Jean-Patrick Brunet

Henri Sohier

Mouadh Yagoubi

Mathieu Bisquay

Pascal Lamothe
email: pascal.lamothe@mpsa.com

Pascal Menegazzi Valeo

Pascal Menegazzi
email: pascal.menegazzi@valeo.com

Simulation architecture definition for complex systems design: A tooled methodology

For the design of complex systems like in the automotive industry, the use of Model Based Systems Engineering (MBSE) is being considered as a promising solution in order to formalize and communicate information. Numerical simulation is also routinely used as a tool to answer potential design questions that arise. However the link between MBSE and simulation still needs further improvement. In this work, a tooled methodology is proposed in order to enhance the link between system architecture and numerical simulation. In a first step, a solicitation package is formalized and implemented in a SysML-based tool to define the simulation needs. In a second step, a tool that allows to define the simulation architecture and to pilot the execution of the simulation is developed. We show that thanks to the proposed process and exchange format between the system and simulation architects, model reuse and agility is improved in a complex systems design.

Introduction

Context

The research presented in this paper was conducted within the AMC project (Agility and Design Margin) which is taking place in the research institute IRT SystemX. One of the main objectives of this project is to propose a tooled methodology for simulation based complex system design. We focus on design questions that can be answered by numerical simulation as this later has been considered as an efficient tool for decision-making in the design of complex systems. The simulation is considered as a complex system that may deviate from the original system it represents. This deviation can be explained by the need to account for different specifications that are specific to the simulation. Such specifications might be the implementation of physical phenomena or balance between accuracy and speed for example. The complex task of linking system and simulation architecture led to the emergence of a fairly new role, the simulation architect [START_REF] Retho | Collaborative methodology for virtual product building to support aerial vehicles with electrical propulsion design[END_REF]. The simulation architect has a key role in the proposed methodology as he interprets the needs from the system architect in order to specify a suitable simulation that would provide the required results for decision. In a standard V-Model (Fig. 1) multiple needs can be expressed during the design phase and the simulation architect will need to answer each of those within the context of the system. The global process begins with the definition of high level stakeholders needs. Then, the system architect defines the system architecture by developing SysML diagrams at the operational, functional and physical levels. Following the need in terms of simulation, the system architect formulates his solicitation to the simulation architect based on a formalism that we have proposed in a previous work [START_REF] Sohier | A Tooled Methodology for the System Architect's Needs in Simulation with Autonomous Driving Application[END_REF]. This solicitation package would contain the system information as well as required initial QCD (Quality Cost Delay) requirements for the simulation. Once received, the simulation architect constructs the elements of the simulation architecture progressively, using the new methodology we propose in this paper. If needed, he will interact with the system architect to alter the initial requirements to better match the simulation possibility and system needs. The resulting simulation is then integrated in a simulation framework connected to a decision support tool. This serves a double purpose, analyzing and post-treating the results of the simulation with a close collaboration between the system and simulation architects and if needed, generating new results with minor modifications of the specification (eg: minor change of actuation distance etc…). Once the collected data are sufficient, the system architect can take the final decision and the process end. In the AMC project different industrial partners (PSA group, Renault, Valeo, Airbus) and software providers (Sherpa engineering, Digital Product Simulation, Siemens) are involved in defining and improving the proposed process. For this, a practical design problem is used: The early design of an autonomous vehicle passing traffic light.

Industrial problem

One of the two industrial problems considered in the project AMC is the design of an autonomous vehicle which can pass traffic lights. Such industrial problem is important as it allows the different partners of the project to share a common language, to identify the current needs in terms of methods and tools, and to validate new solutions. The considered vehicle has a sensor which provides the color and the distance to a traffic light within a certain range subject to uncertainties. The vehicle then follows a control algorithm to adapt the speed to the traffic light and anticipate possible color changes. The speed is reduced when approaching the traffic light to avoid brutal or even unreachable decelerations. Furthermore, when the color of the traffic light changes to orange, the vehicle stops if it cannot pass the traffic light within one second. In the use case considered in this paper, the system architect defined four parameters: two for the characterization of the sensor and two for the control. Setting those parameters impacts two important decision criteria: the cost of the sensor and the total energy consumption of the vehicle. The system architect formulates the design question as a multi-objective optimization problem and transfers the solicitation package to the simulation architect, who will design the simulation architecture, perform said simulation(s) and provide the results to the system architect in order to take a decision regarding the design of the vehicle.

Agility in complex conception cycle

As mentioned in the previous section, the system architect will regularly request simulation to obtain sufficient data in order to take decision. In some cases the link between system engineering and simulation is very direct, with no important change between system and simulation architectures. In the STIMULUS tool (3) formal textual requirement and state diagrams can be tested using random variables, and CIL4Sys applications (4) allow testing of formal sequences and state diagrams through on the shelf or simple plant simulation models and user defined environments. However, complex simulation models are often required for the representation of physical systems with both controls laws and physical phenomena. It usually, then, becomes more difficult to mix system and simulation as they would need to involve different peoples and topologies. MBSE being considered a semi-formal or even informal language, automation and communication between the different stakeholders is complex [START_REF] Rauzy | Foundations for model-based systems engineering and model-based safety assessment[END_REF]. Currently the information needed to be transferred from the system to the simulation actors are ill defined with no specific and readily available format, as SysML in the case of system engineering. It is commonplace to use a documentbased process (emails or written documents) to provide those information leading to loss of traceability and lack of standardization. We presented a new tooled methodology in a previous article [START_REF] Sohier | A Tooled Methodology for the System Architect's Needs in Simulation with Autonomous Driving Application[END_REF], which aims at enhancing the link between system architects and simulation experts by formalizing simulation needs by exploiting the SysML diagrams of the system architecture. Starting from this formalization of the simulation needs by the system architect, a new tooled methodology is proposed in this paper, extending our previous work, enabling the definition of the simulation. The aim of our tooled methodology is to formalize the exchanges between system and simulation architects in order to limit the loss of information, leverage past simulations, formalize requests for new models and give the system architect access to the right data for a supported decision. This methodology is presented in figure 2. The solicitation as described in section 3 is sent to the simulation architect who can then assemble a simulation architecture based on specifications indicated for each system function. In order to do so, he can use past simulations associated to system specifications close to the one requested in the solicitation (with the help of computerized inferences) or, if no existing model match the specification he can generate a new model through the use of the MIC (Model Identity Card) standard [START_REF] Sirin | A Model Identity Card to Support Simulation Model Development Process in a Collaborative Multidisciplinary Design Environment[END_REF][START_REF] Sirin | Multidisciplinary Simulation Model Development: Early inconsistency detection during the design stage[END_REF][START_REF] Fontaine | Modélisation théorique et processus associés pour Architectes Modèle dans un environnement multidisciplinaire[END_REF]. The generated MIC will define all necessary requirement in order for a model provider to create a matching model. Once the simulation architect validates a simulation architecture corresponding to the solicitation, he will generate and link the corresponding simulation to a simulation framework. This framework will allow to generate results for a decision support tool within the specified scope of the model. The system architect, in collaboration with the simulation architect, will then be able to analyze the results and either take a final decision or use the knowledge gained to alter the initial solicitation and repeat the whole process. This allows for fast agile loops within the global V-cycle of the system design process. Agility is also improved on the simulation side, where the simulation is developed iteratively in function of the changing needs from the system side. For each step of said process, we proposed a tooled methodology: one dedicated to the System Architect (described in section 3), another for the Simulation Architect (described in section 4). We believe that through the proposed methodology, true agile cycles can be implemented within a MBSE based design cycle by furnishing a formal framework for exchanges between the stakeholders (system architect, simulation architect, and model providers). Allowing for faster exchange of information, it improves information based decision making and more generally accelerates the cycles. 3 Solicitation package from the system architect

Current practice and alternative

The development of a system, considered here to be supported by a model of the system architecture, can raise various questions requiring simulation. Such question is sometimes manually summarized in a textual document, whether it is a Word document or an email. This textual document occasionally includes diagrams illustrating the system to simulate. It is often challenging to have a complete and clear question. Furthermore, textual documents make it much harder to identify questions which have already been answered and which were similar, preventing from efficiently re-using past simulations. Textual documents finally make it hard to validate that the system is correctly represented in the simulation, and that the simulation results correctly answer question.

Instead of this documental approach disconnected from any model of the system architecture, the process described in this paper first uses the concept of solicitation from the system architect. This concept was introduced in (2). The solicitation is a documented question of the system architect, directly based on the model of the system architecture. Indeed, the model of the system architecture precisely aims at sharing information about the design of the system with different types of people, from mechanical designers to electrical designers, and it is also perfectly suited for a collaboration with simulation exerts.

Content of the solicitation package

The system architect can use the system architecture to communicate three types of information in his solicitation: the part of the system related to the question and which needs to be simulated, the question itself, as well as the environment scenarios to be tested. The part of the system related to the question is important as the question is generally not about the full system. For example, when the question is about the ability of an autonomous vehicle to pass traffic lights, the air conditioning system may typically be ignored. When describing the part of the system related to his question, the system architect should choose the appropriate level of detail. For example, in the case of an autonomous vehicle passing traffic lights, he may choose not to take into account the internal components and the exact way the sensor works to output the color and distance of a traffic light.

The question asked by the system architect in his solicitation can be, amongst others, a request for validation. For instance, a system architect may want to check that the average energy consumption of a car is below a standard reference. The question can also be a request for optimization. A system architect may want to find the best design parameters to minimize one or more performance criteria, like the average energy consumption or the production cost. Whatever the question is, its formulation can be supported by the model of the system architecture. The model includes for example the requirements to be tested in a validation, or the unset parameters whose values are to be explored in an optimization. Furthermore, the system architect may also want to monitor specific properties of the system, like its speed or its instantaneous consumption, which he can also select in the model of the system architecture.

Finally, the environment scenarios are also an important part of the solicitation. The environment scenarios generally start as a rough description of the environment and its evolution during the test. In the case of an autonomous vehicle, it can first be decided to have a straight road with a traffic light at 500m. In the process described here, this rough description is part of the solicitation of the system architect. The environment scenario is then progressively refined when the simulation is developed, up to the exact properties of the road for example. The environment scenario can be defined thanks to the model of the system architecture which characterizes the system's environment, including roads and traffic lights. However, while the model shows that there can be a traffic light on the side of the road, the system architect must specify in his solicitation that the test should include a traffic light at 500m. Thus, the formulation of the solicitation is facilitated by the model of the system architecture as it includes various necessary information. Furthermore, the solicitation is not only generated as a textual document, but also as a model. While numerical continuity is generally lost with a textual document, the model of the solicitation has multiple logical links with the model of the system architecture it comes from.

Implementation of the solicitation package

Using the model of the system architecture to formulate the solicitation requires new software functions. Indeed, it relies on the processing of numerous and complex relationships between the data of the model, from state diagrams to sequence diagrams and bloc diagrams. Some of these software functions have already been prototyped as a plugin in an existing SysML editor called PhiSystem, provided by Sherpa Engineering and based on Papyrus. Fig. 3 shows the menu added to the top bar. The menu first makes it possible to copy the system architecture, which is a practical solution to associate the solicitation to a certain version and configuration of the system. The menu also offers a quick access to diagrams as well as summaries of information at the different levels of the system architecture (operational, functional, and physical) in order to select the perimeter of the question and the necessary level of detail. The menu finally allows to ask a question based on the requirements of the system, and to hide the part of the system which is not related to the question. For example, if the objective of a system architect is to optimize the control and sensor properties of an autonomous car to minimize both the cost and the energy consumption when passing traffic lights, he can first go to "Use case diagram" in the top-bar menu of Fig. 3 and select the use case "To pass traffic light". The implemented software functions automatically find the related data and highlight the perimeter of the question in the menu "Functions connection (IBD -Internal Block Diagram)", as shown in Fig. 4. The level of detail can be set in the menu "Functions hierarchy (BDD-Block Definition Diagram)".

Proposed methodology for the simulation architecture definition

The solicitation defined in the previous section allows to carry the simulation needs (quality, cost and delivery) from the system to the simulation. The simulation architect will then need to define the simulation architecture. This would entail finding existing models, pre/post-processing functions, specify missing models or functions and ensure that the final architecture correctly represents the system and responds to the needs expressed in the solicitation. The simulation architect needs to convert the assembly of system functions into a coherent assembly of models and software functions.

