Jack Bowers
email: jack.bowers@oeaw.ac.atlaurent.romary@inria.fr

Laurent Romary

Bridging the Gaps between Digital Humanities, Lexicography, and Linguistics: A TEI Dictionary for the Documentation of Mixtepec-Mixtec

Keywords: language documentation, digital humanities, Mixtec, TEI, dictionary encoding

This paper discusses the digital dictionary component in an ongoing language documentation project for the Mixtepec-Mixtec language (iso 639-3: mix). Mixtepec-Mixtec (Sa'an Savi 'rain language') is an Otomonguean language spoken by roughly 9,000-10,000 people in the Juxtlahuaca district of Oaxaca and in parts of the Guerrero and Puebla states of Mexico. Creating a digital dictionary for an under-resourced language entails a number of challenges that require unique and nuanced encoding solutions in which a delicate balance between the linguistic content, data structure, potential linked resources, and editorial metadata must be found. Herein we demonstrate how we use TEI to create a reusable, extensible, and machine readable language resource with an emphasis on how our solutions using a combination of novel and established TEI dictionary structures enable us to address our specific needs for Mixtepec-Mixtec and also provide a relevant roadmap for similar under-resourced language projects.

INTRODUCTION

This paper discusses the dictionary component of a larger documentation project of the Mixtepec-Mixtec language (MIX) 1 using the Text Encoding Initiative, or TEI (www.tei-c.org). In addition to the creation of a TEI dictionary, the primary output of the project is an open source body of reusable and extensible multimedia language resources including a text corpus of spoken and written language encoded and annotated in TEI. The language resources created are in turn being used to further knowledge of all aspects of the language itself within the fields of linguistics and lexicography by producing empirical corpus-based descriptions and analyses of aspects of the language's features.

In the process of data collection, annotation, and encoding, we seek to capture content relevant to every linguistic level from phonetic to semantic and etymological, as well as potential sub-dialectal variation. In conjunction with the complexity of the data, given the maximally broad scope of linguistic and lexicographic research being pursued, it is essential to have a means of organizing all the various components of the languages resources (LR) within a dynamic and flexible system. Also, given the lack of any other dictionary resources for the language, it is especially important that what we create is reusable and extensible so that it may continue to evolve, with the possibility of being easily exported or converted to other formats.

In pursuit of these goals, TEI essentially meets all the needs as the primary format for encoding and annotating our corpus, our born-digital dictionary, and our metadata. TEI is widely accepted in the digital lexicographic community as the de facto standard for the encoding of both retro-digitized and born-digital dictionaries and is being increasingly used for annotated lexical text corpora.

While TEI is well established and increasingly more widely adopted for projects and resources dealing with major world languages, 1 Most of the spoken data collected in this project originate from consultation sessions with two native speakers from Yucunani (17.30083, -97.89389), a town that is part of the municipality of San Juan de Mixtepec (http://www.geonames. org/3518634/san-juan-mixtepec.html).

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A TEI Dictionary for the Documentation of Mixtepec-Mixtec 81 particularly those of Europe and North America, it is far less adopted in projects dealing with indigenous languages. Aside from publications related to the current project, Czaykowska-Higgins et al. (2014), describing an application of TEI to the indigenous language Moses-Columbia Salish "Nxaʔamxcín", is a noteworthy example.

Our use of TEI for documentation work requires us to make use of the vocabulary for new or less common applications in order to accommodate the particular nuances of our data. Doing this benefits not only our project: in mapping out how to accommodate new unique combinations of features for a non-Indo-European indigenous language, it also increases the usability of TEI for potential future users and projects seeking to do similar things.

General issues in dealing with an under-resourced language.

While, according to Ethnologue (Simons and Fennig 2018), the status of MIX is "vigorous," it is an under-resourced language, and that gives rise to several significant challenges in our work, including these:

1. MIX lacks established linguistic descriptions beyond those of phonology and morphology upon which to build;

2. given the limited sample size of our data, and the disproportionate amount of data from two primary consultants from a small village outside the main town of San Juan Mixtepec, it is unclear whether certain observations of variation are due to sub-dialectal differences or speaker demographic-specific factors;

3. because speakers attempting to write in MIX are often not aware of phonological minimal distinctions, further consultation is almost always needed when interpreting and annotate such written materials; 4. because the orthography is still under development2 , the work done in creating and editing orthographic data in the project remains subject to revision;

5. the combination of the issues above and the fact that the output of this data will be the first (at least publicly available) set of MIX annotated data means there will be no way to train a learning model3 to automate the corpus annotation work.4

Data: sources, formatting, tools, and markup. Most of the MIX language resources have been collected from recordings of consultation sessions with native speakers5 and a collection of children's text booklets published by the Summer Institute of Linguistics (SIL) Mexico. 6 Other primary sources of data are written material created by native speakers as part of this project; documents about Mixtec that contain examples from others researchers;7 a set of public safety documents published by the Mexican government; excerpts from written personal communication with speakers; a small number of previous academic publications on the language (e.g., Pike and Ibach 1978; Paster and Beam de Azcona 2004a, 2004b; Paster 2005, 2010),8 as well as recordings and videos found online. For our editing and management of the text data we use Oxygen XML editor, and we backup and store our files (excluding soundfiles) o n G itHub.9 A ll o f t he p roject fi les (i ncluding sound and video) are archived in the Harvard Dataverse repository10 and will be registered with the Open Language Archives (OLAC).

Fitting in with the standards. In addition to using TEI for an indigenous language dictionary, Czaykowska-Higgins et al. (2014) cover many other issues relevant to the current project: in particular, questions of standardized lexical terminology, and compatibility with other formats and standards.

In the current project, the issue of deciding upon a theoretically sound fixed lexical terminology is challenging because, given the lack of published linguistic descriptions, there is very little to compare it to, and (as with much else in linguistics) there are multiple overlapping terms to describe the same phenomena.11 On the technical/standards side, these issues are further complicated by the in-flux state of the standardized repositories created to deal with this very question, namely ISOcat and the CLARIN Concept Registry.12 Additionally, there are still numerous linguistic concepts that are a significant part of our description (in both the corpus and the dictionary) for which there is no entry in any of the registries;13 thus, when the given registries are finally stable, as part of this project, we are tasked with submitting the necessary proposals to add entries for certain needed linguistic features.

TEI DICTIONARY

In this section we give an overview of the components of our dictionary and explain the lexicographic, functional, and-in certain cases-theoretical basis of their use.

Aside from the media files and annotated corpus, the main output of the project will be a trilingual TEI dictionary (Mixtepec-Mixtec, English, and Spanish) containing entries for all glossed lexical items observed in our corpus. Entries may contain the orthographic word forms, phonetic forms (and variants), grammar, usage, sense, etymological information, links to relevant external lexical and knowledge resources, and related entries, as well as examples from the corpus. Additionally, a separate inflections d ictionary d ocument s tores v erbal p aradigms.14 I n t he f ollowing sections, we describe these features in more detail and demonstrate their encoding in TEI. 15 The work is still ongoing and the 567 entries in our dictionary (as of October 2018) represent only a small fraction of the total vocabulary in our corpus.

Metadata and linking. In addition to the typical features inherent in dictionaries listed above, through links declared in the header section (TEI Guidelines, The TEI Header) the TEI dictionary acts as a nexus of the linguistic (lexical feature inventories) and other referenced resources (e.g., personographic, bibliographic).

TEI allows numerous ways of linking to important information that may need to be referenced throughout a dictionary, and we make use of several approaches based on the type of reference and the data itself. In the following section we describe several such aspects of the dictionary and how they are relevant within the context of the language documentation. Figure 1 provides an overview of the linked and embedded resources in the Mixtepec-Mixtec dictionary at the heart of this project. Lexical features and terminology inventory. As mentioned above, the project adheres to standards as much as possible in all aspects of the work. Our inventory of lexical terminology is kept in a separate document containing TEI feature structures16 that are used to tag the corpus. Figure 2 shows the declaration of the link to the document contained in the <sourceDesc> of the header in the dictionary (left) and a sample of two particular sets of features (trajector and landmark)17 from the document it links to. External data such as documents by SIL make up a significant portion of our data. Within our dictionary we often need to point to these sources to attribute provenance of an example. To enable this we again declare these sources in the <sourceDesc> of the header. The pathway to the TEI file is declared in @xml:base as shown in Figure 3. Personography. Additionally, as shown in Figure 4, in the header (within <particDesc> embedded in <profileDesc>), we list each person (speakers, editors, and researchers) who may be referred to directly in the dictionary. This list also links to the external TEI personography document "MIX-People.xml" containing detailed information about the participants, the path to which is declared in @xml:base on the element <listPerson>. Linking to related documents in dictionary. In certain entries we link to an external supplementary inflections dictionary containing inflectional paradigms. Within the TEI data structure shown in Figure 5, this is enabled by using the <prefixDef>18 element in the header in which a prefix is declared and serves as a shortcut for a specific path within the inflections dictionary. This enables us to point to entries for a particular paradigm entry in the inflections dictionary, thus linking a lemma with its infections. In Figure 5, the value of @matchPattern is a template for such pointers with the regular expression ([a-zA-Z0-9]+), which is replaced by the specific text of an entry. At the end of the value of @ replacementPattern, #$1 means that any pointer with the prefix "paradigm" should point to the document "../MIX-Paradigms.xml" with the value of the first regular expression: ([a-zA-Z0-9]+). Thus, within the dictionary, the inflections dictionary can be referenced by prefixing "paradigm:" within the string of pointer value. The pointer in Figure 6 links to the entry containing paradigms for the verb kusu 'sleep' (see Figures 12 and13 for the context in which it is used). Other types of links. Additionally, we have other resources such as sound files and some videos for which records, URLs, and metadata are stored in external files in a separate directory. Because of their large and ever-growing numbers, these are not declared in the header but can nonetheless be referenced and linked to within the dictionary using the @source attribute which can be placed in a number of different TEI elements.

STRUCTURE AND CONTENT OF ENTRIES

Forms and grammar. The lemma of a MIX form is given in the orthographic form and, if attested to a high enough degree of certainty, in the phonetic form (IPA) as well. Each entry has a <gramGrp>, with the minimal information of part of speech and other features where applicable. The element containing the form always includes the @ xml:lang attribute, the value of which is the ISO 639 language tag. 19 If an abbreviated value is used, a @norm attribute with the full form of the feature is given in order to align with terminological standards. A typical example is shown in Figure 7. Variation, uncertain, and conflicting forms. As this is a language documentation project, and the language is under-resourced both in its use as a literary language and its linguistic description, we are interested in recording variation and areas of uncertainty of all kinds.

Orthographic variation. Given that the MIX orthography is still under development and significant changes have been made over the last ten years, there are many lexical items in earlier documents with spellings that have since been changed. In these cases, both the old and up-todate forms are represented. In the earlier publications (encoded herein as the variant form), lexical tone was not represented in the orthography; however, this created a large number of homographs (which in some cases were of the same part of speech or even within the same domain). These needed to be distinguished.

The example in Figure 8 shows the updated chuún and antiquated chuun forms of the word meaning 'chicken' [ʧũṹ], which is a tonebased minimal pair with the word meaning 'money' [ʧũũ], the latter retaining the original spelling while the former adds the accent above the second vowel. The old form is labeled with <form type="variant"> and the element <orth> with attribute-value pair of @notation="plain" to indicate that the orthography does not represent tone. Additionally, given that the orthographic standard being developed has not been published, 20 those who write in the language often do not use the working spelling conventions, and thus we are faced with inte-grating all variants into our common system. The example in Figure 9 shows the encoding of a variant spelling of the lexical item meaning 'water' which was observed in a public service publication by the Mexican government. In this orthography, the voiceless alveopalatal affricate is rep-resented as ty instead of the standard ch, and the long word-final vowel is represented only as a single i. The document that is the source of the spelling variant is provided as the value of the @source attribute, which is declared in the bibliography within <listBibl> in the header (see "Linking to related documents" under "Metadata and linking" in TEI DICTIONARY section).

