Repair and reuse networks?

A practical application of a spatiotemporal analysis method Julie Gobert, Romain Allais 1 Introduction

General introduction

In order to go beyond the recycling strategies and consistent with the European directives, different institutional initiatives have recently emerged in France promoting Circular and Cooperative Economy: Energy Transition for Green Growth Act, call of projects from the Environmental Agency… Re-employment, re-use through repair, are taking more and more space. For example, for furniture waste, the rules specify that it is appropriate to "encourage the re-use of elements whose functional and sanitary status is satisfactory or the reuse of waste of furniture." Communities can dedicate a portion of demolition to rehabilitation by creating recyclers or rankings [START_REF] Pacreau | La récupération dans les déchèteries : formes, motivations et devenir[END_REF]. Numerous local and national initiatives are tested and implemented from the associative sector or the Social and Solidarity Economy (SSE) to promote reuse, reemployment and repair, sometimes for a long time (like the Emmaüs network) as pointed by (Horne, Maddrell, 2002), (Defalvard, 2016) or French Environmental agency (ADEME, 2007(ADEME, , 2010(ADEME, , 2012(ADEME, , 2014)). Despite the diversity of the existing solutions, repair and reuse activities remain limited.

Causes of this lack of interest are multiple. First, the practices developed by the ESS and particularly by associations promoting re-employment do not necessarily have a positive image to create a sense of trust and accession [START_REF] Schmidt | Product-Service Systems for influencing customer barriers and customer acceptance[END_REF]. The lack of 'recognition' of institutions and consumers is a brake. In fact, the lack of public sensitivity and the weak institutional recognition of SSE initiatives (often located far away from the urban centres) due to a negative representation (Gregson and Crewe, 2003). In addition, audiences of consumers and/or users remain small due to the lack of awareness and a sometimes negative vision of citizens to donation practices and deposits of objects for repair/reuse (ADEME, 2014) [START_REF] Rumpala | Le réajustement du rôle des populations dans la gestion des déchets ménagers. Du développement des politiques de collecte sélective à l'hétérorégulation de la sphère domestique[END_REF], Anstett et al., 2015). Path dependencies contribute also to the under-deployment of repair and reuse activities. In fact, the current structuring of waste processing and recycling channels leaves little room and legitimacy for these emerging networks [START_REF] Pacreau | La récupération dans les déchèteries : formes, motivations et devenir[END_REF] and the consumer's habits and formatting also: «values associated with ownership and accumulation appear to be important development brakes to new forms of economy» (consumerist habitus, developed in particular by Herbert and Collin-Lachaud, 2017). In addition, traditional business models (i.e. wealth creation thanks to selling of products) constraint the product development process: design to cost implies the use of low quality material or components and the impossibility to repair products. This leads to different types of planned obsolescence (aesthetic, technical, technological…), a core mechanism of the consumption society that create ready-to-dispose devices with the constant increase of waste and their associated socio-ecological issues.

Presentation of the recyluse project

The RECYLUSE project aims to promote the ecological, economic and social transition to the circular economy through the deployment of repair and reuse. In contrast to the current waste management model, which is centralized and subject to the techno-economic imperatives of industrial means of waste processing, it is assumed that the future of waste management is not a unique place but a network of territorialized initiatives based on repair and reuse activities with high social and environmental values.

Recyluse is a research and operational project targeting to better understand the nature of the different resistances against and interests in repair and reuse. This project is based on two case studies to better understand these legitimacy deficits (with questionnaires, observations and semi-directive interviews of waste operators, representatives of recyclers, public authorities, users and non-users) and then create living labs on both territories, integrating repair/reuse professionals, designers and users. This project questions the collective capacity to build circular networks of reuse and repair, while taking into account local characteristics, path dependences and technical lock-in.

The project seeks first to understand the determinants and resistances by analysing the representations of the actors and singularly those of users/non-users. Then, to identify design modalities adapted to product repair. In operational terms, we try to provide answers to the following questions: How can the different consumer profiles be adhered to the repair and reuse of manufactured products? How to better integrate these consumers into repair and reuse networks? How to stimulate collaboration between heterogeneous actors? It is therefore a question of creating tools for decision-making regarding the structuring of "territorial networks of repair" [START_REF] Tyl | Towards synergies between local repairers, citizens, designers, and public actors: the REVALUE project[END_REF].

Regarding design activities, even if co-design (Fuad-Luke, 2006) has been little invested in a repair-reuse logic [START_REF] Tyl | Towards synergies between local repairers, citizens, designers, and public actors: the REVALUE project[END_REF][START_REF] Liedtke | User-integrated innovation in Sustainable LivingLabs: an experimental infrastructure for researching and developing sustainable product service systems[END_REF], its deployment during living labs must suppress some of stakeholders' resistances. Co-design is a participatory design process in which all relevant stakeholders are directly involved. In fact, living labs allow the exchange of knowhow and knowledge between the parties involved. They seem to be an appropriate modality to change representations and thus unblock certain defiance and create paths for territorial innovation. Observations during the living labs provides insights on the motivations of each actor, its integration to the collective, its ability to pass on its skills and collaborate to improve the repair of products.

Consequently, this paper gives a partial and provisional answer to the following research question: Which kinds of resources and stakeholders' configuration do have a positive effect to support reuse and repair circularity? Our main assumption is that repair and reuse activities can enable to develop new local economic schemes if stakeholders succeed in coordinating their respective efforts.

This communication focuses on the implementation of a method to support analyse of the emergence, structuring and operation of territorial projects. This analysis framework considers both temporal and spatial dimensions in addition to stakeholders' network. First, authors present the analysis framework then its application to two cases. While analyzing these two case studies throughout the methodology presented above, we intend to determine which endogenous, exogenous and generic resources are employed to build projects and particularly repair and reuse networks. These cases are then discussed and future work are presented.

Method

Recyluse combines different scientific disciplines: engineering sciences and social sciences with quantitative and qualitative approaches (i.e. survey, semi-directive interviews, observation, and organisation of living labs). This project relies on two French case studies and different work sequences: territorial diagnosis, organisation of living labs.

The first part of the project, the territorial diagnosis aims at determine the socio-genesis of ressourceries/recycleries1 projects thanks to semi-structured interviews and observations. The structures, interlocutors and their functions are detailed in table 1 and2 The second part of the project, the living labs, ambitions to create co-design spaces and supports for a large number of stakeholders as citizens, industrials and community employees or elected representatives. They can design together modus operandi for the emergence of repair territorial networks from the product level (i.e. repair and upcycling of textile or furniture), at the repair workshop level (i.e. values and organisation) and at the territorial level (i.e. resource pooling, creation of actors' network). Living labs are not detailed here but will be subjects of future communications. TEPCV) in July 2016. Semi-directives interviews were conducted to better understand this context and the attempts existing in repair and reuse (Table 1).

BASQUE COUNTRY

The agglomeration community of the Basque Country is a territory comprising 158 municipalities (including Bayonne, Anglet, Biarritz and Mauléon), or 309 723 habitants. Bil Ta Garbi, the mixed waste management union of the Basque Country has shown particular interest in being the project's study area. This territory presents for the RECYCLUSE project two major interests.

