Ismael Cuadrado-Cordero

Felix Cuadrado

Chris Phillips

Anne-Cécile Orgerie

Christine Morin

Microcities: a Platform based on Microclouds for Neighborhood Services

Keywords: Cloud Computing, Internet-of-Things, Mobile Cloud, Networking

The current mobile Cloud computing trend has set the focus on the ubiquity of computation. However, the current architecture confines the cloud to datacenters, which are generally far from the user. Distance leads to increased utilization of the broadband Wide Area Network -WAN -and poor user experience, especially for interactive applications. Decentralized architectures are emerging as an alternative, but they still fail to adapt to situations where files are concurrently modified. A semi-decentralized approach which confines local traffic close to the user while still maintaining centralized characteristics, running on the users and network devices, can provide a better Quality of Experience (QoE) in large urban populations in mobile cloud networks. In this report, we propose a novel semi-centralized cloud architecture based on microclouds. Microclouds are dynamically created and allow users to contribute resources from their computers, mobile and network devices to the cloud. We present the process for building and reconfiguring the microclouds, whilst guaranteeing a high QoE to users of real-time applications. We also provide a description of a realistic mobile cloud use-case. Results from a simulation-based evaluation indicate that the microclouds architecture is able to sustain hundreds of mobile devices and provide a latency significantly lower than regular Clouds based on datacenters.

Microcities: a Platform based on Microclouds for Neighborhood Services

Résumé : La tendance actuelle des applications mobiles dans les nuages informatiques (clouds) a mis láccent sur lómniprésence du calcul. Cependant, lárchitecture actuelle confine le cloud dans les centres de données qui sont généralement loin de lútilisateur. Cette distance conduit á une utilisation accrue de la bande passante des réseaux longue distance (Wide Area Network -WAN) et conduit á une expérience utilisateur médiocre, en particulier pour les applications interactives. Les architectures décentralisées apparaissent comme une alternative, mais elles ne parviennent pas toujours á s'adapter á des situations oú les fichiers sont modifiés de maniére concurrente. Une approche semi-décentralisée, mise en oeuvre sur les équipements des utilisateurs et du réseau et confinant le trafic local á proximité de lútilisateur tout en conservant les caractéristiques centralisées, peut fournir une meilleure qualité déxpérience (QoE) aux grandes populations urbaines utilisant les applications mobiles des clouds. Dans cet rapport, nous proposons une nouvelle architecture de cloud semi-centralisée fondée sur les "microclouds". Les microclouds sont créés dynamiquement et permettent aux utilisateurs de fournir au cloud des ressources de leurs ordinateurs, smartphones et équipements réseau. Nous présentons le processus de construction et de reconfiguration des microclouds, tout en garantissant un haut degré de QoE aux utilisateurs dápplications temps-réel. Nous donnons également une description dún cas d'utilisation réaliste de cloud mobile. Les résultats dúne évaluation menée par simulation indiquent que lárchitecture á base de microclouds est en mesure de gérer des centaines d'équipements mobiles et de fournir un temps de latence nettement inférieur á celui des clouds traditionnels fondés sur les centres de données.

Mots-clés : Informatique en nuage, Internet des objets, informatique mobile, réseau

Introduction

The wide uptake of Cloud architectures have caused global IP traffic to increase five fold [START_REF]Cisco. The zettabyte era: Trends and analysis[END_REF], catalyzed by the ubiquity of mobile devices. According to Cisco, global IP traffic is envisioned to increase threefold over the next five years, with mobile wireless traffic exceeding wired traffic by 2016 [START_REF]Cisco. The zettabyte era: Trends and analysis[END_REF]. This adds to the limitations of mobile devices in terms of resources and connectivity [START_REF] Fernando | Mobile cloud computing: A survey[END_REF], bringing new challenges to the Cloud. In order to address these issues, a new paradigm is emerging: Mobile Cloud computing. This paradigm improves resource-hungry mobile services such as Internet data sharing [START_REF] Marinelli | Hyrax: Cloud Computing on Mobile Devices using MapReduce[END_REF], wearables [START_REF] Amft | From Backpacks to Smartphones: Past, Present, and Future of Wearable Computers[END_REF] or augmented reality [START_REF] Luo | From Augmented Reality to Augmented Computing: A Look at Cloud-Mobile Convergence[END_REF], by offloading data and computation into the Cloud [START_REF] Fernando | Mobile cloud computing: A survey[END_REF].

This offloading of computation requires high-speed connectivity between the clients and the datacenter. However, prevailing highly geographically-centralized Cloud architectures do not properly handle it. Fernando et al. argue that considering data access fees, latency, bandwidth and energy consumption of wireless connectivity, a Mobile local Cloud -constrained to the location of the user -is a better alternative than a remote one [START_REF] Fernando | Mobile cloud computing: A survey[END_REF]. In addition, local Clouds also provide locality-awareness and support for latency-critical interactions, thus providing a better Quality-of-Experience.

In this report, we go a step further into Mobile Cloud computing by integrating the mobile devices themselves into the Cloud architecture. Our proposed architecture, called microclouds, introduces a flexible, semi-decentralized and yet efficient solution for locality-related applications. We build a local cloud on top of networking resources and the mobile devices of users spread across a defined area. This local cloud is managed by lightweight mechanisms which handle the dynamicity of users who can appear/disappear and move.

Our contributions are: 1) a semi-decentralized mobile cloud architecture called microclouds, 2) a description of a realistic mobile cloud use-case in line with smart cities, and 3) a simulation-based evaluation of our architecture. Our results indicate that the microclouds architecture is able to sustain one hundred mobile devices and provide a latency significantly lower than regular Clouds based on large datacenters.

The remainder of this report is structured as follows. Section 2 provides the context and motivation and we establish the case study of this work. In Section 3, we define the design and implementation of our system. In Section 4 we analyze experimentation results. Finally, Section 5 highlights our key findings and draws directions to future work.

Motivation and Scenario

Context and Motivation

The use of centralized datacenters is, nowadays, the most realistic approach regarding the deployment of heavy computation services. This architecture relies on a robust communication infrastructure between distant clients, obtaining a computing power otherwise unattainable. However, centralized architectures suffer downsides such as traffic delays, replication of data and scalability issues related with the physical constraints of datacenter resources. Moreover, it forces other actors involved in the communication (such as Internet Service Providers) to oversize their infrastructure in consequence [START_REF] Bertier | Beyond the Clouds: How Should Next Generation Utility Computing Infrastructures Be Designed?[END_REF].

As studied in [START_REF] Scellato | Track globally, deliver locally: Improving content delivery networks by tracking geographic social cascades[END_REF], [START_REF] Hristova | Keep your friends close and your facebook friends closer: A multiplex network approach to the analysis of offline and online social ties[END_REF] and [START_REF] Cha | A measurementdriven analysis of information propagation in the flickr social network[END_REF], information propagation in real-life is usually not distant. This is because interactions in the Internet are conditioned by our interactions as a society. Groups of interest are generally geographically constrained. This is the case with sports teams (whose fans are generally located in the same area), departments in a company (whose users are located in the same building) or Geographic Information Systems [START_REF] Paul A Longley | Geographic information science and systems[END_REF].

This situation is, by design, approached by Cloud computing, as it is described as a versatile and ubiquitous system. However, in reality, Cloud platforms run on large centralized datacenters, which provide the needed infrastructure. Yet, since existing infrastructures cannot effectively host ever increasing demands, datacenters need to be expanded, which is costly and requires advanced planning. In fact, this situation has already been anticipated by host providers, which have started distributing their datacenters around the globe, balancing the flow of information.

The use of centralized systems in users' inclusive -citizens are both providers and consumers of information -and non-heavy computation services with low propagation is, thus, inappropriate. A distributed approach, where the flow of information is not only produced and consumed, but also processed in the same area is, in this case, desirable. The main advantages of distributed approaches in this scenario are low latency, scalability, and adaptability. This kind of approach fits the mobile clouds created in smart cities initiatives all around the world (Santa Cruz, Amsterdam, Barcelona, etc.) where local authorities deploy wireless platforms to manage traffic or emergency situations.

Even if the proportion of geographically constrained traffic is difficult to estimate, traffic characterization in different areas can be found in literature. For example, in [START_REF] Johnson | Network traffic locality in a rural african village[END_REF] the authors evaluate the consumed and generated traffic in a rural African village. In this scenario, the authors show that most of the generated and consumed traffic is of a local scope, with web and social networking services being the most utilized ones. In addition, in [START_REF] Afanasyev | Usage Patterns in an Urban WiFi Network[END_REF] the authors characterize usage of a freely available outdoor wireless Internet in California (USA). Their results show a peak of smartphone connections in transit areas, while in residential areas the connections are more balanced between static and mobile. Commercial areas show a higher activity than either transit or residential areas.

In this report, we propose a semi-decentralized Cloud architecture for localized communities such as neighborhoods. We propose a mobile Cloud case study based on smart cities initiatives and provide a practical study and evaluation combining mobile and static devices -other existing Cloud infrastructures provide either a mobile peer-to-peer network [START_REF] Marinelli | Hyrax: Cloud Computing on Mobile Devices using MapReduce[END_REF] or local clouds with connectivity to remote cloud servers [START_REF] Kristensen | Scavenger: Transparent development of efficient cyber foraging applications[END_REF]. We show that, through the use of microclouds, latency is greatly reduced -compared to centralized systems -while providing Inria a robust, elastic and adaptive platform of services. Traffic in datacenters, and network providers' broadband utilization and transit costs are also reduced.

Related Work

Two main approaches in the literature are of interest to the case study.

Centralized Clouds

are based on a specific infrastructure to which all clients connect. An issue commonly attributed to these architectures is the excessive distance between clients and the computation infrastructure. Adaptability is also a problem as once the infrastructure has been allocated, it is expensive and complicated to extend. To provide a service "closer to the user" that is better able to handle the increase in demand of computation, Cloud providerssuch as Google [START_REF]Google cloud[END_REF] or Amazon [START_REF]Amazon cloud infrastructure[END_REF] -disperse their datacenters around the globe.

