Amel Bouzeghoub
email: amel.bouzeghoub@int-evry.fr

Abdeltif Elbyed
email: abdeltif.elbyed@int-evry.fr

Ontology Mapping for Learning Objects Repositories Interoperability

Keywords: Learning Resources, Ontology Mapping, Interoperability and Multi-Agent Systems

In order to deal with the need of sharing learning objects within and across learning object repositories most of the recent work argue for the use of ontologies as a means for providing a shared understanding of common domains. But with the proliferation of many different ontologies even for the same domain, it become necessary to provide mapping process to perform interoperability. The main key issue must be addressed is to define educational systems interoperability mechanisms to create a virtual learning space. Although many efforts in ontology mapping have already been carried out, few of them use resources properties to generate semantic relations between local concepts. Our approach uses inference rules to combine several matchers in order to discover mapping dynamically and to improve the results qualities.

Introduction

Ontologies offer a great potential in higher education providing in particular the sharing and reusing of information across educational systems and enabling intelligent and personalized learner support. The increased functionalities that ontologies imply will bring new opportunities to e-learning. Learners will be able to interact with distant educational systems easily and in a personalized way. An overview of ontologies for education field and an initial report on the development of an ontology-driven web portal O4E are presented in [START_REF] Dicheva1 | Ontological Web Portal for Educational Ontologies[END_REF].

We propose in this paper an algorithm which is applied on a Web Based Educational System (WBES) named SIMBAD [START_REF] Bouzeghoub | A RDF Description Model for Manipulating Learning Objects[END_REF]. To facilitate resources exchange with other WBES, it becomes necessary to find solutions allowing the cooperation between various repositories of learning resources. The user may seek resources out of his/her private reference ontology. The problem is that the comprehension of a new classification (a new ontology) is expensive and does not constitute a justified investment. It is thus necessary to propose mechanisms which allow the user to access to resources of other repositories in a transparent way using his/her favourite WBES.

The particularity of this approach is that (i) it focuses on dynamic ontology mapping using a multi-agent system, (ii) it uses the resources properties to enrich the local ontology by generating semantic relations between local concepts (iii) it is based on inference rules to compare the ontologies concepts. These inference rules may be general ones or more specific rules added by an expert. This flexibility allowed the algorithm to be applied to other domains.

In this paper we introduce a dynamic mapping approach for bridging gaps between learning object repositories based on ontologies. Dynamic ontology mapping means that during a user interaction (query), the mapping system receives a sequence of concepts and returns the most specific mapping for each concept.

Algorithm Principal

The objective of our approach is to map ontologies dynamically, and only when needed. The system tries to find semantic relations between the concepts of the user query and the concepts in the target ontologies.

The algorithm is based on a multi-agent system and it combines different similarity measures to find mapping candidates between two ontologies. We distinguish three main categories of similarity: linguistic similarity, structural similarity and rule-based similarity. Using these different similarities may increase the precision of the results.

In this section we describe the global architecture and the agents' behaviours of the mapping process.

Architecture

The ontology interoperability needs to define mapping between ontologies. In our architecture (figure 1) the mapping process is split up into five levels: (i) resources level contains the various resources repositories (ii) ontology level describes the resources classification (iii) interface level where we can find, the users and a set of the ontology agents (OA) which generate new ontologies enriched with semantic relations, (iv) simulation level where we calculate the similarity between candidate concepts and (v) domain expert level allows to a expert to interact with the simulation level.

Mapping Process and Agents Behaviors

The algorithm begins by generating information from the ontology. The ontology agent OA uses the instances (resources) comparisons for deducing semantic relations between concepts of the same ontology to deduce a new ontology O+. The OA agent which intercepts a user query generates all possible relations between the query concepts and sends both concepts and these relations to all other OA agents.

The simulation level contains four agents: SCA (Similarity Computation Agent), GHA (Generation Hypotheses Agent), FHA (Filtration Hypotheses Agent) and CHA (Choice Hypotheses Agent). We describe in the following the general behaviour of each simulation agent:

The SCA agent determines similarity values of candidate mappings via different matchers. The first iteration consists in providing a basic similarity between concepts. In this iteration we use linguistic tools [START_REF] Silva | Ontology mapping for interoperability in semantic web[END_REF]7] to compare concepts' names. In the i th iteration we use the similarity produced in (i-1)

th iteration and we apply the inference rules. These inference rules are either rules inferred from structural similarity or rules proposed by the domain expert. The GHA agent receives all similarities sent by SCA and it generates hypotheses using inference rules. These hypotheses consist of new correspondences between concepts. The generation of a hypothesis at i th iteration is based on either the mapping set or the similarities generated previously. Indeed, depending on the similarity value, we generate mapping hypotheses between the couple of concepts which have a similarity value enough important.

The FHA agent filters all hypotheses generated by GHA. The hypotheses which do not verify certain constraints (e.g. structural constraints) are removed. The subset of filtered hypotheses is sent to CHA agent.

The CHA agent selects the hypotheses which have the best similarity using both existing mapping and user's feedback.

The final mapping is sent to ontology agents. After several user interactions, each OA acquires more knowledge about other OA(s) and defines a set of most relevant OA(s) (i.e. the OA(s) that answer to its needs). This set is called "agent's accountancies".

