Mena B Lafkioui

Codeswitching and artistic performance among multilingual minorities in Flanders

Keywords: multilingual codeswitching, artistry, improvised theatre, interethnicity, glocality

Viewed from an interactional sociolinguistic perspective, this research addresses multilingual codeswitching practices of youngsters with North African roots from the super-diverse Flemish city of Ghent. Particular attention is paid here to their artisticmainly interethnic -performances, as these play a vital role in constructing ethnic and social belonging by voicing "glocal" identities. In so doing, the study provides evidence of the importance of multilingualism and codeswitching in the accommodation, socialization, and emancipation of these youngsters.

Addressing interethnic multilingual interactions

The present study investigates the different formal and functional connections between language use and representation in codeswitched multilingual interactions of young people with a North African -mainly Amazigh-speaking -heritage background living in the Flemish metropole of Ghent. This Belgian city is characterized by both official Dutch monolingualism, as from 1963, and (non-official) multilingualism pertaining to its superdiversity [START_REF] Mena | Le français face à la « super-diversité » dans la ville métropole de Gand[END_REF]).

The study accounts for the importance of heteroglossic practices -i.e., multilingual interactions relating to diverse intersubjective voices construed from diverse sociocultural interactional positions -and hence of codeswitching in the accommodation, socialisation, and emancipation of multilingual speakers, particularly in minority and migration contexts (for more about heteroglossia, see [START_REF] Bakhtine | Discourse in the novel[END_REF][START_REF] Ivanov | Heteroglossia[END_REF]). Accordingly, it provides evidence of how multilingual codeswitching fulfils various and varying pragmatic functions, which pertain to both the linguistic and extralinguistic contexts in which the selected languages are used. It also demonstrates to what extent the different languages -imagined or practiced -contribute to the construction and consolidation of collective "minority" identities of these youngsters, by analysing their multilingual practices in mainly informal and semi-informal contexts (e.g., families, associations, public spaces including shopping malls).

Codeswitching is understood here in its broad sense and includes alternational switches between whole utterances in different languages (generally at clause boundaries), insertional switches of words or multi-words into morphosyntactic configurations of a base language, as well as a combination of the two switch types or even a combination of grammatical configurations from different languages leading into a new hybrid configuration.

A special focus in the study is put on interethnic artistic performances, and in particular on improvised theatre, which often form oppositional vernacular discourses that relate to hybrid urban cultures. Much like hip hop performances, improvised theatre constitutes critical interpretations of society, questioning everything that is inadequate with its structures and organizations considered legitimate. These artistic performances engage interactions based on vehicular and vernacular forms of several languages, such as Dutch, Tamazight (aka Berber), French, English, Arabic, Turkish, Spanish, German, and Serbo-Croatian. They reveal not only the varying multilingual competencies and skilfulness of these adolescents but also their remarkable metadiscursive awareness, playing on all that intertextuality and multimodality (conversation, song, dance) has to offer. The performances investigated are highly heteroglossic and dynamic in nature, but at the same time they are also conventionalized to some extent in that their polyphony is managed jointly, as will be shown in the following sections.

Because globalisation processes and effects take a central place in the study, especially when anchored in multilingual and multicultural local contexts such as those of the city of Ghent, the interactional sociolinguistic approach is adopted here, as it allows to thoroughly understand the complex relationship between language, identity, and power [START_REF] Goffman | Frame Analysis: An Essay on the Organization of Experience[END_REF][START_REF] Goffman | Forms of talk[END_REF][START_REF] Gumperz | Discourse Strategies[END_REF]Gumperz and Hymes 1972;[START_REF] Mena | Le français face à la « super-diversité » dans la ville métropole de Gand[END_REF][START_REF] Rampton | Crossing. Language and Ethnicity among Adolescents[END_REF]among others). Moreover, in this approach, concepts such as identity and ethnicity are considered to be dynamic and hence continually constructed and reconstructed during the interactions. A central position is attributed to the 'interactants', rather than to disembodied language, who jointly construct and reconstruct meaning, and produce and reproduce cultural values, such as identities and ethnicities.

Consequently, interaction is the point where the 'Social' and the 'Subject' meet, their only and unique place of mutual formation.

With regard to the multiplicity (age, class, gender, ethnicity, geopolitical setting, social status, sexual orientation, religion...) and the hybridity of identity and ethnicity, the interactional perspective permits in-depth analyses of an unlimited scale of variables by correlating social variables with linguistic variables in a multidimensional frame.

Accordingly, equal importance is given to linguistic aspects of the interactions as to their extralinguistic aspects, such as the historical, social, cultural, and political context in which they occur and to which they contribute.

The approach is augmented by using notions from the linguistic ethnographical and anthropological perspective. Most notably the focus on the factor power (especially political power) is crucial for this study, in the sense that speech as a social process can be a source of gaining and exercising power, and thus is fundamental for the (re)construction of identities by minorized or oppressed groups and individuals [START_REF] Blommaert | The Sociolinguistics of Globalization[END_REF][START_REF] Bourdieu | Ce que parler veut dire[END_REF][START_REF] Bourdieu | Language and Symbolic Power[END_REF][START_REF] Gal | Language Ideologies, Practices and Polices: Language and the Future of Europe[END_REF][START_REF] Mena | Multilingualism, Multimodality and Identity Construction on French-Based Amazigh (Berber) Websites[END_REF][START_REF] Blackledge | Ideologies of Language in Multilingual Contexts[END_REF].

The study is based on a vast multimodal corpus built following qualitative and ethnographical approaches, which have proved to be adequate for in-depth and socially situated research into multilingual practices. These practices are particularly challenging not only as to their study but also as to how to collect them, since they require increased and improved methodological reflexivity, flexibility, and cooperativeness on behalf of the researcher during the fieldwork investigation [START_REF] Bradley | Liquid Methodologies: using a linguistic ethnographic approach to study multilingual phenomena[END_REF][START_REF] Costley | Methodological Principles for Researching Multilingually: Reflections on Linguistic Ethnography[END_REF][START_REF] Gumperz | The communicative competence of bilinguals: Some hypotheses and suggestions for research[END_REF][START_REF] Gumperz | Discourse Strategies[END_REF][START_REF] Holmes | How to research multilingually: Possibilities and complexities[END_REF][START_REF] Hymes | Ethnography, Linguistics, Narrative Inequality: Towards an understanding of Voice[END_REF][START_REF] Mena | Méthodologie de recherche en géolinguistique[END_REF][START_REF] Rampton | Methodological foundations in linguistic ethnography[END_REF]. The Ghent corpus is part of this multimodal corpus; special attention is paid here to young people, due to the dynamics attested in their relationships between code choice and switching, identity construction, and sociocultural inscriptions [START_REF] Hewitt | White Talk Black Talk[END_REF][START_REF] Lafkioui | Pratiques et représentations linguistiques en contexte multilingue. Le cas des Berbères en Belgique[END_REF][START_REF] Lafkioui | Identity construction through bilingual Amazigh-Dutch "digital" discourse[END_REF][START_REF] Mena | Multilingualism, Multimodality and Identity Construction on French-Based Amazigh (Berber) Websites[END_REF][START_REF] Mena | Le français face à la « super-diversité » dans la ville métropole de Gand[END_REF][START_REF] Rampton | Crossing. Language and Ethnicity among Adolescents[END_REF].

In what follows, the Ghent's youngsters with North African roots -henceforth GYNAR -and their socioeconomic and ethnocultural background will be addressed, along with their language practices and representations, including their artistic performances and in particular those pertaining to improvised theatre. Evidence will be given of the importance of codeswitching in constructing ethnic and social belonging by voicing glocal identities.

Ghent's Youngsters with North African Roots (GYNAR)

North African immigration towards Flanders, and hence towards Ghent, began in the 1950s, the majority of which is Amazigh-speaking and mainly stems from the Rif area (North, Northeast, and Northwest Morocco). Tamazight is a language family which forms a branch of the Afroasiatic phylum and counts about forty languages, which cover entire North Africa, including the Sahara and the northern and western Sahel (Mali, Niger, and Burkina Faso) and the Canary Islands where Amazigh inscriptions (i.e., Libyco-Berber inscriptions) are found that go back to ancient times [START_REF] Mena | Berber languages and linguistics[END_REF].

