Régent Benoît
email: benoit.regent@univ-smb.fr

Favre-Bonté Véronique
email: veronique.favre-bonte@univ-smb.fr

Da Fonseca
email: marie.da-fonseca@univ-perp.fr

PROJETS DE TERRITOIRE STRATEGIC MANAGEMENT AS A KEY TO READING THE DEVELOPMENT OF TERRITORIAL PROJECTS

Keywords: Territoire, projet de territoire, management stratégique, parties prenantes, ressources et compétences Territory, territorial project, strategic management, stakeholders, resources and skills

Une logique de rivalité croissante s'inscrit entre les territoires qui sont amenés à devenir plus attractifs et plus compétitifs. Cette logique de compétition amène les décideurs publics à mettre en place des stratégies et à développer des projets de territoire. Le développement de ces projets de territoire interroge sur les cadres conceptuels mobilisables pour appréhender les actions des parties prenantes du territoire. Dans cette perspective, nous proposons un cadre d'analyse de la stratégie développée à l'échelle des territoires. L'originalité de notre article est de transposer des cadres théoriques, issus du management stratégique, au niveau territorial pour comprendre le développement d'un projet de territoire, projet qui est aujourd'hui un enjeu majeur de leur attractivité.

Introduction

In the era of globalisation, the mobility of factors of production and goods is increasing and, as borders open up and means of transport develop, an increasingly competitive logic [START_REF] Camagni | Compétitivité territoriale : la recherche d'avantages absolus[END_REF] is taking hold between territories (nations, regions, départements, communes, towns, etc.). In this context, territories are led to become more attractive and competitive, and to develop strategies to differentiate themselves in order to understand the elements necessary for their development, such as production factors, capital and know-how. Managerial discourse now has a growing role in the discourse of public decision-makers and the contribution of management to the structuring and development of territorial dynamics has become widely recognised (Zardet and Nougera, 2013). Therefore, public decision-makers increasingly feel the need to unite a wide range of territorial actors around common initiatives with a recurring strategic intention to sustainably strengthen the attractiveness of their territory (Soldo, Arnaud, Keramidas, 2013). In order to respond to the challenges of attractiveness and competitiveness, the actors will then carry out collective actions which usually take the form of a project. This in turn builds connections between them and with their space and organizes their actions.

After being studied for a long time in management sciences through inter-organisational networks, the territory has been set up as an actor [START_REF] Aydalot | L'aptitude des milieux locaux à promouvoir l'innovation[END_REF][START_REF] Assens | Le réseau d'entreprises : vers une synthèse des connaissances[END_REF] who acts through projects and has become a central variable for understanding the development of spaces. The issues underlying these projects go beyond the political dimension alone and include a managerial dimension that remains understudied to this day [START_REF] Gob | Enjeux gestionnaires et politiques des projets territoriaux[END_REF]. Indeed, these projects invite a broader reflection on the strategy developed on the territorial scale and on the conceptual frameworks (Amblard, Berthomé, Houdart, Lardon, 2018) that would make it possible to understand the complexity of collective actions, particularly through territorial projects. Although management tools have gradually entered into territorial action, to our knowledge, the contributions of strategic management to the development of territorial projects remain very little developed. However, strategic management tools could help to ensure that territorial projects lead to real territorial development strategies.

The territory can be perceived as an organisation [START_REF] Casteigts | Le management territorial stratégique[END_REF] which has resources and competences and is composed of interrelated actors. For [START_REF] Casteigts | Le management territorial stratégique[END_REF], "to speak of territorial management is to take note of the status of the territory as a collective actor and as a global organisation, and to associate the consideration of economic and social dynamics with a strategic conception and implementation of public action" (p. 10). Strategic management therefore offers analytical frameworks that allow us to understand the strategies of actors at work in a territory. More particularly, the theory of resources [START_REF] Wernerfelt | A Resource-based View of the firm[END_REF] enriched by the skills approach [START_REF] Barney | Firm resources and sustained competitive advantage[END_REF][START_REF] Hamel | Competence-based competition[END_REF] and the theory of stakeholders [START_REF] Freeman | Strategic Management: A Stakeholder Approach[END_REF] are theoretical frameworks that seem to be entirely appropriate for studying the territorial project and for understanding how it can become a factor of differentiation and sustainable strengthening of attractiveness. Moreover, some authors underline the need for strategic thinking on the development of a competitive advantage and on the allocation of resources and competences on a territorial scale [START_REF]Comprendre le territoire : les apports du management stratégique[END_REF][START_REF] Pupion | Une recherche en management stratégique public en quête d'un cadre théorique[END_REF]. In this context and in order to address these issues, we have formulated the following research question: how can an integrative theoretical framework, derived from strategic management, be mobilised to model the development of a territory project?

The objective of the article is to review the state of the art of strategic management tools by demonstrating those that can be useful for the analysis of territorial projects and ultimately for public decision makers. We therefore propose to approach the notion of the territorial project from a strategic angle in order to identify the managerial tools capable of supporting public or non-public decision-makers in the development of their territorial project. We propose a model which offers a simplified but actionable representation of the development process of a territory project. This type of research is in line with [START_REF] Chamard | La place du marketing territorial dans le processus de transformation territoriale[END_REF] or [START_REF] Colletis | Révélation des ressources spécifiques territoriales et inégalités de développement[END_REF].

After exposing the different conceptions of territory in management sciences and presenting the notion of territory project (1), we will show how a theoretical framework combining the theory of stakeholders and the theory of resources enriched with the approach by competences, proves to be relevant to study the territory and its projects (2). We will conclude by proposing a model of territorial project development. Our objective is to show how an integrative theoretical framework, derived from strategic management, can be adapted and mobilised to model the development of a territory project.

THE NOTION OF TERRITORY IN MANAGEMENT SCIENCES

The notion of territory has grown in importance in the literature, especially in management sciences, where each year more and more articles, workshops and conferences are devoted to it. However, managerial analysis has long considered territory as a simple adjustment variable: "whether we are interested in competition or competitiveness, the role of geographical location is almost absent. (...) Globalisation allows companies to find capital, goods and technology anywhere, and to locate their facilities wherever it will be most profitable" (Porter, 2004, p. 205).

However, many authors now agree that territory in management science emerged with globalisation [START_REF] Longhi | Le développement local[END_REF][START_REF] Casteigts | Le management territorial stratégique[END_REF][START_REF] Pelissier | Étude sur l'origine et les fondements de l'intelligence territoriale : l'intelligence territoriale comme une simple déclinaison de l'intelligence économique à l'échelle du territoire ?[END_REF][START_REF]Comprendre le territoire : les apports du management stratégique[END_REF]) and would even be its opposite [START_REF] Lamara | Les deux piliers de la construction territoriale : coordination des acteurs et ressources territoriales[END_REF]. On the one hand, political factors, such as deconcentration-decentralisation movements and the launch of European regional policies, have led territories to conceive local action in a strategic, collective and concerted manner [START_REF] Casteigts | Le management territorial stratégique[END_REF] and have helped to give them a certain legitimacy in the economic and social sphere [START_REF] Pelissier | Étude sur l'origine et les fondements de l'intelligence territoriale : l'intelligence territoriale comme une simple déclinaison de l'intelligence économique à l'échelle du territoire ?[END_REF]. On the other hand, globalisation puts territories, and no longer nations, at the heart of firms' location strategies. Consequently, for the choice of where to locate companies, territories constitute a key strategic variable. "For global firms, territories are no longer substitutable, since it is the nature of locally produced skills, the functioning of local markets and the nature of the collective learning they allow, and the information they convey, that dictate the choices of firms". (Longhi and Spindler, 1999, p. 19). The territory is then a variable that the company must integrate into its strategic formulation. The territory is therefore no longer neutral and understood as a simple receptacle, but becomes a strategic variable that facilitates the integration of the company into its immediate and more distant environment [START_REF] Torre | Les AOC sont-elles des clubs ? Réflexions sur les conditions de l'action collective localisée, entre coopération et règles formelles[END_REF].

The notion of territory is therefore used more in the managerial literature to understand spatial phenomena. Some authors study the territory in terms of its resources [START_REF] Benko | Les ressources de territoires[END_REF][START_REF] Senil | De l'Evènement Culturel en Régie Directe à l'Attractivité Durable du Territoire : Quelles Conditions Managériales ?[END_REF][START_REF] Landel | Patrimoine et territoire, les nouvelles ressources du développement[END_REF][START_REF] Dissart | Co-construction des capacités et des ressources territoriales dans les territoires touristiques de montagne[END_REF]Chia et al., 2015) while others focus on the relationships developed between actors [START_REF] Loilier | La configuration des réseaux d'innovation une approche par la proximité des acteurs[END_REF][START_REF] Chia | Conclusion : Vers une « technologie » de la gouvernance territoriale ! Plaidoyer pour un programme de recherche sur les instruments et dispositifs de la gouvernance des territoires[END_REF][START_REF] Raveyre | L'entreprise fabrique de réseaux territoriaux ou comment équiper les relations inter organisationnelles ?[END_REF]Gallois, Lecat and Nieddu, 2016). In order to understand this polysemous notion better, we first propose an overview of the notion of territory in management sciences, perceived by some (1. 1) as a geographical space that provides resources [START_REF] Elidrissi | Systèmes régionaux d'innovation : superpositions et articulations[END_REF], a sort of "heritage of specific resources" (Lauriol, Perret and Tannery, 2008, p. 186), and by others (1.2) as a symbolic and relational space [START_REF] Raulet-Croset | La dimension territoriale des situations de gestion[END_REF]) made up of stakeholders, their interrelations and representations. The territory is also sometimes, but rarely, perceived as the sum of the two [START_REF] Loilier | Innovation et territoire[END_REF], i.e. an encompassing structure that houses a set of resources, various actors and forms of coordination and representation (1.3). Finally, we will conclude with a presentation of the territorial project, understood as a unique process that can be assimilated to a collective action, carried out by a varied group of actors which aims to resolve new productive problems (1.4).

The territory, a space that provides resources

Territories (countries, regions, departments, cities) can be defined by their physical characteristics and seen as 'receptacles' for numerous and varied resources. In fact, in a first more physical and economic conception, the territory is understood as a geographical and physical space with its own resources, infrastructures and borders. The territory can then be described as an administratively delimited 'place', endowed with given resources whose competitiveness is essentially linked to the existence of a cost advantage [START_REF] Loilier | Innovation et territoire[END_REF]. The territory is then a data for the actors on which they do not seem to be able to act and where they only have loose relations [START_REF] Loilier | Innovation et territoire[END_REF]. The territory is then "a simple geographical location with directly exploitable physical or economic characteristics" (Torre, 2002, p. 16). The notion of asset differs from the notion of resource in that it constitutes an 'activated' factor, whereas the resource is a latent factor, to be exploited [START_REF] Colletis | Révélation de ressources spécifiques et coordination située[END_REF]. The resource therefore encompasses a wider set of factors that can explain what differentiates the territories. This notion is then privileged in the managerial literature to understand the territories and their strategies. Moreover, it allows us to understand the processes of revelation and valorisation of new factors resulting from the specificity of the spaces within the territory projects.

