Carole Roche-Hawley

Robert Hawley

ON SCRIBAL FAMILIES, TRADITION, AND INNOVATION IN 13 TH CENTURY UGARIT

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

Introduction

Gary Beckman"s contributions to our knowledge and understanding of scribal traditions in Syro-Anatolia in the Late Bronze Age are well known. Even his early work, for example his 1983 study of "Mesopotamians and Mesopotamian Learning at attuša," 1 has proven fundamental, and his interest in the subject has certainly not lagged in the intervening thirty years. 2 It is a pleasure for us to offer him this short essay 3 in his honor, with our gratitude for his learning, friendship and admirable humanity. In the spirit of Gary"s own work, we have attempted here to move beyond habitual preoccupations with epigraphic and philological detail, and to place our inquiry within a broader and more meaningful historical and cultural context. We hope that his vast intellectual curiosity will find some small satisfaction in perusing the essay which followsour own attempt to present a historical scenario that "makes sense" 4 for the development of scribal traditions at Ugarit in the 13 th century.

Colophons as Sources

Scribal Families to Be Expected

It should come as no surprise that the scribal craft in the city of Ugarit in the 13 th century BC was a family affair. Such is the norm not only in the Ancient Near East more generally, 5 but also in northern Syria in the Late Bronze Age in particular, as shown for example by the Emar archives. 6 Furthermore, in more general terms, among members of a specialized craft, that is, among artisans, it is also common cross-culturally to find fathers apprenticing their sons. 7 This makes economic p. 2 Festschrift Gary Beckman : 6,000 words (including biblio and notes), January 31 to bcollin@emory.edu sense, since, among other reasons, the acquisition of a trade or craft, especially of a craft that was coveted or highly valued by the powers that be, was a good means of ensuring the economic and social security not just for a given individual, but also for several generations of his family.

Limited Data for Scribal Families at Ugarit

Because of the limited nature of the textual documentation from Ugarit, however, both in terms of numbers of tablets and fragments, and in terms of the types and amount of information conveyed, only a limited amount of evidence for scribal families is available for study.

This evidence is both direct and indirect. The indirect evidence is the (often assumed) hereditary status of the scribal profession; these data derive from the juridical corpus: several royals acts fall into a category often called "appointments" or "promotions" in which the king effects the legal transfer of a named individual from one professional category into another. 8 The appointment often explicitly involves not only the named individual but also his descendants (i.e., it is hereditary); moreover it is often said to be effective "forever" (i.e., it is perpetual).

Superficially, the presence of these heritability and perpetuity formulas might seem to suggest that professional status or membership was generally hereditary at Ugarit. 9 While it is true that such an inference is consistent with the educational dictum "The son takes the profession of his father", 10 and indeed such a conclusion certainly also makes good intuitive sense, even so, the mere presence of the heritability and perpetuity formulas in particular examples of royal "appointments" does not necessarily demonstrate the general hereditary character of professional status and membership at Ugarit. Such an inference, however likely, is potentially problematic because it is based on data derived from the juridical corpus, and there is no way to know whether the kinds of privileges conveyed in legal contracts were typical or atypical. Indeed, there is good reason to believe that it was the atypical, and not the typical that required the drafting of a royal act. In general, at Ugarit as elsewhere, situations that were consistent with commonly established practice probably required no documentation; it was the exceptional situations that required documentation. 11 For all we know, the "exception" (which triggered the setting down in writing of some or all of the preserved "appointments") may very well have been the hereditary and perpetuity clauses. Thus, in our opinion, the indirect data from the juridical corpus are merely suggestive and corroborative, and do not themselves constitute firm evidence that the scribal craft was hereditary at Ugarit.

The direct evidence, however, is less ambiguous (although still occasionally problematic), and in any case it is the more important for our discussion. It is basically limited to one category of data: scribal colophons in which genealogical information is provided. Outside of scribal colophons, p. 3 Festschrift Gary Beckman : 6,000 words (including biblio and notes), January 31 to bcollin@emory.edu one also frequently encounters patronymics in the administrative corpus, but it is usually not possible to affirm that a person named in the administrative lists was necessarily a scribe.

Types of Colophons

The data used in preparing this study are thus almost exclusively limited to the scribal colophons. Defined broadly, "colophons" in the epigraphic corpus from Ugarit are found appended to two different types of documents:

 "traditional texts"12 in Mesopotamian cuneiform script, mostly scholarly lexical lists and literary compositions; and  documents from daily life, in particular juridical acts, mostly in Mesopotamian cuneiform script, mostly royal, and mostly having to do with land tenure in one way or another.

In the case of the latter group, the juridical acts, the scribal colophon probably has a very practical purpose. 13 In the school texts, however, the presence of a scribal colophon seems not so much practical as traditional: it was part of what one learned to do when one learned the Mesopotamian cuneiform writing system and the languages associated with it. A quick comparison of the formal patterns of the colophons attested at Ugarit14 with some of the Old Babylonian colophons surveyed by Hermann Hunger15 illustrates the extent to which this type of colophon is simply traditional.

Past Scholarship

Fundamental previous work on questions such as those treated here, concerning the scribal craft and scribal education at Ugarit, has been done by Wilfred van Soldt in a series of publications since 1988. Our article is particularly indebted to three of his studies:  a 1988 article containing the fullest available survey of the attested colophons on traditional texts from Ugarit,16  his 1991 published dissertation, with a chapter containing the fullest available prosopographical study of the Ugaritian scribes, [START_REF]and to the recently discovered mythological fragment from the Maison d[END_REF] and  a 1995 article containing the fullest available survey of scribal education at Ugarit.18

Families and Scribal Traditions: The Case of Nuʿ mī-Rašap and Sons

No less that forty-eight individual scribes are listed in van Soldt"s prosopographical survey of the scribes of Ugarit in his published dissertation. 19 Of these, it is possible to use the colophon data to suggest some sort of family connection for at least fourteen cases. Such connections are not to be taken as absolutely certain, but they are probable, as can be seen from the example presented below in fig. 1.

fig. 1. An example of a "probable" family connection in the juridical corpus 20 If we lay out a rough relative chronology of these fourteen scribes linked by such plausible family connections (fig. 2), we observe that the heaviest concentration (and also the most reliable) of the data derive, as one might expect, from the latest documented period at Ugarit, the last fifty years or so prior to the destruction of the city.

fig. 2. Schematic diachronic presentation of Ugaritian scribal families

From this presentation, one family emerges as particularly active in the archives of 13 th century Ugarit: that of a certain Nu mī-Rašap.

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The Prestigious Status of Nu mī-Rašap

The scribe Nu mī-Rašap, son of Abaya, was active in the 2 nd half of the 13 th century BC. 21 The period of activity spans a rather long period: he is attested under three successive reigns. 22 Texts which he signed have been found in and near 23 the Palais royal, in the Ville sud, and in the Quartier résidentiel. 24 By the end of his career, he must have attained a position of high rank or status, since he is responsible for writing an international juridical text in which one of the parties involved is the king of Ugarit himself, Niqmaddu IV. 25 On intellectual and literary levels as well, his prestige (or, at least, pretensions) within local Ugaritian scribal circles is illustrated by the fact that he not only wrote juridical texts (a total of nine legal contracts which he wrote are preserved), but also copied Babylonian Belles-lettres, as shown by his fragmentary copy of the Flood pericope from a local recension of Gilgameš or Atra asīs. 26 Even more striking in this regard is his frequent use of esoteric or "learned" spellings, thus distancing himself from the mundane writings used by other scribes in their daily, non-literary productions. One such example 27 is his use of the logogram SAG (instead of the more banal ÌR) for "servant" in one of his colophons. 28 Another example is found in the legend inscribed on one of his personal seals: 29 he uses the logogram A.BA 30 instead of the more banal DUB.SAR for "scribe". 22 Synchronisms exist with Ammiṯtamru III, Ibbirānu VI, and Niqmaddu IV. 23 For example, between the Palais royal and the "Résidence de Yabnīnu." Here and throughout, references to the archeological contexts of various archives and isolated epigraphic finds are consistent with those by M. Yon, La cité d'Ougarit sur la tell de Ras Shamra, Guides archéologiques de l"Institut Français d"Archéologie du Proche-Orient 2 (Paris: Éditions Recherche sur les Civilisations, 1997). 24 Malbran-Labat and Roche, "Urtēnu Ur-Tešub," 81-82. 27 For some further examples of his "learned", esoteric, or even "precious" writings, see C. Roche, "Jeux de mots, jeux de signes en Ougarit ou de l"influence des textes lexicaux sur les scribes de périphérie," in D'Ougarit à Jérusalem: Recueil d'études épigraphiques et archéologiques offert à Pierre Bordreuil, ed. C. Roche, Orient & Méditerranée 2 (Paris: de Boccard, 2008), 210-211. 28 See above, note 26. 29 RS 29.100, for example, bears the impression this seal; its inscriptions reads: NA4 KIŠIB m nu-mì-d MAŠ.MAŠ LÚ A.BA, "Seal of Nu mī-Rašap, scribe." 30 The use of the logogram A.BA for local *sāpiru (Akkadian ṭupšarru), "scribe", deserves a few comments. V. Hurowitz has recently referred to the Assyriologist S. Parpola"s suggestion "that the logogram LÚ A.BA used for ṭupšarru, "scribe", as early as Ugarit, means "A-BA man", "ABC man", or "alphabet man"" (cited in V. Hurowitz, "Additional Elements of Alphabetical Thinking in Psalm XXXIV," Vetus Testamentum 52 (2002) 333; further bibliography: ibid., note 22, and W. W. Hallo, Origins: The Ancient Near Eastern Background of Some Modern Western Institutions [Leiden: Brill, 1996], 40). Also according to H. Tadmor, "The Aramaization of Assyria: Aspects of Western Impacts," in Mesopotamien und seine Nachbarn. Politische und kulturelle Wechselbeziehungen in alten Vorderasien vom 4. bis 1. Jahrtausend v. Chr. XXV. Rencontre assyriologique internationale Berlin, Juli 1978, ed. H. J. Nissen and J. Renger, Berliner Beiträge zum Vorderen Orient 1 (Berlin: Reimer, 1982), 459, "the "pseudo-logogram" … should be interpreted as "ABC-man" and … it seems to have a western background, since it already occurs in Ugarit … where this "pseudo-logogram" seems to have originated." While this interpretation is in many ways seductive, it should be nuanced by the following three considerations. First, had Nu mī-Rašap himself been aware of p. 6 Festschrift Gary Beckman : 6,000 words (including biblio and notes), January 31 to bcollin@emory.edu

Šamū-Ba lu son of Nu mī-Rašap: Teacher, Diviner, and Priest?