When the first North Africans arrived in Ghent in the 1950s, their social profile was essentially that of the working class. Nowadays, they form a quite heterogeneous group, with an upcoming middle class, thanks to a second generation which is relatively better educated in Dutch, although problems relating to academic success persist and still prevent the social mobility of a significant number of youngsters with North African roots. The numerous North African stores and restaurants -especially Moroccan onescreated in the working-class neighbourhoods where they generally live not only contribute to the formation of this specific middle class but also to the intra-ethnic and inter-ethnic networks that they make up. Intra-ethnic networks were formed fairly quickly upon their arrival in Ghent (cafes, mosques, grocery stores) and allowed to meet their ethnocultural needs and keep a bond with their ancestral culture. Interethnic networks, on the other hand, are much more recent, largely because of the lack of interest on the part of local and national authorities in ethnocultural diversity and its relation to social inequality. Some of these middle class North African families also participate in what is called in Dutch de witte vlucht 'the white fugue' when they move to more affluent neighbourhoods and try to avoid sending their children at what are called "black schools". These latter schools, also commonly called concentratiescholen 'concentration school' in Dutch, generally stand for those schools with a high percentage of pupils with an ethnocultural non-Western minority background.

An example of an intra-ethnic network is the association "Tilelli" (Association Culturelle N'Imazighen asbl / Kulturele Vereniging N'Imazighen vzw), which was founded in 1995 by a small group of Amazigh and French-speaking Kabyles (Amazigh group of North Algeria). It suffices to have an interest in the Amazigh languages and cultures to be able to join this association, which serves for socialization through both Amazigh and intercultural values, as is expressed by its president Amar Sadat and his daughter Luna Sadat in their interview "L'association Tilelli comme exemple de politique d'intégration à Gand. La participation des immigrés à la politique d'intégration", conducted by Stéphane [START_REF] Bienvenue | L'association Tilelli comme exemple de politique d'intégration à Gand. La participation des immigrés à la politique d'intégration[END_REF].

On the other hand, in interethnic associations, generally subsidized by the city of Ghent or the province of East Flanders, Dutch is used as the vehicular language and as a means of intercultural socialization, although practices of codeswitching with other languages are commonplace. In the following sections, I will present some key results of my surveys with the association federation vzw JONG and its two associations "Meisjeswerking -Het Meisjeshuis" (for girls) and "Jeugdhuis Kaarderij" (for boys and girls), which I have been investigating between 2005 and 2019. These two associations are located in the "globalized" district of Rooigem-Brugse Poort in Ghent (see Figure 1 below; western part of Ghent). The association federation vzw JONG is of special interest for this study, because of the socializing efforts it makes, while paying attention to diversity and inequality (https://www.vzwjong.be/over-ons/waarden).

It is worth mentioning that the GYNAR attend more or less assiduously to local Koranic schools, often under parental constraint; the main motivation being learning Standard Arabic -although they are generally Amazigh-speaking -and instruction in the doctrine and practices of Islam.

Figure 1. vzw JONG's associations in Ghent

GYNAR's language and cultural repertoire

The multilingual language repertoire of Ghent's citizens of North African descent generally comprises vehicular and vernacular varieties of Dutch, Tamazight (mostly Tarifit, i.e., the Amazigh language of the Rif area; North, Northeast, and Northwest Morocco), French, Arabic (mostly colloquial varieties), English, and sometimes also Spanish and German, according to the sociohistorical situation and the migratory trajectory of the families concerned. French plays a significant role in the repertoire of these "Gantois", because it remains an interactive resource with a high sociocultural status, a "symbolic capital" [START_REF] Bourdieu | Ce que parler veut dire[END_REF], which promotes both social ascent and "trans-nationalization" of local roots; in other words, it allows to create specific global localities [START_REF] Mena | Le français face à la « super-diversité » dans la ville métropole de Gand[END_REF]. French is mostly spoken among family members and friends, where it contributes to the reproduction of the multilingual intimate framework. Offline, French is regularly encountered in networks with an open language and socioeconomic profile and often settled in the area of residence, such as for instance in the youth associations cited above.

The multilingual practices of the GYNAR also function as powerful vectors in creating and promoting specific group identities. As such, great linguistic creativity is attested among them who, for social and/or ethnic reasons, feel the need to construct and express their "Self" in relation to the "Other" ("othering") and in opposition to hegemonic local (e.g., Ghent, Flemish) or trans-local (e.g., Belgian, Western) forces. This is reflected in particular in artistic creativity, wherein hip-hop culture is an important component.

GYNAR rap, such as the rap performed by Fatih from Ghent, as well as the rap performances of other Belgian rappers and hip-hoppers with North African roots, such as the well-known Flemish performers SLM and Soufiane Eddyani, are marked by their heterogeneous style and make use of various languages like Dutch, French -especially French of young people and of suburbanites of Brussels and France -English, especially in its vernacular forms, as well as Tamazight and North African Arabic, and sometimes also Spanish. For some of these youngsters, young Brussels residents with North African roots represent not only the language norm, as regards vernacular French anyway, but also the socio-cultural norm. The latter norm is manifested for example in cultural modes, such as fashion, having a dog (of the intimidating type), or having a sports motorcycle.

So, these youngsters adopt and contribute to a set of hybrid and hierarchical cultural and language norms, where Dutch predominates as an interactive means (instrumental function), while French, Tamazight, and Arabic serve rather as a marker of social and (sub-)cultural belonging allowing the expression of alternative voices and identities (symbolic function). These indexicalities are primarily reflected as such in the data of my corpus, of which some examples are given here below and in next section. This does not mean, however, that no symbolic functions are ascribed to Dutch or that the other languages do not fulfil instrumental functions. In fact, when it comes to codeswitching, the choice of language is often motivated by several linguistic and extralinguistic factors concurrently (see e.g., example 2 below).

Most of the codeswitches in the hip-hop/rap performances of these youngsters happen in sentence initial or final position and may be of both the insertional and alternational type.

The switches serve various purposes, pertaining to the semantics, pragmatics, and poetics of the texts. For instance, codeswitching is often used for content structuring, which may be instantiated by means of rhyme, line division, and refrain, as well as for content foregrounding, which may be done through repetition, parallelism, and contrast, as is shown in example 1 below (similar mechanisms are found, for example, in Moroccan Arabic rai music, see e.g., [START_REF] Davies | Code switching as a poetic device: Examples from rai lyrics[END_REF] in Canadian rap, see e.g., [START_REF] Sarkar | Multilingual Codeswitching in Quebec Rap: Poetry, Pragmatics and Performativity[END_REF]. These mechanisms are skilfully chosen and always have the discursive coherence and dynamics of the text in mind, which is quite a complex endeavour when combining different languages in a "glocal" context.

Sentence-initial switches are regularly used as vocatives, which is the case exemplified in (1), presenting the refrain from SLM's song Pech 'bad luck'. The Moroccan Arabic vocative ya l'habiba (*ya l-ḥbiba) 'oh darling' introduces the first line and is followed by the vernacular English "I got ya back", from which is switched into Dutch in the second line, where the English word "shine" is inserted. The third line continues in Dutch but ends in Moroccan Arabic rani mhak (*ṛa-ni m3a-k), from which is switched back to Dutch in line four. Lines five and six are both introduced by the common English vocative "baby baby", which is followed by a text in Dutch, wherein the word pech, as in the title, is repeated several times.

( Although the base language of this song is Dutch, its refrain contains some interesting switches, which may cast some doubt on the base language of certain lines of the song.

For instance, the sentence of the first line may be drawn on an English morphosyntactic template, although Dutch would also do fine here and may be regarded as the calquing Apart from the commercialised performances, most hip-hop expressions are generated during spontaneous and playful interactions. Indeed, it is while having fun and being creative that young people generally (re)negotiate the relationship between the languages that make up their multilingual repertoire and the sociocultural groups to which they adhere [START_REF] Hewitt | White Talk Black Talk[END_REF][START_REF] Lafkioui | Les Berbères et leur langue: le cas des immigrés berbères en Belgique[END_REF], 2008[START_REF] Mena | Le français face à la « super-diversité » dans la ville métropole de Gand[END_REF][START_REF] Rampton | Crossing. Language and Ethnicity among Adolescents[END_REF]. That is why these youngsters often voice their group identities in non-explicit ways, by means of all sorts of interactive markers, such as those pertaining to style, gender, and accent, but also those relating to the type of activity in which they participate, whether online or offline.

