Elke Nissen

Lidilem

Combining Classroom-Based Learning and Online Intercultural Exchange in Blended Learning Courses

This chapter poses courses integrating Online Intercultural Exchange (OIE) as a specific type of blended learning (BL) courses. It aims at gaining insight into how experienced OIE course designers combine both learning modes -f2f sessions and the online exchange -into coherent BL courses, and which common denominators this blending shares.

Introduction: From telecollaboration to blended learning

The literature on online intercultural exchange (OIE), also called telecollaboration or virtual exchange, has grown constantly over the last decade. Most of these publications have focused on the actual online exchanges, analyzing how they work, the kinds of tasks used, what students learn, and what has failed (e.g. O 'Dowd & Ritter 2006[START_REF] Müller-Hartmann | Teacher role in telecollaboration: setting up and managing exchanges[END_REF][START_REF] O'dowd | Critical issues in telecollaborative task design[END_REF][START_REF] Guth | Telecollaboration 2.0. Language, literacies and intercultural learning in the 21st century[END_REF], etc. However, most OIE projects are devised as components of blended learning (BL) courses, where BL is defined as a combination of face-to-face (f2f), "classroom education" (Bersin 2004: 85) or "brick-and-mortar setting" [START_REF] Staker | Classifying K-12 blended learning[END_REF] 3) sessions and online learning (e.g. [START_REF] Charlier | Apprendre en présence et à distance: une définition des dispositifs hybrides[END_REF][START_REF] Neumeier | A closer look at blended learning -parameters for designing a blended learning environment for language teaching and learning[END_REF]. As a result, OIE can also be examined from another perspective, that is, in terms of complete courses, rather than in terms of the OIE project and the roles of the different partners. From this perspective, OIE is viewed as a component of a BL course and as a specific online learning mode that differs from other forms of online learning, such as individual online work and online interactions between students and their local teacher and/or their local peers, in that it involves online exchanges with geographically distant partners. As has been frequently pointed out (e.g., [START_REF] Deschryver | HY-SUP -Dispositifs hybrides, nouvelle perspective pour une pédagogie renouvelée de l'enseignement supérieur[END_REF][START_REF] Garrison | Blended learning in Higher Education. Framework, Principles, and Guidelines[END_REF][START_REF] Macdonald | Blended learning and online tutoring. Planning learner support and activity design[END_REF][START_REF] Means | Evaluation of evidence-based practices in online learning : A meta-analysis and review of online learning studies[END_REF], BL offers manifold possibilities for enhancing pedagogy in higher education because it allows learners to benefit from both f2f and online learning modes. Consequently, it has been widely embraced by teachers and learners. However, the broad variety of possibilities to mix both learning modes makes it at the same time difficult to design effective BL courses. Despite (or perhaps because of) this fact, most books describing guidelines and best practice in BL course design simply show the kinds of activities that can be used in each learning mode without suggesting how the learning modes can be linked coherently (e.g., [START_REF] Bersin | The blended learning book: Best practices, proven methodologies, and lessons learned[END_REF]. But, the effectiveness of BL courses cannot be taken for granted as such [START_REF] Chew | Critical Review of the Blended Learning Models based on Maslow's and Vygotsky's Educational Theory, Hybrid Learning and Education[END_REF][START_REF] Means | Evaluation of evidence-based practices in online learning : A meta-analysis and review of online learning studies[END_REF]: they need to have a sound pedagogical design. The crux of BL design is the way the f2f and distant modes are combined. Furstenberg and Levet also highlighted this point, when they stated that what made their BL course including OIE coherent was the "interplay between online and classroom work" (2010: 305).

How are f2f and OIE sessions combined in existing courses? Research question and data collection

Research question

This chapter addresses the question of how the interplay between the f2f and distant modes within such courses works. To date, [START_REF] Furstenberg | Integrating telecollaboration into the language classroom: some insights[END_REF] are the only researchers to have examined this issue specifically with respect to OIE. A small number of publications that have examined other forms of BL in higher education have looked at ways of best combining f2f and online sessions (e.g., [START_REF] Garrison | Blended learning in Higher Education. Framework, Principles, and Guidelines[END_REF]: 105-141, Murphy & Southgate 2011). These publications implicitly consider the f2f sessions the lead (or dominant) learning mode and the online sessions a complementary mode. In fact, in the literature on BL, when one learning mode is considered to be central and dominant this mode is often the f2f mode (e.g., [START_REF] Haeuw | Competice, outil de pilotage des projets TICE par les competences[END_REF][START_REF] Deschryver | HY-SUP -Dispositifs hybrides, nouvelle perspective pour une pédagogie renouvelée de l'enseignement supérieur[END_REF]. Nevertheless, two recent models (Flipped Classroom and Flex model; see Starker & Horne 2012: 10-13) put the online component as the backbone of a BL course. Another taxonomy [START_REF] Bersin | The blended learning book: Best practices, proven methodologies, and lessons learned[END_REF]) -elaborated in professional training contexts -does not focus in the first instance on which mode is the backbone of BL design, but rather on how tight the modes are integrated and interwoven. This chapter examines whether these models apply to BL courses combining f2f and OIE sessions (see 4.1).

The hypothesis underlying the present study is that in BL courses including OIE, the OIE component often becomes the central and lead mode because, first, designing and preparing a course with distant partners is both complex and time-consuming [START_REF] Guth | A Report on the integration of telecollaborative networks in European universities[END_REF]. Teachers who are prepared to invest a lot of time and energy in organizing and coordinating such exchanges do so because they believe it is important for the students' learning and motivation. And, second, since they consider OIE as an effective means to reach the learning aims, it would seem likely that they would give OIE a central role in the courses they design. Moreover, the aim of the present study was to determine whether there are common denominators in the ways teachers with OIE experience design BL courses that combine f2f sessions and virtual exchange. The literature on BL identifies several aspects that have to be considered in order to effectively combine f2f and online learning modes (see [START_REF] Singh | Building effective blended learning programs[END_REF][START_REF] Bersin | The blended learning book: Best practices, proven methodologies, and lessons learned[END_REF][START_REF] Neumeier | A closer look at blended learning -parameters for designing a blended learning environment for language teaching and learning[END_REF][START_REF] Degache | Formations hybrides et interactions en ligne du point de vue de l'enseignant: pratiques, représentations, évolutions[END_REF][START_REF] Garrison | Blended learning in Higher Education. Framework, Principles, and Guidelines[END_REF][START_REF] Deschryver | HY-SUP -Dispositifs hybrides, nouvelle perspective pour une pédagogie renouvelée de l'enseignement supérieur[END_REF]. These aspects, that will structure part 4 of this chapter, include choosing a lead mode that will form the backbone of the course design, and determining the pedagogical function of each mode, the choices students can make within each mode, and the teacher's role in each mode. The teacher's roles can include assessment, but it is essential to identify the mode(s) that give rise to the skills and contents assessed. Another aspect is the sequencing of the learning modes and the amount of time dedicated to each mode (see [START_REF] Nissen | Modéliser le fonctionnement de la formation hybride en langues à travers des recherches ingénieriques[END_REF].

