Flore Villaret
email: flore.villaret@edf.fr

Xavier Boulnat

Pascal Aubry

Yasuhide Yano

Satoshi Ohtsuka

Damien Fabrègue

Yann Decarlan

Yann De Carlan

Laser Beam Direct Energy Deposition of graded austenitic-to-martensitic steel junctions compared to dissimilar Electron Beam welding

Keywords: gradient material, 316L, Fe-9Cr-1Mo, additive manufacturing, mechanical properties, electron beam welding

INTRODUCTION

Dissimilar junctions between austenitic stainless steels and martensitic steels are used in many applications. Examples of such heterogeneous junctions can be found in the nuclear industry where it is required to join 316L and Fe-9Cr-1Mo steel parts. Usually, such junctions are performed by Tungsten Inert Gas (TIG) welding using an Inconel type filling metal. As explained by Pandey et al. [START_REF] Pandey | Some studies on P91 steel and their weldments[END_REF], this process requires preheating before welding to avoid cold cracking and post-welding heat treatments to temper the martensite on Fe-9Cr-1Mo side. These heat treatments allow to obtain the required mechanical properties and especially a good toughness, as shown by Sireesha et al. [START_REF] Sireesha | Microstructure and Mechanical Properties of Weld Fusion Zones in Modified 9Cr-1Mo Steel[END_REF].

Casalino et al. [START_REF] Casalino | Study on the fiber laser/TIG weldability of AISI 304 and[END_REF] demonstrate the feasibility of using hybrid laser/TIG welding as an alternative process to TIG welding. Yano et al. [START_REF] Yano | Weldability of dissimilar joint between PNC-FMS and Type 316 steel under electron beam welding[END_REF] also demonstrate the possibility of using an Electron Beam (EB) welding process to join a ferritic/martensitic steel with a 316L austenitic steel.

The use of chemical composition gradient materials obtained by powder metallurgy is another way to simplify this assembly step. Whether by "conventional" powder metallurgy or additive manufacturing, it is possible to consider the construction of a coupling sleeve which forms a connecting part between the two steels. Additive manufacturing is an attractive process as it opens up the design to new parts with complex geometry and specific functionality. The chemical composition of this connection will gradually change from 316L to Fe-9Cr-1Mo, allowing homogeneous welding at each end to simplify the assembly step. The coefficient of thermal expansion will change gradually through the graded part thereby reducing the mismatch between the two as shown by Sridharan et al. [START_REF] Sridharan | Design , Fabrication , and Characterization of Graded Transition Joints[END_REF]. The reduction of this mismatch allows to reduce the residual stress trough the connector as observed by Woo et al. [START_REF] Woo | Effect of interlayers and scanning strategies on through-thickness residual stress distributions in additive manufactured ferritic-austenitic steel structure[END_REF]. Zuback et al. [START_REF] Zuback | Additive manufacturing of functionally graded transition joints between ferritic and austenitic alloys[END_REF] demonstrate also that the chemical gradient through the connector allows a gradual change in the carbon chemical potential, instead of step change, which can slow carbon diffusion from Fe-9Cr-1Mo to 316L and extend the lifetime of the connection.

Usually, additive manufacturing techniques are used for their superior freedom for shape. In this work it is rather the possibility of changing the material and eventually mixing two alloys during the construction that is investigated. This technique has already been used by some authors such as Yan et al. [START_REF] Yan | Additive Manufacturing of Functionally Graded Metallic Materials Using Laser Metal Deposition[END_REF] to build graded parts and join a variety of different materials successfully. Austenitic steel to ferritic/martensitic steel gradient materials have also been built by direct energy deposition and characterized by Sridharan et al. [START_REF] Sridharan | Design , Fabrication , and Characterization of Graded Transition Joints[END_REF]. The present paper presents another practical case studying a 316L/Fe-9Cr-1Mo graded material obtained by DED-LB with a high dilution rate from one layer to the next affording a smooth gradient. The effect of a martensite tempering heat treatment at 630°C for 8h is also investigated.