Conclusion

Various works [START_REF] Euzenat | State of the art on ontology alignment[END_REF][START_REF] Ehrig | Ontology Mapping -An Integrated Approach April 21[END_REF][START_REF] Kalfoglou | Ontology mapping: the state of the art[END_REF] have been developed for supporting the mapping of ontologies. Most of them are based on syntactic and semantic matching heuristics given by an expert to generate static mapping. None uses inference rules which can be generated for different application domains. In our mapping approach, we try to use as much as possible available information contained in the ontology to determine dynamically and if necessary the relationship between concepts. This information consists of identifier name of concepts, ontology structure, manual/automatic rules and resources properties which generate new semantic relations between concepts of the same ontology.

In future work, we plan to add other match and techniques in order to resolve more complex mapping problems. The implementation of the prototype is in its first phase. It will be followed by a series of the experiments to validate and justify the quality of the approach in sharing learning resources.

Fig. 1 .

 1 Fig. 1. Architecture approach for mapping process


Laurent Libessart
email: laurent.libessart@univ-artois.fr

Mohamed Amine Kenai

Measuring thermal conductivity of green-walls components in controlled conditions

Keywords: Thermal conductivity, green wall, living wall, humidity, substratum, insulation, thermal simulation Highlights

Thermal conductivities are measured in granular state

• Components of green-walls are compared

• The humidity influences more or less thermal conductivities of the substrates • In the dry state the substrates can be considered as isolating (=0.060W/m °C)

• The ivy and the Virginia creeper would be good complements to insulation

Introduction

There exist two types of green-walls: simple green-walls (intensive) and complex green-walls (extensive) [START_REF] Kenai | Impact énergétique d'un mur végétalisé sur les performances thermiques d'une paroi de bâtiment, XIII ème Colloque Interuniversitaire Franco-Québécois sur la thermique des Systèmes[END_REF]. The simple green-walls use climbing plants of standard ivy which come "to invade" the wall. It is necessary to wait several years to get the desired result. The complex green-walls, generally claiming an intensive maintenance technique, are working in terms of design. In fact, there exist several processes of implementation. Several manufacturers dispute the market and propose various construction processes.

But overall, the principle remains the same one: a system of reinforcement is fixed on the wall, leaving a blade of air between this one and the wall. These reinforcements make it possible to place a substrate in the structure to accommodate the plants.

The concept of reconciliation ecology represents one of the major challenges for the future of urban areas [START_REF] Francis | Urban reconciliation ecology: The potential of living roofs and walls[END_REF]. The ecological engineering techniques of living roofs and walls represent a great potential for reconciliation ecology [START_REF] Francis | Urban reconciliation ecology: The potential of living roofs and walls[END_REF]. But, this task is not easy because it faces two main socioeconomic barriers to their constructions: the cost and the cultural perception of nature [START_REF] Francis | Urban reconciliation ecology: The potential of living roofs and walls[END_REF]. Moreover, a French study [START_REF] Sofres | Les français et leur Habitat : perception de la densité et des formes d'habitat[END_REF] showed conflicting interests among citizens. Urban populations express a need for natural spaces and intimacy as well as a need for social contacts with a surrounding environment providing many services such as stores, recreation activities or transport network. Several studies examined the effects of green walls on the temperatures of façades [START_REF] Djedjig | Experimental study of green walls impact son buildings in summer and winter under an oceanic climate[END_REF]. They have a certain cooling efficiency on the immediate environment [START_REF] Wong | Thermal evaluation of vertical greenery systems for building walls[END_REF]: they decrease the temperatures more than on a control facade without vegetation [START_REF] Cheng | Thermal performance of a vegetable cladding system on façade walls[END_REF][START_REF] Sternberg | Evaluating the role of ivy (Hedera helix) in moderating wall surface microclimates and contributing to the bioprotection of historic buildings[END_REF][START_REF] Wong | Acoustics evaluation of vertical greenery systems for building walls[END_REF]. Several factors influence the observed differences of temperatures. They are grouped around three basic mechanisms.

The first is the sunlight ray blocked by the vegetation, which creates a shadow and influences facade temperatures. The importance of this mechanism depends on the density and homogeneity of the foliage.

The second is the thermal insulation induced by plants and substrate. The thickness, the density and moisture content of the substrate, the foliage density, the air movement in the intermediate spaces, and the capacity to stop the wind are all factors that affect this mechanism. The third is the facade protection against wind. The effectiveness of this last mechanism depends on the facades orientation, the direction and speed of the wind, and the foliage density [START_REF] Wong | Thermal evaluation of vertical greenery systems for building walls[END_REF][START_REF] Cheng | Thermal performance of a vegetable cladding system on façade walls[END_REF][START_REF] Alexandri | Temperature decreases in an urban canyon due to green walls and green roofs in diverse climates[END_REF][START_REF] Getter | The Role of Extensive Green Roofs Sustainable Development[END_REF][START_REF] Castleton | Green roofs, building energy savings and the potential for retrofit[END_REF][START_REF] Kontoleon | The effect of the orientation and proportion of a plant-covered wall layer on the thermal performance of a building zone[END_REF]. Precisions on their component's thermal behavior were given by a model on the thermodynamic transmission process of the vertical ecosystem [START_REF] Jim | Estimating heat flux transmission of vertical greenery ecosystem[END_REF]. But no measurements were found in the literature. The screening effect by the plants of a building was studied recently [START_REF] Kenai | Impact of plants occultation on energy balance: Experimental study[END_REF]. The process developed by prediction of the thermal behavior of buildings envelopes in the presence of simple green-walls was applied to a virtual house located at Lille. The green-walls are put up more and more on the buildings frontages. As this background shows, greenwalls are creating a growing interest in research, and they are approached using different methods and scales. Nevertheless, their thermal benefit still not yet completely mastered in thermal simulation. They are always more used for their aesthetic aspect. Many gaps still remain about their ecological and environmental value. More generally, the required design to optimize their ecological, thermal, acoustic and sensory functions still has to be elucidated. Finally the suppliers of green-walls are interested by the determination of the thermal conductivities values to develop their products.