The second generation of this scribal family is attested by no less than four sons, all of them scribes. 31 Two of them 32 are not yet attested as scribes in the corpus of texts from daily life: they appear as scribes only in school texts. In fact, one son, Ewri-muḏu, known from three colophons, 33 presents himself as having been a student of his own brother (whose name is written m BAD-d IM). 34 This scribe whose name is written m BAD-d IM (or m BAD-d U) is attested as a teacher in three colophons, all on extracts from the AR-RA : ubullu thematic lexical series, found in several different archives. 35 Wilfred van Soldt had proposed to read this scribe"s name as Gamir-Haddu: reading BAD as TIL, whose Akkadian reading is gamru "complete", and this sound in turn evoking the Semitic onomastic element *gāmiru, "hero" (which is also a productive Ugaritic noun). 36 We now believe that the name is not to be read as m TIL-d IM, but rather as m IDIM-d IM, that is, Šamū-Ba lu. 37 Indeed, the IDIM sign is attested as a logogram noting the "deified heavens", *šamūma in Ugaritic (šamū in Akkadian), in six sacrificial lists. 38 Both in its older graphic form, 39 and in its contemporary Late Bronze form, 40 which had merged with the BAD sign, IDIM is used interchangeably with AN for denoting "the divine heavens" in these lists. Thus, we allow the potential graphic equivalence of:

(logogram) AN = (logogram) IDIM = ("reading" in Ugaritic) šamūma (šamū in construct)
any putative alphabetic reference for the writing A.BA, it seems unlikely that he would have chosen it for his seal, where prestige seems to have been his concern (Nu mī-Rašap and his family are associated above all with the Mesopotamian cuneiform script, and not the nascent cuneiform alphabet). Secondly, judging from RS 19.159 (see R. 31 Their names are Ilī-Šapšu, Ur-Tešabu, Ewri-muḏu, and a fourth whose name is written m BAD-d IM (or m BAD-d U). 32 Ewri-muḏu and the scribe whose name is written m BAD-d IM (or m BAD-d U). 33 One of the Ras Shamra manuscripts of the Weidner God List carries his colophon. He also signed two extracts of the thematic word list AR-RA : ubullu. Yet another school text might also be his work, but the colophon there is broken: one can read the phrase "student of m BAD-d U", so Ewri-muḏu is a possibility, but not a certainty, since at least one other attested scribe, […]-la-na, also studied with that particular teacher (i.e., he called himself "student of m BAD-d U"); RS 25.453 + ; van Soldt, "The Title Ṯ Y," 316. 34 From one of Ewri-muḏu"s colophons we learn that not only he, but also his teacher m BAD-d IM were both sons of Nu mī-Rašap ( m SIG 5 .GA-d MAŠ.MAŠ).

35 Three colophons contain the phrase "(So-and-so), scribe, student of m BAD-d IM": from the Maison de Rap ānu, from the "Maison aux textes magiques" and from the "Maison aux textes littéraires." 36 Van Soldt, "The Title Ṯ Y," 314, note 15; idem, Studies in the Akkadian of Ugarit, 20, note 179.

It appears plausible, then, that the writings m ša-mud IM and m AN-d U are not only further alternative spellings of the same name (Šamū-Ba lu), but also that these writings both refer to the same person, since both refer to a diviner: m ša-mud IM LÚ MÁŠ.ŠU.GÍD[.GÍD LÚ SANGA ? d U ? ] "Šamū-Ba lu, the div[iner, priest ? of Ba lu ? ]" (RS 17.371 + :14') m AN-d U LÚ MÁŠ.ŠU.GÍD.GÍD LÚ SANGA d U "Šamū-Ba lu, the diviner, priest of Ba lu" (RS 18.002 + :16).

Furthermore, given the overall rarity of the name Šamū-Ba lu, it appears likewise reasonable to identify Šamū-Ba lu the teacher (whose name is written m IDIM-d IM or m IDIM-d U), attested in colophons on school texts, with Šamū-Ba lu the diviner (whose name is written m ša-mud IM and m AN-d U), attested as a witness in two international juridical acts written by Nu mī-Rašap (thus, his father, in our view). In our view, all of these writings refer to the same individual, Šamū-Ba lu son of Nu mī-Rašap.

That a priest and diviner should also be scribe and teacher is not, in fact, surprising. In the Akkadian texts from Ugarit, the word "diviner" is written logographically ( LÚ MÁŠ.ŠU.GÍD.GÍD or LÚ AL); syllabic writings of the Akkadian word bārû are not attested outside of the school tradition. In a lexical list found in the House of Urtênu, for example, we find the following equivalence :

LÚ AL = ba-a-ru = wu-ru-ul-li-ni 41
The Hurrian word wu-ru-ul-li-ni (also written [p]u-ru-l[i ? ]-n[i ? ] in RS 20.201G +42 ) shows up as a loan in alphabetic Ugaritic: the word prln, "diviner", is one of the titles attributed by the scribe Ilīmilku 43 to his teacher, whose name was Attēnu. 44 Another of his titles in scribe Ilīmilku"s longest colophon is rb khnm, "chief of the priests", 45 so the association of teacher-diviner-priest has good precedent at Ugarit.

Moreover, Ilīmilku himself carries not only the title spr, "scribe", but also another title, ṯ y nqmd, "the ṯā iyu-official of (King) Niqmaddu." The precise semantic connotations of the title ṯ y remain uncertain, but what is clear is that the word has definite cultic connotations, notably in the areas of sacrifice and exorcism. 46 Furthermore, the explicit association of the title ṯ y with the royal name Niqmaddu in Ilīmilku"s colophons emphasize the political dimensions of the word. In fact, van Soldt has proposed to identify the logogram SUKKAL 47 as the logographic equivalent in the locally written texts in Mesopotamian script of the Ugaritic title ṯ y, [START_REF] Van Soldt | The Title Ṯ Y[END_REF] a proposal further developed by Malbran-Labat and Roche. 49 Another of the titles worn by Šamū-Ba lu was GAL SUKKAL [LUGAL ? ]. [START_REF][END_REF] Festschrift Gary Beckman : 6,000 words (including biblio and notes), January 31 to bcollin@emory.edu It thus seems reasonable to imagine Šamū-Ba lu as the teacher of his brother, as a diviner ( LÚ MÁŠ.ŠU.GÍD.GÍD) and as a priest ( LÚ SANGA) of the local storm god, Ba lu. Beyond the local parallel in the case of Attēnu, described above, other pertinent parallels for the case of a diviner whose duties included teaching the scribal art may be found in contemporary Emar. 51 3.3 Ur-Tešabu son of Nu mī-Rašap: Scribe in the House of Urtênu Among Nu mī-Rašap"s remaining sons, we may mention not only Ilī-Šapšu, 52 who wrote two juridical acts, 53 but above all Ur-Tešabu, 54 well attested as a scribe in archives from the House of Urtênu.

Ur-Tešabu wrote three juridical acts recovered from those archives. 55 A fourth text, found in the Residential Quarter some years ago but not yet published, 56 has a colophon that is broken at the beginning, but may very well be attributed to him. There, the scribe carries two titles, "scribe" and SUKKAL. In addition, the king of Ugarit calls him "my SUKKAL" in a letter from the House of Urtênu. 57 Ur-Tešabu"s activity is also attested in the administrative corpus. According to the correspondence recovered from the House of Urtênu, Ur-Tešabu was responsible for sealing certain shipments of goods on behalf of the high-ranking administrative official Urtênu. Three labels can be attributed to him. 58 They carry the same seal impression, which, in two of the juridical texts is identified as being the "seal of Ur-Tešabu". Particularly interesting is the fact that one of these labels bears a text written in Ugaritic language and script, while in another the text is written in Mesopotamian cuneiform. 59 The simplest explanation for such a situation is that Ur-Tešabu was trained to write in both scripts. 60 Outside of the archives from the House of Urtênu, the name Ur-Tešabu is only very rarely attested. 61 This low frequency led Florence Malbran-Labat and Carole Roche to group all of the texts in which this name occurs in a single dossier. 62 Finally, we know a little bit of the third generation of Nu mī-Rašap"s family: A ī-Rašap, son of Ur-Tešabu, like his uncle Ewri-muḏu, left his colophon on one of the local manuscripts of the Weidner God List. 63 51 Cohen, Scribes and Scholars of the City of Emar, 38-40. 52 In RS 22.223 the final element of his name is written d UTU-šu, indicating the reading Šapšu.

53 RS 17.036 (an act established in the presence of witnesses, found in the Maison de Rašapabu) and RS 22.223 (a royal act found in the Maison aux textes magiques). 54 In general, see Malbran-Labat and Roche, "Urtēnu Ur-Tešub," 81-98.

55 None of them is royal; two are in the presence of witnesses, and a third is in the presence of "the elders of (the village of) Aru". Cf. Malbran-Labat and Roche, "Urtēnu Ur-Tešub," 84-87. 56 Ibid. 57 Cf. the title ṯ y nqmd, "the ṯā iyu-official of (King) Niqmaddu," in several of Ilīmilku"s colophons. 58 One was found in the Maison d'Urtênu, and the two others near the Palais royal; Malbran-Labat and Roche, "Urtēnu Ur-Tešub," 85-86. 59 One of these two labels (the alphabetic one) has to due with textiles destined for delivery to the professional corporation called the mur ū sākini, perhaps to be translated "the prefect"s guards (or officers)"; the other (in Mesopotamian cuneiform) mentions several large dūdu-measures (written GÙR) of grain. 60 Such an interpretation is preferable to the hypothesis according to which he would have merely impressed his seal on a label written by an alphabetic scribe. 61 Ur-Tešabu is possibly also attested as a witness in a juridical text written by the scribe Rap ānu (found in the Maison de Rap ānu: RS 21.007A). The relevant line reads: [IGI…]x-d IM DUMU m SIG 5 -d MAŠ.MAŠ, perhaps to be reconstructed as "[Witness: Ur]-Tešabu, son of Nu mī-Rašap". 62 Malbran-Labat and Roche, "Urtēnu Ur-Tešub." 63 RS 23.495, found in the Ville sud.

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Summary: The Prominence of Nu mī-Rašap's Family

Nu mī-Rašap and two of his sons carried the title SUKKAL, which in 13 th century Ugarit, as already mentioned, was probably the logographic equivalent in Mesopotamian cuneiform of the local social reality, the official whose name is written ṯ y in alphabetic script. 64 In this capacity they were not only linked with mastery of the scribal craft, but also with the inner circle of the royal government, and (very probably) with the local sacrificial cult and with exorcism. Furthermore, if we are correct in our prosopographic study of Šamū-Ba lu son of Nu mī-Rašap, the family also had clear links with divination and with the priesthood of the storm god. 65 The members of this family thus show up in virtually all aspects of Ugaritian elite life: in the administrative management of the kingdom"s resources, as authorities in legal contracts (even those of international scope), as well as dominant players in the local intellectual (literary) and religious spheres of life. Traces of their activity are found in virtually all of the main archives recovered at Ras Shamra, 66 with the highest concentration of their scribal activities showing up in the southern part of the city.

Equally intriguing is the fact that this period during which Nu mī-Rašap and his family were active as scribes seems to correspond more or less precisely with the period during which the use of the cuneiform alphabet was being developed, assimilated and institutionalized into the scribal bureaucracy of the palace administration.