For comparison, a dissimilar Electron Beam (EB) weld was also performed to join the same two materials. This technique involves high energy density and narrow heat affected zone [START_REF] Bardel | Residual stresses induced by electron beam welding in a 6061 aluminium alloy[END_REF][START_REF] Chowdhury | A Perspective Review on Experimental Investigation and Numerical Modeling of Electron Beam Welding Process[END_REF][11], hence minimizes distortion, which could be favourable for the welding of martensitic steels [START_REF] Albert | Mechanical properties of similar and dissimilar weldments of RAFMS and AISI 316L (N) SS prepared by electron beam welding process[END_REF][START_REF] Wu | Improving impact toughness of heavy section reduced activation ferritic martensitic CLF-1 steel joints with electron beam welding[END_REF]. EB welding and DED-LB additive manufacturing have in particular close cooling rates, which makes the comparison interesting. In order to make a quantitative comparison, both as welded and heat treated samples were characterised in terms of microstructure and mechanical properties at room temperature, at 400°C and 550°C.

MATERIALS AND METHODS

Graded material samples were obtained using 316L and Fe-9Cr-1Mo pre-alloyed steel powders. Chemical composition and size distribution of powders used are given in Table 1 Samples were obtained using the Laser Beam Direct Energy Deposition (DED-LB) process with two powder feeders enabling powder change during building (Figure 1). In this process, the powder is carried from the powder feeders to the constructed part by a carrying gas (argon here). It is possible to control the composition of the deposited layer by controlling the flow of each powder feeders. In this study, Fe-9Cr-1Mo was built over 316L by changing directly from one powder to another without mixing powder in between. Parameters used for construction, listed in Table 2, were optimised for 316L and kept constant during the building despite change of materials. Single tracks walls built with a back-and-forth strategy were used. Heat treatments for martensite tempering were performed at 630°C for 8 h on DED-LB samples. Electron Beam (EB) dissimilar welds were studied for comparison. EB welding conditions were: a focal point of 50 mm, welding speed of 2,000 mm/min and an acceleration voltage of 60 kV. EB welding was performed on 5 mm thick plates with no nickel-base alloy filler. Post-weld heat treatment (PWHT) for all samples was performed at 630°C for 8 h also.

For optical metallographic analysis, samples were polished and successively electro-etched in 10 vol% oxalic acid to reveal the austenitic microstructure and etched with Villela reagent to reveal the martensitic structure. Scanning electron microscopy (SEM), energy dispersive spectroscopy (EDS) and electron backscattered diffraction (EBSD) analyses were undertaken before the two-step etching. Two specimens extracted from the same DED-LB sample were used to study the effect of the tempering heat treatment. The same applies for EB welds.

Vickers microhardness was measured at a load of 100 g for EB welds and 50 g for DED-LB samples. The space between two measurements was set to 100 µm in the X and Y directions for DED-LB samples. This affords a hardness map across the interface between the two materials. However, as the welded area is much larger than the mixed area from additive manufacturing, this is why load used for measurements were different and the space between the two indentations was set to 200 µm in X direction and 500 µm in Y direction for the EB welded samples.

After heat treatments, flat specimens with their graded area as well as with their weld located at the centre of the gauge section were used for tensile tests. Tensile properties were measured at three temperatures: 20 °C, 400 °C and 550 °C. Stress was applied across the weld and the gradient direction (parallel to the build direction for additive manufacturing) with a strain rate of 10 -4 s -1 . DED-LB samples were not surface machined before the test so presented a rough surface. Microtomography with a voxel size of 7.5 µm was used to characterize the gauge length as previously described by [START_REF] Persenot | Enhancing the tensile properties of EBM as-built thin parts: Effect of HIP and chemical etching[END_REF]. The mean gauge section was used to plot tensile curves as the failure does not always occur in the thinnest part of thin specimens and the nature of the material also determines the failure area, depending on the test temperature.

