reproducing the archaic link with the mother is a solid one. The "network" would be a metaphor (even a metonymy of the umbilical then, later, of the mother-child relationship in early childhood, as shown by the table below which takes another look at several issues expressed by the 2006 group during their "self analysis".

Archaic relationship with the mother

Observed relationship with multimedia

A) Winnicott, game, reality and potential space

The domestic multimedia space: a potential gaming space but anchored deeply with in technological and institutional reality Winnicott (1971) defines gaming as "the ability to create an intermediate space between the inside and outside, ability which isn't fulfilled in regulated games, well-constructed as fantasies or rituals, but which are situated at the origin of cultural experience" Fourth summary, (French Translation).

To what extent is this definition applicable to our case? The cultural experience is very original.

Young engineers are good students, gifted for mathematics, who have worked hard in order to integrate into the school and who most often have their first cultural experience of the multimedia space when arriving at the school, aged 20 or 21 years. There is no longer any doubt as to the intermediate nature of this space between the inside and outside.

"MSN allows young engineers to be both inside and outside at the same time: inside because they are "at work", on a work placement or writing a project report and outside because they are permanently connected to their primary social group via MSN".

We have also dealt with a sort of sanctuary where cultural contents are freely made available to the community through the bias of a very dense and private Peer to Peer network. We are neither outside, knowing that these types of cultural practices are illegal in the social world, and neither are we inside, knowing that the explicit functioning rules is the sharing of multimedia contents between pairs of people. Below is an explanation of how interactive socialization between groups of pairs works technically.

"Only very few students buy DVD's, because being connected to the Internet allows them to access all current films on the internal network or to download others from external networks. DVD's can be played on the computer and so nobody owns a DVD player or a VCR (Video Cassette Recorder).

Nobody rents DVD's anymore (2005 Report p. 6)"

These games are certainly adjusted and well organized, especially by technological constraints, but the contents are largely free. It is therefore reasonable to believe that each one can express a certain number of fantasies-only a certain number since the place has an institutional nature and there is a code of practice. All in all, young engineers' domestic multimedia spaces appear to largely conform to Winnicott's definition, but not being totally identical.

By continuing to delve into Winnicott's article (pg. 139) we can read his principal thesis on localizing cultural experiences, knowing that "the place in which one situates cultural experiences is the potential space between the individual and his environment. We can talk about that as much as playing. Cultural experience begins with a creative way of life which is manifested in playing above all."

We can return to our general plan and point out the correspondence with "interactive socialization" linking the subject with his digital world. Winnicott continues (p. 150) "the specific link with this place where playing and cultural experience is as follows: the very existence of this place depends on life experiences". Therefore, this is exactly the radically new situation that can be found occurring amongst young engineers. They are isolated for two years on the edge of Western Europe, in a lace called "Devil's Point" (!), which has a striking resemblance to the insular places such as that described in "Lord of the Flies". We know that in his many psychoanalytical group studies Didier Anzieu considered this novel to constitute a theoretical matrix to the extent that the situation depicted here, where there is no authoritative figure or adult present. Therefore, it is in this place that the majority of young engineers have played the "founding scene" in building their intensive relationship with multimedia, especially within the group of youngsters studied in 2006 (who we 17 years old in 2000, the year of the Internet boom) and who have most often discovered countless playing possibilities on the Internet when starting school.

B) The Interactive Socialization Process

An Amniotic Relationship ?

Students from the ENST College Brittany appear to have an amniotic relationship with the multimedia world. In the way that a fetus is protected by amniotic fluid, the students themselves are immersed in "digitality" by being protected from the outside world and have practically limitless access to multimedia files (music or standard MP3 files or films) which the community makes available to them. The hypothesis is incidentally corroborated by a study by [START_REF] Lardellier | Le pouce et la souris. Enquête sur la nouvelle culture numérique des ados[END_REF], who points out that adolescents are both in front of and inside their screen, especially in the case of second life.

Level of Interactive Socialization that children gain from their parents

The promotional metaphor of the telephone "call of your life" is well known and comes out strengthened by the instances in which young engineers using the internet are interrupted. Acting as a true umbilical cord, the telecommunications network connects them to their friends and family via MSN or at the very least, by E-mail.