In this first acceptance, the territory is therefore a place that provides resources that can take different forms and be grouped into two categories: generic resources and specific resources. A resource will be qualified as generic if it is easily transposable from one territory to another and as specific when it is not [START_REF] Gumuchian | La ressource territoriale[END_REF]. The specific resources will allow territories to differentiate themselves and can be qualified as territorial under certain conditions. The territorial resource can be understood as "a specific resource revealed according to an intentional process, engaging a collective dynamics of appropriation, different according to whether or not it takes the path of market valuation" (François, Hirczak and Senil, 2005, p. 4). For Corrado (2004, p. 23), "a territorial resource represents the discovery and the actualisation of a latent value of the territory by a part of a human society that recognises and interprets it as such, inside a local development project". For Gumuchian and Pecqueur (2007, p. 5), it is "a constructed characteristic of a specific territory and this, in a development perspective". The territorial resource is therefore the result of construction by the actors with a view to local development and particularly within the framework of projects.

This conception of the territory, through resources, leads to consider it as a space where a great number of diverse actors can coordinate to reveal and develop intransferable territorial resources. Its dynamism and development capacities will then be explained by how much the actors will reveal, develop and activate the resources given to them. The challenge is therefore to succeed in combining the strategies of the actors to reveal, develop and activate the resources that can then be qualified as territorial. For Mendez and Marcier (2006), the capacity of a territory to endure comes from its ability to regenerate, to absorb external resources and to develop territorial resources that are intransferable. In a context of global competition [START_REF] Camagni | Compétitivité territoriale : la recherche d'avantages absolus[END_REF], understanding the territory as a set of specific resources can help to understand how a territory can differentiate itself from others in the long term and attract the necessary factors (capital, labour, etc.) for its development.

Companies looking to exploit territorial resources that are valued by the customer or user will then seek to establish themselves in a territory on a long-term basis so that they can exploit the specific resources or assets available locally to solve productive problems in a unique way [START_REF] Lasswell | L'espace et le territoire dans l'agenda de recherche en stratégie[END_REF]. This in turn will allow organisations that set up in a territory to obtain a sustainable competitive advantage, which becomes so when the organisation develops a value-creating strategy that is not simultaneously deployed by another competitor [START_REF] Barney | Firm resources and sustained competitive advantage[END_REF]. In the context of a territory, this notion can be compared to that of 'territorial rent' (Lamara, 2008) or 'sustainable attractiveness' [START_REF] Arezki | Le projet de territoire : une innovation managériale au service de l'action des organisations publiques territoriales[END_REF]. Indeed, the valued specific resources produce a rent which results from the difference between two values or prices that can be observed on the market of the goods and services considered (Mollard, 2001), which will then give the territory a sustainable competitive advantage.

Consequently, "the main factor of differentiation of areas may not result from the relative price of factors or transport costs, but from the potential supply of specific assets or resources, which by definition cannot be put in direct competition on a market" (Colletis and Pecqueur, 1993, p.491). The most resilient territories would then be those in which "companies would be best able to recombine the productive resources at their disposal, when their sectors of specialisation are in decline, in order to remobilise them in other sectors" (Albert-Cromarias and Asselineau, 2016, p.18). The challenge for actors is then to identify resources that are or can become strategic [START_REF] Weppe | Ressources stratégiques, ressources ordinaires et ressources négatives[END_REF] and to renew or activate them.

The territory is therefore a space that provides specific resources that can enable companies that set up in it to differentiate themselves. However, the territory cannot be reduced to a stock of resources; it must also be considered within its own dynamics and in that of its actors.

The territory, a symbolic and relational space

The territory can therefore be understood through its resources, but the dynamics between the actors are essential to understand its development. It is then studied through the interface of the relationships established between the actors and perceived as a social construction. In this more socio-cultural second conception, the territory is considered as an organised space. It can then be described as a link [START_REF] Loilier | Innovation et territoire[END_REF], that is socially delimited and endowed with constructed resources. The territory, whose borders are not geographically delimited, is then studied with regard to the actors and their interrelationships. The link is more important than the place, which means that the actors create the territory from nothing and it is not defined by its physical and spatial characteristics. In this acceptance, the territory has at least two dimensions: a symbolic dimension and a constructed or relational dimension (Vaesken & Zafiropoulou, 2008).

In its symbolic dimension, the territory corresponds to a set of more or less positive representations and images that we have of this space (Meyronin, 2015) and is perceived in its identity component [START_REF] Alaux | Le marketing territorial des Petits et Moyens Territoires : identité, image et relations[END_REF]Rochette, Zumbo-Lebrument and Féniès, 2016). In the main, authors in territorial marketing [START_REF] Benko | Marketing et territoire[END_REF][START_REF] Meyronin | Marketing territorial Enjeux et Pratiques[END_REF]Noisette and Valérugo, 2010) use this approach to understand the territory.

In its relational dimension, the territory is similar to a complex set of actors who organise themselves. It is understood as an organisation combining a location, a cultural heritage, a process of appropriation of space by a group that is aware of an identity, and a management and development process [START_REF] Gillet | Donner collectivement du sens à l'avenir touristique d'un territoire[END_REF]. This organisation is weakly structured and can be compared to a network, in particular because of the cooperative and non-hierarchical inter-organisational relationships that it allows to develop [START_REF] Josserand | Le pilotage des réseaux : fondements des capacités dynamiques de l'entreprise[END_REF][START_REF] Zardet | Distance spatiale et cognitive entre acteurs impliqués dans file management d'un territoire[END_REF]. Indeed, a particularity of the territory is the absence of hierarchical links between actors who are legally independent [START_REF] Zardet | Cas de recherche-intervention de trois territoires-pilotes d'une Région[END_REF]. For some authors, in the current context of globalised competition, efficiency is above all relational: the relational ecosystem has therefore become the key factor of success for territories (Veltz, 2005, p. 45). The territory becomes the privileged supplier of these relational resources, and thus reinforces the capacity of actors to share information and to cooperate in carrying out collective actions. This conception differs from the previous one because the territory is first and foremost the product of the actors' logic. [START_REF] Leloup | Le développement territorial et les systèmes complexes : proposition d'un cadre analytique[END_REF] highlights four dynamics specific to territorial development: the coordination of actors, spatial structuring, the creation of regulation and the implementation of a project. Here the territory is "a construct, resulting from the strategies of actors considered as real stakeholders" (Albert-Cromarias and Asselineau, 2016, p. 6). A territory exists when "actors in a given geographical context meet, identify and formulate a common productive problem and set up a project to solve it" (Pecqueur, 2015, p. 20). The territory can therefore be understood as a complex system, resulting from a feedback loop between a set of actors and the geographical space that they use, develop and manage [START_REF] Moine | Le territoire comme un système complexe : un concept opératoire pour l'aménagement et la géographie[END_REF].

These interactions between a great diversity of actors will lead to the construction of a form of local or territorial governance which allows "thinking upstream of territorial trajectories" and which is "a process of confrontation and adjustment of both systems of representations and actions of groups of geographically close actors" (Mendez and Mercier, 2006, p. 253). The aim of this governance is to make public policies consistent and to coordinate the strategies of public and private partners. For Piraux et al ( 2017) "territorial governance is the product of these coordinations between actors with different objectives, resources and logics but sharing a common project: the construction or consolidation of a territory" (p. 3). Governance should be taken into consideration from a strategic perspective to understand the general orientations of the community grouped around a project. It is above all a question of the methods of construction and implementation of the project, with a particular interest in the involvement of stakeholders in the definition of strategic axes and in the operationalisation of the project (Carassus and Baldé, 2020).

As a construct, the territory is the result of actors who meet, identify and formulate common problems and set up projects to solve them [START_REF] Asselineau | Compétences territoriales et cycle de vie du territoire, socles de la réindustrialisassions ?[END_REF]. As the various actors in the territory do not necessarily share the same representations or interests, competition, rivalry and conflicts arise [START_REF] Mendez | Compétences-clés des territoires. Le rôle des relations inter organisationnelles[END_REF]. In this sense, it is both a space of conflict and discussion where taking into account the interests of different stakeholders is central to decisionmaking.

This vision of the territory raises questions like: How can the stakeholders of a territory be identified (to help in decision-making)?, or What are the resources and skills that can be mobilised on a territorial scale? One way of answering is to consider the territory both as a space that provides resources and also a coordination space for the various stakeholders.

The territory, a structure that encompasses a set of interrelated resources and actors

In a third conception, which reconciles the previous two, the territory is understood as a physical space (the place) and as a set of relations, symbols and projects generated by the actors (the link). It is then an all-encompassing structure that houses a set of resources and actors. [START_REF] Loilier | Innovation et territoire[END_REF] speaks of an "integrated territory". In this acceptance, the actors and the territory are built together by influencing each other. This type of integrated territory is then "rather efficient in terms of innovation thanks to the territorial governance associated with it" but also "thanks to the combination of the two dimensions of the territory (link and place, beyond the mobilisation of the different proximities associated with them) which seems favourable to innovation" (Loilier, 2010, p. 31).