Centralized local clouds reduce the distance between the user and the infrastructure. In [START_REF] Mahadev Satyanarayanan | The case for vm-based cloudlets in mobile computing[END_REF], the authors describe an environment where mobile devices outsource their computation to a Cloudlet -domestic servers which provide cloud services to a relatively small set of users. It reduces the computational load of the devices, the replication of data, and the delay, and provides a service adapted to users' needs. However, they require a, sometimes, prohibitive investment, they are rigid and the dependence on the infrastructure ties the user to the system. Distributed Clouds are not deployed on the same infrastructure but among several independent nodes, providing a more robust and adaptable system compared to a lone infrastructure that offers a single point of failure. A thorough review of distributed technologies is shown in [START_REF] Babaoglu | The people's cloud[END_REF]. One of the more praised benefits of distributed approaches is the low latency experienced by users. Representative examples of distributed cloud systems are Content Distribution Networks (CDNs) [START_REF] Bakiras | Combining replica placement and caching techniques in content distribution networks[END_REF] and Peer-To-Peer (P2P) Clouds [START_REF] Peng | Research of P2P architecture based on cloud computing[END_REF], [START_REF] Xu | A Cloud Computing Platform Based on P2P[END_REF] -systems where individual computers are distributed across the globe and connected through the Internet hosting VMs for parallel computation, content distribution or storage.

Decentralized approaches also have disadvantages, such as network restrictions (distributed approaches are rather network demanding) and excessive replication of content. As a solution, the union of distributed and centralized architectures (semi-centralized) is also covered in literature. In [START_REF] Valancius | Greening the internet with nano data centers[END_REF], authors propose the replication of data in the Internet Provider's datacenter in order to reduce latency and energy consumption. However, it still suffers from higher delay than a totally decentralized approach, in addition to an excessive replication (for example, in between users connecting to different providers).

Scenario: Neighborhood Services

Neighborhood-related applications are a good example of geographically localized services. In a neighborhood, many services are only of interest to the community, like street works, water or electricity cuts or local store information (stocks, opening hours, etc.). These networks are heterogeneous -comprising both mobile and fixed nodes provided by both citizens and city infrastructures that serve up and consume information [START_REF]Orchestrating infrastructure for sustainable Smart Cities[END_REF]. Social networks such as [20] and [11] -where users share and interact information of interest only with their neighbors -are examples of neighborhood-oriented applications. A system appropriate for this environment should adjust to the following characteristics:

Non-Replication: Data can be classified in Announcements, which are immutable and distributed through the network to inform of an event; and collaborative/interactive information (shared documents, forums, etc.), which have several contributors and are prone to version conflicts and extensive replication in non-centralized systems (broadband utilization in multipurpose networks is very dynamic, and information may flood the network if not handled correctly). Therefore, no replication of data and/or management is allowed.

Adaptability, Virtualization and Splitting: Due to a dynamic workload, different communication patterns for different neighborhoods and variability in investment for different neighborhoods, the platform needs to dynamically adapt its topology. Also, virtualization and the ability to split services into users groups allows migration and robustness and enhances the isolation and coexistence of services, providing different services for different neighborhoods.

Network Orientation, Locality Awareness and Disruptive Behavior: Since a neighborhood exists within a relatively small geographical area, the system should take into account the physical location of the nodes to obtain the best possible utilization of the network. Knowing the location of the nodes, the platform is able to manage information dissemination, adapting to the available bandwidth. Due to the interactive nature of real-time information, disruptive behavior (asynchronous communication) is not desirable.

3 Our Solution: Microcities

Architecture Design

Our platform is semi-decentralized and extends the concept of microcloud proposed in [START_REF] Cordero | GRaNADA: A Network-Aware and Energy-Efficient PaaS Cloud Architecture[END_REF] in order to support dynamic interactivity of multiple services with no replication. A microcloud is an overlay network that connects independent users working using the same service -for each service a microcloud is created. Each microcloud hosts a Light Virtual Machine or LVM -a type of operatingsystem-level virtualization which runs the service. Using the LVM, data are centralized for each service, while microclouds distribute the computation across the network. All roles and processes described below are transparent to the user.

Base Node (BN): Stores LVMs -while no client is using them -and security policies. This logical role is taken by a server assumed to be highly available, and provisioned by a public authority in charge of the neighborhood.

Manager: Two types of manager exist:

Inria

• Base Manager (BM): This role is also taken by a resilient and trustworthy device -it can be the same as for the base node -which ensures an inefficiency threshold in the system. It periodically evaluates the QoE of the system and if it exceeds a given threshold -set by the administrator and adapted to the specific characteristics of the infrastructure -requests a new topology and/or an LVM migration to one or more microclouds.

• Service Manager (SM): Controls the topology of the microcloud and distributes the roles (service provider and linking nodes). This role is assigned by the base manager depending on the nodes' computing capacity and robustness. Only one service manager exists per microcloud, and it is preferably assigned to a stable, static node.

Service Provider (SP): Hosts the LVM(s). This role is assigned by the service manager based on the minimum delay between the service provider and every client, reliability in time and hardware capabilities, and it is unique in the microcloud. It is also preferably assigned to a stable, static node.

Linking Node: Links to nodes belonging to the same or different microclouds (noted in Figure 1 using a double line). If a linking node retransmits data to and from a node it is referred to a Bridge. Otherwise, if it forwards non-interactive information from the SP to one or more clients, a Repeater.

Client: Consumer of information, that is, a user. A physical node can take several client roles (one per microcloud), isolated from each other, or even different roles (such as client, service provider and/or service manager). Before using the service, it follows the joining process described in Section 3.2.

To improve the robustness of the system, the roles of SM and SP are replicated across different nodes, called backup nodes, following a hierarchical structure transparent to the user. If either the manager or the service provider is confirmed to have failed, through several reports sent to the BM from nodes in the system which cannot contact it, their role is taken by the next node in the hierarchy. If the dead node is running the SP role, the LVM is relaunched in the node chosen as a backup (to do so, the nodes selected as backups periodically retrieve a snapshot of the LVM). The LVM is migrated if the node acting as a service provider is excluded from the microcloud after the BM requests a reconfiguration. Figure 1 shows the logical distribution of nodes in our use-case.

Join and Detach Processes

The connection process is depicted in Figure 2 and described below. It is launched by the user and extends the DEEPACC protocol presented in [START_REF] Cordero | GRaNADA: A Network-Aware and Energy-Efficient PaaS Cloud Architecture[END_REF] to find the fastest path between the client and the SP. All services are listed in the BM, and the user selects one of them through a graphical interface. Failures in communications and reconfigurations triggered by the BM due to excessive inefficiency cause a reconnection. The process is the same, but is not initiated without a user request.

1. The client obtains the address of the SP through a request to the base manager. After that, a connection request is sent towards the SP, flooding the network through every possible path. In every node, the message is captured, processed and updated with the current Round-Trip Time (RTT).

2. Once the SP obtains all the possible paths, they are sent to the service manager. The SM uses a Branch and Bound algorithm [START_REF] Cordero | GRaNADA: A Network-Aware and Energy-Efficient PaaS Cloud Architecture[END_REF] to plan the microcloudś topology and communicates this topology to the SP and to the BM. The BM uses this information for inefficiency-triggered reconfigurations.

3. The SP sends an acceptance message to the client with its address and a list of the nodes in the route through the chosen route. Before forwarding the message, every node that intercepts it, updates its routing table and extracts itself from the list until the client receives the message.

Disconnections/failures follow the process depicted in Figure 3. This is launched once a user disconnects from the service (properly or due to a failure in communications). This disconnection is either processed or discovered and communicated to the SM, which restructures the network to keep providing the service to the remaining nodes.

Figure 2 shows the process of joining a microcloud. Figure 3 shows the process of properly detaching from a microcloud. If a link is broken, all affected nodes relaunch the DEEPACC protocol, which restarts the joining process.

Dependability

Despite the unreliability of the environment, the design of our system provides a certain resilience to failures. On one hand, the existence of a base infrastructure ensures that, even when no clients make use of a LVM, a working copy of data always exists. On the other hand, during microcloudś runtime, data are backed up in one or more nodes in the same microcloud. If either the SP or the SM are determined to be down, a joining process is repeated using a backup node as the main provider or manager.

Moreover, a microcloud is aware of connectivity and/or physical failures through a keepalive protocol. That way, every client in a microcloud is responsible for the state of its connection with its main neighbor (the first hop between the node and the SP), running a periodic check. If a link dies, a new joining process is launched by the client which detected the failure. Every other participant connecting through this client will remain connected to it, to accelerate the reconnection of the microcloud. Finally, the same service can be provided by a large microcloud or several smaller ones (split from the original microcloud), which offers higher resilience -as a node crash would affect a smaller number of nodes if a large microcloud is split in several smaller ones.

Evaluation

The aim of our experiments is to evaluate the suitability of a microclouds platform for operation within a neighborhood, compared with a centralized approach. To do so, we focus on the user experience in terms of the average packet delay between nodes, the overhead of the routing protocol (extra communication time), packet loss probability and the reliability of the system. Due to scalability issues, we extend the results of a 10 nodes prototype using a simulation on NS3 [START_REF]Ns3 simulator[END_REF]. The simulated network contains a variable number of mobile nodes -with a random mobility in the plane -over a physical network of 45 static nodes.

As a first evaluation, we compare the communication delay perceived by the user, shown in Figure 5. It measures the Round-Trip Time (RTT) both in microclouds and in a centralized cloud approach. For the centralized approach, Amazon EC2 has been chosen due to its representativeness, and we launch the experiments in different availability zones to provide different RTT values. A 45-minute real trace of a shared on-line document session has been used. Due to scale of the network, the results shown have been simulated in an environment comprising 110 nodes (small neighborhood). Data fed to the simulation have been obtained from a 10-client prototype.

Results are explained by the distance between users in the use-case (almost negligible compared to Amazon's world-wide area). As shown, microclouds provide an average delay of about 15 ms, while the centralized experiment shows a RTT several times higher. It is also worth noting that we have not observed packet loss using either microclouds or EC2. Additionally, Figure 6 shows the RTT evolution when the number of nodes in the microcloud increases. As expected, the delay depends on the number of nodes and the distance in between.

Second, we evaluate the robustness of our approach compared to a cen- tralized approach. We assume no infrastructure failures (which affects both approaches equally) and evaluate the probability of service interruptions caused by LVM migrations. An LVM may be migrated after a reconfiguration. Although the migration process adds a very small delay to the overall computation time, frequent migrations are not desirable.

Figure 8 shows the probability of migrating a LVM using either one big or several small microclouds' configurations, compared with the number of clients in the network, shown on Figure 9. For small microclouds (10 or fewer clients), the probability of the LVM being migrated is around 50%. It rapidly decays with the addition of new nodes as the microcloudś topology converges to that of the physical one until all the nodes participate in the overlay network and so no migration is needed. To avoid excessive migration, an inefficiency heuristic is needed to reduce the migrations whilst providing a better QoE than centralized approaches, adapted to the network capabilities based on experimentation.