Susanna: "I taught my parents, who are quite old, how to use the internet and net meetings, because for us Spaniards family is very important. Well, now I live abroad, a thousand kilometers away from them, and these technologies allow us to talk nearly every day".

Regarding our plan, we are concentrating on the socialization domain, on how the process is reversed.

We add here that Latin family structures are considered to be more matriarchal and potentially more amniotic: take Fellini's film, "Mamma", for instance. This reinforces the digital umbilical cord hypothesis.

Are Human Sciences and Psychoanalysis ahead of Ergonomics?

The importance of the early relationship between a mother and her baby is also a strong point developed by Serge Tisseron. He is a French picture psychoanalyst who began his studies by interpreting Hergé's production of Tintin. [START_REF] Tisseron | Conférence à France Telecom R&D, petits déjeuners de la recherche[END_REF] analyses the dialogue between the ego and the computer, and develops the Winnicottian framework in order to explain how the domestic multimedia space is ran in psychoanalytical terms.

"The relationship with the screen reveals a new paradigm. Dialogue with the computer becomes multisensory. Here, human sciences are ahead of ergonomics: man-machine interfaces are going to evolve around a tonic-postural model whose matrix is the mother-child relationship that has been developed before the age of 1 year. This relation was developed in a play mode of mutual understanding around babbling and language learning; it corresponds to the baby's desire to understand it's speaker's mental state, and vice versa. This first positive step in learning, communication and interpersonal language exchange constitutes the matrix for future learning. It is from this largely unconscious, however very meaningful matrix, that multi sensory and multimedia language exchanges with different types of terminals and especially different affective communication network partners are built."

From the Second Self to the Potential Space: Conclusion

The origin of the digital relationship and digitalization of the relation to origins and therefore according to our equivalents to the symbolic plan lead to the idea of an interactive socialization process of an amniotic nature. It is this amniotic nature which apparently gives a particular intensity to the relationship, which we noticed is more predominant amongst boys than girls.

The confrontation between Winnicott's theoretical hypotheses and our empirical material gives meaning and in several respects validates this hypothesis. It is also coherent with Serge Tisseron's study "How the spirit comes to objects".

From a theoretical point of view, the hypothesis of the second self doesn't come out of the confrontation any stronger. Above all, a multimedia space is run holistically and not individually, even if the situation for domestic usage is often individual. Furthermore, the playing dimension appears to be a great deal more interesting to look at in further depth for further research than the split-self dimension. Finally, the idea of a potential space on the contrary doesn't contradict developing research on the plurality of symbolic game systems in the digital world and beyond in post modern society.

General Conclusion

We have put forward a model analysis of the relationships established between young people and their digital world. More precisely, this model analysis concerns young adults' domestic multimedia spaces (young engineers at the ENST Brittany) or 15-18 years olds. Three cases have been envisaged: socialization, interactive socialization and deconstruction.

The socialization hypothesis was quite easy to reveal: Today, it is commonplace to say that young people socialize through extremely varied and constantly recomposed usages. This is the reason why the "on-screen" second self, elaborated by S. Turkle in 1984, seems dated today. For us, it would be much more worthwhile to try to describe and analyze the "selves" who are expressed online or on the screen, by inspiring them with a theoretical plan for an open definition of the "self", sometimes a pure construction of the imagination (Lacan) and sometimes a rhizome budding with the subject's multiple identities (Deleuze & Guattari).

Interactive socialization has been tackled through a description of the domestic space and a primary analysis on young peoples' multimedia potential spaces. Winnicott's definition of potential space is very appropriate as it allows us to emphasize phenomena such as the playful and "amniotic" nature of the relationship with "virtuality", the symbolic power of the digital (umbilical) "cord" and young people's "reversed socialization" with their parents or grandparents. However, this would need to be "tested" in depth on the research which explicitly aims to pinpoint the properties of such a potential space.

Where the deconstruction hypothesis is concerned, three levels have been envisaged: That of the gathered empirical material (i), the research methodology (ii) and the theory (iii).