In order to keep a consistent approach to those simulation blocks, it is chosen to describe them through the properties of the "Model Identity Card" (MIC) which was initially presented in [START_REF] Sirin | Supporting multidisciplinary vehicle modeling towards an ontology-based knowledge sharing in collaborative model based systems engineering environment[END_REF]. The MIC aims at fully specifying a simulation model independently of its execution, as such it can be used to identify an existing simulation model (useful in a reuse case) or to specify requirement for a provider. This approach allows to create consistent architectures mixing existing, reworked, or specifically created models each connected through the ports properties defined in their MIC. The entire workflow of simulation can be validated through the ports properties.

Once assembled the final simulation generated from this simulation architecture will itself be described as a MIC model. The MIC format allows to recursively describe all the internal models of the simulation architecture keeping a traceability of its construction. It is important to note that, at the moment, while a MIC model contains an indication of its validity scope through the indication of range for each variable, it does not contain indication on its accuracy. This might be added in future evolution of the format.

The following sections aim at describing the tools put into place to assist the simulation architect in creating, validating, executing and reporting the simulation architecture that correspond to a specific solicitation.

Developed components a. Solicitation package import module

This tool aims at eliminating the uncertainty in information and traceability incurred in document or email request. The solicitation generated by the system architect is sent, unaltered to the simulation architect tool which will then extract information on the simulation needs, scope and granularity as well as start the traceability of the associated simulation architecture.

b. Models library

Existing simulation models are stored in a library that link both the MIC and the system function (or functions) matching the given model. This aims at facilitating the reuse of previous simulation work by offering visibility of existing models as well as proposing inference type associations to link system function in the new architecture to semantically close ones from previous work. Fig. 5 represents the process of creating a new simulation architecture using the proposed tool. A representation of the filtered system architecture is present in the top with the simulation architecture situated below it. On the right, the models library is displayed. In this figure, the simulation architect already started the creation of the simulation architecture with two system functions (represented in green in the architecture) already associated to some simulation models. Based on the needs, some simulation models have also been added with no direct association to the system functions. At the step shown in the figure, the simulation architect has selected a system function (in red) that he wants to associate to a model. The models library tool (panel on the right of the figure), based on inferences at the system level, proposes a previously used simulation model called "ElectricVehicle" that could be used to simulate this function. The simulation architect selected it (reason for which the simulation model has a green outlay) and the system presented the other system blocs that would also be fulfilled by this model (in orange).

In the case a system function would not have a suitable existing model, the simulation architect can choose to create a specification MIC that would be added to the simulation architecture. Upon having an associated model with that MIC it would be added to the models library for future reuse.

c. Simulation Architecture Verification and creation tool

As presented in Fig. 5 the simulation architecture tool allows the simulation architect to progressively define the simulation architecture while maintaining consistency with the system architecture. While new models are added to the simulation architecture, the tool uses the MIC attributes to check the consistency of the model. This is insured through proposing automated links between similar ports (matching dimensions, units and directions), thus verifying that linked models possess compatible ports. Once the whole architecture is verified by the This methodology also simplify the reuse of prior simulation model by using formal description of those (MIC) and association with prior system function. By presenting those prior model in a standardized and centralized library associated with computer generated suggestion, the workload associated with reuse and validation of prior model can be greatly reduced. Lastly, this proposed methodology also facilitate agile project management during the conception cycle. By allowing centralized, traceable, communication between the shareholders of the project as well as facilitating reuse and simplifying validation of new simulations against the solicitation package, we ensure faster simulation iteration during the conception cycle.

As a future work, we aim to explore the impact of this novel methodology on the whole conception cycle, by considering state-of-the-art agile methods.

Fig. 1 .

 1 Fig. 1. Details of the simulation based decision cycle in a system V-model.

Fig. 2 .

 2 Fig. 2. Proposed methodology for the simulation definition.

Fig. 3 .

 3 Fig. 3. Menus added to the top bar in PhiSystem.

Fig. 4 .

 4 Fig. 4. Perimeter of the system related to the solicitation.

Fig. 5 .

 5 Fig. 5. Screenshot of the tool during the simulation architecture definition process.


Guillaume Touya

Cécile Duchêne

CollaGen

Keywords:

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Approach and Methods

The Collaborative Generalisation Framework

Automatic generalisation research first tried to answer to the questions "why, when and how to generalise?" [START_REF] Mcmaster | Cartographic generalization in digital environment: A framework for implementation in a gis[END_REF]Shea 1988, Brassel and[START_REF] Brassel | A review and conceptual framework of automated map generalization[END_REF]. Inspired from [START_REF] Regnauld | Evolving from automating existing map production systems to producing maps on demand automatically[END_REF] and [START_REF] Duchêne | Combining Three Multi-agent Based Generalisation Models: AGENT, CartACom and GAEL[END_REF], the CG framework we define aims at allowing an answer to the question why, when and how to apply which automatic process? Within this framework, automatic generalisation processes are applied on parts of space where they are expected to be efficient while side effects are likely managed at generalisation neighbourhood (Fig. 1).

Fig. 1. The collaboration principle between generalisation processes. A process 1 is carried out on the town area, a process 2 on the rural area, and then a process 3 on the mountain area and finally a process 4 is carried out on the road network. Side effects are corrected at the neighbourhood (dashed arrows) of application spaces.

Overview of CG Framework Components

Generalising data within the CG framework brings about specific problems like process interoperability, treatment heterogeneity or side effects [START_REF] Touya | First Thoughts for the Orchestration of Generalisation Methods on Heterogeneous Landscapes[END_REF]. The framework function analysis lead to a six main components and three resources groups structure (Fig. 1.2). Partitioning builds the geographic spaces where the available generalisation processes can be applied. The Translator parameterises the processes. The Registry chooses the process to generalise a given space. The Observation provides online evaluation. Side effects are managed by the eponymous component. Finally, the Scheduling Component orchestrates the whole process. Within the CG framework, we developed the CollaGen model that implements all the aspects of collaborative generalisation defined in Fig. 1.2. The next parts describe how each aspect is managed in CollaGen.

The CollaGen Model

Automatic Generalisation Processes

CG consists in making several available automatic processes collaborate to optimise the entire map generalisation. Thus, we consider as an available automatic generalisation process, any process that can be triggered on geographic data from the software platform the CG framework is developed on. A process is a computer program that automatically triggers generalisation operator (e.g. simplification, displacement...) sequences on geographic objects. For instance, in CollaGen, developed on a research platform [START_REF] Renard | Capitalisation problem in research -example of a new platform for generalisation: Cartagen[END_REF], AGENT (Ruas 1999), CartACom [START_REF] Duchêne | The cartacom model: a generalisation model for taking relational constraints into account[END_REF], GAEL [START_REF] Gaffuri | Field deformation in an agent-based generalisation model: the gael model[END_REF], Least Squares [START_REF] Harrie | Simultaneous Graphic Generalization of Vector Data Sets[END_REF] or road selection processes are implemented and thus available for collaboration. Processes published as web services [START_REF] Regnauld | Evolving from automating existing map production systems to producing maps on demand automatically[END_REF][START_REF] Neun | Web service approaches for providing enriched data structures to generalisation operators[END_REF] could also be considered as available processes.

Geographic Spaces

We define a geographic space as a geographically meaningful extract of the data that can be a relevant input for a given generalisation process [START_REF] Touya | Relevant Space Partitioning for Collaborative Generalisation[END_REF]. The use of geographic spaces in CollaGen is useful for both optimising the use of the existing generalisation processes and partitioning the data to avoid the processing of very large datasets. The geographic spaces (Fig. 3) can be areal (e.g. urban or rural area), thematic (e.g. road network or vegetation) or both areal and thematic (e.g. mountain roads). It can be noticed that with such a definition, the geographic spaces do not form a mathematical partition as metric spaces can overlap and thematic spaces cross metric spaces. Spaces can be cut in several portions in order to keep small spaces and minimise processing time.

Moreover, some emerging spaces can be managed by CollaGen: they are sub-spaces where conflicts remain unsolved. During the generalisation of a space by a given process, the observation component can identify conflict clusters (close conflicting objects) that emerge as sub-spaces to be generalised by another process than the one processing the whole space ( §2.3.9).

Formalised Knowledge in CollaGen

Formalised cartographic generalisation knowledge is necessary to allow process collaboration. The model designer (e.g. we are the CollaGen model designer) has to provide a generalisation ontology and sequencing rules; a process developer (the one that makes a new generalisation process available for collaboration) has to provide a process description; the user (the one that generalises data) has to provide generalisation constraints and operation rules (Fig. 4).

-The generalisation ontology is the support for sharing a common vocabulary in the collaboration model. It helps to express that map specification ("buildings smaller than 50 m² must be deleted"), a data type ("BATIMENT" class) or a process requirement ("agent_building" in AGENT process) deal with the same concept of 'building' for instance. -The sequencing rules are guidelines for sequencing applications of processes on geographic spaces. As in the Global Master Plan of [START_REF] Ruas | Strategies for automated generalization[END_REF], it allows to formalise that 'road selection' should be triggered before 'urban generalisation' for instance, or that 'urban areas' should be generalised before 'rural areas'.

-The generalisation constraints and the operation rules formalise the map specifications. The formal model summarises past research [START_REF] Beard | Constraints on rule formation[END_REF][START_REF] Burghardt | Investigations on cartographic constraint formalisation[END_REF][START_REF] Duchêne | Combining Three Multi-agent Based Generalisation Models: AGENT, CartACom and GAEL[END_REF]) and allows to express different constraints like 'building area > 0.2 map mm²', 'building block density should be preserved' or 'very concave buildings should maintain initial concavity with 10% margin'. -A process description formalises the capabilities of a generalisation process. As for web services composition, capabilities are described with pre and post conditions [START_REF] Lutz | Ontology-Based Descriptions for Semantic Discovery and Composition of Geoprocessing Services[END_REF]. In CollaGen, pre conditions are adapted spaces for the process and post conditions are a priori satisfied generalisations constraints and operation rules (after the process has been applied). [START_REF] Touya | Collaborative Generalisation: Formalisation of Cartographic Knowledge to Orchestrate Different Generalisation Processes[END_REF]) describes how each piece of formalised knowledge is modelled in CollaGen and how it can be acquired by a user, a process developer or the model designer.

The CollaGen Workflow

CollaGen proposes a workflow for CG to chain the components actions (Fig. 5). As a simplified example, let us imagine an area that is firstly partitioned (into two rural spaces, an urban space and the road network), and three processes (AGENT, CartACom and Beams). Then, the urban space is chosen and the registry chooses AGENT. The process is parameterised according to the constraints and generalised. Generalisation is evaluated as good and there is no side effect. Then, the next chosen space type is rural and the two instances are ordered. CartACom is chosen and generalises the two instances correctly without side effects. Finally, the Beams generalise the remaining space (road network) but generates side effects by overlapping buildings. Then, the side effects are corrected by a specific process (Least Squares here) and the CollaGen workflow is finished. An implemented version of this example is illustrated in Fig. 15.

Partitioning Component

The partitioning component is responsible for the creation of the geographic spaces as additional data [START_REF] Touya | Relevant Space Partitioning for Collaborative Generalisation[END_REF]. Thus, the component has to be fed with spatial analysis algorithms able to outline the required spaces. For instance, algorithms to identify urban, suburban, rural, costal, mountain areas were implemented, among others.

Translator Component

The translator component provides three kinds of services to translate inputs and outputs of the processes in the language used to convey interoperability: the formal constraints and the ontology [START_REF] Touya | Collaborative Generalisation: Formalisation of Cartographic Knowledge to Orchestrate Different Generalisation Processes[END_REF]. First, the translator allows process interoperability making the constraints the only input and output of every process (Fig. 6). A translating function is provided for every process and transforms the constraints and operation rules into the specific parameters of the process (e.g. numeric thresholds for a river selection, specific constraints for AGENT or CartACom, equations on coordinates for Least Squares or additional attributes for the Elastic Beams). The translator also serves as a registration mapping [START_REF] Lemmens | Lost and found, the importance of modelling map content semantically[END_REF] to tag the geographic data with the corresponding ontology concept: it makes the mapping between the "IGN_BUILDING" data class and the "building" ontology concept for instance [START_REF] Touya | Collaborative Generalisation: Formalisation of Cartographic Knowledge to Orchestrate Different Generalisation Processes[END_REF]. Finally, the translator allows to map the formal constraints with measurement algorithms that compute the constraint current value and satisfaction for a given object as in AGENT [START_REF] Barrault | Integrating multi-agent, object oriented and algorithmic techniques for improved automated map generalization[END_REF]). The use of such mechanism is detailed in §2.3.9.