The territory is considered as an actor and an active development structure, which emerges from "the institutionalisation of the convergence between the strategies of different actors" to build a "collective strategy of territory" (Asselineau and Cromarias, 2010, p. 161). For some authors, it is "an active actor who can reactivate organisational capacities and technical skills" (Aydalot, 1986, p. 3). For [START_REF] Casteigts | Le management territorial stratégique[END_REF], if the territory can be considered as a collective actor, it is then necessary to consider a strategic territorial management, because as a global organisation, the territory can come under management. Territorial management can then be seen as a specific form of strategic management, implemented by territorial organisations to deploy a coherent project [START_REF] Hernandez | Paradoxes et management stratégique des territoires : étude comparée de métropoles européennes[END_REF]. The territory is therefore not considered as a simple framework or support, but as a collective actor, which makes it a complex object of analysis. The territory is then a space that is appropriated by its inhabitants, a bearer of identities and exchanges, which functions as a loop between the actors that shape it and that it shapes (Vaesken & Zafiropoulou, 2008). The territory is therefore not neutral; on the contrary, it actively contributes to development. It is therefore a form of organisation that requires its resources and skills to be strategically managed and the interests of its stakeholders balanced in order to succeed in its development

The territory then constitutes a space for collective projects and structuring global/local relations and pooling of actors and resources [START_REF] Arezki | Le projet de territoire : une innovation managériale au service de l'action des organisations publiques territoriales[END_REF]. Based on the work of [START_REF] Girin | Analyse empirique des situations de gestion : éléments de théorie et de méthode[END_REF], [START_REF] Chia | Conclusion : Vers une « technologie » de la gouvernance territoriale ! Plaidoyer pour un programme de recherche sur les instruments et dispositifs de la gouvernance des territoires[END_REF] describe the territory as a place where "management situations" exist, in which "a set of interacting activities is associated with the idea of collective activity and a result that is subject to judgment" (p. 173). The aim of management sciences is therefore to develop tools that enable "an actor or a group of actors to analyse the processes in which their actions take place and to anticipate their possible evolution" (p. 173). For this purpose, many authors recommend implementing territorial governance [START_REF] Gilly | Enchevêtrement des espaces de régulation et gouvernance territoriale. Les processus d'innovation institutionnelle dans la politique des Pays en France[END_REF][START_REF] Leloup | La gouvernance territoriale comme nouveau mode de coordination territoriale ?[END_REF][START_REF] Chia | Innovations et gouvernance territoriale : une analyse par les dispositifs[END_REF]Torre, 2011;[START_REF] Chia | Comment analyser la gouvernance territoriale ? Mise à l'épreuve d'une grille de lecture[END_REF] that must be based on the two pillars of territorial construction, which are specific resources and the coordination of various actors [START_REF] Lamara | Les deux piliers de la construction territoriale : coordination des acteurs et ressources territoriales[END_REF].

The territory can therefore be considered as an organisation with resources and a set of coordinated actors. Strategic management literature seems to offer a solid theoretical basis to examine the territory. However, understanding it also means studying territorial projects in order to understand how they can contribute to differentiate a territory and make it more attractive and competitive.

The territorial project in the territorial development strategy

The notion of project is widely used today in various contexts. It can be defined in management as: "a collective creation, organised in time and space, with a view to a demand" (Ecosip, 1993, cited by Garel, 2012, p. 18). The Association Francophone du Management de Projet (AFITEP) defines it as: "a single process, consisting of a set of coordinated and controlled activities with start and end dates, undertaken with the aim of achieving an objective in accordance with specific requirements" (AFITEP, 2002, quoted by Garel, 2012, p.15).

A project is therefore a single process, part of an organisation's overall strategy, bounded by time, cost and resource constraints, and which can be broken down into a set of tasks. In any project, therefore, three types of constraints are taken into account simultaneously: time, resources and technical specifications.

The notion of a territorial project emerged in France at the end of the 1980s and was seen as a "tool that enables coordinated actors to provide solutions to new productive problems on a subnational scale" (Pecqueur, 1989, p. 15). This concept has gradually become an essential reference for public action [START_REF] Koop | Pourquoi croire au modèle du développement territorial au Maghreb ? Une approche critique[END_REF][START_REF] Arab | Activité de projet et aménagement urbain : les sciences de gestion à l'épreuve de l'urbanisme[END_REF]. For some authors, it is "the occasion for new forms of public debate involving actors, the state, local authorities, companies, associations and services that express contradictory logics, based on different objects" (Landel and Senil, 2009, p. 13). The territorial project is therefore "everyone's business" (Mauco, Peyrache-Gadeau and Raymond, 2005). [START_REF] Behar | Les nouveaux territoires de l'action publique[END_REF] emphasises the shift for the French public authorities "from a counter logic to a project logic" (p. 5). In fact, according to him, "the government clearly expresses its desire to see the direction of territorial regulation reversed, in favour of a bottom-up approach: the local project takes precedence" (p. 5). This desire is enshrined in the formula: "one territory, one project, one contract" (p. 5). The project covers two dimensions: the outline of an idea or an action (the drawing), and an objective for the future (the design). Indeed, "essentially, a project is defined as a specific, new action that methodically and progressively structures a future reality for which we do not yet have an exact equivalent" (Le Bissonnais, 2000, p. 2). For [START_REF] Arab | Activité de projet et aménagement urbain : les sciences de gestion à l'épreuve de l'urbanisme[END_REF], a territorial project sets out strategic development guidelines for the medium or long term in economic, social, cultural and spatial terms: "the territorial project is therefore not bound by fixed temporal limits and deadlines" and its "purpose is to serve as a guide for operational action [...] necessarily initiated by local public authorities" (p. 148).

To summarise, a territorial project can be understood as a unique process that can be assimilated to a collective action which is led by a varied group of actors and aims to resolve new productive problems. It is characterised by its inclusion in a long-term horizon and by a logic of sustainable development. The territorial project as an instrument of strategic planning [START_REF] Maurel | Signes, Données et Représentations Spatiales : Des éléments de sens dans l'élaboration d'un projet de territoire intercommunal : Application au territoire de Thau[END_REF] is therefore based on a network of coordinated actors [START_REF] Pecqueur | Le développement local[END_REF] and on its specificities which can be considered as territorial resources [START_REF] Corrado | Vers un concept opératoire : la ressource territoriale[END_REF]. The territorial project must therefore deal with the multiplicity of public, private and civil actors [START_REF] Lardon | Chaîne d'ingénierie territoriale : diversité des acteurs dans la conduite d'un projet de territoire[END_REF] and the diversity of resources and competences perceived differently by the actors.

In order to understand a territory project from a management science perspective, we propose a framework from strategic management combining stakeholder theory and resource theory enriched by the skills approach.

STRATEGIC MANAGEMENT TO UNDERSTAND THE TERRITORY PROJECT

In order to cope with the changes that territories are facing in terms of urban planning, employment, environment and economy, the territorial project has gradually become an instrument and a support for territorial mobilisation [START_REF] Gob | Enjeux gestionnaires et politiques des projets territoriaux[END_REF]. By promoting collective action, the project becomes a means of establishing new interactions between actors and public decision-makers and of co-constructing solutions to the challenges they face.

The project thus becomes a response to the difficulty posed by the heterogeneity of actors in seeking collective mobilisation on the territory [START_REF] Mendez | Compétences-clés des territoires. Le rôle des relations inter organisationnelles[END_REF] and is used as an instrument for coordinating collective action on a territory. To analyse the management of a territory project and grasp the complexity of the interactions between actions and the territory, it is necessary to rethink the conceptual frameworks that allow us to understand it [START_REF] Amblard | L'action collective dans les territoires. Questions structurantes et front de recherche[END_REF]). To do this, we propose to combine the stakeholder theory and the Resource Based View (RBV) enriched with the competence approach.

We show here how the stakeholder theory (2.1), as well as the resource theory and the competence approach (2.2) can be mobilised in order to study the territory and its projects. We then build a theoretical framework integrating these theories (2.3) before presenting a general model for the development of territorial projects (2.4).

Stakeholder theory and territory

One of the essential characteristics of territorial projects is their participatory dimension [START_REF] Soldo | L'évaluation stratégique : une démarche qui favorise la gestion démocratique des projets de territoire[END_REF]. The mobilisation of numerous and diverse stakeholders within the framework of a territorial project leads us to question their organisation, according to the logics of influence, power and participation. Indeed, collective action is perceived in a project logic that is organised around the relationships of the stakeholder, which are the key to the collective project's success or failure. It is therefore important to have a very clear vision of the positions of the actors within the project. "Stakeholder theory allows us, by analogy, to reposition this dynamic of actors around the collective project according to an articulation based on the degree of adhesion/opposition and the level of involvement" (Vaesken & Zafiropoulou, 2008, p. 9).

The advantage of stakeholder theory is that it offers tools to identify individuals and groups of individuals who are or may be affected by the actions of the organisation, as well as their motivations and leverage. The objective of this sub-section is to identify which of the stakeholder typologies could be applied to territorial projects.

Many definitions and typologies of stakeholders exist which can be classified into two main categories: by proximity and by power.

The classifications -by proximity- [START_REF] Clarkson | A Stakeholder Framework for Analyzing and Evaluating Corporate Social Performance[END_REF][START_REF] Carroll | Business and society: Ethics and Stakeholder Management[END_REF]Carroll and Näsi, 1997;[START_REF] Dowling | Creating corporate reputations: Identity, Image and performance[END_REF]) emphasise the relationship between the actors. The literature on proximity has developed considerably over the last few decades and a fundamental distinction is made between what is physical proximity and what is organisational or relational proximity [START_REF] Gilly | Proximity relations. Elements for an analytical framework[END_REF]. The first form of proximity -physical or spatial-can be defined as follows: "two units, individuals or organisations can be considered as being close if, given the state of the means of transport, they can meet or exchange at low cost and/or quickly" (Rallet, 2002). The second form "corresponds to a high degree of interactivity between the actors of the exchange, individuals or organisations. The contacts are numerous, frequent and take place over time" [START_REF] Gilly | Proximity relations. Elements for an analytical framework[END_REF]. This organisational or relational proximity can be compared to certain classifications of stakeholders (e.g. taking into account the nature of the relationship (contractual or not; [START_REF] Carroll | Business and society: Ethics and Stakeholder Management[END_REF], or the voluntary exposure to risks; [START_REF] Clarkson | A Stakeholder Framework for Analyzing and Evaluating Corporate Social Performance[END_REF]. It can also be compared to those based on the level of commitment of the stakeholder (Carroll and Näsi, 1997;[START_REF] Dowling | Creating corporate reputations: Identity, Image and performance[END_REF], which can be observed through both the degree of involvement and the responsibilities undertaken. Relational proximity can then be specified in terms of frequency, length of time, diversity or strength of interactions (Dibblle, Levine and Park, 2012), and also commitment (such as the desire to maintain a relationship with the partner and to voluntarily expose yourself to risks - [START_REF] Clarkson | A Stakeholder Framework for Analyzing and Evaluating Corporate Social Performance[END_REF][START_REF] Girard | Pour une cartographie des parties prenantes fondée sur leur engagement : une application aux sociétaires d'une banque mutualiste française[END_REF].

All these stakeholder typologies converge on the characterisation of the physical or organisational interactivity between the actors.

In order to understand the links between stakeholders within a territory project, it seems interesting to us to complete these two types of proximity (physical and organisational) with the typology of proximities proposed by Bouba-Olga and Grosseti (2008) and widely used in the literature [START_REF] Talbot | La dimension politique dans l'approche de la proximité[END_REF][START_REF] Balland | Proximity and innovation: From statics to dynamics[END_REF], particularly to study territory projects [START_REF] Beaurain | L'écologie industrielle comme processus de développement territorial : une lecture par la proximité[END_REF][START_REF] Lardon | Construire un projet territorial : Le" jeu de territoire", un outil de coordination des acteurs locaux[END_REF][START_REF] Chia | Comment analyser la gouvernance territoriale ? Mise à l'épreuve d'une grille de lecture[END_REF]). These authors distinguish two ways of evaluating what brings these individuals together by looking at their levels. The first is based on an analysis of the resources available to them, independently of their organisational mode and their possibility of coordination. The second is based on what brings the individuals together by analysing the obstacles and levers to their coordination. The typology then splits the proximity into two subcategories: resources (with cognitive resources on the one hand and material resources on the other) and coordination (with relational coordination via social networks on the one hand and coordination via the media on the other). The authors then identify five main forms of proximity grouped in figure 1.