(i) From an empirical point of view, none of our material indicates that developing an online life has a significant impact on splitting the psyche, on addictive behaviour or on certain pathologies. On the contrary, the young people's common sense emerges, supported by a certain scepticism of the virtual world. If there are pathologies there, their origin would be to research somewhere else than just in online life, and with another research protocol. In short, the use of information technologies doesn't appear to deconstruct the subject.

(ii) In methodological terms, the researchers' aim was to dismantle (in Derrida's sense) a discourse about a form of technology and it's usages by investigating the blanks and the incoherent components of the discourse. As such, we obtained a possible reconstruction of a young people's multimedia universe, which is neither the description spontaneously produced by them nor the too formatted reading on human sciences. From this point of view, the method looks to be promising. Effectively, it is both totally ethnographical (cartographies, meticulous descriptions, photos…) and supported by a solid hypothesis, knowing that communication from an unconscious to an other unconscious allows us to produce an original research study.

(iii) From a theoretical point of view, we are only at the intuitive stage. It would be convenient look in greater depth at studies by "post-modernity" theorists and to bring them closer to more consistent material; in short, to decline the intuitions of philosophers in human sciences research protocols.

For those who do not know André Gide's opus magna, there are three books in one: the novel, the notes written by the author on the process of writing a novel (which have been finally integrated in the novel itself) and Gide's real life and autobiography which are the main source of the novel.

We apologize if there is any misunderstanding while translating the French "moi" into English. We have used "self" or "ego" without making a clear difference.

« Parlêtre » in the Lacanian vocabulary which could be translated by "speaking being". That implies a philosophy of being (l'être) based on the assumption that any being is defined by its ability to speak (parler). It also implies that speaking comes before being.

by analogy a surgeon does not need to explain his assumptions or theoretical background to his patient for being a "good" surgeon. A strong criteria of his scientific ability is his capacity of diagnosis base on the analysis of a great numbers of data and information.


Emmanuel Moulay

Stability and stabilization of homogeneous systems depending on a parameter

Keywords: Homogeneous systems, nonlinear systems depending on a parameter, stability, stabilization

This paper deals with the Lyapunov stability and stabilization of nonlinear homogeneous systems depending on a parameter. We give a new method for proving global stability and stabilization by using a continuous topological deformation which preserves the stability. The proposed results use a condition based on a well known topological lemma, called the tube lemma. To illustrate these results, we study finite time robust stabilization of the chain of integrators.

I. INTRODUCTION

The stabilization of nonlinear systems is a challenging and difficult subject in control theory. Recent years have seen increasing interest in the stabilization of nonlinear systems. There are several motivations for considering homogeneous systems depending continuously on a parameter. The main one is that it can be seen as a continuous topological deformation which preserves the stability. The proposed framework is based on a topological result known as the tube lemma and leads to a new method for proving global stability and stabilization of homogeneous systems. This idea has been considered in [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF] to study the finite time stabilization of the chain of integrators. The main objective of this paper is to extend the scope of the study to the global stability and stabilization of homogeneous systems depending on a parameter.

One system of particular interest in many practical situations is the chain of integrators. We apply our theoretical results to the finite time robust stabilization of this chain of integrators. A survey on finite time stability of nonlinear systems can be found in [START_REF] Moulay | Finite-time stability and stabilization: state of the art[END_REF] and the problem of finite stabilization of nonlinear systems is considered for instance in [START_REF]Finite time stability and stabilization of a class of continuous systems[END_REF].

The outline of the paper is as follows. In Section II, we fix notation and terminology about control theory. Our main result for the stability and the stabilization of systems depending on a parameter is proved in Section III. In Section IV we discuss the implications of this result to the finite time robust stabilization of the chain of integrators. We make concluding remarks and sketch some directions for future research in Section V.