Migration of LVMs produce, likewise, extra computation due to the joining process and calculation of new routes. While the service is not halted until a new path is computed and established, excessive extra computation affects the QoE and the demands the platform makes of the finite resources. To avoid excessive computational overhead, an inefficiency threshold may be set, so no reconfiguration is launched below this threshold. For our experimentation, we used a heuristic where no reconfiguration is launched until, at least, 50% of the nodes involved in the original overlay network change.

Figure 13 compares the computational overhead of a threshold-triggered redistribution and a reconnection triggered redistribution (launched when a mobile Inria node connects through a new node). As shown on 11, the total computation time added by the joining process exceeds almost 3 times the one used for the remainder of the of the 45-minute trace, while using a threshold triggered approach it remains under 25% of this time. On average -compared to the total execution time -, as shown in Figure 12, the use of a threshold triggered approach is almost imperceptible compared to the rest of the computation time.

Finally, we evaluate the overhead in time that our approach adds due to the computation of the routing path compared with a centralized one for the node acting as the SM. This overhead has been calculated on a MacBook Pro, with a 2.7 GHz Intel Core i7 and 8 GB 1600 MHz DDR3, representing a SM. This overhead is also reduces by splitting the microcloud into smaller onesthus increasing decentralization but reducing the route-planning computation. Figure 16 shows the total and average overhead along the duration of the trace.

We can see that the overhead increases exponentially and is directly related to the number of clients in the microcloud -which affects the size of the network and the number of reconnections of nodes, which causes network planning. As shown, when the network size is around 100 of mobile nodes, the total overhead reaches a 40% of the time. However, due to the ability to split microclouds, this time can be easily reduced by distributing the computation among as many microclouds as necessary. Since the path computation is distributed among different nodes, the average overhead computation on each SM is reduced. Given that each microcloud manages fewer nodes, the total overhead is also reduced. In this report we evaluated the suitability of a microcloud-based platform -a semi-distributed approach to managing geographically constrained data -in the context of mobile clouds. We investigated a neighborhood service-based scenario. Given that microclouds' design is semi-distributed and network-oriented, our approach exploits network resources to reduce unnecessary data movement over long distance networks. We evaluated the RTT, overhead time and robustness of our approach compared to a totally centralized approach. We show that, in the studied scenario, our platform provides better QoE in terms of latency and utilization of the networking resources -in terms of bandwidth utilization and profit from network infrastructure -over the centralized approach, whilst still providing a robust service. Also, we propose the use of an inefficiency-based heuristic to ensure the adaptability of the system to the network, and to reduce the need for extra-computation to deal with network planning.

Inria

We are now working on a pricing model for microclouds, as an extra incentive to adopt a microclouds-based system. The exchange of computational time for monetary revenue would boost the uptake of microclouds. Service creators would benefit from the open market and the increase in competition, while clients would receive a faster and more personalized service, together with financial benefits from making their computational resources available for hosting or relaying service information. Together with this, we plan to work on mobile congestion issues and refine our protocol, as appropriate, and address possible security issues associated with microclouds.

Figure 1 :

 1 Figure 1: Neighborhood overlay services (microclouds)

Figure 2 :

 2 Figure 2: Joining a microcloud

Figure 3 :

 3 Figure 3: Detachment from a microcloud Figure 4: Protocol description

Figure 5 :Figure 6 :Figure 7 :

 567 Figure 5: microclouds vs Amazon EC2 RTT

Figure 8 :

 8 Figure 8: Probability of migration of LVM

Figure 9 :Figure 10 :

 910 Figure 9: Utilization of the networkFigure 10: Probability of migration of a LVM vs. utilization of the network

Figure 12 :Figure 13 :

 1213 Figure 12: Average overheadFigure 13: Overhead comparison (in % of time)

Figure 14 :

 14 Figure 14: Total overhead

Figure 15 :Figure 16 :

 1516 Figure 15: Average overhead Figure 16: Overhead comparison splitting microclouds (in % of time)

RR n°8885

Éditeur Inria Domaine de Voluceau -Rocquencourt BP 105 -78153 Le Chesnay Cedex inria.fr ISSN 0249-6399


Sylvain Sorin

LIMIT VALUE OF DYNAMIC ZERO-SUM GAMES WITH VANISHING STAGE DURATION

We consider two person zero-sum games where the players control, at discrete times {tn} induced by a partition Π of R + , a continuous time Markov state process. We prove that the limit of the values vΠ exist as the mesh of Π goes to 0. The analysis covers the cases of : 1) stochastic games (where both players know the state) 2) symmetric no information. The proof is by reduction to a deterministic differential game.

Introduction

Repeated interactions in a stationary environment have been traditionally represented by dynamic games played in stages. An alternative approach is to consider a continuous time process on which the players act at discrete times. In the first case the expected number of interactions increases as the weight θ n of each stage n goes to zero. In the second case the number of interactions increases when the duration δ n of each time interval n vanishes.

In a repeated game framework one can normalize the model using the evaluation of the stages, so that stage n is associated to time t n = n-1 j=1 θ j , and then consider the game played on [0, 1] where time t corresponds to the fraction t of the total duration. Each evaluation θ (in the original repeated game) thus induces a partition Π θ of [0, 1] with vanishing mesh corresponding to vanishing stage weight. Tools adapted from continuous time models can be used to obtain convergence results, given an ordered set of evaluations, for the corresponding family of values v θ , see e.g. for different classes of games, Sorin [START_REF] Sorin | Big Match" with lack of information on one side (Part I)[END_REF], [START_REF] Sorin | A First Course on Zero-Sum Repeated Games[END_REF], [START_REF] Sorin | Zero-sum repeated games: recent advances and new links with differential games[END_REF], Vieille [START_REF] Vieille | Weak approachability[END_REF], Laraki [START_REF] Laraki | Repeated games with lack of information on one side: the dual differential approach[END_REF], Cardaliaguet, Laraki and Sorin [START_REF] Cardaliaguet | A continuous time approach for the asymptotic value in two-person zero-sum repeated games[END_REF].

In the alternative approach considered here, there is a given evaluation k on R + and one consider a sequence of partitions Π(m) of R + with vanishing mesh corresponding to vanishing stage duration and the associated sequence of values.

In both cases, for each given partition the value function exists at the times defined by the partition and the stationarity of the model allows to write a recursive formula (RF ). Then one extends the value function to [0, 1] (resp. R + ) by linearity and one considers the family of values as the mesh of the partition goes to 0. The next two steps in the proof of convergence of the family of values consist in defining a PDE (Main Equation M E) and proving : 1) that any accumulation point of the family is a viscosity solution of (M E) 2) that (M E) has a unique viscosity solution.

Altogether the tools are quite similar to those used in differential games, however in the current framework the state is basically a random variable and the players use mixed strategies.

Section 2 describes the model. Section 3 is devoted to the framework where both players observe the state variable. Section 4 deals with the situation where the state is unknown but the moves are observed. In both cases the analysis is done by reduction to a differential game. Section 5 presents the main results concerning differential games that are used in the paper. Consider a time homogeneous state process Z t , defined on R + = [0, +∞), with values in a state space Ω and an evaluation given by a probability density k(t) on R + . Each partition Π = {t 1 = 0, t 2 , ..., t n , ...} of R + induces a discrete time game as follows. The time interval L n = [t n , t n+1 [ corresponds to stage n and has duration δ n . The law of Z t on L n is determined by its value at time t n , Ẑn = Z tn and the actions (i n , j n ) ∈ I × J chosen by the players at time t n , that last for stage n. The payoff at time t in stage n (t ∈ L n ) is defined trough a map g from Ω × I × J to R: g Π (t) = g(Z t , i n , j n ) (An alternative choice leading to the same asymptotic results would be g Π (t) = g( Ẑn , i n , j n )). The evaluation along a play is:

Smooth continuous time games and discretization

γ Π,k = +∞ 0 g Π (t)k(dt)
and the corresponding value function is v Π,k . One will study the asymptotics of the family {v Π,k } as the mesh δ = sup δ n of the partition Π vanishes.

Markov process.

From now on we consider the case where Z t , t ∈ R + follows a continuous time Markov process: it is specified by a transition rate q that belongs to the set M of real bounded maps on I ×J ×Ω×Ω with:

q(i, j)[ω, ω ′ ] ≥ 0, if ω ′ = ω,
and

ω ′ ∈Ω q(i, j)[ω, ω ′ ] = 0, ∀i, j, ω.
Let P h (i, j), h ∈ R + be the continuous time Markov chain on Ω generated by the kernel q(i, j):

Ṗh (i, j) = P h (i, j) q(i, j) = q(i, j) P h (i, j)
and for t ≥ 0 : P t+h (i, j) = P t (i, j) e h q(i,j) . In particular, one has:

P h (i, j)[z, z ′ ] = P rob (Z t+h = z ′ |Z t = z), ∀t ≥ 0 = 1 {z} (z ′ ) + h q(i, j)[z, z ′ ] + o(h)

Hypotheses.

One assume from now on: the state space Ω is finite, the evaluation k is Lipschitz continuous on R + . the action sets I, J are compact metric spaces, the payoff g and the transition q are continuous on I × J.

Notations.

If µ is a bounded measurable function defined on I × J with values in a convex set, µ(x, y) denotes its multilinear extension to X × Y , with X = ∆(I) (resp. Y = ∆(J)), set of regular Borel probabilities on I (resp. J). (This applies in particular to g and q). For ζ ∈ ∆(Ω) and µ ∈ R Ω 2 we define :

ζ * µ (z) = ω∈Ω ζ(ω)µ[ω, z]. (When g is a map from Ω to itself and µ[ω, z] = 1 {g(ω)=z} , ζ * g is the usual image measure). In particular, if ζ t ∈ ∆(Ω) is the law of Z t one has, if (i, j) is played on [t, t + h] ζ t+h = ζ t * P h (i, j)
and ζt = ζ t * q(i, j).

Similarly we use the following notation for a transition probability or a transition rate µ operating on a real function f on Ω:

µ[z, .] • f (•) = z ′ µ[z, z ′ ]f (z ′ ) = µ • f [z].