20

The latest known update to the orthography was obtained via personal communication with Mille Nieves of SIL Mexico in June 2017; it is upon this version that all editorial practice is based with regard to spelling normalization. Phonetic variation. In our data there are certain lexical items for which pronunciation variants are observed frequently enough that alternate pronunciations are included in the dictionary entry. In the example shown in Figure 10, the primary pronunciation21 (with the onset voiceless alveo-palatal fricative) is placed as a direct child of <form>, and the variant (with the onset voiced alveo-palatal fricative) is embedded within a separate <form>, also labeled @type= "variant".

Figure 10 Encoding of Phonetic Variants

Integrating data from external sources. Despite there being only a small body of linguistic literature about the language, there are cases where examples of transcribed vocabulary found in such sources are of interest and are thus integrated into the dictionary. Some instances may be the first or only attestation of the word or may diverge in some way from our own characterizations of the item. Additionally, there may be divergence in the transcription conventions used to represent the content.

One such example involves the tone of iin 'nine', where data from our project differs from that of two previous papers. In twenty-four of the twenty-five tokens in our data (from three speakers), the F0 (pitch) analysis shows that the tonal contour is falling, while the twenty-fifth, from a fourth speaker, is ambiguous. Thus, based on our own observations, we characterize the tone as falling. However, both Pike and Ibach (1978) and Paster and Beam de Azcona (2004a) 22 characterize the tone simply as low. 23 Figure 11 shows the encoding of this in the dictionary. Since the statistics support our description we use the certainty attribute with the value "high" on this form. Another noteworthy observation in this example is the treatment of transcription notation. Unfortunately, nearly none of the past studies of MIX phonology used IPA notation in their transcriptions. Fortunately, TEI has the ability both to keep the original forms from the sources in 22 The speaker consultant for the Paster and Beam de Azcona studies is the same individual as the current project's primary speaker (Juan "Tisu'ma" Salazar); we are thus confident that this is an issue of diverging phonological descriptions rather than a difference in pronunciation. Given the extensive body of work we have carried out and the fact that our transcriptions are based on recordings, we can dispute this question with confidence. 23 There are still questions as to the degree of tonal contrasts that remain to be answered via further systematic study of the phonology. In particular it is not yet fully clear whether there are minimal phonological distinctions based on each degree of tonal contour; that is, mid-low vs high-low, or even whether there is a contrast between falling and low tones. For this reason in our IPA transcription of what we characterize as phonologically falling, we currently use the global fall arrow  as a temporary placeholder instead of the combining diacritic iː or the combined tone letters ˥˩ or ˧˩.

@orig and to normalize the notation to IPA in the element values for compatibility.

Inflection and paradigms.

As mentioned above, a separate inflections dictionary contains full inflectional paradigms to which entries can link using the TEI <prefixDef> strategy described earlier. This is done in the primary dictionary24 with the <ptr> element as shown in Figure 12. The pattern of @xml:id values in the paradigm entries is (verb name)-V-paradigm-MIX (e.g., sleep-V-paradigm-MIX). Thus, the pointer in Figure 13 identifies a specific entry in the MIX-Paradigms.xml.

Figure 13 Example of Paradigm Entry Target Referred to in Primary Dictionary

In MIX, inflections can occur on verbs, nouns (for possession), and adverbs (in certain phrasal contexts). 25 Within the form section, full paradigms are represented as embedded blocks of inflected forms in accordance with the recommendations of TEI Lex0 (Bański et al. 2017). Each paradigm is encoded as a sibling of the lemma in <form type="paradigm"> and the primary common feature (tense or mood) is labeled as the value of @subtype, and tense/aspect/voice are encoded in <gramGrp>. In Figure 14, the first two forms of the paradigm for the present/incompletive forms of the verb kusu 'sleep' are shown. Note that there is a <gramGrp> as a direct child of <form type="paradigm">, and this contains the grammatical information common to all the inflected forms in the paradigm and inherited via the inheritance principle (Ide et al. 2000).

SENSE

The <sense> section contains information pertaining to meaning, including definitions, translations, examples of usage in context, domain classification, and a number of other data fields pertaining to semantic relations. An entry can have any number of senses.

Links to external knowledge sources: dbpedia. In order to enrich the content of our dictionary, for each concept (where available) we insert a link to an entry in dbpedia (Auer et al. 2007) or other open knowledge resources in the @corresp within the sense element, as shown in Figure 15. This is done with several benefits in mind. One is that they provide a link between a structured body of human knowledge and the Mixtepec-Mixtec language. Currently there are no Mixtec language wiki resources, and these links to dbpedia could provide a template upon which a MIX version of wiki-type entries could be based. Additionally, the multilingual definitions of the concepts found in the entries could serve as a systematic reference point upon which to base MIX definitions of the senses, which (as discussed below) are currently available for a small number of entries. Finally (with the inclusion of @xml:id) they enable the compatibility of the data to linked data formats such as OntoLex-Lemon (McCrae et al. 2017).

Translations. The most basic facet of the sense section is the multilingual translations into English and Spanish. Translations of lemmas are placed <cit type="translation"> within the <form><orth> element block. If in the translation language the Mixtec item has more than one specific translation, the others are listed in separate <cit> elements. Where possible we may also include links to digital dictionary-external resources for the given translation target languages, as in Figure 16.26 Images. In certain entries (often ones that correspond with certain theoretical interests pertaining to metaphor-driven and metonymy-driven sense change), we may include images showing the concept denoted in the sense. In TEI this is done with <graphic @url>, within which the <desc> element describes the content of the image. As in <def>, we include English and Spanish along with an empty tag for a future Mixtec description to be added. These images could be used for a pictographic or multimedia learning resource (e.g., a children's dictionary) or as examples for our own presentations. Figure 19 shows a visualization of the given sense of the word for 'face' , which in this sense means 'front of' something. Semantics and cultural issues in language documentation. Especially in a language documentation project, it is important and necessary to include other notes on various specifics of an entry. An example is the lexical item sa'an ntavi, one of two terms referring to the Mixtepec-Mixtec language itself and whose components translate literally as 'poor language' . Our project's native speaker consultant (understandably) finds this term offensive and derogatory and wanted it marked as dispreferred in the dictionary and the issue to be recorded in prose. This

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A TEI Dictionary for the Documentation of Mixtepec-Mixtec 97 is achieved with a combination of the TEI <note> and <usg> elements, with the @type value of "attitude" and the @resp specifying the initials of those responsible for recording this information, as shown in Figure 20. The initials are the @xml:id value of the individuals and are declared in the header (as discussed in "Personography" under "Metadata and linking" in TEI DICTIONARY section). Semantic relations and domain. In addition to sense, translations, and definitions, our dictionary includes information on semantic relations and domain. While the former is commonly made use of in structuralist linguistic approaches and computational linguistics such as WordNet (Miller 1995; Fellbaum 2005, 2010), the latter is typical of theoretical approaches based in cognitive linguistics (Langacker 1987; Clausner and Croft 1999).

While within theoretically these features are a mixture of structuralist and encyclopedic models of semantics (Geerarts 2010), for the purposes of the project, including these features in the annotation brings significant benefits. From the point view of potential Mixtec users of this resource, these features can be harnessed to facilitate collection and generation of focused sets of vocabulary to be used for the creation of further, more focused resources such as children's books and thesauri. Below we describe the content and implementation of these features in our dataset.

Semantic relations in the dictionary are encoded within specific senses of an entry within the external relation element <xr>. The given type is encoded in @type with an embedded <ref> that takes the @xml:lang (as we are generally providing both Mixtec and English versions, with English being the metalanguage for computational purposes). Where they point to other entries within the dictionary the @target attribute is used on <ref>. In the dictionary we tag only the members/subclasses but not the top nodes; thus, in the entry for 'fruit' we do not have the semantic relation 'hypernym' for every specific fruit species. Instead, this collection can be inferred and built up from the body of items tagged "hyponym" of 'fruit'.

Hyponymy is realized as <xr type="hyponymOf">. This category is extremely useful for generating taxonomical vocabulary lists. For the semantic relations hyponymy and meronymy, an additional <ref type="sense"> is included with the @corresp, the value of which is the same as occurs on that item's sense element. Thus, for the entry for 'peach' or other type of fruit, the <ref type="sense"> contains the same dbpedia URL as does the <sense @corresp> entry for kui'i 'fruit' itself, as shown in Figure 21. Meronymy is realized as <xr type="meronymOf">28 . As discussed by Geeraerts (2010), meronymy and hypernymy are central to the realiza-tion and analysis of metonymy. Synonymy and antonymy are encoded as <xr type="synonymOf"> and <xr type="antonymOf">. There are limits, however, to semantic relations both functionally and theoretically, as not all relevant semantic correlations in vocabulary or (more importantly) human knowledge can be defined or linked together in terms of hierarchical or pure opposition or identical senses. In order to fill some of that gap, we also use the concept of semantic domain.

In addition to semantic relations, which in lexicography are more immediately useful in computational applications, where applicable we assign semantic domain (Langacker 1987; Clausner and Croft 1999) to the sense of certain entries, a fairly common practice in compiling dictionaries. In lexicographic practice, however, the use of domains in a dictionary is often limited to technical subject classes (e.g., medicine, zoology, literature). Domains are fundamental cognitive concepts according to which humans organize, understand, and represent experience and knowledge of the world (Langacker 1987; Clausner and Croft 1999), and this is a particularly enriching perspective in approaching language documentation.

In cases of polysemy, semantic domain is often a key distinction between the various senses. In Figure 22 we show the senses in the entry for kani 'long' (domain of SPACE), which can be used in the sense of the domain TIME. In TEI, domain is encoded as <usg type="domain">.29 The inclusion of semantic domain potentially enables an alternate system of organization of a dictionary from the typical alphabetical ordering, or a derived domain-specific dictionary, and it can provide assistance with both manual and automatic word sense disambiguation (WSD). 30 Finally, domains enable us to encode and provide more dynamic analyses of sense-based etymological processes in keeping with cognitive linguistic theory. This latter is particularly important to the description of Mixtecan languages, as discussed in the following section.

ETYMOLOGY

In addition to the general documentation of the language, this project's dictionary is being created as a structured database of etymological information. In our data we have observed and encoded the full array of etymological processes, including borrowing (mostly from Spanish, some from Nahuatl), inheritance (from a posited Proto-Mixtecan language inferred by comparing cognates), and form changes: compounding, derivation, phonological change; various types of sense change such as metaphor, metonymy, and grammaticalization, as well as numerous instances of combinations of these processes. Bowers and Romary (2018) discuss such phenomena and their encoding in TEI.

Sense-related etymologies. As mentioned, a major point of emphasis in this project is the semantics, specifically the strategies of lexical innovation, particularly from the perspective of cognitive linguistics. There exists a significant body of literature discussing the evidence of metaphor and metonymy in lexical innovation in related varieties of Mixtecan (Hollenbach 1995; Brugman and Macaulay 1986; Langacker 2002); the dataset for MIX provides ample content that enriches such linguistic discussions (Bowers 2016 and forthcoming).

Figure 23 shows the etymology for MIX kani 'long' in the sense of the domain of TIME (discussed in the previous section). The conventions shown in the encoding of such phenomena in TEI have been derived from the recommendations set forth in Bowers and Romary (2016) and derived from another. When there is one or more embedded <sense> elements, the respective etymologies within should be considered sequential, stemming from the highest sense. In our example, they are also numbered using @n. On the etymology element <etym>, we use the @type to classify the etymological processes. If there is more than one sub-process involved, we can use embedded <etym> element structures to represent them.

We provide a prose description of our analysis of the given process in the <seg type="desc"> element. Given that it is a polysemy and is the same form as the source sense, the etymon <cit type="etymon"> does not have a form in this case. The @corresp attribute points to the source of the sense change that is the first sense. In addition to the @ type="metaphor", the data structure contains the key information for that process in the <usg type="domain">, which are in both senses, and copied within the <cit type="etymon">. Together with the embedding of senses and etymology, the contrast in the domain values from the first sense to the second provides a set of structured data that can be computationally searched and summarized.