II. NOTATION AND DEFINITIONS

Throughout the paper B n denotes the unit open ball of R n . A neighborhood of x in a topological space X is an open subset U of X which contains x. Let A be a subset of a topological space, ∂A and Å denote the boundary and the E. Moulay is with IRCCyN (UMR-CNRS 6597), 1 rue de la Noë, B.P. 92 101, 44321 Nantes CEDEX 03, France (e-mail: Emmanuel.Moulay@irccyn.ec-nantes.fr)

interior of A. A function f : R n → R m is a smooth function if it is at least C 1 .
A matrix A ∈ C n×n is Hurwitz if its eigenvalues have strictly negative real part. We denote by

C (α) =        0 0 • • • 0 α 1 1 0 • • • 0 α 2 0 1 • • • 0 α 3 . . . . . . . . . . . . . . . 0 0 • • • 1 α n        the companion matrix attached to α = (α 1 , . . . , α n ) ∈ C n . A function g : X → Y between two topological spaces is proper if and only if the preimage of every compact set in Y is compact in X.
Let us recall the definitions about weighted homogeneity which generalizes the classical notion of homogeneity and has been first introduced in [START_REF] Rothschild | Hypoelliptic differential operators and nilpotent groups[END_REF]

]. A function V : R n → R is r-homogeneous of degree d if V (∆ r λ (x)) = λ d V (x)
for all λ > 0 where r = (r 1 , . . . , r n ) ∈ R n >0 are the weights and ∆ r λ (x) = (λ r1 x 1 , ..., λ rn x n ) is the dilatation. A vector field f is r-homogeneous of degree d if for all 1 ≤ i ≤ n, the i-th component f i is a r-homogeneous function of degree r i + d, that is

f i (∆ r λ (x)) = λ ri+d f i (x)
for all λ > 0.

Let us consider a continuous vector field f : R n → R n . In the following, we consider that all Cauchy problems associated with the system

ẋ = f (x), x ∈ R n (1)
have unique forward solutions for all positive times denoted by φ x (t) for x ∈ R n and t ≥ 0. Under this assumption on f , the system (1) has a continuous global semi-flow ψ (t, x) = φ x (t) (see [START_REF] Bhatia | Local semi-dynamical systems[END_REF]). A set A ⊆ R n is positively invariant under f if ψ(t, A) ⊆ A for all t > 0. A set A that is positively invariant under f is strictly positively invariant under f if ψ(t, A) ⊂ Å for all t > 0.

The system (1) is r-homogeneous of degree d if the vector field f is r-homogeneous of degree d.

If V : R n → R is a continuously differentiable function then define V (x) = n i=1 ∂V ∂x i (x) f i (x) .
A Lyapunov function V : R n → R ≥0 for the system (1) is a proper continuously differentiable positive definite function such that V is negative definite.

The system (1) is asymptotically stable if the following two conditions hold. 1) the system (1) is stable, that is, for every open neighborhood V ⊆ R n of the origin, there exists an open neighborhood U ⊆ R n of the origin such that ψ(t, U) ⊆ V for every t ≥ 0. 2) the origin is attractive, that is, there exists an open neighborhood W ⊆ R n of the origin such that, for every x ∈ W and every open neighborhood U ⊆ R n of the origin, there exists T ≥ 0 such that ψ(t, x) ∈ U for all t > T . The system (1) is globally asymptotically stable if it is asymptotically stable on the whole space W =R n .

Let us introduce the notion of finite time stability involving the settling time function given in [START_REF] Bhat | Finite time stability of continuous autonomous systems[END_REF].

Definition 1: Let us consider the system (1). The system (1) is finite time stable if it is stable and there exists an open neighborhood N of the origin that is positively invariant under f and a positive definite function T : N → R called the settling-time function such that ψ(T (x), x) = 0 for all x ∈ N and ψ(t, x) = 0 for all x ∈ N \ {0}, t < T (x). The system (1) is globally finite time stable if the it is finite time stable with N = R n .

Various properties of the settling-time function are given in [START_REF] Bhat | Finite time stability of continuous autonomous systems[END_REF].