These different combined forms of proximity will make it possible to characterise the relations and interactions between stakeholders in order to measure their involvement in the development of the project [START_REF] Ingham | Proximité entre parties prenantes et alignement stratégique. Le cas de Triodos Belgique[END_REF]. The classifications [START_REF] Clarkson | A Stakeholder Framework for Analyzing and Evaluating Corporate Social Performance[END_REF]Caroll, 1996;Caroll and Näsi, 1997;[START_REF] Dowling | Creating corporate reputations: Identity, Image and performance[END_REF] can then be used to measure the organisational proximity between the stakeholders of the territory project.

Mediation

Links based on media and intermediaries

Adapted from Bouba-Olga and Grosseti ( 2008) Classifications based on stakeholder power [START_REF] Gardner | Handbook of strategic planning[END_REF][START_REF] Mitchell | Toward a Theory of Stakeholder Identification and Salience: Defining the Principle of Who and What Really Counts[END_REF][START_REF] Chabault | L'apport de la théorie des parties prenantes à la gouvernance des pôles de compétitivité[END_REF]) are based, for example, on the degree of power and dynamism [START_REF] Gardner | Handbook of strategic planning[END_REF] or on the degree of power, urgency and legitimacy [START_REF] Mitchell | Toward a Theory of Stakeholder Identification and Salience: Defining the Principle of Who and What Really Counts[END_REF]. Classifications that relate to power can identify stakeholders who pose a threat to the organisation and with whom relationships should be prioritised over time. These classifications can be used to study the operational governance of a territory network to determine the order of strategic priorities [START_REF] Chabault | L'apport de la théorie des parties prenantes à la gouvernance des pôles de compétitivité[END_REF]. The interest is also to make managers pay particular attention to the stakeholders' sources of influence, legitimacy, degree of involvement and the resources they bring [START_REF] Chabault | L'apport de la théorie des parties prenantes à la gouvernance des pôles de compétitivité[END_REF], to which we can add competences. Here, power is seen as an attribute that enables the person who possesses it to achieve his or her ends, despite possible resistance.

Mitchell, Agle and Wood (1997) identify three types of power: coercive, based on force and constraint, i.e. the capacity of a stakeholder to impose its decisions in a project, and utilitarian, based on the material and immaterial resources as well as the skills that the stakeholder has and uses for the project. The perceptive value of these resources and skills will then influence the level of power of this stakeholder. The third type of power is normative, which is based on symbolic resources such as prestige, notoriety and esteem. For [START_REF] Mitchell | Toward a Theory of Stakeholder Identification and Salience: Defining the Principle of Who and What Really Counts[END_REF], the notions of power and legitimacy partially overlap. We therefore propose here to understand legitimacy as an attribute of power.

In the framework of territorial projects, we have chosen to classify stakeholders by focusing on the two non-exclusive and complementary dimensions of proximity and power. We focus on the five forms of proximity identified by Bouba-Olga and Grosseti (2008) and presented above, and on the coercive, utilitarian and normative dimensions of power identified by [START_REF] Mitchell | Toward a Theory of Stakeholder Identification and Salience: Defining the Principle of Who and What Really Counts[END_REF], adding the legitimacy of the stakeholder. To understand power, we also propose to take into account the resources and skills contributed by the stakeholder to the project, their nature and also the perception of the value of these resources and skills by the other stakeholders (Figure 2).

Figure 2: Stakeholders as perceived through power

Weak power

Strong power

Coercive power

Weak Strong [START_REF] Mitchell | Toward a Theory of Stakeholder Identification and Salience: Defining the Principle of Who and What Really Counts[END_REF] Utilitarian power Weak Strong [START_REF] Mitchell | Toward a Theory of Stakeholder Identification and Salience: Defining the Principle of Who and What Really Counts[END_REF] Normative power Weak Strong [START_REF] Mitchell | Toward a Theory of Stakeholder Identification and Salience: Defining the Principle of Who and What Really Counts[END_REF]

Legitimacy

Weak Strong [START_REF] Mitchell | Toward a Theory of Stakeholder Identification and Salience: Defining the Principle of Who and What Really Counts[END_REF] R&C provided Unique and common Many and rare [START_REF] Chabault | L'apport de la théorie des parties prenantes à la gouvernance des pôles de compétitivité[END_REF]

Perception of R&C provided

Low value High value

Source: the authors

The classification of these stakeholders according to these two proximity and power dimensions allows us to arrive at four main types (Figure 3):

The Dominant stakeholder is a stakeholder that will present a high degree of power for a low degree of proximity, for example, a stakeholder that can validate or disallow a project but is not a partner in the project.

The Central stakeholder is a stakeholder that will have a high degree of power for a high degree of proximity, for example, a founding partner of the project, voluntary and linked to it by a contractual relationship,and has rare and essential resources and skills for its development.

The Dependent stakeholder will have a low degree of power for a high degree of proximity. This is the case, for example, of a contractually committed stakeholder who initiated the project but who fails to contribute resources and skills, or who has few or no resources and skills, or whose value is not seen as strategic to the project.

The Involuntary Stakeholder is a stakeholder who will have a low degree of power for a low degree of proximity, and who will for example be affected by the realisation of the project without having the capacity to participate in its development or the capacity to influence its realisation. It is sometimes a partner in a minor aspect of the project that can be easily interchanged. and Gond, 2005). It must therefore be understood as a 'global framework' that needs to be enriched. We propose to do this by mobilising the theory of resources and the skills-based approach, which, through their complementarity, will make it possible to understand the territory project in a global manner.

Reading the territorial project by the resource and skills approach

The Resource Based View is an approach that appeared in the 1980s. The theory is part of strategic management and is based on the founding work of [START_REF] Penrose | The theory ofthe growth ofthe firm[END_REF], [START_REF] Wernerfelt | A Resource-based View of the firm[END_REF] and [START_REF] Barney | Firm resources and sustained competitive advantage[END_REF]. Penrose first emphasised the importance of resources in analysing the strategy of a firm in 1959. strategic management literature attributes the Resource Based-View approach to Wernerfelt in 1984, who wrote that the heterogeneity of resources and their unequal distribution between firms would explain a competitive advantage. Finally, in 1991, Barney showed that firms are capable of accumulating resources which are transformed into an advantage over competitors if they are rare, create value, are not substitutable and are difficult to imitate. In the model proposed by Barney, resources are assets or potentials permanently owned or controlled by the firm in order to design and implement its strategy. Longevity can also be added to these characteristics as it allows the competitive advantage to be maintained over time [START_REF] Grant | The resource-based theory of competitive advantage: implications for strategy formulation[END_REF], along with appropriation, which refers to the process of activating the resource (Barney, 1995). However, this model was criticised by Weppe, Warnier, Lecocq and Fréry (2012) for the lack of clarity and precision in the definition of the resource concept, the limited nature of its applications and its lack of managerial implications, the positive bias of the resource approach, its focus on a minority of resources and the underestimated role of non-strategic resources. [START_REF] Weppe | Ressources stratégiques, ressources ordinaires et ressources négatives[END_REF] then propose to enrich the theory by introducing the notions of ordinary, strategic and negative resources. Their approach shows that it is not the resources alone that allow an organisation to differentiate itself, but rather the way in which they are perceived and used by the actors that will lead to achieving a sustainable competitive advantage. Indeed, the same resource may be perceived as being negative by some actors and strategic by others.

For [START_REF] Weppe | Ressources stratégiques, ressources ordinaires et ressources négatives[END_REF], an ordinary resource is "a common resource in the market, generally perceived as being neutral in terms of performance, i.e. having an expected productivity equivalent to its cost" (p. 52), a negative resource is "a resource neglected by companies, generally perceived as being negative in terms of performance, i.e. having an expected productivity lower than its cost" (p. 53) and a strategic resource is "a scarce resource, generally perceived by companies in the same sector as being positive in terms of performance, i.e. having an expected productivity higher than its cost" (p. 50). This typology can be combined with the one that [START_REF] Grant | The resource-based theory of competitive advantage: implications for strategy formulation[END_REF] takes from Hofer and Schendel (1978). According to him, it is possible to distinguish resources according to their nature: financial, human, physical, organisational, technological and reputational. The combination of these two classifications will make it possible to identify the nature of the resources mobilised and the perception of their values by the actors.

To study a territory project from a strategic point of view, it seems relevant to identify which of its resources are strategic, ordinary or negative. Thus, within a territory, resources such as identity, heritage, landscapes, etc. have certain characteristics: they are often rare, inimitable and are not substitutable, so can therefore be qualified as strategic. On the other hand, the value of these socalled strategic resources depends on the capacity of the actors in a territory to organise their exploitation by developing skills to identify, reveal and develop them. Moreover, the type of resources (strategic, ordinary and negative) is not fixed in time but evolves, which requires the actors to develop the associated skills. It therefore seems important to integrate the dynamic dimension offered by the competence approach into the resource approach. [START_REF] Grant | The resource-based theory of competitive advantage: implications for strategy formulation[END_REF] proposes the following definition: "a capability is the capacity for a team of resources to perform some task or activity" (p.119). Competence therefore refers to the organisational capacity of a firm to deploy resources to achieve an objective. In strategic management, interest is mainly focused on organisational competencies that can provide a competitive advantage. They are described as 'core' (Prévot, Brulhart and Guieu, 2010), and have the following characteristics: they give access to many markets, they allow activities to be carried out more quickly with greater flexibility and better quality than the competition, they are stable over time, they strongly contribute to the advantage of the final product as perceived by the customer and they are difficult to imitate. "According to [START_REF] Hamel | Competence-based competition[END_REF], "the real novelty of the concept lies not so much in the idea that competition is based on skills as in the distinction between core and non-core skills" (Prévot, Brulhart and Guieu, 2010, p. 92). [START_REF] Quelin | Competence of the firm and strategic alliances: creation of new capabilities and appropriability[END_REF] distinguishes three levels of competences: operational (elementary level), functional (intermediate level) and cross-functional and general (higher level). The identification, revelation and valorisation of a territory's resources can be equated with the development of new competences at these three levels.