On Some Alphabetic Scribes and Their Colophons

The scribal family of Nu mī-Rašap in the 13 th century BC, steeped as they were in Babylonian scribal culture, may be instructively compared, and especially contrasted, with another important contemporary scribal movement at Ugaritindeed, apparently more than a "movement", in many ways a "revolution"involving the implementation of the cuneiform alphabet on the institutional level of the scribal bureaucracy in the palace administration for writing the local vernacular language. 67 Relatively few alphabetic scribes are known by name. 68 Of these, two notable examples, Ilīmilku and Ṯab ilu, are known from colophons preserved on Ugaritic "traditional texts". Taken together, their work provides a useful foil for better understanding not only the birth, development, and gradual "coming-of-age" of the alphabetic tradition over the course of the second half of the 13 th century, but also the reactions of contemporary scribes trained in other writing systems, most especially the prestigious and influential Mesopotamian cuneiform tradition. 64 Van Soldt, "The Title Ṯ Y"; Malbran-Labat and Roche, "Urtēnu Ur-Tešub." Festschrift Gary Beckman : 6,000 words (including biblio and notes), January 31 to bcollin@emory.edu

Ilīmilku

The Ugaritian alphabetic scribe best known to modern scholarship, owing not only to the early publication of tablets containing his colophon, [START_REF] Virolleaud | Fragment nouveau du poème de Môt et d"Aleyn-Baal[END_REF] but also and especially to the extraordinary nature and significance of his work, [START_REF] Eissfeldt | Sanchunjaton von Berut und Ilumilku von Ugarit[END_REF] is Ilīmilku, who was responsible for copying most of the better known mythological texts in the local language and script. [START_REF]Since tablets 4 and 6 of the Ba lu Cycle (= CTA 4, 6) bear versions of Ilīmilku"s colophon (that of Ba lu 4 must be partially restored), and since only one scribal hand seems to be present throughout, it is[END_REF] His work is above all characterized by large, rather carefully executed compositions written in a small script, 72 and laid out in multi-column, "portrait"-format tablets with a colophon at the end. In our opinion, Ilīmilku"s choice of format, [START_REF]All of Ilīmilku"s work is preserved on tablets which would be described as "Type I" multi-column tablets of the Mesopotamian tradition in the tablet typology developed by M. Civil and others. See Civil, The Series lú = ša and Related Texts[END_REF] combined with the structure and content of his colophons, leaves little room for doubt regarding his intentions and aspirations: he was consciously imitating Mesopotamian scribal tradition, modeling the structure and format of his tablets on Mesopotamian-style "textes de bibliothèque".

The dating of Ilīmilku is complicated, however, and has been a matter of much recent discussion. Until recently, scholarly consensus placed Ilīmilku in the 14 th century, during the reign p. 11 Festschrift Gary Beckman : 6,000 words (including biblio and notes), January 31 to bcollin@emory.edu of Niqmaddu III. 74 We agree with Dalix and Pardee 75 that the Ilīmilku colophons most plausibly refer instead to the king Niqmaddu IV who reigned at the end of the 13 th century and at the very beginning of the 12 th century. 76

Ṯab ilu

Until recently Ilīmilku was the only alphabetic scribe whose work was known from multiple manuscripts. Dennis Pardee, however, has recently identified a series of paleographic and other characteristics that may be (cautiously) used as the diagnostic markers in identifying the written work of another scribe, named Ṯab ilu. 77 Accepting the validity these diagnostic markers, 78 it is possible to place a goodly number of alphabetic cuneiform tablets and fragments within a putative "Ṯab ilu corpus". 79 Taken as a whole, the corpus consists mostly of poetic texts of a mythological or otherwise religious content, thus making comparison with Ilīmilku"s preserved work (which is also poetic and mythological) all the 74 See the summary in Smith and Pitard, The Ugaritic Baal Cycle 2, 7-8. 75 Early statements include A.-S. Dalix, "Exemples de bilinguisme à Ougarit : Iloumilkou : la double identité d"un scribe," in Mosaïque de langues, mosaïque culturelle : Le bilinguisme dans le Proche-orient ancien, ed. F. Briquel-Chatonnet (Paris: Maisonneuve, 1996), 81-90, and D. Pardee, "The Ba lu Myth," in The Context of Scripture, vol 1: Canonical Compositions from the Biblical World, ed. W. W. Hallo and K. L. Younger (Leiden: Brill, 1997), 241, note 3. For a thorough presentation of the argument, see now Pardee, "The Ugaritic Alphabetic Cuneiform Writing System in the Context of Other Alphabetic Systems," in Studies in Semitic and Afroasiatic Linguistics Presented to Gene B. Gragg, ed. C. L. Miller (Chicago: The Oriental Institute of the University of Chicago, 2007), 186-189. We believe that the scribe whose name is written DINGIR.LUGAL in two legal texts (RS 17.061 and RS 17.067) must be differentiated from the alphabetic scribe Ilīmilku of the Ugaritic mythological texts. The former was probably active at the end of the 14 th /beginning of the 13 th century, and the latter a century later (such a distinction seems plausible given the high proportion of homonymy at Ugarit and the frequency of the name Ilīmilku in particular). Van Soldt (Studies in the Akkadian of Ugarit, 27-29) had proposed the possibility of identifying the scribe of the two Akkadian juridical texts with the alphabetic scribe Ilīmilku.

76 Not all students of Ugaritic, however, have agreed with such a down-dating. M. Dietrich, "Salmanassar I. von Assyrien, Ibirānu (VI.) von Ugarit und Tud alija IV. von atti," Ugarit-Forschungen 35 (2003) 134-136, for example, suggests a scenario for maintaining a 14 th century date (the tablet RS 92.2016 from the Maison d'Urtênu would be an heirloom from an earlier period); he has been tentatively followed by Wyatt, "The Religious Role of the King in Ugarit," 697, note 7. . Eissfeldt, "Nachtrage zu "Adrammelek und Demarus" und zu "Bist du Elia, so bin ich Isebel" (I Kön. XIX 2)," Bibliotheca Orientalis 26 (1969) 183, note 4. The vocalization of the name is uncertain; we have followed Pardee"s "Ṯab ilu" for the sake of convenience. Other possibilities include "Ṯub ilu" (Hawley, Pardee and Roche-Hawley, "À propos des textes akkadiens alphabétiques conservés au Musée d"Alep," forthcoming in Études ougaritiques III, ed. M. Al-Maqdissi and V. Matoïan, Ras Shamra-Ougarit series [Leuven: Peeters, forthcoming]), or even "Ṯābi ilu" or "Ṯūbi ilu" (postulating the presence of a medial -i-binding vowel between the two onomastic elements; S. Layton, Archaic Features of Canaanite Personal Names in the Hebrew Bible [Atlanta: Scholars Press, 1990], 107-154). 78 A schematic presentation of the data in tabular form will be published in Hawley, Pardee and Roche-Hawley, "À propos des textes akkadiens alphabétiques." 79 Thus far, the putative "Ṯab ilu corpus" includes (listed here in the order of the publication in which an argument as being the work of Ṯab ilu is advanced): RS 19. p. 12 Festschrift Gary Beckman : 6,000 words (including biblio and notes), January 31 to bcollin@emory.edu more pertinent. The Ṯab ilu corpus differs considerably from the Ilīmilku corpus, however, in terms of the remarkable amount of internal diversity to be found within it. This diversity may be observed on various levels: in terms of the find spots of the tablets, [START_REF]of Ilīmilku"s tablets were found on the Acropole, in or near the Maison du Grand-Prêtre; see van Soldt[END_REF] diversity of tablet format and layout, 81 paleographic diversity, [START_REF] Pardee | Deux tablettes ougaritiques de la main d"un même scribe[END_REF] and diversity of language. [START_REF]In addition to Ugaritic, Ṯab ilu used the cuneiform alphabet to copy texts in Akkadian and Hurrian[END_REF] In our opinion, this level of internal diversity is best explained by the hypothesis that, as a whole, Ṯab ilu"s work reflects a period of innovation and experiment with the cuneiform alphabet, and not a period of regular and standardized scribal tradition.

Ṯab ilu"s work can be most plausibly dated to the reign of Ammiṯtamru III. There are two primary arguments in favor of such a date. First, one of his manuscripts was actually found not in the Ras Shamra excavations, but in the excavations at neighboring site of Ras Ibn Hani, for which the excavators have assembled a host of arguments in favor of such a date for the archives recovered there. [START_REF] Lagarce | Remarques sur le matériel égyptien et égyptisant de Ras Shamra ("Maison aux Albâtres") et de Ras Ibn Hani à la lumière de données récentes sur la chronologie de la fin d"Ugarit[END_REF] Secondly, Ṯab ilu"s texts contain an archaic form of the {g} sign (one that evokes the earlier two-stroke form) which is elsewhere explicitly attested in the few preserved Ugaritic documents which may be reliably dated to the reign of Ammiṯtamru III. 85 It should be mentioned of course, that the earliest reliable evidence for the institutionalization of the alphabet among the palace scribes of Ugarit derives from the reign of Ammiṯtamru III. [START_REF]The Ugaritic Alphabetic Cuneiform Writing System," 186-189. On the possibility that RS 15.117 dates to the reign of Niqmêpa VI, the father of Ammiṯtamru III (Márquez Rowe, Royal Deeds of Ugarit, 53-54), see now Pardee, "RS 15.117 et l"origine de l"alphabet cunéiforme d"Ougarit : rapport de collation[END_REF] p. 13 Festschrift Gary Beckman : 6,000 words (including biblio and notes), January 31 to bcollin@emory.edu

From Early Experiments to Scribal Tradition

Ṯab ilu can thus be plausibly dated to the reign of Ammiṯtamru III in the middle of the 13 th century, and Ilīmilku plausibly dated to the reign of that king"s grandson, Niqmaddu IV, at the end of the 13 th /beginning of the 12 th century. fig. 3. Schematic diachronic presentation of Nu mī-Rašap"s family with respect to two alphabetic scribes Such a relative chronology, in our opinion, provides a simple and reasonable explanation for the evolution that can be observed in terms of alphabetic scribal practices in the intervening generation or two. By the time of Ilīmilku, the cuneiform alphabet was less of an oddity, less of an experiment. The scribal habits to be seen in Ilīmilku"s work suggest a certain maturityor, at least, a "coming-of-age"for the alphabetic tradition: the regularity of the sign forms, regularity of the mise-en-page and of tablet shape, the presence of "standard" sign forms (that is, forms which are in general usage by the scribes of the administrative and epistolary tablets). Such is less the case in Ṯab ilu"s work: his tablets show a bewildering variation in sign forms, and considerable latitude in tablet shape and mise-en-page. To these observations we may add the presence of archaic sign forms (and more generally sign forms not in general usage by the scribes of the administrative and epistolary tablets). One must certainly hesitate to convey esthetic judgments, and yet it is difficult to escape the impression of a certain clumsiness in Ṯab ilu"s work, as if he was working without the benefit of well-known, well-established models.

Conclusions: Family Traditions versus Individual Innovators?

By emphasizing the fact that they belonged to prestigious scribal families, the local scribes of Ugarit working in Mesopotamian script, as exemplified by Nu mī-Rašap"s family, were insisting on the fact that they themselves belonged to, and were perpetuating, the prestigious intellectual tradition of the Mesopotamian scribal arts.

Even though the alphabetic scribes, and especially Ilīmilku, appear to have modeled certain aspects of their compositions on Mesopotamian models, judging from the available evidence 87 they nevertheless did not adopt the "son of X" element in their colophons. Rather than insisting on belonging to and perpetuating a tradition, the alphabetic scribes would thus seem instead to be affirming their independence, affirming the fact that they and their work represent not continuity but instead a profound rupture with previous scholarly tradition (a rupture which, incidentally, would turn out to be more or less irreversible for the Levant). This rupture can be observed not only in the 87 Caution is obviously required here since the body of data is so limited: in the literary corpus we have only four colophons attributable to Ilīmilku, and two (very short) colophons for Ṯab ilu. Note that the alphabetic colophons known from non-literary texts, such as those of brqn and bṣmn, however, also omit patronymics.

p. 14 Festschrift Gary Beckman : 6,000 words (including biblio and notes), January 31 to bcollin@emory.edu choice to omit the mention of fathers, but also in the choice to write the local language (instead of the traditional Mesopotamian languages of learning), and to use a locally developed alphabetic script (instead of the traditional Mesopotamian cuneiform script).