FUTURE ENDEAVORS

This project is the subject of an ongoing PhD thesis by the first author, and, moving forward, funding will be sought to further work towards several major goals, including:

■ seeking partnerships to bring in long-term Mixtec contributors and editors, with the goals of expanding the size of the vocabulary and adding MIX iterations of key core contents of the dictionary (particularly the definitions and descriptions of the etymology);

■ inviting additional linguists to make use of the extensive data collected from specific linguistics subfields;

■ integrating other relevant Mixtecan resources into the dictionary and corpus using OCR with GROBID dictionaries (Khemakhem et al. 2017), namely the Colonial Mixtec vocabulary collection by fray Francisco de Alvarado from 1593 (Jansen and Perez 2009).

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A TEI Dictionary for the Documentation of Mixtepec-Mixtec 103

CONCLUSION

In this paper we have described the structure and contents of a multilingual TEI dictionary, which is part of an ongoing language documentation project for the Mixtepec-Mixtec language, an indigenous language of Mexico. We have shown how we are using the TEI dictionary to accommodate a number of key linguistic and metadata-related needs with a particular focus on issues associated with the documentation of an under-resourced language. In addition, we have described how we are integrating the MIX lexical resources with structured external knowledge sources such as dbpedia, and how within this digital dictionary we are working towards building a database of semantic and lexical relations, as well as etymological data. Finally, in using TEI for a language documentation project, we hope that our work may provide a useful reference for prospective projects whose researchers might be seeking guidance for other language and project-specific issues.

As a final note, Figure 24 shows a relatively complete dictionary entry in the print view for antivi 'sky', with various elements present in the entry. The print rendition is done using CSS (Cascading Style Sheets). Note that not all of the contents present in TEI are rendered in the print view: certain features such as the semantic relations are intended for computational purposes and are not necessarily relevant to the dictionary's primary target audience for the print view, principally members of the Mixtec community. The underlined contents signify they are linked to external or internal locations that may be source files or a URL.

Figure 1

 1 Figure 1 Diagram of Linked and Embedded Resources

Figure 2

 2 Figure2Feature Structures Declared in TEI Header and Their Content in Separate Document Bibliographic sources. External data such as documents by SIL make up a significant portion of our data. Within our dictionary we often need to point to these sources to attribute provenance of an example. To enable this we again declare these sources in the <sourceDesc> of the header. The pathway to the TEI file is declared in @xml:base as shown in Figure3.

Figure 3

 3 Figure 3 <sourceDesc> and <listBibl>

Figure 4

 4 Figure 4 <particDesc> and <listPerson> for Persons

Figure 5

 5 Figure 5 Declaration of the <prefixDef> Pattern for Linking between Documents

Figure 6

 6 Figure 6 Using Prefix Definition to Reference Verbal Paradigm Entry in Inflections Dictionary

Figure 7

 7 Figure 7 Typical <form> and <gramGrp> Section of an Entry

Figure 8

 8 Figure 8 Entry with Alternative Spelling without Tonal Diacritics

Figure 9

 9 Figure 9 Variant Orthography from MIX Language Publication

Figure 11

 11 Figure 11 Entry with Conflicting Phonological Descriptions in External Sources

Figure 12

 12 Figure 12 Verbal Lemma Entry with Link Using <prefixDef> Pattern to Link to Paradigms

Figure 14

 14 Figure 14 Partial Verbal Paradigm for MIX Verb kusu

Figure 15

 15 Figure 15 Linking Sense to dbpedia

Figure 18

 18 Figure 18 Multilingual Definitions with Empty Element for Mixtec to be Added

Figure 19

 19 Figure 19 FIGURE 19 Image Modified to Illustrate an Extended Sense of the Word for 'face'

Figure 20

 20 Figure 20 Components of Entry Detailing Dispreferred Status of Lemma with Pointer to Synonym 27

Figure 21

 21 Figure 21 <xr> for Hyponym Relations as in Entries for Types of Fruit

Figure 22

 22 Figure 22 Two Senses in the Entry for kani 'long'

Figure 24

 24 Figure 24 Example of Print View of Entry antivi as Rendered Using CSS

The MIX orthography is under development by Mille Nieves and Gisela Beckmann of SIL Mexico in consultation with native speakers.

An exception to this might be in automatic annotation of phonetic units, but this would not apply to tones because in the data contained so far there is a large degree of ambiguity in tonal classification, owing in part to a lack of diversity in speakers and, in some cases, poor recording quality.

This issue is significant because MIX is tonal and the orthography in previous SIL Mexico publications in the language (which make up most of the text data sources of this project) did not represent tone, resulting in a large number of homographs.

Recorded speech data are transcribed using Praat(Boersma and Weenik 2017) and then converted to TEI-XML using XSLT, where the text transcription output is integrated into the corpus. In accordance with the recommendations of[START_REF] Austin | Data and language documentation[END_REF], original recordings are stored in uncompressed 44kHz wav files.

http://www.mexico.sil.org/es/lengua_cultura/mixteca/mixteco-mixtepec-mix

Such sources are unpublished notes, obtained from personal communication with Mille Nieves of SIL Mexico.

The primary speaker consultant for the Pastor and Beam de Azcona papers was Juan "Tisu'ma" Salazar, the same primary speaker consultant for the current project.

https://github.com/iljackb/Mixtepec_Mixtec

As of the time of submission, the repository is still in preparation. When published it will be available at https://doi.org/10.7910/DVN/BF2VNK

The SIL-Mexico researchers in Oaxaca have provided a partial inventory of their working terminology; other than that, the only available publications are linguistic descriptions of related Mixtec varieties, which is limited in its utility in certain cases owing to theoretical differences.

12 See[START_REF] Ide | A registry of standard data categories for linguistic annotation[END_REF] for an initial discussion of this that has led to the (now stalled) ISOcat registry. The data corresponding to the latest active state of ISOcat are still statically available from https://old.datahub.io/dataset/isocat in various formats. The CLARIN Concept Registry (see https://www.clarin.eu/ccr) has taken up some of these to provide them as linked open data, but the status of many of these concepts is not yet

finalized. 13 This is also an issue with GOLD (General Ontology for Linguistic Description[START_REF] Farrar | A linguistic ontology for the semantic web[END_REF]), a high quality and dynamic vocabulary that is no longer under development. Consequently, where there are missing concepts needed for our work, there is no way to add them, thus necessitating the usage of other vocabularies as well.

MIX nouns and, in particular phrasal contexts, certain adverbs may be inflected for possession or other morpho-semantic features, and thus noun paradigms may also appear in the separate document.

Note that the dictionary is still undergoing editing and at the time of submission the formatting discussed herein is not yet universally applied.

http://www.tei-c.org/release/doc/tei-p5-doc/en/html/FS.html

Currently there exist no registered entries for these concepts in any public terminological repository, and they are among the list of proposals to be submitted for inclusion in the future.

http://www.tei-c.org/release/doc/tei-p5-doc/en/html/ref-prefixDef.html

The values of English and Spanish used are ISO 639-2; that of Mixtepec-Mixtec is from ISO 639-3 as that is the only option.

The primary pronunciation, where present, is determined by weighing the factors of observation frequency and knowledge of the language's phonology.

"Primary dictionary" refers to the main Mixtepec-Mixtec dictionary as distinct from the inflections dictionary.

To our current knowledge, only the adverb ncho'a can be inflected (for person) and only in certain lexicalized phases in which it occurs with a copular adjective or verb phrase.

The use of Wiktionary as a source of enhanced translations is a work in progress and is not yet systematically applied to each instance. In the future we may seek an automated means of gathering and adding this data.

See following section (Semantic relations) for explanation of the use of <xr type="synomymOf">.

While meronymy can be and in a number of theoretical sources is subtyped according to different conceptual paradigms, there are theoretical conflicts[START_REF] Geeraerts | Theories of Lexical Semantics[END_REF] as to the soundness of these distinctions; until further research and evaluation of this question can be carried out, then, we will not assign such sub-typologies.

Where available, like <sense> and <xr>, domain may also include URLs from external ontologies or sources such as dbpedia.

Word sense disambiguation is particularly important given that the MIX orthography represents tone only on a small percentage of words.

The TEI-Lex0 project is taking place within the DARIAH lexical data working group.

AUTHOR COPY AUTHOR COPY

A TEI Dictionary for the Documentation of Mixtepec-Mixtec 95 Examples. Any number of examples of the usage of an item in the context of the source data can be included within sense; these are also encoded as <cit> with the @type="example" and the wrapper <quote>. The form in question is wrapped within an orthographic reference element <oRef>. A pointer to the source of the example is included as the value of @target in the pointer <ptr/> element, as in Figure 17. Definitions. Entries can include definitions in Mixtec, Spanish, and English. A major goal is to have definitions for senses in all three languages to allow for the creation of monolingual, bilingual, and trilingual dictionaries. At the present stage, however, most entries do not have a Mixtec-language definition. In such cases we create a simple Spanish or English definition that can be used as a template for a future Mixtec one. In those cases we include separate <def> elements for each language. Figure 18 shows a sub-sense of nuu 'face' , which is used to express the sense 'front of (something)' .

AUTHOR COPY

A TEI Dictionary for the Documentation of Mixtepec-Mixtec 101 are in accordance with those in the etymology section of TEI-Lex0 31 (Bowers and al. 2018). Despite having no written evidence of this lexical item in earlier stages of the language, we are able confidently to assert the directionality of this relationship between these senses, as the metaphorical process of SPACE > TIME is a predictable mapping that follows the general pattern of utilizing concrete conceptual structures to describe and understand abstract concepts (Kövecses 2010; Gentner et al. 2002; Boroditsky 2000).

Herein the sense of 'long' (TIME) is embedded within the first-spatial-sense, which in this dictionary is done where one sense is clearly AUTHOR COPY


Anna Golovkina
email: a.golovkina@spbu.ru

Dmitri Ovsyannikov
email: d.a.ovsynnikov@spbu.ru

Sorin Olaru
email: sorin.olaru@centralesupelec.fr

PERFORMANCE OPTIMIZATION OF RADIOACTIVE WASTE TRANSMUTATION IN ACCELERATOR DRIVEN SYSTEM

Keywords: Optimization, optimal control, ADS, transmutation, radioactive waste

Long-lived actinides transmutation in accelerator driven system (ADS) is considered in this paper. The objective is to improve the transmutation performance from the overall radiotoxity of nuclear waste point of view by means of an optimization approach based on control theory. Parameters subjected to determination are the initial values of actinides nuclear concentration in the loaded fuel. The change in time of actinides' atomic concentration is described by a system of ordinary differential equations with initial conditions, that can be given with some inaccuracy. For such dynamical system a gradient-based optimization algorithm taking into account robustness of initial loading will be constructed. Moreover, constraint on value of effective multiplication factor is considered during numerical optimization. Based on this approach initial atomic concentration values providing the best indicators of actinides burnup can be obtained. Calculation results of actinides volume fraction and effective multiplication factor dynamics are presented.

Introduction

The level of ecological impact of nuclear power engineering is measured by number of radionuclides (fission products (FP) and actinides (MA)), which are formed during nuclear reactors operation. The radiotoxicity of spent nuclear fuel is firstly dominated by short-lived fission products, and later, by actinides [nuc, 1996]. A few hundred-thousand years after the discharge, the radiotoxicity of the unprocessed fuel drops to the natural toxicity level for traditional thermal reactors, i.e. the equilibrium radiotoxicity of the natural uranium ore [Mai, 2002]. The separation (and intermediate storage) of uranium and plutonium could reduce the radiotoxicity of the remaining highlevel radioactive waste by an order of magnitude, but it would still take about twenty-thousand years for the radiotoxicity of this waste to reach the natural toxicity level [oec, 2006].