Remark 2: If the system (1) is finite time stable, then it cannot possess uniqueness in backward time at the origin, in particular f cannot be locally Lipschitz at the origin. Consider the control system

ẋ = f (x, u) , x ∈ R n , u ∈ R m (2) with f ∈ C 0 (R n × R m , R n ) and f (0, 0) = 0.
The control system (2) is continuously stabilizable (respectively globally continuously stabilizable) if there exists a feedback control u ∈ C 0 (R n , R m ) such that:

1) u (0) = 0, 2) the closed-loop system

ẋ = f (x, u(x)) , x ∈ R n . (3)
is asymptotically stable (respectively globally asymptotically stable). The control system (2) is finite time stabilizable (respectively globally finite time stabilizable) if the closed-loop system (3) is finite time stable (respectively globally finite time stable).

III. STABILITY AND STABILIZATION OF SYSTEMS DEPENDING ON A PARAMETER

For the convenience of the reader, we first recall a result dedicated to homogeneous vector fields which can be found in [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF]Theorem 6.1].

Lemma 3: Suppose that the vector field f is homogeneous. If A ⊂ R n is a compact set that contains the origin and is strictly positively invariant under f , then the system (1) is globally asymptotically stable.

Let us recall a Lyapunov theorem dedicated to the homogeneous systems which appears in [7, Theorem 2].

Lemma 4: Suppose that the vector field f is continuous on R n and r-homogeneous of degree d. Let p be a positive integer. If the system (1) is asymptotically stable, then there exists a C p -Lyapunov function for the system (1) r-homogeneous of degree k > p max 1≤i≤n r i .

Now, let us consider a continuous function

f : Λ × R n → R n
where Λ is a topological space, and for ω ∈ Λ the system

ẋ = f (ω, x), x ∈ R n . (4)
For every ω ∈ Λ, we suppose in the following that all Cauchy problems associated with the system (4) have unique forward solutions for all positive times. The following theory is based on a well known topological lemma called the tube lemma. Let us recall this lemma whose proof can be found in [START_REF] Munkres | Topology[END_REF]Lemma 26.8].

Lemma 5 (tube lemma): Let X and Y be topological spaces with Y compact and consider the product space X ×Y .

If N is an open set of X × Y containing the slice {x 0 } × Y of X × Y , then N contains some tube W × Y about {x 0 } × Y , where W is a neighborhood of x 0 in X.
We can now state the main result of this paper. Theorem 6: Let Λ be a topological space and suppose that f is a continuous vector field on

Λ×R n such that x → f (ω, x) is (r 1 (ω), . . . , r n (ω)) -homogeneous 1 ∈ R n >0 . If the system ẋ = f (ω 0 , x), x ∈ R n (5)
is globally asymptotically stable for ω 0 ∈ Λ, then there exist a neighborhood Λ ′ of ω 0 in Λ such that for all ω ∈ Λ ′ , the system (4) is globally asymptotically stable. Moreover, if the weights r = (r 1 , . . . , r n ) ∈ R n >0 do not depend on ω, then for any r-homogeneous smooth Lyapunov function V for the system (5) there exists a neighborhood Λ ′′ of ω 0 in Λ such that V is also a Lyapunov function for the system (4) for all ω ∈ Λ ′′ .

Proof: As the system ( 5) is globally asymptotically stable, the theorem of Kurzweil given in [9, Theorem 7] ensures there exists a smooth Lyapunov function V : R n → R ≥0 for the system [START_REF] Bhatia | Local semi-dynamical systems[END_REF]. Let us denote by

Vω (x) = n i=1 ∂V ∂x i (x) f i (ω, x) . As V is proper, S = V -1 ({1})
is compact. Define the continuous function

ϕ : Λ × S → R (ω, x) → Vω (x) then ϕ -1 (R <0
) is an open subset of Λ×S containing the slice {ω 0 } × S. Since S is compact, there exists a neighborhood Λ ′ ⊆ Λ of ω 0 such that ϕ -1 (R <0 ) contains some tube Λ ′ × S about {ω 0 } × S (see Lemma 5). We have Vω (x) < 0 for all (ω, x) ∈ Λ ′ × S. The compact set

A = V -1 ([0, 1])
is strictly positively invariant under x → f (ω, x) for all ω in Λ ′ , due to the fact that S = ∂A.