Data collection

As stated above, the aim of this exploratory study was to identify any common approaches to designing BL courses involving an OIE component. To do this, I analyzed and compared the designs of six BL courses that combine an OIE component with f2f sessions in what their designers consider a coherent way, i.e. both modes being integrated and interdependent. In order to ensure the courses analyzed would have comparable characteristics, I focused on higher-education language-training courses. All six course designers had experience in OIE (minimum experience: 3 years, mean experience: 9.5 years) and were involved in the course as teachers, as well as designers. Consequently, they all had in-depth knowledge of the course they presented. In addition, they all teach and/or do research into OIE, so they had the awareness and expertise needed to reflect on the courses they had designed. Data on five of the six courses were gathered through semi-guided interviews (average length: 49 minutes), whereas data for the sixth course (21F.303) were obtained from a book chapter [START_REF] Furstenberg | Integrating telecollaboration into the language classroom: some insights[END_REF] and via answers to questions sent to G. Furstenberg by email (Table 1). The interview questions focused on the parameters that have to be taken into account in order to ensure the f2f and OIE components of a course are combined in a purposeful way (see 2.1.). The interviews were analyzed using content analysis. All the teachers also filled in a questionnaire giving factual data about their course, such as the course title, the name of the institution, the degree the course was delivered in, the number of course hours, and the number of ECTS credits. In addition, course materials were obtained for four of the courses (see table 1), so their designs could be examined in more detail.

New Trends in Education (NTE)

The NTE course consists of nine units, all of which have the same structure. Each unit (see figure 1) starts with a video clip the students watch online before individually answering comprehension questions and filling in a learning log. This is followed by a classroom, teacher-led pre-discussion about the clip to prepare the students for their online discussions about the clip, carried out in small groups, with distant peers. The students finish the unit offline with a reflective questionnaire and a reflective panel discussion. Every three units, the students use a new and more difficult tool for the OIE, starting with a written asynchronous tool (forum), followed by a written quasi-synchronous tool (chat), and then an oral-synchronous tool combined with a chat tool (Skype).

Spoken English (SE)

The SE course is run in parallel to 13 other courses with the same title at the same university and with the same assessment criteria, but integrates an OIE component and is offered to students with a higher level of English (higher intermediate level, B2, to bilingual). During the first six weeks, the students work exclusively f2f and do preparation work in small groups. They start by choosing a topic, together with a number of related video clips in English and in French. Then, they prepare an English and French study guide, focusing on language, and prepare discussion questions. During the final six weeks, they have online discussions with distant partners (in groups of four) based on these questions and the study guides. In parallel, they use a Voice Forum to record reflective messages focusing on language and intercultural issues.

21F.303

21F.303 is a course based on the Cultura project [START_REF] Furstenberg | Integrating telecollaboration into the language classroom: some insights[END_REF] in which f2f sessions add a language component to the project's intercultural focus. In the initial phase of the course, the students use an online forum to complete three anonymous questionnaires on concepts (through word association), relationships (through sentence completion), and attitudes and values (through reactions to hypothetical situations). In the second phase, they compare the answers given by course members and distant partners to the same questionnaires. As homework, and with help from a detailed worksheet, every student chooses an item on one of the questionnaires and compares the answers given by French and American students. In the classroom, students who chose to analyze the same item work together, using a whiteboard, to identify what is said in the forums and how it is said. Then, they compare their results with those of the other local groups before discussing their findings online with the distant partners. The partners' answers are analyzed as another homework assignment and the results are shared in the classroom. In the next project phase, the students carry out a search for specific data, and then summarize, discuss, and compare their findings on a forum, before comparing and discussing them f2f. At the end of the course, the students give a reflective oral presentation (see figure 1).

English language 1 (EL1)

The EL1 course follows the OIE project phases described by [START_REF] Müller-Hartmann | Teacher role in telecollaboration: setting up and managing exchanges[END_REF] and O 'Dowd and Ware (2009). After an introductory and presentation phase, the students create a blog in order to exchange information with their distant partners. Then, with help from questions they had prepared beforehand, they interview their partners on cultural topics, mostly through a written asynchronous tool, but they can also choose to use other tools (e.g., Skype). Because of the limited overlap with the partners' university calendars, the OIE then stops. Until this time, the f2f and OIE sessions alternate throughout the course. The semester ends with an f2f oral presentation based on critical reflection on the online exchange.

English language 3 (EL3)

The OIE component of the EL3 course follows similar phases to EL1, but it is divided into two parts. In the first part, the students work f2f and use a wiki as a virtual learning environment (VLE) in order to read articles, analyze two films that address intercultural issues, and then prepare questions for an online discussion with the distant partners. This online discussion is carried out one-to-one using an oralsynchronous tool (Skype). The f2f work and the wiki then form the basis for a Skype debriefing.

The second part of the course (weeks 7 to 9) is a final collaborative project to write an essay. Students and their distant partners work in small groups, dividing up the roles and work required to complete the essay. Then, each student looks for resources, writes his/her part of the essay, corrects his/her peers, and edits the text on the wiki. The course ends with an f2f reflective discussion about their online experiences.

Intercomprehension between Romance languages (IRL)

The aim of the IRL course is to enhance intercomprehension, that is to say, improve understanding of other languages within a language family. During the first semester, the students assess their skills in different Romance languages, and work on similarities and differences between these languages in order to increase their ability to understand all of them. F2f sessions are combined with individual online learning activities in a VLE. The OIE project, called Galanet, consists of writing a press file and starts in the third week of the second semester. It enables the students to practice intercomprehension by interacting with distant peers in many different countries. It is divided into four phases: 1) icebreaking and choosing a topic for the press file, 2) discussing the topic, 3) choosing the columns to include and looking for resources and information, and 4) writing and publishing a press file in four Romance languages [START_REF] Degache | Aspects du contrat didactique dans une formation plurilingue ouverte et à distance, Le Français dans le monde[END_REF]). F2f sessions during the second semester are devoted to tasks connected to each of the OIE phases. These tasks have the same objectives as the OIE phases.

Results and discussion

These short descriptions of the six courses and figure 1 bring to the fore the variations in how they are structured, how the two learning modes are sequenced, the kind of learning activities they include, and the tools they use. The descriptions also highlight differences in the "main threads" that form the backbones of the courses in terms of the combination and sequencing of f2f and OIE sessions. A main thread is determined by the learning objectives but can also relate to pedagogical approaches such as TBL or collaborative learning that influence largely the course's architecture. Some of the courses have more than one main thread, which makes both their design and their analysis more complex. The main threads of the six courses analyzed here can be summarized as follows: specific language skills (aimed at communication and involving task-or project-based learning, or intercomprehension), intercultural skills (relationship building, intercultural awareness, intercultural learning), digital literacy skills (multiliteracies), and transversal academic skills. These main threads are summarized in table 3.