Ecological impact of short-lived radionuclides can be reduced till the admissible level using organizational and technical measures in storage and fuel cycle facilities. However, in case of geological disposal, longlived nuclear waste has a potential risk over a few ten thousands years. The geological disposal site is difficult to select, mainly because it is hard to predict the changes in the structure of ground layers for the coming ten thousands years. Due to these facts, the problem of long-lived radioactive waste elimination arises, as an important technological research topic. The de-cision for this problem can find a pertinent solution in transmutation [START_REF] Coquelet | Comparison of different options for transmutation scenarios studied in the frame of the french law for waste management[END_REF].

Transmutation is a transformation of longed-lived radionuclides into short-lived or stable isotopes under influence of neutral or charged particle flows.

From the viewpoint of the radiotoxicity, transmutation must first be concerned with the minor actinides, particularly americium and neptunium, because the toxicity of the fission products lays at least two orders of magnitude below that of the actinides after a few hundred years [Mai, 2002].

For the transmutation of actinides, the key reaction is the fission reaction which transforms long-lived, highly radiotoxic actinides into mostly short-lived, less toxic fission products. Other reactions such as capture and (n, 2n) reactions just transform actinide species into other actinide species without a significant effect on long-term radiotoxicity [START_REF] Tucek | Neutronic and Burnup Studies of Accelerator-driven Systems Dedicated to Nuclear Waste Transmutation[END_REF]. However, these reactions are useful insofar as they transform fertile actinides with a low fission probability into fissile actinides with a high fission probability.

Fast-spectrum systems are required to handle the minor actinides. Conventional fast reactors can burn selfgenerated minor actinides in a closed cycle, but they are not suited for burning pure minor actinides. In this regard accelerator-driven systems are considered to provide a promising way of actinides transmutation.

The concept of accelerator-driven systems (frequently called hybrid systems) combines a particle accelerator with a sub-critical core [START_REF] Nifenecker | Accelerator Driven Subcritical Reactors[END_REF]. The principal advantages that accelerator-driven subcritical systems have relative to critical reactors are twofold: greater flexibility with respect to fuel composition, and potentially enhanced safety. Accelerator driven systems are ideally suited to burning fuels which are problematic from the standpoint of critical reactor operation, namely, fuels that would degrade neutronic characteristics of the critical core to unacceptable levels due to small delayed neutron fractions and short neutron lifetimes [START_REF] Golovkina | Simplified dynamics model for subcritical reactor controlled by linear accelerator[END_REF]. It should also be noted that the proportionality of the reactor power to the accelerator current simplifies the reactor power maintenance [START_REF] Golovkina | The problem of ads power-level maintenance[END_REF].

In order to achieve better transmutation performance the initial fuel loading should be chosen in an optimal way. Most works devoted to this problem use combinatorial approach where a number of x fuel assemblies are permuted in l locations in the core [START_REF] Rahman | Deterministic methods for multi-control fuel loading optimization[END_REF], [START_REF] Washington | Optimization of plutonium and minor actinide transmutation in an ap1000 fuel assembly via a genetic search algorithm[END_REF], [START_REF] Ishida | Finding the best fuel assemblies shuffling scheme of ads for ma transmutation using dynamic programming[END_REF]. In this paper a different approach implementing optimization of MA composition in the fuel is proposed. This problem statement and model equations are presented in the section 2. Section 3 contains the description of optimization approach, with some calculations results placed in section 4.

Problem Statement

Evolution of fuel concentration in ADS is described by a system of ordinary differential equations (also known as Bateman equations [START_REF] Bateman | Solution of a system of differential equations occurring in the theory of radioactive transformations[END_REF]):

dN i dt = ∑ j̸ =i f j→i λ j N j + ∑ k̸ =i ∑ x g k→i x N k ⟨σ k x ϕ⟩ + (1) + ∑ l γ l→i F l - ( λ i + ⟨σ k a ϕ⟩ ) N i ,
with given initial conditions:

N i (0) = N i 0 , (2)
where N i -atomic concentration of isotope i; the first term on the right-hand side expresses the production rate of nuclide i due to radioactive decay of another nuclide j on the burnup chain; λ j is the decay constant of nuclide j and f j→i is the probability of decay to nuclide i; the second term is the production rate of nuclide i due to the nuclear reaction x of another nuclide k the major nuclear reaction is the neutron capture reaction and other reactions such as the (n, 2n) reaction can be considered if necessary; g k→i x is the probability of transmutation into nuclide i for nuclear reaction x of nuclide k; the third term is for the production of FPs; F l is the fission rate of heavy nuclide l and γ l→i is the production probability (yield fraction) of nuclide i for the fission reaction; the last term is the loss rate of nuclide i due to radioactive decay and absorption reactions.

⟨σ k

x ϕ⟩ -the microscopic reaction rate of nuclide k integrated over all neutron energies -is defined by the following relation:

⟨σ k x ϕ⟩ = Emax ∫ Emin σ k x (E)ϕ(E)dE (3) ≈ G ∑ i=1 σ k x (E i )ϕ(E i )∆E i ,
where ϕ(E i ) represents the value of neutron flux normalized to the reactor heat power in the i-th energy group, G -number of considered energy groups.

In general, neutron flux is a volume-averaged solution of stationary neutron transport equation. Its diffusion Figure 1. Relations of the most important actinides via different types of decay and neutron reactions approximation will be considered in the present paper [START_REF] Okumura | Nuclear reactor design, chapter 2[END_REF]:

D∇ 2 ϕ -Σ a ϕ + S = 0. ( 4
)
with the notation

D -diffusion coefficient, Σ a = n ∑ i=1
N i σ i a -macroscopic absorption cross section in the core (n -total number of considered isotopes), S -neutron supply from fission, scattering and external neutron source.

In should be noted that flux value in (1) is considered to be constant during the step of integration. Equation ( 4) is solved again with new atomic concentrations obtained after one step of (1) integration. Neutron flux from (4) is calculated in multi-group diffusion approximation using the program code MDSR [START_REF] Golovkina | Program for multi-group diffusion calculation of subcritical reactor neutron-physical characteristics[END_REF]. System (1) can be rewritten in the compact matrix form:

ẋ = f (t, x) = A(t)x, (5)
where we assume that x = [ N 1 , N 2 , . . . , N n ] , A(t) -matrix of n × n dimensions, constructed on the basis of burnup chain (see Fig. 1 [START_REF] Okumura | Nuclear reactor design, chapter 2[END_REF]) for considered actinides and n is the number of considered isotopes [START_REF] Golovkina | Nuclear constants database for fuel isotope change estimation[END_REF]. Initial conditions for system (5), determining the initial loading of fuel to the reactor core are given

x(0) = x0 . ( 6
)
The aim of optimization process is to find x * 0 values providing the best performance of transmutation: minimal radiotoxity of discharged fuel. The initial values determining dynamics of system (5) can be effectively realized in the process with some error that can lead to the different performance of waste transmutation. In order to take into account such possibility it is supposed that initial values of nuclear concentrations x i (0) can be taken from some neighborhood of point x0i . Then the initial values of nuclear concentrations can be taken from the interior part of the ellipsoid

(x 0 -x0 ) * G 0 (x 0 -x0 ) = 1, (7)
where G 0 is a positive definite symmetric matrix, x0the ellipsoid center. Ellipsoid shift along the solutions of a linear system is also ellipsoid with center in x(t), bounding nuclear concentration initial values set in time moment t:

(x -x(t)) * G(t)(x -x(t)) = 1. (8)
Here x(t) = x(t, x0 ) -the solution of system (5) with initial conditions (6), matrix G(t) -determined by the solution of the following equation [START_REF] Ovsyannikov | Mathematical methods of beam control[END_REF]

dG dt = -A * (t)G(t) -G(t)A(t) (9)
with initial conditions G(0) = G 0 .

The inverse matrix G -1 (t) satisfies the similar to (9) equation [START_REF] Ovsyannikov | Mathematical methods of beam control[END_REF], [START_REF] Ovsyannikov | Modeling and optimization of charged particle beam dynamics[END_REF]

dG -1 dt = AG -1 + G -1 A * . ( 10
)
It's clear from the matrix equation ( 9) that initial condition (6) (considered here as a control actions) doesn't influence the matrix G(t), determining the ellipsoid shape at some time moment t. Because ADS is considered in the paper as transmutation facility, the requirement of its reactor to stay subcritical throughout the operation time should be taken into account. That means the effective multiplication factor should be less than one

k eff < 1. ( 11
)
k eff is the lowest eigenvalue of translation operator and the corresponding eigenfunction φ is neutron flux density in the reactor core [START_REF] Vladimirov | Equations of Mathematical Physics[END_REF]. In this paper we use the diffusion approximation of translation equation in order to simplify calculations. In this case k eff is the lowest eigenvalue of equation ( 4), where

S = νΣ f k eff φ, Σ f = n ∑ i=1
N i σ i f -macroscopic fission cross section in the core (n -total number of isotopes). Condition ( 11) puts additional constraint on x(t, x0 ) values in time interval [0, T ].

Optimization Approach

The aim of optimization is to provide the minimal output waste radiotoxity after ADS operation ends in time T . The corresponding performance criterion in this case is

J(p) = n ∑ i=1 ω(x 2 i (T )), (12)
where ω(x 2 i (T )) = x2 i (T )λ i -function, translating isotopes nuclear concentrations [1/cm 3 ] in their specific radiotoxity [Bq/cm 3 ], λ i -radioactive decay constant [1/s], p -control parameter subjected to optimization (in our case the initial values x0 ).

In this formulation the similar problem for advanced fuel cycle was solved in [Golovkina et al., 2018a]. In this paper it's proposed to consider the problem of robust optimization, stipulatory to the possible errors in fuel loading to the ADS reactor core.

In order to provide the robustness to the optimal solution besides minimization of ( 12) the problem of ellipsoid (8) radius constriction in moment T should be taken into account. Let us denote S = G -1 , z = x -x(t), then ellipsoid with matrix S is described by the equation

z * S(t)z = 1.
The objective is then to find the maximal value of z on the ellipsoid (8) surface. Using the standard approach with Lagrangian multipliers we will get the following estimation [START_REF] Ovsyannikov | Modeling and optimization of charged particle beam dynamics[END_REF]

max z * G(t)z=1 z i = √ s ii (t), (13)
where s ii -diagonal elements of matrix S, z i -the i-th component of vector z. Thus functions √ s ii (t) are the envelopes on the variables x -x(t) of system (5) [START_REF] Ovsyannikov | Mathematical modeling and optimization of beam dynamics in accelerators[END_REF]. Thereby the optimized criterion, taking into account the system robustness is

J(p) = n ∑ i=1 ω ( xi (T ) + √ s ii (T ) ) 2 . ( 14
)
Gradient based optimization methods can be used to minimize functional ( 14). In this case the explicit expression of its gradient should be obtained

∇J(p) = dJ dp = ( ∂J ∂p 1 , . . . ∂J ∂p n ) * .
Let us denote

y i = xi (T ) + √ s ii (T ), φ(•) = ∑ ω(• 2
), then the expression ( 14) can be rewritten in the form

J(p) = n ∑ i=1 ω(y 2 i (T )) = φ(y(T )). ( 15
)
dJ dp = dφ dy • dy dp = ( ∂φ ∂y 1 . . . ∂φ ∂y n ) • ( ∂y ∂p 1 . . . ∂y ∂p n ) * dy dp i = ∂ xi (T ) ∂p i + ∂ √ s ii (T ) ∂p i . ( 16
)
In the expression (16) the first term can be obtained using variation equation [Golovkina et al., 2018a] d dt

( ∂x(t, p) ∂p i ) = A(t) ∂x(t, p) ∂p i , ( 17
) ∂x(0, p) ∂p i = E.
The second term in ( 16) is equal to zero, because as shown above in the case of dynamical process described by a linear system (5) the elements of matrix S do not depend on the parameter p. Their values at each moment of time t are completely determined by the initial values from S 0 and matrix of the system (5) A.

Thus the robustness of initial loading (uncertainty limits of the parameter p) can be estimated by matrix S 0 .