In this approach, the authors are therefore interested in how the organisation's competences are exploited and how they change over time. Sustainable competitive advantage can therefore only be explained by a combination of resources and competencies (Figure 4). The resources of a territory such as heritage, identity and culture are rare, inimitable, not substitutable resources whose value will depend on the capacity of the actors to organise their exploitation. Moreover, the collaborative process which is a particularity of how territories function around projects presents characteristics which make them difficult to imitate. Territorial resources can often therefore be considered as strategic. Organisational resources such as the presence of a town council or a town hall are considered as ordinary at the scale of a town. Finally, for most of the

Competitive advantage

Competences

Operational

Functional

Cross-functional & general

Sustainable competitive advantage

Resources

Ordinary

Strategic

Negative

Value

Rare

Inimitable

Not substituable n territories, industrial wastelands appear to be negative, particularly regarding their impact on the attractiveness of tourism.

The skills for territorial managers that are directly linked to current activities, such as the organisation of elections or the distribution of newsletters, are qualified as operational. At a more intermediate level, specialised skills are aggregated in functional skills such as financial management, territorial marketing or cultural development policy. Finally, at a higher level, we find cross-functional competences that influence the whole organisation, such as decision-making processes or coordination processes.

We therefore propose to study the territory project in light of the nature of its strategic, ordinary and negative resources, and of the operational, functional, inter-functional and general skills mobilised.

The territory can then be observed with regard to its stakeholders, their power and proximity, but also via the approach by resources and competences which allows to understand how a territory project can become differentiating and contribute to reinforcing the economic, residential and tourist attractiveness.

Management of a territorial project: between resource and skills management and stakeholder management

Territory projects can help to differentiate a territory and make it more attractive and competitive. To do this, it is necessary to analyse the nature of the resources and skills mobilised, as well as their management, and to determine the role played by the various actors involved. The objective of this third sub-section is to understand how actors (political decision-makers, companies or individuals) can rely on the tools of strategic management to make their territorial projects successful. We combine the theory of resources and competences, essentially focused on internal aspects, with the analysis of stakeholders [START_REF] Gherra | Stratégies de développement durable[END_REF] focused on external aspects in order to obtain a complete vision of a territory project.

The Project Management Institute, founded in 1969, defines project management as: "the application of knowledge, tools, and methods in projects to meet or exceed the needs and expectations of project stakeholders" (Garel 2012, p. 17). It is important to understand the differences between 'managing a project', which refers to the instrumental functions of project steering, and 'project management', which concerns all the actions undertaken by one or more organisations to determine, launch and carry a project out. Here, we are particularly interested in the management of a territorial project during its development.

Although the project can be analysed as a process, the modelling of it is a subject for debate between project management researchers. Nevertheless, the one constant is that the process is often expressed in terms of a sequence of phases or stages [START_REF] Langley | A process study of new technology adoption in smaller manufacturing firms[END_REF][START_REF] Poole | Organizational change and innovation processes: Theory and methods for research[END_REF]. [START_REF] Lasswell | L'espace et le territoire dans l'agenda de recherche en stratégie[END_REF] was the first to propose an analysis of public policy in terms of process. Subsequently, [START_REF] Jones | An Introduction to the Study of Public Policy[END_REF] proposed a five-phase policy process: problem identification followed by programme development, implementation, evaluation and termination. This division was then very often used to analyse the development of public policies, and the model inspired other authors who adapted the framework to territorial action in general and to the territorial project in particular. [START_REF] Teisserenc | Les politiques de développement local : une approche sociologique[END_REF] proposes a model of local development, seen as a process of transformation and mobilisation, which is composed of three stages: learning, diffusion and development. This division is adapted to reflections on the territory but not to the project logic. Guillemet's concept (2002) was able to overcome this problem by proposing to understand the project as a process in four phases. The first phase concerns the preliminary studies and the specification of the project. The second concerns design and planning. The third phase concerns implementation and execution, while the fourth and final phase relates to validation and evaluation (Figure 5).

From these two models, we can propose a division of territorial project management by phases. The first is a learning and generation of new ideas phase, during which the actors exchange and become informed about the context and the multiple development initiatives that, in turn, leads to generating the first ideas. The second phase is the design, information dissemination and allocation of resources for the development of the project. The third phase is the development of the project and its concrete implementation. The fourth phase is the evaluation of the project, i.e. the match between objectives and results. Even though, presented as a sequential approach, this phasing gives a global vision, it rarely corresponds to r territ ory. The challenge is to know which resources are strategic for the territorial project, and when and how they are used by the actors.

To participate in the sustainable attractiveness of a territory, the project must be part of a strategic territorial process and requires the collective action of all the stakeholders. Its success is therefore conditioned by the more or less sustained participation of all the stakeholders in the territory and by the identification and consideration of the interests of all these stakeholders. In the context of territorial projects, we propose to study the roles and interactions of stakeholders as well as the resources and skills mobilised at each phase of the development of a territorial project.

As the literature in strategic management offers a solid theoretical basis for examining the territory and studying territorial projects in order to understand how they can contribute to differentiate a territory and make it more attractive and competitive, we can then propose a model of the development process of a territory project.

Modelling the development process of a territorial project

Based on our synthesis of the literature, we propose the following model of the development of a territorial project (Figure 6). The resources and skills needed to carry out the project will determine the type of stakeholders who will be involved in the project. It is then possible to identify management methods adapted to the project phase, the types of stakeholders and the resources and skills mobilised (see illustration in Box 1).

The contribution of such a model is multiple. First of all, it is designed to follow the evolution of a project longitudinally and provide a global and in-depth vision of the territory project observed. Secondly, it also helps to identify the determining elements of the project's success; in this sense, it is an aid tool so that decision-makers can have a complete understanding of the territorial transformations caused by the implementation of a territorial project. Finally, it can be used to compare several territorial projects, especially at the time of project selection, in order to identify whether the territory has the necessary resources and stakeholders to carry out the project successfully.

Box 1: An illustrative view of modelling _ The Only Lyon territory project

The Only Lyon development programme was launched in 2007. However, the reflection process started in the early 2000s to federate local economic players. During the elaboration phase of the economic development plan it became clear that the territory needed a territorial marketing approach and a strong network of actors. In 2007, the Lyon agglomeration decided to communicate on a national and international scale by launching the Only Lyon territory brand. Thirteen founding partners from the private and public sectors collectively launched this initiative to promote the city of Lyon throughout the world. Several networks were created behind this Only Lyon territorial brand with the aim of bringing together local players around a common development project for their territory.

During the first phase, the various players in the governance process began by asking Ernst&Young to conduct a survey to measure Lyon's reputation and compare its economic and tourist appeal with those of other European cities of similar size. Ernst&Young played a central role in this first phase by providing an objective diagnosis of the region's strengths and weaknesses, which was essential to the launch of the Only Lyon project.

During the second phase, the governance looked for ways to give meaning to the Only Lyon project by broadening the number of active partners. Integrating new private partners was a way to diversify the financing sources as, up till then, the development of Only Lyon had mainly been dependent on public funds. These new partners identified technological and reputational resources as strategic.

Residents were then integrated into the strategy to become very active volunteer ambassadors on social networks and considerably increased Lyon's influence at an international level. Only Lyon operates through different circles of stakeholders who play different roles and contribute or mobilise various strategic resources, or become strategic resources. The number of ambassadors is a powerful communication tool for Only Lyon. However, not all members agreed with this strategic choice to open the programme to "civilian" stakeholders.

During the third phase, the founding partners helped to maintain a good network dynamic while also ensuring that the economic players were involved as relays, by allowing them to appropriate the tools developed. This led to an important skill being revealed: the ability to organise open meetings based on the diversity of the players and lead them to new innovative actions that have transformed this territory project into a sustainable fixture for the development of the city of Lyon. Indeed, the appropriation for the economic actors of tourism (souvenir shops, press, restaurants, hotels) has increased the local visibility of the development programme, and Universities have also taken up the tools made available by Only Lyon. The programme makes it possible to involve various stakeholders, which serves their development as much as it serves the influence of Only Lyon.

As we can see from this illustration, our theoretical model emphasises the complex and dynamic relationships that are established around stakeholders, resources and skills, and management methods during the development of a territorial project. The objective for the decisionmakers is multiple: it involves not only distinguishing the stakeholders who will have a more or less important influence on the realisation of the project according to the phases, but also identifying the resources and competences that will prove crucial to determine the strategic priorities.

Depending on the phases of the project, specific managerial arrangements will have to be put in place for each type of stakeholder, to ensure both the smooth running of the project and to optimise collecting and using the resources and skills.

The managerial interest of this model is to provide public decision-makers with a tool for monitoring the development of territorial projects. Its particularity is to emphasise the role of stakeholders and the use of resources and skills at each phase of the process.

Conclusion

The study of territorial projects is increasingly important to understand the development of territories. However, very few works have theoretically looked into the question of analytical frameworks to study territorial projects [START_REF]Comprendre le territoire : les apports du management stratégique[END_REF]. The objective of this article was to show that the joint mobilisation of the stakeholder theory, the RBV amended by [START_REF] Weppe | Ressources stratégiques, ressources ordinaires et ressources négatives[END_REF] and the competency-based approach makes it possible to identify the determinants of the success of a territory project and to formulate managerial recommendations for their management.

The methodological contribution of this research is a new analysis grid that offers to consider the territory project through the stakeholders, by looking at their type, and through the resources and skills, by focusing on their values.

The theoretical contribution of this research is to define the notion of territory project from a strategic point of view as well as to provide a synthesis of works in strategic management and territory project management. This theoretical research reconciles the stakeholder approach and RBV to provide an analysis grid to understand the development of territorial projects. This grid allows for a global understanding of a territory project through the dynamics of its stakeholders and resources in order to differentiate a territory and strengthen its attractiveness and competitiveness. This research also leads to managerial contributions. We have identified tools capable of accompanying public or non-public decision-makers in the development of their territorial project and have also provided a grid that identifies the key actors as well as the key resources to be mobilised at each phase of a territorial project. Moreover, this research proposes to present the territory project as an instrument that federates the actors. It emphasises the need to strengthen the participation of civil society and the importance of enhancing territorial resources. Finally, the grid provides a strategic analysis that goes beyond the traditional frameworks of public management and project management and can help public decision-makers integrate new thinking into their decisionmaking processes.

Our research is not without its limitations. First of all, this theoretical research aims to achieve a synthesis of knowledge and to propose a model of the development of a territory project, but this model must be tested through case studies via the analysis of territory projects. Moreover, although this research is based solely on a literature review, to our knowledge there are no articles in management sciences that identify the managerial modalities of territorial projects. A promising avenue of research would be to identify both the management methods at work in territorial projects (cultural, sports, etc.) and their specificities based on the modelling proposed in this research.