Course title Main thread New Trends in Education (NTE)

From oral comprehension and focus on form to oral fluency & multiliteracy Spoken English (SE)

Conversation on specific themes, prepared during f2f tasks 21F.303 (BL course based on Cultura)

Intercultural learning: "the students journey together through both cultural 'lands'" [START_REF] Furstenberg | Integrating telecollaboration into the language classroom: some insights[END_REF]. Language skills English Language 1 (EL1)

Blending through projects/tasks. Intercultural awareness (finding the link between language & culture). Academic skills English Language 3 (EL3)

From discussion to collaboration. Intercultural awareness Intercomprehension between Romance Languages (IRL) Intercomprehension skills Table 3. Main threads of the six courses.

Despite these apparent differences, analysis of the inner course design showed that the courses have a number of points in common in the way they are designed and in the way they combine the two learning modes. For example, all the courses follow Biggs' "aligned teaching" principle in that their teaching methods, activities, and assessment methods are coherent with the courses' learning objectives (Biggs 2003: 27), as far as this is possible in the given context (see 4.3.).

Lead mode

Analyzing the courses in terms of BL models also reveals similarities. For example, all six courses correspond to [START_REF] Bersin | The blended learning book: Best practices, proven methodologies, and lessons learned[END_REF] "program flow model", with quite rigid course scheduling compared to several courses run in business contexts, as all the learners on a course start and finish at the same time, and most activities are obligatory. Bersin contrasts this model with a "core-and-spoke model" in which optional activities and media are added to a single main mode whenever the learner needs them. In the six courses examined in the present study, neither the f2f sessions nor the OIE components are optional; they are both integrated parts of the course. This is in line with Furstenberg and Levet's affirmation that in their course OIE "is not, as is often the case, merely an add-on" (2010: 308). In addition, since the courses have a predominately common schedule for all students of one local class, they do not correspond to the Flex model in which "students [would] move on an individually customized, fluid schedule among learning mod[es]" (Staker & Horn 2013: 12). Even if both learning modes are integral parts of a BL course, one mode may still be dominant and more central to the course design. As stated in the introduction, in non-OIE BL courses, f2f is often considered the lead mode. In contrast, the designers of all six OIE-BL courses analyzed here stated that OIE is the lead mode. For example, the first semester of the IRL course, which does not include any virtual exchanges, is considered preparation for the OIE sessions in the second semester; in the SE course, "the f2f part is just the lead-up to the OIE"; and in the 21F.303 course, "the whole language course is built around the [OIE] project" (Furstenberg & Levet 2010: 308). If the OIE has synchronous and asynchronous parts, it is the synchronous part that everything is centered on: "The focus, particularly for the students, were the synchronous sessions" (EL3).

Nevertheless, although the distant OIE mode is the dominant one, these courses match neither with the Flex model (see above) nor with the flipped classroom model. In a flipped classroom, "the primary delivery of content and instruction is online" (Staker & Horn 2013: 10), whereas f2f, students put into practice what they have learned at a distance. In the six OIE-BL courses, it is to a large extent the OIE-online part that is dedicated to "putting into practice certain things that you want the students so learn about" (EL1; see 4.2). It is interesting to note that, in terms of the time allocated to each mode, OIE would appear to be the subsidiary mode in three of the courses (NTE, SE, and IRL), where fewer hours are allocated to online exchanges than to f2f (see table 4). However, the estimated workloads for the OIE components of all the courses are at least as high as the estimated workloads for the f2f components. Although all of the teachers accepted to give such an estimation (see table 4), some of them found it difficult to give percentages for the workload, because 1) it depends on how well the students do the online component and, more importantly, because 2) it is often difficult to differentiate between f2f sessions, homework, and participation in virtual exchanges. A possible explanation for this is that synchronous OIE usually occurs during f2f sessions (NTE, EL3, IRL) in order to ensure a common time slot and to be able to provide the students with technical help if needed. The only course where this is not the case is EL1, which has a large number of students. In this case, small groups of distant students agree on which communication tool and which mode (synchronous or asynchronous) to use. 3) And, most importantly, as the teachers have stated, OIE is not just an integral part of course design, it is the hub. This is another possible reason why the teachers' had difficulty distinguishing between f2f and OIE modes, and homework. Several reasons were given for the dominance of the online component:

Course title

No

-The aims of all the courses are to improve communication skills or fluency, and/or intercultural skills.

The best mode for practicing these skills is the online mode. "The telecollaboration is a way of putting into practice certain things that you want the students to learn about" (EL1). -An OIE project is complex and needs explaining, for example, intercultural differences, troubleshooting, and task instructions. This is time consuming and does not leave time for many other things (NTE). -OIE clearly fosters student motivation: "The telecollaboration is the driving bit. It's the motivational bit" (EL1). -Most of the f2f work and homework is oriented toward preparing the virtual exchanges (see 4.2.2.).

Function of each mode

The above specifications show that OIE, in terms of estimated workload and of targeting the learning aims, is at the center of the design of all six courses; "even th[e] f2f sessions were telecollaboration-oriented", as the NTE-teacher puts it. However, the OIE and f2f facets of the courses are tightly interwoven. The following section shows that the role of the f2f sessions in all six courses is to prepare for and/or reflect on the OIE sessions (see figure 2). It is interesting to note that this applies regardless of their timely sequencing, either in rather regular intervals (NTE, 21F.303, EL1, IRL), in separate blocks (SE), or in a combination of both (EL3). The particular types of activities included in each mode nevertheless depend on the main focus of the course, for example, language or intercultural awareness.

OIE online discussions: chosen language and aims

In four of the six courses (NTE, SE, EL1, EL3), at least part of the time, the students use a second language when interacting with their distant peers. All four courses aim at fostering oral fluency, developing communicative competence, and/or enabling the students to carry on discussions in their L2. As a result, it is the OIE online discussions and interactions that the teachers feel are most important. In the other two courses (IRL and 21F.303), the students express themselves in the Romance language they know best (IRL) or in their 'mother' tongue (21F.303). In the IRL course, the aim of the OIE component is to allow students to practice plurilingual written interactions, that is, to read and understand other Romance languages and to interact effectively. In 21F.303, the focus of the online exchanges is both expression -the content and form of which is used afterwards as input from local peers for a reflective activity -and discussion, with the aim of fostering intercultural awareness1 . Other courses also use OIE to improve intercultural skills (EL1, EL3). Additionally, if exchanges include one-to-one synchronous oral sessions, they are considered opportunities to "develop some kind of very personal relationship" (EL3).

