S Schaffert

F Bry

P Besnard

H Decker

S Decker

C Enguix

A Herzig

Position Paper: Paraconsistent Reasoning for the Semantic Web

Due to the Semantic Web's decentralised and distributed management, contradictory information is and will remain frequent. However, classical reasoning systems fail to work properly in the presence of inconsistencies, because they implicitly or explicitly assume the ex contradictione quod libet (ECQL) principle stating that anything follows from contradictory premises. Paraconsistent reasoning challenges this ECQL principle. Stressing practical cases of reasoning on the Web, this position paper first argues that paraconsistent reasoning is likely to become a key issue for successful deployment of the Semantic Web. Then, it briefly introduces the main approaches to date to paraconsistent reasoning.

Introduction

Classical and other logic, upon which modern computing is based, requires the complete absence of contradictions. With the classical ex contradictione quod libet (ECQL) rule (or principle of explosion), everything, and thus nothing useful at all, can be inferred from a contradiction. For instance, from a contradiction in a train information system can be derived that the moon is made of green cheese. Nonetheless, inconsistencies play an important role in practice (Section 2).

Paraconsistent logics are a rather novel direction in mathematical logics that challenge the ECQL principle in order to allow "reasonable" reasoning in the presence of inconsistencies without introducing more problems than are already present in the data. Several different approaches to paraconsistent logics exist and are briefly outlined in Section 3.

We conclude this article with a perspective for paraconsistent reasoning on the Semantic Web (Section 4).

Cases for Paraconsistent Reasoning

Distributed Information Systems

In distributed information systems, like the online information systems of European railways companies, contradictory information is frequent. For example, the German railway company might give different arrival times for trains to Paris than the French railway company, because construction works on the track in France have not been entered into the German system. Human beings can easily cope with such inconsistencies in various ways (e.g. identify which information is more likely or "don't care"). Reasoning systems on the (Semantic) Web must equally be able to derive useful conclusions from the "inconsistency-free" premises.

Coping with Change

Belief change is the field of artificial intelligence devoted to the rational change of belief in the light of new evidence. E.g., a train timetable might be updated with new train connections that have to be taken into account in further reasoning. Likewise, train connections might have been removed making previously drawn conclusions invalid.

In practice, changes like updates to an information system may cause inconsistencies that cannot be discarded. Standard methods for belief change are based on classical logic and hence accept the ECQL principle. As a consequence, they cannot be used for deriving useful conclusions in presence of updates causing contradictions.

Inconsistencies Welcome!

In some situations, inconsistencies are even desirable. This is, e.g., the case when contradictory viewpoints are present and need to be reconciled. For instance, two ontologies describing appartment rental offers and appartment sale offers might well inconsistently describe preferences and prices for city areas. This obviously should neither prevent considering both ontologies nor deriving meaningful conclusions in the same reasoning context (like helping in taking a decision for buying or renting an appartment). Obviously, human beings are capable of doing so without applying the ECQL principle, and so should automated reasoning systems on the Web.

Another example is policy reasoning. At the beginning of a negotiation towards selling/buying a Web service, the policies of the buyer and seller might be contradictory. Instead of applying the ECQL principle, a reasoning system should strive to overcome the inconsistencies, i.e. find a way to pass a contract acceptable for both the service buyer and seller without requiring them to change their policies.

"Dialetheias"

In practice, there are cases where contradictions are inherent to the problem, so-called "dialetheias". Since such cases arise in knowledge modelling, they will also arise on the Semantic Web. This is in particular the case with the well known Liar's Paradox where a sentence states its own falsity ("this sentence is not true").

On the Semantic Web, dialetheias might easily arise though reification, especially of RDF statements, and through modalities -such as "A believes B" or "A does not believe what B states" -that are needed e.g. for policy reasoning. Liar sentences can also be indirect consequences of statements that are themselves unproblematic, e.g. when combining knowledge from different Web resources.

Approaches to Paraconsistent Reasoning

Most approaches to paraconsistent logic and reasoning allow a formula F and its negation ¬F to hold in an interpreation (or "model"). Major approaches of paraconsistent logics and reasoning are stressed below:

Relevant Logics

Relevant logics have been first proposed by Anderson and Belnap. Semantics for such logics based on "different worlds" have been developed by Routley and Meyer. Conjunction and disjunction behave in the usual way, but each world w has an associated world w * such that ¬A is true in w iff A is false in w * (not in w). As a consequence, if A is true in w and false in w * , then A ∧ ¬A is true in w. Note that requiring w * = w yields the standard classical logic.

Many-Valued Systems

A multi-valued logic is a logic with more than two truth values. The formulas that hold in a multi-valued interpretations are those which have a specific truth-value, the so-called designated formulas. A multi-valued logic is paraconsistent if it allows both a formula and its negation to be designated.

The simplest approach uses three truth values: true and false, like in classical logic, and a third truth-value denoting "both truth and false" such that if a formual F has this third truth-value in an interpreation, then so does also ¬F . Considering the real numbers between 0 and 1 instead of discrete values results in a paraconsistent fuzzy logic.

Non-Adjunctive Systems

A non-adjunctive logic is a logic in which one cannot conclude A from A ∧ B. The first non-adjunctive logic, and also the first paraconsistent logic, ever proposed is the discussive (or discursive) logic of Jaskowski. In dicussive logic, several contributors state "opinions". Each opinion is consistent in itself but might be inconsistent with another opinion. A modal logic (S5) is used to define interpretations: a world corresponds to a contributor, and in it, all the contributor's sentences are true. Thus, A ∧ ¬A can hold in an interpretation consisting of several worlds, but not in a single world.

Non-Truth-Functional Logics

Non-truth functional logics have been introduced by da Costa. Their idea is to make negation "non-truth-functional" while keeping the other connectives like in standard, e.g. classical, logics. Seeing an interpretation as a function mapping formulas to 0 (false) or 1 (true), a non-truth funtional logic gives rise to defining the truth-value of ¬A independently of that of A (while keeping the usual functional dependencies of the truth-value of A ∧ B, A ∨ B, A ⇒ B, etc. to the truth values of A and B).

Paraconsistency on the Semantic Web

We believe that dealing with inconsistencies will play a central role in the emergence of the Semantic Web. Paraconsistent reasoning provides foundations and techniques that will allow future applications to function properly in the presence of inconsistencies. In particular, we think that paraconsistent reasoning will influence the following areas:

Paraconsistency in Ontology Reasoning

Ontology reasoning (e.g. instance checking) on the Semantic Web is usually based on reasoning techniques, e.g. the tableaux calculus, developed for description logics. Therefore, a first step towards an "inconsistency-aware" Semantic Web will be to adapt existing reasoning algorithms using techniques from paraconsistent reasoning.

Paraconsistency in Query Languages

Querying data plays a very important role on the Semantic Web, as indicated by the multitude of existing Semantic Web query languages. Building upon ontology reasoning, Semantic Web query languages will likely need to be adapted so as to work in the presence of inconsistencies.

Paraconsistency and Trust

In a distributed environment like the Semantic Web, where anyone can author content, trust is a key issue. Conflicts with classical logic are apparent: for example, different sources might make conflicting assertions about the trustworthiness of a site, and users might be interested in more fine-grained levels of trust besides the binary "trusted" or "not trusted".


Evaluating multimodal literacy: Academic and professional interactions around student-produced instructional video tutorials

Keywords: workplace-based multimodal literacies, multimodal evaluation grids, instructional video tutorials, collaborative needs-based analysis, technical communication, EMI

This paper offers a reflection on how academic and professional interactions can help guide best practices for constructing viable evaluation grids to assess multimodal literacy. Preparing English-language learners for today's digitally and culturally complex workplace environment is a central concern in English as a second language (L2) teaching environments. It requires meeting specific teaching goals, such as supporting traditional print and multimodal literacies as well as increasing learners' English-language fluency and appropriateness. Our study focuses on an underexplored professional multimodal genre -instructional video tutorials -and proposes a multimodal evaluation grid incorporating theoretical concepts and empirical results from multimodal linguistics and multimedia learning. We examine how four video communication professionals use the grid to measure the effectiveness of students' video tutorials and identify areas for improvement. We present results for three areas for which the experts considered students did not meet expected professional standards: information organization, timing, and L2 spoken language narration. Our findings suggest possibilities for introducing appropriate forms of action or intervention into teaching multimodal design projects to better prepare L2 English students to meet workplace multimodal literacy requirements.

Introduction

Increasingly, students in higher education must develop domain competence in a foreign language (L2) to accommodate face-to-face and computer-mediated workplace interactions to navigate the complex technology-mediated environments they are training for [START_REF] Hewett | Professional Development in Online Teaching and Learning in Technical Communication: A Ten-Year Retrospective[END_REF]. Accordingly, language students' pedagogical needs extend beyond school-based literacy for L2 writing and speaking instruction, toward harnessing the multiliteracy skills more appropriate to current workplace environments. To become competent contributors in their workplace cultures, students need to develop 'language-techno cultural competence' [START_REF] Sauro | Toward Langua-technocultural Competence[END_REF] whereby they appropriate the modes of communication, ways of doing and professional identities specific to the workplace culture, in two or more languages.

In response, the L2 teaching focus has moved beyond grammar and vocabulary toward a literacies approach [START_REF] Ware | Technology, new literacies and language learners[END_REF], which posits that students are better served if their L2 courses are designed around workplace-based and academic literacy practices integrating a variety of literacy types: computer, information, visual, multimodal. In this sense, the print-dominant view of literacy has shifted to a 'multiliteracies' perspective (Cope & Kalantzis, 2000, 2015) emphasizing "a combination of one or more elements of digital, multimodal, communicative and multilingual practices" (Ware, 2017: 267).