(1) The syndicate responsible for waste treatment, Bil Ta Garbi, deals with the waste of the agglomeration, is thinking about a policy favouring the repair and reuse of products at the end of life. In 2013, it opened two recycling centers, Canopia and Mendixka, which reduce landfill by half. It is a particularly dynamic territory with regard to the number of actors operating in the field of 3Rs.

(2) the project carried by the union was labelled in 2015 Territory "Zero Rubbish, Zero Waste" by the Ministry of the Environment. This project aims to integrate all the actors of the territory (project leaders, citizens, businesses, communities) in the management of waste in the Basque territory. For the research different kinds of stakeholders were interviewed: companies developing specific actions for waste valorisation, public institutions, associations and repair and reuse activities (Table 2).

Table 2. Interviews conducted on the Basque Country conurbation In order to support the analysis of these two territories and understand their specificities, authors develop a framework that considers the stakeholder network development by the prism of resource mobilization. This framework is presented below.

2 Presentation of the spatio-temporal analysis framework

The concept of perimeter of action, defined by the evolution of stakeholders' network and their mobilization of resources, was presented in previous articles [START_REF] Allais | A multidisciplinary method for sustainability assessment of PSS: challenge and developments[END_REF][START_REF] Allais | Conceptual framework for spatio-temporal analysis of territorial projects[END_REF]. These resources may be tangible (e.g. wood or machine tool) or intangible (e.g. patent, knowledge), territorial (i.e. specific to a territory), exogeneous or generic but they are all socially built. These resources are assessed thanks to the framework for (im-)material flow analysis that consider resources at multiple analytical levels, described as micro, meso and macro levels [START_REF] Gobert | La mobilisation du capital territorial pour le développement d'une logique d'Ecologie industrielle territoriale[END_REF].

Since our framework concerns the mobilisation of territorial resources over time by the actors, the project shapes the perimeter of action through the stakeholder areas of competence (e.g. the firm for entrepreneurs, the administrative constituencies for public authorities), their common readiness and capacity to influence project implementation [START_REF] Allais | A multidisciplinary method for sustainability assessment of PSS: challenge and developments[END_REF] and the scalar projection of their activities on the project level [START_REF] Allais | Conceptual framework for spatio-temporal analysis of territorial projects[END_REF].

The perimeter of action is dynamic and embedded in a history: both inherited (i.e. previously constructed networks, geographical perimeter, sector ...) and built during the structuring and realization phases of a project. Amongst the four steps of territorial project development, our case study analysis focuses here on the three first steps described below.

• Time 0: upstream of the project. Specific local resources, stakeholders, networks that can be mobilized but are still 'latent', inactivated for the project, define the initial perimeter of action. • Time 1: project launch. A need emerges or one or more stakeholders build a strategy (cf. time 0). They will seek to activate resources, create other relationships to find new partners possessing assets necessary for the launch of the project, such as founders (e.g. government, banks), external expertise (e.g. university, consulting firms). The perimeter of action will evolve in accordance with the project objective and the input of these new partners.

• Time 2: the project. The activation of territorial resources and the recruitment of the missing external skills (subcontractors…) for the realization of the project will concretize the perimeter of action.

• Time 3: downstream of the project. This is an evolution of time 0: the network developed becomes a latent network that may be activated for a new project, or conversely, in the event of conflict or failure, may be broken up.

Case studies

We analysed the data obtained by interviews to identify for each case the evolution of the perimeter of actions, the role and the number of stakeholders and the resources mobilised at the different steps of the project.

COEUR DE SAVOIE CONURBATION

Coeur de Savoie is a new conurbation; the executive board is then trying to define coherent and consistent public policies on its new area. However, this spatial change requires time before being effective and creating efficient synergies. Moreover, concerning the waste management, this administrative area inherits from the past a specific configuration, which can hamper projects promoting repair and reuse.

On the one hand, the waste collection and treatment are divided in two areas and managed differently: for one side, directly by the administration of the conurbation, for the other side by a specific organisation (SIBRECSA), a syndicate of municipalities putting together some municipalities of CdS, as well municipalities of the bordering department. This division creates difficulties to harmonize public policies and further the progress of repair and reuse initiatives. On the other hand, on the area managed by the syndicate of municipalities, an incinerator was built in the 1970's and a renovation was recently financed. Consequently, one of the objectives of the administrative board is to make profitable this investment. This could create a technological lock-in, which could prevent elected representatives to be active for encouraging repair and reuse.

We will relate the different steps explaining how the project of recyclerie emerged and became a project designed by different stakeholders. Time 0: An initial idea studied in an associative structure Creating a recyclerie was at the beginning an idea supported by a local association (régie de quartier), whose ambition is to enable cultural, social and economic development and to create job opportunities for inhabitants (non-competitive activities). The project was partly inspired by the experience of a mayor of one municipality of CdS. Indeed, he is a former researcher, was president of another Régie and participated in the early 2003 in the creation of a recyclerie in Grenoble. However, regarding the investments in terms of time, human resources and funds, the initial idea was temporally rejected.

Few specialised initiatives have emerged in CdS. A craftsman (Api'R Bois) working from wood second-hand commodities and transforming them in other products (upcycling) has freshly set up at Saint-Pierre-de-Soucy. The Régie de territoire created a few years ago a sewing activity (Fibr'ethik) from canvas sheet, advertising and truck tarps waste; the objective was firstly to socially and professionally integrate female job seekers. They are part of departmental, regional and national networks for administrative and creative collaborations, and they find inspiration and partly sell their products online (e.g. upcycling market place…). An activity of self-repair in the main city of CdS (Montmélian) opened in 2018. In addition, a project holder, who frequented a large fab lab, La Casemate, in Grenoble (50km from CdS -16.0000 inhabitants) is trying to create a repair café and fab lab in La Rochette, with the support of a local association.

Time 1: TZCLD, the recyclerie emergence

This association, in relation with local charities (ADMR and ATD Quart monde), took the opportunity offered by a national call for tenders (Territoire Zéro Chômeur Longue Durée): local communities were invited to propose new local solutions for long-term job seekers. The stakeholders created a specific network of associations, citizens, local officials and national institutions (Pole Emploi and la Dirrecte). A dynamic emerged from this brainstorming and the tender writing. The consortium proposed nine development axes (e.g. proximity services development, improve the quality and cleanliness of the living environment or reduce and recover waste). Amongst the numerous solutions to create economic activity for long-term job seekers, a suggestion was to develop a recyclerie on the CdS area.

Numerous resources were pooled together for the writing of to the TZCLD project. Initial project holders (charities) collected local data on long-term unemployment while evaluating the potential opportunities of the TZCLD call. CdS supported the project by mandating its management team to participate in the committee's work and visit national and regional exemplary territories. The consortium was finally composed of 22 associations and structures from local, regional and national levels, 63 citizens, including 25 unemployed and 6 local officials from 27 communes of the conurbation CdS.

The TZCLD project was not funded but the some of the actors are still involved into the recyclerie project.