As a matter of fact, even during football sessions, for instance, codeswitching practices are part of their interactive routines and serve to signify specific contents and values.

Example (2) accounts for this and stems from one of the many football sessions that I have recorded at Jeugdhuis Kaarderij.

( The utterance starts with the common Moroccan Arabic expression wa 3ayyeqti 'you exaggerate > come on', which is repeated in the hybrid switch wa HANDICAP 3ayyeqti

'oh you nitwit come on', where it is broken up by the insertion of the Dutch noun handicap, which is prosodically emphasised. Interestingly, in the GYNAR's informal verbal interactions (especially of the male youngsters), the noun handicap is frequently employed as a frozen interjection with all sorts of discursive values, mostly pertaining to the expression of emotion; it has high indexical value. Consequently, the noun handicap has been grammatically reanalysed as a discursive marker, which endows it with much more leeway for both syntactic and pragmatic flexibility, at least when occurring in codeswitched text. The insertion of handicap in preverbal position does not accord well with Moroccan Arabic's word order, which would give preference to wa 3ayyeqti a handicap (preceded by the vocative a), if one would use handicap as a vocative interjection here. As a matter of fact, it would not be a felicitous configuration in Dutch (e.g., Flemish of Ghent) either, but it would be in Tarifit, which may hint to the latter language as the base language of the hybrid switch. Note that the interjection handicap

has not yet developed into a discursive filler (see [START_REF] Poplack | Sometimes I'll start a sentence in Spanish y termino en español: Toward a typology of code-switching[END_REF], as it is not entirely independent of the syntax of the codeswitched configurations wherein it occurs. While for the GYNAR their multilingualism and their heteroglossia index symbolic capital (internationalization, emancipation, agency), this is not necessarily the case for all Ghent residents. I was able to observe during my fieldwork investigations that these alternative language practices are also perceived, especially by "outsiders", as forms of non-assimilation to the official Dutch-speaking standard, and therefore as an index of refusal of integration into Flemish society. Nevertheless, some recent efforts of opening up by the official Belgian media towards more diverse and inclusive language use is recorded. For instance, the expression ewa drerrie 'come on guys' (< *iwa d-drari, VOC DEF-boys.PL; Moroccan Arabic) has been chosen as the Children's Word of the Year 2020 on Ketnet, a youth channel of the national broadcasting radio and television network VRT (Vlaamse Radio en Televisie), whose language advisor (Ruud Hendrickx) says the following about the expression:

"De keuze voor "ewa drerrie" laat zien dat kinderen één grote diverse en inclusieve groep vormen. … Kinderen pikken woorden en uitdrukkingen van elkaar op en breiden zo de Nederlandse omgangstaal uit met taalelementen uit de hele wereld."

'The choice for "ewa drerrie" shows that children form one large diverse and inclusive group. Children pick up words and expressions from each other and thus expand colloquial Dutch with language elements from all over the world.'

(Source: Kinderen kiezen 'Ewa drerrie' als woord van het jaar, 14/12/2020 om 16:52 door wle, Bron: BELGA; https://www.standaard.be/cnt/dmf20201214_95719610)

The rap group SLM (see also example 1) and their song "Drarrieville" has certainly had a hand in this state of affairs. This song title is a bilingual compound based on the American English "X-ville", like e.g., in Nashville, wherein the first part (X) is filled in by the Moroccan Arabic "Drarrie" (*drari), which means 'guys'. This rap group also had a huge success with their previous song "Ewa ja" (the North African vocative interjection 'oh yeah'), whose title was chosen as the favourite expression by Ketnet in 2017.

Given the evidence discussed above and in what follows, codeswitching practices among the GYNAR contribute, in its diverse forms, to the phenomenon of "vernacular globalization" (Appadurai 1996: 10;Blommaert 2010: 75-77), that is to say -it contributes to the multiple ways in which the processes of globalization are combined with specific local contexts, such as those of the multilingual North African minorities, the majority of whom live in "globalized" neighbourhoods -the so-called "black" neighbourhoods -deeply marked by super-diversity. GYNAR's creative codeswitching not only echoes their everyday language use but also the particular local and trans-local ideologies the different languages in use index. Power is of great importance in the multilingual interactions of these youngsters, as these interactions point to the relevance of social, cultural, and political minorization and oppression that they experience in their daily life. Their codeswitching also invites the local and trans-local authorities to rethink society by accepting language and cultural diversity. The GYNAR take pride in the ability to blend linguistic and cultural resources, which mirrors their super-diverse identities. In so doing, they question the establishment and its traditional ideologies about language (monolingualism) and culture (monoculturalism) by calling for more symmetrical interactional power relations based on multilingualism and multiculturalism. The evidence discussed in next section accounts for this.

Improvised interethnic theatre

Theatre is one of the artistic performance types in which the GYNAR also regularly partake, especially in its improvised format. Improvised theatre is often offered and even encouraged in superdiverse socialisation networks, such as VZW JONG, because of its reflexive and learning quality, as well as its dynamic and inclusive approach. Improvised theatre is regarded here as a kind of interactive staged process that leads to an official performance which relates to Goffman's "platform format", corresponding to "the arrangement found universally in which an activity is set before an audience" [START_REF] Goffman | The Interaction Order. American Sociological Association, 1982, Presidential Address[END_REF]. Improvised theatre engages specific interactive practices pertaining to the planning, rehearsal, and official presentation of the play, for which I refer to Johnstone (1981) for more details. Central to this kind of theatrical performance is to jointly create a set of improvisation routines by means of certain interactive techniques pertaining to language choice and use, as well as to the extralinguistic context, which may engage bodily expression (e.g., mimes, music, dance), decoration, and theatre paraphernalia, for instance. In other words, by mutually exploring and learning how to improvise, the play comes into being progressively. There is always a conductor present, who gently steers the improvisation process and gives advise not only regarding the artistic and technical aspects of the play but also regarding ethical and sociocultural values and conduct.

The case presented here concerns the interethnic play Ben ik dat nu? 'Is this really me?', which is performed by a group of girls of various ethnic origins -among which the Amazigh ethnical background -attending the Meisjeshuis association in Ghent in the "globalized" district of Rooigem-Brugse Poort, that I followed up and recorded (audio The first excerpt is from the thread of "Madame Patoe", which concerns social isolation, death, illness, and individualism, all aspects of life that the girls consider to be typical of Western society.

(3) So: s' Zomers schijnt de zon in het hoekje van Sibel haar huis.

'In summer, the sun shines in the corner of Sibel's house.'

All but So: Wie is Sibel? 'I don't know who Mrs. Patoe was. She is dead.'

The intersentential switch to Turkish by Si and Gu at the end of this excerpt reformulates the Dutch sentence by So in order to draw attention to its content and to emphasise the social issue of elderly people dying alone, even in the globalised neighbourhood of the girls.

The pragmatic functions of drawing attention and emphasis, instantiated through codeswitched reformulation, are also involved in the next excerpt, which stems from the thread of "Seppa" -from French sais pas < *je ne sais pas ('not know' < 'I do not know').

This thread deals with marginalization, ethnicization, sexuality, and the tensions that all these sociocultural matters may generate.

( Spatial anchoring, in both its geographical and social dimensions, therefore plays a primordial role in language practices and representations. And, therefore, it is at this level that the inequalities are located and not at the level of languages, which are only a symbolic reflection of it.

The play also shows how globalized multilingual interactions and hence codeswitching participate in the deconstruction of the concept of ethnicity as a monolithic and static construction, and underline its hybrid and dynamic nature, the study of which requires multidimensional approaches taking into account the plurality of individual affiliations and groups, which is important for adequately understanding superdiversity, especially that experienced by minorities [START_REF] Schiller | Beyond the Ethnic Lens: Locality, Globality, and Born-Again Incorporation[END_REF][START_REF] Lafkioui | Identity construction through bilingual Amazigh-Dutch "digital" discourse[END_REF][START_REF] Mena | Multilingualism, Multimodality and Identity Construction on French-Based Amazigh (Berber) Websites[END_REF][START_REF] Mena | Le français face à la « super-diversité » dans la ville métropole de Gand[END_REF].