A crucial component of multiliteracies is multimodal literacy, a combination of skills which allows for "reading, viewing, understanding, responding to and producing and interacting with multimedia and digital texts" (Walsh, 2010: 213). As [START_REF] Ware | Technology, new literacies and language learners[END_REF] observes, demonstrating multimodal literacy involves producing texts that blend together various combinations of semiotic resources (e.g., language, speech, sound, graphics, animation. Such texts' multimodal composition challenges traditional L2 competency assessment practices (Crawford-Camiciottoli & Campoy-Cubillo, 2018). This paper explores the challenges posed by professional multimodal literacies in the context of an English medium of instruction (EMI) graduate program in technical communication, a field which requires a diverse range of skills. Europe-based professional technical communicators must possess good writing skills in two or more languages, including English. They must also demonstrate the ability to coordinate and repurpose several semiotic resources -linguistic, visual, spatial, aural, gestural -across numerous professional genres including print-based user-guides, online help and instructional video tutorials. Preparing students for such multilingual and multimodal work environments can be challenging. This paper seeks to fill a gap in the literature: despite growing interest [START_REF] Hafner | Embedding digital literacies in English language teaching: Students' digital video projects as multimodal ensembles[END_REF][START_REF] Hafner | Genre innovation and multimodal expression in scholarly communication: Video methods articles in experimental biology[END_REF][START_REF] Walsh | Multimodal literacy: What does it mean for classroom practice[END_REF], there is limited guidance on teaching three-dimensional multimodal professional genres such as instructional video tutorials, i.e., professional 'how-to' videos. Our study proposes a framework teachers may use to help students better master the multimodal skills required for the workplace, particularly with regard to video tutorials. Students may also use the framework for self-evaluation. To this end, we devised an evaluation grid created specifically for teaching and assessing this professional genre. It combines a multimodal (Cope & Kalantzis, 2000;[START_REF] Kress | Multimodality: A Social Semiotic Approach to Contemporary Communication[END_REF][START_REF] Kress | Reading Images: The Grammar of Visual Design[END_REF] and a multimedia-learning approach (Mayer, 2014a;[START_REF] Van Der Meij | Eight guidelines for the design of instructional videos for software training[END_REF] to ensure video tutorial effectiveness for a professional setting. While combining two very different epistemologies -multimodality and multimedia learning -within a single evaluative framework may raise concern, we argue that they are complementary and mutually reinforcing for the purposes of teaching and assessing instructional video tutorials. In addition, our evaluative framework integrates procedural genre-based features [START_REF] Ganier | Factors affecting the processing of procedural instructions: Implications for document design[END_REF][START_REF] Steehouder | Designing and evaluating procedural instructions with the four components model[END_REF] and L2 spoken language criteria from the Common European Frame of Reference (CEFR). Although the CEFR references many of the above issues, research juxtaposing all these points is rare.

We first describe instructional video tutorials and establish our theoretical framework: multimodal linguistics, social semiotics, and multimedia learning. We then present our multimodal evaluation grid before addressing two research aims: (1) to determine which criteria are most problematic for L2 students when designing video tutorials in English, according to domain professionals; (2) to present a methodology for designing and validating future evaluative frameworks through collaboration between academics and professionals. We tested the evaluation grid with four video communication professionals for usability as an assessment tool, i.e., whether it describes actual professional practice. We propose the grid can be used for student self-evaluation of video tutorial productions, and by instructors for training and assessment purposes. In conclusion, we discuss the implications of our findings for designing and validating evaluative frameworks for multimodal student projects.

Theoretical background

Instructional video tutorials

Over the past 20 years, video has become a primary means of delivering 'how-to' information, thus contributing to the emergence of hybrid forms of specialized genres. In education, the number of howto educational videos has progressed steadily [START_REF] Bateman | The communicative effectiveness of education videos: Towards an empirically-motivated multimodal account[END_REF][START_REF] Bateman | Explanation videos unravelled: Breaking the waves[END_REF][START_REF] De Koning | Developments and trends in learning with instructional video[END_REF]. Industry and the professions are no exception to this trend. Traditional print-based procedural genres (e.g., user manuals) have given way to instructional video tutorials in response to user demands for greater interactivity (informality, entertainment, encouragement, support) and exposure to richer channels of communication including spoken narration, animation and sound (van der Meij & van der Meij, 2013).

Instructional video tutorials are a dynamic "three-dimensional" [START_REF] Lotherington | Teaching multimodal and digital literacy in L2 settings: New literacies, new basics, new pedagogies[END_REF] multimodal genre. Given the video media's affordances, designers must pay careful attention to viewers' engagement [START_REF] Mayer | Principles based on social cues in multimedia learning: Personalization, voice, image, and embodiment principles[END_REF][START_REF] Mayer | A personalization effect in multimedia learning: Students learn better when words are in conversational style rather than formal style[END_REF], working memory and cognitive load [START_REF] Paas | Implications of cognitive load theory for multimedia learning[END_REF]. For example, because viewers often find video tutorials on the Web, they can click off at any time if video content or accessibility does not suit their purposes. Professionally-produced video tutorials thus need to convince viewers they will efficiently demonstrate how to carry out a task while inciting them to accept future interactions with the company. To devise an evaluation grid for teaching and assessing this professional multimodal genre, we have drawn on relevant literature summarized as follows.

Multimodal linguistics and social semiotics

Multimodal linguistics investigates the ways in which semiotic resources interact. It draws on semiotics, "the systematic study of systems of signs" (Lemke, 1990: 183), a field which traces back to Saussure's (1916) and Peirce's (1867-1871) work on signs and social meaning [START_REF] Jewitt | Introducing Multimodality[END_REF][START_REF] Tan | Multimodality[END_REF]. One of its present-day corollaries, social semiotics [START_REF] Halliday | Language as Social Semiotic[END_REF][START_REF] Kress | Social Semiotics[END_REF], builds on the premise that meaning-making is organized and codified as sociocultural practices. In this perspective, all meaning is considered to materialize and be interpreted through the realization of semiotic resources [START_REF] Kress | Multimodality: A Social Semiotic Approach to Contemporary Communication[END_REF], which van Leeuwen (2005: 3) defines as "the actions and artefacts we use to communicate", ranging from the vocal tract, to the muscles allowing for facial expression and gestures, to technologies, etc.

Scholars in this field consider that the semiotic resources within a system comprise options chosen among to make meaning. Although language has received the most attention, scholars have also investigated how other systems of semiotic resources -visual, aural, gestural, spatial -similarly make meaning. As Kress et al. (2000: 44) note, "visual communication, gesture, and action have evolved through their social usage into articulated or partially articulated semiotic systems in the same way that language has." They are thus resources for meaning-making in their own right, and depend on situational context and culture for salience and relevance [START_REF] Van Leeuwen | Rhythm and multimodal semiosis[END_REF].

The multiplicity of semiotic resources and the ways in which they combine to make meaning has been termed 'multimodality' [START_REF] Kress | Multimodality: A Social Semiotic Approach to Contemporary Communication[END_REF], as reflected in various social semiotic approaches. These approaches emphasize different aspects of meaning-making, e.g., focusing more on grammar as in O'Halloran & Lim's (2014) approach to systemic functional multimodal discourse analysis, or like [START_REF] Kress | Multimodality: A Social Semiotic Approach to Contemporary Communication[END_REF] and [START_REF] Kress | Reading Images: The Grammar of Visual Design[END_REF], attending to a more general social semiotic stance (see also [START_REF] Jewitt | Introducing Multimodality[END_REF].

In our multimodal evaluation grid, we adopt the New London Group's (Cope & Kalantzis, 2000) fivepart modal classification scheme, reproduced in Figure 1. As explained in the following sections, we have adapted this scheme slightly: we do not include the gestural mode given the focus of this paper. Likewise, we include the temporal mode to better account for the specific features of instructional video tutorials.

Empirical guidelines for designing video tutorials

A multimodality framework accounts for the modes and semiotic resources that comprise video tutorials. Studies drawing on multimedia learning principles, however, provide empirically-driven specifications for how to marshal specific semiotic resources for effectiveness in video tutorials.

Multimedia learning principles, developed in cognitive psychology (Mayer, 2014a), offer general guidance on how to reduce cognitive load in working memory and improve learning when instructional messages combine linguistic and visual modes, like in video tutorials. Focusing on the operational limitations of human learning and memory, Mayer (2014a) describes these principles, e.g., extraneous, non-semantically related material should be avoided (coherence principle); information related spatially or temporally should be presented near one another (contiguity principle); visual and linguistic cues should draw attention to relevant information (signaling principle); information should not be repeated using resources from the same mode, such as writing and speech (redundancy principle); information should be integrated to avoid competing areas of attention (split attention principle); combining animation and spoken language helps avoid split attention (modality principle); in dynamic formats, information must be learner-paced (segmentation principle); in narration, informal conversational styles are preferred (personalization principle). Another effect not explicitly described in Mayer (2014a), but of relevance to our study on video tutorials, relates to coherence. [START_REF] Mayer | A coherence effect in multimedia learning: The case for minimizing irrelevant sounds in the design of multimedia instructional messages[END_REF] state that irrelevant sounds including music should be excluded from the aural mode to avoid split attention and support coherence, thus reducing cognitive load. In sum, multimedia learning research examines how these principles impact people's ability to comprehend and remember information, e.g., in video tutorials, fast-paced, loud music can interfere with viewers' ability to concentrate on narration, and thus learn to carry out a task.