Time 2: La Salamandre

On the basis of the TZCLD proposition, a long-term unemployed person (LTUP) took over the portage of the project of recyclerie. In association with CdS conurbation and Positive planet, they study the business model and technical feasibility of La Salamandre, the recyclerie of Coeur de Savoie. Positive planet is a national association implanted in La Rochette, to accompany project holders free of charge in the creation of businesses. They ambition to create economic activity for eight persons and two technical and administrative managers. First, both administrative and technical management was the responsivity of LTUP but they realize he was not able to manage these two tasks, lacking time and skills for the administrative part. They propose the administrative support was taken in charge by the R de Récup' manager but there were two difficulties. First, she also misses time to engage in this new project and secondly, the Salamandre is considered as a territorial project. Consequently, it raised administrative difficulties as the DIRRECTE (a deconcentrated service of the State which distributes financial aids for the creation of insertion jobs) works on a specific area (Département). Besides CdS and the Régie de Territoire were reluctant to give this project to an actor outside the conurbation. The Régie de Territoire finally employed LTUP and is now managing the Salamandre project.

Considering the technical aspects, the location and the access to a surface of 950 and 1000m² is still a difficulty for making true the project. An old industrial building in a commercial area was first analysed but heavy investments were required to fulfil safety standards; the proposition was then rejected. Another opportunity appeared at Saint-Pierre d'Albigny through the creation of a third place with multiple activities in a former industrial site. As the business model for this third place became more precise, the area available for the recycling project decreased until the project was no longer feasible. The analysis of the different interviews does not show a cooperation between the different reuse and repair initiatives. Actually, the spatial proximity of these initiatives is not a catalyst of cooperation, as the valleys deeply divide the landscape and lengthens the routes. The perimeter for action of the Salamandre project is less the whole conurbation, that one part of this, depending of available premises and the stakeholder's involvement.

The mobilised resources and the perimeter for action

Different initiatives, located outside the conurbation area, mainly influence, inspire and/or support the recyclerie project in CdS and play a role of referrers. They were solicited at different steps of the Salamandre, particularly for the time 1 and 2. R de récup is a recyclerie in Pontcharra, a city near but outside CdS conurbation but located in the SIBRECSA waste management perimeter. It is considered as a model to replicate on the conurbation area (inspiration). It also provided both technical skills and insights on the management of a recyclerie. As well, the development of an activity of self-repair was the result from a cooperation between a local association promoting cycling mobility and an activist association for cycling in Savoie (Roue Libre) located in Chambery (20km from CdS -60.000 inhabitants). The Emmaus community, les Chantiers valoristes and Trialp3 were also referrers for CdS stakeholders. These actors are particularly close, because they share the same objectives: facilitating the insertion of vulnerable persons while empowering them, enabling them to develop skills and giving them possible step by step to integrate the labour market. The holders of the Salamandre shared this core value as they intend to use waste valorisation as a means to valorise people.

We note that the perimeter for action of the repair reuse initiatives are mainly defined by the affinities and worldviews shared between the actors. Thus, the different initiatives of the territory CdS consistent with the initiatives of Chambéry or Grenoble (e.g. Roue libre implantation in Montmélian or the FabLab at La Rochette) or more distant or digital inspirations for API'R bois or Fibr'Ethik.

The involvement of the public authorities is both a lever and a difficulty. The help of the conurbation was decisive for Roue libre -Montmélian to find the premises, as well for La Salamandre. Generally, the conurbation administration displays ambitious objectives to reduce waste flows and enable their transformation. However, its view of repair and reuse workshops is not systematically shared, particularly when the initial impulse comes from the conurbation4 . Some national actors (ADEME or DIRRECTE) provide expertise and fund social jobs or investments.

THE BASQUE COUNTRY

The Pays Basque is administratively a new conurbation, but it comes from a very long process and a strong involvement of local actors, promoting the Basque identity.

Time 0: A fertile territory for repair and reuse activity

Numerous initiatives have emerged the last years, notably specialized in repair/reuse services, as well upcycling undertakings. The territory was the basis of multiple projects that failed, under construction or already operating (listed below):

-AIMA is a charity that runs several second hand shops; it does not propose repair workshops.

-Recycl'Arte is a project of repair and reuse workshop, but the access to available premises impedes the initiative to grow as planned. -Emmaüs Tarnos is an "old" and well-established stakeholder for the selling of second-hand commodities.

-Txirrind'Ola is a participative repair shop, which enables people to fix by themselves their bikes and promotes cycling in the public space. It provides users with advice and tools adapted for bike repair. Moreover, cycling classes are given for children. The bikes that are employed for these lessons or sold are collected in the centers for waste sorting. -Le Hangar du Sport is a second-hand shop dedicated to sport products. This initiative tries to combine selling, repair activities and sport center. -Orratzetik Hari is a group of couturiers and fashion designers who creates new models or accessories from scraps of clothes. They propose couture lessons for children. -Api'up is an association and a brand that works with firms to collect their office waste and transform them into new furniture. It is not located on the Basque Country conurbation. It is perceived sometimes as a model, sometimes as a counter model.

Each initiative tries to enlarge its main activity (repair or second-hand selling) with other actions to strengthen its business model. Even if the different holders of initiatives know each other and that they are not yet in competition, they have not formed a territorial formal network.

Time 1 -Attempt to create a local network gathering repair and reuse stakeholders

Bil ta Garbi, the syndicate managing waste in the Basque Country, was interested in creating local dynamics for reuse and repair. This objective was the results of different stakes:

-Firstly, the syndicate was recognized in 2015 "Territory -Zero rubbish, zero waste" and aimed at implementing a global policy for reducing and managing waste while promoting circular economy. Encouraging repair and reuse is one of the stated objectives. Contrary to the above case study, whose motor was human insertion, they focus their project on waste management and reduction.

-Secondly, as it has the responsibility of the household waste centres, it was face an increasing demand of associative actors to obtain an access for collecting commodities (bicycles or other items, which can be fixed, cleaned and then reused). It appears impossible to build or to leave the dedicated places/skips for each of them. From the point of view of Bil ta Garbi representatives, it would have been more coherent, the association mutualise their means for the transport of commodities, etc.

As a consequence, Bil Ta Garbi organised different meetings and events, trying to give the impulse for structuring a local network for reuse and repair. Nevertheless, this has encountered resistance from some stakeholders involved in repair and reuse. Moreover, Bil ta Garbi legitimacy is regularly questioned by other stakeholders. On the one hand, some actors guessed Bil ta Garbi's main objective is to make their recent investment (Canopia) profitable and not to encourage larger waste sorting. On the other hand, as the leader of "Zero rubbish, zero waste", the ways the projects concerning repair and reuse are chosen and financed are criticized.

The large size of the territory and the distribution of the various initiatives seems not to require formal collaboration in the Bask country. However, the recent emergence of recycling projects in the city of Bayonne (one of which is carried out by BIZI) raises the question of how these actors coordinate to avoid regulation by competition and market.