The aim of this study is to develop the integration of the green-walls on the building and to show its benefits on the building frontage. Firstly, the measurement of the thermal conductivity of different substrates for extensive green-wall (sphagnum moss (Sphagnum cristatum), clay balls and substrate of a green-wall CVF are made for manufacturers. Secondly, many walls in France are covered with ivy (Hedera helix) or Virginia creeper (Parthenocissus quinquefolia), it is interesting to look at their thermal interest. Can the green-wall replace a standard thermal insulation? Which thickness of green-walls is it necessary to allow a profitable insulation? Finally, the thermal conductivities knowledge is an asset to develop the databases of the software and thus create more precise thermal simulations.

Methodology and theory

Conduction and conductivity

The thermal transfer through an environment is made possible thanks to three different heating transfer reactions: conduction, convection, and radiation [START_REF] Cherif | Superficial heat transfer by forced convection and radiation in a horizontal channel[END_REF]. For the conductivity measurements, the transfer reaction by conduction was used.

The term of conduction describes the distribution inherent to the internal shocks without transport of material. For example, when two solids with different temperatures enter into contact, the element which the highest temperature is placed into the one with a lower temperature. Such a transfer always shows itself when a material is subject to a gradient of temperature. A heat flux goes across the region where the temperature is the highest towards the region where the temperature is the lowest. The conduction theory rests on the Fourier law. The flow density is proportional in the temperature gradient:

𝜑 ⃗ = -𝜆𝑔𝑟𝑎𝑑 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ (𝑇) (1)
Where

 = Flux density (W/m 2 )  = Thermal conductivity (W/mK) T = temperature (Kelvin)
The energy preservation depends on thermal properties: the thermal conductivity, the thermal diffusivity, and the geometry of the environment. Heat transfer in an opaque, homogeneous solid, without term of source and isotropic is translated by the following equation:

1 𝑎 𝜕𝑇 𝜕𝑡 = ∇²T (2)
Such as: 'a' represents the thermal diffusivity which is only a function of  (thermal conductivity) and Cp (heat volume).

To resolve the heat equation of any environment, it is indispensable to know two parameters; the thermal conductivity and the heat volume. They characterize the thermal transfer in the environment. This project relied on the measurement of thermal properties, especially thermal conductivity of the porous media.

The measure is complex because it depends on more parameters as the material, the contact resistance, the measurement time and the thickness.

Thermal conductivity of porous media

From the resolution of the energy equation [START_REF] Filali | Conductivité thermique apparente des milieux granulaires soumis à des contraintes mécaniques : modélisation et mesures[END_REF][START_REF] Delacre | Identification by thermal impedance: Application to wet materials[END_REF] many models of calculations of thermal conductivity were developed (Table 1). The models found in literature can give good results for the calculation of the apparent thermal conductivity of porous media. They are predictions tools developed to know the mechanisms of thermal transfer in porous media. It is necessary to specify that the models suggested relate to only the diphasic granular mediums. The solid phase consists of only one phase represented by solid particles. In addition, the fluid is represented by pores filled exclusively with air.

λ app min λ f = λ s λ f + ε 1 1- λ s λ f ⁄ + α 3λ s λ f ⁄ λ app max λ f = 1 + α 1 λ s λ f ⁄ -1 + ε 3
The medium is isotropic

Maxwell λ app = ελ f (2λ f + λ s ) + 3αλ s λ f ε(2λ f + λ s ) + 3αλ f
Particles are spherical and no mutual interaction between the particles

Hamilton 𝜆 𝑎𝑝𝑝 𝜆 𝑓 = λ s + (n h -1)λ f -(n h -1)ε(λ f -λ s ) λ s + (n h -1)λ f + α(λ f -λ s )
An

2.3.Experimentation

To characterize materials, the used device is a horizontal system by the guarded hot plate method [START_REF] Lassue | Instrumentation for experimental study on heat transfer coefficient for external surface of building[END_REF][START_REF] Martin | Méthode d'identification de la conductivité thermique des matériaux isolants[END_REF].

It is an experiment in permanent transfer mode (ISO 8302) where one imposes on a plane sample a oneway heat flow, and measures the difference in temperature at the boundaries of the sample. This method rests on the use of a thin plate compared to the dimensions of the sample. The implementation of materials tested is easier. The purpose of this experimentation is to measure thermal conductivities. It is necessary to prevent the thermal transfer by convection by the creation of closed wood box (Fig. 1). Its composition is detailed below. To measure the temperature difference in the borders of the samples, K type thermocouples are used. Its composition is Chromel (alloy nickel + chrome) and Alumel (alloy nickel + aluminum (5%) + siliceous).