Such research serves as a basis for the detailed guidelines on software training videos proposed by van der Meij and van der [START_REF] Van Der Meij | Eight guidelines for the design of instructional videos for software training[END_REF][START_REF] Van Der Meij | Demonstration-based training (DBT) in the design of a video tutorial for software training[END_REF]. Incorporating research on usability and minimalism (e.g., [START_REF] Lazonder | The minimal manual: Is less really more?[END_REF][START_REF] Van Der Meij | Three decades of research and professional practice on printed software tutorials for novices[END_REF][START_REF] Van Der Meij | Principles and heuristics for designing minimalist instruction[END_REF], their eight guidelines for video tutorial design and demonstration-based training summarize criteria for designing effective video tutorials: facilitating ease of access, combining narration and animation, enabling viewer interactivity with the video, previewing tasks, and allowing users to practice and review content. They also emphasize the need to present procedural rather than conceptual information, make tasks clear and simple, and keep videos short. Detailed heuristics, based on an extensive literature review, provide research-based explanations to help designers achieve these guidelines, and describe how to determine what makes effective titles, task-based content, screen interfaces, pacing, narration, animation, and designer-viewer interaction.

Multimodal evaluation grid

We integrate the above into our multimodal evaluation grid, organized around five modes -linguistic, aural, visual, spatial, temporal -and their semiotic resources (cf. Figure 1). We did not include the gestural mode in the present version of the grid; because narrators' on-screen presence in video tutorials is optional, our teaching did not focus on gestures. Also, as far as we are aware, time has not been singled out as a specific design mode in multimodal analysis, although it is a crucial element of video tutorials. Without claiming to modify the New London Group's (Cope & Kalantzis, 2000) fivemode schema, we have opted for time's inclusion as a mode in our grid inasmuch as it is a social construct through which meaning is organized and created in mode-specific ways (Andersen et al., 2015;[START_REF] Jewitt | Multimodality literacy in school classrooms[END_REF][START_REF] Kress | Multimodality: A Social Semiotic Approach to Contemporary Communication[END_REF]. In video tutorials, time and its semiotic resources -length, speed, tempo, narration pacing, narration pauses, temporal cueing -are of utmost importance. Designers must balance meaningful sequences of simultaneously occurring resources (e.g., animation, spatial and visual content, spoken narration) within manageable segments of information delimited by length to keep viewers interested while avoiding cognitive overload [START_REF] Paas | Implications of cognitive load theory for multimedia learning[END_REF]. Given the importance of time in video tutorials, we have subsequently turned to research on multimedia learning principles for further indications about relevant temporal resources.

Multimedia learning-based studies fill this gap, for example by providing research-based recommendations about suitable video length. While Plaisant and Schneiderman (2005) propose 15-60 seconds and [START_REF] Chan | Advantages of video trigger in problem-based learning[END_REF] three minutes, van der Meij and van der Meij (2013: 221) underscore that length should be decided in terms of information organization: "arbitrary time limits … [are] hardly satisfactory. What matters more is that the user perceives a video as having a clear beginning and end." They suggest designers look for physical screen changes, indicating goal or sub-goal completion, as meaningful moments for segmentation, or that they divide "the stream of information into smaller units with identifiable beginning and end points" (p. 221). They further recommend using narration pauses and temporal cues (e.g., then, next, now) to help viewers identify meaningful segments.

Other temporal resources identified in the video tutorial literature include speed, organized by tempo and narration pacing which allow the designer to match the "scenario of the unfolding instructional events" with the viewer's cognitive "resources and capabilities" (van der Meij & van der Meij, 2013: 212). Tempo, for example, is controlled by synchronizing actions through graphic animation and narration. Determining "the right speed for the user" (p. 212) is also linked to narration pacing: extending natural pauses and adopting a "conversational pace". In turn, narration pace can be controlled through verbal-pausing interactions, rate of speech and prosody.

Getting a sense of these complex interactions is a significant challenge for novice designers. Considering these observations, we ascribe to Alhadeff-Jones' (2017: 2) view that the interpretation of time, whether "fast/slow, early/late, retarded/advanced, or mature/immature... remains socially constructed" and is contingent upon gaining professional experience through extensive contact with viewers' genre expectations [START_REF] Schriver | What do technical communicators need to know about information design[END_REF]. With regard to video tutorials, we consider that time is a socially-situated and learned concept, whose communicative purpose is to divide the video into segments and balance the amount of information by using temporal resources.

We transpose five modes, including the temporal mode, into a multimodal evaluation grid comprising 28 modal competency criteria (Figure 2), or statements about how to render semiotic modes and resources (cf. Cope & Kalantzis, 2000) specifically operational for instructional video tutorials, following multimedia learning principles including Mayer (2014a), van der Meij and van der Meij (2013, 2016), and others. For example, to minimize cognitive load and motivate viewers, videos should orally narrate demonstrated tasks rather than provide on-screen written text [START_REF] Ayres | The split-attention principle in multimedia learning[END_REF]. Furthermore, human rather than machine narrators should address viewers using simple but appealing language [START_REF] Mayer | Principles based on social cues in multimedia learning: Personalization, voice, image, and embodiment principles[END_REF]. Narration should be delivered using a "conversational pace" (Morain & Swarts, 2010) to allow for appropriate video tempo and speed. Also, ensuring sound quality and avoiding extraneous auditory information (e.g. music, sound effects) is essential [START_REF] Mayer | A coherence effect in multimedia learning: The case for minimizing irrelevant sounds in the design of multimedia instructional messages[END_REF]. Animations' spatial content and task sequences should be displayed as viewers would see them [START_REF] Tversky | Animation: Can it facilitate?[END_REF]. Additionally, designers must create congruence between screen capture animation and real-life task execution, help viewers perceive temporal changes, and provide easy-to-follow models enabling viewers to mimic observed actions [START_REF] Bétrancourt | The animation and interactivity principles[END_REF][START_REF] Van Der Meij | A comparison of paper-based and video tutorials for software learning[END_REF]. Similarly, cropping the window to show only content relevant to the immediate message helps avoid cognitive overload (van der Meij & van der Meij, 2013).

Given our objective of devising a multimodal evaluation grid for teaching and assessing studentproduced video tutorials in an EMI technical communication graduate program, our grid also integrates L2 spoken language as a subset of the aural mode using CEFR spoken language criteria, targeting B2+ level descriptors (Council of Europe, 2018).1 This choice is motivated by the observation that narration is a crucial part of video tutorials. Effective narration includes appropriate word choice and syntax but also phonological control, accent and prosodic features such as word stress, rhythm, and intonation. These features pose particular challenges to L2 speakers, requiring careful attention in an EMI teaching context. Finally, given video tutorials' inherently instructional nature, we also incorporate features of instructional-procedural genres [START_REF] Ganier | Factors affecting the processing of procedural instructions: Implications for document design[END_REF][START_REF] Steehouder | Designing and evaluating procedural instructions with the four components model[END_REF]. We present our evaluation framework in Figure 2.

Although it may be expected for L2 spoken language criteria to be included in the linguistic mode, we have opted to incorporate them under the aural mode following Cope and Kalantzis' (2000) categorization schema, to highlight their role in creating effective narration. To address our study's two research aims, in what follows, we examine which of the aforementioned modal competency criteria, and interactions between them, students had most difficulty with, according to domain professionals. We also test the validity of our multimodal evaluation grid using a small-scale reception study with experts in the field.

Study approach and methods

Research design

In our reception study [START_REF] Tardy | The construction of author voice by editorial board members[END_REF], the effectiveness of student-produced video tutorials was assessed by professional technical communicators using an online multimodal evaluation survey, which replicates the five modes and 28 categories shown in Figure 2. Insight into the professionals' thinking during the assessment was collected using online written feedback and stimulated recall interviews. Additionally, the interviews were used to gauge the professionals' observations about the evaluation grid's viability as a professional design tool.

Many applied linguists and writing researchers use reception evidence to improve analysis validity [START_REF] Ceccarelli | A hard look at ourselves: A reception study of rhetoric of science[END_REF][START_REF] Paul | Moving beyond the moment: Reception studies in the rhetoric of science[END_REF][START_REF] Tardy | The construction of author voice by editorial board members[END_REF], thereby adopting the fundamental assumption that the meaning of a message -including all forms of media -is not pre-given but interpreted by its recipient; meaning emerges from the context-dependent interaction between a text and an interpretative reader [START_REF] Fiske | Television Culture[END_REF]. To determine the socio-cognitive relevance of a text for a target audience, reception studies incorporate empirical and social scientific methods, including quantitative assessments (e.g., citation histories) and qualitative methods, e.g., peer commentary and observational studies of reader response using think-aloud protocols, where readers are recorded as they verbalize their reactions to what they are reading. Subsequently, our study employs written feedback and stimulated recall interviews, in addition to an evaluative survey, as a means of eliciting professionals' reception of the student-produced videos. Stimulated recall interviews were chosen as an introspective research procedure to invite participants to recall their concurrent thinking during the event (Mackey, cited in [START_REF] Lyle | Stimulated recall: A report on its use in naturalistic research[END_REF], allowing participants to explain their decision-making while allowing for fairly unstructured responses [START_REF] Lyle | Stimulated recall: A report on its use in naturalistic research[END_REF].

Participants

Four student project teams participated in the study. Of the ten student participants (8 female, 2 male), seven were L1 French speakers, two were L1 Mandarin Chinese speakers and one L1 Cantonese. Their level of English ranged from CEFR B1+ to C12 (Council of Europe, 2018).

Four L1 English-speaking technical communication professionals (P1-P4) evaluated the studentproduced videos' effectiveness. All informants worked in Europe with 15-to-30+ years job experience. Aged 50-72, one informant was female, three were male. All held Masters' degrees in technical communication or journalism, had significant experience in producing and/or evaluating video tutorials, and worked daily with non-native English speakers. Participant names have been changed.