Conclusion

This study has demonstrated how codeswitching practices in multilingual glocal interactions, and in particular those pertaining to artistry, contribute to questioning and renegotiating certain sociocultural relationships, including the relationship between languages and social and ethnocultural affiliations. In other words, creative multilingual codeswitching not only reflects everyday sociolinguistic hierarchies but also challenges them by calling for a revision of established language and cultural ideologies and policies.

The study has also shown how the concept of ethnicity in the culturalist and therefore static and essentialising sense can be deconstructed through joint playful action, making it an object of artistic creativity. It therefore calls into question the perspective of "ethnic absolutism" [START_REF] Gilroy | There Ain't No Black in the Union Jack[END_REF], which prevents us from understanding the diverse sociocultural relationships of minorities in their local and trans-local spatial anchoring.

In this way, artistic performance takes into account the capacity for action (agency) of these young people, which is essential for any constructive socialization.

  source for the English sentence. The code choice and switching of the first line allow to clearly set off the refrain from the rest of the text, which is mostly in Dutch. At the same time, the linguistic divergence of the expressions of the refrain permits to introduce the addressee (ya l'habiba) and the theme of the song (I got ya back). With the last word of the first line, the English "back", rhymes mhak from Moroccan Arabic in line three, which is part of the insertional switch rani mhak. This latter switch is interesting because it completes as object complement the sentence started in Dutch with the duplicated verb phrase gij weet gij weet 'you know you know'. It is difficult to distinguish in this line which of the two languages, Dutch or Moroccan Arabic, serves as the base language, since the grammar of both works fine here, even though they are genealogically (Germanic versus Semitic) and typologically (SVO versus VSO) unrelated. The Moroccan Arabic component which ends the line repeats the theme of the song, expressed in English in the first line, i.e., 'I've got your back', which is reformulated in the subsequent line in Dutch, ik blijf plakken aan je zij 'I'll stick by your side' (again at the end of the line), and even in the following lines five and six by means of gij hebt pech 'you're out of luck', framed here ironically.Dutch-based hip-hop culture is very popular and productive among youngster with North African roots in Flanders, and even more in the Netherlands, where rappers such as the well-established Ali B. and the much fresher Riffi are very successful. Riffi has also songs wherein Tamazight (Tarifit) is the principal language, such as in the song "Playa", which indexes the summer holidays at the beaches of the Rif.

  More grammatical hybridity is witnessed in the subsequent switch zijde gij ṯeddaγreḏ of zo jong, wherein the conjugated verb phrase in Tarifit ṯeddaγreḏ 'you are blind' is used in a morphosyntactic configuration of vernacular Dutch, which usually takes an adverb in this position, such as blind, for instance. It is worth mentioning that the switches containing Moroccan Arabic and Tarifit function here as a means for channelling and expressing intense emotions, such as excitement and frustration.Codeswitching cases like in (2) are regularly recorded in my corpus. They are remarkable in that they engage structurally incongruent languages and hence challenge the validity of the concept of congruent lexicalization (Muysken 2014), wherein switching would preferably occur between genetically or typologically congruent languages. My findings rather indicate intense codeswitching within specific conventionalised multilingual formats (called "conventionalised heteroglossia" in Lafkioui 2019) as significant for enabling codeswitching of the type exemplified in (2). This latter type implies of course a certain fluency in the languages involved, which is in line with the findings presented in[START_REF] Hofweber | How different codeswitching types modulate bilinguals' executive functions: a dual control mode perspective[END_REF], as well as in Backus and Demirçay's contribution in this volume.

  and video) from the first rehearsal to the final official performance. The conductor of this play paid much attention to the individual dispositions of each of the girls and encouraged them to express themselves without holding back out of fear to be criticised or reprimanded. The whole improvisation process took place in a close setting, wherein the rehearsals consisted of a warm-up phase, followed by some performance exercises (including the joint construction of the decoration and of the outfits), and ended by an appraisal of the exercises completed in order to adjust their content and form according to what the performers thought the audience might expect. This play is of particular interest for this study as it engages multilingual repertoires of adolescent girls with various heritage languages pertaining to their different ethnic background and migration history. The multilingual practices witnessed in this play chiefly include vehicular and vernacular forms of Dutch, French, English, Tamazight (Tarifit mainly), Moroccan Arabic, Turkish, and Serbo-Croatian. During the improvisation performances, these girls have shown an extraordinary ability to manage their multilingual repertoire and to employ it creatively by drawing on intertextual and multimodal contents and techniques. By so doing, the girls reviewed and incorporated along different narrative threads all sorts of ethnic, social, and cultural stereotypes of their daily lives, of which I will present some excerpts in what follows.

'

  Who is Sibel?' So: Door het gat in het zijstraatje waar Madame Patoe woont. 'Through the hole in the side street where Ms. Patoe lives.' Sa: Wie is Mevrouw Patoe? 'Who is Ms. Patoe?' So: Ik weet niet wie Mevrouw Patoe WAS. Ik weet dat ze dood is. 'I don't know who Ms. Patoe "was". I just know she is dead.' Si & Gu: Bayan Patou'nun kim olduğunu bilmiyorum. O öldü.

  1) Ya l'habiba I got ya back VOC DEF.darling.FS 1SG have.PART POSS.2SG back 'Oh, darling I've got your back' Je moet niet kijken naar de shine, dat hoort erbij 2SG have.PRES.2SG NEG look.INF to DEF shine DEM.DIST belong.PRES.3SG it.by 'You shouldn't look at the shine, that's part of it' Je hebt voor mij gekozen, ik blijf plakken aan je zij 2SG have.2SG for 1SG choose.PART 1SG stay.PRES.1SG stick.INF to POSS.2SG side 'You chose me, I'll stick by your side' Baby baby gij hebt pech pech pech pech pech pech baby baby you have.2SG bad luck bad luck bad luck bad luck bad luck bad luck 'Baby baby, you're out of luck bad luck bad luck bad luck bad luck' Baby baby gij hebt pech pech pech baby baby you have.2SG bad luck bad luck bad luck 'Baby baby, you're out of luck bad luck bad luck'

	Gij weet gij weet rani mhak
	2SG know.PRES.2SG 2SG know.PRES.2SG DEM.1SG with.2SG
	'You know you know I am with you'

  2) wa 3ayyeqti, die bal, die bal, daar moete zijn, wa HANDICAP 3ayyeqti zijde gij ṯeddaγreḏ of zo jong INTJ exaggerate.PFV.2SG, DIST ball, DIST ball, DIST must.AUX.PRES.2SG be, INTJ handicap exaggerate.PFV.2SG be.PRES.2SG 2SG be.blind.PFV.2SG or what young 'Come on, that ball, that ball, there you should be, oh you nitwit come on, are you blind or what, man.'

  These metalinguistic insertions occur in Dutch-based sentences, which alternate with French sentences, which introduce the new thread of "Seppa". The next switch to English is used to put emphasis on the command to "Shut up!" expressed by the other girls and at the same time also to announce So's song in English.This interethnic play, and similar ones that I have recorded, are reflections of society and more precisely of its system of social stratification. It represents social categories of the Other who delimit what is socially legitimate, accepted. The multilingual language practices engaged in these interethnic artistic performances index intercultural interactions, but yet exist in a sociocognitive space delimited by a differentiation in ethnic groups and social classes. Because, for instance, the French of young girls like Sa would hardly be welcomed in French-speaking families of Flemish origin, if she is not part of their offline social networks, which are very closed. And conversely, the French of a French-speaking "Gantoise" with Flemish roots would hardly be considered legitimate in this interethnic theatre group if she does not live in the neighbourhood. My surveys indicate a clear separation between these two social groups, which live different lives in different neighbourhoods (see Lafkioui 2019 for a study on francophonie in Flanders).