Preparatory and reflective f2f activities

The f2f mode is complementary to the OIE component, but necessary and irremovable. The two main functions of the f2f sessions are to prepare for the central mode of the course, that is, the OIE sessions, and to analyze the virtual exchanges in terms of what was said, how it was said, and what happened. In both cases (i.e. in preparative and reflective sessions), the main focus of the course -whether it is language skills (e.g., listening skills, intercomprehension skills and strategies, vocabulary, form, academic discourse, etc.), intercultural awareness, academic skills, technical issues, or several of these -, determines the types of learning activities and the aspects of the OIE that the f2f sessions deal with. For example, a course with a focus on intercultural awareness, such as EL3, concentrates f2f on behavior during the OIE, on misunderstandings and hasty judgements, whereas a course with a focus on intercomprehension (IRL) includes f2f work on how to express ideas, related to the topics the students wish to choose for the collaborative OIE project, in different romance languages. F2f sessions are often based on various, interactive, learning activities: they include guided discussions (NTE, EL1, EL3), either as preparation for distant discussions or to review things students noticed during their virtual exchanges, and small group activities (SE, 21F.303, EL1, IRL), such as brainstorming, comparing ideas, negotiating, and collaborating. Several teachers emphasize that f2f sessions within these courses are very interactive, as the following two quotes illustrate. "The classroom is a highly interactive place where students, taking center stage and interacting with their classmates, develop insights and co-construct and expand their own knowledge and understanding of the subject matter" (Furstenberg & Levet 2010: 333).

In a less interactive part of the f2f sessions, teachers give instructions, for example, on how to accomplish a given task, or on how to deal with technical problems (see 4.2.4.).

A supplementary distant mode

Some courses combine the f2f and OIE modes with another distant mode (IRL first semester, NTE, SE), such as a VLE, a wiki, or videos or activities on other websites. This second distant mode is not directly integrated in the OIE, although it also is an integrated part of the course. It provides links to online videos, listening comprehension activities and hosts learning logs (NTE), makes available instructions and links to online language exercises (IRL), or allows the student to record and post an oral reflective production expressing a personal, metacognitive and cultural point of view, in order to get a mark for his course (SE). According to the course designers, these distant learning activities using the supplementary mode could or do exist independently of the OIE sessions; nevertheless they provide input for the virtual exchanges or allow students to work on the skills needed for these exchanges. These courses hence contain a double distant mode (one primarily dedicated to the virtual exchange with distant peers, the other one not), and a f2f mode.

The teacher's role

Unsurprisingly, the teacher's role in the OIE mode is almost exclusively proactive (organizing and coordinating online exchanges with distant partners). During the actual exchanges, their main role is to provide technical assistance. They also listen to or read students' online discussions in order to identify points they think are interesting and important to discuss afterwards in class. Teachers' interventions during virtual exchanges are mostly f2f. Their declarations show that these interventions may fulfill a number of roles, depending on whether the function of the f2f mode is to prepare the OIE session (give instructions, e.g., on how to use a tool, encourage students to interact), react to the OIE session (reassure students, identify critical points of the OIE), encourage students to reflect on the OIE session (discuss what happened during the OIE or problems during the OIE, act as a cultural moderator), or evaluation. F2f sessions are largely dedicated to interactive activities such as discussions and group work (see 4.2.2.). Nevertheless, the amount of guidance and input a teacher gives during discussions and group work depends on the teacher's style.

Assessment

Another question that has to be addressed when analyzing the combination of two learning modes into a BL course is how assessment takes into account the contributions of the two different learning modes. In many cases, teachers' freedom to choose assessment methods is restricted, at least partly, by institutional constraints (e.g., SE, where the tasks assessed have to be the same as those assessed in parallel courses). Other factors constraining choices of assessment methods include general teaching practice, with students being unused to, for example, formative assessment (NTE), the number of students to be assessed (EL1), the different roles students may play in the online project, which makes it difficult to evaluate their contribution to the final collaborative task (IRL), and the difficulty of assessing skills such as intercultural awareness (EL3). In addition, the skills and tasks assessed depend on the courses' main objectives. Despite this long list of inherent differences, the items on which assessment is based generally bring together both modes of a BL course. For example, a portfolio may integrate reflection on what happened in both learning modes (21F.303). Improvements in language skills (assessed in IRL) or analytical skills (21F.303) will also be due to both modes. Other aspects that may be assessed include participation, task completion, and reflection, as illustrated in table 5.

Student choices

The courses analyzed here follow Bersin's "program flow model" and would therefore be expected to be quite rigid. This raises the question of whether or not students have the opportunity to make choices, for example, about which learning activities to use or time management. Giving students choices is a way of keeping transactional distance low [START_REF] Moore | Theory of transactional distance: the evolution of theory of distance education[END_REF]) and helping them make the course their own. Of course, scheduling an OIE session with one or more distant partners requires tight planning, of the OIE component but also of the related f2f sessions. Analysis of the teachers' statements on flexibility for learners showed that flexibility tends to be related mostly to the presence of larger tasks or projects that last longer than a week and that include several steps, where the students are able to make choices. This is the case for both the OIE and f2f components of the courses. In three of the courses, these tasks form the hub of the virtual exchange (EL1, EL3, IRL), with students being able to choose distant partners, online discussion questions, when to hand in work within a given deadline, topics, resources, tools, and their role in the online project. In one course (SE), rather than the OIE component, it is the f2f component that is task-based, with students able to choose their local partners, deadlines, topics, and resources. Thus, whether a task is accomplished f2f or via OIE, it is within the task that students have the most choices. Two of the courses (NTE and 21F.303) do not include long tasks that can be divided into several steps and completed using a project-based approach. In these two courses, students are given fewer choices: they choose either deadlines or the topic to be addressed. Both courses are based round small tasks or activities that are repeated from week to week, with the aim of progressively developing certain skills either in the field of multiliteracy and oral fluency (NTE) or in order to take the students on an "intercultural journey" (21F.303).

Conclusion

This chapter addresses the questions of how experienced OIE course designers combine OIE and f2f sessions into coherent BL courses, and of the common denominators this blending shares. An analysis of six higher education language-learning courses in different European countries revealed several major differences between these courses, including study year, degree program, course objectives -that result, together with constraints due to university calendars of the different project partners, different teaching styles, and different methodological approaches, in various course structures, and in various sequencing of the two learning modes. However, the courses also had numerous similarities including the facts that most of them lasted only one term and involved (with one exception) relatively small numbers of students (between 7 and 20). Like most HE courses, all six courses followed a linear structure with learners starting and finishing the course simultaneously. As such, they fit [START_REF] Bersin | The blended learning book: Best practices, proven methodologies, and lessons learned[END_REF] "program flow model", in which learners tend to have little freedom to choose their learning activities. In the courses analyzed, students were free to make choices, particularly when the course was based on a longer task and a project-based approach. However, this freedom to choose is not necessarily contradictory to Bersin's "program flow model", as it extended only to aspects such as local or distant partners, online discussion questions, when to hand in work within a given deadline, the topics studied, and the resources or tools the students used. Working on a joint topic or project, together with one or several other classes at a distance, which have their own constraints in terms of assessment, scheduling, etc., makes it difficult to give larger choices to the students. In only one case (IRL) were students free to make choices about learning activities. Nevertheless, student's choice is only one means to reduce transactional distance [START_REF] Moore | Theory of transactional distance: the evolution of theory of distance education[END_REF]. Interaction is another one, which is central in these courses.