The mobilised resources for the project

Even if "there is no consensus on what being Basque means or on what grounds people can define themselves as Basque or "something else"", political leaders and many inhabitants recognize themselves as Basque, because they have a common language and history, which were the cause for confrontation. This Basque identity profoundly pervades the initiatives we investigated, not only because many stakeholders want to promote the economic development of the Basque Country, but also because they consider their initiative could become a place for speaking Basque. The Basque identity is then an important resource strongly mobilized by different stakeholders, but for all that, it does not facilitate the consensus between stakeholders. Despite the absence of coordination, the trajectories of these leaders show that the informal resource constituted by past or present involvements in the same organisations are decisive. We can guess Bizi (an environmental organisation) played a role of incubator promoting new representations over waste and resources. However, different persons, now committed to the creation of initiatives, considered that the role of environmental objector and lobbyist towards public decision-makers was not sufficient. They wanted to be involved in very practical actions and not just in theoretical reflexions. Txinrrind'Ola or Recycl'arte were both build by previous BIZI members. Likewise, Txirrind'Ola was more or less a space of exchange and prefiguration of other initiatives. It has acted as a referrer, because it was the first specialised repair and reuse workshop and it has supported some new initiatives by giving advice. Some leaders were volunteers at the workshop, as well.

The public sphere provides different resources for these initiatives (e.g. administrative support, low cost rental or loan of premises, funds…). The conurbation of the Basque Country collects waste and the syndicate Bil ta Garbi manages the households waste centers, where the repair and reuse initiatives can collect commodities. In addition, these entities are currently reflecting of the best way to promote reuse and repair on the territory, for example while transforming waste centers, improving public awareness. However, this institutional asset is not always accepted.

Some public-private structures can have a role of intermediary like the social and solidarity economy territorial incubator Tube à ESS'ai that helps project holders to sharpen and administratively, organisationally strengthen their territorial proposition.

Discussion

Methodological reflexion

At this moment of the research, different difficulties have been raised. The distance from one case study brings about difficulties to put into practice the planned methodology, to organise workhops and to conduct surveys.

The purpose of the study must be clearly defined: The Salamander project in CdS, the territorial network project conducted by Bil ta Garbi in the Basque Country because it imposes to differently apprehend the context, the geographical area or the involved actors. Besides, each of these dimensions are in interactions and co-evolve during the study, as repair and reuse are at the heart of global dynamics. Inasmuch the studied set of stakeholders and the projects quickly evolve, i was consequently not easy to plan meetings and workshops, particularly when open or discreet conflicts are revealed or emerge during the research time. However, we succeeded in outlining the essential resources needed for repair and reuse projects, their spatial and organisational projection and explaining why some stakeholders' coordination can encounter difficulties or be facilitated.

Material and immaterial resources essential for repair and reuse activities

From the two case studies we can draw a first picture of the needed resources for developing repair and reuse workshops and maybe territorial networks between initiatives and actors involved to waste management. The converging results coming from the two cases reveal that some resources are decisive so that the holders of repair and reuse initiatives could succeed in.

To have an access to material resources (waste collected in rubbish dumps or directly in retail shops; places/business premises where the different reuse and repair activities can take place), the initiatives often require the help of public actors, because they are in charge of critical competencies like economic development promotion, waste collection and treatment. The public support may have a real positive impact to accelerate the development processes, if the public governments or actors respect the independence and the way of thinking of nongovernmental organisations. Otherwise some mistrust can appear between them. This institutional resource depends on the political dynamics (which appetence has the executive board of a conurbation to go forward) and the past experiences, which may have drawn a trajectory and already found, activated and mobilised helpful resources.

On the other hand, repair activities cannot be developed without a specific immaterial resource: skills for fixing, sewing, etc. The lack of this technical resource may hamper or stop a project. As well, it was noticeable that, the role of "models" or referrers, which can guide the holders while anchoring their project on the territory, are determining, and can be "employed" or called at very different times of a project.

Confronting perimeters of action and customer/user catchment area?

The mobilisation of external intangible resources (experience in repair and reuse, expertise in waste management…) shows the necessity to overcome the territorial limits of the conurbation. The perimeter for action evolves in relation to the resources mobilised and the capacity to act and to convince the different stakeholders.

The shared values play a large role in the choice of referees and partners. For CdS, initiatives holders engage with actors from places outside the CdS territory (Chambery, Grenoble, Pontcharra…), national, regional or departmental networks (e.g. national ressourceries network or upcycling networks) or online resources (e.g. online market places…). They could glean information, formations, advise and expertise. They could also obtain financial aids for different elements of their project (funding of job creation, of investment to transform and adapt technical and commercial premises). The resort to referrers is possible because people share common values. Conversely, the recourse to subsidizers, providing financial support, is more pragmatic; according to calls of tender, opportunities, the initiatives holders.

In order to facilitate the collaboration between actors, the value system co-construction and the referees' identification are integrated into the living lab tools and methodology and tested within the two territories.

The perimeter for action has not to be confused with the competencies/customer/user catchment area. For the conurbation Coeur de Savoie, it appears that the Salamandre's supposed catchment area does not correspond to the administrative boundaries. This is due partly to the natural landscape that constraint the infrastructures and communications between the two valleys of the territory. People do not imagine to have a motorised journey, lasting more than 20 min.

In the case of Basque Country, Bil ta Garbi has a territorial view matching its competencies and area of action, that is to say the whole Basque Country. The studied repair and reuse initiatives have a perimeter of action more limited as their project attempts to sensibilise inhabitants living close to the initiative location. That is one reason for this two kinds of actors do not find a common ground for instituting a repair and reuse network with organisations motivated by different objectives.

Shared values: a lever of coordination and resources pooling

Inspiration, worldviews and "orders of values" or cities [START_REF] Boltanski | De la justification. Les économies de la grandeur[END_REF] play a major role in the possible coordination between actors for this kind of environmental and human-centred activities. Each stakeholder involved in a project would like to make progress repair and reuse, but different core values stir them. Public actors and syndicates are mainly driven by the principles of efficacy and their values essentially match the industrial city. They attempt to defend their investments because they are aware of not being able to rapidly adapt their technical infrastructure and administration. Conversely, numerous project holders consider that society has to radically change and particularly the ways of consuming, producing and wasting have to be profoundly modified. Their main objective is to diminish the human footprint on the environment. Their values correspond to the "civic city" (as they refer to participation, adhesion…) and the "inspired city" (creativity). Two ways of considering repair and reuse activities bring into opposition: Those who considered that initiatives must lead to traditional economic activities and become independent on public subsidies, those who prefer building an alternative social economy.

Conclusion and future work

This paper aimed at displaying a current research on repair and reuse activities and the first results obtained, enabling to understand the resources necessary for building initiatives and networks. Even if there are sometimes some distortions between values and misunderstandings between stakeholders, local arrangements can progressively be negotiated.

As we intend to more deeply work on the data gathering during our research, we will attempt to answer the following questions: What impact does the compliance of orders of values have on the ways of revealing, considering (valuing) and employing resources? Which compromises (structurally justified) or punctual arrangements (feasible and pragmatic action) can be found between the stakeholders as they are imbedded in very different frames of reference? Can repair and reuse take part to the definition of an order of values corresponding to the "environmental city" (Lafaye and Thévenot, 1994)?