Registry Component

The registry serves as yellow pages service to choose a relevant process to generalise a given space at a given time. To a request like 'what are the best available processes to generalise rural space n°x?' the registry responds with a list of applicable processes sorted by relevance (e.g. 'CartACom, Least Squares, AGENT').

To build the list, the registry first selects the relevant processes according to their description pre-condition: if the pre-condition, a list of space types with relevance rate (e.g. 'urban area 4/5') contains the request space type, the process is selected; then it is rated according to relevance rate.

In a second step, the processes are reordered according to the description post-condition, the list of a priori satisfied constraints after generalisation: a ratio is computed between the post-condition and the occurrences of constraints inside the space (Fig. 7). The ratio weights the relevance rate, reordering the processes. Fig. 7. Illustration of the registry response according to the constraints inside a given space: process 2 matches 14/15 constraints against 5/15 for process 1.

Scheduling Component

The scheduling component orchestrates the generalisation of spaces by processes. After every generalisation, it decides what to do next: it chooses the next type of geographic space to generalise and then orders the instances of this type. As in a Global Master Plan [START_REF] Ruas | Strategies for automated generalization[END_REF] but here rule-based, the space type (urban, rural...) is chosen according to the active sequencing rules. If it is not enough to choose, the space type whose instances have the highest conflict mean is selected first. Once the type chosen, the instances are ordered by conflict importance: the most conflicting ones are peeked (Fig. 8).

Generalisation is considered in CollaGen as a four step operation: geometry changes (e.g. area-to-point collapses), selection, cartographic and graphic generalisation [START_REF] Harrie | Simultaneous Graphic Generalization of Vector Data Sets[END_REF]. The processes are described as contributing to one or more of these steps and can be used only during the right steps. The scheduling component chains these steps according to the rules. Moreover, the scheduling component provides state management to allow local and global corrections (Fig. 9). When a generalisation is badly evaluated, the scheduling may cancel it and go back to previous states of data. To allow this try/error mechanism, as in [START_REF] Zhou | Towards a data model for update propagation in mr-dlm[END_REF], the initial state stores the attribute data while all the geometry states are kept linked to the generalisation pair (space/process) that led to the following step.

Observation Component

Generalising a space with the best available process does not guarantee a complete success. Conflicts clusters may emerge during generalisation and the observation component allows to detect them as conflicting areas [START_REF] Duchêne | Emergence de zones de conflits dans deux modèles de généralisation cartographique multi-agents[END_REF]. Therefore, the component observes the generalisations online and evaluates the progress. The observation component analyses the conflict areas, pauses the process when some are too big, extends the areas to subserve solving, triggers the registry component to propose a local solution and once the local conflicts are solved, resumes to the interrupted generalisation.

The online evaluation embedded in the emergence mechanism, as well as the global evaluation performed on a geographic space after generalisation rely on the constraints to guarantee homogeneous evaluation for every process. To enable this monitoring of the constraints against the data, located constraints monitors (LCM) are added on each object concerned by constraints as in [START_REF] Barrault | Integrating multi-agent, object oriented and algorithmic techniques for improved automated map generalization[END_REF]: if "granularity" and "size" constraints are defined on buildings, granularity and size LCMs are added for every building. LCMs are able to give the satisfaction of the constraint for the given building. The located constraints are provided with a geometry (Fig. 10) that allows the spatial clustering necessary for the emergence mechanism. The geometry also allows to quantify the LCMs inside a geographic space so as to evaluate globally the space generalisation from the distribution of individual evaluation. Progress means less unsatisfied LCMs in the distribution while good generalisation means few very unsatisfied LCMs and lots of very satisfied LCMs.

Side Effects Component

Within the CG Framework, generalising a space may cause additional conflicts just outside the space: for instance, a building is moved too close to a building that was just outside (thus not managed by generalisation). We call such additional conflicts side effects. In order to detect and correct side effects, the neighbourhood of each space is defined depending on the topological relation shared by two spaces (adjacency or overlap) [START_REF] Touya | Relevant Space Partitioning for Collaborative Generalisation[END_REF].

Following the principle of the LCM evaluation, the side effects management is based on consistency constraints located in each space's neighbourhood. Consistency constraints are kind of integrity constraints that guarantee the consistency of data before and after the generalisation of a space. Three types of consistency constraints can be identified: the inter-space relational constraints, the non-existence relational constraints and the operation consistency constraints.

The inter-space relational constraints are relational LCMs (LCM on a geographic relation between two objects) concerning an object inside the space and an object outside the space (Fig. 11). The figure "relative position relation" has a building in rural space 1 and the other in rural space 2. The "relative orientation relation" is an inter-space relation in the road network space point of view (building outside the space) but not in the rural space 2 point of view as the road is inside the space. Then, non-existence relational constraints check that no additional inter-space relation has been created by generalisation.

The operation consistency constraints affect the intersection neighbourhood. If the intersecting space has already been generalised, these constraints check that the second generalisation is not inconsistent with the first one. For instance, if a building has been moved one way, it should not be moved the other way round. The operation consistency constraints are based on the previous states of the objects stored by the scheduling component.

Finally, if consistency constraints are violated, side effects correcting processes are triggered. Side effects correctors are monitored by consistency constraints as processes are monitored by constraints. GAEL [START_REF] Gaffuri | Field deformation in an agent-based generalisation model: the gael model[END_REF], Elastic Beams [START_REF] Bader | Building displacement over a ductile truss[END_REF] or Least Squares [START_REF] Harrie | Simultaneous Graphic Generalization of Vector Data Sets[END_REF] can be used to correct side effects (Fig. 12) as well as diffusion processes [START_REF] Legrand | Propagation of the displacements and deformations during a generalisation process[END_REF]. If no balanced solution can be found, the component can arbitrate by choosing a solution (before or after 2 nd generalisation) like a legislator [START_REF] Ruas | The roles of meso objects for generalisation[END_REF] or by relaxing some less important constraints like a controller [START_REF] Ruas | The roles of meso objects for generalisation[END_REF].

Results

The CollaGen model is not fully implemented but some experiments were carried out on French topographic data (1:15k reference scale) to produce a 1:50k map on a large area containing heterogeneous landscapes (shore, city, country, mountain). A standard set of constraints and rules, extracted from NMA experience, is used as specification (80 constraints and rules). The available processes are the ones implemented on CartAGen platform [START_REF] Renard | Capitalisation problem in research -example of a new platform for generalisation: Cartagen[END_REF]): AGENT, CartACom, GAEL, Least Squares, Elastic Beams and network selection processes. Fig. 13 shows the steps of a rural area generalisation: the rural space is generalised by CartACom and the road network by the Beams generating a side effect conflict. Fig. 14 shows the registry answer to a rurban space request: three processes match the pre-condition but CartACom is ranked first as it better matches the constraints (a 10 constraints caricatural sample is used) a priori (97% against 59 for Agent and 6% for the Least Squares).

Fig. 14. The registry answer to a request from the delineated rurban space. CartACom is ranked first as it better suits the constraints than Agent or the Least Squares processes.

Fig. 15 shows a CollaGen result (without side effect correction) on a small area with urban, suburban and rural spaces. Fig. 16 illustrates the importance of choosing the best process and the best order. The remaining conflicts in picture (2) show that the choices are not as good as pictures ( 3) and ( 4) ones. Moreover, the order of picture ( 4) is better than in picture (3) as it did not result in side effects. Thus, the sequencing component has to be careful in choosing the order! Finally, Fig. 17 shows observed emerging conflicting areas in a space too dense for CartACom and the conflict correction using Least Squares. In order to evaluate CollaGen contribution, we used it to generalise the benchmark dataset from EuroSDR generalisation state-of-the-art project [START_REF] Stoter | Methodology for evaluating automated map generalization in commercial software[END_REF]) and compared the results to the best ones obtained during the tests with CPT and Clarity™ software (Fig. 18). Although the seven processes used could be tuned to improve individual results, the comparison shows the pros of CG and CollaGen, particularly in the southwest suburban part that has to be generalised differently from the town area.

Conclusion and Future Plans

This paper introduced a new framework to perform automatic cartographic generalisation by making optimised use of past research processes: Collaborative Generalisation. Within this framework, the CollaGen model allows to sequence interoperable generalisations of geographic spaces by the relevant available process. Among Colla-Gen contributions, a formal constraints model, a generalisation domain ontology, online evaluation and side effects detection mechanisms can be noticed.

A lot can still be made to improve both CollaGen and the Collaborative Generalisation Framework. The side effects correction really need to be more investigated to know how far corrections can be made without undoing what was previously generalised. In-depth testing (different data and scale change) of CollaGen and each component is also necessary to identify remaining issues. Moreover, rather than being implemented on a platform, the available processes could be called as web services as proposed by [START_REF] Regnauld | Evolving from automating existing map production systems to producing maps on demand automatically[END_REF].

Fig. 1 . 2 .

 12 Fig. 1.2. The main Components (rectangles) and Resources (ellipses) of a Collaborative Generalisation framework and how the components act on the resources (plain arrows). Dashed arrows show that formal knowledge is used by each component.

Fig. 3 .

 3 Fig. 3. (a) buildings in black, urban areas in red and rurban areas in blue. (b) vegetation thematic space on the same area. (c) mountains roads space (in the rectangle).

Fig. 4 .

 4 Fig. 4. A diagram of the 5 parts of formalised knowledge and their use in the collaborative model. The dashed arrows show that the Ontology provides shared concepts to every part.

Fig. 5 .

 5 Fig. 5. Simplified view of the sequence of CollaGen generalisation (emergence mechanism is not included). Each box is the action of one component.

Fig. 6 .

 6 Fig. 6. CollaGen allows interoperability with the use of a single output, constraints and the translator.

Fig. 8 .

 8 Fig. 8. Interactions of the scheduling component with other components and resources.

Fig. 9 .

 9 Fig. 9. UML class diagram of the state management system of the scheduling component.

Fig. 10 .

 10 Fig. 10. Conflicting LCM are clustered by triangulation to observe 'conflicting area' emergence.

Fig. 11 .

 11 Fig. 11. Two examples of inter-space relations with two rural space portions separated by a road: a relative position relation that should be preserved after each rural space is generalised and relative orientation relation (shared by rural space 2 and the road network space) that should be preserved if the road network is generalised.

Fig. 12 .

 12 Fig. 12. Example of side effect correction by least squares adjustment : a balance between both generalisations.

Fig. 13 .

 13 Fig. 13. (a) initial rural area. (b) rural space generalised by CartACom. (c) Roads generalised by the Beams with a side effect on the circled building.

Fig. 15 .

 15 Fig. 15. (a) initial data. (b) urban space generalised by AGENT, then rurban and rural spaces by CartACom and roads by the Beams (urban space is delineated with the thick outline).

Fig. 16 .

 16 Fig. 16. Illustration of order and registry importance: (1) before generalisation. (2) generalisation with AGENT then Least Squares process. (3) generalisation with CartACom then the Beams. (4) generalisation with the beams then CartACom.

Fig. 17 .

 17 Fig. 17. Conflict emergence from CartACom generalisation. (a) ungeneralised data. (b) emerging conflict areas are detected after generalization. (c) zoom on a conflict area extended to better solve the conflict. (d) conflicts solved by least squares generalization.

Fig. 18 .

 18 Fig. 18. A mountainous French dataset from EuroSDR tests (Stoter et al. 2009) generalized with CollaGen compared to the best results from the tests.