	Next excerpt comes from the thread mon rêve 'my dream', inspired by teenage girls'
	dreams and the problems they encounter in achieving them (e.g., singing, dancing,
	wealth), and this in relation to certain personality traits, such as vanity, the lack of self-
	esteem, and materialism, which they present as both challenging and epitomised by the
	Western world.
	(5)
	All but Sa: Shut up!
	So: I don't wanna be a murderer […]
	In what follows, the girls sing Rihanna's pop song Unfaithful: "story of my life,
	'Moroccan' they mean Moroccan Arabic. The character of Seppa -an adolescent of searching for the right, but it keeps avoiding me […]"
	Moroccan (Rif Amazigh) origin -presents a typification of the Other in its unacceptable
	forms, including, for example, that of ezel, seppa, ḥmar 'donkey'. These multilingual The alternational reformulation switch from French to the Dutch kijk maar, ik kan goed
	insertional switches are metalinguistic reformulations of the concept of "being zingen 'just look, I can sing well' by Sa draws attention to what follows, which is literally
	ignorant/stupid", developed out of the French expression *(je) sais pas 'I don't know', singing.

) Yo: Seppa? Qui est Seppa? D'où vient Seppa? 'Seppa? Who is Seppa? Where does Seppa come from?' All except Yo: Je ne sais pas 'I don't know.' Gu: Weet ge da seppa in het Joegoeslaafs 'ezel' betekent? 'Do you know that "seppa" means "donkey" in Yugoslavian?' So + Yo: en seppa in het Marokkaans is ḥmar […] 'And "seppa" in Moroccan (Arabic) is ḥmar.' Note that by Joegoeslaafs 'Yugoslav' the girls mean Serbo-Croatian and by Marokkaans which perfectly matches with Serbo-Croatian seppa 'donkey' and its index, reproduced here also in Dutch (ezel) and in Moroccan Arabic (ḥmar). Sa: Mon rêve, c'est d'avoir un laptop, un permis, une voiture, un Gsm, mm… je disais un Gsm, mais mon plus grand rêve, c'est d'avoir une belle voix. Je sais très bien chanter, regardez, kijk maar, ik kan goed zingen, […] 'My dream is to have a laptop, a license, a car, a mobile phone, mm ... I said a mobile phone, but my biggest dream is to have a beautiful voice. I can sing very well, look, just look, I can sing well, […]'


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

Michael Malisoff
email: malisoff@lsu.edu

New Versions of Halanay's Inequality with Multiple Gain Terms

Keywords: Stability, delays, time-varying. Supported

Halanay's inequality in its standard form is a widely used tool for the analysis of systems with delays and uncertainties, but the assumptions needed to use it are sometimes too restrictive to cover applications of interest. This paper provides new generalizations of Halanay's inequality where, instead of the usual supremum in the gain term in standard versions of Halanay's inequality, we use a weighted sum of suprema over different intervals. This allows us to derive sufficient conditions for asymptotic and input-to-state stability. We apply our results to linear systems with switched delays and other examples that illustrate how our results are less restrictive than those of other contributions available in the literature.

I. INTRODUCTION

Delayed systems play an essential role in control theory and control applications, owing to time delayed information transmission (leading to sensor delays) and time delays in control actuation (which can be represented by input delays). Lyapunov-Krasovskii techniques are well suited for systems with known constant delays, but usually are not as easily applied to systems with time-varying or unknown delays. By contrast, Halanay's inequality technique (which was initiated in [START_REF] Halanay | Differential Equations: Stability, Oscillations, Time Lags[END_REF]) is a useful stability analysis tool for the stability analysis of systems with poorly known time-varying delays. This inequality and variants of it, which complement Razumikhin's theorem (which was used in contributions such as [START_REF] Hale | Introduction to Functional Differential Equations[END_REF] and [START_REF] Zhou | Time-varying Razumikhin and Krasovskii stability theorems for time-varying delay systems[END_REF]), have been studied in several papers, such as [START_REF] Grifa | On stability analysis of discrete-time systems with constrained time-delays via nonlinear Halanay-type inequality[END_REF], [START_REF] Hien | New generalized Halanay inequalities with applications to stability of nonlinear non-autonomous time-delay systems[END_REF], [START_REF] Mazenc | Stability analysis using generalized sup-delay inequalities[END_REF], [START_REF] Mazenc | Stability analysis using new variant of Halanays inequality[END_REF], [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF], [START_REF] Pepe | A nonlinear version of Halanay's inequality for the uniform convergence to the origin[END_REF], and [START_REF] Wen | A new generalization of Halanaytype inequality and its applications[END_REF], to extend the domain of application of Halanay's stability analysis strategy.

In particular, the paper [START_REF] Ruan | Novel stability results for Halanay inequality and applications to delay neural networks[END_REF] presents notable results for time-varying Halanay inequalities of the type

v(t) ≤ -a(t)v(t) + b(t) sup ∈[t-T,t] v( ) (1)
where T ≥ 0 is a constant, v is nonnegative valued, and the term containing b is called the gain term. In [START_REF] Ruan | Novel stability results for Halanay inequality and applications to delay neural networks[END_REF], sufficient conditions for lim t→+∞ v(t) = 0 to hold are given for cases where [START_REF] Fridman | Introduction to Time-Delay Systems[END_REF] holds and where a(t) can take both positive and negative values and where b(t) is larger than a(t) on some arbitrarily large time intervals. This contrasts with the standard Halanay's inequality (e.g., [1, Lemma 4.2, p. 138]) having the form [START_REF] Fridman | Introduction to Time-Delay Systems[END_REF] where a > 0 and b ∈ [0, a) are constants. Motivated by the fact that inequalities of the type (1) can be used to study time-varying systems with time-varying delays, we revisit the main result of [START_REF] Ruan | Novel stability results for Halanay inequality and applications to delay neural networks[END_REF]. To obtain less restrictive conditions and establish input-to-state stability (or ISS) like inequalities, we consider generalized Halanay's inequalities with multiple gain terms of the type

v(t) ≤ -a(t)v(t) + k i=1 b i (t) sup ∈[t-T2,i,t-T1,i] v( ) + δ(t) (2)
where a, the b i 's, and δ (which can represent an uncertainty) are nonnegative scalar valued piecewise continuous functions; see [START_REF] Sontag | The ISS philosophy as a unifying framework for stabilitylike behavior[END_REF] for a presentation of the ISS notion. The b i 's are the coefficients in our multiple gain terms that are summed in [START_REF] Grifa | On stability analysis of discrete-time systems with constrained time-delays via nonlinear Halanay-type inequality[END_REF]. Later in Section II, we will describe persistence of excitation relations involving a and the b i 's that will ensure ISS conditions. The key idea which guides us consists of taking advantage of the knowledge of the constants T 1,i and T 2,i ≥ T 1,i , which can be derived from information on the delays of a system with poorly known time-varying delays. As in [START_REF] Ruan | Novel stability results for Halanay inequality and applications to delay neural networks[END_REF], our assumptions are satisfied by functions a which take both positive and negative values (which contrasts with our earlier results on generalized Halanay's inequalities from [START_REF] Mazenc | Stability analysis using generalized sup-delay inequalities[END_REF]- [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF], which required a to be nonnegative valued) and the largest value of b i can be arbitrarily large over arbitrarily large intervals, provided that the values a(t) of the function a are positive and large on sufficiently long time intervals. Also, our Halanay inequality generalizations [START_REF] Mazenc | Stability analysis using generalized sup-delay inequalities[END_REF]- [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF] did not use the comparison function approach that we use here, and they use only one gain term, instead of the multiple gain terms that we use here. By contrast with the main result of [START_REF] Ruan | Novel stability results for Halanay inequality and applications to delay neural networks[END_REF], we establish ISS like inequalities. By the definition of ISS, this ensures that lim t+∞ v(t) = 0 when δ is the zero function. Our examples in Section III below show that the results we obtain are less restrictive than the latest existing results about Halanay's inequality, in several circumstances. For instance, see Section III for an example where our use of multiple gain terms is shown to be useful for obtaining less conservative results than [START_REF] Mazenc | Stability analysis using generalized sup-delay inequalities[END_REF]- [START_REF] Mazenc | Stability and observer designs using new variants of Halanay's inequality[END_REF]. By contrast with [START_REF] Mazenc | Extension of Razumikhin's theorem and Lyapunov-Krasovskii functional constructions for time-varying systems with delay[END_REF], the technique we propose does not use the strictification technique from [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF].

We use standard notation, which is simplified when no confusion would arise. The dimensions of our Euclidean spaces are arbitrary unless otherwise noted. The standard Euclidean norm and induced matrix norm are denoted by |•|, |•| ∞ is the corresponding sup norm, and |•| S is the supremum over a set S. We set Z ≥0 = {0, 1, . . .}, N = Z ≥0 \{0}, R is the set of all real numbers, [0,+∞) is the set of all nonnegative real numbers, and I is the identity matrix.