State controlled and publicly observed

This section is devoted to the case were the process Z t is controlled by both players and observed by both (there is no assumptions on the signals on the actions). A stage n (time t n ) both players know Z tn . This corresponds to a stochastic game G in continuous time analyzed trough a time discretization along Π, G Π . Previous related papers to stochastic games in continuous time include Zachrisson [START_REF] Zachrisson | Markov Games, Advances in Game Theory[END_REF], Tanaka and Wakuta [START_REF] Tanaka | On continuous Markov games with the expected average reward criterion[END_REF], Guo and Hernandez-Lerma [START_REF] Guo | Zero-sum games for continuous-time Markov chains with unbounded transition and average payoff rates[END_REF], [START_REF] Guo | Zero-sum continuous-time Markov games with unbounded transition and discounted payoff rates[END_REF], Neyman [START_REF] Neyman | Continuous-time stochastic games, DP 616, Center for the Sudy of Rationality[END_REF]. The approach via time discretization is related to similar procedures in differential games, see Section 5, Fleming [START_REF] Fleming | A note on differential games of prescribed duration, Contributions to the Theory of Games[END_REF], [START_REF] Fleming | The convergence problem for differential games[END_REF], [START_REF] Fleming | The convergence problem for differential games II, Advances in Game Theory[END_REF], Scarf [START_REF] Scarf | On differential games with survival payoff, Contributions to the Theory of Games[END_REF] and Neyman [START_REF] Neyman | Stochastic games with short-stage duration[END_REF].

3.1. General case. Consider a general evaluation k. Since k is fixed during the analysis we will write v Π for v Π,k , defined on R + × Ω.

Recursive formula.

The hypothesis on the data implies that v Π exists, see e.g. [START_REF] Mertens | Repeated Games[END_REF], Chapters IV and VII, or [START_REF]Stochastic Games and Applications[END_REF], and in the current framework the recursive formula takes the following form:

Proposition 3.1.
The game G Π has a value v Π satisfying the recursive equation:

v Π (t n , Z tn ) = val X×Y E z,x,y [ t n+1 tn g(Z s , i, j)k(s)ds + v Π (t n+1 , Z t n+1 )] = val X×Y [E z,x,y ( t n+1 tn g(Z s , i, j)k(s)ds) + P δn (x, y)[Z tn , .] • v Π (t n+1 , .)] (1)
Proof This is the basic recursive formula for the stochastic game with state space Ω, action sets I and J and transition kernel P δn (i, j), going back to Shapley [START_REF] Shapley | Stochastic games[END_REF].

Recall that the value v Π (., z) is defined at times t n ∈ Π and extended by linearity to R + . Denote thus by V the (non empty) set of accumulation points of the family {v Π } Π (for the uniform convergence on compact subsets of R + × Ω) as the mesh δ vanishes. Definition 3.1. A continuous real function u on R + × Ω is a viscosity solution of:

(2) 0 = d dt u(t, z) + val X×Y {g(z, x, y)k(t) + q(x, y)[z, .] • u(t, •)}, if for any real function ψ, C 1 on R + × Ω with u -ψ having a strict maximum at ( t, z) ∈ R + × Ω: 0 ≤ d dt ψ( t, z) + val X×Y {g(z, x, y)k( t) + q(x, y)[z, .] • ψ( t, •)}
and the dual condition.

Proposition 3.3. Any u ∈ V is a viscosity solution of (2).
Proof Let ψ(t, z) be a C 1 test function such that u -ψ has a strict maximum at ( t, z). Consider a sequence V m = v Π(m) converging uniformy locally to u as m → ∞ and let (t * (m), z(m)) be a minimizing sequence for

{(ψ -V m )(t, z), t ∈ Π m }. In particular (t * (m), z(m)) converges to ( t, z) as m → ∞. Given x * m optimal for V m (t * (m), z(m)) in (1), one obtains, with t * (m) = t n ∈ Π m : V m (t n , z(m)) ≤ E z(m),x * m ,y [ tn t n-1 g(Z s , i, j)k(s)ds] + P δn (x * m , y)[z(m), .] • V m (t n+1 , .), ∀y ∈ Y,
so that by the choice of (t * (m), z(m)):

ψ(t n , z(m)) ≤ E z(m),x * m ,y [ tn t n-1 g(Z s , i, j)k(s)ds] + P δn (x * m , y)[z(m), .] • ψ(t n+1 , .)] ≤ δ n k(t n ) g(z(m), x * m , y) + ψ(t n+1 , z(m)) + δ n q(x * m , y)[z(m), .] • ψ(t n+1 , .) + o(δ n ). This implies: 0 ≤ δ n k(t n ) g(z(m), x * m , y) + δ n d dt ψ(t n , z(m)) + δ n q(x * m , y)[z(m), .] • ψ(t n+1 , .) + o(δ n )
hence dividing by δ n and taking the limit as m → ∞ one obtains, for some accumulation point x * in the compact set ∆(I):

(3) 0 ≤ k( t) g(z, x * , y) + d dt ψ( t, z) + q(x * , y)[z, .] • ψ( t, .), ∀y ∈ Y so that: 0 ≤ d dt ψ( t, z) + val X×Y [ g(z, x, y) k( t) + q(x, y)[z, .] • ψ( t, .)].

Convergence.

A first proof of the convergence of the family {v Π } Π would follow from the property: (P ) Equation ( 2) has a unique viscosity solution.

An alternative approach is to relate the game to a differential game on an extended state space ∆(Ω). Define V Π on R + × ∆(Ω) as the expectation of v Π , namely:

V Π (t, ζ) = ζ, v Π (t, .) = ω∈Ω ζ(ω)v Π (t, ω) and denote X = X Ω and Y = Y Ω . Proposition 3.4. V Π satisfies: V Π (t n , ζ tn ) = val X×Y [ ω ζ tn (ω)E ω,x(ω),y(ω )( t n+1 tn g(Z s , i, j) k(s)ds) + V Π (t n+1 , ζ t n+1 )] ( 4
)
where ζ t n+1 (z) = ω ζ tn (ω)P δn (x(ω), y(ω))(ω, z).
Proof (4) follows from (1), the definition of V Π and the formula expressing ζ t n+1 . By independence the optimization in X at each ω can be replaced by optimization in X and one uses the linearity in the transition.

Equation (1) corresponds to the usual approach following the trajectory of the process. Equation (4) expresses the dynamics of the law ζ of the process, where the players act differently at different states ω.

3.1.4. Related differential game. We will prove that the recursive equation ( 4) is satisfied by the value of the time discretization along Π of the mixed extension of a deterministic differential game G (see Section 5) on R + , defined as follows: 1) the state space is ∆(Ω), 2) the action sets are I = I Ω and J = J Ω , 3) the dynamics on ∆

(Ω) × R + is: ζt = f (ζ t , i, j) with f (ζ, i, j)(z) = ω∈Ω ζ t (ω)q(i(ω), j(ω))[ω, z].
4) the flow payoff function is given by: ζ, g(., i(.), j(.))

= ω∈Ω ζ(ω)g(ω, i(ω), j(ω)).
5) the global outcome is:

+∞ 0 γ t k(t)dt
where γ t is the payoff at time t.

In G Π the state is deterministic and at each time t n the players know ζ tn and choose i n (resp. j n ). Consider now the mixed extension G II Π (Section 5) and let V Π (t, ζ) be the associated value. Proposition 3.5. The value V Π (t, ζ) satisfies the recursive equation ( 4).

Proof

The mixed action set for player 1 is X but due to the separability in ω one can work with X. Then it is easy to see that equation [START_REF] Sorin | Big Match" with lack of information on one side (Part I)[END_REF] corresponds to (4).

The analysis in section 5 thus implies that : -any accumulation point U of the sequence V Π is a viscosity solution of

(5) 0 = d dt U (t, ζ) + val X×Y [ ζ, g(., x(.), y(.)) k(t) + f (ζ, x, y), ∇U (t, ζ) ]
-Equation ( 5) has a unique viscosity solution.

In particular let

U(t, ζ) = ζ, u(t, .) = ω ζ(ω)u(t, ω) where u ∈ V. Proposition 3.6. U(t, ζ)
is the viscosity solution of (5).

Proof

Follows from the fact that V Π and V Π satisfy the same recursive formula, hence U is an acumulation point of the sequence V Π .

This leads to the convergence property.

Corollary 3.1. Both families V Π and v Π converge to some V and v with

V (t, ζ) = ω ζ(ω)v(t, ω).
V is the viscosity solution of (5). v is the viscosity solution of (2).

Proof

One has ∇V(t, ζ) = {v(t, .)} hence:

0 = ζ, d dt v(t, .) + val X×Y [ ζ, g(., x(.), y(.)) k(t) + z [ ω ζ(ω)q(x(ω), y(ω))[ω, z]v(t, z)] = ζ, d dt v(t, .) + val X×Y [ ζ, g(., x(.), y(.)) k(t) + ω ζ(ω)[ z q(x(ω), y(ω))[ω, z]v(t, z)]
This gives:

0 = ζ, d dt v(t, .) + val X×Y [g(., x, y)k(t) + q(x, y)[., •] • v(t, •)]
which is equivalent to:

0 = d dt v(t, z) + val X×Y [g(z, x, y)k(t) + q(x, y)[z, .] • v(t, •)].
and this is (2).

Stationary case.

We consider the case k(t) = ρe -ρt and again the game along the partition Π.

Recursive formula.

The general recursive formula (1) takes now the following form:

Proposition 3.7. v Π,ρ (t n , Z tn ) = val X×Y E z,x,y [ t n+1 tn g(Z s , i, j)ρe -ρ ds + v Π,ρ (t n+1 , Z t n+1 )] = val X×Y [E z,x,y ( t n+1 tn g(Z s , i, j)ρe -ρ ds) + P δn (x, y)[Z tn , .] • v Π,ρ (t n+1 , .)] (6)
and if Π is uniform, v Π,ρ (t, z) = e -ρt ν δ,ρ (z) with: (7) ν δ,ρ (Z 0 ) = val X×Y [E z,x,y ( δ 0 g(Z s , x, y)ρe -ρ ds) + e -ρδ P δ (x, y)[Z 0 , .] • ν δ,ρ (.)]

Main equation.

The next result is standard, see e.g. Neyman [START_REF] Neyman | Stochastic games with short-stage duration[END_REF], Prieto-Rumeau and Hernandez-Lerma [START_REF] Prieto-Rumeau | Selected Topics on Continuous-Time Controlled Markov Chains and Markov Games[END_REF], p. 235. We provide a short proof for convenience.