In at least two of his colophons,88 for example, Ilīmilku fills the expected "son of X" slot with a gentilic adjective, as if to convey that he is not the son of anyone particular, rather he is from the village of Šubbānu. The repeated presence of this element in his colophons suggests that the scribe was apparently proud of his provincial origins. Is it legitimate to see here an intentional desire on the part of Ilīmilku to detach himself from the scribal traditions of the capital, as represented by Nu mī-Rašap"s family?

A second example comes from the most recently discovered Ilīmilku colophon, that of RS 92.2016, [START_REF] Caquot | Un texte mythico-magique (no 53)[END_REF] in which the scribe seems to be parodying another traditional Mesopotamian colophon motif: that of the copied work being checked, verified, and complete. [START_REF]found in the colophons of locally copied Mesopotamian traditional texts: for example[END_REF] The most obvious reading for the phrase w ınd ylmdnn in the third line of the colophon is "nobody taught it (to me)". 91 In other words, Ilīmilku seems to be conveying that the text does not contain traditional material that could be checked, but rather original new material of his own creation.

Still further illustrations of a potential tension between the old prestigious scribal families steeped in age-old Mesopotamian lore on the one hand, and the alphabetic scribes, the "young turks" as it were, on the other, may be found in the Mesopotamian cuneiform corpus. Perhaps, for example, we might interpret the proliferation of "learned" spellings, esoteric writings, and graphic puns, 92 all of which are especially well attested with Nu mī-Rašap, as a kind of reaction to the alphabetic enterprise on the part of local scribal establishment. Most of these "learned" writings are best characterized not as archaisms, but as examples of attempted archaizing: intentionally "oldlooking" spellings, even intentionally obscure spellings. Is it coincidence that these kinds of writings are particularly visible in the mid-13 th century, just when the palace has more or less officially adopted the alphabetic cuneiform script as the vehicle of domestic administration, that is, just when the Mesopotamian tradition"s age-old monopoly on scribal privilege has been threatened? However we answer that question, a few decades later, by the end of the 13 th /beginning of the 12 th century, the tension seems much less apparent, and perhaps has even been resolved altogether. [START_REF]For the example of Ur-Tešabu"s seal on a label with a Ugaritic inscription, see Malbran-Labat and Roche[END_REF] Bi-graphic compositions are more common, [START_REF] Roche | Classification de l"utilisation du cunéiforme mésopotamien dans les textes ougaritiques[END_REF] as if nothing could be easier that the

25 RS 17 .

 17 371 + . 26 RS 22.421 (Ugaritica V, n o 167); see J. Nougayrol, "Textes suméro-accadiens des archives et bibliothèques privées d"Ugarit," in Ugaritica V: Nouveaux textes accadiens, hourrites et ugaritiques des archives et bibliothèques privées d'Ugarit, Commentaires des textes historiques, première partie, ed. J.-C. Courtois, Mission de Ras Shamra 16, Bibliothèque Archéologique et Historique 80 (Paris: Imprimerie Nationale, 1968), 300-304; W. G. Lambert and A. Millard, Atra-asīs: The Babylonian Story of the Flood (Oxford: Clarendon, 1969), 131-133.

  Hawley, "Apprendre à écrire à Ougarit : Une typologie des abécédaires," in D'Ougarit à Jérusalem: Recueil d'études épigraphiques et archéologiques offert à Pierre Bordreuil, ed. C. Roche, Orient & Méditerranée 2 [Paris: de Boccard, 2008], 220, 224-225, with further bibliography), the first two signs of the local alphabetic inventory were pronounced a-be, not a-ba. Finally, since examples of this logogram are known from 13 th century Assyria, dated from Tiglath-Pileser I or even Tukulti-Ninurta (Hunger, Babylonische und assyrische Kolophone, 32 [n° 52]; O. Pedersén, Archives and Libraries in the City of Assur: A Survey of the Material from the Geman Excavations, Part I, Studia Semitica Upsaliensia 6 [Uppsala: Almqvist & Wiksell, 1985], 31-42; ibid., Part II, Studia Semitica Upsaliensia 8 [Uppsala: Almqvist & Wiksell, 1986], 24 [n° 89]), chronology also poses a problem. Is it feasible to imagine the direct, almost immediate influence of Ugaritian scribal habits in 13 th century Assur? Influence in the opposite direction appears to us much more likely (Arnaud, Corpus des textes de bibliothèque, 9-11)?

  77 D. Pardee, "Deux tablettes ougaritiques de la main d"un même scribe, trouvées sur deux sites distincts : RS 19.039 et RIH 98/02," Semitica et Classica 1 (2008) 9-38; idem, "Un chant nuptial ougaritique (RS 5.194 [CTA 24]): Nouvelle étude épigraphique suivie de remarques philologiques et littéraires," Semitica et Classica 3 (2010) 13-46. On Ṯab ilu"s colophons, see already the suggestion of O

  039 and RIH 98/02 [Pardee, "Deux tablettes ougaritiques de la main d"un même scribe"]; RS 5.194 (CTA 24) and several small fragments (RS 5.219-223) [Pardee, "Un chant nuptial ougaritique"]; RS 5.229 (CTA 9) and RS 5.259 (CTA 25) [Pardee, "RS 5.229 : restitution d"une nouvelle signature du scribe Ṯab ilu," in Scribes et érudits dans l'orbite de Babylone, ed. C. Roche-Hawley and R. Hawley, Orient & Méditerranée 9 (Paris: de Boccard, 2012), 31-49]; the entire corpus of alphabetic Akkadian texts (RS 5.156 + , RS 5.199, RS 5.213, RS 5.303bis, and possibly RS 5.216) [Hawley, Pardee and Roche-Hawley, "À propos des textes akkadiens alphabétiques."]; and part of the corpus of alphabetic Hurrian texts (for example, RS 1.004, RS 5.182 and RS 5.200; perhaps also RS 24.285) [provisionally, see the description of the hands in Herdner, CTA, 255, 266-267]. We have followed Pardee in provisionally grouping these texts as the work of a single scribe named Ṯab ilu, but it is also possible that this corpus represents not merely the work of one individual, but the work of several individuals who were part of the same "school".

  21 Earlier studies of Nu mī-Rašap"s family include van Soldt, Studies in the Akkadian of Ugarit, 181, and especially the more detailed presentation in Malbran-Labat and Roche, "Urtēnu Ur-Tešub," in Le royaume d'Ougarit de la Crète à l'Euphrate : Nouveaux axes de recherche Actes du Congrès Internationale de Sherbrooke 2005, ed. J.-M. Michaud (Sherbrooke, Québec: Éditions G.G.C. Itée, 2007), 81-98.

  65 It is tempting to identify the person mentioned in RS16.186 ([…] LÚ SUKKAL LÚ UGULA SANGA, dated from Ibbirānu VI) with study of Šamū-Ba lu son of Nu mī-Rašap, who is also SUKKAL and priest. 66 These include the Palais royal, the Maison de Rap ānu, the Maison de Rašapabu, the Maison aux textes littéraires, the Maison aux textes magiques, and last but not least, the Maison d'Urtênu. In addition to Ilīmilku, van Soldt (Studies in the Akkadian of Ugarit, 19-32) also lists bṣmn (RS 15.116, KTU 4.183) and brqn (RS 19.066, KTU 3.8). Only the first letter of the scribe"s name (b[…]) is preserved in RS 15.111 (KTU 3.2; see B. Kienast, "Rechtsurkunden in ugaritischer Sprache," Ugarit-Forschungen 11 [1979] 447). On the alphabetic scribe Ṯab ilu, see below. ROCHE-HAWLEY and HAWLEY, "Scribal Families, Tradition, and Innovation in 13 th Century Ugarit"

	p. 10

67

On the date of the cuneiform alphabet, see below, note 86.

68

Y. Cohen, The Scribes and Scholars of the City of Emar in the Late Bronze Age, Harvard Semitic Studies 59 (Winona Lake: Eisenbrauns, 2009).

For Mesopotamia, for example, van der Toorn, Scribal Culture, 62, cites the dictum "The son takes the profession of his father" from a Sumerian literary text (for a transcription, see Å. Sjöberg, "Der Vater und sein missratener Sohn," JCS

(1973) 112; for a translation, ibid., p. 117: "Das, was Enlil den Menschen bestimmt hat, ist, Dass der Sohn dem Beruf seines Vaters folge").

In other words, these are texts which represent the Oppenheimian "stream of scribal tradition", those which Daniel Arnaud has called "textes de bibliothèque" (as in Arnaud, Corpus des textes de bibliothèque de RasShamra-Ougarit (1936[START_REF] Pardee | Les textes rituels, Ras Shamra-Ougarit[END_REF] en sumérien, babylonien et assyrien, Aula Orientalis-Supplementa 23 [Barcelona: Sadabell, 2007], for example).

 13 The scribe essentially presents himself at the end of the list of witnesses (probably presenting himself as one of them, despite the frequent lack of the sign IGI preceding the scribe"s name) who guarantee and could potentially vouch for the authenticity of the transaction recorded, and to whom recourse could potentially be made should there be a subsequent dispute. See also Márquez Rowe, Royal Deeds ofUgarit,[206][207][208]

.14 For a convenient typology of their compositional elements, see W. van Soldt, "The Title Ṯ Y," Ugarit-Forschungen 18 (1988) 314

-318. 15 H. Hunger, Babylonische und assyrische Kolophone, AOAT 2 (Kevelaer and Neukirchen-Vluyn: Butzon & Bercker and Neukirchener Verlag, 1968)

, 25-34.[START_REF] Bietak | Contributions to the Chronology of the Mediterranean 17[END_REF] Van Soldt, "The

Title Ṯ Y." 17 Idem, Studies in the Akkadian of Ugarit: Dating and Grammar, AOAT 40 (Kevelaer and Neukirchen-Vluyn: Butzon & Bercker and Neukirchener Verlag,

1991), 19-32.18 Idem, "Babylonian Lexical, Religious and Literary Texts and Scribal Education at Ugarit and Its Implications for the Alphabetic Literary Texts," in Ugarit: ein ostmediterranes Kulturzentrum, Ergebnisse und Perspektiven der Forschung, I: Ugarit und seinem altorientalische Umwelt, ed. M. Dietrich and O. Loretz, Abhandlungen zur Literatur Alt-Syrien-Palästinas 7 (

Münster: Ugarit-Verlag, 1995), 171-212.

Festschrift Gary Beckman : 6,000 words (including biblio and notes), January 31 to bcollin@emory.edu

Idem, Studies in the Akkadian of Ugarit, 19-26. We have not taken into consideration the fragmentary names grouped under n° 49.

RS 16.206 is actually one of the rare examples in which genealogical information can be gleaned from the actual body of a juridical text, and not merely from the colophon (as in the two other cases cited). Despite the high proportion of homonymy in Ugaritian onomastics, given the limited number of scribes likely to have been active at any given time, and given the fact that uṣānu and Ya ḏirānu were both scribes and were both active during the reign of Niqmêpa VI (Ya ḏirānu remained active also during the following reigns), it is highly probable that Ya ḏirānu"s father (whose name was uṣānu) and the scribe uṣānu who was a slightly earlier contemporary, were in fact one and the same person. Here and throughout, our numbering of the Ugaritian homonymous kings follows Arnaud, "Prolégomènes à la rédaction d"une histoire d"Ougarit II: Les bordereaux de rois divinisés," Studi Micenei ed Egeo-Anatolici 41 (1999) 153-173.

This is a change of opinion with respect to Malbran-Labat and Roche, "Urtēnu Ur-Tešub," 83, and Roche, "Jeux de mots, jeux de signes," 212-213, where different interpretations were suggested.