Taking into account the previous considerations gradient components can be calculated using the following formula

∂J ∂p i = 2 (x i (T ) + √ s ii (T )) ∂ xi (T ) ∂p i . ( 18
)
Thus optimization problem is narrowed down to the solution of equation ( 17) and the following use of one of the gradient-based methods constructed with the obtained expression for gradient components (18). It should be noted that taking into account the constraint (11) now is carried out in the following way. At each optimization step the value of ( 11) is computed, if it exceeds 1 then the step size in optimization method should be reduced. More comprehensive way of considering this condition is being developed.

Calculation Results

Program code for calculation of fuel isotopes composition dynamics and optimization of initial fuel loading is developed in order to implement the described techniques. It is shown that the burnup capability of ADS is better in comparison with conventional nuclear reactors. Transmutation performance criterion ( 14), taking into account robustness of the system is used as an objective function. As was described earlier control of constraints values ( 11) is carried out on each of the step during numerical optimization. Fig. 2 shows the evolution of the neutron multiplication factor during the operation time. In table 1 volume fractions of transmutated actinides in initial ADS loading and their fractions by the end of operation cycle are presented. Calculation also showed that overall radioactivity of the charged fuel was reduce in approximately 5 times.

Conclusion

Optimization approach based on control theory applied to the problem of long-lived actinides transmutation in accelerator driven system is considered in the paper. The transmutation performance criterion taking into account the system robustness is constructed. Gradient-based optimization algorithm is designed and implemented as a program tool. Constraint on value of effective multiplication factor is taken into account during numerical optimization. Based on this approach some calculations results were obtained. After optimization radioactivity of fuel charged in the core was reduced in approximately 5 times.

Figure 2 .

 2 Figure 2. k eff time evolution during transmutation cycle (dotted line is an approximation curve)

Table 1 .

 1 Comparison of actinides volume fraction in charged and discharged fuel in ADS


Souad Amrane

Fazia Bedouhene

Islam Boussaada

Silviu-Iulian Niculescu

On Qualitative Properties of Low-Degree Quasipolynomials: Further remarks on the spectral abscissa and rightmost-roots assignment by

Keywords: Time-delay systems, Stability, Spectral abscissa, Control design, Pole assignment, Non oscillation 2010 Mathematics Subject Classification: 34C60, 34K06, 35B35, 70J25

The present study concerns a frequency domain approach in the stability analysis and stabilization via reduced-order controller design for linear time-invariant retarded Time-delay systems. More precisely, we address the problem of the spectral abscissa characterization and the coexistence of non oscillating modes for such functional differential equations. The design approach we propose is merely a delayed-output-feedback where the candidates' parameters result from the manifold defined by the coexistence of an exact number of negative spectral values, which guarantees the asymptotic stability of the system's solutions. The dominancy of such non oscillating modes is analytically shown for the considered reduced order Time-delay systems. Finally, using the W-Lambert function, further description of the spectrum distribution is presented.

Introduction

Investigation of dynamical systems with time-delay is an active research area that connects a wide range of scientific disciplines including mathematics, physics, engineering, biology, economics etc. The present paper focuses on stability and stabilizing-controllers design for linear time-invariant retarded time-delay systems. The study of conditions on the equation parameters that guarantees the exponential stability of solutions is a question of ongoing interest and remains an open problem especially when the systems are of high order or having multiple and/or distributed delays. In particular, in frequency-domain, the problem reduces to the analysis of the distribution of the roots of the corresponding characteristic equation, see for instance [2,[START_REF] Cooke | On zeroes of some transcendental equations[END_REF][START_REF] Walton | Direct method for tds stability analysis[END_REF][START_REF] Stépán | Retarded Dynamical Systems: Stability and Characteristic Functions[END_REF][START_REF] Hale | Introduction to functional differential equations[END_REF][START_REF] Michiels | Stability and stabilization of time-delay systems[END_REF][START_REF] Olgac | An exact method for the stability analysis of time delayed linear time-invariant (lti) systems[END_REF][START_REF] Sipahi | Stability and stabilization of systems with time delay[END_REF].

The starting point of the present work is an interesting property, discussed in recent studies, called multiplicity induced-dominancy. As a matter of fact, it is shown that multiple spectral values for Time-delay systems can be characterized using a Birkhoff/Vandermondebased approach; see for instance [START_REF] Boussaada | Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: A Vandermonde-based approach[END_REF][START_REF] Boussaada | Characterizing the codimension of zero singularities for time-delay systems[END_REF][START_REF] Boussaada | Computing the codimension of the singularity at the origin for delay systems: The missing link with Birkhoff incidence matrices[END_REF][START_REF] Boussaada | Multiplicity and stable varieties of timedelay systems: A missing link[END_REF]. More precisely, in [START_REF] Boussaada | Characterizing the codimension of zero singularities for time-delay systems[END_REF], it is shown that the admissible multiplicity of the zero spectral value is bounded by the generic Polya and Szegö bound denoted P S B , which is nothing but the degree of the corresponding quasipolynomial (i.e the number of the involved polynomials plus their degree minus one), see for instance [START_REF] Pólya | Problems and Theorems in Analysis[END_REF]. In [START_REF] Boussaada | Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: A Vandermonde-based approach[END_REF], it is shown that a given crossing imaginary roots with non vanishing frequency never reaches P S B and a sharper bound for its admissible multiplicities is established. Moreover, in [START_REF] Boussaada | Multiplicity and stable varieties of timedelay systems: A missing link[END_REF], the variety corresponding to a multiple root for scalar Time-delay equations defines a stable variety for the steady state. The multiplicity of a root itself is not important as such but its connection with the dominancy of this root is a meaningful tool for control synthesis. An example of a scalar retarded equation with two delays is studied in [START_REF] Boussaada | Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: A Vandermonde-based approach[END_REF] where it is shown that the multiplicity of real spectral values may reach the P S B . In addition, the corresponding system has some further interesting properties: (i) it is asymptotically stable, (ii) its spectral abscissa (rightmost root) corresponds to this maximal allowable multiple root located on the imaginary axis. Such observations enhanced the outlook of further exhibiting the existing links between the maximal allowable multiplicity of some negative spectral value reaching the quasipolynomial degree and the stability of the trivial solution of the corresponding dynamical system. This property induced from multiplicity appears also in optimization problems since such a multiple spectral value is nothing but the rightmost root, see also [START_REF] Vanbiervliet | A nonsmooth optimisation approach for the stabilisation of time-delay systems[END_REF][START_REF] Michiels | Continuous pole placement for delay equations[END_REF]. Also notice that the property was already observed in [START_REF] Ramirez | Design of proportional-integralretarded (pir) controllers for second-order lti systems[END_REF], where a tuning strategy is proposed for the design of a delayed Proportional-Integral controller by placing a triple real dominant root for the closed-loop system. However, the dominancy is only checked using a Mikhailov curve and QPmR toolbox, see for instance [START_REF] Vyhlídal | Mapping based algorithm for large-scale computation of quasi-polynomial zeros[END_REF]. To the best of our knowledge, the first time an analytical proof of the dominancy of a spectral value for the scalar equation with a single delay was presented in [START_REF] Hayes | Roots of the transcendental equation associated with a certain differencedifferential equation[END_REF]. The dominancy property is further explored and analytically shown in the case of second-order systems and a rightmost root assignment based design using delayed state-feedback is proposed in [START_REF] Boussaada | On the coalescence of spectral values and its effect on the stability of time-delay systems: Application to active vibration control[END_REF][START_REF] Boussaada | Further remarks on the effect of multiple spectral values on the dynamics of time-delay systems. application to the control of a mechanical system[END_REF] where its applicability in damping active vibrations for a piezo-actuated beam is proved. See also [START_REF] Boussaada | Toward a decay rate assignment based design for time-delay systems with multiple spectral values[END_REF][START_REF] Boussaada | On the dominancy of multiple spectral values for time-delay systems with applications[END_REF] which exhibit an analytical proof for the dominancy of the spectral value with maximal multiplicity for second-order systems controlled via a delayed proportional-derivative controller.

By this paper, we would like to extend such an analytical characterization of the spectral abscissa for retarded time-delay system with real spectral values which are not necessarily multiple. The effect of the coexistence of such non oscillatory modes on the asymptotic stability of the trivial solution will be explored. In particular, the coexistence of P S B real spectral values makes them rightmost-roots of the corresponding quasipolynomial. Furthermore, if they are negative, this guarantees the asymptotic stability of the trivial solution. The study of oscillatory/non oscillatoty solutions of delay differential equation has been the focus of numerous contributions, see for instance [1,2,[START_REF] Gopalsamy | Stability and oscillations in delay differential equations of population dynamics[END_REF], and [START_REF] Guo | Existence of nonoscillatory solutions for a second-order nonlinear neutral delay differential equation[END_REF] where the strongest deal is with linear delay differential equations with constant coefficients and constant delay. In this case necessary and sufficient conditions are given for all solutions to be oscillatory, for the case of non constant coefficient and non constant delay and for nonlinear equations, typical results give conditions sufficient for all solutions to be oscillatory, or conditions sufficient for some solutions to be non oscillatory see for instance [START_REF] James | Oscillation and nonoscillation of solutions of second order linear differential equations with integrable coefficients[END_REF] and [START_REF] Jack | Introduction to functional differential equations[END_REF].

The remaining paper is organized as follows; in Section 2 we recall some important facts on the spectrum distribution for retarded Time-delay systems. Section 3 is dedicated to investigate the coexistence non oscillatory modes for the scalar differential equation with a single delay. In Section 4 second order delay equation with three real spectral values is considered. In Section 5, some concluding remarks ends the paper.

Prerequisites and Problem Statement

In this section, we present some important facts and properties of the spectrum distribution of Time-delay systems. Let consider the generic n-order system with a single time delay:

ẋ(t) = A 0 x(t) + A 1 x(t -τ ). (2.1)
Here τ is a positive constant delay and the matrices A j ∈ M n (R) for j = 0 . . . 1. It is well known that the asymptotic behavior of the solutions of (2.1) is determined from the spectrum designating the set of the roots of the associated characteristic function (denoted in the sequel ∆(s, τ )). Namely, the characteristic function corresponding to system (2.1) is a quasipolynomial ∆ : We start by introducing a proposition which plays an important role in the study of continuity properties of the spectrum of retarded Time-delay systems. For more details see [19, page 10].

C × R + → C of the form: ∆(s, τ ) = det(sI -A 0 -A 1 e -τ s ). ( 2

Proposition 1 ([19]

). If s is a spectral value corresponding to system (2.1) then it satisfies

|s| ≤ A 0 + A 1 e -sτ
2 .

(

The above proposition allows to construct an envelope curve around the characteristic roots of the quasipolynomial (2.2).

The following result was first introduced and claimed in the problems collection published in 1925 by G. Pólya and G. Szegö. In the fourth edition of their book [START_REF] Pólya | Problems and Theorems in Analysis[END_REF]Problem 206.2, page 144 and page 347], G. Pólya and G. Szegö emphasize that the proof was obtained by N. Obreschkoff in 1928 using the principle argument, see [START_REF] Obreschkoff | Nullstellen linearer kombinationen von exponentialfunktionen[END_REF]. Such a result gives a bound for the number of quasipolynomial's roots in any horizontal strip. As a consequence, a bound for the number of quasipolynomial's real roots can be easily deduced.

Theorem 1 ([22]

). Let τ 1 , . . . , τ N denote real numbers such that τ 1 < τ 2 < . . . < τ N and d 1 , . . . , d N positive integers such that d 1 + d 2 + . . . + d N = D. Let f i,j (s) stand for the function f i,j (s) = s i-1 exp(τ j s), for 1 ≤ i ≤ d j and 1 ≤ j ≤ N . Let be the number of zeros of the function

f (s) = 1≤j≤N 1≤i≤dj c i,j f i,j (s) (2.4)
that are contained in the horizontal strip α ≤ Im(z) ≤ β. Assuming that

1≤k≤d1 |c k,1 | > 0 and 1≤k≤d N |c k,N | > 0 then (τ N -τ 1 )(β -α) 2π -D + 1 ≤ ≤ (τ N -τ 1 )(β -α) 2π + D + N -1. (2.5)
Setting α = β = 0, the above theorem yields P S ≤ D + N -1 where D stands for the sum of the degrees of the polynomials involved in the quasipolynomial function f and N designates the associated number of polynomials. This gives a sharp bound for the number of f s real roots.