Materials

To address our two research aims, we employed one type of learning material (student-produced video tutorials) and created three measurement tools for the reception study (online multimodal evaluation survey, online written feedback form, stimulated recall interview guide). This section details each.

Student-produced instructional video tutorials

Prior to designing the videos, the students participated in a 12-week workshop during a third semester EMI graduate-level program at a French university. The 80-hour project-based workshop was designed to help them assimilate knowledge and skills about technical communication. Working in self-elected groups of 2-4 students to create a set of user-support genres for the presentation software Prezi, the project teams produced, in English, a paper-based user-guide, an online help system and a set of instructional video tutorials. Each of the four teams determined video content employing a user analysis conducted for their paper-based user-guide. Because the evaluation grid described in this study was developed subsequent to the workshop, it was not available to guide the teams' video designs.

After identifying a theme common to all four teams' video productions ('adding animations'), we selected three teams' productions for analysis; one video, produced by the L1 Chinese team, was excluded due to incomparable length (over 8 minutes). 3 The remaining three videos ran between 1'15 to 2 minutes, and were produced by L1 French teams (Appendix B).

Online multimodal evaluation survey

To address our first research aim of determining which modal competency criteria are most problematic for L2 students when designing video tutorials, an online multimodal evaluation survey reproduced the 28 criteria from Figure 2 to gauge the informants' immediate reactions after viewing the video tutorials. For each criteria, the domain professionals indicated their (dis)agreement using a six-point Likert scale.

Online written feedback form

The 28 modal competency criteria (Figure 2) were also reproduced in an online written feedback form, used by informants to comment on the three selected video tutorials.

Stimulated recall interview guide

Stimulated recall interviews were held with the informants post-evaluation. An interview guide (Appendix A) elicited information about training and professional background, experience with video tutorials, and working language(s). In addition to gaining insight into informants' assessments of whether the videos met professional workplace standards, it also sought their opinions about the evaluation grid's relevance and ease-of-use. This addressed our second research aim, presenting a methodology for designing and validating future evaluative frameworks through collaboration between academics and professionals.

Data collection procedures and analysis

Before viewing the three videos, informants were asked to preview the 28 modal competency criteria in the written feedback form. As we were interested in their appreciation of the grid's usability as a professional assessment tool, no prior training was provided to avoid influencing their reaction. After previewing the criteria, informants viewed the video tutorials, completed the survey and provided written feedback using the online form. Informants were contacted within 24 hours of completing the evaluation to organize the stimulated recall interviews, which occurred between 6 and 30 days later and were conducted via videoconferencing and recorded. Interviews lasted between 31 and 49 minutes and were automatically transcribed using Otter.ai.

The numbers obtained in the survey were merged to produce average values for each modal competency criteria. 4 We targeted the most negatively rated criteria for analysis, seeking insight from specific comments in the written feedback form. These comments and the stimulated recall interviews were analyzed thematically to gain a deeper understanding of the professionals' reasoning. Our analysis draws together the data sources.

Results and discussion

This section describes the four professionals' assessment of students' preparedness for meeting professional expectations regarding workplace-based multimodal literacies and English-language fluency.

Here we target three modes for analysis: the linguistic mode, the temporal mode, and L2 spoken language within the aural mode. We chose to focus on the first two modes because students' enactment of them was rated most negatively in the online survey. Moreover, although informants rated the L2 spoken language positively, written feedback and interviews pointed to underlying issues with students' spoken proficiency in English in a professional setting, which could impact the overall perceived quality of videos produced in the workplace.

Linguistic mode

Below, we discuss the scores obtained from the 28-criteria survey in terms of how students managed the linguistic mode (i.e., Criteria 1-6). Figure 3 indicates that while informants appreciated the student videos' title presentation around an overall goal (C1=4.08), they felt the video titles needed more clarity (C2=3.08) and that the videos' sub-parts should be better identified (C3=3.25). Similarly, although the videos were segmented around a clear goal and sub-goal (C4=4.5), they lacked a navigational layer (C5=0, C6=0), e.g., displaying segmentation visually in the video timeline to help the audience better navigate within the video and view progress. This would have helped students better adhere to the principles of segmentation and contiguity (Mayer, 2014a). Specifically, designers must keep in mind Mayer's (2014a) principle of spatial contiguity: the title and opening phrases will immediately impact viewers' decision to watch the video or not. However, informants reported that the students' video titles were too generic and lacked a distinctive search term identifiable by Web search engines. In videos 2 and 3, for example, the YouTube titles did not mention the software Prezi by name (Figure 4), nor did the students state the software's name in their introduction. Although video 2 (Figure 4) visually displays the software name, the meaning established through proximity between the word 'Prezi', the logo, and the opening-screen title occurs only after viewers access and begin watching the video. Tutorial viewers are not a captive audience, however; they can click off and look for answers elsewhere. This affordance reflects the need to create congruence between the tutorials and real-life tasks [START_REF] Tversky | Animation: Can it facilitate?[END_REF]. Professional informants focused on this aspect during stimulated recall interviews: Similarly, informants commented that the narration did not contextually preview the video's purpose, necessary for convincing viewers that the video tutorial is relevant to their needs. Instead, students skipped directly to a demonstration of steps. This is illustrated in Figure 5 (lines 1-4), which shows the students' use of the linguistic mode (word choice, text) across visual and aural modes; the aural mode is further broken down into narration and sound/voice. Indeed, all informants emphasized how much video tutorial viewers rely on the contextual setting, which must 'hook into' whatever moment viewers are watching the video: P2: "Video 1 starts off with, today we will be presenting to you x y z, right. So, given you don't know when this video is going to be viewed, it's not today when you're recording it that someone's going to be watching it, you just have to disassociate it with any particular moment in time because it's supposedly ever-green."

Two other issues raised for the linguistic mode include the need for a clear task-progression throughout the video, and to conclude the video by reviewing the steps, as described in van der Meij and van der Meij's (2013) guidelines: P1: "There is no, what do you do next? So the end of the video is kind of like 'Thanks for watching'. And, you know, that was great, but I need to know what to do next." P2: "The instruction bit should be broken down into one, two, three or four actions and then some kind of feedback loop on what those actions have accomplished. … And that's one thing I felt was missing, like one of the videos just seemed to be a random collection of three or four different things you can do with the software, but there was no progression towards the goal…."

Temporal mode

Another mode flagged by professional informants is the temporal mode, in terms of length, speed, pacing and narration pauses (Criteria 24-28). Figure 6 shows that informants favorably evaluated students' synchronization of animated actions and voice (C25=5.25), a feature of tempo used to manage speed and enhance comprehension in multimedia presentations [START_REF] Mayer | Principles based on social cues in multimedia learning: Personalization, voice, image, and embodiment principles[END_REF]. However, students' skill in managing other temporal resources was less productive. Video tutorial speed (C27=3.83) and viewer control of content (C28=3.66) raised concern. Moreover, informant P4 highlighted the need to pause after each tutorial step (C26=4.0) so that viewers can assimilate the information and/or stop the video to execute the step.

Informants were sensitive to conversational pace (Morain & Swarts, 2010). They considered that video 2's (cf. Figure 5) overall speed, pacing and tempo could have been faster. Conversely, video 1 felt rushed; its speed was too fast to allow viewers to follow the steps. Figure 7 shows the difficulties students had managing the temporal mode in video 1, as illustrated by the short timestamp and narration pauses, and fast-paced music. Accentuating narration pauses are important for not overburdening viewers' cognitive load [START_REF] Paas | Implications of cognitive load theory for multimedia learning[END_REF] and in allowing for viewers' understanding to emerge: P4: "When you're doing video and particularly a tutorial, you have to slow down and take what we would feel like are unnatural pauses, but for the viewer, they're necessary. Anything you're doing on video, you have to over articulate and over pause. ... [Timing] is very important, even though it may feel unnatural, it doesn't feel unnatural necessarily when you're watching it. It also gives the user the opportunity to hit the pause button. And it's difficult to hit the pause button when someone jumps from one step to the next as though it's one continuous sentence." P3: "Time is critical, and people need time to absorb. … You have to assume that people are only going to get one exposure. So you have to make sure that the density of your information is just right."

The temporal mode was challenging for students, in terms of choosing relevant length, managing information density by segmenting tasks and sub-tasks through pausing and temporal cues, as well as controlling speed and narration pacing. Students appeared to systematically apply stringent length criteria (Plaisant & Schneiderman, 2005;[START_REF] Chan | Advantages of video trigger in problem-based learning[END_REF], rather than construct it using adaptive temporal resources (van der Meij & van der Meij, 2013). These difficulties also impacted their L2 spoken language, as discussed in the next section.

L2 spoken language within the aural mode

In this section, we identify specific L2 spoken language concerns from the professional informants' point of view (Criteria 14-18). Our decision to focus on this set of modal competency criteria is motivated by comments informants made in written feedback and during stimulated recall interviews. Figure 8, reporting survey results for L2 spoken language, shows that informants rated the five spoken language criteria favorably. Flow and intonation (C14=4.58, C18=4.83) were rated somewhat more positively than style and grammar (C15=4.25, C16=4.33). L2 accent, although detectable, was not considered overly distracting (C17=4.16):

P3: "The person speaking was saying 'template' [*templet]. Because it was repeated and because it was close enough, you get it after about the second or third time."

Although hesitant to criticize the students' language proficiency, all four informants revealed concerns in written feedback and during interviews about how prosody interacts multimodally with other modal competency criteria: narration pace and pauses (temporal mode), video length and speed (temporal mode), animation (visual mode) and conversationality (aural mode).