Roche-Hawley, "Procédés d"écriture des noms de divinités ougaritaines en cunéiforme mésopotamien," in Scribes et érudits dans l'orbite de Babylone, ed. C. Roche-Hawley and R. Hawley, Orient & Méditerranée 9 (Paris: de Boccard, 2012), 151 (2.1.1), 169.

The older form consists of two sequential AŠ, the second one being "reversed" (that is, pointing to the left).

RS 2.[009] + viii 53 (CTA 6); RS 92.2016 (RSO XIV 53). The word šbny should also be reconstructed in the colophon on the left edge of RS 2.[004] (CTA 17).


caching policy (LRU, LFU, popularity, etc.

CCN an old idea ?

A MOBICOM 2000 paper [IGE00] [AY05] "Directed Diffusion" describes the "Data-Centric" paradigm "Directed Diffusion is an important milestone in the data-centric routing research of sensor networks. The idea aims at diffusing data through sensor nodes by using a naming scheme for the data" "The main reason behind using such a scheme is to get rid of unnecessary operations of network layer routing" "In order to create a query, an interest is defined", "The interest is broadcast" "Each node receiving the data can do caching for later use" "Hence, by utilizing interest and gradients, paths are established between sink and sources. Several paths can be established" "all communication is neighbor-to-neighbor with no need for a node addressing mechanism"

Why focusing on CCN ?

Why is CCN so popular among ICN solutions ? Some use cases do not make sense in CCN, others must be adapted.

CCN-specific use cases

CCN UC1 :

 UC1 Filtering on content provider (Example : known untrustworthy or banned) CCN UC2 : Filtering on bad signature CCN UC3 : Filtering on content name and semantic (Example : excluding contents named with a given keyword) CCN UC4 : Composition (content provider & content name) CCN UC5 : Filtering on content direction (Example : avoid leakage of certain documents) CCN UC6 : Filtering on heavy traffic (Preservation of QoS) CCN UC7 : Filtering of stored data (Example : caching only storing specific content) Sequences of more than 3 characters are extracted Segmented as real human-readable words For each word, x semantically similar words are found... ... and included into an extended regular expression Implementation Integration within the CCN stack Firewall directly processes content chunks : captures packets arriving on a face, applies rules on it, eventually calls standard CCN process Impact of the number of rules on the transfer time Increasing step 100 MB Requested files size : 500 MB and 1 GB Shows small to no impact on transfer time Impact of a 1000-rules firewall on the transfer time Repeated experiment (500 MB file transfer) to obtain significant results Applied Chi-square and KS-test on obtain result Overhead of the firewall is insignificant

  Private data [ACG + 13] a Ephemeral data [CPW11] b a. "Cache Privacy in Named-Data Networking" Acs & al. b. "Content-based publish/subscribe networking and information-centric networking" Carzaniga & al.What is available in CCNx commands ?

	Network management Network management Traffic Management Nodes Management Security Management CCNx commands Monitoring of ccnd Experimentation for management activities IRTF Information-Centric Networking Research IRTF Information-Centric Networking Research Plan Motivation IP firewall main use cases
	Simple architecture based on simple ideas ACM CoNEXT 2009 paper [JST + 09] : Good educational introduction, most architectural aspects covered Many research questions clearly highlighted (routing, key management, etc.) ... even if not all (scalability regarding number of contents or updates frequencies, enforcement of unique names at the Internet scale, mobility while providing content, etc.) CCNx implementation / community Lucky factor : right time / right research community ? What is network management ? Wikipedia attempt : "Network management refers to the activities, methods, procedures, and tools that pertain to the operation, administration, maintenance, and provisioning of networked systems" NMRG functional areas : Fault management, Performance management, Security management, Configuration management, Accounting management, Service management, Event management Configure and control a set of resources that ensure the network is running well Means : monitoring (centralized, distributed, autonomous) coupled with control/optimization/economic/machine learning/stochastic theories What is relevant for CCN ? Traffic management Cache management Content management Nodes management Security management Key challenges Traffic control for QoS [ORS12] [FRRS12] Fair queuing, congestion avoidance (Interest discard, etc.) Traffic control for business/political purposes (easier filtering / censorship ?) Traffic differentiation : several hacks of the CCN architecture to handle specific traffic types Key challenges Monitoring of CCN nodes : collect information and status of CCN nodes, detection of anomalies Strategy layer needs information Define relevant information to be monitored + right granularity (per prefix, per face) ? Define efficient architecture (CCN ready) for collection of information Design and implementation of new network tools (ping, traceroute, etc.) Design of new metrics (CCN/ICN flows, etc.) Remote configuration (no push mechanism) Group Group How to enforce security policies in CCN ? Key challenges Identification of new threats [WSV12] a DoS by resource exhaustion of stateful routers On PIT : Interest flooding attack On FIB : Announcing conflicting domains, non-existing content, non-aggregable names On CS : Privacy issues (cache probing) [LLR + 12] b [ACG + 13], cache pollution Cryptography attacks (long-lived content, many encryptions with the same key) a. "Threats to Stability and Security in Information-Centric Networking" Wahlisch & al. b. "Privacy risks in named data networking : what is the cost of perfor-mance ?" Lauinger & al. Content management ccngetfile : retrieve a file published as CCNx content and save it to a local file ccnputfile : publish a file as CCNx content (local file filename or url to content with the ccnxname), manage segmentation, key signature, etc. ccnrm : mark as stale any locally cached content objects matching the given prefix (no further Interest response) ccnls : list name components available at the next level for a given CCNx name prefix repositories) Limitation : sliding time window (avg of last minute) provides Implementation in CCNx and performance evaluation Deployment and interoperability (with BGP, OSPF) inaccurate results Solution 2 : PlanetLab nodes (cf NEPI talk) preprocessing for semantic enhancement Business, legal and regulatory frameworks Design of a semantic CCN firewall : grammar definition, ccnlsrepo : explore content stored under a given prefix (one or more Tools ccn ccnbtoxml : convert ccn binary encoded data into XML form ccn xmltoccnb : convert XML into ccn binary encoded data (ccnb) ccndsmoketest : testing of communications, send and receive data on sockets Monitoring commands Need of better tools Main topics Goal : prevent users from downloading malicious/forbidden contents IP UC1 : Filtering based on the protocol (Example : http, smtp, ccndstatus : display the status a running ccnd (nb of active faces, stat of each face, etc) ccnpeek : generates an Interest, get one content item matching the name prefix and write it to stdoud (eq to IP ping) ccnponk : read data from stdin, send it as a single ContentObject in response to an interest Web Interface http ://localhost :9695/ , similar to ccndstatus More monitored parameters Better accuracy New metrics Need of a large scale testbed to support experiments Based on CCNx enabled nodes young Internet) Application/application-protocol design and APIs Use case analysis : Identification of security needs for CCN Solution 1 : federated testbed between academic partners (like the Authentication of content possible (lower layers : simple etc.) Naming schemes for ICN and scalable name resolution for flat names 1 Key Challenges for the management of CCN verification) but real security tools missing IP UC2 : Filtering based on status of the connection (new, Scalable routing schemes Inheritance of IP firewalls limited : no filter on IP addresses or ports established, etc.) Congestion control, QoS approaches, and caching strategies 2 A firewall for CCN New security features enabled by the CCN paradigm IP UC3 : Filtering based on a list of known blacklisted IP addresses Metrics that make it possible to evaluate ICN implementations in a consistent manner access control to them Content firewall : considering content name and signature Security and privacy, including scoping of information objects and Other related IRTF working groups : RRG (Routing Research Group), NMRG (Network Management Working Group) IP UC4 : Filtering unusual inbound traffic pattern (from a denial of Contribution service attack attempt)
	Management of Content-Centric Networking Management of Content-Centric Networking Management of Content-Centric Networking Management of Content-Centric Networking
	18 / 34 20 / 34 21 / 34 22 / 34

Very active group, mailing list : icnrg@irtf.org, web : irtf.org/icnrg

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Management of Content-Centric Networking

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Cache Management

Key challenges

What types of contents should be cached ? Where ? With which priority ? [FRRS12] Video on demand Long lived multimedia contents (file sharing or user generated) Web What types of contents should not be cached ?

Conversational (two parties) Ephemeral events (notifications from IoT world, online games) Private communications (email, VOIP, etc.)

Syntax definition

Syntax based on iptables for ease of use and readability


Humberto Stein Shiromoto
email: humberto.shiromoto@gmail.com.

Vincent Andrieu

Christophe Prieur

Interconnecting a System Having a Single Input-to-State Gain With a System Having a Region-Dependent Input-to-State Gain

For an ISS system, by analyzing local and nonlocal properties, it is obtained different input-to-state gains. The interconnection of a system having two input-to-state gains with a system having a single ISS gain is analyzed. By employing the Small Gain Theorem for the local (resp. non-local) gains composition, it is concluded about the local (resp. global) stability of the origin (resp. of a compact set). Additionally, if the region of local stability of the origin strictly includes the region attraction of the compact set, then it is shown that the origin is globally asymptotically stable. An example illustrates the approach.

I. INTRODUCTION

The use of nonlinear input-output gains for the study of the stability of nonlinear was introduced in [START_REF] Zames | Functional analysis applied to nonlinear feedback systems[END_REF], [START_REF] Zames | On the input-output stability of time-varying nonlinear feedback systems part I: Conditions derived using concepts of loop gain, conicity, and positivity[END_REF] by considering a system as an input-output operator. The condition that ensures stability, called Small Gain Theorem, of the resulting interconnected system is based on the contraction principle ( [START_REF] Zames | On the input-output stability of time-varying nonlinear feedback systems part I: Conditions derived using concepts of loop gain, conicity, and positivity[END_REF]).

The works [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF] and [START_REF] Sontag | Further facts about input to state stabilization[END_REF] introduce a new concept of gain relating the input to system states. This notion of stability, called Input-to-State Stability (ISS), combines Zames and Lyapunov approaches ( [START_REF] Sontag | The ISS philosophy as a unifying framework for stability-like behavior[END_REF], [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF]). Characterizations in terms of dissipation and Lyapunov functions are given in [START_REF] Sontag | On characterizations of the input-to-state stability property[END_REF] and [START_REF] Sontag | New characterizations of input-to-state stability[END_REF].

In [START_REF] Jiang | Small-gain theorem for ISS systems and applications[END_REF], the contraction principle is used in the ISS notion to obtain an equivalent Small Gain Theorem. A formulation of this criteria in terms of Lyapunov functions may be found in [START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF] and [START_REF] Liberzon | Small-gain theorem of LaSalle type for hybrid systems[END_REF].

Besides stability analysis, the Small Gain Theorem may also be used for the design of dynamic feedback laws satisfying robustness constraints. The interested reader may see [START_REF] Doyle | Feedback Control Theory[END_REF], [START_REF] Freeman | Robust Nonlinear Control Design: State-Space and Lyapunov Techniques[END_REF], [START_REF] Isidori | Nonlinear Control Systems II[END_REF] and [START_REF] Sastry | Nonlinear Systems[END_REF] and references therein.

Other versions of the Small Gain theorem do exist in the literature, examples of which can be found in [START_REF] Astolfi | A weak version of the small-gain theorem[END_REF]. See also [START_REF] Angeli | A tight small-gain theorem for not necessarily ISS systems[END_REF], [START_REF] Ito | State-dependent scaling problems and stability of interconnected iISS and ISS systems[END_REF] and [START_REF] Ito | Necessary and sufficient small gain conditions for integral input-to-state stable systems: A Lyapunov perspective[END_REF] for the interconnection of possibly non-ISS systems.