It allows a measure in a range of wide temperature from -250°C to 1372°C. It is stuck on the surface to measure the temperature (T1).

To allow a unidirectional flow and to avoid the flow loss on the sides, a guard zone was installed around each sensor (Fig. 3).

Fig. 3. Principal plan

After the second thermocouple is arranged on its surface (T2), the temperature difference can be measured for the Fourier Law resolution.

To generate the heat flow two devices are necessary. Heating plates heat the surface of concrete to a maximal temperature of 40°C. And a heating resistance homogenizes the temperature of the surface.

Furthermore, it will serve for the delicate measurements of plants to obtain a lower temperature (20°C) to avoid the drying of the plants. As a reminder, the measure of conductivity is based on the principle of the guarded warm plate method and the fluxmetric method. It is realized in steady state. The heat flow must be constant in time and uniform in space.

The conductivity value () is calculated by using Fourier Law (3). For it, two temperatures have been measured on the borders of the sample (T) for a surface (S) of 1m² and with a thickness (e) gives the following equation [START_REF]EN 12664 Normes caractérisation de la conductivité thermique des matériaux[END_REF][START_REF] De Pont | Conductivité thermique des isolants[END_REF]:

𝜆 = 𝜙.𝑒 Δ𝑇.𝑆 (W/mK) (3)
The measurements are carried out with a portable multimeter and are correct when the device reached the steady state.

The experimental protocol is carried out in the following manner:

➢ To put on the hotplates and it is necessary to wait a few hours to reach the steady state.

➢ To make sure of the uniformity of the temperature of the concrete slab. The ΔT should not exceed 1°C. Three thermocouples are distributed on the concrete slab.

➢ To place the fluxmeter in the center of the concrete slab (a layout is envisaged)

➢ To place material to be studied with the top of the fluxmeter.

o For a solid material To place the material directly at the top of the fluxmeter.

o For a granular material

To place the polyurethane framework of which the interior is covered with geotextile.

To put the material to be studied within the framework, to fill it to the brim without packing (the material should beforehand be weighed).

To cover the framework with a film of geotextile in order to avoid resistances of contact ➢ To place the second fluxmeter at the top ➢ Lastly, to close the box and to expect the steady state.

Green-wall components

The substrate characterization for complex green-wall are presented in table 2 and in table 3 for the two climbing plants (Ivy, Virginia creeper).

Table 2. Substrate characterization

2.5.Thermal simulation

To complete this study, a modeling was carried out in order to look at the influence of the thermal conductivity of the ivy on consumption energy of a standard house. To determine the influence of the green-wall of a dwelling, the thermal software of simulation instigates TRNSYS 17 has been used. The study case considered thereafter is a house on under floor space. This house is called "House MOZART": it is a "standard" house representing the average of the French houses in main home. It was defined by EDF in 1994 in co-operation with the CSTB (French Scientific and Technical Centre for Building) and GDF.

The thermal simulation is based on the thesis of Mr. Gossard [START_REF] Gossart | Impact de l'enveloppe sur la performance énergétique des bâtiments : de la caractérisation jusqu'à l'optimisation[END_REF] and a study carried out by the CSTB [START_REF] Cstb | Inertie thermique des logements et confort d'été[END_REF] (for comparison purposes) and to have the data necessary to model the "House Mozart". The total space of this house is of 100.86 m² with a height under ceiling of 2.3 meter. The results are presented in paragraph 3.5.

Results and discussion

Device validation

To validate the device, two materials have been used: a plain plate extruded polystyrene and sand. The extruded polystyrene has a conductivity value (supplied by the manufacturer) of  = 0.036 W/m.K. The measurement gives a conductivity value of λmeasured = 0.0377 W/m.K. For this validation, an error of the conductivity value of the polystyrene extruded is about 4 %. Furthermore, T must be higher for 20°C as required by the trial standard [START_REF]EN 12664 Normes caractérisation de la conductivité thermique des matériaux[END_REF]. For the sand, Fillali [START_REF] Filali | Conductivité thermique apparente des milieux granulaires soumis à des contraintes mécaniques : modélisation et mesures[END_REF] found a value of thermal conductivity of dry sand of  = 0.225 W/m.K. The measurement gives a conductivity value of λmeasured= 0.229 W/m.K. The thermal conductivity of sand was measured. The experimental device is validated for porous media.

3.2.Dry substrate

Every material was allowed to dry for 48 hours in a steam room. The basic materials of green-walls were chosen as well as a composition used directly as substrate.

Table 3 presents the results obtained by the measurement of thermal conductivity for an average of three tests. The thickness of the samples is 4 centimeters. The steady state was obtained after 10 hours. According to the French standards, an insulating material is recognized as a heat insulator if its thermal conductivity is lower than 0.06 W/m.K. In our case and in the dry state, the outdoor planting mix, the sphagnum moss and the substrate CVF can be considered as heat insulators. But, numerous heat insulators with lower thermal conductivity are available on the market. In fact, the traditional heat insulators have a thermal conductivity included generally between 0.035 and 0.045 W/m.K. Furthermore, if the substrate is dry, it is very likely that no plant can push above.