Figure

  Figure 1: Stakeholders apprehended through proximity

Figure 3 :

 3 Figure 3: Classification of stakeholders in a territorial project

Figure 4 :

 4 Figure 4: The combination of resources and competences for sustainable competitive advantage

Figure 5 :

 5 Figure 5: The different phases of a territorial project

Figure 6 :

 6 Figure 6: Modelling the development of a territorial project


Franz Chouly
email: franz.chouly@u-bourgogne.fr

Tom Gustafsson
email: tom.gustafsson@aalto.fi

Patrick Hild
email: patrick.hild@math.univ-toulouse.fr

A Nitsche method for the elastoplastic torsion problem

Keywords: variational inequalities, elastoplastic torsion problem, finite elements, Nitsche, error estimates. 2020 MSC: 65N15, 65N30, 74C05

This study is concerned with the elastoplastic torsion problem, in dimension n ≥ 1, and in a polytopal, convex or not, domain. In the physically relevant case where the source term is a constant, this problem can be reformulated using the distance function to the boundary. We combine the aforementioned reformulation with a Nitsche-type discretization as in [Burman, Erik, et al. Computer Methods in Applied Mechanics and Engineering 313 (2017): 362-374]. This has two advantages: 1) it leads to optimal error bounds in the natural norm, even for nonconvex domains; 2) it is easy to implement within most of finite element libraries. We establish the wellposedness and convergence properties of the method, and illustrate its behavior with numerical experiments.

Introduction

Problems written with weak formulations involving variational inequalities represent various nonlinear phenomena which occur in mechanics and physics [START_REF] Duvaut | Grundlehren der Mathematischen Wissenschaften[END_REF][START_REF] Kinderlehrer | An introduction to variational inequalities and their applications[END_REF]. We focus on the elastoplastic torsion problem, as presented in, e.g., [START_REF] Glowinski | Lectures on numerical methods for nonlinear variational problems[END_REF] (see also [START_REF] Caffarelli | The free boundary for elastic-plastic torsion problems[END_REF][START_REF] Glowinski | Numerical analysis of variational inequalities[END_REF]). In the aforementioned reference, a direct piecewise affine Lagrange finite element approximation of the variational inequality is also presented, as well as a convergence result (Theorem 3.3), and two error estimates in the H 1 -norm, in dimension one (Theorem 3.4) and in dimension two (Theorem 3.5). The error estimate in one dimension is optimal (O(h)), whereas it remained suboptimal in dimension two, as it is of order O(h 1 2 -1 p ) for a source term in L p , p > 2. Among the first and few existing results are weak and strong convergence results [START_REF] Mouallif | Approximation du problème de la torsion élasto-plastique d'une barre cylindrique par régularisation et discrétisation d'un problème inf-sup sur H 1 0 (Ω) × L ∞ + (Ω)[END_REF], and error estimates of O(h) for the L 2 -norm of the gradient of the solution and under suitable restrictive assumptions, for mixed finite element approximations, using P 1 /P 0 finite elements [START_REF] Falk | Error estimates for elasto-plastic problems[END_REF] or Raviart-Thomas finite elements [START_REF] Bermúdez De Castro López | A mixed method for the elastoplastic torsion problem[END_REF].

In this paper, we focus on the torsion problem with a positive constant source term, corresponding to constant shear modulus and angle of twist. In this case the variational inequality can be reformulated as an "obstacle" problem where the constraint involves the distance to the boundary, so the obstacle is nonsmooth and the usual techniques from the obstacle problem cannot be directly applied: for instance in [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF]Theorem 5.1.2], the obstacle is supposed of Sobolev regularity H 2 . In a previous paper [START_REF] Chouly | On a finite element approximation for the elastoplastic torsion problem[END_REF], a direct finite element approximation of the variational inequality has been proposed, that makes use of piecewise affine, continuous, Lagrange finite elements, and in which the constraint involving the distance function is imposed at each node. When the domain is convex, error estimates have been established in any dimension n = 1, 2, 3, with an optimal error bound of O(h), for a regular enough continuous solution. In the case of a nonconvex domain, an error bound of O(h 3 4 ) has been proven for a solution of Sobolev regularity H α , α ≥ 7/4. In the present paper, we propose a new method that combines both the reformulation with the distance function, as in [START_REF] Chouly | On a finite element approximation for the elastoplastic torsion problem[END_REF], and a Nitsche term that allows to incorporate weakly the inequality constraint, following [START_REF] Burman | Galerkin least squares finite element method for the obstacle problem[END_REF] and related works on Nitsche's method for variational inequalities, see, e.g., [START_REF] Chouly | An overview of recent results on Nitsche's method for contact problems[END_REF] and references therein. For this discretization, we manage to derive optimal error estimates, for linear and quadratic finite elements, and even in the nonconvex situation, which improves the result of [START_REF] Chouly | On a finite element approximation for the elastoplastic torsion problem[END_REF]. Moreover, this method is easy to implement into modern finite element librairies, and we provide also some numerical experiments, that allow to confirm the expected theoretical convergence rates.

As usual, we denote by H s (•), s ∈ R, the Sobolev spaces. The usual norm of H s (D) is denoted by ∥•∥ s,D , and the corresponding semi-norm is denoted by |•| s,D . The space H 1 0 (D) is the subspace of functions in H 1 (D) with vanishing trace on ∂D. The letter C stands for a generic constant, independent of the mesh size, which value can changes at different occurences.

The elastoplastic torsion problem

Let Ω ⊂ R n , n ≥ 1, be an open bounded polytope, connected and with Lipschitz boundary. We consider the variational inequality which, for n = 2, models the torsion of an infinitely long elastoplastic cylinder of cross section Ω and plasticity yield r > 0. To simplify we assume that r = 1. The problem is to find the stress potential u such that

u ∈ K 1 : a(u, v -u) ≥ L(v -u) ∀ v ∈ K 1 , (1)
where a : H 1 0 (Ω) × H 1 0 (Ω) → R is the bilinear form given by:

a(u, v) := Ω ∇u • ∇v, ∀ u, v ∈ H 1 0 (Ω),
and

L(v) := Ω f v, ∀ v ∈ H 1 0 (Ω),
with f ∈ L 2 (Ω). The notation K 1 represents the nonempty closed convex set of admissible stress potentials:

K 1 := v ∈ H 1 0 (Ω) : |∇v| ≤ 1 a.e.
in Ω , where | • | denotes the euclidian norm in R n . From Stampacchia's theorem we deduce that Problem (1) admits a unique solution (see also, e.g., [START_REF] Duvaut | Grundlehren der Mathematischen Wissenschaften[END_REF][START_REF] Glowinski | Lectures on numerical methods for nonlinear variational problems[END_REF][START_REF] Glowinski | Numerical analysis of variational inequalities[END_REF][START_REF] Kinderlehrer | An introduction to variational inequalities and their applications[END_REF]).

Remark 2.1. We recall some regularity results for (1): if Ω ⊂ R n is open, bounded and convex, with Lipschitz boundary, and for f ∈ L p (Ω) with n < p < +∞, then u ∈ W 2,p (Ω) ∩ C 1,α (Ω), where α = 1 -n/p [START_REF] Brezis | Sur la régularité de la solution d'inéquations elliptiques[END_REF]. When the domain is nonconvex the W 2,p (Ω) regularity can be obtained but the boundary needs to be more regular (C 1,1 more precisely, see [START_REF] Gerhardt | Regularity of solutions of nonlinear variational inequalities with a gradient bound as constraint[END_REF]) so reentrant corners of polytopes are not allowed. When reentrant corners of polytopes are considered, the loss of W 2,p -regularity is only located near these corners [START_REF] Caffarelli | The free boundary for elastic-plastic torsion problems[END_REF].

Next we suppose that f = C is a constant function. In this case and according to [START_REF] Brézis | Équivalence de deux inéquations variationnelles et applications[END_REF] (see also [START_REF] Idone | Variational inequalities and the elastic-plastic torsion problem[END_REF]) the problem (1) can be rewritten as follows: find the stress potential u such that

u ∈ K : a(u, v -u) ≥ C Ω (v -u) ∀v ∈ K, (2)
with

K := v ∈ H 1 0 (Ω) : |v| ≤ d ∂Ω a.e.
in Ω , and d ∂Ω denotes the (interior) distance function with respect to the boundary ∂Ω:

d ∂Ω (x) := inf y∈∂Ω |x -y|, ∀ x ∈ Ω.
Note that (2) still admits a unique solution from Stampacchia's theorem. To lighten the discussion we can suppose without loss of generality that C > 0 (see [START_REF] Chouly | On a finite element approximation for the elastoplastic torsion problem[END_REF]Remark 2.2]), so problem (2) can be rewritten as follows: find the stress potential u such that

u ∈ K : a(u, v -u) ≥ C Ω (v -u) ∀ v ∈ K, (3)
with

K := v ∈ H 1 0 (Ω) : v ≤ d ∂Ω a.e.
in Ω . Again (3) admits a unique solution from Stampacchia's theorem. Moreover, Problem (2) and Problem (3) are equivalent, when C > 0 [9, Proposition 2.1 and Remark 2.3]. So the torsion problem can be seen as an obstacle problem where the distance function plays the role of the obstacle. Generally speaking, for a polytope, the distance function does not lie in H 2 (Ω). This implies that the classical finite element error analysis for the obstacle problem can not be directly applied.

Problem (3) in strong form, reads: find u : Ω -→ R solution to:

         -∆u ≤ C in Ω, u = 0 on ∂Ω, u ≤ d ∂Ω in Ω, (u -d ∂Ω )(∆u + C) = 0 in Ω. ( 4
)
We reformulate (4) using a Lagrange multiplier λ, and get:

               -∆u + λ = C in Ω, u = 0 on ∂Ω, λ ≥ 0 in Ω, u ≤ d ∂Ω in Ω, (u -d ∂Ω )λ = 0 in Ω. (5)
We introduce the following notation for the positive part: [a] + := max(0, a), for a ∈ R, and recall the relationship

([a] + -[b] + )(a -b) ≥ ([a] + -[b] + ) 2 , (6)
for a, b ∈ R.

Following [START_REF] Burman | Galerkin least squares finite element method for the obstacle problem[END_REF], the Kuhn-Tucker condition (5) 3-5 can equivalently be reformulated as

λ = γ -d ∂Ω + u + γ -1 λ + , (7)
with γ an arbitrary positive function on the domain Ω.