Another result concerns the coherence of BL courses. Concordantly to [START_REF] Guth | A Report on the integration of telecollaborative networks in European universities[END_REF] finding that virtual exchanges tend to be integrated rather than add-on activities to other courses, both learning modes were inherent to the designs of all six courses analyzed. This is maybe due to the fact that the teachers were asked to present a course that combines both modes in a way they considered coherent, which probably excludes simple add-ons. In addition, a common feature of all six courses is that the OIE mode is the central part of the design, with the f2f sessions being mostly devoted to preparatory and analytical activities relating to the virtual exchange. These courses' design differs from existing BL models that consider the online mode to be central (flipped classroom and flex model). But, these courses integrating OIE fully meet a key design principle for BL courses that several authors claim with reference to Vygotsky's Cultural-Historical Theory [START_REF] Chew | Critical Review of the Blended Learning Models based on Maslow's and Vygotsky's Educational Theory, Hybrid Learning and Education[END_REF][START_REF] Garrison | Blended learning in Higher Education. Framework, Principles, and Guidelines[END_REF][START_REF] Nissen | Modéliser le fonctionnement de la formation hybride en langues à travers des recherches ingénieriques[END_REF]): facilitation through online interaction with peers and a tutor. The aim of this study was to examine the designs of different BL courses involving virtual exchanges in order to determine the factors that produce a coherent course design. This is an issue that, to date, has been little studied. The identification of parallels between these courses, regarding aspects such as using the OIE part as a lead mode, orienting the f2f part towards the OIE part, assessing most often jointly both modes, and student choices, should help course designers and teachers develop effective BL courses that include OIE.

New

  Design of the 6 courses.

Figure 2 .

 2 Figure 2. Relationship between the f2f and OIE modes in the six BL courses

"

  In class, [students] spend most of the time negotiating what they are going to use [within their collective task]" (SE).

Table 1 . Data collection 3. Brief presentation of the courses

 1

	Course title	Person interviewed	Data
	New Trends in Education (NTE)	Malgorzata Kurek	Interview 59 min., questionnaire
	Spoken English (SE)	Ciara Wigham	Interview 46 min., questionnaire, example and
			assessment grid of study guide
	21F.303 (BL course linked to Cultura)	Gilberte Furstenberg Book chapter Furstenberg and Levet (2010) &
			questions / answers via e-mail, questionnaire,
			open access to past sessions of Cultura
	English Language 1 (EL1)	Robert O'Dowd	Interview 34 min., questionnaire, task description
			and assessment grid
	English Language 3 (EL3)	Sarah Guth	Interview 53 min., questionnaire, access to wiki
	Intercomprehension between Romance	Yasmin Pishva	Interview 55 min., questionnaire
	Languages (IRL)
	An overview of the six courses (table 2) shows that they displayed a number of differences and similarities.
	Differences included the fact that they were delivered in different countries, within different degree
	programs, and for different study years; similarities included the fact that all these courses including an OIE
	component have the same duration of one semester -even though the IRL course incorporates a first
	semester of preparatory activities -, they were assigned relatively few ECTS credits, and involved classes with
	relatively small numbers of students (from 7 to 20), except for EL1, which involved two classes of 40
	students.

Course title Institution (country) Degree / study year Duration / official no. hours ECTS No. stud ents OIE component Language(s) Partner countries

	New Trends in	WSL (Poland)	Teacher training,	1 term	3	7-8	Formulaic	English	Taiwan
	Education (NTE)		MA 4th year	(30h)			language use in	lingua franca
							virtual academic
							discussions
	Spoken English	Univ. Blaise	Applied foreign	1 term	2.5	12	Open Learning	English, French	UK
	(SE)	Pascal	languages, 1st &	(20h)			Project (Tandem)
		(France)	2nd year
	21F.303 (BL	MIT (USA)	L2 course,	1 term	(12 units) 15	Cultura	French, English	France
	course based on		3rd semester					(L1)
	Cultura)
	English Language	Univ. of León	English studies &	1 term	(part of)	40	(no name)	English (&	1) USA
	1 (EL1)	(Spain)	Spanish studies,		6	x 2		Spanish in the	2) Israel,
			1st year					forums)	USA
	English Language	Univ. of	Modern	10 weeks	3 (pass-	20	Padova-Dickinson	English, Italian	USA
	3 (EL3)	Padua (Italy)	languages, 3rd		fail
			year		course)
	Intercomprehensi	Univ.	L2 course, any	2 terms	6 (3 per	10	Galanet	French, Italian,	Brazil,
	on between	Grenoble	level & any	(48h)	term)			Spanish,	France,
	Romance	Alpes	degree					Portuguese	Mauritius,
	Languages (IRL)	(France)						(L1)	Italy, etc.

Table 2 .

 2 Overview of the six courses

Table 4 .

 4 Hours allocated to the f2f and OIE modes, and the estimated workloads for the two modes.

Table 5 .

 5 Assessed items and the modes they relate to (SDM = supplementary distant mode; see 4.2.3).

In this course, the language objectives are addressed in the f2f sessions and as homework, not online.


Maria Bekcheva
email: maria.bekcheva@lss.supelec.fr

Luca Greco
email: luca.greco@lss.supelec.fr

Hugues Mounier
email: hugues.mounier@lss.supelec.fr

Alban Quadrat
email: alban.quadrat@lss.supelec.fr

Euler-Bernoulli beam flatness based control with constraints

Keywords: Flexible structures, infinite dimensional systems, rings, modules, controllability, Euler-Bernoulli partial differential equation, Gevrey-Roumieu functions, input constraints

The control of infinite dimensional systems with constraints is a notoriously difficult task. We consider a general class of linear systems governed by partial differential equations with boundary control. This problem is here treated in a quite natural manner through the freeness property, the analogue of differential flatness for linear systems. Any variable is then expressed as infinite order differential operators applied to the basis components, the analogue of the flat output components. The specialisation of the basis components are functions which are both of Gevrey regularity (in order for the infinite order differential operators to be convergent) and pertaining the flexibility of polynomial splines. An illustration is made through an Euler Bernoulli beam example.

I. INTRODUCTION

Control of infinite dimensional systems with constraints is known to be a difficult task. We here consider the case of systems giverned by partial differential equations with a control at the boundary. We shall study the class of free systems, the linear analogue of nonlinear finite dimensional differentially flat systems, and make use of algebraic techniques. The differential flatness (see [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and applications[END_REF]) gives a solution to systems governed by ordinary differential equations. As noticed in [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and applications[END_REF], this method is not restricted to ordinary differential equations and can be adapted to delay differential systems and partial differential equations with boundary control.