• Mixture of outdoor planting mix and clay balls

Formulations of substrate of green-walls can contain clay balls. A mixture of outdoor planting mix and clay balls has been realized in the dry state. The results obtained are presented in the following figures. The graph gives us as relation ( 4):

𝜆 𝑚𝑖𝑥𝑡𝑢𝑟𝑒 = 0.00025 (% 𝐶𝑙𝑎𝑦𝑏𝑎𝑙𝑙𝑠 -% 𝑃𝑙𝑎𝑛𝑡𝑚𝑖𝑥 ) + 0.08675 (4)

3.3.Wet substrate

The next stage consists of studying the substrate in the wet state, here are the results obtained for the outdoor planting mix (Fig. 6) and the sphagnum moss for a 4-centimeter thickness (Fig. 7). steady state was obtained after 13 hours.

And for the sphagnum moss relation [START_REF] Sternberg | Evaluating the role of ivy (Hedera helix) in moderating wall surface microclimates and contributing to the bioprotection of historic buildings[END_REF] :

𝜆 𝑠𝑝ℎ𝑎𝑔𝑛𝑢𝑚_𝑚𝑜𝑠𝑠 = 0.000206 × 𝑤 𝑠𝑝ℎ𝑎𝑔𝑛𝑢𝑚_𝑚𝑜𝑠𝑠 (%) + 0.071892

And for the substrate CVF relation [START_REF] Wong | Acoustics evaluation of vertical greenery systems for building walls[END_REF]:

𝜆 𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒𝐶𝑆𝑉 = 0.000540 × 𝑤 𝑠𝑢𝑏𝑠𝑡𝑟𝑎𝑡𝑒𝐶𝑉𝑆 (%) + 0.04328 (8)
Despite the important water presence, the sphagnum moss preserves a low thermal conductivity. Indeed, for a 890 % moisture content, its conductivity is equal to 0.28 W/m°C, compared with the outdoor planting mix, it corresponds approximately to a 150 % moisture content. These results show the importance of the choice of the substrate for the green-wall creation. It is important to control moisture content.

With the relations (3 and 4), the thermal conductivity of materials can be determined with a measure of moisture content (Fig. 8). The separation between the measured and calculated conductivities can result from the measure of the moisture content, where a sampling is taken from the global mass.

3.4.Plant palette

To avoid the dry out of plants and to avoid the loss of moisture, the paving concrete was warmed slightly to create a temperature difference of about 10°C. After measurement, the plant was always in the same state. plant arrangement was realized so as to reproduce a green-wall thickness of 10 centimeters.

The foliage implementation is realized to feign a green-wall consisted of sheets and air. Measured thermal conductivities are presented table 4. help to reduce overheating on the walls in summer [START_REF] Kenai | Impact of plants occultation on energy balance: Experimental study[END_REF]. In order to quantify the contribution of a greenwall on a dwelling a modeling on TRNSYS is carried out. To take into account the green-wall (Fig. 10), TRNBuild determined these specific properties (λ= 0.25 W/m°C, = 16 kg/m 3 and a heat capacity of 150 Wh/m 3 .K).

3.5.Thermal simulation

Conclusion

This study gives concrete values of thermal conductivity of the components of the complex green-walls for manufacturers. This preliminary study enabled us to find conductivity apparent being able made up a green-wall. This approach remains in conditions controlled with measurements in the laboratory. The measures bring thermal knowledge for the manufacturing of complex green-walls. The manufacturers can develop their product. The green-walls made up of sphagnum have the best thermal conductivities in wet condition. Moreover, it is important to control the moisture of the substrates to obtain the best thermal conductivity. The manufacturers can add sensors moisture in the substrate to find a balance between thermal conductivity, moisture and growth of the plants.

The green-wall materials cannot be used as a replacement to classic types of insulation. On the other hand, they can be used to complement other types of insulation. Their presence controls the solar thermal radiations and thus to avoid overheating in summer.

Concerning the simple green-wall, to obtain a wall respecting the wall insulation (R = 3 m 2 .K/W) with 20 cm of ivy or Virginia creeper and with a veil in concrete of 20 cm would be to add a complementary insulation ( = 0.04 W/mK) of:

➢ For the Virginia creeper : 8.2 centimeters ➢ For the Ivy : 8 centimeters

The thermal insulation investigation shows that 5 cm of ivy makes it possible to reduce the needs for heating. The increase of the ivy thickness to 10 cm does not make real improvement of the insulation. The first results of thermal simulation show the interest of the green-walls on the building. It is important to take them into account in thermal simulations in design offices.

Furthermore, the measure of thermal conductivity is realized on rough materials and provides an intrinsic value of the material; it does not take into account its use. It would be necessary to pursue experiments on exposed sites to measure its influence on the thermal comfort of the house.

  extension of the Maxwell model to particles ellipsoidal Where n = Number of contact per particle  = Porosity of the granular layer  = Relative density i, app = Apparent thermal conductivity of porous media, W/mK s = Thermal conductivity of the solid phase, W/mK f = Thermal conductivity of the fluid to the atmospheric pressure, W/mK  = Sphericity referring to the spherical geometry with nh=3/ The problem is complex. It is necessary to develop models, bringing in different physical phenomena (transfer / condensation, contact law between particles, conduction, and convection) and to verify the microstructure of the environment (size, shape of the particles) and to study the state of the constituents of the environment (conductive phase). Not knowing the thermal conductivity of materials, the bibliographical references validated the experimental models. The measurement of apparent thermal conductivity is interested in taking into account two phases: solid and fluid. It is necessary to consider the problem of resistance to contact which can intervene during the measurement.