Alexandre Thorel
email: alexandre.thorel@univ-lehavre.fr

Operational approach for biharmonic equations in L p -spaces

Keywords: Operational differential equations, functional calculus, analytic semigroups, interpolation spaces, maximal regularity. 2010 Mathematics Subject Classification: 35B65, 35C15, 35J40, 35R20, 47A60, 47D06. or

In this work, we study the existence, uniqueness and maximal L p -regularity of the solution of different biharmonic problems. We rewrite these problems by a fourth order operational equation and different boundary conditions, set in a cylindrical n-dimensional spatial region Ω of R n . To this end, we give an explicit representation formula, using analytic semigroups, and invert explicitly a determinant operator in L p -spaces thanks to E ∞ functional calculus and operator sums theory.

Introduction

In this article we develop an operator approach to study some properties of the solutions of the biharmonic problem

∆ 2 u = f, (1)
set on a straight cylinder Ω = (a, b) × ω where ω ⊂ R n-1 denotes a bounded domain with a C 2 boundary and n ≥ 2 is a given integer while the function f ∈ L p (Ω) with p ∈ (1, ∞).

Here we are concerned with the existence, uniqueness and L p -maximal regularity of the solutions of (1) when this problem is supplemented with various types of non-homogeneous boundary conditions (see below).

Biharmonic problems arise in various applicative fields, including physics, engineering and biology. In [START_REF] Costabel | On boundary integral equations of the first kind for the bi-Laplacian in a polygonal plane domain[END_REF], [START_REF] Jaswon | Integral equation methods in potential theory and elastostatics[END_REF], or [START_REF] Saker | On the bilaplacian problem with nonlinear boundary conditions[END_REF], biharmonic equations are used to model elasticity problem where displacement, stress, and strain satisfy [START_REF] Bourgain | Some remarks on Banach spaces in which martingale difference sequences are unconditional[END_REF]. One may also refer to [START_REF] Cakoni | On the boundary integral equation methodfor a mixed boundary value problem of thebiharmonic equation[END_REF], [START_REF] Guo | Revisiting the biharmonic equation modelling electrostatic actuation in lower dimensions[END_REF] or [START_REF] Lin | Nonlinear non-local elliptic equation modelling electrostatic actuation[END_REF] for biharmonic models in electrostatic, to [START_REF] Favini | Resolution and Optimal Regularity for a Biharmonic Equation with Impedance Boundary Conditions and Some Generalizations[END_REF], [START_REF] Geymonat | Analyse asymptotique du comportement en flexion de deux plaques collées[END_REF] or [START_REF] Titeux | Conditions de transmission pour les jonctions de plaques minces[END_REF] for applications in plate theory where various different boundary conditions arise. We also refer to [START_REF] Cohen | A generalized diffusion model for growth and dispersal in population[END_REF], [START_REF] Labbas | Generalized linear models for population dynamics in two juxtaposed habitats[END_REF], [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF] or [START_REF] Ochoa | A generalized reaction-diffusion model for spatial structures formed by motile cells[END_REF] for models in population dynamics.

Coming back to [START_REF] Bourgain | Some remarks on Banach spaces in which martingale difference sequences are unconditional[END_REF], we now specific the boundary conditions that will be studied along this work. First we assume the homogeneous Dirichlet boundary conditions on the lateral boundary of Ω, namely (a, b) × ∂ω. This reads as

u(x, ζ) = ∆u(x, ζ) = 0, (x, ζ) ∈ (a, b) × ∂ω.
(

) 2
As far as the boundary conditions on {a, b} × ω, we shall study the following five different types u(a, y) = ϕ 1 (y), u(b, y) = ϕ 2 (y), y ∈ ω, ∂ xx u(a, y) = ϕ 3 (y), ∂ xx u(b, y) = ϕ 4 (y), y ∈ ω, (3.1) or ∂ x u(a, y) = ϕ 1 (y), ∂ x u(b, y) = ϕ 2 (y), y ∈ ω, ∆u(a, y) = ϕ 3 (y), ∆u(b, y) = ϕ 4 (y), y ∈ ω, (

or u(a, y) = ϕ 1 (y), u(b, y) = ϕ 2 (y), y ∈ ω, ∂ x u(a, y) = ϕ 3 (y), ∂ x u(b, y) = ϕ 4 (y), y ∈ ω, (3.3) or u(a, y) = ϕ 1 (y), u(b, y) = ϕ 2 (y), y ∈ ω, ∆u(a, y) = ϕ 3 (y), ∆u(b, y) = ϕ 4 (y), y ∈ ω.

(3.5)

In the above boundary conditions, the functions ϕ 1 , ϕ 2 , ϕ 3 and ϕ 4 are assumed to belong to L p (ω) As mentioned above, in this article we develop an operator theoric approach to handle the biharmonic equation ( 1)-( 2) and supplemented with one of the previous boundary conditions. This work can be viewed as a continuation of the analysis proposed in [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF] where the equation

k 2 ∆ 2 u -k 1 ∆u = f,
has been studied using a suitable operational framework. Note that in the aforementioned work, the condition k 1 = 0 plays a crucial role in this analysis. So that the present work extends this previous analysis by taking into account this limit case k 1 = 0.

Here, to handle (1)-( 2) and one of the previous boundary conditions, we reformulate this problem as a suitable fouth order abstract differential equation that allow us to derive existence, uniquess and also to provide necessary and sufficient conditions on the boundary functions ϕ i arising in the previous boundary conditions, ensuring maximal L p -regularity.

More specifically let us define A 0 , the Laplace operator in R n-1 , n ∈ N \ {0, 1}, on the Banach space L q (ω), q ∈ (1, +∞), with homogeneous Dirichlet boundary conditions on ∂ω. Then, A 0 reads as:

D(A 0 ) := {ψ ∈ W 2,p (ω) : ψ = 0 on ∂ω} ∀ψ ∈ D(A 0 ), A 0 ψ = ∆ y ψ.
(

Thus, equation ( 1) could be reformulated as the fourth order operational differential equation,

u (4) (x) + 2A 0 u (x) + A 2 0 u(x) = f (x), for a. e. x ∈ (a, b),
where f ∈ L p (a, b; L q (ω)), p ∈ (1, +∞), with u(x) := u(x, •) and f (x) := f (x, •). Then, we consider the generalization of this equation with (A, D(A)), instead of (A 0 , D(A 0 )):

u (4) (x) + 2Au (x) + A 2 u(x) = f (x), for a. e. x ∈ (a, b), (4)
where "-A" is a BIP operator of angle θ A ∈ (0, π) on a UMD space X, see Section 2 below for the definitions of BIP operator and UMD spaces, and f ∈ L p (a, b; X), p ∈ (1, +∞).

More precisely, we will study equation ( 4) under the following boundary conditions:

u(a) = ϕ 1 , u(b) = ϕ 2 , u (a) = ϕ 3 , u (b) = ϕ 4 , (BC1) or u (a) = ϕ 1 , u (b) = ϕ 2 , u (a) + Au(a) = ϕ 3 , u (b) + Au(b) = ϕ 4 , (BC2) or u(a) = ϕ 1 , u(b) = ϕ 2 , u (a) = ϕ 3 , u (b) = ϕ 4 , (BC3) or u (a) = ϕ 1 , u (b) = ϕ 2 , u (a) = ϕ 3 , u (b) = ϕ 4 , (BC4) or u(a) = ϕ 1 , u(b) = ϕ 2 , u (a) + Au(a) = ϕ 3 , u (b) + Au(b) = ϕ 4 , ( BC5
)
where

ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 ∈ X.
Recall that u is a classical solution of equation ( 4) if u is a solution of (4) satisfying

u ∈ W 4,p (a, b; X) ∩ L p (a, b; D(A 2 )) and u ∈ L p (a, b; D(A)),
Moreover, for i ∈ {1, 2, 3, 4, 5}, we say that u is a classical solution of problem ( 4)-(BCi), if u is a classical solution of (4) satisfying (BCi).

Our main result is given by Theorem 2.8 in which we give necessary and sufficient conditions for the existence and uniqueness of a classical solution u of problem ( 4)-(BCi).

As a byproduct of our analysis, we also obtain an explicit representation formula of the solutions of (4) and also a regularity result of the terms composed by a polynomial function and an analytic semigroup.

The paper is organised as follows:

In Section 2, we detail our hypotheses on a UMD Banach space X and on the operator A. Then, we give our main results. As a consequence, we obtain Corollary 2.9 which establishes the existence and the uniqueness of the solution u in W 4,p (Ω) of problem ( 1)-( 2)- (3.3).

In Section 3, we recall some well-known results useful for the next sections.

In Section 4, we establish some preliminary results. First of all, in Section 4.1, we make explicit, in Proposition 4.1, a general representation formula for classical solutions. In Section 4.2, we give a very useful technical result in Proposition 4.2 concerning the regularity of the product between a polynomial function and an analytic semigroup. In Section 4.3, we give some traces result in Proposition 4.3. Then, in Section 4.4, we use functional calculus to prove an invertibility result in Proposition 4.5 which is very useful for the last section.

Finally, in the last section, we prove our main result, Theorem 2.8, which completes Theorem 2.6 by showing the existence and the uniqueness of the classical solution u of problems ( 4)-(BC2), ( 4)-(BC3) and ( 4)-(BC4).

Assumptions and statement of results

Hypotheses

Throughout the article, (X, • ) is a complex Banach space. We first give some useful definitions before stating our hypotheses. Definition 2.1. A Banach space X is a UMD space if and only if for all p ∈ (1, +∞), the Hilbert transform is bounded from L p (R, X) into itself (see [START_REF] Bourgain | Some remarks on Banach spaces in which martingale difference sequences are unconditional[END_REF] and [START_REF] Burkholder | A geometrical characterisation of Banach spaces in which martingale difference sequences are unconditional[END_REF]).

Definition 2.2. A closed linear operator

T 1 in X is called sectorial of angle θ ∈ [0, π) if i) σ(T 1 ) ⊂ S θ , ii) ∀ θ ∈ (θ, π), sup λ(λ I -T 1 ) -1 L(X) : λ ∈ C \ S θ < +∞,
where

S θ := {z ∈ C : z = 0 and | arg(z)| < θ} if θ ∈ (0, π), (0, +∞) if θ = 0, (5)
see [START_REF] Haase | The functional calculus for sectorial Operators[END_REF], p. 19.

Definition 2.3. We denote by BIP(X, θ), θ ∈ [0, π), see [START_REF] Prüss | On operators with bounded imaginary powers in Banach spaces[END_REF], p. 431, the class of sectorial injective operators

T 1 in X such that i) D(T 1 ) = R(T 1 ) = X, ii) ∀ s ∈ R, T is 1 ∈ L(X), iii) ∃ C ≥ 1, ∀ s ∈ R, ||T is 1 || L(X) ≤ Ce |s|θ .
We are now in position to describe our assumptions:

(H 1 ) X is a UMD space, (H 2 ) A is a closed linear operator in X and 0 ∈ ρ(A),

(H 3 ) -A ∈ BIP(X, θ A ) for some θ A ∈ (0, π), Remark 2.4.
1. The operator A 0 , given by ( 3), satisfies all the previous hypotheses with X = L q (ω), q ∈ (1, +∞). From [START_REF] Rubio De Francia | Martingale and integral transforms of Banach space valued functions[END_REF], Proposition 3, p. 207, X satisfies (H 1 ) and from [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF], Theorem 9.15, p. 241 and Lemma 9.17, p.242, A 0 satisfies (H 2 ). Moreover, (H 3 ) is satisfied for every θ A ∈ (0, π), from [START_REF] Prüss | Imaginary powers of elliptic second order differential operators in L p -spaces[END_REF], Theorem C, p. 166-167.

2. If we consider the real case (with -A > 0), then we can solve equation ( 1) with the help of the roots ± √ -A of the characteristic equation

x 4 + 2Ax 2 + A 2 = 0,
this is why, in our operational case, we consider the operator

M := - √ -A.
From (H 3 ), since -A is a sectorial operator, then the existence of M is ensured, see [START_REF] Haase | The functional calculus for sectorial Operators[END_REF], p. 61.

3. Due to (H 2 ) we deduce that 0 ∈ ρ(M ), from [START_REF] Haase | The functional calculus for sectorial Operators[END_REF], Proposition 3.1.1, p. 62.

4. From [START_REF] Haase | The functional calculus for sectorial Operators[END_REF], Proposition 3.2.1, p. 71 and (H 3 ), we deduce that -M ∈ BIP(X, θ A /2).