Drawing on the features of conversational English to create a dialogue with viewers is important for establishing the video's legitimacy, as captured by the personalization principle [START_REF] Mayer | Principles based on social cues in multimedia learning: Personalization, voice, image, and embodiment principles[END_REF][START_REF] Mayer | A personalization effect in multimedia learning: Students learn better when words are in conversational style rather than formal style[END_REF]. It allows for a motivational relationship to emerge with viewers, who intuitively become more receptive to the message conveyed by the video's narrator. Prosody, which includes rhythm, intonation and word stress, plays a crucial role in this process. Because video tutorial narration should largely mirror natural speech, getting the "rhythms, inflections and intonations" of conversational English right was identified by informants as a key feature of effective video tutorials.

The genre frame appeared to be a source of difficulty in this regard. Professionally-produced video tutorials should be perfectly calibrated performances that seamlessly orchestrate a balance between verbal and visual content, timing, audience awareness, cognitive load and social interaction. Students found themselves having to project spoken narration for an audience they had difficulty imagining while trying to conform to procedural-genre expectations, such as being clear and brief [START_REF] Ganier | Factors affecting the processing of procedural instructions: Implications for document design[END_REF][START_REF] Steehouder | Designing and evaluating procedural instructions with the four components model[END_REF]. Consequently, their efforts to strive for clarity (van der Meij & van der Meij, 2013) and not overload working memory [START_REF] Paas | Implications of cognitive load theory for multimedia learning[END_REF] seem to have caused them to lose control over some prosodic features through 'hyper-correct' narration (Figure 9, video 2).

In video 2, students applied themselves to making the narration easy to follow: the narrator spoke slowly, over-enunciated, and used short, simple sentences (Figure 9, showing the L2 spoken language focus in the aural mode). This hyper-correctness led informant P2 to state in written feedback that the prosody was inappropriate: the narration was "quite formal and monotone" and should have been "more conversational, faster, and expressive." As informant P4 explained: P4: "It's okay to have inflection and emotion. When you're doing a video tutorial, you don't have to have your voice really flat. [Being] passionate ... comes across in your voice [and] subconsciously makes [viewers] very interested in the subject."

This issue coincided with the narration pace being too slow and long pauses occurring between step demonstrations (Figure 9: lines 2-3). Students used the pauses to animate narrated steps, causing a perceived redundancy (Mayer, 2014a) between animation and narration: P1: "They would read the screen to me, they would read the options to me, they would say okay, click here and here are the options that you have and I go, I can see that, give me something else." Similarly, over-enunciation occasionally accentuated the narrator's L1 French phonology (Figure 9: lines 1, 3). Finally, contrary to [START_REF] Mayer | A coherence effect in multimedia learning: The case for minimizing irrelevant sounds in the design of multimedia instructional messages[END_REF] who recommend not overloading the audio with extraneous information including music, the inclusion of a voice-based sound track disconcertingly drowns out the narrator's more subdued tone (Figure 9: lines 2-4).

Keeping the video 'short' can also affect a narrator's control of rhythm, intonation and word stress. We see this in video 1: P2: "[T]he narration goes too quickly. In one case, the last task starts without any break after The narrator's phonological control in English ("Accent is rather strong, it's sometimes difficult to capture what is being said", P3) further compounds these issues, potentially interfering with the conversational tone narrators are expected to use in video tutorials [START_REF] Mayer | Principles based on social cues in multimedia learning: Personalization, voice, image, and embodiment principles[END_REF][START_REF] Mayer | A personalization effect in multimedia learning: Students learn better when words are in conversational style rather than formal style[END_REF]).

Informants stressed the importance of having a native English speaker do the narration for professional video tutorials, when possible. If not, the narrator must have a good sense of the language's 'music and rhythm':

P3: "You have to get the tonic accent, right? You have to get the music of the language, right? In sum, our informants' observations about how narration pace, including rate of speech, narration pauses and prosody (rhythm, intonation, word stress) affects tutorial viewers' perception of effectiveness align closely with research on social cues in multimedia presentations. Video designers can enhance their viewers' motivational commitment by using social cues, communicated through the narrator's attention to conversational style [START_REF] Mayer | Principles based on social cues in multimedia learning: Personalization, voice, image, and embodiment principles[END_REF][START_REF] Mayer | A personalization effect in multimedia learning: Students learn better when words are in conversational style rather than formal style[END_REF]. Therefore, attending to the key features of conversational English is of vital importance in video tutorials because it "primes the activation of a social response in [viewers], such as the commitment to try to make sense of what the speaker is saying" (Mayer, 2014b: 346).

Conclusion: Implications and practical applications

Our study establishes a set of research and workplace-based criteria to determine what students and teachers could attend to when designing or providing feedback on dynamic multimodal genres. Our purpose-made multimodal evaluation grid includes 28 modal competency criteria organized around five modes (linguistic, aural including L2 spoken language, visual, spatial, temporal), allowing for a better targeting of critical design areas.

To address our first research aim, this paper has described professionals' assessment of studentproduced video tutorials using the grid, and their identification of areas of difficulty for students.

While prior work has identified the modes and resources (e.g., Cope & Kalantzis, 2000) which can be used to produce video tutorials, and other studies have provided specific design recommendations using cognitive principles and end-user appreciations of usability (Mayer, 2014a;[START_REF] Van Der Meij | Eight guidelines for the design of instructional videos for software training[END_REF], our study combines these research areas to better examine the issues novice designers encounter when producing video tutorials. Results show students' difficulties in three areas.

Regarding the linguistic mode, choosing identifiable titles, previewing the task, and specifying task progression and step review were problematic. Moreover, establishing appropriate length through tempo, pacing and narration pauses proved elusive. Spoken language was also a concern in terms of prosody, with a breakdown of intonation control due to perceived genre expectations.

While relying on empirical research to describe workplace-based genre expectations is essential, validating those expectations using a guided reception study with domain specialists is equally important. Thus addressing our second aim, our evaluation grid has been tested by field experts to engage a dialogue with the targeted professional community [START_REF] Tardy | The construction of author voice by editorial board members[END_REF] and foster the development of more authentic teaching materials, thereby improving pedagogical and design guidance for instructional video tutorials. Notably, the experts offered insight into how to make feedback meaningful for the professional workplace. They helped refine criteria wording and identify other criteria to include. Moreover, written feedback and interviews allowed us to understand some of the more sensitive issues at play, such as unspoken attitudes about L2 proficiency which professionals may be hesitant to go on record with when assessing student-produced video tutorials, but which affect overall quality nonetheless.

In terms of study limitations, we take pains to emphasize that, given our goal of showing how empirically-driven recommendations from multimedia learning can be integrated as modal competency criteria within a multimodal framework, and of making this information accessible to novice designers, the evaluation grid described in this paper is intentionally non-exhaustive in terms of the numerous semiotic resources that give form to instructional video tutorials. It has undoubtedly paid less attention to resources like grammar and word choice more common in systemic functional multimodal analysis [START_REF] O'halloran | Systemic functional-multimodal discourse analysis (SF-MDA): Constructing ideational meaning using language and visual imagery[END_REF][START_REF] O'halloran | Systemic functional multimodal discourse analysis[END_REF] and to visual and spatial semiotic resources, such as color, proximity and visual cueing direction, as addressed in a social semiotics approach to multimodality [START_REF] Kress | Reading Images: The Grammar of Visual Design[END_REF]. Further research could elaborate on these and other outstanding issues in criteria determination, including the temporal resources which strongly impact the enactment of other modal competency criteria.

Undoubtedly, the students from our study would have benefitted from an articulated set of guidelines to guide their design process [START_REF] Goodrich Andrade | Teaching with rubrics: The good, the bad, and the ugly[END_REF][START_REF] Brookhart | The quality and effectiveness of descriptive rubrics[END_REF]. This and the foregoing observations lead us to consider appropriate forms of pedagogical action that could be introduced into course design to better prepare students to meet workplace requirements regarding multimodal literacies and L2 fluency, and better accompany their integration into the professional community by raising their awareness of professionals' standards and expectations.

In ongoing work (Dressen-Hammouda & Wigham, 2022a[START_REF] Dressen-Hammouda | Aligning the teaching of multimodal literacies with students' professional needs[END_REF], we reemploy the research data collected as pedagogical materials. Awareness-raising activities integrating our data include a course task in which students analyze the strengths and weaknesses of the student-produced videos reported on in this study, and confront their opinions with professionals' by showing them segments from stimulated recall interviews. We anticipate that this form of a posteriori peer review could help students-in-training become more conscious of the unspoken codes of professional practice learned mostly through on-the-job experience, in turn better allowing for a more critical awareness of multimodal literacies (Dressen-Hammouda & Wigham, 2022a).

In another study, we employ the revised multimodal evaluation grid as a self-assessment tool to guide students' video tutorial productions and support their capacity to produce professional-standard level work. A replication study with the same informants compares student video tutorial productions with and without access to the grid during design [START_REF] Dressen-Hammouda | Aligning the teaching of multimodal literacies with students' professional needs[END_REF]. Other research avenues could investigate how the grid may help teach students the differences between paper-based and related three-dimensional genres.

Our ultimate aim is to render the various skills involved in successful multimodal design more "teachable" by articulating them with more explicit accounts [START_REF] Bateman | Multimodality and Genre. A Foundation for the Systematic Analysis of Multimodal Documents[END_REF], thus helping students build a clearer mental image of users and tasks. The ability to call on strong mental images is one of the foundations of information design expertise [START_REF] Schriver | What do technical communicators need to know about information design[END_REF], a learned skill for which students require varied types of pedagogical support.

Figure 1 .

 1 Figure 1. Modes and resources for designing video tutorials (based on Cope & Kalantzis, 2000)

Figure 2 .