Fig. 1 .

 1 Fig. 1. Overview of the wood box and the device of measure.

Fig. 2 .

 2 Fig. 2. Detail of the device of measure.

  The expanded clay is industrially made from natural clay which will successively be dried, reduced to flour, will in some water then will in ovens what gives after all balls or blocks with clay balls. This material has a density between 350 and 700 kg/m 3 . Sphagnum moss It possesses a big absorption capacity and retention of the water, the sphagnum moss is a moss which can store 20× its volume in water grace to its very light texture. It is antibacterial what allows a protection against the decay, the diseases and the parasites and it has a pH naturally acid of 4.8.

Fig. 4 .

 4 Fig. 4. Mix ball of clay and plant mix.

Fig. 5 .

 5 Fig. 5. Variation of the conductivity according to the % of plant mix and clay balls.

Fig. 6 .Fig. 7 .

 67 Fig. 6. Apparent densities of the plant mix according to moisture content

Fig. 8 .

 8 Fig. 8. Comparison between thermal conductivity measured and calculated.

Fig. 9 .

 9 Fig. 9. Modeling of the house Mozart on SketchUp.

Fig. 10 .••

 10 Fig. 10. Modeling of the green-wall of the house Mozart on SketchUp. The assumptions of simulation are Heating: • 19°C except for garage, roofs and underfloor space • Power = limited to 7.5kW per zone • Radiative share = 70%Occupation:• 3 people present of 17:00 at 8:00 the week and all the day the safe weekend in the garage, roofs and underfloor space

Table 1 .

 1 Models developed for the simple granular mediums

	Parallel	λ app = ∑ ε i λ i	Medium made up in several states parallel between them
	Series	λ app = (∑	ε i λ i	) -1	Medium made up in several states in series between them
	Hashin-
	Shtrikman

Table 3 .

 3 The vegetal palet

	Vegetal (%)	Pictures
	Ivy
	Virginia creeper

Table 3 .

 3 Values of thermal conductivity for substrate testing

	Substrate	Thermal conductivity (W/mK)
	Sand	0.225
	Universal plant mix	0.062
	Clay balls	0.105
	Sphaignum	0.060
	Substrate CVF	0.051

Table 4 .

 4 Measure of the vegetal palet

	Vegetal (%)	Thermal conductivity (W/mK)
	Ivy	0.239 -0.265
	Virginia creeper	0.220 -0.274
	It is not easy to reproduce the test with the plants. Five tests were carried out to have this measurement
	range. The results show that the plants don't bring real thermal insulation. On the other hand, they can

By modeling the green-wall, a reduction in the production of heating of 3% has been noted which remains low. An ivy addition of 5 cm does not make it possible to have a better performance in insulation. It is necessary to do a more complete study of the thermal performances of the green-walls and integrated the influences of the other phenomena related to the presence of the vegetation [START_REF] Djedjig | Experimental study of green walls impact son buildings in summer and winter under an oceanic climate[END_REF][START_REF] Kenai | Impact of plants occultation on energy balance: Experimental study[END_REF][START_REF] Kenai | Caractérisation, analyse et modélisation des échanges énergétiques entre un mur végétalisé intensif et son environnement[END_REF].


K V Sklyar
email: sklar@univ.szczecin.pl

R Rabah
email: rabah@emn.fr

G M Sklyar

On a vector moment problem appearing in controllability of neutral type systems

Keywords: vector moment problem, neutral type system, spectral assignment

We consider solvability of a vector moment problem by means of its correspondence to controllability problem for a certain delayed system of neutral type. In this way we succeded to determine exsctly the minimal interval on which the moment problem is solvable.

Studying controllability problem for the following neutral type system ż(t) = A-1 ż(t-1)+

Z 0 -1 A2(θ) ż(t+θ) dθ+ Z 0 -1
A3(θ)z(t+θ) dθ+Bu(t), [START_REF] Avdonin | Families of exponentials. The method of moments in controllability problems for distributed parameter systems[END_REF] where A-1 is a constant n × n-matrix, det A-1 = 0, A2, A3 are n × nmatrices whose elements belong to L2(-1, 0) and B is a constant n × rmatrix, we reduced it to an equivalent moment problem [START_REF] Rabah | The Analysis of Exact Controllability of Neutral-Type Systems by the Moment Problem Approach[END_REF]. We consider the operator model of the neutral type system (1) introduced by Burns and al. in product spaces. The state space is M2(-1, 0; C n ) = C n × L2(-1, 0; C n ), shortly M2, and (1) is rewritten as d dt " y(t) zt(•)

« = A " y(t) zt(•) « + Bu (2)
where the operator A is given by

A " y(t) zt(•) « = " R 0 -1 A2(θ) żt(θ)dθ + R 0 -1 A3(θ)zt(θ)dθ dzt(θ)/dθ « with the domain D(A) = ˘(y, z(•)) : z ∈ H 1 (-1, 0; C n ), y = z(0) -A-1z(-1) ¯⊂ M2.
The operator A is the infinitesimal generator of a C0-group. The operator B is defined by Bu = (Bu, 0). The relation between the solutions of the neutral type system (1) and the system (2) is given by the substitutions

y(t) = z(t) -A-1z(t -1), zt(θ) = z(t + θ).
Complete desription of the spectral properties of operator A is given in [START_REF] Rabah | Stability analysis of neutral type systems in Hilbert space[END_REF]. Consider for simplicity the case when all eigenvalues of A-1 are simple with different arguments. Denote these eigenvalues by µm, m = 1, . . . , n. We will use the notation à for A in the particular case A2(θ) = A3(θ) = 0. Operator à has a simple eigenvalues