The main results

We first need some definitions and results concerning real interpolation spaces, the definition of which is given in [START_REF] Da Prato | Sommes d'opérateurs linéaires et équations différentielles opérationnelle[END_REF], [START_REF] Grisvard | Spazi di tracce e applicazioni[END_REF], [START_REF] Lions | Sur une classe d'espaces d'interpolation[END_REF] or in [START_REF] Triebel | Interpolation theory, function Spaces, differential Operators[END_REF]. We consider here the following particular case: Definition 2.5. Let T 2 : D(T 2 ) ⊂ X -→ X be a linear operator such that (0, +∞) ⊂ ρ(T 2 ) and ∃ C > 0 :

∀ t > 0, t(T 2 -tI) -1 L(X) C. ( 6
)
Let k ∈ N \ {0}, θ ∈ (0, 1) and q ∈ [1, +∞]. We will use the real interpolation spaces (D(T k 2 ), X) θ,q = (X, D(T k 2 )) 1-θ,q , defined, for instance, in [START_REF] Lions | Sur une classe d'espaces d'interpolation[END_REF], or in [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF].

In particular, for k = 1, we have the following characterization

(D(T 2 ), X) θ,q := ψ ∈ X : t -→ t 1-θ T 2 (T 2 -tI) -1 ψ X ∈ L q * (0, +∞) ,
where L q * (0, +∞) is given by

L q * (0, +∞) := f ∈ L q (0, +∞) : +∞ 0 f (t) q dt t 1/q < +∞ , for q ∈ [1, +∞),
and for q = +∞, by L ∞ * (0, +∞) := sup t∈(0,+∞)

f (t) ,
see [START_REF] Da Prato | Sommes d'opérateurs linéaires et équations différentielles opérationnelle[END_REF] p. 325, or [START_REF] Grisvard | Spazi di tracce e applicazioni[END_REF], p. 665, Teorema 3, or section 1.14 of [START_REF] Triebel | Interpolation theory, function Spaces, differential Operators[END_REF], where this space is denoted by (X, D(T 2 )) 1-θ,q . Note that we can also characterize the space (D(T 2 ), X) θ,q taking into account the Osservazione, p. 666, in [START_REF] Grisvard | Spazi di tracce e applicazioni[END_REF].

We set also, for any

k ∈ N \ {0} (X, D(T 2 )) k+θ,q := ψ ∈ D(T k 2 ) : T k 2 ψ ∈ (X, D(T 2 )) θ,q . ( 7
)
We recall the following Theorem obtained in [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF] (this result is given for k = 0 but clearly remains true if k = 0). Theorem 2.6 (see [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF]). Let f ∈ L p (a, b; X) with a, b ∈ R, a < b and p ∈ (1, +∞). Assume that (H 1 ), (H 2 ) and (H 3 ) hold. Then 1. there exists a unique classical solution u of (4)-(BC1) if and only if

ϕ 1 , ϕ 2 ∈ D(A) and Aϕ 1 , Aϕ 2 , ϕ 3 , ϕ 4 ∈ (D(A), X) 1 2p ,p . ( 8
)
This unique classical solution is called F Φ,f with Φ := (ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 ) and is explicitly described by

F Φ,f (x) := e (x-a)M Zϕ 1 + e (b-x)M Zϕ 2 - 1 2 e (x-a)M ZM -1 b a e (s-a)M v 0 (s) ds - 1 2 e (b-x)M ZM -1 b a e (b-s)M v 0 (s) ds + 1 2 M -1 x a e (x-s)M v 0 (s) ds + 1 2 M -1 b x e (s-x)M v 0 (s) ds -e (b-x)M e (b-a)M Zϕ 1 -e (x-a)M e (b-a)M Zϕ 2 + 1 2 e (x-a)M Ze (b-a)M M -1 b a e (b-s)M v 0 (s) ds + 1 2 e (b-x)M Ze (b-a)M M -1 b a e (s-a)M v 0 (s) ds, x ∈ [a, b], ( 9
)
where

v 0 (x) := e (x-a)M Z (ϕ 3 + Aϕ 1 ) + e (b-x)M Z (ϕ 4 + Aϕ 2 ) - 1 2 e (x-a)M ZM -1 b a e (s-a)M f (s) ds - 1 2 e (b-x)M ZM -1 b a e (b-s)M f (s) ds + 1 2 M -1 x a e (x-s)M f (s) ds + 1 2 M -1 b x e (s-x)M f (s) ds -e (b-x)M e (b-a)M Z (ϕ 3 + Aϕ 1 ) -e (x-a)M e (b-a)M Z (ϕ 4 + Aϕ 2 ) + 1 2 e (x-a)M Ze (b-a)M M -1 b a e (b-s)M f (s) ds + 1 2 e (b-x)M Ze (b-a)M M -1 b a e (s-a)M f (s) ds, x ∈ [a, b], (10)
and Z := I -e 2(b-a)M -1 . The existence of Z is ensured from [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF], Proposition 2.3.6, p. 60.

there exists a unique classical solution u of (4)-(BC5) if and only if

ϕ 1 , ϕ 2 ∈ D(A) and Aϕ 1 , Aϕ 2 , ϕ 3 , ϕ 4 ∈ (D(A), X) 1 2p ,p . ( 11
)
In this case the unique solution is

u = F (ϕ 1 ,ϕ 2 ,ϕ 3 -Aϕ 1 ,ϕ 4 -Aϕ 2 ),f .
Remark 2.7.

1. The previous results use the works of [START_REF] Da Prato | Sommes d'opérateurs linéaires et équations différentielles opérationnelle[END_REF] and [START_REF] Dore | On the closedness of the sum of two closed operators[END_REF].

2. In the previous Theorem, statement 2. is obtained from statement 1. in the following manner: condition (BC5) could be written as follows

u(a) = ϕ 1 , u(b) = ϕ 2 , u (a) = ϕ 3 -Aϕ 1 , u (b) = ϕ 4 -Aϕ 2 .
From statement 1., there exists a unique classical solution u of problem ( 4)-(BC5) if and only if

ϕ 1 , ϕ 2 ∈ D(A) and Aϕ 1 , Aϕ 2 , ϕ 3 -Aϕ 1 , ϕ 4 -Aϕ 2 ∈ (D(A), X) 1 2p
,p , which is equivalent to [START_REF] Geymonat | Analyse asymptotique du comportement en flexion de deux plaques collées[END_REF]. Thus, we deduce that u = F (ϕ 1 ,ϕ 2 ,ϕ 3 -Aϕ 1 ,ϕ 4 -Aϕ 2 ),f . Now, we give our main results.

Theorem 2.8. Let f ∈ L p (a, b; X) with a < b, a, b ∈ R and p ∈ (1, +∞). Assume that (H 1 ), (H 2 )
and (H 3 ) hold. Then 1. there exists a unique classical solution u of (4)-(BC2) if and only if

ϕ 1 , ϕ 2 ∈ D(A), Aϕ 1 , Aϕ 2 ∈ (D(A), X) 1 2 + 1 2p ,p and ϕ 3 , ϕ 4 ∈ (D(A), X) 1 2p ,p . ( 12
)
Moreover, if, in (H 3 ), θ A < π/2, then 2.
there exists a unique classical solution u of (4)-(BC3) if and only if

ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 ∈ D(A), Aϕ 1 , Aϕ 2 ∈ (D(A), X) 1 2p ,p and Aϕ 3 , Aϕ 4 ∈ (D(A), X) 1 2 + 1 2p ,p , (13)
3. there exists a unique classical solution u of (4)-(BC4) if and only if

ϕ 1 , ϕ 2 ∈ D(A), Aϕ 1 , Aϕ 2 ∈ (D(A), X) 1 2 + 1 2p ,p and ϕ 3 , ϕ 4 ∈ (D(A), X) 1 2p ,p . (14)
As a consequence of Theorem 2.8, we can make explicit our problems ( 4)-(BCi) with A = A 0 and, for instance, we do it for (4)-(BC3) and solve problem (1)-( 2

∈ W 4,p (Ω) of                            ∆ 2 u(x, y) = f (x, y), (x, y) ∈ Ω u(x, ζ) = ∆u(x, ζ) = 0, (x, ζ) ∈ (a, b) × ∂ω u(a, y) = ϕ 1 (y), y ∈ ω u(b, y) = ϕ 2 (y), y ∈ ω ∂ x u(a, y) = ϕ 3 (y), y ∈ ω ∂ x u(b, y) = ϕ 4 (y), y ∈ ω, if and only if            ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 ∈ W 2,p (ω) ∩ W 1,p 0 (ω) ∆ϕ 1 , ∆ϕ 2 ∈ W 2-1 p ,p (ω) ∩ W 1,p 0 (ω) ∆ϕ 3 , ∆ϕ 4 ∈ W 1-1 p ,p (ω) ∩ W 1,p 0 (ω).
Proof. The proof is quite similar to the one of Corollary 2.7 in [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF], see also Corollary 3.6 in [START_REF] Labbas | Generalized linear models for population dynamics in two juxtaposed habitats[END_REF].

Taking into account the result of Theorem 2.8, we can also obtain anisotropic result by considering f ∈ L p (a, b; L q (ω)) with p, q ∈ (1, +∞).

Prerequisites

In this section, we recall some well-known facts, useful in our proofs. Lemma 3.1 ([13]). Let T 2 be a linear operator satisfying [START_REF] Da Prato | Sommes d'opérateurs linéaires et équations différentielles opérationnelle[END_REF]. Let u such that

u ∈ W n,p (a 1 , b 1 ; X) ∩ L p (a 1 , b 1 ; D(T k 2 )),
where

a 1 , b 1 ∈ R with a 1 < b 1 , n, k ∈ N \ {0} and p ∈ (1, +∞).
Then for any j ∈ N satisfying the Poulsen condition 0 < 1 p + j < n and s ∈ {a 1 , b 1 }, we have

u (j) (s) ∈ (D(T k 2 ), X) j n + 1 np ,p .
This result is proved in [START_REF] Grisvard | Spazi di tracce e applicazioni[END_REF], Theorem 2'.

Lemma 3.2. [START_REF] Triebel | Interpolation theory, function Spaces, differential Operators[END_REF] Let ψ ∈ X and T 2 be a generator of a bounded analytic semigroup in X with 0 ∈ ρ(T 2 ). Then, for any n ∈ N \ {0} and p ∈ [1, +∞], the next properties are equivalent:

1. x → e (x-a)T 2 ψ ∈ W n,p (a, b; X) 2. x → T n 2 e (x-a)T 2 ψ ∈ L p (a, b; X) 3. x → e (x-a)T 2 ψ ∈ W n,p (a, +∞; X) 4. x → T n 2 e (x-a)T 2 ψ ∈ L p (a, +∞; X) 5. ψ ∈ D(T n-1 2 ) and T n-1 2 ψ ∈ (D(T 2 ), X) 1 p ,p
Obviously, statement 1. is equivalent to statement 2. and also statement 3. is equivalent to statement 4. Furthermore, 4. is equivalent to 5., see [START_REF] Triebel | Interpolation theory, function Spaces, differential Operators[END_REF], Theorem, p. 96. Finally, since T 2 is a generator of a bounded analytic semigroup with 0 ∈ ρ(T 2 ), we have ω T 2 := sup{Re(λ) : λ ∈ σ(T 2 )} < 0, so from Proposition 2.3.1. in [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF], p. 56, we have

∃ M 1, ∃ δ > 0, ∀ x b, T n 2 e (x-a)T 2 M (x -a) n e -δ(x-a) ,
from which we deduce that statement 2. is equivalent to statement 4.

Lemma 3.3 ([20]

). Let V ∈ L(X) such that 0 ∈ ρ(I + V ). Then, there exists W ∈ L(X) such that

(I + V ) -1 = I -W, and W (X) ⊂ V (X). Moreover, if T is a linear operator in X such that V (X) ⊂ D(T ) and for ψ ∈ D(T ), T V ψ = V T ψ, then ∀ ψ ∈ D(T ), W T ψ = T W ψ.
This result is proved in [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF], Lemma 5.1.