Proposition 3.8. 1) For any R ∈ M and any ρ ∈ (0, 1] the equation, with variable ϕ from Ω to R:

(8) ρ ϕ(z) = val X×Y [ρ g(z, x, y) + R(x, y)[z, .] • ϕ(.)]
has a unique solution, denoted W ρ .

2) For any δ ∈ (0, 1] such that δR/(1 -δρ) ≤ 1 the solution of ( 8) is the value of the repeated stochastic game with payoff g, transition P = I + δR/(1 -δρ) and discounted factor δρ.

Proof

Recall from Shapley [START_REF] Shapley | Stochastic games[END_REF], that the value W ρδ of a repeated stochastic game with payoff g and discounted factor δρ satisfies:

(9) W ρδ (z) = val X×Y [δρ g(z, x, y) + (1 -δρ)E z,x,y {W ρδ (.)}].
Assume the transition to be of the form P = I + δq with q ∈ M. One obtains:

(10) W ρδ (z) = val X×Y [δρ g(z, x, y) + (1 -δρ){W ρδ (z) + δ q(x, y)[z, .] • W ρδ (.)}]
which gives:

(11) δρ W ρδ (z) = val X×Y [δρ g(z, x, y) + δ(1 -δρ) q(x, y)[z, .] • W ρδ (.)]
so that:

(12) ρ W ρδ (z) = val X×Y [ρ g(z, x, y) + (1 -δρ) q(x, y)[z, .] • W ρδ (.)].
Hence with q = R/(1 -δρ) one obtains:

(13) ρ W ρδ (z) = val X×Y [ρ g(z, x, y) + R(x, y)[z, .] • W ρδ (.)]. 3.2.3. Convergence.
Again the following result can be found in Neyman [START_REF] Neyman | Stochastic games with short-stage duration[END_REF], Theorem 1, see also Guo Hernandez-Lerma [START_REF] Guo | Zero-sum games for continuous-time Markov chains with unbounded transition and average payoff rates[END_REF][START_REF] Guo | Zero-sum continuous-time Markov games with unbounded transition and discounted payoff rates[END_REF].

Proposition 3.9.

As the mesh δ of the partition Π goes to 0, v Π,ρ converges to the solution W ρ of (8) with R = q:

(14) ρ W ρ (z) = val X×Y [ρ g(z, x, y) + q(x, y)[z, .] • W ρ (.)]
Proof Consider the strategy σ of Player 1 in G Π defined as follows: at state z, use an optimal strategy x ∈ X = ∆(I), for W ρ (z) given by ( 14). Let us evaluate, given τ , strategy of Player 2, the following amount:

A 1 = E σ,τ [ t 2 t 1 g Π (s)ρe -ρs ds + e -ρt 1 W ρ (Z t 1 )].
Let x 1 the mixed move of Player 1 at stage one given Z 0 = Ẑ1 . Then if y 1 is induced by τ , there exists a constant L such that:

A 1 ≥ δ 1 ρ g( Ẑ1 , x 1 , y 1 ) + (1 -δ 1 ρ)[W ρ ( Ẑ1 ) + δ 1 q(x 1 , y 1 )[ Ẑ1 , .] • W ρ (.)] -δ 1 Lδ ≥ δ 1 ρ g( Ẑ1 , x 1 , y 1 ) -δ 1 ρ W ρ ( Ẑ1 ) + δ 1 q(x 1 , y 1 )[ Ẑ1 , .] • W ρ (.) + W ρ ( Ẑ1 ) -2δ 1 Lδ ≥ W ρ ( Ẑ1 ) -2δ 1 Lδ.
Similarly let:

A n = E σ,τ [ t n+1 tn g Π (s)ρe -ρs dsds + e -ρtn W ρ ( Ẑn+1 )|h n ]
where

h n = ( Ẑ1 , i 1 , j 1 , • • • , i n-1 , j n-1 , Ẑn ).
Then, with obvious notations:

A n ≥ e -ρt n-1 [δ n ρ g( Ẑn , x n , y n ) + (1 -δ n ρ)[W ρ ( Ẑn ) + δ n q(x n , y n )[ Ẑn , .] • W ρ (.) -δ n Lδ] ≥ e -ρt n-1 [δ n ρ g( Ẑn , x n , y n ) -δ n ρ W ρ ( Ẑn ) + δ n q(x n , y n )[ Ẑn , .] • W ρ (.) + W ρ ( Ẑn ) -2δ n Lδ] ≥ e -ρt n-1 [W ρ ( Ẑn ) -2δ n Lδ].
Taking the sum and the expectation, one obtains that the payoff induced by (σ, τ ) in G Π satisfies:

E σ,τ [ +∞ 0 g Π (s)k(s)ds] ≥ W ρ ( Ẑ1 ) -2( n δ n e -ρt n-1 )Lδ
and ( n δ n e -ρt n-1 )Lδ → 0 as δ → 0.

Comments:

The proof in Neyman [START_REF] Neyman | Stochastic games with short-stage duration[END_REF] is done, in the finite case, for a uniform partition but shows the robustness with respect to the parameters (converging family of games).

This procedure of proof is reminiscent of the "direct approach" introduced by Isaacs [START_REF] Isaacs | Differential Games[END_REF]. To show convergence of the family of values of the discretizations v Π : i) one identifies a tentative limit value v and a recursive formula RF (v) and ii) one shows that to play in the discretized game G Π an optimal strategy in RF (v) gives an amount close to v for δ small enough.

For an alternative approach and proof, based on properties of the Shapley operator, see Sorin and Vigeral [START_REF] Sorin | Operator approach to values of stochastic games with varying stage duration[END_REF].

Remark that if k(t) = ρe -ρt , v(t, z) = e -ρt ν(z) satisfies (2) iff ν(z) satisfies ( 14).

State controlled and not observed

This section studies the game G where the process Z t is controlled by both players but not observed. However the past actions are known: this defines a symmetric framework were the new state variable is the law of Z t , ζ t ∈ ∆(Ω). Even in the stationary case there is no explicit smooth solution to the main equation hence a direct approach for proving convergence, as in the previous Section 3.2, is not feasible.

Here also the analysis will be trough the connection to a differential game G on ∆(Ω) but different from the previous one G, introduced in Section 3.

Given a partition Π denote by G Π the associated game and again, since k is fixed during the analysis we will write V Π for its value V Π,k defined on R + × ∆(Ω). Recall that given the initial law ζ tn and the actions (i tn , j tn ) = (i, j) one has: [START_REF] Fleming | The convergence problem for differential games[END_REF] ζ ij t n+1 = ζ tn+δn = ζ tn * P δn (i, j) and that this parameter is known by both players. Extend g(., x, y) from Ω to ∆(Ω) by linearity: g(ζ, x, y) = ζ(z)g(z, x, y).

Recursive formula.

In this framework the recursive structure leads to: Proposition 4.1. The value V Π satisfies the following recursive formula:

V Π (t n , ζ tn ) = val X×Y E ζ,x,y [ t n+1 tn g(ζ s , i, j)k(s)ds + V Π (t n+1 , ζ ij t n+1 )] (16)
Proof Standard, since G Π is basically a stochastic game with parameter ζ.

Main equation.

Consider the differential game G on ∆(Ω) with actions sets I and J, dynamics on ∆(Ω) × R + given by: ζt = ζ t * q(i, j), current payoff g(ζ, i, j) and evaluation k.

As in Section 5, consider the discretized mixed extension G II Π to X × Y and let V Π be its value. Proposition 4.2. V Π satisfies [START_REF] Fleming | The convergence problem for differential games II, Advances in Game Theory[END_REF]. [START_REF] Sorin | Big Match" with lack of information on one side (Part I)[END_REF] which is, using [START_REF] Fleming | The convergence problem for differential games[END_REF], equivalent to [START_REF] Fleming | The convergence problem for differential games II, Advances in Game Theory[END_REF].

Proof V Π satisfies
The analysis in Section 5, Proposition 5.12 thus implies:

Proposition 4.3.
The family of values V Π converge to V unique viscosity solution of :

(17) 0 = d dt u(t, ζ) + val X×Y [g(ζ, x, y)k(t) + ζ * q(x, y), ∇u(t, ζ)].

Stationary case.

Assume k(t) = ρe -ρt . In this case one has V(ζ, t) = e -ρt v(ζ) hence ( 17) becomes ( 18)

ρv(ζ) = val X×Y [ρ g(ζ, x, y) + ζ * q(x, y), ∇v(ζ) ] 4.4. Comments.
A differential game similar to G where the state space is the set of probabilities on some set Ω has been studied in full generality by Cardaliaguet and Quincampoix [START_REF] Cardaliaguet | Deterministic differential games under probability knowledge of initial condition[END_REF], see also As Soulaimani [START_REF] Soulaimani | Viability with probabilistic knowledge of initial condition, application to optimal control[END_REF]. Equation ( 18) is satisfied by the value of the Non-Revealing game in the framework analyzed by Cardaliaguet, Rainer, Rosenberg and Vieille [START_REF] Cardaliaguet | Markov games with frequent actions and incomplete information: the limit case[END_REF] see Section 6.

Discretization and mixed extension of differential games

We study here the value of a continuous time game by introducing a time discretization Π and analyzing the limit behavior of the associated family of values v Π as the mesh of the partition vanishes. This approach was initiated in Fleming [START_REF] Fleming | A note on differential games of prescribed duration, Contributions to the Theory of Games[END_REF], [START_REF] Fleming | The convergence problem for differential games[END_REF], [START_REF] Fleming | The convergence problem for differential games II, Advances in Game Theory[END_REF], and developped in Friedman [START_REF] Friedman | Differential games[END_REF], [START_REF] Friedman | Differential games[END_REF], Eliott and Kalton [START_REF] Elliot | The existence of value in differential games[END_REF].

A differential game γ is defined trough the following components: Z ⊂ R n is the state space, I and J are the action sets of player 1 (maximizer) and 2, f from Z × I × J to R n is the dynamics kernel, g from Z × I × J to R is the payoff-flow function and k from R + to R + determines the evaluation. Formally the dynamics is defined on [0, +∞) × Z by :

(19) żt = f (z t , i t , j t )
and the total payoff is :

+∞ 0 g(z s , i s , j s ) k(s) ds.
We assume: I and J metric compact sets, f and g continuous and uniformly Lipschitz in z, g bounded, k Lipschitz with +∞ 0 k(s)ds = 1. Φ h (z; i, j) denote the value at time t + h of the solution of ( 19) starting at time t from z and with play {i s = i, j s = j} on [t, t + h].