A Nitsche finite element method

Let V k h be a family of Lagrange finite element spaces of degree k ≥ 1 indexed by h, and coming from a family T h of simplicial meshes of the domain Ω (h := max T ∈T h h T where h T is the diameter of T ∈ T h ). The family of meshes is assumed regular. More precisely we have:

V k h = {v h ∈ C (Ω) ∩ H 1 0 (Ω) : v h | T ∈ P k (T ) ∀ T ∈ T h }.
Each simplex T of the mesh T h is supposed to be closed, and we denote by T the interior of T . We define a piecewise polynomial discrete Laplacian as follows, for every v h in V k h , and every simplex

T ∈ T h : (∆ h v h )| T := ∆(v h | T ).
The value of ∆ h v h on the facets of the mesh is of no importance, and can be set in practice to 0, for instance. We define also:

R h (v h ) := ∆ h v h + C. Remark that, for k = 1, ∆ h v h = 0 and R h (v h ) = C.
The Nitsche-type method proposed for the discretization of the elastoplastic torsion problem (5) reads: find

u h ∈ V k h such that a(u h , v h ) + γ h -d ∂Ω + u h + γ -1 h R h (u h ) + , v h = (C, v h ) (8)
for all v h ∈ V k h . Above the notation (•, •) stands for the L 2 (Ω)-scalar product, and the function γ h is defined cell-wise as follows:

γ h | T := γ 0 h 2 T ,
where γ 0 > 0 is the Nitsche parameter. Again, the value of γ h on the facets of the mesh is of no importance, and can be set in practice to 0, for instance.

Remark 3.1. As in [START_REF] Chouly | Symmetric and non-symmetric variants of Nitsche's method for contact problems in elasticity: theory and numerical experiments[END_REF][START_REF] Hu | Skew-symmetric Nitsche's formulation in isogeometric analysis: Dirichlet and symmetry conditions, patch coupling and frictionless contact[END_REF], for any parameter θ ∈ R, we can write a whole family of methods:

a(u h , v h ) -θ(γ -1 h ∆ h u h , ∆ h v h ) + γ h -d ∂Ω + u h + γ -1 h R h (u h ) + , v h + θγ -1 h ∆ h v h = (C, v h + θγ -1 h ∆ h v h ). (9)
Method (8) corresponds to θ = 0 and can be called an incomplete method, using the terminology widespread for discontinuous Galerkin methods. This method involves less terms and is the easiest to extend to more complex problems [START_REF] Mlika | An unbiased Nitsche's formulation of large deformation frictional contact and self-contact[END_REF]. A symmetric method is recovered when θ = 1, that corresponds to the Galerkin Least Squares technique of [START_REF] Burman | Galerkin least squares finite element method for the obstacle problem[END_REF] and the Nitsche method of [START_REF] Gustafsson | Mixed and stabilized finite element methods for the obstacle problem[END_REF]: this symmetric method can be recovered thanks to a minimization argument, and the tangent system has a symmetric Jacobian. Provided that the Nitsche parameter γ 0 is large enough, the analysis below, for θ = 0, can be extended without difficulty to other values of θ.

Remark 3.2. Remark that, for the distance function, there holds d ∂Ω ∈ H 1 (Ω) ∩ C 0,1 (Ω), see [START_REF] Chouly | On a finite element approximation for the elastoplastic torsion problem[END_REF] and references therein. In [START_REF] Burman | Galerkin least squares finite element method for the obstacle problem[END_REF] the assumption made on the obstacle function is stronger and this function is supposed to be C 1,1 (Ω). In fact, such Nitsche or Galerkin Least Squares formulations do not require so much regularity on the obstacle function.

The following local inverse inequality will be helpful in the sequel, that holds for an arbitrary v h ∈ V k h and every T ∈ T h :

∥∇v h ∥ 0,T ≤ C I h -1 T ∥v h ∥ 0,T , (10)
where C I > 0 is a constant that depends on the shape regularity of the mesh and of the polynomial order k, but not of h and T ∈ T h . See, e.g., [START_REF] Brenner | The mathematical theory of finite element methods[END_REF][START_REF] Ern | Finite elements I-Approximation and interpolation[END_REF] for the proof.

Numerical analysis: well-posedness and error estimate

We first state a preliminary consistency result:

a(u, v h ) + γ h -d ∂Ω + u + γ -1 h λ + , v h = (C, v h ) ∀ v h ∈ V k h . (11)
The above result is a direct consequence of ( 5)-( 7) and the inclusion V k h ⊂ H 1 0 (Ω).

Well-posedness

We make use of the results from Brezis (see, e.g., [START_REF] Brezis | Équations et inéquations non linéaires dans les espaces vectoriels en dualité[END_REF]) for M-type and pseudo-monotone operators in vector spaces. It consists in showing that the operator associated to Problem ( 8) is one-to-one. Proof: We introduce the nonlinear operator B h : V k h → V k h defined as follows:

B h z h , v h 1,Ω def = a(z h , v h ) + γ h -d ∂Ω + z h + γ -1 h R h (z h ) + , v h (12)
for all z h ∈ V k h and where (•, •) 1,Ω is the scalar product in H 1 (Ω). Note that the right-hand side of ( 12) is linear with respect to v h and, hence, B h is well defined thanks to Riesz' Theorem. The existence and uniqueness of solution for problem ( 8) is equivalent to the property of B h to be one-to-one. To this purpose, according to [START_REF] Brezis | Équations et inéquations non linéaires dans les espaces vectoriels en dualité[END_REF], it suffices to show that by B h is monotone and hemicontinuous.

For the monotonicity we first note that, owing to (12), we have

B h z h -B h v h , z h -v h 1,Ω =∥∇(z h -v h )∥ 2 0,Ω + γ h -d ∂Ω + z h + γ -1 h R h (z h ) + -γ h -d ∂Ω + v h + γ -1 h R h (v h ) + , z h -v h T 1 (13)
for all z h , v h ∈ V k h . On the other hand, by adding and subtracting suitable terms into T 1 , we get

T 1 = γ h -d ∂Ω + z h + γ -1 h R h (z h ) + -γ h -d ∂Ω + v h + γ -1 h R h (v h ) + , -d ∂Ω + z h + γ -1 h R h (z h ) -(-d ∂Ω + v h + γ -1 h R h (v h )) + γ 1 2 h -d ∂Ω + z h + γ -1 h R h (z h ) + -γ 1 2 h -d ∂Ω + v h + γ -1 h R h (v h ) + , γ -1 2 h (∆ h v h -∆ h z h ) .
Hence, using the inequality (6), Cauchy-Schwarz, Young inequalities and the inverse inequality [START_REF] Chouly | Symmetric and non-symmetric variants of Nitsche's method for contact problems in elasticity: theory and numerical experiments[END_REF], it follows that

T 1 ≥ γ 1 2 h -d ∂Ω + z h + γ -1 h R h (z h ) + --d ∂Ω + v h + γ -1 h R h (v h ) + 2 0,Ω - 1 2 γ 1 2 h -d ∂Ω + z h + γ -1 h R h (z h ) + --d ∂Ω + v h + γ -1 h R h (v h ) + 2 0,Ω - 1 2 ∥γ -1 2 h (∆ h z h -∆ h v h )∥ 2 0,Ω ≥ 1 2 γ 1 2 h -d ∂Ω + z h + γ -1 h R h (z h ) + --d ∂Ω + v h + γ -1 h R h (v h ) + 2 0,Ω - C 2 I 2γ 0 ∥∇(z h -v h )∥ 2 0,Ω .
The monotonicity of B h then follows by inserting this estimate into (13) under the condition γ 0 ≥ C 2 I . For the hemicontinuity, we must show that the real function φ : [0, 1] → R defined as

φ(t) def = (B h (z h -tv h ), v h ) is continuous for all z h , v h ∈ V k h . We bound φ(t) -φ(s) using the inequality |[a] + -[b] + | ≤ |a -b|, for all a, b ∈ R. This gives |φ(t) -φ(s)| = (s -t)a(v h , v h ) + γ h -d ∂Ω + (z h -tv h ) + γ -1 h R h (z h -tv h ) + -γ h -d ∂Ω + (z h -sv h ) + γ -1 h R h (z h -sv h ) + , v h ≤|t -s|a(v h , v h ) + γ h (z h -tv h ) + γ -1 h R h (z h -tv h ) -(z h -sv h ) -γ -1 h R h (z h -sv h ) , v h =|t -s| a(v h , v h ) + γ h v h + γ -1 h ∆ h v h ) , |v h |
which means that φ is Lipschitz and, thus, B h is hemicontinuous. □

A priori error estimate

We provide first an abstract estimate, as in [START_REF] Burman | Galerkin least squares finite element method for the obstacle problem[END_REF].

Theorem 4.2. Assume that the solution (u, λ) (λ = ∆u + C) to the elastoplastic torsion problem (5) belongs to K × L 2 (Ω). For γ 0 > 0 large enough, the approximation u h provided by (8) satisfies the following error estimate:

∥u -u h ∥ 1,Ω + γ 1 2 h -d ∂Ω + u + γ -1 h λ + --d ∂Ω + u h + γ -1 h R h (u h ) + 0,Ω + γ -1 2 h (∆u -∆ h u h ) 0,Ω ≤ C inf v h ∈V k h ∥u -v h ∥ 1,Ω + γ 1 2 h (u -v h ) 0,Ω + γ -1 2 h (∆u -∆ h v h ) 0,Ω . (14)
Proof: Let v h ∈ V k h . We first use the V -ellipticity and the continuity of a(•, •), as well as Young's inequality, to obtain:

α∥u -u h ∥ 2 1,Ω ≤ a(u -u h , u -u h ) = a(u -u h , (u -v h ) + (v h -u h )) ≤ C∥u -u h ∥ 1,Ω ∥u -v h ∥ 1,Ω + a(u -u h , v h -u h ) ≤ α 2 ∥u -u h ∥ 2 1,Ω + C 2 2α ∥u -v h ∥ 2 1,Ω + a(u -u h , v h -u h ), (15)
with α > 0 the ellipticity constant. We can transform the last term using the consistency property [START_REF] Ciarlet | The finite element method for elliptic problems[END_REF] for u and the finite element formulation (8) for u h . This yields

a(u -u h , v h -u h ) =a(u, v h -u h ) -a(u h , v h -u h ) = γ h -d ∂Ω + u + γ -1 h λ + -γ h -d ∂Ω + u h + γ -1 h R h (u h ) + , u h -v h . ( 16
)
We now transform the expression u h -v h :

u h -v h =u h -v h + γ -1 h R h (u h ) -γ -1 h R h (u h ) --d ∂Ω -u -γ -1 h λ + d ∂Ω + u + γ -1 h λ = -(-d ∂Ω + u + γ -1 h λ) -(-d ∂Ω + u h + γ -1 h R h (u h )) -v h + u -γ -1 h (R h (u h ) -λ).
Hence, by inserting this expression into ( 16), we get

a(u -u h , v h -u h ) = -γ h -d ∂Ω + u + γ -1 h λ + -γ h -d ∂Ω + u h + γ -1 h R h (u h ) + , T 1 (-d ∂Ω + u + γ -1 h λ) -(-d ∂Ω + u h + γ -1 h R h (u h )) T 1 + γ h -d ∂Ω + u + γ -1 h λ + -γ h -d ∂Ω + u h + γ -1 h R h (u h ) + , -v h + u -γ -1 h (R h (u h ) -λ) T 2 . ( 17
)
The first term is estimated by using the inequality [START_REF] Caffarelli | The free boundary for elastic-plastic torsion problems[END_REF], which yields