RESULTS: DED-LB GRADIENT MATERIAL

MICROSTRUCTURAL STUDY

Cross-sectional optical images of the 316L/Fe-9Cr-1Mo gradient sample after etching are presented in Figure 2 (c) and (d). A continuous and coherent interface between the two materials is seen. No large defects are observed in these samples with only a few pores. The 316L side is characterized by long elongated grains lying in build direction (BD) (Figure 2 (b)) while the Fe-9Cr-1Mo side is martensitic (Figure 2 (a) and (e)). Due to the parameters used the dilution rate from one layer to another is large at around 80% (measured by metallography). Even if, after the last 316L layer, a 100% Fe-9Cr-1Mo powder layer is deposited, the composition of that last deposited layer will be 80% of re-melted 316L + 20% of the new Fe-9Cr-1Mo, assuming complete mixing of the two alloys in the melted pool. For this reason, a smooth, graded area, not etched by any of the two etchants, is observed between 316L and Fe-9Cr-1Mo in Figure 2 (c) and (d). The length of the unetched zone is around 1.5 mm. The morphology of some of these centred cubic regions suggests that they may be ferritic rather than martensitic. Distinguishing these two phases by EBSD is not easy and the BCC domains are relatively small, so it is not possible to separate them here. However, the intermediate compositions between 316L and Fe-9Cr-1Mo pass through the three-phase F+M+A zone of the Schaeffler diagram, so it is possible to observe ferrite for these unusual compositions. Indeed, it is possible to maintain delta ferrite from the high temperature domain in additive manufacturing, due to the very high cooling rate undergone [START_REF] Villaret | Modelling of delta ferrite to austenite phase transformation kinetics in martensitic steels: application to rapid cooling in additive manufacturing[END_REF]. EBSD maps of the graded transition before and after heat treatment are presented in Figure 5. Away from the graded area the microstructure (SEM scale) of base metals is not modified by the heat treatment (Figure 5 (a) and (c)). Cr and Ni composition profiles are not modified by this tempering heat treatment as both temperature and duration are insufficient to allow homogenization of these elements [START_REF] Ahsan | Heat-treatment effects on a bimetallic additively-manufactured structure (BAMS) of the low-carbon steel and austenitic-stainless steel[END_REF]. In the transition area the formation of an austenitic phase on the martensitic side and a ferritic phase at the grain boundaries on the austenitic side are induced by heat treatment (Figure 5 (b) and (d)). The austenitic/martensitic mixed area seems to be wider after heat treatment and the austenite formed on the martensitic side is finer and more homogeneously distributed after heat treatment. These BCC domains are also finer after heat treatment, this suggest that this area could be austenitized partially during the heat treatment because its specific composition lowers Ae1 and Ae3 in comparison with Fe-9Cr-1Mo.

MECHANICAL PROPERTIES

Hardness

Changes in microstructure and alloy chemistry can lead to changes in mechanical properties. Vickers microhardness tests were performed to evaluate these changes across the junction and assess the results of heat treatment.

Figure 6 (b) and (c) show the microhardness maps obtained for DED-LB samples. Hardness on the 316L side is around 220 HV, the usual hardness found in as-built 316L DED-LB samples, for example by Ma et al. [START_REF] Ma | A comparison on metallurgical behaviors of 316L stainless steel by selective laser melting and laser cladding deposition[END_REF]. On the Fe-9Cr-1Mo side, the hardness is around 300 HV, a characteristic value of a slightly tempered martensite. In the graded area hardness rises progressively in the austenitic part. This smooth increase can be explained by the increase of carbon content coming from Fe-9Cr-1Mo steel. Hardness then rises sharply to 430 HV when the microstructure becomes martensitic. Kim et al. [START_REF] Kim | Microstructure and mechanical characteristics of multilayered materials composed of 316L stainless steel and ferritic steel produced by direct energy deposition[END_REF] observed such hardness changes in direct energy deposited graded materials. These high hardness values could be due to chemical enrichment in alloying elements (such as Ni and Cr) of the martensite from the 316L in this area in compared to the Fe-9Cr-1Mo steel. The fine and dispersed austenite/BCC phase mix observed previously (Figure 5) also cause this hardness increase. After this peak, hardness progressively decreases until the Fe-9Cr-1Mo value is reached.

Applying a 630°C/8 h tempering heat treatment to the graded parts reduces the hardness of the material globally (Figure 6 (a)). On 316L and Fe-9Cr-1Mo sides, hardness is reduced to 200 and 210 HV, respectively. Maximum hardness in the graded area is reduced to around 300 HV (Figure 6 (c)).

Tensile tests

Tensile tests were performed after heat treatment on both on EB welds and DED-LB samples to evaluate mechanical properties of such junctions at a larger scale. Tensile test in EB base metal were also performed for comparison (Table 3). Flat specimens with their graded area as well as with their weld located at the centre of the gauge section were used for tensile tests. Tensile properties were measured at three temperatures: 20 °C, 400 °C and 550 °C. Stress was applied across the weld and the gradient direction (parallel to the build direction for additive manufacturing) with a strain rate of 10 -4 s -1 . Average gauge section determined by microtomography was used to plot the tensile stress-strain curves for DED-LB in Figure 7. Each tensile test was performed one time.