3 On the coexistence of real spectral value and their dominancy for scalar delay equation

Consider the simple scalar differential equation with one delay representing a biological model discussed by K.L.Cooke in [START_REF] Cooke | Stability analysis for a vector disease model[END_REF]. It describes a vector disease dynamics where the infected host population x(t) is governed by:

ẋ(t) + ax(t) + bx(t -τ ) = 0, (3.1)
here b > 0 designates the contact rate between infected and uninfected populations and it is assumed that the infection of the host recovery proceeds exponentially at a rate b > 0, see also [START_REF] Ruan | Delay differential equations in single species dynamics[END_REF] for more insights on the modeling and stability results. The characteristic equation associated to (3.1) is as follows:

∆(s, τ ) := s + a + b exp(-sτ ) = 0. (3.2)
Theorem 2. For a given delay τ > 0, the system (3.1) admits two distinct real spectral values at s = s 2 and s = s 1 , with s 2 < s 1 , if and only if

     a = a(s 1 , s 2 , τ ) := s 2 exp(-s 1 τ ) -s 1 exp(-s 2 τ ) exp(-s 2 τ ) -exp(-s 1 τ ) ; b = b(s 1 , s 2 , τ ) := s 1 -s 2 exp(-s 2 τ ) -exp(-s 1 τ ) . (3.3)
• Moreover, both spectral values s 2 and s 1 of (3.1) are negative, if and only if equation

a(s 1 , s 2 , τ ) = 0
is solvable with respect to τ > 0. Furthermore, the zero solution of (3.1) is asymptotically stable.

• The spectral value s 1 is nothing but the spectral abscissa corresponding to (3.1).

Proof. According to the Theorem 1, see also [START_REF] Pólya | Problems and Theorems in Analysis[END_REF], the number of real roots for (3.2) is two, hence we are interested to investigate the existence of two distinct negative spectral values, s 2 and s 1 with s 2 < s 1 . The values of a and b are calculated by solving the system:

s 2 + a + b exp(-s 2 τ ) = 0; s 1 + a + b exp (-s 1 τ ) = 0. (3.4)
We obtain immediately the values of a and b given in (3.3), as functions of s 2 and s 1 and τ . These values are unique for each fixed τ > 0. Observe that b(s 1 , s 2 , τ ) > 0 for every τ > 0, while the parameter a(s 1 , s 2 , τ ) changes the sign according to the delay τ and the position of s 1 and s 2 on the real line. On the other hand, both s 1 + a and s 2 + a are not null, otherwise, b = 0 and then s 1 = s 2 . Which is impossible. Now, from (3.4), we have

s 1 + a s 2 + a = exp(-τ (s 1 -s 2 )).
This means that s 2 + a and s 1 + a have necessarily same sign, and the following expression of τ :

τ = - ln(s 1 + a) -ln(s 2 + a) s 1 -s 2 (3.5)
is well-defined. Applying the Mean Value Theorem to the function

t → ln (t + a), t ∈ [s 2 , s 1 ]. This ensures the existence of c ∈]s 2 , s 1 [ such that τ = - 1 c + a . (3.6)
Since τ > 0 so c + a < 0. Consequently s 2 + a < a + c < 0. This means that s 2 < -a.

Likewise for s 1 , since sign(s 1 + a) = sign(s 2 + a), we get s 1 < -a.

To show the negativeness of s 1 , we use the variations of the mapping τ → a(s 1 , s 2 , τ ). An easy calculation shows that a is a continuous function with respect to τ , and increasing from -∞ to -s 1 . So, if equation a(s 1 , s 2 , τ ) = 0 admits a root τ * * > 0, then such root is necessarily

τ * * = ln |s 2 | -ln |s 1 | s 1 -s 2 (3.7)
and a(s 1 , s 2 , τ ) ≥ 0 holds for all τ ≥ τ * * . Consequently, we get s 1 < 0. The converse implication is obvious.

The study of the stability of the system (3.1) is based on the dominancy of s 1 . To do this, we use an adequate factorization of the quasipolynomial ∆(s, τ ) from the characteristic equation (3.2). This later can be written as follows:

∆(s, τ ) = (s -s 1 ) 1 - (a + s 1 ) -exp(-(s -s 1 )τ ) s -s 1 = (s -s 1 ) 1 -(a + s 1 ) -exp(-(s -s 1 )τ ) s -s 1 + 1 s -s 1 . (3.8) Thus, ∆(s, τ ) = (s -s 1 ) 1 -τ (a + s 1 ) 1 0 exp(-τ (s -s 1 )t)dt . (3.9)
To prove that s 1 is the right most root (dominancy), we assume that there exists some

On the spectral abscissa and rightmost-roots of TDS

s 0 = ζ + jη, a root of (3.2) such that ζ > s 1 . Then 1 = τ (a + s 1 ) 1 0 exp(-τ (s 0 -s 1 )t)dt = Re τ (a + s 1 ) 1 0 exp(-τ (ζ + jη -s 1 )t)dt) ≤ τ |a + s 1 | 1 0 exp(-τ (ζ + jη -s 1 )t)dt ≤ τ |a + s 1 | 1 0 exp(-τ (ζ -s 1 )t)dt. (3.10)
From (3.6), we have

τ < 1 -(a + s 1 ) = 1 |a + s 1 | . (3.11) Thus τ | a + s 1 |< 1. (3.12)
Moreover, since ζs 1 > 0, we have

1 0 exp(-τ (ζ -s 1 )t)dt < 1. (3.13)
Inequality (3.10) can not be satisfied simultaneously with (3.12) and (3.13). Which proves that the hypothesis ζ > s 1 is inconsistence. So s 1 is the spectral abscissa corresponding to (3.1), guarantying the asymptotic stability of the system (3.1).

Remark 1. Theorem 2 shows that the co-existence of two spectral negative values s 1 and s 2 is guaranteed by the positivity of the mapping τ → a(s 1 , s 2 , τ ), in some interval. Inequality a(s 1 , s 2 , τ ) ≥ 0 is satisfied if and only if τ ≥ τ * * . From a control theory point of view, the assignment of such real negative spectral values subjects to choose them in the interval ] -∞, -a[.

Remark 2.

1. In recent results [START_REF] Boussaada | Computing the codimension of the singularity at the origin for delay systems: The missing link with Birkhoff incidence matrices[END_REF][START_REF] Boussaada | Tracking the algebraic multiplicity of crossing imaginary roots for generic quasipolynomials: A Vandermonde-based approach[END_REF], the question of the effect of multiple spectral values on the dynamics of time-delay systems is investigated. So, consider s 0 to be the corresponding double real root, we are interested in the behavior and the geometric structure of the corresponding envelope. In such case, the coefficients a and b satisfy the following relations a(s 0 , τ ) = -s 0 -1 τ ; and b(s 0 , τ ) = 1 τ exp(τ s 0 ). Furthermore, there exists a critical value τ = -1 s0 , which corresponds to the vanishing of a(s 0 , τ ). The connected structure of the envelope is then observed, with the appearance of an invariant node with respect to the delay, at the spectral abscissa s 0 , for every τ ≥ τ . see Figure 1 τ , see [START_REF] Boussaada | Multiplicity and stable varieties of timedelay systems: A missing link[END_REF].

delay τ * , the connected structure of the envelope is preserved, only in a single point (a node), see Figure 2 (left). Exceeding this critical value τ * , there is appearance of two disconnected components, which moving away more and more until a constant distance that is reached for every delay τ ≥ τ * * , see Figure 2 (right). 3. Such a geometry is encountered in [START_REF] Qiao | A linear time-invariant consensus dynamics with homogeneous delays: Analytical study and synthesis of rightmost eigenvalues[END_REF], where an analytical study and synthesis of rightmost eigenvalues of ẋ(t) = Ax(tτ ) is considered.

The description of this kind of curve, when considering two simples spectral values s 2 , s 1 , with s 2 < s 1 < 0, is given in the following theorem.

Theorem 3. Let C τ := C (s 2 , s 1 , τ ) = (x, y) ; x 2 + y 2 = (|a (s 2 , s 1 , τ )| + |b (s 2 , s 1 , τ )| exp(-xτ ))
(3.14) be the envelope curve associated to the quasipolynomial (3.2).

By browsing the delay τ from a small value until a large value, the following assertions holds for (3.2):

1. For any τ > 0, the envelope curve C τ intersects one times the real axis, in a positive abscissa, x + (τ ).

2. There exists τ * , τ * * > 0, with τ * < τ * * , such that:

• When τ = τ * , the curve C τ * contains a node, (x(τ * ), 0), in the region x < 0.

• For τ < τ * , there is only one intersection of the envelope curve with the real axis. • For τ ∈]τ * * , τ * [, the curve C τ splits into two disjoint segments C τ 1 and C τ 2 , which intersect the the region ]s 2 , s 1 [ at points (x - 1 (τ ), 0) and (x - 2 (τ ), 0), respectively.

• There exists x (τ * * ) < s 2 such that for every τ < τ * * ,

• C τ * * ⊂ • C τ in the region
x > x (τ * * ), and for every τ > τ * * ,

• C τ ⊂ • C τ * * . • For τ ≥ τ * * , the curve C τ splits into two disjoint segments C τ 1 and C τ 2 , with the property that (s 2 , 0) ∈ τ ≥τ * * C τ 1 and (s 1 , 0) ∈ τ ≥τ * * C τ 2 and d(C τ 1 , C τ 2 ) = |s 2 -s 1 |. 3. As τ → ∞, • x - 1 (τ ) → s 2 , x - 2 (τ ) → s 1 and x + (τ ) → -s 1 .
The two first limits are reached from τ = τ * * , see Figure 4.

• C ∞ , the envelope limit, is the circle centered at the origin with the rayon r = |s 1 |.

Proof.

To find the point of intersection of the curve with the x-axis, we solve this system of two equation obtained from (3.14) using the W-Lambert function [START_REF] Yi | Analysis and control of time delayed systems via the lambert W function[END_REF] |a|

+ |b|e -xτ -x = 0 |a| + |b|e -xτ + x = 0. (3.15)
Using a change of variable z = -xτ we obtain

|s 1 -s 2 |e z - z τ |e -s2τ -e -s1τ | + |s 2 e -s1τ -s 1 e -s2τ | = 0 (3.16
)

|s 1 -s 2 |e z + z τ |e -s2τ -e -s1τ | + |s 2 e -s1τ -s 1 e -s2τ | = 0. (3.17)
The above two equation are under the following form γe z + β + σ = 0, where γ = |s 1s 2 |, β = ± e -s 2 τ -e -s 1 τ τ and σ = |s 2 e -s1τs 1 e -s2τ | The existence of real solutions depends on the sign of the discriminant ∆(τ ) = γ β e -σ β and its position with respect to -e -1 .

-

3 -2 -1 0 1 2 -3 -2 -1 0 1 2 3 x -3 -2 -1 0 1 2 -3 -2 -1 0 1 2 3
The segment C τ 1. For τ > 0, we search the positive solution, so we study the equation (3.17). The discriminant of this equation is given by:

∆ + (τ ) = τ |s 1 -s 2 | |e -s2τ -e -s1τ | exp - τ |s 2 e -s1τ -s 1 e -s2τ | |e -s2τ -e -s1τ | . (3.18)
Hence equation (3.17) has a positif real solution

z + 1 (τ ) = -W 0 (∆ + ) - |s 2 e -s1τ -s 1 e -s2τ | |e -s2τ -e -s1τ | , thus x + (τ ) = 1 τ W 0 (∆ + ) + |s 2 e -s1τ -s 1 e -s2τ | |e -s2τ -e -s1τ | , (3.19)
which implies that the envelope curve intersects only one times the real axis in the right half plan at x + (τ ).