The freeness of a certain modules, whose properties are obtained through homological arguments (see, e.g., [START_REF] Eisenbud | Commutative Algebra with a View Toward Algebraic Geometry[END_REF], [START_REF] Lang | Algebra[END_REF]; We especially use the resolution of Serre's conjecture by Quillen [START_REF] Quillen | Projective modules over polynomial rings[END_REF] and Suslin [START_REF] Suslin | Projectives modules over a polynomial ring are free (in russian)[END_REF], already exploited in [START_REF] Mounier | Propriétés structurelles des systèmes linéaires à retards : aspects théoriques et pratiques[END_REF]; see also [START_REF] Fliess | Quelques propriétés structurelles des systèmes linéaires à retards constants[END_REF], [START_REF] Fliess | Interpretation and comparison of various types of delay system controllabilities[END_REF], [START_REF] Fabiaska | Applications of the Quillen-Suslin theorem to multidimensional systems theory[END_REF], [START_REF] Quadrat | An introduction to constructive algebraic analysis and its applications, les cours du CIRM[END_REF]), enables, by assigning a trajectory to a basis of this module, to obtain the desired trajectory tracking.

We shall, as systematized in [START_REF] Rudolph | Motion planning and open loop control design for linear distributed parameter systems with lumped controls[END_REF], envision the equations of the system as a Cauchy problem in the spatial variable, a problem which is well posed in some suitable spaces of generalized functions: the desired convolutional system is obtained by first solving the Cauchy problem and plugging its solution into the boundary conditions, i.e., the equations imposed by the boundary conditions further restrict the Cauchy data.

Differentially flat systems with constraints are generally attacked through the use of optimal control problems (see, e.g. [START_REF] Faiz | Differentially Flat Systems with Inequality Constraints: An Approach to Real-Time Feasible Trajectory Generation[END_REF], [START_REF] Faulwasser | Optimal Exact Path-Following for Constrained Differentially Flat Systems[END_REF], [START_REF] Oldenburg | Flatness and higher order differential model representations in dynamic optimization[END_REF], [START_REF] Petit | Inversion based constrained trajectory optimization[END_REF], [START_REF] Sira-Ramírez | Differentially flat systems[END_REF]). We here propose to embed the constraint fulfillment in the trajectory design. We thus specialize the basis (or flat output) to the convolution of a so called Gevrey function with a polynomial B-spline.

In Section II we briefly recall general definitions about Rlinear systems. In the next Section, bounday value problems are modeled as modules over a ring parametrized by space. In Section IV various controllability definitions are recalled. Section V is the main section of the paper, dealing with constraints fulfillment. Application to an Euler-Bernoulli beam is the subject of Section VI.

II. R-LINEAR SYSTEMS

We shall consider in this section quite general definitions for linear systems viewed as modules over a ring. In the next section, we shall be more specific in order to describe boundary value problems as modules over a ring parametrized by space.

Definition 1: An R-system Λ, or a system over R, is an R-module. A presentation matrix of a finitely presented Rsystem Σ is a matrix

P such that Σ ∼ = [v]/[P v] where [v]
is free with basis v. An output y is a subset, which may be empty, of Λ. An input-output R-system, or an input-output system over R, is an R-dynamics equipped with an output. The next definition allows, by extension of scalars, to obtain much nicer algebraic properties when needed.

Definition 2: Let A be an R-algebra and Λ be an R-system. The A-module A ⊗ R Λ is an A-system, which extends Λ.

III. BOUNDARY VALUE PROBLEMS AS SYSTEMS PARAMETRIZED BY SPACE

We shall here consider boundary value PDE systems as modules over rings. A space parametrization is embedded in the chosen rings.

A. Model class

Models are here considered as space dynamics with time differential operator coefficients.

1) Distributed equations: The envisioned model equations are based on a Cauchy-Kowalevski form:

∂ x w i = A i w i + B i u, w i : Ω i → (D * ) p , u ∈ (D * ) m , A i ∈ (R[∂ t ]) pi×pi , B i ∈ (R[∂ t ]) pi×m , i ∈ {1, . . . , l} (1a)
where w 1 , . . . , w l are the distributed variables, u = (u 1 , . . . , u m ) the lumped variables, and D * denotes a space of (ultra -) distributions.

2) Assumptions: We shall make two assumptions:

• The intervals Ω 1 , . . . , Ω l are given by an open neighbor-

hood of Ωi = [x i,0 , x i,1 ], i = x i,1 -x i,0 = q i q i ∈ Q, ∈ R (1b)
Without loss of generality, assume x i,0 = 0. • The characteristic polynomials of the matrices A 1 , . . . , A l can be written

P i (λ) := det(λI -A i ) = pi ν=0 a i,ν λ ν , (1c)
a i,ν = ν+µ≤pi a i,ν,µ ∂ µ t (1d)
with a i,j,k ∈ R, a i,pi,0 = 1. Moreover, their principal parts µ+ν=pi a i,µ,ν ∂ µ t λ ν are hyperbolic w.r.t. the time t, i.e. the roots of µ+ν=pi a i,µ,ν λ j are real. Remark 1. -Note that the above assumptions apply to most spatially one-dimensional boundary controlled evolution equations including Euler-Bernoulli or Timoshenko beam equations, more general parabolic diffusion-reaction-convection equations, damped and undamped wave-equations, etc. The only notable exception is the case where the maximal order derivative is a mixed one, such as, e.g. models of structural damping (α∂ t + 1)(∂ 2

x -∂ 2 t )w = 0 3) Boundary conditions: The models are completed by the following boundary conditions

l i=1 L i w i (0) + R i w i ( i ) + Du = 0 (1e) with D ∈ (R[∂ t ]) q×m and L i , R i ∈ (R[∂ t ]) q×pi .

B. Solution of the Cauchy Problem

Some properties of the solution of the Cauchy problem (1a) with initial conditions given by x = ξ, i.e.