Preliminary results

We first note that F 0,f defined in Theorem 2.6 is a particular solution of equation ( 4), this will be used, in the following subsection, to build a general representation formula of the solution of equation ( 4). Then, in the next subsection, we give a result which allow us to study the regularity of this representation formula. In the third subsection, we give a result on traces of classical solution of (4). Finally, in the last subsection, we give some invertibility results using functional calculus.

Representation formula

Theorem 2.6 furnishes results of existence and uniqueness of a classical solution of problem ( 4)-(BC1) and ( 4)-(BC5). With other boundary conditions, the equation ( 4) is not easy to study: the first step is to state a representation formula of the solution of equation ( 4). A similar representation formula has been given in [START_REF] Favini | Resolution and Optimal Regularity for a Biharmonic Equation with Impedance Boundary Conditions and Some Generalizations[END_REF] without details, this formula gives a formal solution of equation ( 4), nevertheless, it has not been proved that this formula leads to formal solution of all the classical solutions of equation ( 4).

Proposition 4.1. Under (H 1 ), (H 2 ), (H 3 ), if u is a classical solution of (4), then there exist K i ∈ X, i = 1, 2, 3, 4, such that, for all x ∈ [a, b], we have

u(x) = e (x-a)M K 1 + (x -a)e (x-a)M K 2 + e (b-x)M K 3 + (b -x)e (b-x)M K 4 + F 0,f (x), ( 15
)
where F 0,f is defined in Theorem 2.6.

Proof. Let u be a classical solution of (4). From Theorem 2.6, there exists a classical solution F 0,f of ( 4)-(BC1). Thus

F 0,f (a) = F 0,f (b) = F 0,f (a) = F 0,f (b) = 0. ( 16
)
Set Then, for a. e. x ∈ (a, b), we have

u h := u -F 0,f . Then, u h is a classical solution of u (4) h (x) + 2Au h (x) + A 2 u h (x) = 0, a. e. x ∈ (a, b).
v (x) + Av(x) = u (4) h (x) + 2Au h (x) + A 2 u h (x) = 0.
Using the proof of Theorem 5, p. 173, in [START_REF] Favini | A simplified approach in the study of elliptic differential equations in UMD spaces and new applications[END_REF] or [START_REF] Kunstmann | New criteria for the H ∞ -calculus and the Stokes operator on bounded Lipschitz domains[END_REF] in [START_REF] Favini | Complete abstract differential equations of elliptic type in UMD spaces[END_REF], we obtain that the solution v of the previous homogeneous equation reads as

v(x) = e (x-a)M C 0 + e (b-x)M C 1 , for all x ∈ [a, b],
where C 0 , C 1 ∈ X. Now, we solve the whole equation

u h (x) + Au h (x) = v(x), for all x ∈ [a, b].
Using again the proof of Theorem 5, p. 173 in [START_REF] Favini | A simplified approach in the study of elliptic differential equations in UMD spaces and new applications[END_REF] or [START_REF] Kunstmann | New criteria for the H ∞ -calculus and the Stokes operator on bounded Lipschitz domains[END_REF] in [START_REF] Favini | Complete abstract differential equations of elliptic type in UMD spaces[END_REF], we obtain that, for all x ∈ [a, b], the solution u h reads as

u h (x) = e (x-a)M C 2 + e (b-x)M C 3 + 1 2 M -1 x a e (x-s)M v(s) ds + 1 2 M -1 b x e (s-x)M v(s) ds where C 2 , C 3 ∈ X. Then u h (x) = e (x-a)M C 2 + e (b-x)M C 3 + 1 2 M -1 x a e (x-s)M e (s-a)M C 0 + e (b-s)M C 1 ds + 1 2 M -1 b x e (s-x)M e (s-a)M C 0 + e (b-s)M C 1 ds = e (x-a)M C 2 + e (b-x)M C 3 + 1 2 M -1 x a e (x-a)M C 0 ds + 1 2 M -1 x a e (x+b-2s)M C 1 ds + 1 2 M -1 b x e (2s-x-a)M C 0 ds + 1 2 M -1 b x e (b-x)M C 1 ds = e (x-a)M C 2 + (x -a)e (x-a)M 1 2 M -1 C 0 - 1 4 M -2 e (b-x)M -e (x-a)M e (b-a)M C 1 +e (b-x)M C 3 + 1 4 M -2 e (b-x)M e (b-a)M -e (x-a)M C 0 + (b -x)e (b-x)M 1 2 M -1 C 1 = e (x-a)M C 2 + 1 4 M -2 e (b-a)M C 1 -C 0 + (x -a)e (x-a)M 1 2 M -1 C 0 +e (b-x)M C 3 + 1 4 M -2 e (b-a)M C 0 -C 1 + (b -x)e (b-x)M 1 2 M -1 C 1 .
So, we set

K 1 = C 2 + 1 4 M -2 e (b-a)M C 1 -C 0 , K 2 = 1 2 M -1 C 0 ,
and

K 3 = C 3 + 1 4 M -2 e (b-a)M C 0 -C 1 , K 4 = 1 2 M -1 C 1 .
We conclude that, for all x ∈ [a, b], we have

u h (x) = e (x-a)M K 1 + (x -a)e (x-a)M K 2 + e (b-x)M K 3 + (b -x)e (b-x)M K 4 .
Thus, u satisfies [START_REF] Haase | The functional calculus for sectorial Operators[END_REF], which gives the result.

Regularity of polynomial exponential terms

We state below a result which concerns the regularity of the terms composed by a polynomial function and an analytic semigroup. This result improves the ones obtained in [START_REF] Favini | Resolution and Optimal Regularity for a Biharmonic Equation with Impedance Boundary Conditions and Some Generalizations[END_REF], Lemma 3.2 and Proposition 1.

Proposition 4.2. Let T be the infinitesimal generator of a bounded analytic semigroup, p ∈ [1, +∞]

with 0 ∈ ρ(T ). For all x ∈ [a, b] and ψ ∈ X, we set

v ψ (x) = (x -a)e (x-a)T ψ and w ψ (x) = (b -x)e (b-x)T ψ.
Then, we have

1. v ψ ∈ L p (a, b; D(T )). 2. For n ∈ N \ {0, 1}, v ψ ∈ L p (a, b; D(T n )) if and only if ψ ∈ D(T n-2 ) and T n-2 ψ ∈ (D(T ), X) 1 p ,p . 3. For n ∈ N \ {0}, if v ψ ∈ L p (a, b; D(T n )) then v ψ ∈ W n,p (a, b; X).
Moreover, these three statements hold if we replace v ψ by w ψ . Proof.

1. It is clear that v ψ ∈ L p (a, b; X) and from [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF], Proposition 2.1.1, statement (iii), p. 35 or [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF], equation (6.6), p. 70, we obtain the result.

2. From [START_REF] Triebel | Interpolation theory, function Spaces, differential Operators[END_REF], Theorem, p. 96, for θ ∈ (0, 1), we know that

ψ ∈ (X, D(T n )) θ,p ⇐⇒ +∞ 0 t n(1-θ) T n e tT ψ p dt t < +∞ ⇐⇒ +∞ 0 t n(1-θ)-1 p T n e tT ψ p dt < +∞.
Moreover, for n 2, we set

θ = 1 - 1 n - 1 np ∈ (0, 1), since n(1 -θ) - 1 p = 1, we obtain that ψ ∈ (X, D(T n )) θ,p ⇐⇒ +∞ 0
tT n e tT ψ p dt < +∞ ⇐⇒ t -→ tT n e tT ψ ∈ L p (0, +∞; X).

Replacing t by x -a, we have

t -→ tT n e tT ψ ∈ L p (0, +∞; X) ⇐⇒ v ψ ∈ L p (a, +∞; D(T n )),
and from Lemma 3.2, we deduce that

v ψ ∈ L p (a, +∞; D(T n )) ⇐⇒ v ψ ∈ L p (a, b; D(T n )).
From the reiteration Theorem and Theorem in section 1.14.3 in [START_REF] Triebel | Interpolation theory, function Spaces, differential Operators[END_REF], we obtain

(X, D(T n )) θ,p = (X, D(T n )) 1-1 n -1 np ,p = (X, D(T )) n-1-1 p ,p = (X, D(T )) (n-2)+1-1 p ,p .
Thus, it follows from (7) that

ψ ∈ D(T n-2 ) and ψ ∈ (D(T ), X) 1 p ,p ⇐⇒ v ψ ∈ L p (a, b; D(T n )). (17)
3. For statement 3., we begin by the case

n 2: if v ψ ∈ L p (a, b; D(T n )) then from statement 2., ψ ∈ D(T n-2
) and T n-2 ψ ∈ (D(T ), X) 1 p ,p , this ensure by Lemma 3.2 that we have

x -→ T n-1 e (x-a)T ψ ∈ L p (a, b; X).
We conclude that v

(n) ψ (•) = nT n-1 e (•-a)T ψ + T n v ψ (•) ∈ L p (a, b; X).

Now, we consider the case

n = 1. If v ψ ∈ L p (a, b; D(T )) then we have v ψ (•) = e (•-a)T ψ + T v ψ (•) ∈ L p (a, b; X).
To prove that these three statements hold if we replace v ψ by w ψ , it sufficient to use a variable change:

w ψ (x) = v ψ (b + a -x).

Traces result

Proposition 4.3. Assume that (H 2 ), (H 3 ) hold. Let u be a classical solution of (4), then, for s ∈ {a, b}, we have

           u(s) ∈ D(M 3 ) and M 3 u(s) ∈ (D(M ), X) 1 p ,p , u (s) ∈ D(M 2 ) and M 2 u (s) ∈ (D(M ), X) 1 p ,p u (s) ∈ D(M ) and M u (s) ∈ (D(M ), X) 1 p ,p . (18)
Moreover, we have

                   ψ ∈ D(M 3 ) and M 3 ψ ∈ (D(M ), X) 1 p ,p ⇐⇒ ψ ∈ D(A) and Aψ ∈ (D(A), X) 1 2p ,p , ψ ∈ D(M 2 ) and M 2 ψ ∈ (D(M ), X) 1 p ,p ⇐⇒ ψ ∈ D(A) and Aψ ∈ (D(A), X) 1 2 + 1 2p ,p , ψ ∈ D(M ) and M ψ ∈ (D(M ), X) 1 p ,p ⇐⇒ ψ ∈ (D(A), X) 1 2p ,p , ψ ∈ (D(M ), X) 1 p ,p ⇐⇒ ψ ∈ (D(A), X) 1 2 + 1 2p ,p . ( 19
)
Proof. Since A 2 = M 4 , we have u ∈ W 4,p (a, b; X) ∩ L p (a, b; D(M 4
)). Then, from Lemma 3.1, for s ∈ {a, b}, we obtain

u(s) ∈ (D(M 4 ), X) 1 4p ,p , u (s) ∈ (D(M 4 ), X) 1 4 + 1 4p ,p and u (s) ∈ (D(M 4 ), X) 1 2 + 1
4p ,p , which leads to (18) by using the reiteration theorem and Theorem in section 1.14.3 in [START_REF] Triebel | Interpolation theory, function Spaces, differential Operators[END_REF]. Noting that A = M 2 , again by the reiteration theorem and Theorem in section 1.14.3 in [START_REF] Triebel | Interpolation theory, function Spaces, differential Operators[END_REF], we obtain [START_REF] Labbas | Generalized linear models for population dynamics in two juxtaposed habitats[END_REF].

Note that the result remains true if (H 3 ) is replaced by a sectoriality assumption on A.

Functional calculus

Throughout the paper, we set c = b -a > 0.