By using Lemma 3, we deduce that the system (4) is globally asymptotically stable for every ω in Λ ′ . Now, suppose that the weights r = (r 1 , . . . , r n ) ∈ R n do not depend on ω. By using Lemma (4), we know there exists a smooth Lyapunov function V : R n → R ≥0 for the system (5) r-homogeneous of degree k. As before, there exists a neighborhood Λ ′′ of ω 0 in Λ such that Vω (x) < 0 for all (ω, x) ∈ Λ ′′ × S. As x → f (ω, x) is r-homogeneous of degree d (ω) and V is r-homogeneous of degree k, it follows that Vω (x) is r-homogeneous of degree d (ω) + k for all ω ∈ Λ ′′ (see for instance [START_REF] Hermes | Homogeneous ccordinates and continuous asymptotically stabilizing feedback controls[END_REF], [START_REF] Kawski | Geometric homogeneity and stabilization[END_REF]). Thus Vω (x) < 0 for all (ω, x) ∈ Λ ′′ × R n * and V is a Lyapunov function for the system (4) for all ω ∈ Λ ′′ .

Remark 7: Theorem 6 gives the existence of a neighborhood Λ ′ and not a precise specification of the knowledge of the domain of stability. Numerical techniques could be used to determine a more precise domain of stability. Nevertheless, when the degree of homogeneity of the system (4) depends on the constant weights (r 1 , . . . , r n ) ∈ R n >0 for all ω ∈ Λ, the knowledge of a Lyapunov function can be useful to specify the neighborhood Λ ′′ .

For systems without any homogeneity property, we still have the following result:

Corollary 8: If the system ( 5) is globally asymptotically stable for ω 0 ∈ Λ, then there exists a compact set A and a neighborhood Λ ′ of ω 0 in Λ such that A is strictly positively invariant under x → f (ω, x) for all ω in Λ ′ .

The next result is a corollary of Theorem 6 which deals with the problem of stabilization.

Corollary 9: Consider the control system (2) and the continuous function

u : Λ × R n → R m (ω, x) → u (ω, x) .
Suppose that for every ω ∈ Λ the vector field x → f (x, u (ω, x)) is homogeneous and all Cauchy problems associated with the closed-loop system

ẋ = f (x, u(ω, x)) , x ∈ R n (6)
have unique forward solutions for all positive times. If the control system (2) is globally continuously stabilized under the feedback control u (ω 0 , x) with ω 0 ∈ Λ, then there exists a neighborhood Λ ′ of ω 0 in Λ such that for all ω ∈ Λ ′ , the system (2) is globally continuously stabilized under the feedback control u (ω, x).

IV. APPLICATION TO FINITE TIME ROBUST STABILIZATION

An application of the previous theory is in studying the global finite time robust stabilization of the chain of integrators (see [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF]Proposition 8.1]). In this example, a nonlinear robust feedback control is derived from a linear one by using Corollary 9.

The chain of integrators is given by

         ẋ1 = x 2 . . . ẋn-1 = x n ẋn = u (7)
denoted in short by

ẋ = Γ (x, u) .
The following characterization of the homogeneity property of the chain of integrators will be useful. Lemma 10: The system ( 7) is r-homogeneous of degree d if and only if

r i+1 = r i + d, 1 ≤ i ≤ n -1 with d ∈ -rn n-1 , r n and u is r-homogeneous of degree r n + d.
In [1, Proposition 8.1], a control algorithm is built for the chain of integrators [START_REF] Rosier | Homogeneous Lyapunov function for homogeneous continuous vector field[END_REF]. In the following, we extend this result to a robust feedback control in the sense that we obtain a sector margin for the choice of the control gain. Indeed, we built a closed-loop chain of integrators which is robust to perturbations of the feedback control gains. Moreover, a more precise specification of the domain of stability is given by using Corollary 9 and Lemma 10.