∂ x w = Aw + Bu, w(ξ) = w ξ (2) with A ∈ (R[∂ t ]) p×p , B ∈ (R[∂ t ]
) p×q as assumed in the previous section for A i , B i , will be used. The notation of the previous section is used in what follows, dropping the index i ∈ {1, . . . , l}. Consider the initial value problem

P (∂ x )v(x) = 0, (∂ j x v)(0) = v j ∈ E * (R), j = 0, . . . , p-1 (3)
associated with the characteristic equation

P (λ) := det(λI -A) = p j=0 a j λ j , a j = j+µ≤p a j,µ ∂ µ .
According to [24, Thrm. 12.5.6] or [49, Thrm 2.5.2,Prop. 2.5.6] the initial value problem (3) has a unique solution which may be written as

v(x) = p-1 j=0 C j (x)v j ,
where juxtaposition of symbols means convolution and C 0 , . . . , C p-1 are smooth functions mapping Ω to the space of compactly supported Beurling ultradistributions E * (R) := E (p/(p-1)) (R) of Gevrey order p/(p -1). The functions C 0 , . . . , C p-1 satisfy (k, j ∈ {0, . . . , p -1})

∂ k x C j (0) = 1, k = j 0, k = j (4)
and

∂ x C j = C j-1 -a j C p-1 , j = 1, . . . , p -1, (5)
∂ x C 0 = -a 0 C p-1 . (6)
With this preparatory steps, the unique solution x → Φ(x, ξ) of the initial value problem (2) can be expressed as

w(x) = Φ(x, ξ)w ξ + Ψ(x, ξ)u. (7)
Therein, Φ(x, ξ) ∈ E * (R) p×p is the fundamental matrix of the initial value problem w(x) = Φ(x, ξ)w(x), w(ξ) = w ξ and Ψ(x, ξ) ∈ E * (R) p×m corresponds to the particular solution of (2) with vanishing data w ξ = 0. Explicit expressions for Φ and Ψ can be given using the ultradistribution-valued functions C 0 , . . . , C p-1

Φ(x, ξ) = p-1 j=0 A j C j (x -ξ), Ψ(x, ξ) = x ξ Φ(x, ζ)dζB. (8)
Substituting the general solutions of the initial value problems into the boundary conditions, one obtains the following linear system of equations:

w(x) = W ξ (x)c ξ , P ξ c ξ = 0. (9)
Here, ξ

= (ξ 1 , . . . , ξ n ), c T ξ = (w T 1 (ξ 1 ) • • • w T l (ξ l ), u T ), W ξ =    Φ 1 (x, ξ 1 ) 0 0 Ψ 1 (x, ξ 1 ) 0 . . . 0 . . . 0 • • • Φ l (x, ξ l ) Ψ l (x, ξ l )    , P ξ = P ξ,1 • • • P ξ,l+1 with P ξ,i = L i Φ i (0, ξ i ) + R i Φ i ( i , ξ i ), i = 1, . . . , l P ξ,l+1 = D + l i=1 L i Ψ i (0, ξ i ) + R i Ψ i ( i , ξ i ).
A possible choice for the coefficient ring is the ring

R I R [s, S, S I ].
Here, for any X ⊆ R, R I X = [S X , S I X ], with S = {C, S}, S X = {C(z ), S(z )|z ∈ X}, S I = {C I , S I }, S I X = {C I (z ), S I (z )|z ∈ X}, defined as in (1b), and

S I (x) = x 0 S(ζ)dζ, C I (x) = x 0 C(ζ)dζ.
Inspired by the results given in [START_REF] Mounier | An algebraic interpretation of the spectral controllability of a linear delay system[END_REF], [START_REF] Loiseau | A result that could bear fruit for the control of delay-differential systems[END_REF], [START_REF] Üerssen | A behavioral approach to delay-differential systems[END_REF], and in view of the simplification of the analysis of the module properties, instead of the ring R I R , we shall use a slightly larger ring,

given by R R = C(s)[S R ] ∩ E * .
Definition 3: The convolutional system Σ associated with the boundary value problem (1) is the module generated by the elements of c ξ over R = R R [S, S I ] with presentation matrix P ξ . By Σ R (resp. Σ Q ) we denote the same system but viewed as a module over R R (resp. R Q ).

IV. SYSTEM CONTROLLABILITIES A. General controllabilities

In this section we emphasize several controllability notions which are defined directly without referring to a solution space. Let us start with some purely algebraic definitions: Definition 4 (see, e.g. , [20, Def. 2.4.]): Let A be an R algebra. An R-system Λ is said to be A-torsion free controllable (resp. A-projective controllable, A-free controllable) if the A-module A ⊗ R Λ is torsion free (resp. projective, free). An R-torsion free (resp. R-projective, R-free) controllable Rsystem is simply called torsion free (resp. projective, free) controllable.

Elementary homological algebra (see, e.g., [START_REF] Rotman | An Introduction to Homological Algebra[END_REF]) yields Proposition 1: A-free (resp. A-projective) controllability implies A-projective (resp. A-torsion free) controllability.

Proposition 2: R-free controllability implies A-free controllability for any R-algebra A. More generally, given any Rsystem Σ that is a direct sum of a torsion module tΣ and a free module Λ, the extended system A ⊗ R Σ is a direct sum of the torsion module A ⊗ R tΣ and the free module A ⊗ R Λ.

The importance of the notions of torsion free and free controllability is intuitively clear: While the first one refers to the absence of a nontrivial subsystem which is governed by an autonomous system of equations, the latter refers to the possibility to freely express all system variables in terms of a basis of the system module. For this reason, and, secondarily, in reminiscence to the theory of nonlinear finite dimensional systems, we have the following:

Definition 5: Take an A-free controllable R-system Λ with a finite output y. This output is said to be A-flat, or A-basic, if y is a basis of A ⊗ R Λ. If A ∼ = R then y is simply called flat, or basic.

V. CONTROL WITH CONSTRAINTS A. Gevrey functions

For an A-free controllable R-system Λ , the basis y is introduced in order to express all system variables through infinite order differential operators:

w(x, t) = ∞ i=0 a i (x)y (i) (t) (10a) u(t) = ∞ i=0 b i y (i) (t) (10b)
This representation make sense only if the series (10a) and (10b) can be made convergent. When an appropriate basis y is chosen, these series lead in particular to an open loop control.

Definition 6 (Gevrey Class): [see, e.g., [START_REF] Komatsu | Structure theorems and a characterization[END_REF], [START_REF] Ramis | Séries divergentes et théories asymptotiques[END_REF], [START_REF] Rodino | Linear Partial Differential Operators in Gevrey Spaces[END_REF]] A smooth function φ : R → R is of Gevrey order α if on any compact subset

K ⊂ R ∃m K , γ K ∈ R + , ∀k ∈ N, sup t∈K |φ (k) (t)| ≤ m K γ k K (k!) α .
The functions of Gevrey order α < 1 are entire, while analytic for α = 1 and non-analytic if α > 1.

The Taylor expansion of a smooth function is not convergent, unless the function is analytic. The Gevrey order α estimates this divergence. Gevrey functions of order α > 1 have divergent Taylor expansion; the larger α , the more divergent the Taylor expansion. Important properties of analytic functions generalize to Gevrey functions of order α > 1: the scaling, addition, multiplication and derivation of Gevrey functions of order α ≤ 1 is of order α. But contrary to analytic functions, functions of order α > 1 may be constant on an open set without being constant everywhere.