We will need the invertibility of U and V given by

   U := I -e 2cM + 2cM e cM V := I -e 2cM -2cM e cM . ( 20
)
To this end, we need a result from functional calculus. To state it, we have to set some notations and give some technical lemmas. Let θ ∈ (0, π), we denote by H(S θ ) the space of holomorphic functions on S θ , the sector of angle θ (see [START_REF] Costabel | On boundary integral equations of the first kind for the bi-Laplacian in a polygonal plane domain[END_REF]) with values in C. Moreover, we consider the following subspace of H(S θ ):

E ∞ (S θ ) := {f ∈ H(S θ ) : f = O(|z| -s ) (|z| → +∞) for some s > 0} .
In other words, E ∞ (S θ ) is the space of polynomial decreasing holomorphic functions at +∞. Let T be an invertible sectorial operator of angle θ T ∈ (0, π) with bounded inverse. If f ∈ E ∞ (S θ ), with θ ∈ (θ T , π), then we can define, by functional calculus, f (T ) ∈ L(X), see [START_REF] Haase | The functional calculus for sectorial Operators[END_REF], p. 45. In this work, we use functional calculus, as classicaly done, see for instance [START_REF] Haase | Functional calculus for groups and applications to evolution equations[END_REF], [START_REF] Kunstmann | New criteria for the H ∞ -calculus and the Stokes operator on bounded Lipschitz domains[END_REF], [START_REF] Merdy | A sharp equivalence between H ∞ functional calculus and square function estimates[END_REF] or [START_REF] Limam | On Some Transmission Problems Set in a Biological Cell, Analysis and Resolution[END_REF].

For z ∈ C \ R -, we set

   ũ(z) = 1 -e -2c √ z -2c √ ze -c √ z ṽ(z) = 1 -e -2c √ z + 2c √ ze -c √ z . ( 21
)
Note that, we can write formally U = ũ(-A) and V = ṽ(-A). We set

C + := {z ∈ C : Re(z) 0}.
We obtain the following results:

Lemma 4.4. The complex functions ũ and ṽ, given by [START_REF] Merdy | A sharp equivalence between H ∞ functional calculus and square function estimates[END_REF] do not vanish on C + \ {0}.

Proof. Let z ∈ C \ R -such that Re(z) 0. Then, we have

ũ(z) = 1 -e -2c √ z -2c √ ze -c √ z = e -c √ z e c √ z -e -c √ z -2c √ z , and ṽ(z) = 1 -e -2c √ z + 2c √ ze -c √ z = e -c √ z e c √ z -e -c √ z + 2c √ z .
Moreover, one has

|ũ(z)| = e -c √ z e c √ z -e -c √ z -2c √ z e -c √ z e c √ z -e -c √ z -2 c √ z ,
and |ṽ(z

)| = e -c √ z e c √ z -e -c √ z + 2c √ z e -c √ z e c √ z -e -c √ z -2 c √ z .

Now, we focus on e c

√ z -e -c √ z . To this end, we set

√ z = α c + i β c , where α, β ∈ R. Thus e c √ z -e -c √ z 2 = e α+iβ -e -α-iβ 2 = 2 (cosh(2α) -cos(2β)) ,
and

2 (cosh(2α) -cos(2β)) = 2 +∞ n=0 2 2n (2n)! α 2n -(-1) n β 2n = 4 α 2 + β 2 + 2 +∞ n=2 2 2n (2n)! α 2n -(-1) n β 2n > 4 α 2 + β 2 + 2 +∞ n=2 2 2n (2n)! α 2n -β 2n .
Since Re(z) 0, we deduce that for all n ∈ N \ {0}, we have α 2n β 2n . Then, we obtain

4 α 2 + β 2 + 2 +∞ n=2 2 2n (2n)! α 2n -β 2n 4 α 2 + β 2 = 4|α + iβ| 2 .
Finally, we have

e c √ z -e -c √ z 2 = e α+iβ -e -α-iβ 2 > 4|α + iβ| 2 = 4 c √ z 2 , hence e c √ z -e -c √ z > 2 c √ z .
This implies that |ũ(z)| > 0 and |ṽ(z)| > 0, for z ∈ C + \ {0}.

Proposition 4.5. Assume that (H 2 ) and (H 3 ) with θ A < π/2 hold. Then U and V , defined by [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF], are invertible with bounded inverse.

Proof. From Lemma 4.4, we know that ũ and ṽ, given by ( 21), do not vanish on (R + × R) \ {0}.

Then, setting f = ũ or ṽ, we have [START_REF] Haase | The functional calculus for sectorial Operators[END_REF], pp. 28 and 45, in this case we can set:

1 f = 1 + h, where h := 1 -f f ∈ H(S θ A ). Since 1 -f ∈ E ∞ (S θ A ), there exists R A > 0 such that, if z ∈ S θ A and |z| > R A , then |1 -f | < C |z| -α where α > 0, C > 0. Moreover, there exists R 0 > 0 such that, for all z ∈ S θ A with |z| > R 0 we have |1 -f (z)| < 1/2, we deduce that |f (z)| > 1/2. So, for all z ∈ S θ A with |z| > max(R A , R 0 ) we obtain |h(z)| ≤ 2 C |z| -α . It follows that h ∈ E ∞ (S θ A ). Indeed, since σ(-A) ⊂ S θ A , from
1 f (-A) = I + h(-A) ∈ L(X),
and the invertibility of f (-A) = U or V is obtained by writing

1 f (-A)f (-A) = 1 f × f (-A) = 1(-A) = I,
and similarly, f

(-A) 1 f (-A) = I.
5 Proof of Theorem 2.8

Note that from Proposition 4.3, in Theorem 2.8, we just have to prove the reciprocal implications.

For each boundary conditions, the proof of Theorem 2.8 is divided in two steps. First, using the representation formula obtained in Proposition 4.1, we show the uniqueness of the classical solution.

Then, in a second step, we state that the solution given by the representation formula is a classical solution.

Proof of statement 1. (Boundary Conditions (BC2))

Assume that (H 1 ), (H 2 ), (H 3 ) and ( 12) hold. First of all, we will show that the solution of problem ( 4)-(BC2) is unique by determining constants K i , i = 1, 2, 3, 4 of the representation formula [START_REF] Haase | The functional calculus for sectorial Operators[END_REF]. Then, we will prove that this formula is a classical solution.

First step: Uniqueness. If u is a classical solution of problem ( 4)-(BC2), then from Proposition 4.1, u reads as

u(x) = e (x-a)M K 1 + (x -a)e (x-a)M K 2 + e (b-x)M K 3 + (b -x)e (b-x)M K 4 + F 0,f (x).
To conclude, it is sufficient to show that K 1 , K 2 , K 3 and K 4 are uniquely determined. Since F 0,f satisfies [START_REF] Haase | Functional calculus for groups and applications to evolution equations[END_REF], by making explicit the boundary conditions with M -1 u(x), M -1 u (x) and M -1 u (x), we obtain the following system:

                 K 1 + M -1 K 2 -e cM K 3 -cI + M -1 e cM K 4 = M -1 ϕ 1 -F 0,f (a) e cM K 1 + cI + M -1 e cM K 2 -K 3 -M -1 K 4 = M -1 ϕ 2 -F 0,f (b) 2K 2 + 2e cM K 4 = M -1 ϕ 3 2e cM K 2 + 2K 4 = M -1 ϕ 4 .
Note that we have considered M -1 u(x), M -1 u (x) and M -1 u (x), because we do not know if

K i ∈ D(M ), i = 1, 2, 3, 4.
We deduce that this system can be written as two uncoupled sub-systems:

     K 1 -e cM K 3 = M -1 ϕ 1 -F 0,f (a) -K 2 + cI + M -1 e cM K 4 e cM K 1 -K 3 = M -1 ϕ 2 -F 0,f (b) + K 4 -cI + M -1 e cM K 2 (22)
and

       K 2 + e cM K 4 = 1 2 M -1 ϕ 3 e cM K 2 + K 4 = 1 2 M -1 ϕ 4 , (23)
which we have to solve. We begin by system [START_REF] Lin | Nonlinear non-local elliptic equation modelling electrostatic actuation[END_REF]. We calculate the determinant of the associated matrix called Λ 1 . Hence det(Λ 1 ) = I -e 2cM , which is invertible with bounded inverse from [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF], Proposition 2.3.6, p. 60. Thus

       K 2 = 1 2 M -1 I -e 2cM -1 ϕ 3 -e cM ϕ 4 K 4 = 1 2 M -1 I -e 2cM -1 ϕ 4 -e cM ϕ 3 .
To determine K 1 and K 3 (as function of K 2 and K 4 ), it is sufficient to remark that system [START_REF] Limam | On Some Transmission Problems Set in a Biological Cell, Analysis and Resolution[END_REF] reads as

     K 1 -e cM K 3 = M -1 ϕ 1 -F 0,f (a) -K 2 + cI + M -1 e cM K 4 e cM K 1 -K 3 = M -1 ϕ 2 -F 0,f (b) + K 4 -cI + M -1 e cM K 2 .
Then we compute the determinant of the associated matrix called Λ 2 .

det

(Λ 2 ) = -I -e 2cM ,
which is invertible with bounded inverse from [START_REF] Lunardi | Analytic semigroups and optimal regularity in parabolic problems[END_REF], Proposition 2.3.6, p. 60. Thus K 1 and K 3 are uniquely determined. Moreover, we have

     K 1 = M -1 ϕ 1 -F 0,f (a) -K 2 + cI + M -1 e cM K 4 + e cM K 3 K 3 = -M -1 ϕ 2 -F 0,f (b) + K 4 + cI + M -1 e cM K 2 + e cM K 1 , (24)
where e cM K 1 and e cM K 3 are regular terms, see [START_REF] Ochoa | A generalized reaction-diffusion model for spatial structures formed by motile cells[END_REF] below.

Second step: Existence. From the previous step, if the classical solution exists, it is unique and is given by the representation formula [START_REF] Haase | The functional calculus for sectorial Operators[END_REF]. Note that since F 0,f ∈ W 4,p (

) 25
We have to make explicit constants K i , i = 1, 2, 3, 4 and to determine their regularity. For c > 0 and ψ ∈ X, one has

e cM ψ ∈ D(M ∞ ) := k 0 D(M k ). (26)
Thus, from Lemma 3.3, we have

I ± e 2cM -1 = I + R ± , where R ± (X) ⊂ D(M ∞ ) and R ± M = M R ± .
Then, from ( 24), ( 26) and Lemma 3.3, there exist

R i ∈ D(M ∞ ), i = 1, 2, 3, 4, such that                        K 2 = 1 2 M -1 ϕ 3 + R 2 K 4 = 1 2 M -1 ϕ 4 + R 4 K 1 = M -1 ϕ 1 -F 0,f (a) -K 2 + R 1 K 3 = -M -1 ϕ 2 -F 0,f (b) + K 4 + R 3 .
So, from ( 12) and ( 25), we have

K 2 , K 4 ∈ D(M 2 ) and M 2 K 2 , M 2 K 4 ∈ (D(M ), X) 1 p ,p , hence, we deduce that K 1 , K 3 ∈ D(M 3 ) and M 3 K 1 , M 3 K 3 ∈ (D(M ), X) 1 p ,p .
Thus, from Lemma 3.2, we obtain that u M : x -→ e (x-a)M K 1 + e (b-x)M K 3 , satisfies

u M ∈ W 4,p (a, b; X) ∩ L p a, b; D(A 2 ) and u M ∈ L p (a, b; D(A)) . ( 27
)
From Proposition 4.2, we deduce that v K

2 : x -→ (x -a)e (x-a)M K 2 satisfies v K 2 ∈ W 4,p (a, b; X) ∩ L p a, b; D(A 2 ) and v K 2 ∈ L p (a, b; D(A)) . ( 28
)
In the same way, v

K 4 : x -→ (b -x)e (b-x)M K 4 satisfies v K 4 ∈ W 4,p (a, b; X) ∩ L p a, b; D(A 2 ) and v K 4 ∈ L p (a, b; D(A)) . ( 29
)
Since F 0,f is a classical solution of (4), from ( 27), ( 28) and ( 29), u is a classical solution of problem ( 4)-(BC2).

Proof of statement 2. (Boundary Conditions (BC3))

Now, we assume that (H 1 ), (H 2 ) and (H 3 ) with θ A < π/2 hold. For the statement 1., the representation formula was easily obtained by taking into account the boundary conditions. Nevertheless, for the following statements, to build the representation formula, we need the invertibility of determinants operators, defined by [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF]. This invertibility has been proved in Proposition 4.5 .

Proof of 2. of Theorem 2.8. Assume that (13) holds. We will show that there exists a unique classical solution of ( 4)-(BC3). The proof is divided in two step. First, we prove the uniqueness of the solution by determining constants K i , i = 1, 2, 3, 4, of the representation formula [START_REF] Haase | The functional calculus for sectorial Operators[END_REF]. Then, we show that the formula obtained is a classical solution.