 2 Figure 2. Multimodal evaluation grid with modal competency criteria

Figure 3 :

 3 Figure 3: Survey results: Linguistic mode (C=Criteria) 4

Figure 4 .

 4 Figure 4. Disconnect between viewer search strategies and video findability

  P2: "[In] the first five seconds, if you realize that, oh, you didn't understand the title correctly, or that's not what you were looking to do, you can go off and find something else. [It's] really important, the first five or ten seconds. … [T]he word Prezi doesn't appear anywhere in the title .... So if I was searching, I'd be specifically looking for something on Prezi."

Figure 5 .

 5 Figure 5. Lack of appropriate task preview and contextualization in video 2

Figure 6 .

 6 Figure 6. Survey results: Temporal mode (C=Criteria) 4

Figure 7 .

 7 Figure 7. Difficulties in pacing and tempo (video 1)

Figure 8 .

 8 Figure 8. Survey results: Aural mode L2 spoken language (C=Criteria) 4

Figure 9 .

 9 Figure 9. Narration hyper-correctness affects intonation and prosodic features 5

5

  Key: / rising intonation; \ falling intonation; * incorrect pronunciation or word stress shown in transcription; pause in (ss;x/30s) the previous one finishes: you don't have time to absorb what words have been spoken."The quick video speed caused significant problems for the narrator's word stress and intonation, which rose and fell in unexpected ways (Figure10, video 1). Normally, a falling intonation signals the end of a turn or step, so viewers would expect a screen change. However, falling intonation occurred repeatedly mid-sentence without an accompanying step or screen change (Figure10: lines 1, 5, 7, 8).

Figure 10 .

 10 Figure 10. Attention to video length affects intonation and prosodic features 5

  And no, if you don't get that right, very often people won't understand what you're saying. And why make it hard for them? Especially since they can just click off your video anytime they want. I would always say if … you can get a native speaker to do the narration, that's always best. Second best is somebody who really gets the rhythm."

The CEFR criteria were adapted for use by non-language-teaching specialists, and were intended to reflect native-speaker intuition.

CEFR B1 level is the equivalent to intermediate level, TOEFL results of 57-86 or IELTS level 4 and CEFR C1 level is the equivalent to advanced English, TOEFL results of 110-120 and IELTS level 7.

Although eliminated for this study, the inclusion of the video in future analysis may help us to determine whether conditions related to length or L1 may influence effective video tutorial design, which in turn may help us refine our evaluation grid.

Each criteria was rated on a six-point Likert-scale: 1=strongly disagree, 2=disagree, 3=somewhat disagree, 4=somewhat agree,

5=agree, 6=strongly agree.

Appendix A. Stimulated Recall Interview Guide

Stage

Step Questions

Introducing the research

Researcher introductions

Brief presentation (depending). Check technical features.

Thank participant

Thanks for participating in our study and agreeing to a follow up interview online with me today. The purpose of this meeting is to get a little more background information about you and have a short exchange about your professional evaluation of the videos you watched.

Permission to record

Would it be okay with you if I record our conversation from here on in using the screen capture software XXX?

We'll make all information identifying you and the companies you've worked for anonymous so that in the final research study a reader would not be able to identify you.

Any initial questions


Xinyong Wang
email: xinyong.wang@centralelille.fr

Ying Tang
email: ying.tang@univ-lille.fr

Christophe Fiter
email: christophe.fiter@univ-lille.fr

Laurentiu Hetel
email: laurentiu.hetel@centralelille.fr

Stability Analysis for a Class of Linear Hyperbolic System of Balance Laws with Sampled-data Control ⋆

Keywords: Linear hyperbolic system, sampled-data control, LMIs methods, Lyapunov function, Rε-stability

The stability for a class of linear hyperbolic systems with distributed sampled-data controllers is discussed in this paper. The original sampled-data system is firstly transformed into a new equivalent system by modelling the sampling induced error as a reset integrator operator. Then we construct an appropriate Lyapunov function and obtain sufficient conditions for the Rε -stability of the system based on linear matrix inequalities (LMIs). A numerical example illustrates our results: when the sampling interval is within the allowable range, the solution of the system converges from the domain of attraction to a positive invariant set.

INTRODUCTION

Nowadays, control systems are usually implemented on digital platforms. The properties of digital implementations are studied in the domain of sampled-data systems [START_REF] Åström | Computercontrolled systems: theory and design[END_REF]; [START_REF] Chen | Optimal sampled-data control systems[END_REF]). For finite-dimensional systems (linear or nonlinear), the stability and control design problems have been extensively studied (see the references in [START_REF] Hetel | Recent developments on the stability of systems with aperiodic sampling: An overview[END_REF]; [START_REF] Laila | sampleddata control of nonlinear systems[END_REF]; Monaco andNormand-Cyrot (2001, 2007)). Here we are interested in the sampled-data control problem of partial differential equations (PDEs) which involves an infinite dimensional state space. More precisely we focused on sampled-data control of systems governed by hyperbolic PDEs since many typical examples of processes can be described by this kind of system [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF], such as the transmission of electrical energy, the flow of liquid in a pipe, the light propagation in optical fibers, the road traffic, etc. Different from the research method of finite dimensional systems, fewer results are available for hyperbolic PDEs.

For the case of idealized sample-and-hold process, [START_REF] Logemann | Stability of infinite-dimensional sampled-data systems[END_REF][START_REF] Logemann | Generalized sampled-data stabilization of well-posed linear infinite-dimensional systems[END_REF] provided sampled-data control design methods which stabilized an infinite dimensional system for sufficiently small sampling periods. E. Fridman and co-authors have addressed several problems related to the sampled-data control of parabolic PDEs using a time-delay method [START_REF] Kang | Distributed sampleddata control of kuramoto-sivashinsky equation[END_REF]; Selivanov andFridman (2016, 2017)). For hyperbolic PDEs, event-triggered sampled-data control with controller on the boundaries was considered in [START_REF] Espitia | Event-based control of linear hyperbolic systems of conservation laws[END_REF][START_REF] Espitia | Stabilization of boundary controlled hyperbolic pdes via lyapunov-based event triggered sampling and quantization[END_REF]. In [START_REF] Karafyllis | Sampled-data boundary feedback control of 1-d linear transport pdes with non-local terms[END_REF], the application ⋆ This work was partially supported by the French "Rgion Hautsde-France" grant and by the project H2020 UCOCOS.

results of Zero-Order-Hold boundary feedback control in one-dimensional linear hyperbolic systems with non-local terms on bounded domains were given. The boundary feedback control of a 2 × 2 hyperbolic system was implemented by backstepping method in [START_REF] Davó | Stability analysis of a 2 × 2 linear hyperbolic system with a sampled-data controller via backstepping method and looped-functionals[END_REF], and the global asymptotic stability was realized. As we can see from the literature review, the analysis of sampled-data controller for hyperbolic PDEs is a wide open research field and there is still the place for important contributions.

In this paper, we address the stability problem for sampled-data hyperbolic PDEs. Differently from the existing works where boundary control has been considered, here we study distributed sampled-data controllers. The problem is studied from an Input-Output point of view, extending the approaches for finite dimensional systems [START_REF] Fujioka | Stability analysis of systems with aperiodic sample-and-hold devices[END_REF]; [START_REF] Kao | Stability analysis of systems with uncertain time-varying delays[END_REF]; [START_REF] Mirkin | Some remarks on the use of timevarying delay to model sample-and-hold circuits[END_REF]; [START_REF] Omran | Stability analysis of bilinear systems under aperiodic sampled-data control[END_REF]). The main idea is to represent the sampling induced error as a perturbation for a continuous time hyperbolic equation. By converting the original sampled-data system into an interconnection of a continuous-time PDEs and a reset-integral operator, constructive local stability results are derived.

The layout of this paper is as follows. Section 2 introduces the systems we will study and the problem we have to deal with. In Section 3, we give the equivalent transformation form of the system including sampling error and the concrete stability analysis process. A numerical example in Section 4 shows that our method is feasible. Finally, the summary and prospect of the paper are presented in Section 5.

Notations: N is the set of nonnegative integers from 0 to infinity, R + is the set of nonnegative reals, R n is used to denote the set of n-dimensional Euclidean space with the norm | • |. L 2 (0, L) stands for the Hilbert space of square integrable scalar functions on (0, L) with the corresponding norm

• L 2 (0,L) , defined by e L 2 (0,L) = L 0 |e (x)| 2 dx. The associated norm to Sobolev space H 1 (0, L) is defined as e H 1 (0,L) = L 0 |e (x)| 2 + |e x (x)| 2 dx. The set of functions ϕ : [0, L] → R n such that L 0 |ϕ (x)| 2 dx < ∞ is denoted by L 2 ([0, L]; R n ) and given a functional V : H 1 ([0, L]; R n ) → R + such that L V ≤C = y ∈ H 1 ([0, L]; R n ) : V (y) ≤ C .
The notation M ≤ 0 denotes that M is symmetric and negative semidefinite. The symmetric elements are denoted by * in the symmetric matrix. The identity matrix is denoted by I and λ min (Q) and λ max (Q) are the minimum and maximum eigenvalues of the matrix Q. C 0 is the continuous function and C 1 is the continuously differentiable function.

SYSTEM DESCRIPTION AND PROBLEM FORMULATION

System Description

We consider the sampled-data controlled hyperbolic system (1) given below

       ∂ t y (t, x) + Λ∂ x y (t, x) + Υy (t, x) + u (t, x) = 0, u (t, x) = Ky (t k , x) , ∀t ∈ [t k , t k+1 ), k ∈ N, y(t, 0) = 0, ∀t ≥ 0, y(0, x) =y 0 (x), ∀x ∈ [0, L] , (1a) (1b)
(1c) (1d) where y : [0, +∞) × [0, L] → R n , Λ = diag {λ 1 , λ 2 , ..., λ n } with λ 1 , λ 2 , .
.., λ n > 0, K and Υ are real n × n constants matrices. The sequence is defined as {t k } k∈N where t 0 = 0, t k+1 -t k ∈ (0, h],

(2) and h > 0.