In order to apply the Small Gain Theorem, it is required that the composition of the nonlinear gains is smaller than the argument for all of its positive values ( [START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF], [START_REF] Liberzon | Small-gain theorem of LaSalle type for hybrid systems[END_REF]). Such a condition, called Small Gain Condition, restricts the application of the Small Gain Theorem to a composition of well chosen gains.

In this work, it is made use of the Small Gain Theorem in a less conservative way. This new condition ensures the asymptotic stability of a system by showing that if there exist two different gains compositions such that they satisfy the Small Gain Condition, not for all values of the arguments, but in two different regions, and if these regions cover the set of all positive values, then the resulting interconnected system is globally asymptotically stable. Thus, this approach may be seen as a composition of two different small gain conditions that hold in different regions: a local and a global.

The use of a unifying approach is well known in the literature, see [START_REF] Andrieu | Uniting two control Lyapunov functions for affine systems[END_REF] for the combination of control Lyapunov functions and [START_REF] Chaillet | Strong iISS: combination of iISS and ISS with respect to small inputs[END_REF] for a stability concept uniting ISS and the integral variant of ISS (namely, iISS [START_REF] Sontag | Comments on integral variants of ISS[END_REF]) properties.

This paper is organized as follows. In Section II, the basic concepts of Input-to-State Stability and the Dini derivative are presented. Also, the system under consideration, the problem statement and a motivational example are presented. In Section III, the assumptions to solve the problem under consideration, as well as the main results are presented. Section IV presents an example that illustrates the assumptions and main results. Section V contains the proofs of the main results. Concluding remarks are given in Section VI. Finally, in Section VII, auxiliary results are stated. Due to space limitations, some of the proofs were omitted.

Notation. Let S be a subset of R n containing the origin, the notation S =0 stands for S \ {0}. The closure of S is denoted by S. Let x ∈ R n , the notation |x| stands for the Euclidean norm of

x. A function f : S → R defined in a subset S of R n containing 0 is positive definite if, ∀x ∈ S =0 , f (x) > 0 and f (0) = 0. It is proper if f (|x|) → ∞ as |x| → ∞. By C k it
is denoted the class of k-times continuously differentiable functions, by K it is denoted the class of continuous and strictly increasing functions γ : R ≥0 → R ≥0 such that γ(0) = 0; it is denoted by K∞ if, in addition, they are unbounded. Let c ∈ R>0, the notation Ωc(f ) stands for the subset of R n defined by {x ∈ R n : f (x) < c}. Let x, x ∈ R ≥0 , the notation x ր x (resp. x ց x) stands for x → x with x < x (resp. x > x).

II. BACKGROUND AND PROBLEM STATEMENT Consider the system

ẋ(t) = f (x(t), u(t)), (1)
where, ∀t ∈ R ≥0 , x(t) ∈ R n and u(t) ∈ R m , for some positive integers n and m. The map f : R n × R m → R n is assumed to be continuous, locally Lipschitz on x and uniformly in u on compact sets. A solution of (1) with initial condition x, and input u at time t is denoted by X(t, x, u). Assume that the origin is an equilibrium point for the system (1), i.e., f (0, 0) = 0.

→ R such that, ∀(x, u) ∈ R n × R m , |x| ≥ α x (|u|) ⇒ D + f V (x, u) ≤ -λ x (x) (2) holds.
•

From now on, V will be assumed to be an ISS-Lyapunov function for [START_REF] Andrieu | Uniting two control Lyapunov functions for affine systems[END_REF].

Consider the system

ż(t) = g(v(t), z(t)), (3)
where, ∀t ∈ R ≥0 , v(t) ∈ R n and z(t) ∈ R m , for some positive integers n and m. The map g : R n × R m → R n is assumed to be continuous, locally Lipschitz on z and uniformly in v on compact sets. A solution of (3) with initial condition z, and input v at time t is denoted by Z(t, z, v). Assume that the origin is an equilibrium point for the system (3), i.e., g(0, 0) = 0. Consider also the following Assumption 1. There exists a continuous locally Lipschitz function W : R m → R that is an ISS-Lyapunov function for the z-subsystem. More precisely, there exist class K ∞ functions β and β satisfying, ∀z ∈ R m , β(|z|) ≤ W (z) ≤ β(|z|). Furthermore, there exist a class K function δ and a continuous positive definite function

λ z : R m → R such that, ∀(x, z) ∈ R n × R m , W (z) ≥ δ(V (x)) ⇒ D + g W (x, z) ≤ -λ z (z), (4)
where V is the ISS-Lyapunov function of x-subsystem. • System under consideration. Interconnecting systems (1) and (3) by linking the state of (1) with the input of (3) and vice versa leads to the following system

ẋ = f (x, z) ż = g(x, z). (5)
Since f (0, 0) = 0 and g(0, 0) = 0, the origin is an equilibrium point for [START_REF] Doyle | Feedback Control Theory[END_REF]. Considering the ISS-Lyapunov inequalities, after the interconnection the following implications

V (x) ≥ γ(W (z)) ⇒ D + f V (x, z) ≤ -λ x (x), W (z) ≥ δ(V (x)) ⇒ D + g W (x, z) ≤ -λ z (z)
are obtained with suitable class K functions γ and δ.

A sufficient condition that ensures stability of ( 5) is given by the following Theorem 1. [START_REF] Jiang | A Lyapunov formulation of the nonlinear small-gain theorem for interconnected ISS systems[END_REF] If, ∀s ∈ R >0 , γ • δ(s) < s. (6) Then, the origin is globally asymptotically stable for [START_REF] Doyle | Feedback Control Theory[END_REF].

Problem statement. At this point, it is possible to explain the problems that are dealt with, in this work.

• ISS gains computation. Although the use of ISS gains renders the analysis of stability easy to work with, it is not a trivial task to compute those gains;

• Small gain condition. Since the ISS gain is not unique, it might not be an easy task to find two ISS gains: one for the x-subsystem of ( 5) and another for the z-subsystem of ( 5) such that their composition satisfies [START_REF] Freeman | Robust Nonlinear Control Design: State-Space and Lyapunov Techniques[END_REF], for all positive values of the argument. An illustration of the problem that is dealt with is presented in the following EXAMPLE 1. Let the functions f, g : R × R → R and consider the system

ẋ = f (x, z) = -ρ(x) + z ż = g(x, z) = -sign(z) δ(|z|) + x, (7)
where δ will be defined below.

Let, ∀x ∈ R, V (x) = |x|, ρ(x) = 5x/ 4 -2x 2 + x 3 and, ∀z ∈ R, W (z) = |z|. Taking the Dini derivative of V in the f -direction, ∀(x, z) ∈ R × R, it yields D + f V (x, z) ≤ -ρ(V (x)) + W (z). (8) This implies that ∃ε x ∈ (0, 1) such that, ∀(x, z) ∈ R × R, ρ(V (x)) ≥ W (z) 1-εx ⇒ D + f V (x, z) ≤ -λ x (x), (9)
where λ x (•) := ε x ρ(V (•)). From now on, let ε x = 0.05. Note also that, in the interval [1/ 2 , 5/ 6 ], ρ is decreasing.

Consider the piecewise continuous function Γ defined by

Γ(s) = ρ -1 s 0.95 , if s ∈ 0, 0.95ρ 5 6 , ρ -1 + s 0.95 , if s ∈ 0.95ρ 5 6 , ∞ , (10)
where

[5/ 6 , ∞) ∋ s → ρ + (s) = ρ(s) ∈ [ρ(5/ 6 ), ∞). Remark 1.
The function Γ can be viewed as a discontinuous input-to-state gain of the x-subsystem of (7). More preciselly,

∀(x, z) ∈ R × R, V (x) ≥ Γ(W (z)) ⇒ D + f V (x, z) ≤ -λ x (x)
. Furthermore, the function Γ is "optimal", in the sense that if there exist a function Γ * : R → R and a value

s * ∈ R >0 such that Γ * (s * ) < Γ(s * ), then ∃(x * , z * ) = (0, 0) such that V (x * ) ≥ Γ * (W (z * )) and D + f V (x * , z * ) > 0.
• It follows from Remark 1 that an ISS gain for the xsubsystem of ( 7) is any class

K function γ such that, ∀s ∈ R >0 , Γ(s) ≤ γ(s). A local gain. Consider the function [0, 1/ 2 ) ∋ s → ρ -(s) = ρ(s) ∈ [0, ρ(1/ 2 )). Since ρ -is strictly increasing on its domain, it is invertible. Let γ ℓ be a class K function such that, ∀s ∈ [0, 0.95ρ(1/ 2 )), γ ℓ (s) = ρ -1 - s 0.95 . (11) Note that, γ ℓ satisfies the following inequalities ∀s ∈ 0, 0.95ρ 1 2 , γ ℓ (s) ≤ Γ(s), ∀s ∈ 0.95ρ 5 6 , 0.95ρ 1 2 , γ ℓ (s) < Γ(s). Moreover, ∀(x, z) ∈ Ω ρ(1/2) (V ) × R, V (x) ≥ γ ℓ (W (z)) ⇒ D + f V (x, z) ≤ -λ x (x). ( 12
)
Let the constant values M ℓ = 0.236 and M g = 0.245. At this point, it is possible to define the function δ of the z-subsystem of [START_REF] Isidori | Nonlinear Control Systems II[END_REF]. It is a function of class K ∞ satisfying the following inequalities ∀s ∈ (0,

M ℓ ], γ ℓ (s) < δ(s), ( 13
) ∀s ∈ [M g , ∞), Γ(s) < δ(s), ( 14
) ∀s ∈ ρ 5 6 , M ℓ , δ(s) < Γ(s). (15)
Equations ( 13) and ( 14) correspond to two different small gain conditions, the first may be seen as a small gain condition for small values of the argument while the last as a small gain condition for large values of the argument. Note that [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF] implies that Theorem 1 cannot be applied. 1Fig. 1 shows a plot of the functions ρ, id, Γ, γ ℓ and δ. • In this work, it will be shown that, if • there exist two ISS gains γ ℓ and γ g , for the x-subsystem of (5); • there exists one ISS gain δ, for the z-subsystem of (5);

• the compositions γ ℓ • δ and γ g • δ satisfy the Small Gain Condition, not for all values of the arguments, but for two different intervals (I ℓ , I g ⊂ R ≥0 ). In other words, ∀s ∈ I ℓ \{0}, γ ℓ •δ(s) < s and, ∀s ∈ I g \{0}, γ g •δ(s) < s; • these intervals are such that I ℓ ∩I g = ∅ and I ℓ ∪I g = R ≥0 ; then, the origin is globally asymptotically stable for [START_REF] Doyle | Feedback Control Theory[END_REF]. See Theorem 2 below for a precise statement of this result.

III. ASSUMPTIONS AND MAIN RESULTS

In this section, it is specified the assumptions on the system (5) necessary to solve the problem under consideration. The proof of the stabilization results are provided from Section V-A to Section V-C.

A. Local set of assumptions on the x-subsystem

In this section, it is introduced the set of assumptions to ensure that the origin is locally asymptotically stable for (5). Assumption 2. There exist a class K function γ ℓ and a strictly positive constant M ℓ such that,

M ℓ < lim s→∞ γ ℓ (s) = b ℓ . ( 16
) Moreover, ∀(x, z) ∈ Ω M ℓ (V ) × R m , V (x) ≥ γ ℓ (W (z)) ⇒ D + f V (x, z) ≤ -λ x (x). (17)
• Assumption 3. The composition of the functions γ ℓ and δ is such that, ∀s ∈ (0,

M ℓ ], γ ℓ • δ(s) < s. (18)
• Proposition 1. Under Assumptions 1, 2 and 3 the origin is locally asymptotically stable for system [START_REF] Doyle | Feedback Control Theory[END_REF].

no class K function γ such that (6) holds.