Proposition 11:

Let k = (k 1 , . . . , k n ) ∈ R n such that C (k) is Hurwitz. There exists ǫ ∈ 1 -1 n-1 , 1 and ǫ i > 0 for 1 ≤ i ≤ n such that for all α ∈ (1 -ǫ, 1) , k ′ i ∈ (k i -ǫ i , k i + ǫ i ) ,
the system ( 7) is globally finite time stabilizable under the continuous feedback control

u (α, k ′ 1 , . . . , k ′ n , x) = -k ′ 1 ⌊x 1 ⌉ α1 -. . . -k ′ n ⌊x n ⌉ αn (8)
with

⌊x⌉ α = |x| α sgn (x) , x ∈ R and    α i-1 = αiαi+1 2αi+1-αi , 2 ≤ i ≤ n α n = α, α n+1 = 1 (9)
Moreover, the closed-loop system

ẋ = Γ (x, u (α, k ′ 1 , . . . , k ′ n , x)) (10)
is 1 α1 , . . . , 1 αn -homogeneous of degree d = α-1 α .

Proof: Homogeneity property: With Lemma [START_REF] Hermes | Homogeneous ccordinates and continuous asymptotically stabilizing feedback controls[END_REF], it is easy to verify that the system (10) is homogeneous of degree d if and only the the following inequalities hold

- 1 α (n -1) < α -1 α < 1 α , n > 1.
So the system must satisfy

0 < 1 - 1 n -1 < α < 2
which is equivalent to the fact that

- 1 n -1 < d < 1 2 .
Asymptotic stability: It is well known that the system

ẋ = Γ (x, u (1, k 1 . . . , k n , x))
is globally asymptotically stable. Corollary 9 implies that there exists

(a, b) ∈ R 2 >0 with 1 - 1 n -1 < a < 1 < b < 2
and ǫ i > 0 with 1 ≤ i ≤ n such that the system (10) is globally asymptotically stable for all α ∈ (a, b) and k

′ i ∈ (k i -ǫ i , k i + ǫ i ).
Finite time convergence: It follows from [1, Theorem 7.1] that the closed-loop system (10) is finite time stable if the degree of homogeneity d is negative. Finally, we have

1 - 1 n -1 < α < 1.
Remark 12: Due to the fact that the function x → ⌊x⌉ α is not differentiable for 0 < α < 1, the closed-loop system [START_REF] Hermes | Homogeneous ccordinates and continuous asymptotically stabilizing feedback controls[END_REF] is not equivalent to the closed-loop system

ẋ = Γ (x, u (1, k 1 . . . , k n , x))
or a well known stabilizable system under a diffeomorphic change of coordinates in the state space. This is the reason why we had to find a new way of doing.

Example 13: Let us give the example of the triple integrator

   ẋ1 = x 2 ẋ2 = x 3 = u . For α = 11 20 , k 1 = -1, k 2 = k 3 = -3 2 , the feedback control (8) becomes u 11 20 , -1, - 3 2 , - 3
2 , x = -⌊x 1 ⌉ 11 38 - 3 2 ⌊x 2 ⌉ 11 29 - 3 2 ⌊x 3 ⌉ 11 20 .
Then, the system is 38 11 , 29 11 , 20 11 -homogeneous of degree - 9 11 . A simulation of the system leads to Figure 1 which gives an estimate of the settling time.

With Proposition 11, we know that for α = 1 2 we lose the finite time property as displayed by Figure 2.

V. CONCLUSION

This paper analyzes the stability and stabilization of systems depending on a parameter. The main goal of this paper is to develop a topological tool to obtain the global stability and stabilization of a system class depending on a parameter. The tube lemma plays a key role in the presented theoretical results. These are particularly suited for deriving a nonlinear feedback control from a linear one as for the chain of integrators.

A possible direction for future research is to build a state space varying parameter ω (x), with values in the neighborhood Λ ′ , in order to optimize the properties of the feedback controls u (ω (x) , x), as for instance the speed of convergence. It will be interesting to find a more precise specification of the neighborhood Λ ′ , or at least of ∂Λ ′ , in order to bound the state space varying parameter ω (x).

Fig. 1 .Fig. 2 .

 12 Fig. 1. Simulation with α = 11 20

In general, the weights (r 1 (ω), . . . , rn(ω)) depend on ω.

ACKNOWLEDGMENT

The author gratefully acknowledge discussions on homogeneous systems with Laurent Praly.