First step: Uniqueness. If u is a classical solution of problem ( 4)-(BC3), then from Proposition 4.1, the solution u reads as

u(x) = e (x-a)M K 1 + (x -a)e (x-a)M K 2 + e (b-x)M K 3 + (b -x)e (b-x)M K 4 + F 0,f (x).
To simplify our computations, we set

α 1 := K 1 -K 3 2 , α 2 := K 2 -K 4 2 , α 3 := K 1 + K 3 2 and α 4 := K 2 + K 4 2 . ( 30
)
Then, u is given by

u(x) = e (x-a)M -e (b-x)M α 1 + (x -a)e (x-a)M -(b -x)e (b-x)M α 2 + e (x-a)M + e (b-x)M α 3 + (x -a)e (x-a)M + (b -x)e (b-x)M α 4 + F 0,f (x). (31)
As F 0,f satisfies ( 16), to make explicit the boundary conditions, we consider M -1 u(x) and M -1 u (x), since we do not know if α 1 , α 3 ∈ D(M ). Therefore, we obtain the following equations:

I -e cM M -1 α 1 -ce cM M -1 α 2 + I + e cM M -1 α 3 + ce cM M -1 α 4 = M -1 ϕ 1 ( 32
) -I -e cM M -1 α 1 + ce cM M -1 α 2 + I + e cM M -1 α 3 + ce cM M -1 α 4 = M -1 ϕ 2 (33)
I + e cM α 1 + I + (cM + I) e cM M -1 α 2 + I -e cM α 3 + I -(cM + I) e cM M -1 α 4 = M -1 ϕ 3 -F 0,f (a)
(34)

I + e cM α 1 + I + (cM + I) e cM M -1 α 2 -I -e cM α 3 -I -(cM + I) e cM M -1 α 4 = M -1 ϕ 4 -F 0,f (b) (35)
Summing and subtracting [START_REF] Titeux | Conditions de transmission pour les jonctions de plaques minces[END_REF] with [START_REF] Triebel | Interpolation theory, function Spaces, differential Operators[END_REF] and summing and subtracting (34) with (35), we obtain the following system

                               I + e cM α 3 + ce cM α 4 = ϕ 1 + ϕ 2 2 I -e cM α 3 + I -(cM + I) e cM M -1 α 4 = M -1 φ1 -I -e cM α 1 + ce cM α 2 = ϕ 2 -ϕ 1 2 I + e cM α 1 + I + (cM + I) e cM M -1 α 2 = M -1 φ2 , where φ1 := ϕ 3 -ϕ 4 + F 0,f (b) -F 0,f (a) 2 and φ2 := ϕ 3 + ϕ 4 -F 0,f (a) -F 0,f (b) 2 . ( 36
)
We deduce that this system can be written as two uncoupled sub-systems:

       I + e cM α 3 + ce cM α 4 = ϕ 1 + ϕ 2 2 I -e cM α 3 + I -(cM + I) e cM M -1 α 4 = M -1 φ1 , (37)
and

     -I -e cM α 1 + ce cM α 2 = ϕ 2 -ϕ 1 2 I + e cM α 1 + I + (cM + I) e cM M -1 α 2 = M -1 φ2 , (38)
we have to solve. We begin by system (37). By summing both lines, we obtain

         I + e cM α 3 + ce cM α 4 = ϕ 1 + ϕ 2 2 2α 3 + I -e cM M -1 α 4 = M -1 φ1 + ϕ 1 + ϕ 2 2 .
(39)

Then, we calculate the determinant of the associated matrix Λ

1 , det(Λ 1 ) = I + e cM I -e cM M -1 -2ce cM = M -1 I -e 2cM -2cM e cM = M -1 V,
where V is defined by [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF]. Now, we consider system (38). By subtracting the first line to the second, we obtain

       -I -e cM α 1 + ce cM α 2 = ϕ 2 -ϕ 1 2 2α 1 + I + e cM M -1 α 2 = M -1 φ2 - ϕ 2 -ϕ 1 2 . ( 40
Since F 0,f is a classical solution of (4), from (41), ( 42) and (43), we deduce that u is a classical solution of ( 4)-(BC3).

Proof of statement 3. (Boundary Conditions (BC4))

We proceed as in the previous proof and we also assume that (H 1 ), (H 2 ) and (H 3 ) with θ A < π/2 hold. We only point out the differences between the two proofs.

Proof. Assume that (14) holds.

First step: Uniqueness. If u is a classical solution of ( 4)-(BC4), then from Proposition 4.1, u is given by [START_REF] Saker | On the bilaplacian problem with nonlinear boundary conditions[END_REF]. Since F 0,f satisfies [START_REF] Haase | Functional calculus for groups and applications to evolution equations[END_REF], by using M -2 u (x) and M -2 u (x) and following the same computations as done before to make explicit the boundary conditions, we obtain the two following uncoupled sub-systems: (49)

In the same way, from ( 14) and ( 25), we deduce Therefore, since F 0,f is a classical solution of (4), from ( 49) and (50), we deduce that u is a classical solution of ( 4)-(BC4).

Conclusion

In this work, the biharmonic equation with boundary conditions is studied. This model which is described by a fourth order operational equation, within a cylindrical n-dimensional spatial region Ω of R n is analysed using mathematical tools borrowed from interpolation spaces and Dore-Venni sums theory. The main result could be applied, for instance, to study a transmission problem in two juxtaposed habitats. In the next important step, we will focus on the nonlinear case.

Corollary 2 . 9 .

 29 )-(3.3): Consider a cylindrical domain Ω = (a, b) × ω of R n , where n 2, a, b ∈ R, a < b and ω is a bounded open set of R n-1 , where n ∈ N \ {0, 1}, with C 2 boundary. Let f ∈ L p (Ω) with p ∈ (1, +∞) and p > n. Then, there exists a unique solution u

Now, it remains

  to determine the expression of u h . To this end, we have to solve the previous homogeneous equation. For all x ∈ [a, b], we set v(x) = u h (x) + Au h (x). Since u h ∈ W 4,p (a, b; X) ∩ L p (a, b; D(A 2 )) with u h ∈ L p (a, b; D(A)), we deduce that v ∈ W 2,p (a, b; X) ∩ L p (a, b; D(A)).

  (a, b; X) ∩ L p (a, b; D(A 2 )), then from Lemma 3.1, (16), we have F 0,f (a), F 0,f (b) ∈ D(M 2 ) and M 2 F 0,f (a), M 2 F 0,f (b) ∈ (D(M ), X) 1 p ,p .

  )Then, we calculate the determinant of the associated matrix Λ 2 , det(Λ 2 ) = -I -e cM I + e cM M -1 -2ce cM = -M -1 I -e 2cM + 2cM e cM = -M -1 U,Then, from[START_REF] Grisvard | Spazi di tracce e applicazioni[END_REF], since ϕ 1 , ϕ 2 ∈ D(M 3 ) and M 3 ϕ 1 , M 3 ϕ 2 ∈ (D(M ), X) 1 p ,p , we haveK 1 = ϕ 1 + R 1 ∈ D(M 3 ) with M 3 K 1 ∈ (D(M ), X) 1 p ,p ,andK 3 = ϕ 2 + R 3 ∈ D(M 3 ) with M 3 K 3 ∈ (D(M ), X) 1 p ,p . From Lemma 3.2, u M : x -→ e (x-a)M K 1 + e (b-x)M K 3 satisfies u M ∈ W 4,p (a, b; X) ∩ L p a, b; D(A 2 ) with u M ∈ L p (a, b; D(A)) . (41)In the same way, from[START_REF] Grisvard | Spazi di tracce e applicazioni[END_REF], since ϕ 1 , ϕ 2 ∈ D(M 3 ) and ϕ 3 , ϕ 4 ∈ D(M 2 ) such thatM 3 ϕ 1 , M 3 ϕ 2 , M 2 ϕ 3 , M 2 ϕ 4 ∈ (D(M ), X) 1 p ,p ,we haveK 2 = ϕ 3 -M ϕ 1 + R 2 ∈ D(M 2 ) with M 2 K 2 ∈ (D(M ), X) 1 p ,p , and K 4 = -M ϕ 2 -ϕ 4 + R 4 ∈ D(M 2 ) with M 2 K 4 ∈ (D(M ), X) 1 p ,p . From Proposition 4.2, v K 2 : x -→ (x -a)e (x-a)M K 2 satisfies v K 2 ∈ W 4,p (a, b; X) ∩ L p a, b; D(A 2 ) and v K 2 ∈ L p (a, b; D(A)) ,(42) moreover, v K 4 : x -→ (b -x)e (b-x)M K 4 satisfies v K 4 ∈ W 4,p(a, b; X) ∩ L p a, b; D(A 2 ) and v K 4 ∈ L p (a, b; D(A)) .

II

  + e cM α 1 + I + (I + cM ) e cM M -1 α 2 = M -1 φ1 2α 1 + 3I -e cM M -1 α 2 = M -1 M -1 ϕ 3 --e cM α 3 + I -(I + cM ) e cM M -1 α 4 = M -1 φ2 2α 3 + 3I + e cM M -1 α 4 = M -1 M -1 ϕ 3 + ϕ 4 2 + φ2 ,(45)Then, from (14) and (25), we haveK 1 , K 3 ∈ D(M 3 ) and M 3 K 1 , M 3 K 3 ∈ (D(M ), X) 1 p ,p .From Lemma 3.2, u 1 : x -→ e (x-a)M K 1 and u 3 : x -→ e (b-x)M K 3 , satisfyu 1 , u 3 ∈ W 4,p (a, b; X) ∩ L p a, b; D(A 2 )and u 1 , u 3 ∈ L p (a, b; D(A)) .

K 2 , K 4 ∈

 24 D(M 2 ) and M 2 K 2 , M 2 K 4 ∈ (D(M ), X) 1 p ,p . From Proposition 4.2, v 2 : x -→ (x -a)e (x-a)M K 2 and v 4 : x -→ (b -x)e (b-x)M K 4 satisfy v 2 , v 4 ∈ W 4,p (a, b; X) ∩ L p a, b; D(A 2 )and v 2 , v 4 ∈ L p (a, b; D(A)) .(50)

Acknowledgments

This research is supported by CIFRE contract 2014/1307 with Qualiom Eco company. The author would like to thank the referee for the valuable and useful comments and corrections which help to improve this paper.

where U is defined by [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF].

From Proposition 4.5, U and V are invertible with bounded inverse. Thus, if u is a classical solution of ( 4)-(BC3), then u is uniquely determined by [START_REF] Saker | On the bilaplacian problem with nonlinear boundary conditions[END_REF], where α i , i = 1, 2, 3, 4 (which the explicit expression is given in the second step) are the unique solutions of systems (39) and (40).

Second step: Existence.

From the previous step, if the classical solution u exists, it is unique and it is given by ( 31), (39) and ( 40). Now we have to make explicit constants α i and K i , i = 1, 2, 3, 4 to determine their regularity. From (36) and (40), we have

In the same way, from (36) and (39), we have

From [START_REF] Ochoa | A generalized reaction-diffusion model for spatial structures formed by motile cells[END_REF], there exist

Then, from Lemma 3.3, we have

We deduce that there exist

From ( 25) and [START_REF] Rubio De Francia | Martingale and integral transforms of Banach space valued functions[END_REF], there exist

where

We calculate the determinant of the associated matrix of system (44

where V is defined by [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF]. Then, we do the same for Λ 2 , the associated matrix of system (45),

where U is defined by [START_REF] Labbas | On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain[END_REF]. From Proposition 4.5, U and V are invertible with bounded inverse. Thus, from (44), ( 45) and ( 46), we obtain

and

Therefore, u is a classical solution of ( 4)-(BC4), then u is uniquely determined by [START_REF] Saker | On the bilaplacian problem with nonlinear boundary conditions[END_REF], where coefficients α i , i = 1, 2, 3, 4, are given by ( 47) and (48).

Second step: Existence. From the previous step, if the solution u exists, it is unique and it is given by ( 31), (44) and (45). From ( 26), [START_REF] Rubio De Francia | Martingale and integral transforms of Banach space valued functions[END_REF] and Lemma 3.3, we deduce that there exist R i ∈ D(M ∞ ), i = 1, 2, 3, 4, such that