To integrate users' point of view, we will strive to see how these kinds of short local systems of waste valorization can be implemented and whether they meet the users' expectations, representations and appropriations (Akrich, 2016). This should give inputs for decision takers at the local and regional levels and enable to understand what kinds of new services and systems can be brought for improving territorial circularities [START_REF] Schmidt | Product-Service Systems for influencing customer barriers and customer acceptance[END_REF].

Figure

  Figure 1. Methodology

  manager-territorial development organisation Deputy mayor of Hendaye -Community counselor of Agglomeration community Sustainable development and social and solidarity economy project managers -Agglomeration Community Mission manager -Social and solidarity economy territorial incubator Director in charge of wastedirector -humanitarian and charity association Recyclerie project holder -repair, upcycling, multi-product Director -insertion association Recyclerie project holder -under development Sport recyclerie director -repair, second hand, upcycling

Figure 2 .

 2 Figure 2. The internal and external stakeholders involved in the recyclerie project

Figure .

 . Figure . The internal and external stakeholders involved in repair and reuse activities (Basque Country)

Table 1 .

 1 Interviews conducted at Coeur de Savoie

	Struture	Function of the interlocutor
	Companies	Carpenter & designer (upcycling activity)
	Structures working in the	Co-director -Insertion association
	waste recovery	Site director -waste recycling company
		Designer and site director -ressourcerie
		Director -waste collection company
		Director -intercommunal union for household waste
		Director -insertion association
	Public institutions	Mayor and project leader Zero long-term unemployed, former project holder of
	Community	recyclerie
	representatives	Waste service manager -Coeur de Savoie
	Elected officials	Circular economy project manager in charge of the zero waste territory
		initiative -regional agency
		Business development project manager-regional agency
	Repair -reuse initiatives	Recyclerie project holder -La Salamandre
		Bike workshop manager
		Textile workshop manager
		Repair café manager
	Territorial association	Director -animation
		Member -Restoration and Maintenance Trail paths
	Digital tool developers

(not included in the study) Recyclerie consultant -training for project holders and stock management software Market place founder specialized in reuse and upcycled products

Workshops which integrates the selling of second hand commodities and often repair activities, and then encourages reuse.

A public institution of intercommunal cooperation (EPCI) is a French administrative structure combining several communes in order to exercise some of their competences in common.

Thanks to the revenues earned by reuse and waste management activities, an Emmaus community established since 1970 in Chambery accommodates vulnerable persons (homeless, refugees…) and "employs" them for different tasks. Another insertion company, Les Chantiers valoristes, assists persons without employment thanks to economic activities related to the dismantling of products, the recovery and resale of materials, recycling and wood workshops, etc. Trialp is a company collecting and revalorising waste with an objective of social and professional insertion.

As an example, it launched a call of project for organising repair café, but was unfruitful, even though some citizens are willing to be committed to this kind of projects. Moreover, as we organised creative workshops with the conurbation (particularly for finding a location, for inviting people…), we felt a silent resistance.

Acknowledgments

The authors wish to thank the region Nouvelle-Aquitaine for their financial support as well as the French national agency ADEME for its funding of the RECYLUSE project.

They are also grateful to their colleagues participating in this research: Benjamin Tyl, José-Frédéric Deroubaix and Cyril Baldacchino.


Nouara Habrache
email: nouara.habrache90@gmail.com

Ahmed Maidi
email: ahmed.maidi@gmail.com

Jean-Pierre Corriou
email: jean-pierre.corriou@univ-lorraine.fr

Born In 1974 In Algeria

Feedback control of bilinear distributed parameter system by input-output linearization

Keywords: partial differential equation, bilinear distributed parameter system, geometric control, characteristic index, exponential stability, co-current heat exchanger

In this paper, a control law that enforces the tracking of a boundary controlled output for a bilinear distributed parameter system is developed in the framework of geometric control. The dynamic behavior of the system is described by two weakly coupled linear hyperbolic partial differential equations. The stability of the resulting closed-loop system is investigated based on eigenvalues of the spatial operator of a weakly coupled system of balance equations. It is shown that, under some reasonable assumptions, the stability condition is related to the choice of the tuning parameter of the control law. The performance of the developed control law is demonstrated, through numerical simulation, in the case of a co-current heat exchanger. The control objective is to control the outlet cold fluid temperature by manipulating its velocity. Both tracking and disturbance rejection problems are considered.

Introduction

The early lumping approach consists in approximating the PDE model by a set of ordinary differential equations (ODEs), i.e., the DPS is reduced to a lumped parameter system (LPS). This approach can be seen as model reduction. The equivalent LPS model is obtained by approximating either the PDEs or their solutions [START_REF] Wang | Model-based predictive control for spatially-distributed systems using dimensional reduction models[END_REF][START_REF] Ray | Advanced Process Control[END_REF]Li and Qi, 2010). The resulting reduced model is then used to design the controller in the framework of the control theory of LPSs. However, the early lumping approach presents significant drawbacks. First, reducing the DPS to LPS does not preserve the fundamental control properties (controllability, observability and stability) of the original DPS [START_REF] Ray | Advanced Process Control[END_REF][START_REF] Singh | Effect of finite-dimensional approximation on observability analysis of distributed parameter models[END_REF]. Secondly, ensuring a good approximation of the distributed behavior of DPS requires a high order reduced model, which makes the control design step more difficult. Thirdly, the dimension of the resulting controller is important, which makes it difficult to implement. In addition, neglecting the distributed nature of the DPS often leads to a controller with a poor performance (Christofides, 2001a).

The late lumping approach represents the effective alternative to the early lumping approach [START_REF] Christofides | Feedback control of hyperbolic PDE systems[END_REF]Christofides, 2001a). This approach directly uses the PDEs model, without any reduction, for the design of the controller. Therefore, the distributed nature of the DPS and its fundamental control properties are preserved [START_REF] Ray | Advanced Process Control[END_REF]Christofides, 2001a). This approach yields a controller of distributed nature that enhances the performance in closed loop [START_REF] Christofides | Feedback control of hyperbolic PDE systems[END_REF].

In this work, following the late lumping approach, a control law that enforces both output tracking and disturbance rejection is developed for a bilinear DPS in the framework of geometric control. By assuming a boundary output, it is shown that the obtained control law, which is of finite dimensional nature, yields a time-varying bilinear system in closed loop. Then, based on the mathematical properties of the spatial operator of the resulting closed-loop system, the stability condition is established. The performance of the developed control law is evaluated in the case of a cocurrent heat exchanger. The problem is to control the outlet temperature of the cold fluid by manipulating its inlet velocity.

The paper is structured as follows: The control problem of a bilinear DPS is presented in Section 2. Section 3 gives the main results concerning the stability of coupled linear hyperbolic PDEs. Section 4 is devoted to the control law design and to the stability analysis of the resulting closed-loop system. Numerical simulation results that show the performance of the control law, in the case of a co-current heat exchanger, are reported in Section 5. Section 6 concludes the paper.