To deal with the problem under consideration, we need the following compatibility condition:

Condition 1. The initial condition y 0 (x) for ∀x ∈ [0, L], satisfies y 0 (0) = 0. ( 3
)
Remark 1. Let us discuss the notion of solution used in the present work. The system (1)-( 2) can be rewritten as a first order system

dy(t) dt = Ay (t) + f (y (t k )) , t ∈ [t k , t k+1 ) , k ∈ N, y (0) = y 0 , where f (y(t k )) = -ky(t k ), and the operator A is defined by Ay = -Λ∂ x y (t, x) -Υy (t, x) , (4) with domain D(A) = y ∈ H 1 (0, L; R n ) | y(0) = 0.
(5) The operator A generates a stable C 0 semigroup (see the proof of theorem A.1. in [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF]). Moreover, we note that

f k : H 1 (0, L) → H 1 (0, L) is continuously differentiable for t ∈ [t k , t k+1 ). If y 0 ∈ D(A),
then according to Theorem 6.1.5 of [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF], there exists a classical solution for each t ∈ [t k , t k+1 ), k ∈ N. Therefore, we can construct a solution by choosing the last value of the previous sampling interval as the initial condition of the following sampling interval such that it is continuous at each sampling instant.

Problem Formulation

In our work, we prove Rε-stability, which is defined in Definition 1 below.

Definition 1. Rε-stability [START_REF] Polyakov | Practical stabilization via relay delayed control[END_REF] Consider positive scalars R and ε, such that 0 < ε < R, and a candidate Lyapunov function V :

H 1 ([0, L]; R n ) → R + .
If for all solutions of system (1) with y 0 (x) ∈ L V ≤R , the trajectory of the state y(t, x) converges to L V ≤ε as t goes to infinity and the set L V ≤ε is positive invariant, then, system (1) is said to be Rε-stable from L V ≤R to L V ≤ε .

In this work, our main goal is to guarantee the Rε-stability of the closed loop system (1)-( 2) based on an Input-Output approach.

MAIN RESULT

This section is divided into two parts. First, we represent the sampled-data system as an equivalent continuous hyperbolic PDE where the sampling induced error appears in the input, as a disturbance. Secondly, based on the provided model, constructive Rε-stability criteria are provided.

System Remodeling

System (1) can be equivalently re-expressed as

       ∂ t y (t, x) + Λ∂ x y (t, x) + (Υ + K)y (t, x) + Kω (t, x) = 0, ∀t ∈ [t k , t k+1 ), k ∈ N, y(t, 0) = 0, ∀t ≥ 0, y(0, x) = y 0 (x), ∀x ∈ [0, L] . (6a) (6b) (6c)
with the sampling induced error ω (t, x) = y (t k , x) -y (t, x) .

(7) Define the function ϕ as

ϕ (t, x) = ∂y (t, x) ∂t , ∀t ≥ 0, x ∈ [0, L] . (8)
Note that

ω (t, x) = - t t k ∂y (θ, x) ∂θ dθ = - t t k ϕ (θ, x) dθ, ∀t ∈ [t k , t k+1 ) , k ∈ N, x ∈ [0, L] . (9)
As a result, the closed-loop system can be regarded as the interconnection of two systems G and Ψ shown in Figure 1, where the operator G : L 2 (0, L) → L 2 (0, L) is defined by

G :      ∂ty (t, x) = -Λ∂xy (t, x) -(K + Υ) y (t, x) -Kω (t, x), y(t, 0) = 0, ∀t ≥ 0, y(0, x) =y0(x), ∀x ∈ [0, L] , ϕ (t, x) = -Λ∂xy (t, x) -(K + Υ) y (t, x) -Kω (t, x) = ∂ty (t, x) ,
(10) and the operator Ψ : L 2 (0, L) → L 2 (0, L) is defined by

Ψ : ω (t, x) = (Ψy)(t, x) = - t t k ϕ (θ, x) dθ, ∀t ∈ [t k , t k+1 ) , k ∈ N, x ∈ [0, L] . (11

Stability Analysis

In the following, we introduce our main result. Theorem 1. Consider systems ( 10)-( 11) with (2)(or equivalently (1) with ( 2)) and the initial condition satisfying (3):

(i) Let λ = min i∈{1,...,n} λ i . Assume that there exist µ 1 , γ 1 , µ 2 , γ 2 > 0, and symmetric positive matrices

Q 1 ∈ R n×n , Q 2 ∈ R n×n satisfying M (0) ≤ 0, M (L) ≤ 0, ( 12
) with M (x) defined for all x ∈ [0, L] as M (x) =    Ω -e -2µ1x Q 1 K 0 0 * -γ 1 I 0 0 * * E -e -2µ2x Q 2 K * * * -γ 2 I    , (13) where
Ω = -e -2µ1x (K + Υ) T Q 1 + Q 1 (K + Υ) , E = -e -2µ2x Υ T Q 2 + Q 2 Υ + βQ 2 . ( 14
) (ii) If ∃ε ∈ R + ; R ∈ R + s.t. 0 < ε < R and γ 1 3 h |Λ| 2 Φ + |Υ| 2 + |K| 2 Ψ + γ 2 Φ < min(2ν 1 , 2ν 2 -β)ε, (15) with Φ
= R λ min (Q 2 ) e -2µ2L , Ψ = R λ min (Q 1 ) e -2µ1L , where ν 1 = µ 1 λ, ν 2 = µ 2 λ, 0 < β < 2ν 2 .
Then the considered system (1) is Rε-stable from L V ≤R to L V ≤ε for any sampling sequence satisfying (2). Moreover, a Lyapunov function is defined as

V (y) = V 1 (y) + V 2 (y), (16) with V 1 (y) = L 0 y T e -2µ1x Q 1 ydx, ( 17
) V 2 (y) = L 0 y T x e -2µ2x Q 2 y x dx. ( 18
)
Proof. Consider the Lyapunov function ( 16)-( 18). It can be bounded as follows: Θ y (t, •)

2

H 1 ( [0,L] ; R n ) ≤ V (y (t, •)) ≤ Ξ y (t, •) 2 H 1 ( [0,L] ; R n ) , (19) where Θ = min(λ min (Q 1 ) , λ min (Q 2 ))e -2µL , µ = max(µ 1 , µ 2 ), Ξ = max(λ max (Q 1 ) , λ max (Q 2 )).
Step 1: In this step we study the continuity of the function V defined in ( 16).

(1) Since y(t, x) is continuous with respect to t for all t ∈ [t k , t k+1 ), k ∈ N, and continuous at sampling instants by construction (see Remark 1) , then V 1 is continuous for all t ≥ 0. (2) From system (1), we can get

y x (t, x) = Λ -1 (-y t (t, x) -Υy (t, x) -Ky (t k , x)) , (20)
for all t ∈ [t k , t k+1 ), k ∈ N. Since all the terms on the right of the equation ( 20) are continuous in t on (t k , t k+1 ), ∀k ∈ N, then y x (t, x) and thus V 2 are also continuous in t for all (t k , t k+1 ), k ∈ N. Now we consider the time interval [t k , t k+1 ), for some k ∈ N and an initial condition y k (x). The solution of ( 1) is defined as y(t, x) on the time interval [t k , t k+1 ), and is such that y and y x are both C 0 in t ∈ [t k , t k+1 ).

Next, we prolong the solution to C 1 in t on [t k , t k+1 ]. We denote z(t, x) the solution on [t k , t k+1 ] with initial condition y k (x). z(t, x) and z x (t, x) are C 0 in t on [t k , t k+1 ]. We got the following properties on [t k , t k+1 )

y(t, x) = z(t, x), y x (t, x) = z x (t, x).
(21)

Then the left limit can be calculated as lim

t→t - k+1 y x (t, x) = lim t→t - k+1 z x (t, x) = z x (t k+1 , x) . ( 22
)
For the next time interval [t k+1 , t k+2 ), we set the initial condition y k+1 (x) = z(t k+1 , x). Then the solution y(t, x) of system (1) on [t k+1 , t k+2 ) satisfies y x (t, x) is C 0 in t on [t k+1 , t k+2 ). Therefore, we have the right limit property

lim t→t + k+1 y x (t, x) = y x (t k+1 , x) = z x (t k+1 , x) . ( 23
)
According to ( 22) and ( 23), we can see that by construction, y x (t, x) is continuous in t at time instant t k+1 . Similarly, we can show that the function y x (t, x) is continuous at all sampling instants, which shows both the continuity of y x (t, x) with respect to time for all t ≥ 0 and the continuity of V 2 .

Remark 2. Here V 1 is used in order to bound y, and V 2 is used to deal with the term y x that appears in the derivative of V 1 .