To define the strategies we have to specify the information: we assume that the players know the initial state z 0 , and at time t the previous play {i s , j s ; 0 ≤ s < t} hence the trajectory of the state {z s ; 0 ≤ s ≤ t}. The analysis below will show that Markov strategies (i.e. depending only, at time t, on t and z t ) will suffice.

Deterministic analysis.

Let Π = ({t n }, n = 1, ...) be a partition of [0, +∞) with t 1 = 0, δ n = t n+1 -t n and δ = sup δ n . We consider the associated discrete time game γ Π where on each interval [t n , t n+1 ) players use constant moves (i n , j n ) in I × J. This defines the dynamics on the state. At time t n+1 , (i n , j n ) is announced and the corresponding value of the state, z t n+1 = Φ δn (z tn ; i n , j n ) is known. The associated maxmin w - Π satisfies the recursive formula:

(20) w - Π (t n , z tn ) = sup I inf J [ t n+1 tn g(z s , i, j)k(s)ds + w - Π (t n+1 , z t n+1 )]
The fonction w - Π (., z) is extended by linearity to [0, +∞) and note that: [START_REF] Isaacs | Differential Games[END_REF] ∀ε > 0, ∃T, such that t ≥ T implies |w - Π (t, .)| ≤ ε and that all "value" functions that we will consider here will satisfy this property.

The next four results follow from the analysis in Evans and Souganidis [START_REF] Evans | Differential games and representation formulas for solutions of Hamilton-Jacobi equations[END_REF], see also Barron, Evans and Jensen [START_REF] Barron | Viscosity solutions of Isaacs's equations and differential games with Lipschitz controls[END_REF], Souganidis [START_REF] Souganidis | Approximation schemes for viscosity solutions of Hamilton-Jacobi equations[END_REF] and the presentation in Bardi and Capuzzo-Dolcetta [START_REF] Bardi | Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations[END_REF], Chapter VII, Section 3.2.

Obviously similar properties hold for the minmax w + Π and w + ∞ .

Finally define Isaacs's condition on I × J by : (26) sup

I inf J [g(z, i, j) k(t)+ f (z, i, j), p ] = inf J sup I [g(z, i, j) k(t)+ f (z, i, j), p ], ∀t ∈ R + , ∀z ∈ Z, ∀p ∈ R n ,
which, with the notation [START_REF] Laraki | Repeated games with lack of information on one side: the dual differential approach[END_REF], corresponds to :

h -(t, z, p) = h + (t, z, p).
Proposition 5.5. Assume condition [START_REF] Neyman | Stochastic games with short-stage duration[END_REF]. Then the limit value exists, in the sense that:

w -= w + (= w - ∞ = w + ∞ )
Note that the same analysis holds if the players use strategies that depend only at time t n on t n and z tn .

Mixed extension.

We define two mixed extensions of γ as follows: for each partition Π we introduce two discrete time games associated to γ Π and played on X = ∆(I) and Y = ∆(Y ) (set of probabilities on I and J respectively). We will then prove that their asymptotic properties coincide.

Deterministic actions.

The first game Γ I is defined as in subsection 5.1 were X and Y are now the sets of actions (this corresponds to "relaxed controls") replacing I and J. The main point is that the dynamics f (hence the flow) and the payoff g are extended to X × Y by taking the expectation w.r.t. x and y: f (z, x, y) = I×J f (z, i, j)x(di)y(dj) [START_REF]Stochastic Games and Applications[END_REF] żt = f (z t , x t , y t ) g(z, x, y) = I×J g(z, i, j)x(di)y(dj).

Γ I Π is the associated discrete time game where on each interval [t n , t n+1 ) players use constant actions (x n , y n ) in X × Y . This defines the dynamics: Φh (z; x, y) denotes the value at time t + h of the solution of [START_REF]Stochastic Games and Applications[END_REF] starting at time t from z and with play {x s = x, y s = y} on [t, t + h]. Note that Φh (z; x, y) is not the bilinear extension of Φ h (z; i, j). At time t n+1 , (x n , y n ) is announced and the current value of the state, z t n+1 = Φδn (z tn ; x n , y n ) is known. The maxmin W - Π satisfies the recursive formula:

W - Π (t n , z tn ) = sup X inf Y [ t n+1 tn g(z s , x, y)k(s)ds + W - Π (t n+1 , z t n+1 )].
The analysis of the previous paragraph applies, leading to:

Proposition 5.6.

The family {W - Π (t, z)} is uniformly equicontinuous in both variables. The HJI equation corresponds here to the Hamiltonian:

(28) H -(t, z, p) = sup X inf Y [g(z, x, y)k(t) + f (z, x, y), p ].
Proposition 5.7. 1) Any accumulation point of the family {W - Π }, as the mesh δ of Π goes to zero, is a viscosity solution of:

(29) 0 = d dt W -(t, z) + sup X inf Y [g(z, x, y)k(t) + f (z, x, y), ∇W -(t, z) ]
2) The family {W - Π } converges to W -, unique viscosity solution of [START_REF] Rosenberg | An operator approach to zero-sum repeated games[END_REF].

Finally let W - ∞ be the maxmin of the differential game Γ I played (on X × Y ) using non anticipative strategies with delay. Then: Proposition 5.8. 1) W - ∞ is a viscosity solution of (29).

2)

W - ∞ = W -.

As above, similar properties hold for W + Π and W + ∞ .

Due to the bilinear extension, Isaacs's condition on X × Y which is, with the notation ( 28):

H -(t, z, p) = H + (t, z, p) ∀t ∈ R + , ∀z ∈ Z, ∀p ∈ R n , (30)
always holds. Thus one obtains: Proposition 5.9. The limit value W exists:

W = W -= W + ,
and is also the value of the differential game played on X × Y . It is the unique viscosity solution of :

(31) 0 = d dt W (t, z) + val X×Y [g(z,
x, y)k(t) + f (z, x, y), ∇W (t, z) ].

Random actions.

We define now another game Γ II Π where the actions (i n , j n ) ∈ I × J are chosen at random at time t n according to x n ∈ X and y n ∈ Y , then constant on [t n , t n+1 ) and announced at time t n+1 . The new state is thus, if (i n , j n ) = (i, j), z ij t n+1 = Φ δn (z tn ; i, j). It is clear, see e.g. [START_REF] Mertens | Repeated Games[END_REF] Chapter 4, that the next dynamic programming property holds: Proposition 5.10. The game Γ II Π has a value W Π , which satisfies the recursive formula:

(32) W Π (t n , z tn ) = val X×Y E x,y [ t n+1 tn g(z s , i, j)k(s)ds + W Π (t n+1 , z ij t n+1 )]
and given the hypothesis one obtains as above:

Proposition 5.11.

The family {W Π (t, z)} is equicontinuous in both variables.

Moreover one has:

Proposition 5.12. 1) Any accumulation point U of the family {W Π }, as the mesh δ of Π goes to zero, is a viscosity solution of [START_REF] Shapley | Stochastic games[END_REF].

2) The family {W Π } converges to W, unique solution of (31).

Proof 1) Let ψ(t, z) be a C 1 test function such that U -ψ has a strict maximum at ( t, z). Consider a sequence W m = W Π(m) converging uniformy locally to /U as m → ∞ and let (t * (m), z(m)) be a minimizing sequence for (ψ -W m )(t, z), t ∈ Π(m). In particular (t * (m), z(m)) converges to ( t, z) as m → ∞ . Given x * (m) optimal in (32) one has with t * (m) = t n ∈ Π(m):

W m (t n , z(m)) ≤ E x * (m),y [ t n+1 tn g(z s , i, j)k(s)ds + W m (t n+1 , z ij t n+1 )],
∀y ∈ Y so that by the choice of (t * (m), z(m)):

ψ(t n , z(m)) ≤ E z,x * (m),y [ t n+1 tn g(z s , i, j)k(s)ds + ψ(t n+1 , z ij tn+1 )] ≤ δ n k(t n ) g(z(m), x * (m), y) + ψ(t n+1 , z(m)) + δ n E x * (m) y f (z(m), i, j), ∇ψ(t n+1 , z(m)) + o(δ n ).
This implies:

0 ≤ δ n d dt ψ(t n , z(m)) + δ n k(t n ) g(z(m), x * (m), y) + δ n E x * (m) y f (z(m), i, j), ∇ψ(t n , z(m)) + o(δ n )
hence dividing by δ n and taking the limit as m → ∞ one obtains, for some accumulation point x * ∈ ∆(I):

(33) 0 ≤ d dt ψ( t, z) + k( t) g(z, x * , y) + E x * y f (z, i, j), ∇ψ( t, z) , ∀y ∈ Y.
Thus U is a viscosity solution of :

0 = d dt u(t, z) + val X×Y I×J [g(z, i, j)k(t) + f (z, i, j), ∇u(t, z) ]x(di)y(dj)
which by linearity, reduces to [START_REF] Shapley | Stochastic games[END_REF].

2) The proof of uniqueness follows from Proposition 5.9.

Note again that the same analysis holds if the players use strategies that depend only at time t n on t n and z tn .

Comments.

Both games lead to the same limit PDE (31) but with different sequences of approximations: In the first case (Γ I ), the evolution is deterministic and the state (or (x, y)) is announced. In the second case (Γ II ), the evolution is random and the state (or the actions) are announced (the knowledge of (x, y) would not be enough).

The fact that both games have same limit value is a justification for playing distribution or mixed actions as pure actions in continuous time and for assuming that the distributions are observed, see Neyman [START_REF] Neyman | Continuous-time stochastic games, DP 616, Center for the Sudy of Rationality[END_REF].

Remark also that the same analysis holds if f and g depend in addition continuously on t.

A related study of differential games with mixed actions, but concerned with the analysis trough strategies can be found in [START_REF] Buckdahn | Value function of differential games without Isaacs' conditions. An approach with nonanticipative mixed strategies[END_REF], [START_REF] Buckdahn | Differential games with asymmetric information and without Isaacs' condition[END_REF], [START_REF] Jimenez | Differential games with incomplete information on a continuum of initial positions and without Isaacs condition[END_REF].

The advantage of working with discretization is to have a well defined and simple set of strategies hence the recursive formula is immediate to check for the associated maxmin or minmax W ± Π . On the other hand the main equation (HJI) is satisfied by accumulation points. The use of mixed actions in extensions of type II allows to have values in the associated game.