B. Non-local set of assumptions on the x-subsystem

In this section, it is introduced the set of assumptions to ensure that a neighborhood of the origin is globally attractive for (5). Assumption 4. There exist a class K function γ g and a strictly positive constant M g such that

M g < lim s→∞ γ g (s) = b g . (19)
Moreover, ∀(x, z)

∈ (R n \ Ω Mg (V )) × R m , V (x) ≥ γ g (W (z)) ⇒ D + f V (x, z) ≤ -λ x (x). (20)

C. Main result

In this section, it is introduced the assumption to ensure that the origin is globally asymptotically stable for [START_REF] Doyle | Feedback Control Theory[END_REF]. Assumption 6. The positive constants M ℓ and M g given, respectively, by Assumptions 2 and 4 satisfy M g < M ℓ . • Theorem 2. Under Assumptions 1-6, the origin is globally asymptotically stable for system [START_REF] Doyle | Feedback Control Theory[END_REF].

IV. ILLUSTRATION EXAMPLE 2. [Example 1 revisited.]

Verifying Assumption 1. Let the function δ be given by the inverse of δ. It follows that,

∀(x, z) ∈ R × R, W (z) ≥ δ(V (x)) ⇒ D + g W (z) ≤ -λ z (z)
, where for a given ε z ∈ (0, 1) and ∀z ∈ R, λ z (z) = ε z W (z). Thus, Assumption 1 holds. Verifying Assumption 2. The function γ ℓ is given by [START_REF] Jiang | Small-gain theorem for ISS systems and applications[END_REF] and M ℓ = 0.236. Moreover, it follows from (12) that Assumption 2 holds. Verifying Assumption 3. It follows from inequality (13) that Assumption 3 holds.

From Proposition 1, it follows that the origin is locally asymptotically stable for [START_REF] Isidori | Nonlinear Control Systems II[END_REF]. Figure 2 shows some solutions of [START_REF] Isidori | Nonlinear Control Systems II[END_REF].

• -0.2 -0.1 0 0.1 0.2
M g = 0.245. It follows from Remark 1 that, ∀(x, z) ∈ (R \ Ω Mg (V )) × R, V (x) ≥ γ g (W (z)) ⇒ D + f V (x, z) ≤ -λ x (x)
. Thus, Assumption 4 holds.

Verifying Assumption 5. It follows from inequality (14) that Assumption 5 holds.

From Proposition 2, it follows that a neighborhood of the origin is globally asymptotically stable for [START_REF] Isidori | Nonlinear Control Systems II[END_REF]. Figure 2 shows some solutions of [START_REF] Isidori | Nonlinear Control Systems II[END_REF].

•

V. PROOFS

A. Proof of Proposition 1

Proof. This proof is divided into three parts. In the first one, it is obtained a function σ ℓ that is class K ∞ and C 1 with strictly positive derivative. This function is used in the second part, where a class C 0 proper positive definite function U ℓ is defined and its Dini derivative is studied. In the last part, it is shown that U ℓ is locally Lipschitz and the local asymptotical stability of the origin is concluded by using Lemma 2. First part. Consider the class K functions δ and γ ℓ from Assumptions 1 and 2. Under Assumption 3, δ and γ ℓ are such that, ∀s ∈ (0, M ℓ ], δ(s) < γ -1 ℓ (s). Since γ ℓ is class of K, from Lemma 1, there exists a class

K ∞ function γℓ such that, ∀s ∈ R >0 , δ(s) < γℓ (s) (22) and, ∀s ∈ (0, M ℓ ], γℓ (s) < γ -1 ℓ (s). (23)
Since δ is of class K and γℓ is of class K ∞ satisfying, ∀s ∈ R >0 , inequality [START_REF] Zames | Functional analysis applied to nonlinear feedback systems[END_REF], from Lemma [10, Lemma A.1], there exists a class K ∞ and C 1 function σ ℓ whose derivative is strictly positive and satisfies, ∀s ∈ R >0 , δ(s) < σ ℓ (s) < γℓ (s).

(24)

Second part. Let, ∀(x, z) ∈ R n × R m , U ℓ (x, z) := max{σ ℓ (V (x)), W (z)}. Note that the function U ℓ is proper positive definite. Pick (x, z) ∈ R n × R m , one of three cases is possible: σ ℓ (V (x)) < W (z), W (z) < σ ℓ (V (x)) or W (z) = σ ℓ (V (x)
). The proof follows by showing that the Dini derivative of U ℓ is negative. For each case, assume that 24), the following inequality δ(V (x)) < σ ℓ (V (x)) < W (z) holds. Together with (4), it follows that

(x, z) ∈ Ω M ℓ (V ) × R m . Case 1. Suppose that σ ℓ (V (x)) < W (z). This implies that U ℓ (x, z) = W (z) and D + f,g U ℓ (x, z) = D + g W (x, z). From (
D + g W (x, z) ≤ -λ z (z). This concludes Case 1. Case 2. Suppose that W (z) < σ ℓ (V (x)). This im- plies that U ℓ (x, z) = σ ℓ (V (x)) and D + f,g U ℓ (x, z) = σ ′ ℓ (V (x))D + f V (x, z). From (24), the following inequality W (z) < σ ℓ (V (x)) < (V (x)) holds. Since V (x) ≤ M ℓ , it follows that W (z) < σ ℓ (V (x)) < γℓ (V (x)) < γ -1 ℓ (V (x)), (25)
where the last inequality follows from [START_REF] Zames | On the input-output stability of time-varying nonlinear feedback systems part I: Conditions derived using concepts of loop gain, conicity, and positivity[END_REF]. Equation [START_REF] Sontag | Comments on integral variants of ISS[END_REF] together with (25) yields

D + f V (x, z) ≤ -λ x (x). Since, ∀s ∈ R >0 , σ ′ ℓ (s) > 0, it follows that D + f,g U ℓ (x, z) = σ ′ ℓ (V (x))D + f V (x, z) ≤ -σ ′ ℓ (V (x))λ x (x). This concludes Case 2. Case 3. Let W (z) = σ ℓ (V (x)) := U * ℓ (x, z). This implies D + f,g U * ℓ (x, z)= lim sup tց0 1 t (max{σ ℓ (V (X(x, z, t))), W (Z(z, x, t))} -U * ℓ (x, z)) = lim sup tց0 max σ ℓ (V (X(x,z,t)))-σ ℓ (V (x)) t , W (Z(z,x,t))-W (z) t = max{σ ′ ℓ (V (x))D + f V (x, z), D + g W (x, z)}.
The analysis of D + f,g U * ℓ is divided in two sub cases. In the first one, the function D + g W is analyzed while in the last, the function D + f V is analyzed. Case 3.a. The analysis of D + g W . From (24), the following inequality δ(V (x)) < σ ℓ (V (x)) = W (z) holds. Together with Equation ( 4), it yields

D + g W (x, z) ≤ -λ z (z). Case 3.b. The analysis of D + f V . From (24), the following inequality W (z) = σ ℓ (V (x)) < γℓ (V (x)) holds. Since V (x) ≤ M ℓ , it follows that W (z) = σ ℓ (V (x)) < γℓ (V (x)) < γ -1 ℓ (V (x)), (26)
where the last inequality is due to [START_REF] Zames | On the input-output stability of time-varying nonlinear feedback systems part I: Conditions derived using concepts of loop gain, conicity, and positivity[END_REF].

Equation [START_REF] Sontag | Comments on integral variants of ISS[END_REF] together with (26) yields

D + f V (x, z) ≤ -λ x (x).
To conclude Case 3, Third part. To conclude local asymptotical stability of the origin, it remains to show that U ℓ is locally Lipschitz.

W (z) = σ ℓ (V (x)) ⇒ D + f,g U * ℓ (x, z) ≤ -min{σ ′ ℓ (V (x))λ x (x), λ z (z)} holds, since (x, z) ∈ Ω M ℓ (V ) × R m . Let Mℓ := max{c ∈ R >0 : Ω c (U ℓ ) ⊂ Ω M ℓ (V ) × {0} and Ω c (U ℓ ) is connected}. To sum up all the above cases, ∀(x, z) ∈ Ω Mℓ (U ℓ ), U ℓ (x, z) ≤ Mℓ ⇒ D + f,g U ℓ (x, z) ≤ -E ℓ (x, z), (27)
Since σ ℓ (V (•)) (resp. W ) is locally Lipschitz, U ℓ is locally Lipschitz in the region W (•) ≤ σ ℓ (V (•)) (resp. σ ℓ (V (•)) ≤ W (•)).
Since the hypotheses of Lemma 2 (in Section VII) below are verified with U (•) = U ℓ (•) and E(•) = E ℓ (•), the origin is locally asymptotically stable for [START_REF] Doyle | Feedback Control Theory[END_REF]. This concludes the proof of Proposition 1.

B. Proof of Proposition 2

Proof. This proof is analogous to the proof of Proposition 1 and divided into three parts. In the first one, it is obtained a function σ g that is class K ∞ and C 1 with strictly positive derivative. This function is used in the second part, where a class C 0 proper and positive definite function U g is defined and its Dini derivative is studied. In the last part, it is used Lemma 4 to show that the set Ω Mg (V ) × {0} is globally asymptotically stable.

First part. Consider the class K functions δ and γ g from Assumptions 1 and 4. The function γ -1 g is defined on [0, b g ) and satisfies lim sրbg γ -1

g (s) = ∞. Assumption 5 implies that, ∀s ∈ [M g , b g ), δ(s) < γ -1 g (s). Since γ g is of class K, from Lemma 1 (in Section VII), there exists a class K ∞ function γg such that, ∀s ∈ R >0 , δ(s) < γg (s) (28) and, ∀s ∈ [M g , b g ), γg (s) < γ -1 g (s). ( 29
)
Since δ is of class K and γg is of class K ∞ satisfying, ∀s ∈ R >0 , the inequality (28), from Lemma [10, Lemma A.1], there exists a function σ g that is of class K ∞ and C 1 whose derivative is strictly positive and satisfies, ∀s ∈ R >0 , δ(s) < σ g (s) < γg (s).

(30) Second part. Let, ∀(x, z) ∈ R n × R m , U g (x, z) := max{σ g (V (x)), W (z)}. Note that the function U g is proper positive definite. Pick (x, z) ∈ R n × R m , one of three cases is possible: σ g (V (x)) < W (z), W (z) < σ g (V (x)) or W (z) = σ g (V (x)
). The proof follows by showing that the Dini derivative of U g is negative. For each case, assume that (x, z)

∈ (R n \ Ω Mg (V )) × R m . Case 1. Suppose that σ g (V (x)) < W (z). Analogously to the Case 1 of proof of Proposition 1, σ g (V (x)) < W (z) ⇒ D + f,g U g (x, z) ≤ -λ z (z). This concludes Case 1. Case 2. Suppose that W (z) < σ g (V (x)). This im- plies that U g (x, z) = σ g (V (x)) and D + f,g U g (x, z) = σ ′ g (V (x))D + f V (x, z). From (30), the following inequality W (z) < σ g (V (x)) < γg (V (x)) (31)
holds. At this point, two regions of x will be analyzed:

b g ≤ V (x) and M g ≤ V (x) < b g .
Case 2.a. In the region where b g ≤ V (x), Equation (20) together with [START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF]

yields D + f V (x, z) ≤ -λ x (x). Case 2.b. In the region where M g ≤ V (x) < b g , from (29) and (31), it yields W (z) < σ g (V (x)) < γg (V (x)) < γ -1 g (V (x)). (32)
Equation ( 20) together with (32) yields

D + f V (x, z) ≤ -λ x (x). Since, ∀s ∈ R >0 , σ ′ g (s) > 0, it follows that D + f,g U g (x, z) = σ ′ g (V (x))D + f V (x, z) ≤ -σ ′ g (s)λ x (x)
. This concludes Case 2.