Step 2: In this step we study the time derivative of the function of V (y) defined in ( 16). Thanks to commutativity condition: ΛQ 1 = Q 1 Λ, we first compute the time derivative of V 1 (y) along the solutions to ( 10)-( 11),∀t

∈ [t k , t k+1 ) , k ∈ N, V1 (y) = L 0 ∂ t y T e -2µ1x Q 1 y + y T e -2µ1x Q 1 ∂ t y dx = L 0 (-Λ∂ x y -(K + Υ)y -Kω) T e -2µ1x Q 1 y +y T e -2µ1x Q 1 (-Λ∂ x y -(K + Υ)y -Kω) dx = L 0 -∂ x y T Λe -2µ1x Q 1 y dx + L 0 -y T (K + Υ) T e -2µ1x Q 1 y -y T e -2µ1x Q 1 (K + Υ)y -y T e -2µ1x Q 1 Kω -2µ 1 y T Λe -2µ1x Q 1 y -ω T K T e -2µ1x Q 1 y dx = -y T Λe -2µ1x Q 1 y L 0 + L 0 -y T (K + Υ) T e -2µ1x Q 1 +e -2µ1x Q 1 (K + Υ) y -ω T K T e -2µ1x Q 1 y -y T e -2µ1x Q 1 Kω dx -2µ 1 L 0 y T Λe -2µ1x Q 1 ydx. (24)
In order to get the time derivative of y x in V 2 , we refer to the original system (1). Since y

: [0, +∞) × [0, L] → R n has consecutive partial derivatives in [0, +∞) × [0, L],
according to Schwartz's theorem [START_REF] James | Advanced calculus belmont[END_REF]) we can obtain

∂ xt y (t, x) = ∂ tx y (t, x) = -Λ∂ xx y (t, x) -Υ∂ x y (t, x) -K∂ x y (t k , x) , ∀t ∈ (t k , t k+1 ), k ∈ N. (25)
For the next calculation of the time derivative of V 2 , we use Lemma 1 in the appendix. According to (25) and Lemma 1, we have

∂ x y(t, 0) = 0, ∀t ∈ [t k , t k+1 ), k ∈ N, y 0 (0) = 0, ∂ x y 0 (0) = 0. (26a) (26b)
Similarly to the computation of V1 , by using the commutativity condition: ΛQ 2 = Q 2 Λ, the time derivative of V 2 (y) along the solutions to ( 25)

-(26), ∀t ∈ (t k , t k+1 ) , k ∈ N is shown as follows V2 (y) = -∂ x y T Λe -2µ2x Q 2 ∂ x y L 0 + L 0 -∂ x y T Υ T e -2µ2x Q 2 + e -2µ2x Q 2 Υ ∂ x y -∂ x y T (t k , •) K T e -2µ2x Q 2 ∂ x y -∂ x y T e -2µ2x Q 2 K∂ x y T (t k , •) dx -2µ 2 L 0 ∂ x y T Λe -2µ2x Q 2 ∂ x ydx. (27)
Adding 27) for some γ 1 > 0, γ 2 > 0, β > 0, and using boundary condition (6b) and (26b) we have V (y) = V1 (y) + V2 (y)

γ 1 ω (s, •) 2 L 2 ( [0,L] ; R n ) -γ 1 ω (s, •) 2 L 2 ( [0,L] ; R n ) to (24) and γ 2 ∂ x y (t k , •) 2 L 2 ( [0,L] ; R n ) -γ 2 ∂ x y (t k , •) 2 L 2 ( [0,L] ; R n ) , β L 0 y T x e -2µ2x Q 2 y x dx -β L 0 y T x e -2µ2x Q 2 y x dx to (
≤ -2ν 1 V 1 (y) -(2ν 2 -β)V 2 (y) + L 0 η T M (x)ηdx + γ 1 ω (s, •) 2 L 2 ( [0,L] ; R n ) + γ 2 ∂ x y (t k , •) 2 L 2 ( [0,L] ; R n ) . (28)
with

ν 1 = µ 1 λ, ν 2 = µ 2 λ, η = [y T ω T (∂ x y) T (∂ x y) T (t k , •)] T (
29) and M (x) defined in (13).

Since M (x) satisfies LMIs (12), by convexity we have η T M (x)η ≤ 0, ∀x ∈ [0, L] and thus

L 0 η T M (x)ηdx ≤ 0, (30)
we deduce from ( 28)

V (y) ≤ -2ν 1 V 1 (y) -(2ν 2 -β)V 2 (y) + γ 1 ω (s, •) 2 L 2 ( [0,L] ; R n ) + γ 2 ∂ x y (t k , •) 2 L 2 ( [0,L] ; R n ) . (31)
Step 3: The negative properties of function V (y) for t ∈ [t k , t k+1 ), k ∈ N will be discussed in this step. Consider some k ∈ N and let us first assume that y (t,

•) ∈ L V ≤R \ L V ≤ε and ∀θ ∈ [t k , t], y (θ, •) ∈ L V ≤R . L 0 y T e -2µ1x Q 1 ydx ≤ R, ∀y(θ, •) ∈ L V ≤R , L 0 ∂ x y T e -2µ2x Q 2 ∂ x ydx ≤ R, ∀∂ x y(θ, •) ∈ L V ≤R . ( 32
)
Then the following inequalities are further derived

y (θ, •) 2 L 2 ([0,L];R n ) ≤ R λ min (Q 1 ) e -2µ1L , ∀θ ∈ [t k , t], (33) ∂ x y (θ, •) 2 L 2 ([0,L];R n ) ≤ R λ min (Q 2 ) e -2µ2L , ∀θ ∈ [t k , t]. (34) Recalling y (t, •) ∈ L V ≤R \ L V ≤ε and ∀θ ∈ [t k , t], y (θ, •) ∈ L V ≤R , for t ∈ [t k , t k+1 ), k ∈ N. The bound of ω (s, •) 2 L 2 ([0,L]; R n ) can be calculated by (9) ω (s, •) 2 L 2 ([0,L]; R n ) = L 0 |ω (s, x)| 2 dx = L 0 t t k ∂y (θ, x) ∂θ dθ 2 dx ≤ 3 h |Λ| 2 R λ min (Q 2 ) e -2µ2L + |Υ| 2 + |K| 2 R λ min (Q 1 ) e -2µ1L = ̟. (35) In addition, since y (t, •) , ∂ x y (t, •) / ∈ L V ≤ε , we have -2ν 1 V 1 (y) -(2ν 2 -β)V 2 (y) ≤ -min(2ν 1 , 2ν 2 -β)(V 1 (y) + V 2 (y)) < -min(2ν 1 , 2ν 2 -β)ε.
(36) Therefore, instituting (35) and ( 36) into (31), we have that for all

t ∈ [t k , t k+1 ), k ∈ N, V (y) < -min(2ν 1 , 2ν 2 -β)ε + γ 1 ̟ + γ 2 R λ min (Q 2 ) e -2µ2L , (37)
and thus from assumption (15), we have shown that if which means that since V is continuous, that y will remain in L V ≤R during the whole sampling interval [t k , t k+1 ],

y (t, •) ∈ L V ≤R \ L V ≤ε and ∀θ ∈ [t k , t], y (θ, •) ∈ L V ≤R for ∀t ∈ [t k , t k+1 ), k ∈ N V (y) < 0, (38)
and by recursion, we can see that it will always remains there. As a consequence, L V ≤R is positively invariant. Furthermore, since V < 0 wherever y / ∈ L V ≤ε , that means that L V ≤ε is attractive, which ends the proof of Rε-stability.

NUMERICAL SIMULATION

In this section, we present a numerical example to illustrate the stability we proposed in section 3. Consider system (1) One can see that the Lyapunov function V deceases when ε < V (y) < R.

     ∂ t y (t, x) + Λ∂ x y (t, x) + Υy (t, x) + u (t, x) = 0, u (t, x) = Ky (t k , x) , ∀t ∈ [t k , t k+1 ), k ∈ N, y(t, 0) = 0, ∀t ≥ 0, y (

CONCLUSION

The main work of this paper is to construct a sampling controller for distributed control of linear hyperbolic balance laws. The closed-loop system is reformulated from an Input-Output point of view. In addition, we prove the local stability of the system by means of the Lyapunov method. This is somewhat conservative because the sampling interval needs to be small enough. In the future, we will expand the sampling interval by virtue of dissipativity theory, and sample the controller in space not only in time, so as to propose a more general stability theory.

Fig. 1 .

 1 Fig. 1. Alternative representation of the closed-loop system.3.2 Stability Analysis

Fig. 2 .

 2 Fig. 2. Response of state y 1 .

  0, x) =y 0 (x), ∀x ∈ [0, , y 0 (x) = (1 -cos x) sin 2x 3(1 -cos x) sin x .The parameters in Theorem 1 are selected as: h = 0.0001, β = 0.01, µ 1 = 0.1, µ 2 = 0.08,γ 1 = γ 2 = 0.01, R = 10, ε = 3.063, conditions (12), (13), and (15). The results of numerical simulations are presented in Figs.2-4. Figs.2-3show the states converge to the vicinity of the origin with the controller. The initial conditions satisfying the compatibility condition (3) are given in Figs.4-5. The time-evolution of Lyapunov function V is shown in Fig.6.

Fig. 3 .Fig. 4 .Fig. 5 .

 345 Fig. 3. Response of state y 2 .

Fig. 6 .

 6 Fig. 6. Time-evolution of function V.

ACKNOWLEDGEMENTS

We would like to thank A. Polyakov and N. Espitia, for their advice and inspiration on a draft of this paper.

Appendix A. PROOF OF LEMMA 1 Lemma 1. Consider the system (1)-( 2) with initial condition y 0 satisfying Condition 1. Then ∀t ∈ [t k , t k+1 ), k ∈ N, ∂ x y(t, 0) = 0.

Proof. We recall system (1)

The time derivative of the boundary condition leads to

2) Combining (A.1a) with (A.2), we obtain 0 = ∂ t y (t, 0) = -Λ∂ x y (t, 0) -Υy (t, 0) -Ky (t k , 0) . (A.3) Since y(t, 0) = 0, ∀t ≥ 0, we have ∂ x y(t, 0) = 0, ∀t ∈ [t k , t k+1 ), k ∈ N.

(A.4)