λm k = ln |µm| + i(arg µm + 2πk), m = 1, . . . , n, k ∈ Z with corresponding eigenvectors φm k = " 0 e λm k θ Φm « ∈ M2,
where A-1Φm = µmΦm, m = 1, . . . , n. Besides à has the eigenvalue λ0 = 0 that corresponds to an n-dimensional subspace of generalized eigenvectors: φ0 1 , . . . , φ0 n . All eigenvectors { φm k } ∪ { φ0 j } constitute a Riesz basis in M2. In general case if A2(θ) and A3(θ) are chosen in such a way that all eigenvalues of A, λ m k , m = 1, . . . , n, k ∈ Z, are still simple then they satisfy the condition

X k,m ˛λm k -λm k ˛2 < ∞ (3)
and the corresponding eigenvectors {ϕ m k } satisfy

X k,m ϕ m k -φm k 2 < ∞.
Eigenvectors {ϕ m k } together with {ϕ 0 j } form a Ries basis in M2 quadratically close to the spectral basis { φ} of Ã.

Let { ψ} and {ψ} are biorthogonal basises for { φ} and {ϕ} respectively. It is easily checked that { ψ} and {ψ} are spectral basises of adjoint operators à * and A * ,

à * ψm k = λm k ψm k , A * ψ m k = λm k ψ m k .
Besides these basises are also quadraticaly close [START_REF] Gokhberg | Introduction to the theory of linear nonselfadjoint operators[END_REF]:

X k,m ' ' 'ψ m k -ψm k ' ' ' 2 < ∞.
Finally note that as it is shown in [START_REF] Rabah | On strong regular stabilizability for linear neutral type systems[END_REF] vectors ψm k , k = 0, are of the form

ψm k = " 1 k+ 1 2 Ψm * « ∈ M2 ψ m k = " 1 k+ 1 2 Ψ mk * « , (4)
where Ψm, m = 1, . . . , n are eigenvectors of A-1,

A-1Ψm = μmΨm, and X k,m Ψ mk -Ψm 2 < ∞, m = 1, . . . , n. ( 5
) Let now x ∈ M2. Then x = X ϕ∈{ϕ} x, ψ ϕ. A state x = " y z(•) « ∈ M2 is reachable at time T by a control u(•) ∈ L2(0, T ; C r ) iff the steering condition x = " y z(•) « = Z T 0 e At Bu(t)dt. (6)
holds. This steering condition may be expanded using the basis {ϕ}. A state x is reachable iff

X ϕ∈{ϕ} x, ψ ϕ = X ϕ∈{ϕ} Z T 0 e At Bu(t), ψ dtϕ,
for some u(•) ∈ L2(-1, 0; R r ). Then the steering condition ( 6) can be substituted by the following system of equalities

x, ψ = Z T 0 e At Bu(t), ψ dt, ψ ∈ {ψ}. (7)
Let {b1, . . . , br} be columns of B and bi =

" bi 0 « ∈ M2, i = 1, . . . , r.
Then the right hand side of (7) takes the form

Z T 0 e At Bu(t), ψ dt = r X i=1 Z T 0 e At bi, ψ ui(t)dt.
Taking into accout the fact that {ψ} is a spectral basis of A * we rewrite the steering conditions as

s m k = " k + 1 2 « X, Ψ m k = Z T 0 e λ m k t `b1 k,m u1(t) + . . . + b r k,m ur(t) ´dt, (8) m = 1, . . . , n, k ∈ Z, where b j k,m = `k + 1 2
´ bj, ψ m k . Equations (8) pose a vector moment problem with respect to unknown uj(t), j = 1, . . . , r. Using (4) we observe that

" k + 1 2 « bj, ψm k = fi" bj 0 « , " Ψm * «fl = bj, Ψm = b j m ,
So this value does not depend on k. Then from (5) it follows that the coefficients b j k,m satisfy the condition

X k,m ˛bj k,m -b j m ˛2 = X k,m | bj, ψ mk -ψm | 2 < ∞ j = 1, . . . , r. (9)
In the scalar case r = 1, under the conditions

b 1 m = b1, ψm = 0, m = 1, . . . , n (10)
(approximate controllability of system (2)) the solvability of (8) can be studied by application of the methods given in [START_REF] Avdonin | Families of exponentials. The method of moments in controllability problems for distributed parameter systems[END_REF]. In the vector case the problem is essentially more complicated. We showed [START_REF] Rabah | The Analysis of Exact Controllability of Neutral-Type Systems by the Moment Problem Approach[END_REF] that the minimal interval of solvability of ( 2) is related with the first controllability index n1(A-1, B) of the system ẋ = A-1x+Bu (see [START_REF] Wonham | Linear multivariable control: A geometric Approach[END_REF]). This result encouraged us to consider the inverse problem: to study moment problems of the form (8) (of course, under some special assumptions) by means of their correspondence to the controllability problems for neutral type systems. We show that the conditions (3), (9) together with some conditions analogous to (10), (11) are not only necessary but also sufficient in order, for the moment problem (8), to be generated by a neutral type system (1).