Case 3. Let W (z) = σ g (V (x)) := U * g (x, z). Analogously to the Case 3 of proof Proposition 1 and together with the analysis of Cases 1 and 2, the implication

W (z) = σ g (V (x)) ⇒ D + f,g U g (x, z) ≤ -min{σ ′ g (s)λ x (x), λ z (z)} holds, since (x, z) ∈ (R n \ Ω Mg (V )) × R m . Let Mg = min{c ∈ R >0 : Ω Mg (V ) × {0} ⊂ Ω c (U g ) and Ω c (U g ) is connected}. To sum up all the above cases, ∀(x, z) ∈ (R n × R m ) \ Ω Mg (U g ), Mg < U g (x, z) ⇒ D + f,g U g (x, z) ≤ -E g (x, z), ( 33
) where E g (•, •) = min{σ ′ g (V (•))λ x (•), λ z (•)} is continuous and positive definite.
Third part. Analogously to the third part of the proof of Proposition 1, it follows that U g is locally Lipschitz. From Lemma 3 and (33), it follows that, ∀(x, z) ∈ R n × R m and ∀t ∈ R ≥0 , along solutions of (5), D + U g (X(t, x, z), Z(t, z, x))=D + f,g U g (X(t, x, z), Z(t, z, x)). Since solutions of (5) are absolutely continuous functions and, along solutions of ( 5), E g is a continuous positive definite function, from Lemma 4, ∀(x, z) ∈ (R n × R m ) \ Ω Mg (U g ) and ∀t ∈ R ≥0 , the function t → U g (X(t, x, z), Z(t, z, x)) (34) is strictly decreasing. Pick (x, z) ∈ (R n × R m ) \ Ω Mg (U g ), it will be proven that U ∞ g := lim t→∞ U g (X(t, x, z), Z(t, z, x)) ≤ Mg .

To see the above suppose, by contradiction, that

U ∞ g > Mg . From the continuity of U g , ∃ε > 0 such that U ∞ g -ε > Mg and U ∞ g -ε ≤ U g (x, z) ≤ U ∞ g + ε. Since U g is proper, the constant ξ = min{E g (x, z) > 0 : (x, z) ∈ U g (x, z) and U ∞ g -ε ≤ U g (x, z) ≤ U ∞ g + ε} exists.
Recalling the definition of U g , ∃T > 0 such that, ∀t ≥ T , U g (X(t, x, z), Z(t, z, x)) -U ∞ g < ε. Moreover, from the definition of the constant ξ, U g (X(t, x, z), Z(t, z, x)) -U g (X(T, x, z), Z(T, z, x))

= t T D + U g (X(s, x, z), Z(s, z, x)) ds ≤ -ξ • (t -T ). Then, U ∞ g = lim t→∞ U g (X(t, x, z), Z(t, z, x))
= U g (X(T, x, z), Z(T, z, x))

+ lim t→∞ t T D + U g (X(s, x, z), Z(s, z, x)) ds ≤ -∞
which contradicts the fact that U g is positive definite. Therefore, U ∞ g ≤ Mℓ . In summary, the following facts hold for the function U g : 1) U g is a proper positive definite function; 2) U g decreases along solutions of (5) having initial conditions in (R n × R m ) \ Ω Mg (U g ). From facts 1) and 2), the set Ω Mg (U g ) is globally asymptotically stable for [START_REF] Doyle | Feedback Control Theory[END_REF]. This concludes the proof of Proposition 2.

C. Proof of Theorem 2

Proof. Under Assumption 6, ∃M > 0 such that M g < M < M ℓ . Under Assumptions 1, 2, 3 and Proposition 1, it follows that the origin is locally asymptotically stable. From the proof of Proposition 1, there exists a proper positive definite function given, ∀(x, z) ∈ R n × R m , by U ℓ (x, z) = max{σ ℓ (V (x), W (z)}, where σ ℓ is of class K ∞ and C 1 satisfying (24). Moreover, letting Mℓ := max{c ∈ R >0 : c > M, Ω c (U ℓ ) ⊂ Ω M ℓ (V ) × {0} with Ω c (U ℓ ) connected} every solution starting in Ω Mℓ (U ℓ ) converges to the origin.

Together with Assumptions 1, 4, 5 and the proof of Proposition 2, it is possible to define, ∀s ∈ R ≥0 , a class K ∞ function γg (s) = min{γ g (s), σ ℓ (s)} satisfying (28) and (29). Then, it is obtained a class K ∞ and C 1 function σg whose derivative is strictly positive and satisfies, ∀s ∈ R >0 , δ(s) < σg (s) < γg (s).

(30.new) Defining a proper positive definite function given

, ∀(x, z) ∈ R n × R m , by Ûg (x, z) = max{σ g (V (x)), W (z)} and the constant Mg = min{c ∈ R >0 : c < M, Ω Mg (V ) × {0} ⊂ Ω c ( Ûg ) with Ω c ( Ûg ) connected}, it follows from the proof of Proposition 2 that the set Ω Mg ( Ûg ) is globally asymptotically stable. Since, ∀s ∈ R >0 , σg (s) < σ ℓ (s), it follows that, ∀(x, z) ∈ (R n × R m ) \ {(0, 0)}, Ûg (x, z) < U ℓ (x, z). This inequality implies that, ∀c ∈ R >0 , Ω c ( Ûg ) ⊂ Ω c (U ℓ ). Then, the following inclusion holds Ω Mg ( Ûg ) ⊂ Ω M ( Ûg ) ⊂ Ω M (U ℓ ) ⊂ Ω Mℓ (U ℓ ). (35
) Thus, every solution of (5) starting in (R n × R m ) \ Ω Mℓ (U ℓ ) converges to Ω Mg ( Ûg ), in finite time. Then, due to (35), Ω Mg ( Ûg ) ⊂ Ω Mℓ (U ℓ ) holds, and thus solutions will converge to the origin, as t → ∞.

From the above, combining the local asymptotical stability of the origin with its global attractivitty it is concluded that the origin is globally asymptotically stable for [START_REF] Doyle | Feedback Control Theory[END_REF].

VI. CONCLUSION AND PERSPECTIVES

In this work, the authors shown that it is possible to make use of local and non-local input-to-state properties of an ISS system, in order to derive an "optimal" ISS gain. As a result of such approach, it is possible to apply the Small Gain Theorem in a less conservative way by deriving local and non-local small gain conditions to ensure the stability of an interconnected system.

In a future work, the authors will generalize the above results for the case in which there exist four ISS gains: two for each subsystem. Moreover, the authors also intend to use the region-dependent gain condition to develop a methodology for the design of feedback laws under different gains constraints. (36)

VII. AUXILIARY RESULTS

Let also p, q be two constants and α be a class K function such that, 0 < p < q and, ∀s ∈ [p, q], β • α(s) < s. , if q + ε < b and s ∈ [p, q], A + B(sq), if q + ε < b and s ∈ [q, q + ε), α(s) + s, if q + ε ≥ b or s ∈ [q + ε, ∞), (38) is such that, ∀s ∈ R >0 , α(s) < β(s).

(39) Moreover, ∀s ∈ [p, q], it also satisfies β(s) < β -1 (s).

(40) Due to space constraints, the proof of Lemma 1 is not provided in this paper. Due to space constraints, the proof of Lemma 4 is not provided in this paper.

Fig. 1 .

 1 Fig. 1. Plot of the functions id (dotted black line), Γ (dashed blue line), γ ℓ (dash dotted red line) and the continuous function δ (solid blue line), in the interval [0.225, 0.25]. The vertical lines are the values M ℓ = 0.236 and Mg = 0.245, respectively.

Fig. 2 .

 2 Fig.2. Solutions to[START_REF] Isidori | Nonlinear Control Systems II[END_REF] for initial condition starting at a ball centered at the origin with radius, respectively, given by M ℓ and 5.EXAMPLE 3. [Example 1 revisited.]Verifying Assumption 4. Let a class K ∞ function γ g be such that, ∀s ∈ [0.95ρ(5/6), ∞), γ g (s) = Γ(s). Moreover,

  where E(•, •) := min{σ ′ ℓ (V (•))λ x (•), λ z (•)} is continuous and positive definite.

Lemma 1 .

 1 Let β be a class K function with b = lim s→∞ β(s).

  Then, the class K ∞ function β given by β) + min{s, K}, if p = 0 and s ∈ [0, p), α(s) + min s, β -1 (s)-α(s) 2

Lemma 2 .

 2 [START_REF] Praly | Fonctions de Lyapunov, stabilité, stabilisation et atténuation de perturbations[END_REF] Théorème 2.133] Let S ⊂ R k be a be a neighborhood of the origin. Let also the class C 0 function h : R k → R k and consider the system ẏ = h(y). If there exist a positive definite and locally Lipschitz function U : S → R and a positive definite function E : S → R such that, ∀y ∈ S, D + h U (y) ≤ -E(y). Then, the origin is locally asymptotically stable for ẏ = h(y).

Lemma 3 .

 3 [13, Lemme 1.28] Let the measurable and essentially bounded function d : R → R p and the classC 0 function h : R k × R p → R k . If U : R k → Ris locally Lipschitz, then, for all maximal solutions Y (t, y, d) of the system ẏ = h(y, d(t)) defined in the interval (t -, t + ), the function t → U (Y (t, y, d)), defined over (t -, t + ), is locally Lipschitz and, for almost every t ∈ (t -, t + ), ∂U (Y ) ∂t (t, y, d) = D + U (Y (t, y, d)) = D + h U (Y (t, y, d)). Moreover, if d is continuous, the above equality holds, ∀t ∈ (t -, t + ).

Lemma 4 .

 4 Let Y : R → R k be an absolutely continuous function, U : R k → R be a locally Lipschitz proper positive definite function and E : R k → R be a continuous positive definite function. Define, ∀t ∈ R,U (t) = U • Y (t) and E(t) = E • Y (t). If, ∀t ∈ R, D + U (t) ≤ -E(t), then, ∀t ∈ R, U (t) is strictly decreasing.

•

  Assumption 5. The composition of the functions γ g and δ is such that, ∀s ∈ [M g , ∞), γ g • δ(s) < s.

	(21)
	•
	Proposition 2. Under Assumptions 1, 4 and 5, there exist a
	proper definite positive function U

g and a positive constant Mg such that the set Ω Mg (U g ) is globally asymptotically stable for system

[START_REF] Doyle | Feedback Control Theory[END_REF]

.

To see this fact, note that M ℓ < 0.95ρ(1/

). Since δ is of class K∞ and from[START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF], ∀s ∈ (0.95ρ(5/ 6 ), M ℓ ), s < δ-1 • Γ(s). Thus, there exists

The works of the first and the third authors are partly supported by HYCON2 Network of Excellence Highly-Complex and Networked Control Systems, grant agreement 257462.