Although several interesting control strategies have been developed for nonlinear distributed parameter systems (DPSs) [START_REF] Chen | Control of Nonlinear Distributed Parameter System[END_REF][START_REF] Dai | Iterative learning control for discrete parabolic distributed parameter systems[END_REF][START_REF] Wang | Model-based predictive control for spatially-distributed systems using dimensional reduction models[END_REF]Christofides, 2001a,b;[START_REF] Dubljevic | Distributed nonlinear control of diffusion-reaction processes[END_REF][START_REF] Padhi | An account of chronological developments in control of distributed parameter systems[END_REF][START_REF] Maidi | Boundary control of a parallel-flow heat exchanger by input-output linearization[END_REF]Corriou, 2011, 2014;[START_REF] Ding | Boundary predictive control of second-order linear modulus-vary distributed parameter systems based on wavelets transformation[END_REF], control design for this class remains a challenging problem (Christofides, 2001b). If, for linear DPSs, a general control theory is developed by means of semigroup theory [START_REF] Curtain | An Introduction to Infinite-Dimensional Linear Systems Theory[END_REF], for nonlinear DPSs, this is difficult and the investigation is made by assuming certain particular classes. Among these classes, can be cited the first-order hyperbolic PDEs (Hanczyc and Palazoglu, 1995;[START_REF] Christofides | Feedback control of hyperbolic PDE systems[END_REF]Garc´ıa-Sandoval et al., 2008;Gundepudi and Friedly, 1998), the quasilinear parabolic partial differential equations (PDEs) [START_REF] Dubljevic | Distributed nonlinear control of diffusion-reaction processes[END_REF] and the nonlinear parabolic PDE (Maidi and Corriou, 2014).

An important class of DPSs encountered in a wide variety of practical applications is described by coupled linear hyperbolic PDEs issued from first principles [START_REF] Bartecki | Modeling and Analysis of Linear Hyperbolic Systems of Balance Laws[END_REF]. In this case, two kinds of DPSs can be distinguished: strongly and weakly coupled [START_REF] Bartecki | Modeling and Analysis of Linear Hyperbolic Systems of Balance Laws[END_REF]. When both temporal and spatial derivatives of different state variables are involved in each equation of the model, the system is said to be strongly coupled. In the other case, when each equation of the model contains only the temporal and spatial derivatives of the same state variable, the system is weakly coupled. Note that, under certain assumptions, a strongly coupled system can be transformed into a weakly one by means of a decoupling procedure [START_REF] Bartecki | Modeling and Analysis of Linear Hyperbolic Systems of Balance Laws[END_REF].

Heat exchangers, electrical transmission lines, irrigation channels and transportation pipelines are some examples of DPSs whose dynamic behavior is described by coupled hyperbolic PDEs [START_REF] Xu | The state feedback servoregulator for countercurrent heat exchanger system modelled by sytem of hyperbolic PDEs[END_REF][START_REF] Bartecki | Modeling and Analysis of Linear Hyperbolic Systems of Balance Laws[END_REF]. For these DPSs, for certain practical considerations, some parameters of the model are taken as control variables. As an example, in the case of heat exchanger, manipulating a fluid velocity is more practical than manipulating its inlet temperature. Thus, the system is turned into a bilinear one, which represents another interesting particular class of nonlinear DPSs.

When a parameter is taken as a manipulated variable to control a boundary output, a weakly coupled DPS is characterized by a finite characteristic index [START_REF] Christofides | Feedback control of hyperbolic PDE systems[END_REF], which is a generalization of the notion of the relative degree of lumped parameter systems (LPSs) (Isidori, 1995). In addition, the stability analysis of a weakly coupled DPS can be done easily by examining the eigenvalues of the spatial operator. These interesting properties motivate the use of geometric control for bilinear DPSs to design a control law following the late lumping approach.

Designing a controller for a DPS can be done using two possible approaches [START_REF] Ray | Advanced Process Control[END_REF]Christofides, 2001a): early lumping and late lumping.

Problem formulation

Let us consider a linear 2 × 2 system issued from first principles balances described by the following hyperbolic coupled PDEs [START_REF] Bartecki | Modeling and Analysis of Linear Hyperbolic Systems of Balance Laws[END_REF]

∂x 1 (z, t) ∂t = -u 1 ∂x 1 (z, t) ∂z + α 1 (x 2 (z, t) -x 1 (z, t))
(1)

∂x 2 (z, t) ∂t = -u 2 ∂x 2 (z, t) ∂z + α 2 (x 1 (z, t) -x 2 (z, t)) (2)
with the following boundary conditions

x 1 (0, t) = x 10 (3) x 2 (0, t) = x 20 (4)
and initial conditions

x 1 (z, 0) = x 0 1 (z) (5) x 2 (z, 0) = x 0 2 (z) (6)
In this model, t ∈ [0, ∞) and z ∈ [0, l] represent the time and spatial variables, respectively. α 1 , α 2 , x 10 and x 20 are constant positive parameters. The variables u 1 and u 2 can be assumed as manipulated variable and disturbance, respectively and vice versa, depending on the considered control configuration. x 1 and x 2 are the state variables assumed to be in Hilbert space L 2 ([0, l]) while x 0 1 and x 0 2 are the initial spatial profiles. To simplify the presentation, it is assumed in the following that our aim is to control the following boundary output

y(t) = x 1 (l, t) (7)
by manipulating the variable u 1 whereas u 2 is a disturbance. Assuming this control configuration, the model ( 1)-( 6) represents a bilinear DPS [START_REF] Ouzahra | Approximate and exact controllability of a reaction-diffusion equation governed by bilinear control[END_REF][START_REF] Bühler | Topics in Identification and Distributed Parameter Systems[END_REF]. This particular class of DPSs is termed weakly coupled or decoupled systems since the terms that contain derivatives are not coupled [START_REF] Bartecki | Modeling and Analysis of Linear Hyperbolic Systems of Balance Laws[END_REF].

To solve this formulated control problem, the following assumptions are made.

Assumption 1: The spatial profile x 1 (z, t) is a monotonic function with respect to z at given t, furthermore assumed increasing [START_REF] Ayres | Theory And Problems Of Differential Equations. Schaum's Outline[END_REF].

Assumption 2: At z = 0, the boundaries conditions are such that x 2 (0, t) -x 1 (0, t) > 0, that is, x 20 -x 10 > 0.

Remark 1: For the distributed parameter system (1-2), since u 1 , u 2 , α 1 and α 2 are positive parameters, assumption 2 implies that x 2 (z, t) -x 1 (z, t) > 0 for z ∈ [0, l].

Before addressing the control law design, the stability issue of the system (1)-( 6) is discussed in the following section.

Stability of linear hyperbolic systems of balance laws

The stability of the semigroup or of its generator (operator), i.e. the stability of DPS, can be investigated using some tools from spectral theory [START_REF] Engel | A Short Course on Operator Semigroups[END_REF]. Thus, if all the eigenvalues of the operator are negative real or complex with negative real part, the semigroup is stable, i.e. the DPS is stable.