B. Identity approximation

We shall make use of identity approximation whose definition we recall: Definition 7: An identity approximation is a family

φ ε in L 1 (R) indexed by ε > 0 such that: 1) φ ε 1 is bounded, independently of ε > 0. 2) ∀ε > 0, R φ ε = 1.
3) ∀η > 0, lim

ε→0 |t|<η |φ ε | = 0.
The following regularisation result explains the previous terminology Proposition 3: Consider an identity approximation (φ ε ). For any function f in L p (1 p < ∞) the sequence (φ ε * f ) converges towards f in L p . Thus, the sequence (φ ε ) can be seen to converge to the Dirac distribution. The following lemma is a useful construction of an identity approximation.

Lemma 1: Let φ : R → C be a function in D(R) with non zero integral, then the sequence (φ ε ) with

φ ε (t) = φ (t/ε) ε R φ(τ )dτ
is an identity approximation. 1) Gevrey identity approximation: Consider the following function

g(τ ) = exp(-(1 -τ 2 ) -σ ), if τ ∈ [-1, 1]
0, otherwise which is Gevrey of order 1 + 1/σ and with compact support

[-1, 1].
The previous construction leads to an identity approximation which is Gevrey of order 1 + 1/σ (see [START_REF] Lynch | Flatness-based boundary control of a class of quasilinear parabolic distributed parameter systems[END_REF], [START_REF] Rudolph | Motion planning and open loop control design for linear distributed parameter systems with lumped controls[END_REF]:

g ε (t) = g (t/ε) ε R g(τ )dτ

C. Basis as a B-spline Gevrey approximation

We shall consider the following functions for the basis y:

y = g ε * S ( 11
)
where S is a polynomial B-spline curve of order m (see [START_REF] De Boor | A pratical quide to spline[END_REF]) and g ε is the previously defined Gevrey identity approximation.

Since y is a convolution with the Gevrey function g ε it is Gevrey of the same order as g ε . Since g ε is an identity approximation:

lim ε→0 g ε * S = S
Hence, to design a reference trajectory y r for y, one has the same flexibility as in the B-spline curve design.

The variables w(x, t) and u(t) are thus expressed as

w(x, t) = ∞ i=0 a i (x)g (i) ε (t) * S(t) A ε (x, t) * S(t) (12) u(t) = ∞ i=0 b i g (i) ε (t) * S(t) B ε (t) * S(t) (13)
in virtue of the identity ḟ * g = f * ġ.

D. Constraints fulfillment

By Young's inequality, one obtains

u 1 = B ε * S 1 B ε 1 S 1 (14)
Since g ε is Gevrey of order 1 + 1/σ, one has the following estimates for its derivatives:

g (i) ε ∞ m I γ i K (i!) 1+ 1 σ
Considering the preceding inequality for i = 0, one gets this possible choice m I :

m I = g ε 1 = g ε (0) = 1 eεI g where I g = R g(τ )dτ .
Then, for i = 1, one obtains a possible γ

K γ K = g ε ∞ ġε ∞ = - (1 -t * 2 ) σ+1 2σt * g(t * )eεI g Note that t * ∈ [-ε, 0] and ∀i ∈ I, ġ(t * ) > ġ(t i ).
Then, the L 1 norm for B ε can be estimated as

B ε 1 = ∞ i=0 b i g (i) ε (t) 1 ∞ i=0 b i g (i) ε (t) 1 = ∞ i=0 |b i | g (i) ε (t) 1 ∞ i=0 2ε |b i | g (i) ε (t) ∞ ∞ i=0 2ε |b i | m I γ i K (i!) 1+ 1 σ (15)
Then, S 1 is estimated since an approximating B-spline curve is always contained in the convex hull defined by its control points. Thus, one has to choose the highest control point of S such that

u 1 = B ε * S 1 B ε 1 S 1 (16)
is bounded by the constraint on the control. Remark 2. -In practice, the infinite series (13) will be truncated (for implementation reasons) to a sufficiently high order. Since the bound we will consider in ( 15) is of uniform type (i.e. we take norms inside the sum), the inequality (14) will still be fulfilled.

VI. APPLICATION TO AN EULER-BERNOULLI BEAM

A. The model

The model of an Euler-Bernoulli beam is described below (see [START_REF] Aoustin | Theory and practice in the motion planning control of a flexible robot arm using Mikusiǹski operators[END_REF]).

∂ 2 w ∂t 2 = - ∂ 4 w ∂ 4 x (17a) w(0, t) = 0, ∂w(0, t) ∂x = Lu(t) (17b) ∂ 2 w(1, t) ∂x 2 = -λ ∂ 3 w ∂x 2 ∂t (1, t), (17c)
∂ 3 w(1, t) ∂x 3 = µ ∂ 2 w ∂t 2 (1, t) (17d) with λ = J ρSL 3 , µ = M ρSL

B. Open loop control

It has been shown in [START_REF] Aoustin | Theory and practice in the motion planning control of a flexible robot arm using Mikusiǹski operators[END_REF] that the system corresponding to eqs. ( 17) is free over a suitable ring.

The state and input parametrizations can be determined as: If y(t) is of class Gevrey, with α < 2, the following series, which corresponds to cosh( √ 2s)y is absolutely convergent

n≥0 2 n (2n)! y (n) (t)
For all specialisation of the basis y to a function of the class Gevrey, α < -λ(4n + 3)(4n + 4) (1 + i -x) 4n+2 y(t) where (resp. ) denotes the real (resp. imaginary) part. In other words, the two relations above define a family of trajectories for the hybrid system [START_REF] Fliess | Flatness and defect of non-linear systems: introductory theory and applications[END_REF].

Then, the constraint depicted in ( 16) can be satisfied when replacing the b i 's by the ones found in the expression above.

VII. CONCLUSION

We have outlined a framework for tackling boundary controlled flat PDE systems open loop trajectory tracking with contraints. We have made use of Gevrey identity approximation convolved with a polynomial B-spline for the basis. This retains the flexibility of approximating polynomial B-splines while maintaining the Gevrey character. Constraints fulfillment is ensured through young's inequality and Gevrey estimates. An application to the Euler-Bernoulli beam is outlined.

  2, equation (??) yields

	u(t) =	-J m Lα 2 1+	∞ n=0	2 2n+1 (4n + 4)!	(1 + λµ)	d 2 dt 2 +
					(4n + 4)(µ +	4n + 3 2	λ)	d 2n+4 dt 2n+4 y(τ )+
	EI L 2	∞ n=0	2 2n+1 (4n + 4)!	(4n + 4)	1 2	+	λµ 2	d 2 dt 2 +
		(4n + 3)(µ +	(4n + 1)(4n + 2) 2	λ)	d 2n+2 dt 2n+2 y(t)
									(18)
	and with (??),
	w(x, t) =	∞ n=0	(-1) n s 2n (4n)!	x 4n+1 2(4n + 1)	+
	( -)(1 + i -x) 4n+1 2(4n + 1)	+ µ (1 + i -x) 4n y(t)+
			∞ n=0	(-1) n s 2n+2 (4n + 4)!	λµ 2	+
		(4n + 2)!
		(4n + 4)!

( -)(1 + i -x) 4n+1 -x 4n+1