T 1 ≤ -γ 1 2 h -d ∂Ω + u + γ -1 h λ + --d ∂Ω + u h + γ -1 h R h (u h ) + 2 0,Ω . ( 18
)
For the term T 2 we use Cauchy-Schwarz and Young's inequality, to obtain

T 2 ≤ 1 2 γ 1 2 h -d ∂Ω + u + γ -1 h λ + --d ∂Ω + u h + γ -1 h R h (u h ) + 2 0,Ω + γ 1 2 h (u -v h ) 2 0,Ω + γ -1 2 h (λ -R h (u h )) 2 0,Ω . (19)
There remains to bound the last term above. Let T ∈ T h be a mesh cell. Using a triangular inequality, we bound first

∥R h (u h ) -λ∥ 0,T = ∥∆ h u h -∆u∥ 0,T ≤ ∥∆ h u h -∆ h v h ∥ 0,T + ∥∆u -∆ h v h ∥ 0,T .
With the inverse inequality [START_REF] Chouly | Symmetric and non-symmetric variants of Nitsche's method for contact problems in elasticity: theory and numerical experiments[END_REF] and triangular inequality, we bound

∥∆ h u h -∆ h v h ∥ 0,T ≤ C I h -1 T ∥∇u h -∇v h ∥ 0,T ≤ C I h -1 T (∥∇u h -∇u∥ 0,T + ∥∇u -∇v h ∥ 0,T ).
Therefore the last term in [START_REF] Gustafsson | Mixed and stabilized finite element methods for the obstacle problem[END_REF] can be bounded as

γ -1 2 h (R h (u h ) -λ) 2 0,Ω ≤ C γ -1 0 ∥∇u h -∇u∥ 2 0,Ω + ∥∇u -∇v h ∥ 2 0,Ω (20)
+C γ

-1 2 h (∆u -∆ h v h ) 2 0,Ω .
We combine estimates ( 15)-( 17)-( 18)-( 19)- [START_REF] Hu | Skew-symmetric Nitsche's formulation in isogeometric analysis: Dirichlet and symmetry conditions, patch coupling and frictionless contact[END_REF], which yields

α 2 - C γ 0 ∥u -u h ∥ 2 1,Ω + 1 2 γ 1 2 h -d ∂Ω + u + γ -1 h λ + --d ∂Ω + u h + γ -1 h R h (u h ) + 0,Ω ≤ C 2 2α + C γ 0 ∥u -v h ∥ 2 1,Ω + γ 1 2 h (u -v h ) 0,Ω + C γ -1 2 h (∆u -∆ h v h ) 0,Ω . (21)
Choosing γ 0 large enough, we obtain the desired bound on the first two terms in [START_REF] Falk | Error estimates for elasto-plastic problems[END_REF]. The bound on the error on the multiplier λ comes from combination of ( 20) and [START_REF] Idone | Variational inequalities and the elastic-plastic torsion problem[END_REF]. □

The optimal convergence of the method for P 1 and P 2 Lagrange finite elements is stated below, for a Sobolev regularity α ≤ 2.

Theorem 4.3. Suppose that k = 1, 2, and that the solution u belongs to H 1 0 (Ω) ∩ H α (Ω), with the Sobolev regularity α that satisfies max(1, n/2) < α ≤ 2. Suppose that λ belongs to L 2 (Ω). Suppose finally that the Nitsche parameter γ 0 is large enough. The solution u h to Problem (8) satisfies the following error estimate:

∥u -u h ∥ 1,Ω + γ 1 2 h -d ∂Ω + u + γ -1 h λ + --d ∂Ω + u h + γ -1 h R h (u h ) + 0,Ω + γ -1 2 h (∆u -∆ h u h ) 0,Ω ≤ C(h α-1 |u| α,Ω + h∥∆u∥ 0,Ω ) (22)
with C > 0 a constant, independent of h and u, but not of γ 0 .

Proof: We consider first the case k = 1. We start from the estimate [START_REF] Falk | Error estimates for elasto-plastic problems[END_REF]. We take

v h = I 1 h u, the Lagrange interpolant of u onto V 1
h . So, first, from standard interpolation estimates, we obtain

u -I 1 h u 1,Ω ≤ Ch α-1 |u| α,Ω ,
and γ

1 2 h u -I 1 h u 0,Ω ≤ Ch α-1 |u| α,Ω .
And then, for k = 1, we can simply proceed as follows:

γ -1 2 h ∆u -∆ h (I 1 h u) 0,Ω ≤ γ -1 2 0 h∥∆u -∆ h (I 1 h u) =0 ∥ 0,Ω = γ -1 2 0 h∥∆u∥ 0,Ω .
For k = 2, we proceed exactly as above. We start from the estimate [START_REF] Falk | Error estimates for elasto-plastic problems[END_REF].

Since V 1 h ⊂ V 2 h , we take v h = I 1 h u ∈ V 1 h (⊂ V 2 h ), the Lagrange interpolant of u onto V 1 h
, and get the same estimates as above for the three terms. This ends the proof. □

The next statement is for P 2 Lagrange finite elements, where a better convergence rate than O(h) can be expected if the solution u is regular enough.

Theorem 4.4. Suppose that k = 2, and that the solution u belongs to H 1 0 (Ω) ∩ H α (Ω), with the Sobolev regularity α that satisfies max(2, n/2) < α ≤ 3. Suppose that the Nitsche parameter γ 0 is large enough. The solution u h to Problem (8) satisfies the following error estimate:

∥u -u h ∥ 1,Ω + γ 1 2 h -d ∂Ω + u + γ -1 h λ + --d ∂Ω + u h + γ -1 h R h (u h ) + 0,Ω + γ -1 2 h (∆u -∆ h u h ) 0,Ω ≤ Ch α-1 |u| α,Ω , (23)
with C > 0 a constant, independent of h and u, but not of γ 0 .

Proof: We proceed as previously, in Theorem 4.3, but with the following modifications. We start from the estimate [START_REF] Falk | Error estimates for elasto-plastic problems[END_REF]. We take v h = I 2 h u, the Lagrange interpolant of u onto V 2 h . Still from standard interpolation estimates, we obtain

∥u -I 2 h u∥ 1,Ω ≤ Ch α-1 |u| α,Ω ,
and γ

1 2 h u -I 2 h u 0,Ω ≤ Ch α-1 |u| α,Ω .
For the last term, we need the following local error bound, for each T ∈ T h :

γ -1 2 h ∆u -∆ h (I 2 h u) 0,T = γ -1 2 0 h T ∥∆u -∆(I 2 h u| T )∥ 0,T ≤ Cγ -1 2 0 h T ∥u -I 2 h u| T ∥ 2,T
which is possible since α > 2. Then we use standard (local) interpolation error estimates to get:

γ -1 2 h ∆u -∆ h (I 2 h u) 0,T ≤ Cγ -1 2 0 h T h α-2 T |u| α,T ≤ Ch α-1 T |u| α,T .
By summation on the simplices T of the mesh T h , we get finally

γ -1 2 h ∆u -∆ h (I 2 h u) 0,Ω ≤ Ch α-1 |u| α,Ω .
This ends the proof. □

Numerical experiments

The following results are computed with the help of scikit-fem [START_REF] Gustafsson | scikit-fem: A Python package for finite element assembly[END_REF] for finite element assembly and autograd [START_REF] Maclaurin | Autograd: Effortless Gradients in Numpy[END_REF] for automatic differentiation. We consider two experiments introduced in [START_REF] Glowinski | Numerical analysis of variational inequalities[END_REF] with convex and nonconvex domains. The parameters are chosen as C = γ 0 = 10. First, a convergence study is performed using linear elements and the mesh sequences are depicted in Figure 1. Error in the stress potential u is computed against a reference solution which is obtained using quadratic finite elements and a sufficiently refined mesh. The resulting discrete solutions u h and the magnitudes of the gradient |∇u h | are depicted in Figures 2 and4, and the error in Figures 3 and5 for the convex and nonconvex examples, respectively. The convergence rate of the H 1 error is O(h) which is also an expected consequence of Theorem 4.3. Moreover, the convergence rate of the L 2 -error is O(h 2 ), a rate we are not able to prove. It is well-known that L 2 -error estimates are difficult to prove for problems modelled by variational inequalities.

We continue by solving the same experiments using quadratic elements. For this purpose, a reference solution is computed using cubic elements and a suitably refined mesh. The resulting discrete solutions are given in Figure 6 with the convergence rates in Figure 7. This time we observe that the H 1 error is approximately O(h 1.5 ). This is in agreement with Theorem 4.4. However this is lower than the rate O(h 2 ) for quadratic elements and a smooth solution. The reduction in the convergence rate is explained by the Sobolev regularity of the exact solution as u ∈ H 2 (Ω) while u ̸ ∈ H 3 (Ω). In one dimension, it is easy to check that the second derivative is expected to have a step discontinuity at the free boundary between elastic and plastic zones which suggests u ∈ H s (Ω), s < 5/2 (see also Remark 2.1). This also means that adaptive techniques could be used to improve the convergence rate with respect to the number of degrees-of-freedom. Figure 3: The error between u h and a reference solution as a function of the mesh parameter h for the convex example. The reference solution is obtained using a quadratic finite element method on a suitably refined mesh. Figure 5: The error between u h and a reference solution as a function of the mesh parameter h for the nonconvex example. The reference solution is obtained using a quadratic finite element method on a suitably refined mesh. Figure 7: The error between u h and a reference solution as a function of the mesh parameter h for the convex example using quadratic elements. The reference solution is obtained using a cubic finite element method on a suitably refined mesh.

Theorem 4 . 1 .

 41 For γ 0 ≥ C 2 I , Problem (8) admits one unique solution u h ∈ V k h .

Figure 1 :Figure 2 :

 12 Figure 1: The first three meshes for the convex (top) and nonconvex (bottom) examples from the uniform mesh sequences. The side length of the larger square is 1 while the smaller square in the nonconvex example has a side length of 0.2.

Figure 4 :

 4 Figure 4: The first three discrete solutions u h (top) and gradient magnitudes |∇u h | (bottom) for the nonconvex example.

Figure 6 :

 6 Figure 6: The first three discrete solutions u h (top) and gradient magnitudes |∇u h | (bottom) for the convex example using quadratic elements.

Acknowledgements

FC's work is partially supported by the I-Site BFC project NAANoD and the EIPHI Graduate School (contract ANR-17-EURE-0002). FC is grateful of the Center for Mathematical Modeling grant FB20005. The work of TG was supported by the Academy of Finland (decision nr. 338341).