Let us write the PDEs model (1-6) under the following operator form

∂x(z, t) ∂t = H x(z, t) (8)
where x(z, t) = [x 1 (z, t), x 2 (z, t)] T is the vector of state variables and the spatial operator H is defined as follows

H = A ∂ ∂z + B (9)
with

A = -u 1 0 0 -u 2 (10)
and

B = -α 1 α 1 α 2 -α 2 (11)
The stability of the linear hyperbolic systems issued from balance laws has been investigated in the literature [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions[END_REF][START_REF] Diagne | Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws[END_REF]. The aim of this section is to give the condition that ensures the exponential stability developed based on the spectral theory [START_REF] Engel | A Short Course on Operator Semigroups[END_REF]. This theory provides the qualitative behavior of the semigroup generated by the spatial operator H [START_REF] Engel | A Short Course on Operator Semigroups[END_REF].

Given the boundary conditions ( 3)-( 4), since the eigenvalues of the matrix A are negative, the eigenvalues of the spatial operator H are complex and equal to [START_REF] Christofides | Feedback control of hyperbolic PDE systems[END_REF][START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions[END_REF]

λ k = -∞ + k π i, k = -∞, . . . , +∞ (12)
with i is the imaginary unit, which implies that the system (8) is exponentially stable, i.e., the spatial operator H generates a stable semigroup. Consequently, the PDEs model ( 1)-( 6) is exponentially stable if the parameters u 1 and u 2 are positive.

Control law design and closed loop stability analysis

In this section, the control problem formulated above is solved in the framework of geometric control using the characteristic index concept introduced by [START_REF] Christofides | Feedback control of hyperbolic PDE systems[END_REF], which is a generalization of the relative degree of a finite dimensional system (Isidori, 1995).

Control law design

The calculation of the first time derivative of the controlled output (7) yields

dy(t) dt = ∂x 1 (z, t) ∂t z=l (13) = -u 1 ∂x 1 (z, t) ∂z z=l + α 1 (x 2 (l, t) -x 1 (l, t)) (14)
hence, from ( 14) it follows that the manipulated variable u 1 appears linearly in the first time derivative of the controlled variable y(t). This suggests requesting a firstorder dynamical behavior between an external variable y d (t) and the controlled output y(t), i.e.,

τ dy(t) dt + y(t) = y d (t) ( 15
)
where τ is the desired time constant. Now, Assumption 1 ensures that

∂x 1 (z, t) ∂z z=l = 0 (16)
thereafter, the of the first time derivative of the controlled output y(t) by its expression ( 14) into (15), and by solving the resulting equation with respect to the manipulated variable u 1 , the following control law results

u 1 = y(t) -y d (t) + τ α 1 (x 2 (l, t) -x 1 (l, t)) τ ∂x 1 (z, t) ∂z z=l (17)
with y(t) = x 1 (l, t).

Remark 2: The developed control law (17) involves the first spatial derivative of the state x 1 at z = l, which can be approximated by the following finite difference at right

∂x 1 (z, t) ∂z z=l = x 1 (l, t) -4 x 1 (l -∆z, t) + 3 x 1 (l -2 ∆z, t) 2 ∆z ( 18
)
where ∆z is the discretization spatial step. Hence, for practical implementation, one needs the measurements at both positions z = l -∆z and z = l -2 ∆z. These measurements can be provided from the measurement of the controlled output y(t) = x 1 (l, t) using an observer.

Closed-loop stability

The control law (17) yields, in closed-loop, the finite dimensional system (15), which is externally stable. In this subsection, the internal stability of the resulting closed-loop system (15) is analyzed.

The internal representation of the resulting closedloop is obtained by substituting the manipulated variable u 1 by its expression (17) into the system to be controlled (1)-( 6). In this case, the closed-loop can be written under the form operator (8) with

A =       -     y(t) -y d (t) + τ α 1 (x 2 (l, t) -x 1 (l, t)) τ ∂x 1 (z, t) ∂z z=l     0 0 -u 2       (19)
and B remains unchanged. Note that, since the obtained control law ( 17) is of finite dimensional nature, hence the obtained closed-loop can be assumed as a bilinear parameter-varying system [START_REF] Briat | Linear Parameter-Varying and Time-Delay Systems: Analysis, Observation, Filtering and Control[END_REF]. Thus, according to the development given in Section 3, if the eigenvalues of the matrix (19) are negative, consequently the closed-loop ( 15) is internally stable. The condition that ensures the stability of the matrix ( 19) is provided by Proposition 1.

Proposition 1: If τ ≥ 1/α 1 , the eigenvalues of the operator (19) are negative.

Proof. Since u 2 > 0, it remains to show that

y(t) -y d (t) + τ α 1 (x 2 (l, t) -x 1 (l, t)) τ ∂x 1 (z, t) ∂z z=l (20) is positive. Assumption 1 implies that ∂x 1 (z, t) ∂z z=l > 0 (21)
and since the time constant τ > 0, consequently the denominator of ( 20) is positive.

To have the condition (20) satisfied, it remains to show that the numerator of ( 20) is also positive, that is,

τ α 1 x 2 (l, t) -x 1 (l, t) > y d (t) -y(t) (22)
Now, if y d (t) -y(t) > 0, according to Remark 1, it follows that the condition (22) holds since both τ and α 1 are positive, consequently the numerator of (20) is positive.

If y d (t) -y(t) < 0, that is, y d (t) -x 1 (l, t) < 0, the numerator of ( 20) is positive if the following condition holds

τ α 1 x 2 (l, t) -x 1 (l, t) > y d (t) -y(t) (23)
According to Remark 1, we have

x 2 (l, t) > x 1 (l, t) (24)
and since

x 1 (l, t) ≥ y d (t) (25) therefore x 2 (l, t) -x 1 (l, t) > y d (t) -y(t) (26)
If we take τ such as τ α 1 ≥ 1, that is, τ ≥ 1/α 1 , then

τ α 1 x 2 (l, t) -x 1 (l, t) ≥ x 2 (l, t) -x 1 (l, t) (27)
and according to ( 26), one concludes that

τ α 1 x 2 (l, t) -x 1 (l, t) > y d (t) -y(t) (28)
which implies that the numerator of ( 20) is positive.

Remark 3: From Proposition 1, it follows that the internal stability condition is related to the choice of the tuning parameter τ of the control law (17).

Application example

In this section, the performance of the designed control law is evaluated in the case of the co-current heat exchanger depicted in Figure 1. A cold fluid enters at temperature T c,0 and flows through the internal tube, of length l, with a velocity v c . This cold fluid exchanges heat with the hot fluid that flows in the external tube, in the same direction as the cold fluid, with a velocity v h . The hot fluid enters at temperature T h,0 . The cold and hot fluids leave at temperatures T c,l and T h,l , respectively. The dynamic behavior of the co-current heat exchanger is described by the following system of one dimensional linear hyperbolic PDE energy balances [START_REF] Maidi | Boundary control of a parallel-flow heat exchanger by input-output linearization[END_REF] Cold fluid Hot fluid

0 l z T c,0 (t) T c (z, t) T c,l (t) T h,0 (t) T h (z, t) T h (z, t) T h,l (t) Figure 1: Co-current heat exchanger. ∂T h (z, t) ∂t = -v h ∂T h (z, t) ∂z + α h (T c (z, t) -T h (z, t)) (29) ∂T c (z, t) ∂t = -v c ∂T c (z, t) ∂z + α c (T h (z, t) -T c (z, t)) (30) T h (0, t) = T h,0 (t) (31) T c (0, t) = T c,0 (t) (32) T h (z, 0) = T 0 h (z) (33) T c (z, 0) = T 0 c (z) ( 34
)
where T h (z, t) and T c (z, t) are the temperatures of the hot and cold fluids, respectively. α h and α c are the heat transfer coefficients. T 0 h and T 0 c are the initial spatial temperature profiles of the hot and cold fluids, respectively.