2. Now we study the equation (3.16), where the discriminant is written as follows:

∆ -(τ ) = - τ |s 1 -s 2 | |e -s2τ -e -s1τ | exp τ |s 2 e -s1τ -s 1 e -s2τ | |e -s2τ -e -s1τ | . (3.20)
Since ∆ -(τ ) < 0, we need to evaluate the sign of ∆ -(τ ) + e -1 . Note that the function τ → ∆ -(τ )+exp (-1) is continuous on R + . In addition, as τ → 0, ∆ -(τ )+exp (-1) → exp (1) + exp (-1) < 0. We have then x + (τ)

∆ -(τ * * ) = -ln s 1 s 2 s 1 s 2 - s 1 s 2 -s 1 - s 1 s 2 - s 2 s 2 -s 1 -1 + e -1 .

The variations of

F (t) := -ln (t) t -t 1-t -t -(1-t) -1 -1 + e -1 , t ∈ R * + -{1} 0.5 1
x - 1 (τ)

x - 2 (τ)

τ * * = ln(2)
x - • For τ = τ * , ∆ -(τ * ) = -e -1 , so (3.16) admits a (double) negative real solution

1 (τ) = s 1 = -1, ∀τ ≥ τ * * x - 2 (τ) = s 2 = -2, ∀τ ≥ τ * * τ * ≃ 0.67288 x - d (τ * ) ≃ -1.5280
x - d (τ * ) = 1 τ * W 0 (-e -1 ) - |s 2 e -s1τ -s 1 e -s2τ | |e -s2τ -e -s1τ | = - 1 τ * - |s 2 e -s1τ -s 1 e -s2τ | |e -s2τ -e -s1τ | ,
which allows to the appearance of a node at the point (x(τ * ), 0). This equation admits also a positive solution (x + (τ ), 0) given by (3.19), from which we deduce that there are two intersections with the real axis.

• For τ < τ * , there exists one positive solution (x + (τ ), 0) given by (3.19), which corresponds to unique intersection with the half real axis, x > 0.

• For τ ∈]τ * , τ * * [, we have ∆ -(τ ) + e -1 > 0, then equation (3.16) admits two negative real solutions

z 1 (τ ) = -W 0 ∆ -+ τ |s 2 e -s1τ -s 1 e -s2τ | |e -s2τ -e -s1τ | , z 2 (τ ) = -W -1 ∆ -+ τ |s 2 e -s1τ -s 1 e -s2τ | |e -s2τ -e -s1τ | , thus x - 1 (τ ) = 1 τ W 0 ∆ -- |s 2 e -s1τ -s 1 e -s2τ | |e -τ s2 -e -s1τ | , (3.21)
x .22) this implies that the curve C τ intersects two times the real axis, from which we deduce that the curve splits into two disjoints segments C τ 1 and C τ 2 which inter-sect the region ]s 2 , s 1 [ at points (x - 1 (τ ), 0) and (x - 2 (τ ), 0) respectively. Details about the property C τ 2 ∩ C τ 2 = ∅ will be presented bellow. • Here we consider the case where τ < τ * * . Recall that the sign of a(s 2 , s 1 , τ ) depends on the parameter τ > 0, while b(s 2 , s 1 , .) > 0. So

- 2 (τ ) = 1 τ W -1 ∆ -- |s 2 e -s1τ -s 1 e -s2τ | |e -s2τ -e -s1τ | , ( 3
|s 2 exp(-s 1 τ ) -s 1 exp(-s 2 τ )| =    0 if τ = τ * * s 2 exp(-s 1 τ ) -s 1 exp(-s 2 τ ) if τ > τ * * s 1 exp(-s 2 τ ) -s 2 exp(-s 1 τ ) if τ < τ * * Hence d dτ (|a (s 2 , s 1 , τ )|) =          e -τ s2 e -τ s1 (s 2 -s 1 ) 2 (e -τ s2 -e -τ s1 ) 2 if τ > τ * * -e -τ s2 e -τ s1 (s 2 -s 1 ) 2 (e -τ s2 -e -τ s1 ) 2 if τ < τ * * d dτ (|b (s 2 , s 1 , τ )| exp(-xτ )) = e -xτ (s 1 -s 2 ) (e -τ s2 -e -τ s1 ) s 2 e -τ s2 -s 1 e -τ s1 e -τ s2 -e -τ s1 -x Note that s 2 e -τ s1 -s 1 e -τ s2 = (1 -θτ ) e -θτ (s 2 -s 1 ), for some θ ∈ ]s 2 , s 1 [ , so x(τ ) ∆ = s 2 e -τ s1 -s 1 e -τ s2
e -τ s2e -τ s1 < 0, ∀τ > 0.

In addition, τ → x(τ ) is continuous and increasing on ]0, τ * * [, with

lim τ →τ * * x (τ ) = s 2 -(s 1 -s 2 ) s 2 s 1 s 1 s 2 -s 1 s 2 s 1 s 2 s 2 -s 1 - s 2 s 1 s 1 s 2 -s 1 = x (τ * * ) < s 2 .
Thus τ → |b (s 2 , s 1 , τ )| exp(-xτ ) is decreasing on R + * , for every x > x (τ * * ) . This means that when x > x (τ * * ) , the function

τ → |a (s 2 , s 1 , τ )| + |b (s 2 , s 1 , τ )| exp(-xτ )
is continuous and decreasing on ]0, τ * * [ . We deduce that

∀τ ∈ ]0, τ * * [ ; C (s 2 , s 1 , τ * * ) ⊂ C (s 2 , s 1 , τ ) .
On the other hand, the variations of (3.23) from which we deduce that

x → |a(s 2 , s 1 , τ )| + |b(s 2 , s 1 , τ )| exp (-xτ ) show that for all x ∈ R -[s 2 , s 1 ] and for all τ > τ * * |b(s 2 , s 1 , τ * * )| exp (-xτ * * ) -|a(s 2 , s 1 , τ )| + |b(s 2 , s 1 , τ )| exp (-xτ ) < 0,
∀ τ > τ * * ; C (s 2 , s 1 , τ * * ) ⊂ C (s 2 , s 1 , τ ) .
(3.24)

• Now we study the case of τ > τ * * . We have

lim τ →τ * * C (s 2 , s 1 , τ ) = τ <τ * * C (s 2 , s 1 , τ ) = C (s 2 , s 1 , τ * * ) = (x, y) ; x 2 + y 2 = M (s 2 , s 1 , x) where M (s 2 , s 1 , x) = s 1 -s 2 s 2 s 1 - s 2 -x s 1 -s 2 - s 2 s 1 - s 1 -x s 1 -s 2 .
Observe that

M (s 2 , s 1 , s 2 ) = -s 2 ; M (s 2 , s 1 , s 1 ) = -s 1 .
Hence the boundary of C (s 2 , s 1 , τ * * ) contains both (s 1 , 0) and (s 2 , 0). We get more, that is

(s 2 , 0) ∈ τ ≥τ * * C τ 1 and (s 1 , 0) ∈ τ ≥τ * * C τ 2 . (3.25)
On the other hand, since

M (s 2 , s 1 , αs 1 + (1 -α) s 2 ) = -s 2 s 1 s 2 α
for every α ∈ ]0, 1[, we can show that

C (s 2 , s 1 , τ * * ) ∩ ]s 2 , s 1 [ = ∅. Indeed, if not there exists z ∈ ]s 2 , s 1 [ , such that |z| ≤ M (s 2 , s 1 , z) . Writing z = αs 1 + (1 -α) s 2 , for some α ∈ ]0, 1[ , we have -s 1 < |z| ≤ -s 2 s 1 s 2 α < -s 2 s 1 s 2 = -s 1 .
This implies that -s 1 < -s 1 allowing to a contradiction. This reasoning allows us to show that

C (s 2 , s 1 , τ * * ) ∩ (x, y) ∈ R 2 , s 2 < x < s 1 = ∅. (3.26)
In fact, in the contrary case, we will get

-s 1 ≤ s 2 1 + y 2 < x 2 + y 2 = |z| ≤ -s 2 s 1 s 2 α < -s 1 .
Impossible. This means that the envelope curve C (s 2 , s 1 , τ * * ) is splits into two disjointed parts C x<s2 (s 2 , s 1 , τ * * ) and C x>s1 (s 2 , s 1 , τ * * ), separated by the band (x, y) ∈ R 2 , s 2 < x < s 1 . As a consequence, we get that distance

d C x<s2 (s 2 , s 1 , τ * * ) , C x>s1 (s 2 , s 1 , τ * * ) = d C x<s2 (s 2 , s 1 , τ ) , C x>s1 (s 2 , s 1 , τ ) = inf z1∈C x<s 2 (s2,s1,τ * * ) z2∈C x>s 1 (s2,s1,τ * * ) d (z 1 , z 2 ) = s 1 -s 2 , ∀τ > τ * * .
d being the Euclidean distance.

3. Now, we deal with the asymptotic behavior of the envelope curve, x - 1 (τ ), x - 2 (τ ), and x + (τ ) as τ → ∞.

• Since ∆ -(τ ) ∈ -e -1 , 0 , we have W 0 (∆ -(τ )) satisfies:

-1 = W 0 -e -1 < W 0 ∆ -(τ ) < W 0 (0) = 0, thus lim τ →∞ 1 τ W 0 ∆ -= 0.
From this we deduce that lim

τ →∞ x - 1 (τ ) = s 1 .
On the other hand, since we have

d(C τ 1 , C τ 2 ) = s 1 -s 2 , ∀τ > τ * * ,
we deduce that lim

τ →∞ x - 2 (τ ) = s 2 . 1 (3.27)
Similarly, we prove that lim τ →∞

x + (τ ) = -s 1 . Indeed, there is always

τ 0 > 0 such that |s 2 e -s1τ -s 1 e -s2τ | |e -τ s2 -e -s1τ | > - s 1 2
, for all τ ≥ τ 0 .

This means that x + (τ ) in (3.19) given via the WLambert function satisfies

x + (τ ) > - s 1 2 , ∀τ ≥ τ 0 .
Using the fact that

0 < e -τ x + (τ ) |e -τ s2 -e -s1τ | < e s 1 τ 2 |e -τ s2 -e -s1τ |
, ∀τ ≥ τ 0 .

1 As a consequence, using (3.22), we get the following property of the WLambert function:

limτ→∞ 1 τ W -1 ∆ -(τ ) = 0.
we deduce that e -τ x + (τ ) |e -τ s2e -s1τ | = 0. Letting τ tend to ∞ in the following expression:

x + (τ ) = e -τ x + (τ ) |e -τ s2 -e -s1τ | + |s 1 e -τ s2 -s 2 e -s1τ | |e -τ s2 -e -s1τ | (3.28)
we obtain lim Hence, we deduce that as τ → ∞ the envelope curve (3.14) becomes

x 2 + y 2 = s 2 1 ,
which corresponds to a circle C ∞ with the ray r = |s 1 |. Concerning the region x < s 2 , simulation results show that for τ sufficiently large, the segment C τ 1 disappears. The question of finding the upper bound of τ for which C τ 1 is exactly the half-plan x ≤ s 2 remains open.

Second-order systems

Second-order linear systems capture the dynamic behavior of many natural phenomena, and have found wide applications in a variety of fields, such as vibration and structural analysis. The equation of this system is given as follows:

ẍ(t) + a ẋ(t) + bx(t) + αx(t -τ ) = 0. (4.1)
The characteristic equation associated to (4.1) is given by: ∆(s, τ ) = s 2 + as + b + α exp(-sτ ) = 0. (4.2)

2 As a consequence, we get the following property lim

τ →∞ 1 τ W 0 ∆ + = 0.
Theorem 4. The system (4.1) admits three distinct real spectral values s 3 , s 2 and s 1 with s 3 < s 2 < s 1 if and only if the parameters a, b and α satisfy

                       a = a(s 1 , s 2 , s 3 , τ ) := 1 Q i,j,k∈Λ i<j,i =j =k (-1) i+j (s 2 i -s 2 j ) exp(-s k τ ) b = b(s 1 , s 2 , s 3 , τ ) := - 1 Q i,j,k∈Λ i<j,i =j =k (-1) i+j s i s j (s i -s j ) exp(-s k τ ) α = (s 1 , s 2 , s 3 , τ ) := - 1 Q i,j∈Λ i<j (s i -s j ) (4.3)
where

Q := Q(s 1 , s 2 , s 3 , τ ) = i,j,k∈Λ i<j,k =i,j (-1) i+j (s i -s j ) exp(-s k τ ).
In this case, α is necessarily negative.