II. GENERALIZED HALANAY'S INEQUALITIES

This section provides our main theorems, which we apply to systems with switched delays and other cases in Section III. Our first theorem covers cases of the form (2) with multiple suprema, and is based on a novel comparison approach. Our second theorem only allows one supremum term (i.e., k = 1 in (2)), but it provides very different sufficient conditions as compared with our first theorem.

A. Halanay's result with several sup terms

Consider any k ∈ N and constants T 1,i and T 2,i such that To analyze the behavior of the function v, we fix constants > 0 and t 0 ≥ 0, we define the functions S i by

0 ≤ T 1,i ≤ T 2,i for all i ∈ {1, ..., k} and T 2,1 ≤ .... ≤ T 2,k-1 ≤ T 2,k . (3)
S i (t) = max 0, sup ∈[t-T2,i,t-T1,i] t-T1,i a(m)dm (4)
for i = 1, . . . , k, and we introduce the comparison system

                 ẏ (t) = -a(t)y (t) + δ(t) + k i=1 [b i (t) + ]e Si(t) y (t -T 1,i ) for all t ≥ t 0 y (t) = sup ∈[t0-T 2,k ,t0] v( ) + for all t ∈ [t 0 -T 2,k , t 0 ]. (5)
We prove the following result, whose conclusion ensures that v satisfies an ISS estimate for all t ≥ t 0 when y satisfies such an ISS estimate, and where we will provide sufficient conditions for y to satisfy the required ISS estimates in our ISS corollary below, which also provides explicit ISS estimates for the function v with ( , δ) viewed as the perturbation and which gives an ISS result with respect to δ as → 0:

Theorem 1: Consider the functions v and y defined above. Then the inequality

v(t) ≤ y (t) (6)
holds for all t ≥ t 0 -T 2,k .

Proof: The second equality in ( 5) and the positiveness of imply that v(t) < y (t) for all t ∈ [t 0 -T 2,k , t 0 ]. Next, we proceed by contradiction. Bearing in mind that v and y are continuous, let us suppose that there were a t > t 0 such that v(t ) = y (t ) and v(t) < y (t) for all t ∈ [t 0 -T 2,k , t ), i.e., t is the first time t when v(t) = y (t). Then from (2) and ( 5), it follows that the inequality

v(t ) -ẏ (t ) ≤ k i=1 b i (t ) sup ∈[t -T2,i,t -T1,i] v( ) - k i=1 [b i (t )+ ]e Si(t ) y (t -T 1,i ) (7)
is satisfied. By the second equality in the comparison system [START_REF] Hien | New generalized Halanay inequalities with applications to stability of nonlinear non-autonomous time-delay systems[END_REF] A( )d y (t 0 ) > 0 for all t ≥ t 0 . In the second case, we can apply variation of parameters to the same inequality in a similar way to get y (t) > 0 for all t ∈ [t 0 , t 0 + min i∈N T 1,i ]. Repeating this process in a method of steps in the case where N = ∅ gives y (t) > 0 for all t ∈ [t 0 + min i∈N T 1,i , t 0 + ( + 1) min i∈N T 1,i ] for all ∈ Z 0 . By combining both cases, it now follows that y (t) > 0 for all t ≥ t 0 -T 2,k . Hence, for all i ∈ {1, ..., k}, we have e S i (t ) y (t -T 1,i ) > 0, so

v(t ) -ẏ (t ) < k i=1 b i (t ) sup ∈[t -T2,i,t -T1,i] v( ) -e Si(t ) y (t -T 1,i ) . ( 8
)
Since v is continuous, it follows that for each i ∈ {1, ..., k}, there is an

s i ∈ [t -T 2,i , t -T 1,i ] such that v(s i ) = sup ∈[t -T2,i,t -T1,i] v( ). Then (8) implies that v(t )-ẏ (t ) < k i=1 b i (t ) v(s i )-e Si(t ) y (t -T 1,i ) . (9)
Next consider any i ∈ {1, ..., k}, and three cases. Case 1: s i > t 0 . Recalling the nonnegative valuedness of y and δ, it follows from (5) that we have ẏ (t) ≥ -a(t)y (t) [START_REF] Mazenc | Stability and observer designs using new variants of Halanay's inequality[END_REF] for all t ≥ t 0 . By integrating [START_REF] Mazenc | Stability and observer designs using new variants of Halanay's inequality[END_REF], we obtain

y (t -T 1,i ) ≥ e - t -T 1,i s i a(m)dm y (s i ). (11)
Since s i ∈ (t 0 , t ], we deduce from the definition of t that v(s i ) ≤ y (s i ), which we can combine with [START_REF] Mazenc | Vector extensions of Halanay's inequality[END_REF] to obtain

y (t -T 1,i ) ≥ e - t -T 1,i s i a(m)dm v(s i ). (12)
It follows that [START_REF] Mazenc | Stability and observer designs using new variants of Halanay's inequality[END_REF] gives

y (t -T 1,i ) ≥ e -Si(t ) v(s i ). Case 2: s i ≤ t 0 ≤ t -T 1,i . Integrating
y (t -T 1,i ) ≥ e - t -T 1,i t 0 a(m)dm y (t 0 ) = e - t -T 1,i t 0 a(m)dm sup ∈[t0-T 2,k ,t0] v( ) + . ( 13
) Since t 0 -T 2,i ≤ t -T 2,i ≤ s i ≤ t 0 , it follows that v(s i ) ≤ sup ∈[t0-T 2,k ,t0] v( ).
Also, [START_REF] Ruan | Novel stability results for Halanay inequality and applications to delay neural networks[END_REF] gives

y (t -T 1,i ) ≥ e - t -T 1,i t 0 a(m)dm v(s i ). ( 14
)
Since t -T 2,k ≤ s i * ≤ t 0 , we can use [START_REF] Sontag | The ISS philosophy as a unifying framework for stabilitylike behavior[END_REF] to conclude that

y (t -T 1,i ) ≥ e -Si(t ) v(s i ) (15)
is satisfied. Case 3: s i ≤ t -T 1,i ≤ t 0 . Recalling that our formula (4) for S i ensures that S i is nonnegative valued, we get

v(s i * ) -e Si(t ) y (t -T 1,i ) ≤ v(s i * ) -y (t -T 1,i ) = v(s i * ) -sup ∈[t0-T 2,k ,t0] v( ) -≤ -. (16)
Hence, in all three cases, we have e Si(t ) y (t -T 1,i ) ≥ v(s i ) for i = 1, . . . , k. By combining the preceding inequalities with [START_REF] Mazenc | Stability analysis using new variant of Halanays inequality[END_REF], it follows that v(t ) -ẏ (t ) < 0, so there is t ∈ [t 0 -T 2,k , t ) such that v(t ) -y (t ) > 0, because v(t ) -y (t ) = 0. This contradicts the definition of t . This concludes the proof.

Remark 1: Theorem 1 implies that if there is a constant > 0 such that y satisfies an ISS inequality, then v satisfies an ISS inequality too that is valid for all t ≥ t 0 . Hence, Theorem 1 provides a way to conclude ISS properties from its generalized Halanay's inequalitites; see Corollary 1.

Remark 2: Theorem 1 has the following two crucial advantages. First, when several poorly known delays are present in a studied system, it improves on the stability conditions for the results available in the literature by taking advantage of the information on the delays; see our illustrations below. Second, when the constants T 2,i converge to T 1,i , then the stability conditions we obtain converge to those of the case where T 2,i = T 1,i for all i ∈ {1, ..., k}. This is in sharp contrast with the conditions of [13, Theorem 1].