Concluding comments and extensions

This research is part of an analysis of asymptotic properties of dynamic games through their recursive structure : operator approach [START_REF] Shapley | Stochastic games[END_REF], [START_REF] Rosenberg | An operator approach to zero-sum repeated games[END_REF].

Recall that the analysis in terms of repeated games may lead to non convergence, in the framework of Section 3 with compact action spaces, see Vigeral [START_REF] Vigeral | A zero-sum stochastic game with compact action sets and no asymptotic value[END_REF], or in the framework of Section 4 even with finite action spaces, see Ziliotto [START_REF] Ziliotto | Zero-sum repeated games: counterexamples to the existence of the asymptotic value and the conjecture maxmin = lim vn[END_REF] (for an overview of similar phenomena see Sorin and Vigeral [START_REF] Sorin | Reversibility and oscillations in zero-sum discounted stochastic games[END_REF]).

The approach in terms of vanishing duration of a continuous time process allows, via the extension of the state space from Ω to ∆(Ω) to obtain smooth transition and nice limit behavior as δ vanishes.

A similar procedure has been analyzed by Neyman [START_REF] Neyman | Stochastic games with short-stage duration[END_REF], in the finite case, for more general classes of approximating games and developed in Sorin and Vigeral [START_REF] Sorin | Operator approach to values of stochastic games with varying stage duration[END_REF].

The case of private information on the state variable has been treated by Cardaliaguet, Rainer, Rosenberg and Vieille [START_REF] Cardaliaguet | Markov games with frequent actions and incomplete information: the limit case[END_REF] in the stationary finite framework: the viscosity solution corresponding to (ME) involves a geometric aspect due to the revelation of information that makes the analysis much more difficult. The (ME) obtained here in Section 3 corresponds to the Non Revealing value that players can obtained without using their private information.

Let us finally mention three directions of research: the study of the general symmetric case i.e. a framework between Section 3 and Section 4 where the players receive partially revealing symmetric signals on the state, [START_REF] Sorin | Limit value of dynamic games zero-sum games withvanishing stage duration: the symmetric case[END_REF], the asymptotic properties when both the evaluation tends to +∞ and the mesh goes to 0: in the stationary case this means both ρ and δ vanishes. In the framework of Section 3, with finite actions spaces this was done by Neyman [START_REF] Neyman | Stochastic games with short-stage duration[END_REF] using the algebraic property of equation ( 14), the construction of optimal strategies based at time t on the current state z t and the instantaneous discount rate k(t)/ +∞ t k(s)ds.

2. 1 .

 1 Discretization of a continuous time process and associated game. Date: March 2016. Dedicated to the memory of L.S. Shapley. A fisrt version of this work was presented at the conference "Mathematical Aspects of Game Theory and Applications", Roscoff, June 30 -July 4, 2014. This research was partially supported by PGMO 2014-LMG.

  The family of values {v Π } is uniformly Lipschitz w.r.t. t ∈ R + .

	3.1.2. Main equation.
	The first property is standard in this framework.
	Proposition 3.2.

Proposition 5.1. The family {w - Π (t, z)} is uniformly equicontinuous in both variables.

Hence the set U of accumulation points of the family {w - Π } (for the uniform convergence on compact subsets of R + × Z), as the mesh δ of Π goes to zero, is non empty.

We first introduce the notion of viscosity solution, see Crandall and Lions [START_REF] Crandall | Condition d'unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre[END_REF]. Definition 5.1. Given an Hamiltonian H from R + × Z × R n to R, a continuous real function u on R + × Z is a viscosity solution of:

and the dual condition holds.

We can now introduce the Hamilton-Jacobi-Isaacs (HJI) equation that follows from [START_REF] Guo | Zero-sum continuous-time Markov games with unbounded transition and discounted payoff rates[END_REF], corresponding to the Hamiltonian:

Any accumulation point u ∈ U is a viscosity solution of:

Note that in the discounted case, k(t) = λe -λt , with the change of variable u(t, z) = e -λt φ(z), one obtains: [START_REF] Neyman | Continuous-time stochastic games, DP 616, Center for the Sudy of Rationality[END_REF] λφ(z) = sup

The main property is the following:

Proposition 5.3. Equation ( 24) has a unique viscosity solution.

Recall that this notion and this result are due to Crandall and Lions [START_REF] Crandall | Condition d'unicité pour les solutions généralisées des équations de Hamilton-Jacobi du premier ordre[END_REF], for more properties see Crandall, Ishii and Lions [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF].

The uniqueness of accumulation point implies:

Corollary 5.1. The family {w - Π } converges to some w -. An alternative approach is the consider the game γ in normal form on R + . Let w - ∞ be the maxmin (lower value) of the continuous time differential game played using non anticipative strategies with delay. Then from Evans and Souganidis [START_REF] Evans | Differential games and representation formulas for solutions of Hamilton-Jacobi equations[END_REF], extended in Cardaliaguet [START_REF] Cardaliaguet | Introduction to Differential Games[END_REF], Chapter 3, one obtains: Proposition 5.4. 1) w - ∞ is a viscosity solution of (24). 2) Hence:

w - ∞ = w -.


A Chiocca
email: alexis.chiocca@univ-montp2.fr

F Soulié

F Deschaux-Beaume

C Bordreuil

In-situ observation and modelling of solidification and fluid flow on GTAW process

. All the tests are done thin plate of Cu-30wt.%Ni.

Introduction

The microstructure of welded joints mainly determines their mechanical properties. The weld bead microstructure results from the solidification process occurring in the weld pool at the solid/liquid interface. This process of solidification includes mechanisms that are driven by heat and fluid flow. The understanding of solidification mechanisms in relation with weld pool behaviour is then a key point to analyse microstructure formation and to control joint strength.

Compared to the common case of solidification such as casting, welding processes induce high thermal gradients and high solutal gradients at the solid/liquid interface and the front velocity is much faster. Due to these solidification conditions, columnar and equiaxed dendritic microstructures are broadly observed. The characteristic length of the weld pool is around 10mm. In the weld pool, fluid flow and heat transfer can be complex and can vary from point to point along the interface [START_REF] Kou | Welding Metallurgy[END_REF]. Depending on the position on the solid/liquid interface, the thermal gradient and the solidification speed change. At the trailing edge, the solidification speed is maximal and the thermal gradient is minimal but at the fusion line, it is the opposite. To understand the microstructural evolution along the interface, information from two scales are necessary: (i) macroscopic fields around the interface and (ii) microscopic observations. Macroscopic fields (temperature gradient, for example) gives driven force for the solidification. On in-situ observations, Zhao et al. [START_REF] Zhao | ASM Proceedings of the International Conference: Trends in Welding Research[END_REF] proves that the fluid flows have complex shapes that can influence solidification. Microscopic observations tells us the behaviour of the interface (stability and/or fragmentation). Delapp et al. [START_REF] Delapp | Proceedings of the 7th Internatioanl Conference on Trends in Welding Research[END_REF] based on microscopic observations showed that fluid flow could produce fluctuation of the front speed giving ripples on the weld bead. 8.1 Gibbs-Thomson coefficient, Γ (K.m) [2] 1.96e-7 Stability constant, σ * [2] 0.0253 Dynamic viscosity, µ (P a.s) [9] 3.1e-3 Liquidus temperature, T l (K) [8] 1520 Diffusion coefficient D l (m 2 /s) [9] 1e-9 Solidification range ∆T r (K) [10] 1472-1520

Modelling dendritic growth under weld solidification conditions is still a major issue. Different works introduced important phenomena observed during solidification. Hunziker et al. [START_REF] Hunziker | [END_REF] gives, for the case of welding, an exponential relation between the solidification speed and the total undercooling. Gandin et al. [2] have simulated a dendritic growth with fluid flow showing the interactions between the growth velocity and the fluid flow ahead the interface.

The present work wants to develop a solidification model in welding conditions with the help of different experiments. Experiments are done on Cu30wt.%Ni plate and the front velocity, the fluid flow velocity and the thermal field are analysed under different thermal conditions. Then the comparison is done with the model which is adapted from two different models [2,[START_REF] Hunziker | [END_REF].

In situ Observations

In this part, experimental setup is described with the two scale devices to investigate solidification under welding conditions. One purpose is to compare microstructure evolution with different solidification conditions induced by different process parameters. The main process parameters are the intensity of welding and the down-slope. The intensity has a direct control on the power input and the down-slope allow the solidification to be slower by decreasing the intensity before the end of welding with a negative slope. In order to estimate the kinetic of solidification, front velocity, the fluid flow velocity and the thermal fields will be measured in-situ.

Details of the experimental setup and errors made in the measurement can be found in an other article [START_REF] Chiocca | Science and Technology of Welding and Joining[END_REF].

Material

For an efficient comparison with the theoritical models, the simple binary alloy Cu30Ni has been chosen. This alloy is composed of 70% of copper and 30% of nickel and its phase diagram has only a solid solution in the solid part. The physical parameters useful for the model are listed on table 1. The partition coefficient, the diffusion coefficient and the liquidus slope are supposed constant. The liquid density is taken at the liquidus temperature.

Experimental setup

The experimental setup design is illustrated on figure 1. The observations are done on a static weld pool after or during the arc extinction. The weld pool generated by a GTAW torch is full-penetrated to allow the observation on the back side. On this side, The liquid metal is protected from oxidation by a chamber full of argon. The coupon is 1.6 mm thick and 70 mm of diameter.

Observations of the weld pool for solidification front and fluid flow measurements are obtained thanks to an high speed camera. This camera is placed under the plate and observe the area represented by a square on figure 2. The observation area is lighted up by a LASER in order to insure the segmentation of the liquid and solid phase. The back side of the plate is filmed with the IR camera and two thermocouples are placed below the plate for the calibration of the temperature field. The observation area represent the whole weld pool and a large solid part.

2.3. Field measurements 2.3.1. Thermal fields IR camera gives access to temperatures in the solid phase and near the interface on the back surface, after a calibration step with the thermocouples.

The solid part temperature of the sample along a the dashed line on figure 3 is plotted on figure 4. The temperature increases from the edge to the centre of the sample. Due to the calibration procedure, the measured temperature is only available in the solid phase. Nevertheless, the temperature gradient in the solid phase can be approximated. With this setup configuration, the gradients measured are about 2 * 10 5 K/m and the temperatures at the interface are about 1300 K (Fig. 3 and4).