Let us assume that the objective is to control the outlet cold fluid temperature, that is,

y(t) = T c (z, t)| z=l (35) = T c,l (t) (36)
by manipulating the velocity v c . T h,0 , T c,0 and v h represent external disturbances that affect the heat exchanger.

In this case, both assumptions 1 and 2 hold for the co-current heat exchanger since T c (z, t) is a monotonic increasing function with respect to z and, ∀ t ∈ [0, ∞), T h (z, t) -T c (z, t) > 0. Consequently, the proposed design methodology can be applied.

According to the development given in the Subsection 4.1, the following state feedback results

v c = T c,l (t) -T d c,l (t) + τ α c (T h (l, t) -T c (l, t)) τ ∂T c (z, t) ∂z z=l (37)
The performances are evaluated by simulation and the closed-loop system is simulated using the method of lines [START_REF] Vande Wouwer | Adaptive Method of Lines[END_REF] by assuming a number of discretization points equal to 100. The spatial derivatives are approximated using finite differences.

The parameters of the heat exchanger are summarized in Table 1. The initial temperature profiles T 0 h (z) and T 0 c (z) are the spatial profiles at steady-state obtained by considering the boundary conditions and the velocities reported in Table 2. The controller tuning parameter τ that satisfies the condition of Proposition 1 is taken equal to 6 s.

In the performed simulation runs, to avoid sudden variations of the controlled output T c,l (t), the desired set-point T d c,l (t) is smoothed using a first-order filter that yields the filtered set-point or reference trajectory T f d c,l (t), i.e. where τ f is the time constant of the filter taken equal to 5 s. Thus, considering the filtered set point T f d c,l (t) instead of the set point T d c,l (t), the control law (37) takes the following form

τ f dT f d c,l (t) dt + T f d c,l (t) = T d c,l (t) (38)
v c = T c,l (t) -T f d c,l (t) + τ α c (T h (l, t) -T c (l, t)) τ ∂T c (z, t) ∂z z=l (39)
Remark 4: According to proposition 1, the condition of the choice of τ ensures that both v h and v c have the same sign (both positive). This means physically that both cold and hot fluids flow in the same direction, which is characteristic of a co-current heat exchanger.

Set point tracking

The first simulation run deals with the tracking problem. Thus, in order to evaluate the tracking capability of the control law (39), two step set points T d c (t) = 50 • C and T d c (t) = 35 • C have been specified at t = 5 s and t = 60 s, respectively. Figure 2 shows clearly that the outlet cold fluid temperature T c,l (t) tracks perfectly the imposed set point. The output-tracking is achieved with a smooth evolution of the velocity v c (Figure 3). The performance of controller is confirmed by the 3D temperature profiles given by Figure 4.

Disturbance rejection

The second performed test deals with disturbance rejection. For this test, a sudden change of -60% of the hot fluid velocity v h is applied at t = 10 s. Figure 5 shows variations of the cold fluid temperature at some positions along the heat exchanger. From the obtained results, one clearly notices that the disturbance effect observed at z = 0 is attenuated and becomes invisible at the outlet z = 1 of the heat exchanger (Figure 5), that is, on the controlled variable. This attenuation is achieved by a slight variation of the cold fluid velocity v c as shown by Figure 6.

Parameter Value

α h 0.3 [s -1 ] α c 0.2 [s -1 ] l 1.0 [m]

Variable Value

T h,0 80 [ • C] T c,0 25 [ • C] v h 10 [m • s -1 ] v c 5 [m • s -1 ] 0
Remark 5: From a practical point of view, the velocity or flow rate variation is achieved by a valve that is characterized by a limited range. This means that some desired temperatures cannot be achieved if they need variations that exceed the specified range of the valve or an important inlet hot fluid temperature. Thereby, the analysis of the controllability and reachability properties of the heat exchanger is an important step to specify the set of the reachable set points. These questions, that have been investigated in the literature [START_REF] Alotaibi | Controllability of cross-flow heat exchangers[END_REF][START_REF] Sano | Observability and reachability for parallelflow heat exchanger equations[END_REF], are out of the scope of the present work.

Conclusion

The control of a linear 2 × 2 system of hyperbolic PDEs issued from first principles balances, based on the input-output linearization approach, is investigated in this paper. The manipulated variable is taken as a parameter of the system to control a boundary output. This control configuration transforms the system equations as a bilinear DPS. Then, based on the notion of characteristic index that is a generalization to the infinite dimensional systems of the well-known concept of relative degree, a control law, of finite dimensional nature, that ensures both tracking output and disturbance rejection is developed. This control law yields a linear time-varying 2 × 2 system of hyperbolic balance laws in closed loop. The stability condition of the closed-loop is derived based on the stability of the eigenvalues of the spatial operator in closed loop.

The tracking and disturbance rejection capabilities of the control law developed are evaluated through numerical simulation runs in the case of a co-current heat exchanger. The obtained results show that the stated feedback achieves good performance in both output tracking and disturbance rejections.

Figure 2 :

 2 Figure 2: Set point tracking: evolution of the outlet cold fluid temperature T c l (t).

]Figure 3 :

 3 Figure 3: Set point tracking: evolution of the cold fluid velocity v c (t).

Figure 4 :

 4 Figure 4: Set point tracking: 3D temperature profiles. Top: hot fluid temperature profile. Bottom: cold fluid temperature profile.

Figure 5 :]Figure 6 :

 56 Figure 5: Disturbance rejection: effect of the disturbance v h (t) on the cold temperature T c (z, t) at some positions in the internal tube.

Table 1

 1 Heat exchanger parameters.

Table 2

 2 Boundary conditions and steady-state velocities.

  ). P. K. Gundepudi and J. C. Friedly. Velocity control of hyperbolic partial differential equation systems with single characterstic variable. Chemical Engineering Science, 53(24):4055-4072, 1998. E. M. Hanczyc and A. Palazoglu. Sliding mode control of nonlinear distributed parameter chemical processes. Industrial & Engineering Chemistry Research, 34(2): 557-566, 1995. A. Isidori. Nonlinear Control Systems. Springer-Verlag, New York, 1995. H. X. Li and C. Qi. Modeling of distributed parameter systems for applications-A synthesized review from time-space separation. Journal of Process Control, 20 (8):891-901, 2010. A. Maidi and J. P. Corriou. Boundary control of nonlinear distributed parameter systems by inputoutput linearization. In the Proceedings of the 18th IFAC World Congress, Milan, Italy, pages 10910-10915, August 28-September 02, 2011. A. Maidi and J. P. Corriou. Distributed control of nonlinear diffusion systems by input-output