Remark 3: The preceding theorem is new, even in the special case where the T 1,i 's are all zero and when a and the b i 's all have the same period P > 0 and δ = 0. In that special case, y will exponentially converge to zero provided This follows by first noting that, in this case, g also has period P . Hence, for any N ∈ N such that N P ≥ t 0 , condition (17) gives y ((N + j)P ) ≤ e -jI * y (N P ) for all integers j ≥ 0, where I * is the integral in (17); this follows by applying variation of parameters to the first equation in (5) on the interval [N P, (N + j)P ]. For t ≥ N P , this gives

y (t) = e -t t f P g( )d y (t f P ) ≤ B * e -I * (t f -N ) y (N P ) → 0 as t → +∞, (18)
where t f = Floor(t/P ) and

B * = sup e -t s g( )d : s ≥ 0, 0 ≤ t -s ≤ P , ( 19
)
and Floor is the floor function, i.e., Floor(s) is the largest integer j such that j ≤ s. Similar reasoning shows that (17) implies that v satisfies an ISS estimate with disturbance (δ, ). We illustrate the use of the criteria (17) in Section III.

B. ISS corollary

We next present a consequence of Theorem 1, which provides an ISS estimate with respect to (δ, ) that converges to an ISS estimate with respect to δ as → 0 + . The linear ISS Lyapunov-Krasovskii functional (25) in its proof motivated us to propose its persistence of excitation condition (20) (but see Remark 3 for alternative sufficient persistence of excitation conditions that ensure the required stability property for (5) under periodicity assumptions).

Corollary 1: Let the assumptions of Theorem 1 hold and t 0 ≥ 0 be given. Assume that a and the b i 's are bounded, and that there is a constant c > 0 such that

a(t) - k i=1 [b i (t + T 1,i ) + ]e Si(t+T1,i) ≥ c (20)
holds for all t ≥ t 0 . Then we can find constants

ζ i > 0 for i = 1, 2 such that v(t) ≤ ζ 1 e -ζ2t |v| [t0-T 2,k ,t0] + ζ 2 (δ(t) + e -ζ2t
) for all t ≥ t 0 .

Proof:

Consider the functional V 1, (t, y ,t ) = y (t) + k i=1 t t-T1,i [b i ( + T 1,i ) + ]e Si( +T1,i) y ( )d . ( 21
)
Its time derivative along all solutions of (5) satisfies

V1, (t) = -a(t) + k i=1 [b i (t+T 1,i )+ ]e Si(t+T1,i) y (t) + δ(t) (22)
for all t ≥ t 0 . Then, along solutions of ( 5),

V1, (t) ≤ -cy (t) + δ(t) (23)
for all t ≥ t 0 . Then let

V 2, (t, y ,t ) = V 1, (t, y ,t ) + c 2µ t t-µ t m y (s)ds (24)
with µ = max i∈{1,...,k} {T 1,i }. Then, along solutions of ( 5

v(t) ≤ y (t) ≤ V 2, (t, y ,t ) ≤ e -ςt (1 + cµ)|y | [t0-T 2,k ,t0] + δ(t) ς = e -ςt (1 + cµ)|v| [t0-T 2,k ,t0] + δ(t) ς + e -ςt (1 + cµ) (28)
for all t ≥ t 0 , proving the corollary.

C. Halanay's result with only one sup term

Let T ≥ 0. Consider the inequality

v(t) ≤ -a(t)v(t) + b(t) sup ∈[t-T,t] v( ) + δ(t) (29)
where a : R → R, b : [0, +∞) → [0, +∞), and δ : [0, +∞) → [0, +∞) are locally bounded and piecewise continuous, and where v : [-T, +∞) → [0, +∞) is of class C 1 . We use any constants > 0 and t 0 ≥ 0, the function

Γ(t) = max 0, sup ∈[t-T,t] t [a(r) -b(r)]dr , (30)
and the comparison system

       ẏ (t) = -a(t) + (b(t) + )e Γ(t) y (t) + δ(t) for all t ≥ t 0 y (t) = sup ∈[t0-T,t0] v( )+ for all t ∈ [t 0 -T, t 0 ]. (31)
The system (31) differs from system (5), because (a) there is only one (instead of k) overshoot terms in the y dynamics in (31), (b) the exponential term (4) in ( 5) having the integrand a has been replaced by a new exponential term (30) with the integrand a -b, and (c) the delays T 1,i in ( 5) are not used in (30)-(31). See Section III-C for an example where (30)-(31) lead to less conservative conditions than Theorem 1. We prove the following, which provides ISS properties for v when y satisfies ISS, and where the required stability properties for y can be checked using classical methods like in the preceding subsection, which can lead to explicit ISS like estimates for v as in Corollary 1:

Theorem 2: Consider the v and y defined above. Then

v(t) ≤ y (t) (32)
holds for all t ≥ t 0 -T . Proof: First note that the second equality in (31) implies that v(t) < y (t) for all t ∈ [t 0 -T, t 0 ]. Now, we proceed by contradiction. Bearing in mind that v and y are continuous, suppose that there is a t > t 0 such that v(t ) = y (t ) and

v(t) < y (t) (33)
for all t ∈ [t 0 -T, t ) by choosing t to be the first time t ≥ t 0 when the inequality (33) is violated as before. Then

v(t ) -ẏ (t ) ≤ b(t ) sup ∈[t -T,t ] v( ) -(b(t ) + )e Γ(t ) y (t ). (34)
Since e Γ(t ) y (t ) > 0, we have [a(r)-b(r)]dr y (t 0 )

v(t )-ẏ (t ) < b(t ) sup ∈[t -T,t ] v( )-e Γ(t
≥ e - t t 0 [a(r)-b(r)]dr sup ∈[t0-T,t0] v( ), (40)
by (31). Since s ∈ [t -T, t 0 ] and t ≥ t 0 , we deduce that

y (t ) ≥ e - t t 0 [a(r)-b(r)]dr v(s ) ≥ e - sup ∈[t -T,t ] t [a(r) -b(r)]dr v(s ) ≥ e -Γ(t ) v(s ). (41)
In both cases, we get e Γ(t ) y (t ) ≥ v(s ), which we can combine with (36) to obtain v(t ) -ẏ (t ) < 0. Hence, there is a t ∈ [t 0 , t ) such that v(t ) -y (t ) > 0, because v(t ) -y (t ) = 0. This contradicts the definition of t , so no such t can exist. This concludes the proof.

III. ILLUSTRATIONS A. Systems with switching delays

Systems with switching delays commonly arise when controls need to switch between different sensors or different actuators that have different latencies, and so are of considerable research interest in the control theory community. Therefore, we first illustrate Theorem 1 using the class of systems with switching delays from [START_REF] Mazenc | Stability analysis using generalized sup-delay inequalities[END_REF] and [START_REF] Mazenc | Stability and observer designs using new variants of Halanay's inequality[END_REF], using our less restrictive new generalized Halanay's conditions [START_REF] Grifa | On stability analysis of discrete-time systems with constrained time-delays via nonlinear Halanay-type inequality[END_REF].

Let T > 0 and T ≥ T be any constants, and consider any sequence t i satisfying T ≤ t i+1 -t i ≤ T for all i ≥ 0. As in [8, Section III.B], let τ l and τ s be any constants such that

T > 5(τ l + τ s ) (42)
and τ l > τ s ≥ 0, and we consider the system

ẋ(t) = M x(t) + N x(t -τ (t)) (43)
with x valued in R n , where τ is a time-varying piecewise continuous unknown delay such that

0 ≤ τ (t) ≤ τ s if t / ∈ E, and 0 ≤ τ (t) ≤ τ l if t ∈ E, where E = ∪ i∈N [t i , t i + T ) and T = τ s + τ l , (44)
and M ∈ R n×n and N ∈ R n×n are constant (which includes cases where τ does not switch, e.g., with T = t i+1 -t i for all i). Following [8, Section III.B] and [10, Section 4.2], we also assume the following, where M 1 ≥ M 2 for square matrices means that M 1 -M 2 is nonnegative definite: Assumption 1: There are a symmetric positive definite matrix Q ∈ R n×n and a constant q > 0 such that

Q(M + N ) + (M + N ) Q ≤ -qQ (45)
and I ≤ Q are satisfied.

In terms of the notation

L = 2|N QN |(|M |+|N |) 2 q , (46)
we also use the following lemma, whose proof consists of the first part of the proof of [10, Proposition 1]: Lemma 1: With the preceding notation and under Assumption 1, the time derivative of the function U (x) = x Qx along all solutions of (43) is such that

U (t) ≤ -q 2 U (x(t)) + Lτ 2 s sup m∈[t-τ l -τs,t] U (x(m)) (47)
for all t ∈ [0, +∞) \ E and

U (t) ≤ -q 2 U (x(t)) + 8|N QN | q sup l∈[t-τ l ,t] U (x(l)) (48)
for all t ∈ E.