Theorem 1. Moment problem (2) corresponds to the controllability problem for an exact controllable neutral type system of the form (1) if and only if it satisfies the condition i)

X k,m ˛λm k -λm k ˛2 < ∞, where λm k = ln|µm| + i(arg µm + 2πk), m = 1, . . . , n, k ∈ Z, µ1, .
. . , µr are some nonzero different complex numbers;

ii)

X j,k,m ˛bj k,m -b j m ˛2 < ∞
, where b j m (m = 1, . . . , n, j = 1, . . . , r) are some real numbers such that

X j ˛bj m ˛> 0, m = 1, . . . , n; iii) X j ˛bj k,m ˛> 0, m = 1, . . . , n, k ∈ Z.
The n × n and n × r-matrices A-1 and B being given by A-1 = diag{µ1, . . . , µn}, B = {b j m } j=1,...,r m=1,...,n .

Our proof of Theorem 1 is based on the following concept. We seek a system (1) in the form

ż(t) = A-1 ż(t -1) + +B Z 0 -1 [F2(θ)z(t + θ) + F3(θ) ż(t + θ)] dθ + Bu(t),
where matrices A-1, B are chosen above, F2, F3 are (r ×n)-indetermined matrices with elements from L2(-1, 0). In infinite-dimensional form this corresponds to the system

d dt " y(t) zt(•) « = A " y(t) zt(•) « + Bu = " Ã + BP " " y(t) zt(•) « + Bu, where P = 0 B @ p1 . . . pr 1 C
A, and pj, j = 1, . . . , r are functionals from space X-1, where

X-1 = 8 < : X ψ∈{ ψ} d ψ ψ, X k,m (d m k /k) 2 < ∞ 9 = ; ,
{ ψ} is a spectral basis of à * . One needs to choose an operator P in such a way that

1) σ( Ã + BP) = {λ m k }, k ∈ Z, m = 1, . . . , n; 2) b j , ψ m k = b j km , ψ m k are eigenvalues of A * .
We reduce this problem to the following one. Denote by Q(λ) an (r × r) matrix-function of the form

Q(λ) = PR( Ã, λ)B, R( Ã, λ) is resolvent of Ã.
It turns out that all the eigenvalues are the roots of the equation

det(I + Q(λ)) = 0
and for and m = 1, . . . , n, k ∈ Z, vector w * mk = (b 1 k,m , . . . , b r k,m ) can be found from the condition:

w * mk (I + Q(λ m k )) = 0.
Finally we give a choice of P providing the required values of parameters {λ m k }, {b j k,m }. Note that Theorem 1 is a direct generalization of pole assignement theorem for neutral type systems with simple control from [START_REF] Rabah | On strong regular stabilizability for linear neutral type systems[END_REF]. Denote by RT the set of moment sequences {s m k } for which (8) is solvable, this means that there exist functions uj(t), j = 1, . . . , r from L2[0, T ] such that (8) is satisfied. It follows from general moment problem theory that RT is a subset of te space l2. Reduction to the controllability problem allows to determine the exact values of T for which RT may be equal l2.

Combining Theorem 1 and the results of [START_REF] Rabah | The Analysis of Exact Controllability of Neutral-Type Systems by the Moment Problem Approach[END_REF] (section 7) we obtain Theorem 2. Let the conditions i) -iii) be satisfied. We have:

1. If T > n1(A-1, B) then RT = h1. 2. If T = n1(A-1, B) then RT is a subspace of h1 of finite co-dimension in h1 (it is possible that RT = h1). 3. If T < n1(A-1, B) then RT is a subspace of h1 of infinite co- dimension.
Example. Consider two vector moment problems:

s 1 k = Z T 0 e (i(π+2πk)+ε 1 k )t (11u1(t) + u2(t))dt, s 2 k = Z T 0 e (i2πk+ε 2 k )t (u1(t) + u2(t))dt, s 3 k = Z T 0 e (log 2+i2πk+ε 3 k )t (u1(t) -u2(t))dt, s 4 k = Z T 0 e (log 3+i2πk+ε 4 k )t (u1(t) -u2(t))dt, (12)
and then the moment problem ( 12) is solvable for all {s j k } ∈ l2 if T > 2 while the problem (13) if T > 3.

s 1 k = Z T 0 e (i(π+2πk)+ε 1 k )t (6u1(t) -u2(t))dt, s 2 k = Z T 0 e (i2πk+ε 2 k )t (4u1(t) -u2(t))dt, s 3 k = Z T 0 e (log 2+i2πk+ε 3 k )t (3u1(t) -u2(t))dt, s 4 k = Z T 0 e (log 3+i2πk+ε 4 k )t (2u1(t) -u2(t))dt, (13)
k ∈ Z, P k (ε j k ) 2 < ∞, j = 1, . . . , 4

(

  controllability of the pair (A-1, B)) and b1, ψ m k = 0, m = 1, . . . , n; k ∈ Z

  . The both moment problems can be reduced to the controllability problem for systems of the form (1, µ3, µ4} = {-1, 1, 2, 3}, corresponding eigenvectors of A * choice of A2(θ), A3(θ). Sincen1(A-1, B1) = 2, n1(A-1, B2) = 3