Velocities fields

Two kind of velocities are extracted from the high speed films: the velocities of the solidification front and the velocities of the fluid flow. The velocities of the solidification front are obtained by detecting contours of solidification front in images. The field of velocities in the liquid are computed by Particle Image Velocimetry (PIV) processing [11]. Velocities of particles at the vicinity of the front are estimated around 50 mm/s (Fig. 5 and6) and front speed are between 1.5 mm/s and 4 mm/s (Fig. 7 and8) depending on the adjustments of the power source.

Fluid flow curves (Fig. 6) show a peak at the vicinity of the front. The projection of the velocity of the fluid particle is high when it goes out the front and decreases while approaching the weld pool centre. Only few eddies are observed on videos telling us that the flow is mainly laminar. For the back side, a vortex is observed from the interface to the center of the weld pool. Fluid velocity are more important at the extinction of the arc and then tends to decrease. The decreasing of the radius of the weld pool (Fig. 8) is almost linear with a decrease of the slope at the end of the curve. Irregularity all along the curve might be caused by oscillation of the weld pool.

Observations

A complete observation test is illustrated on figures 3, 5 and 7. For this test the intensity applied on the electrode is 70 A during 5.5 seconds without down-slope. After extracting and processing data, the solidification speed is 3.43 mm/s (Fig. 8) and the velocity of the fluid flow at the vicinity of the interface is 39 mm/s (Fig. 6). The field of temperature allow us to estimate the gradient near the interface at 1.82 * 10 5 K/m and the total undercooling at 191 K (Fig. 4). The undercooling is equal to the difference between the temperature at the interface and the liquidus temperature.

During high speed imaging, small particles are ejected from the interface. The videos tell us that the solidification is mainly columnar and that at the very end some equiaxed grains seems to be observed. All the dendrite tips are submerged during growth. The tip grows below the liquid surface, the dendrite is then visible when the primary branches grow, becoming fatter and joining with its neighbour dendrites.

With measurements, dimensionless numbers can be estimated in order to appreciate the heat and mass transfer in this configuration. In particular, the Péclet numbers show the dominance of convection over diffusion P v = R * v D l and P u = R * u D l with v the solidification speed, R the radius of the dendrite tip and u = u piv -v the velocity of the liquid with respect to the front. These Péclet numbers are respectively estimated at 10 -1 and 10 0 in order of magnitude, demonstrating that the liquid is well homogenized. The value of the Reynolds number Re = ρuL µ with the experimental velocity is estimated at 1300 confirming the laminar state.

Welding solidification modeling

It exists a lot of theoretical models on solidification. The variations in the models depend on the conditions of solidification. For example, with slow or rapid solidification. In the case of slow solidification or low Péclet number [12] assumptions can accelerate computations of the model. For welding solidification, the conditions taken for this study are the one found in the KGT model [1]. Kurz et al. have developed a model for rapid solidification with assumptions that can be applied to welding. This model results in two equations with two unknowns:

∆T = mC 0 1 - 1 1 -(1 -k)Ω + 2Γ R R = 2π 2Γ mG c ξ -G (1)
with R the radius of the dendrite tip, ξ a factor present in the model [1] and Ω = Iv() the supersaturation equal to the Ivantsov function. Even if it is suitable for welding, it doesn't take into account the fluid flows. It is only a diffusive model.

Fluid Velocities

To implement the fluid flow without using a fully numerical model, one solution is to add a boundary layer δ ahead the front. It is the solution presented in the article of Gandin et al. [2]. It takes into account the intensity and the direction of the flow by changing the thickness of the layer. Changes are made on the supersaturation:

Ω = P v exp(P v )(E 1 (P v ) -E 1 (P v (1 + 2δ R ))) δ R = 2 ARe B 2r Sc C sin(θ/2)
(2)

E 1 (P v ) = ∞ Pv exp(-τ ) τ dτ (3)
with A, B and C parameters defined in the article of Gandin et al. [2], Re number of Reynolds and Sc number of Shmidt. However this model, as it is described, is made for low Péclet numbers corresponding to slow solidification.

Welding growth

To take into account high thermal gradients and velocities measured during the experiment, a model is developed. The system to solve is the same one as in equation 1 but the definition of the supersaturation is the one of equation 2. This model shows an exponential relation between the undercooling and the front velocity (Fig. 9). Fluid flow velocity has the strongest influence when it is in the opposite direction of the front growth. But when it is in the same direction, the influence on solidification is weakest and is found null as it is shown in the study of Gandin et al. [2]. In this model, the temperature gradient has no influence on front velocity but a big influence on the stability limit of the planar front. As the gradient increase, the stability limit velocity increases too [1].

Results and discussions

The models presented previously show relations between intensity and direction of the fluid flow, velocity of the front velocity and thermal parameters. In this section, the purpose is to know if the results coming from the models can be observed experimentally. Different tests are summarized on table 2.

Macroscopic results

An increase of power input (70 to 90A) leads to larger and hotter weld pool. This can explain the two main differences in the tests : a lower front velocity for 90A and lower fluid velocity for 70A. With higher temperature in the weld pool, it is expected that the front needs to transform more energy to change phase. With higher temperature on the arc surface due to more energetic plasma, the gradient in temperature can be more important leading to higher Marangoni forces inducing higher fluid velocities. The undercooling at the interface, corresponding to the difference between liquidus temperature and the temperature measured at the interface, does not seem to be influenced by the power input of the welding process. Looking at the energy increasing from test 1 to test 4, the undercooling values between test 1 and test 4 have not big variations.

According to the model, the solidification front velocity increase with the level of undercooling at the interface. Between the different test, the same level of undercooling gives different value of solidification speed. With the experimental values applied on the adapted model (Fig. 9), the undercooling found vary between 5 K and 8 K. It is very few compared to the experimental results about 200 K. The values might be wrong because the limit between liquid and solid is blur. The temperatures measured at the interface are underestimated because it is not the temperature of dendrite tip. The dendrites tip might be submerged in the liquid while the measurement is done at the surface, corresponding to a point before the tip where the dendrite has already cooled.

Interface behaviour

At the microscopic level, several variations have to be highlighted. The direction of solidification and the direction of the fluid flow are mainly in the same direction. In the presented model, the fluid flow has in that case no influence on dendritic growth velocity.

From a test to an other, the morphology is dendritic columnar in the window of observation of the high speed camera. If the observation area is placed on the centre of the weld pool, at the end of the solidification, grains which do not come from the front grow inside the liquid bulk. These grains are either equiaxed grains that have nucleated in the bulk or columnar grains coming from the upper side of the plate. The first assumption could be qualitatively explained. At the end of the solidification, the gradient can be lower, the solidification speed accelerate and impurities driven by the flow might be blocked in the centre. If the top of the weld is solidified at first, columnar grains can nucleate and grow downward until the bottom of the weld.

The dendrite spacing measured on all the tests do not show great variations over the dendrites velocity. The spacing between dendrites is more sensitive to the temperature gradient. The changes of spacing between the different measurements have been found coherent with the law of David et al. [13]: d = a(G 2 v) -0.25

Growth modeling during solidification

During the solidifcation after arc extinction, it is found that the undercooling keep a quasi constant value while the gradient decrease. These results are coherent with the theory. A constant undercooling verify a constant value of velocity according to the model. The decreasing of the gradient show the flattening of the temperature field and may cause the growth of the supposed equiaxed grains in the centre. It has already been found that discrepancy are observed for fluid flow direction in the range of [ 7π 4 ; π 4 ] [2]. The model presented in this case is finally found not pertinent for the direction of fluid flow. However the fluid flow does not seems to have an influence on front velocity speed. For three equal velocities of fluid, the front velocities are different.

Conclusion

Experimental results have been presented on solidification of a binary alloy (Cu-wt.%30Ni) processed by GTAW. Several welding conditions have been carried out to vary solidification conditions. Assumption have been made to explain the results and to compare them to the modelling data. A lot of data, that was not measured yet, are determined in this study: the solidification front velocity, the fluid flow velocity and the thermal field.

The macroscopic observations explain the solidification behaviour. The velocities are driven by the thermal conditions, the undercooling only plays a role for the position of the dendrite front between solidus and liquidus.

The lack of accuracy of the thermal measurements are the main problem of the difficulty to set experiments against modelling. It is not possible to measure the temperatures of dendrite tip inside the liquid. A way of improvement could be a prediction of the dendrite temperature tip with the available informations on the gradient and the temperatures.

Figure 1 .

 1 Figure 1. Drawing of the set-up showing the different measurements done around the plate.

Figure 2 .

 2 Figure 2. Geometrical description of the observed system, at the top a cross-section view and at the bottom a bottom view.

Figure 3 .

 3 Figure 3. Thermal Field under the plate. The plain contour --circle the weld pool.

Figure 4 .

 4 Figure 4. The --curve correspond to temperature along the dashed line ----on figure 3. The dashed line ----is the temperature gradient at the interface. The approximative limit between solid and liquid is represented by the dotted line • • • • • •.

Figure 5 .

 5 Figure 5. Velocity field ahead the front. On the left the dendrite tips growing from the left to the right and on the left the liquid pool.Figure 6. Evolution of the velocity vectors from the left to the right along the straight lines on figure 5.

Figure 6 .

 6 Figure 5. Velocity field ahead the front. On the left the dendrite tips growing from the left to the right and on the left the liquid pool.Figure 6. Evolution of the velocity vectors from the left to the right along the straight lines on figure 5.

Figure 7 .

 7 Figure 7. Contour detection of the front with in plain line --the raw contour and in dashed line ----the estimated circle.

Figure 8 .

 8 Figure 8. In plain line --the decreasing of the radius of the weld pool across time and in dashed line ----the mean slope.

Figure 9 .

 9 Figure 9. Solidification speed with respect to the undercooling.

Table 1 .

 1 Parameter used for the analytical model.

	Parameters	Values
	Liquidus slope, m (K/wt.%N i)	1.40
	Partition coefficient, k	3.89
	Nominal alloy composition C 0 (wt.%N i) 30
	Liquid density, ρ (g/cm 3 )[8]

Table 2 .

 2 Results on four tests.

	Test number Intensity Down-Slope v	u piv	G	∆T
	1	70 A	0 s	3.46 mm/s 39 mm/s 182.4 K/mm 191 K
	2	80 A	0 s	3.03 mm/s 49 mm/s 184.6 K/mm 185 K
	3	90 A	0 s	2.90 mm/s 50 mm/s 187.9 K/mm 215 K
	4	90 A	2 s	1.38 mm/s 50 mm/s 123.8 K/mm 226 K