In terms of the constant a * and the function b * in a * = e -q(τs+τ l )/2 and b * ( ) = 2(1-a * ) q e 2(R-q/2)(τ l +τs) (49)

where R = 8|N QN |/q, the main result in [START_REF] Mazenc | Stability analysis using generalized sup-delay inequalities[END_REF] for (43) is: Proposition 1: With the above notation, let Assumption 1 hold, and assume that

Lτ 2 s ≤ q 2 < 8|N QN | q and a * + b * (Lτ 2 s ) a * + b * 8|N QN | q < 1. (50)
Then the origin of ( 43) is a globally exponentially stable equilibrium point on R n . On the other hand, we can apply Theorem 1 from Section II-A above in the preceding case, by choosing v(t) = U (x(t)), a(t) = q 2 , k = 2, b 1 (t) = 8 q |N QN |χ Sa (t), b 2 (t) = Lτ 2 s χ S b (t), T 1,1 = T 1,2 = 0, T 2,1 = τ l , and T 2,2 = τ l + τ s (51) under suitable conditions on the parameters, where χ S is the indicator (or characteristic) function for each set S ⊆ R, meaning χ S ( ) = 1 if ∈ S and χ S ( ) = 0 if ∈ R \ S, and where S a = E and S b = R \ E. For example, if we choose τ s = 0, τ l = 0.01, t i = i for all i ≥ 0, the matrices

M = 1 2 -2 -1
, Q = 2.68767 0.789041 0.789041 1.70137 ,

and N = -2I, and q = 1.5, then the assumptions of Proposition 1 would not hold, but the assumptions of Theorem 1 would hold with = 0.001, where Q and q can be found by solving the Lyapunov equation Q a (M + N ) + (M + N ) Q a = -I for Q a , and then scaling Q a to obtain a Q that satisfies the requirements of Assumption 1 for a small enough constant q > 0. The fact that the requirements from Theorem 1 hold for any t 0 ≥ 0 in this case follows from Remark 3 above with P = 1.

In Fig. 1, we plot the solutions of (43) with the preceding values and τ (t) = 0.01 for all t ∈ E for three sets of constant initial functions (i.e., initial states), using Mathematica. They show convergence to the desired equilibrium. This illustrates how Theorem 1 can provide less restrictive conditions than the conditions in [START_REF] Mazenc | Stability analysis using generalized sup-delay inequalities[END_REF] for systems with switched delays.

P

  0 g( )d > 0, where g( ) = a( )-k i=1 [b i ( )+ ]e Si( ) . (17)

Fig. 1 .

 1 Fig. 1. Simulations of (43) Showing x 1 (t) (Red) and x 2 (t) (Blue) for Initial States (1, 1) (Solid), (-2, 2) (Dotted) and (3, -3) (Dashed-Dotted)

  Hence, since δ and the b i 's are nonnegative valued, we can collect terms on the right side of the first equality in[START_REF] Hien | New generalized Halanay inequalities with applications to stability of nonlinear non-autonomous time-delay systems[END_REF] to find a continuous function A such that ẏ (t) ≥ A(t)y (t) for all t ≥ t 0 if each T 1,i is zero, or such that ẏ (t) ≥ A(t)y (t) on [t 0 , t 0 + min i∈N T 1,i ] where N ⊆ {1, . . . , k} is the index set of all i values such that T 1,i > 0 if not all of the T 1,i 's are 0. In the first case, we can apply the method of variation of parameters to the inequality ẏ (t) ≥ A(t)y (t) on any interval of the form [t 0 , T

	t t 0

and the positiveness of and the nonnegativity of the v values, we have y (t) > 0 for all t ∈ [t 0 -T 2,k , t 0 ]. * ) to get y (t) ≥ e

  Then we can apply the method of variation of parameters to (26) and recall the structure of V 2, to obtain the required ISS estimate for y . In fact, Theorem 1 gives

), V2, (t, y ,t ) ≤ -c 2 y (t) -c 2µ t t-µ y (s)ds + δ(t) (25) for all t ≥ t 0 . Since the functions a and b i are bounded, there is a constant ς > 0 such that V2, (t, y ,t ) ≤ -ςV 2, (t, y ,t ) + δ(t) (26) for all t ≥ t 0 . For instance, since V 2, (t, y ,t ) ≤ y (t) + c t t-µ y (s)ds (27) holds for all t ≥ t 0 with the choices c = k(|b| ∞ + )e |S|∞ + c 2 and b = (b 1 , . . . , b k ), we can choose ς = c 2 min{1/µ, 1}/ max{c, 1}.

by NSF Grants 1711299 and 2009659 (Malisoff).

F.

B. Example with comparison with

 [START_REF] Ruan | Novel stability results for Halanay inequality and applications to delay neural networks[END_REF]Consider the special case

sup

of ( 2) with k = 2 and δ = 0, and with the choices

100 sin 2 t -49π 100 , T 2,2 = 2π, and b 2 (t) = 4e -π 100 sin 2 t-199π 100 , and

Then the sufficient condition (20) for lim t→+∞ v(t) = 0 to hold is that there is a constant c > 0 such that

for all t ≥ 0. Let us check that this inequality is satisfied for sufficiently small positive values and c.

To this end, notice that using the notation from Theorem 1, it follows that for all t ≥ 0, we have

where the equalities in (56) follow because a is nonnegative valued. Hence, the left side of (55) is bounded below by

for all t ≥ 0. Since

it follows that when > 0 is small enough, we have

which ensures that lim t→+∞ v(t) = 0, by Corollary 1.

On the other hand, [START_REF] Ruan | Novel stability results for Halanay inequality and applications to delay neural networks[END_REF]Theorem 1] would not apply to the preceding example. To see why, first observe that (53)

for all t ≥ 0. Then, with the choice H( ) = sin 2 ( -0.49π) + sin 2 ( -1.99π), the function

satisfies

which follows by using sin 2 ( ) = 1 2 (1 -cos( 2)) to evaluate and then bound the integrals in (62). Since lim t→+∞ G(t) = +∞, it follows that [13, Theorem 1] does not allow us to prove that lim t→+∞ v(t) = 0. By covering the preceding example which is not covered by [13, Theorem 1], it follows that Theorem 1 is less restrictive than [13, Theorem 1].

C. Illustration of Theorem 2

Using the notation from Theorem 2, and considering any constant b 0 ∈ (0, 1), consider the special case where a(t) = sin 2 (t), b(t) = b 0 sin 2 (t), and T = 2π. Then

for all t ≥ 0. It follows that sup

To apply Theorem 2, we use the comparison system

Bearing Theorem 2 and Remark 3 in mind, we obtain the following sufficient condition for stability:

for all t ≥ 0. Condition (66) holds if and only if b 0 e π(1-b0) + 2 e π(1-b0) < 1. By choosing > 0 small enough, we obtain the condition b 0 e -πb0 < e -π .

For a comparison, we next apply Theorem 1, with k = 1, T 1,1 = 0, and T 2,1 = 2π, using the comparison system

Reasoning as in Remark 3, we obtain the stability condition 2π 0

-sin 2 (s) + (b 0 sin 2 (s) + )e π ds < 0 (69)

which is equivalent to -1 + b 0 e π < -2 e π . Since > 0 is arbitrarily small, we obtain the stability condition

Note that e -π e -πe -π < e -π , so if (70) holds, then so does (67) (because xe -πx increases over x ∈ [0, e -π ]), and (67) holds with b 0 = e -π . Hence, (67) is less restrictive than (70). This example shows how Theorem 2 is less restrictive than Theorem 1, because it can cover the case b 0 = e -π that was not covered by Theorem 1.

IV. CONCLUSIONS

We proposed new stability analysis results for functions that satisfy generalized time-varying inequalities of Halanay's type. We illustrated how our results can provide less restrictive conditions than ones in the literature. Since earlier generalizations of Halanay's inequality have been shown to be effective for solving observer design problems that were beyond the scope of the observers literature (e.g., in [START_REF] Mazenc | Stability and observer designs using new variants of Halanay's inequality[END_REF]), we aim to apply our work to observer designs for continuousdiscrete, event-triggered, and switched systems.