• The spectral value s 1 is negative if and only if there exists τ 0 > 0 such a(s 1 , s 2 , s 3 , τ 0 ) + s 2 = 0.

This guarantees the asymptotic stability of the system.

• The root s 1 is the spectral abscissa of (4.1).

Proof. The parameters given in (4.

α = - (s 1 -s 2 ) (s 2 -s 3 ) (s 1 -s 3 ) Q (s 1 , s 2 , s 3 , τ ) . (4.5)
We apply twice the Mean Value Theorem to the denominator Q (s 1 , s 2 , s 3 , τ ), we get:

Q (s 1 , s 2 , s 3 , τ ) = s 2 e -τ s3
s 1 e -τ s3 + s 1 e -τ s2s 3 e -τ s2 + s 3 e -τ s1s 2 e -τ s1 = (s 2s 1 ) e -τ s3e -τ s1 + (s 3s 1 ) e -τ s1e -τ s2

= -τ (s 2s 1 ) (s 3s 1 )

1 0 e -τ (ts1+(1-t)s3) dt +τ (s 3 -s 1 ) (s 2 -s 1 ) 1 0 e -τ (ts1+(1-t)s2) dt = -τ (s 2 -s 1 ) (s 3 -s 1 ) 1 0 e -τ (ts1+(1-t)s3) -e -τ (ts1+(1-t)s2) dt = τ 2 (s 2 -s 1 ) (s 3 -s 1 ) (s 3 -s 2 ) 1 0 1 0 (1 -t) e -τ (ts1+(1-t)(θs3+(1-θ)s2)) dθdt.
On the spectral abscissa and rightmost-roots of TDS This means that

α (s 1 , s 2 , s 3 , τ ) = -1 τ 2 Q (s 1 , s 2 , s 3 , τ ) (4.6)
where

Q (s 1 , s 2 , s 3 , τ ) = 1 0 1 0 (1 -t) e -τ (ts1+(1-t)(θs3+(1-θ)s2)) dθdt > 0.
Now, let us show the negativeness of s 1 , s 2 , s 3 . From (4.4) and (4.6) , we get that

   (s 3 -s 2 ) [s 3 + s 2 + a] = -α [e -τ s3 -e -τ s2 ] > 0 (s 2 -s 1 ) [s 2 + s 1 + a] = -α [e -τ s2 -e -τ s1 ] > 0 (s 3 -s 1 ) [s 3 + s 1 + a] = -α [e -τ s3 -e -τ s1 ] > 0.
Since s 3 < s 2 < s 1 , we obtain the following equations:

   s 3 + s 2 < -a s 2 + s 1 < -a s 3 + s 1 < -a (4.7)
System (4.7) is reduced to the single inequality s 2 + s 1 < -a, from which we deduce that s 3 < s 2 < -a 2 , and s 1 < -as 2 . Since the mapping τ → a(s 1 , s 2 , s 3 , τ ) + s 2 is continuous and increasing from -∞ to -s 1 when τ varies in R + * , this means that a(s 1 , s 2 , s 3 , τ ) + s 2 takes positive values if and only if s 1 < 0. This means that the equation a(s 1 , s 2 , s 3 , τ ) + s 2 = 0 has a unique root, τ 0 > 0. Now, to study the stability of the system, we need to study the dominancy of s 1 by using an adequate factorization of the quasipolynimial (4.2), so we have:

s 2 + as + b + α exp (-τ s) = (s -s 2 ) (s -s 1 ) s 2 + as + b + α exp (-τ s) (s -s 2 ) (s -s 1 ) = (s -s 2 ) (s -s 1 ) P (s, s 1 , s 2 , τ ).
The factor P (s) := P (s, s 1 , s 2 , τ ) can be rewritten under the more suitable form

P (s) = 1 + (a + s 2 + s 1 ) s + b -s 2 s 1 (s -s 2 ) (s -s 1 ) + α exp (-τ s) (s -s 2 ) (s -s 1 ) = 1 - s 2 2 + as 2 + b (s 1 -s 2 ) (s -s 2 ) + s 2 1 + as 1 + b (s 1 -s 2 ) (s -s 1 ) + α exp (-τ s) (s 2 -s 1 ) (s -s 2 ) - α exp (-τ s) (s 2 -s 1 ) (s -s 1 ) = 1 + α exp (-τ s) (s 2 -s 1 ) (s -s 2 ) - α exp (-τ s 2 ) (s 2 -s 1 ) (s -s 2 ) + α exp (-τ s 1 ) (s 1 -s 2 ) (s -s 1 ) - α exp (-τ s) (s 2 -s 1 ) (s -s 1 )
.

Using the integral form of the remainder in Taylor's Theorem, we get

P (s) = 1 + αe -τ s2 e -τ (s-s2) -1 (s 2 -s 1 ) (s -s 2 ) + αe -τ s1 1 -e -τ (s-s1) (s 2 -s 1 ) (s -s 1 ) (4.8) = 1 + αe -τ s2 -τ (s -s 2 ) 1 0 e -τ (s-s2)t dt (s 2 -s 1 ) (s -s 2 ) - αe -τ s1 -τ (s -s 1 ) 1 0 e -τ (s-s1)t dt (s 2 -s 1 ) (s -s 1 ) = 1 + ατ (s 2 -s 1 )
1 0 e -τ ts e -τ (1-t)s1e -τ (1-t)s2 dt.

Using again the following integral representation

exp (-τ (1 -t) s 1 )-exp (-τ (1 -t) s 2 ) = -τ (s 1 -s 2 ) 1 0 exp (-τ (1 -t) (θs 2 + (1 -θ) s 1 )) dθ we get P (s, s 1 , s 2 , τ ) = 1+ατ 2 1 0 1 0 (1 -t) exp (-τ t (s -s 1 )) exp (-τ [s 1 + θ (1 -t) (s 2 -s 1 )]) dθdt.
To prove the dominancy property, namely s 1 is the rightmost root of (4.2), let us assume that there exists some s 0 = ζ + jη a root of (4.2) such that ζ > s 1 . Then P (s 0 , s 1 , s 2 , τ ) = 0 and for any t > 0, one has: exp

(-τ [t (ζ -s 1 )]) < 1. So 1 = -ατ 2 1 0 1 0 (1 -t) exp (-τ t (s 0 -s 1 )) exp (-τ [s 1 + θ (1 -t) (s 2 -s 1 )]) dθdt = Re -ατ 2 1 0 1 0 (1 -t) exp (-τ t (s 0 -s 1 )) exp (-τ [s 1 + θ (1 -t) (s 2 -s 1 )]) dθdt ≤ |α| τ 2 1 0 1 0 (1 -t) exp (-τ t (ζ -s 1 )) exp (-τ [s 1 + θ (1 -t) (s 2 -s 1 )]) dθdt < |α| τ 2 1 0 1 0 (1 -t) exp (-τ [(1 -θ (1 -t)) s 1 + θ (1 -t) s 2 ]) dθdt. (4.9) Using the relation 1 -θ (1 -t) = t + (1 -t) (1 -θ) , we get exp (-τ (1 -θ (1 -t)) s 1 ) = exp(-τ ts 1 ) exp (-τ (1 -t) (1 -θ) s 1 ) .
Moreover, since s 3 < s 1 , we obtain, for every t ∈ (0, 1) and every θ ∈ (0, 1) , the following estimation exp

(-τ (1 -t) (1 -θ) s 1 ) < exp (-τ (1 -t) (1 -θ) s 3 ) .
On the spectral abscissa and rightmost-roots of TDS Summing up, using (4.6) , inequality (4.9) becomes

1 < |α| τ 2 1 0 1 0 (1 -t) exp(-τ ts 1 ) exp (-τ (1 -t) [(1 -θ) s 3 + θs 2 ]) dθdt = |α| τ 2 Q = 1,
which is inconsistent. This proves the dominancy of s 1 . The proof of Theorem 4 is achieved. 2. If s 3 > -R, the connected structure of the envelope is preserved independently of the delay τ . For τ sufficiency large, the envelope takes the shape of a "vertical Omega" Ω never closing, positioned in s 3 , and which is carried by the centered circle of radius R, see Figure 6 (left).

3. If s 3 = -R, the connected structure of the envelope is still preserved as in the second case. Here the "vertical Omega" structure is also observed. This structure is deformed when the delay increases. Indeed, the angular points are getting closer and closer, until forming a node at s 3 , see Figure 6 (right).

Concluding remarks

The property of multiplicity induced-dominancy for spectral value of time-delay systems of retarded type is extended through this work. More precisely, it is shown, for reduced order Time-delay systems, that assigning P S B real spectral values (not necessarily multiple root) make them the rightmost roots of the corresponding quasipolynomial. Furthermore, if they are set to be negative, this guarantees the asymptotic stability of the trivial solution. This new result emphasizes a new delayed controller-design based on the trivial solution's decay rate assignment.

  .

2 .τ 5 Figure 1 :

 251 Figure 1: Envelope curve of the characteristic equation (3.2). Case when s 0 = -2 is a double root. The critical delay τ = 0.5 corresponds to the value that vanishes the coefficient a = 2 -1τ , see[START_REF] Boussaada | Multiplicity and stable varieties of timedelay systems: A missing link[END_REF].

Figure 2 :

 2 Figure 2: Envelope curve of the characteristic equation (3.2). Case of co-existence of two simple real roots s 1 = -1, s 2 = -2.

2 τ = 2 τ = 10 τ = 1 τ = 0. 8 τC τ 1 Figure 3 :

 2101813 Figure 3: (Left) Decreasing behavior of the envelope curve versus τ < τ * * . (Right) Increasing behavior of the envelope curve versus τ ≥ τ * * . Case s 1 = -1 and s 2 = -2

Figure 4 :

 4 Figure 4: Asymptotic behavior of the roots of the envelope curve. Case s 1 = -1, s 2 = -2 show that ∆ -(τ * * ) + e -1 > 0, ∀t ∈ R * + -{1}. Thanks to the intermediate values theorem, there exists τ * ∈]0, τ * * [ such that ∆ -(τ * ) + e -1 = 0.

τ 2 •

 2 →∞ x + (τ ) = |s 1 |. Let us consider the equation of the envelope curve (3.14). In the region x > s 1 , we have lim τ →∞ |b(s 1 , s 2 , τ )| exp (-xτ ) = 0, and lim τ →∞ |a(s 1 , s 2 , τ )| = |s 1 |.

3 ) s 2 3 + 2 + 1 +

 3321 can be easily obtained by solving the system   as 3 + b + αe -s3τ = 0 s 2 as 2 + b + αe -s2τ = 0 s 2 as 1 + b + αe -s1τ = 0. (4.4) Now, let us study the sign of α. Recall that

Remark 3 ., 2 2 1 . 2 + 2 Figure 5 :

 321225 Figure 5: Envelope curve of the characteristic equation (4.2). Case s 1 = -2, s 2 = -3, s 3 = -13.

6 x 2 + y 2 yFigure 6 :

 6226 Figure 6: Envelope curve of the characteristic equation (4.2). Case s 1 = -2, s 2 = -3, and s 3 ≥ -R.

Acknowledgement. IB and SIN are partially financially supported by a grant from Hubert Curien (PHC) BRANCUSI 2017, project number 38390ZL and a grant from PHC BALATON 2018, project number 40502NM.