For all tests failure never occurred in the weld area nor in the graded area and the global tensile behaviors looked similar. At room temperature and at 400°C failure occurred in 316L base metal both in EB welds and DED-LB samples. For these temperatures, DED-LB samples showed yield strength and ultimate tensile strength slightly higher than EB welds, while the elongation was almost equal. At 550°C DED-LB samples and EB welds had similar properties with a failure in the Fe-9Cr-1Mo base metal. Such change in failure location with temperature is quite common in austenitic/martensitic welded assemblies, it has been observed for example by Yano et al. [START_REF] Yano | Weldability of dissimilar joint between PNC-FMS and Type 316 steel under electron beam welding[END_REF]. At room temperature and 400 °C the yield strength and ultimate tensile strength of 316L are usually lower than those of Fe-9Cr-1Mo so strain is mainly on the 316L side (Table At 550°C Fe-9Cr-1Mo strengths were lower and closer to those of 316L; strain was homogeneous and failure occurred in Fe-9Cr-1Mo as it is less ductile.

YS (MPa

MICROSTRUCTURE

Microstructural characterization of EB welds was performed for comparison with DED-LB transitions. The starting materials have the same compositions but different forms. For DED-LB, steel powder was used, for EB welding laminated sheets were used. In EB welds, only the weld area is melted at the junction which means 316L and Fe-9Cr-1Mo sides away from the junction present typical equiaxed microstructure of laminated and heat treated sheets.

Figure 8 (a) presents an overview of the EB weld. This microstructure can be divided into several areas.

-On each side are the two base metals (BM), the austenitic side with some delta ferrite corresponding to 316L stainless steel and the tempered martensitic microstructure of the Fe-9Cr-1Mo side. -The welded metal in the middle is the area where the molten metal is mixed during welding.

-Two heat-affected zones (HAZ) surround the welded metal where the microstructure is changed by thermal input during welding but the metal is not melted.

Large EBSD maps and EDS scans presented in Figure 8 (b)-(e) demonstrate that weld metal is mainly martensitic with few austenitic areas. Weld metal composition obtained by EDS is 13% Cr and 6% Ni corresponding to a 50/50 mix (approx.) between 316L and Fe-9Cr-1Mo.

At the interface between weld metal and 316L HAZ (Figure 8 (b)) martensitic and austenitic strips can be observed (indicated by an arrow). In the austenitic strips chemical composition is equivalent to 316L, while martensitic strips have a composition closer to the weld metal containing less Ni and Cr than 316L (Figure 8 (c)). This unmixing at the weld metal/HAZ boundary is not usually observed in dissimilar welds, for example Abburi Venkata et al. [START_REF] Venkata | Applying electron backscattering diffraction to macroscopic residual stress characterisation in a dissimilar weld[END_REF] did not highlight any such phenomenon in their observations. The high speed used in EB welding can be responsible for instabilities of the melt pool during welding leading to unmixed areas after cooling. The weld metal/Fe-9Cr-1Mo HAZ boundary (Figure 8 (d)) does not present similar austenitic and martensitic strips; segregations observed on the EDS profile (Figure 8 (e)) are not important enough to promote austenite formation. The EBSD maps in Figure 10 show the mixed area between austenite and martensite in the weld metal. EBSD observations in this area should interpreted with caution because of the large heterogeneity that can occur due to the welding process. It is interesting to note that, in comparison, the DED-LB graded area seems to be fully martensitic apart from a small two-phase area. For this weld metal composition, the Schaeffler diagram predicts a tri-phased microstructure with austenite, ferrite and martensite. Such microstructure has already been observed by Abburi Venkata et al. [START_REF] Venkata | Applying electron backscattering diffraction to macroscopic residual stress characterisation in a dissimilar weld[END_REF]. In this area, heat treatment does not affect the very fine lath morphology of martensite. In the weld metal, austenite content seems to increase after heat treatment (Figure 10 (c)), this was also observed in the DED-LB graded area. Such an increase in austenite content due to heat treatment has been reported previously by Liu et al. [START_REF] Liu | Microstructural evolution of dissimilar welded joints between reduced-activation ferritic-martensitic steel and 316L stainless steel during the post weld heat treatment[END_REF]. . In DED-LB, delta ferrite was not observed on the 316L side unlike in EB welds. Depending on build parameters, for example with a lower energy density, as experienced by Guo et al. [START_REF] Guo | Study on microstructure, mechanical properties and machinability of efficiently additive manufactured AISI 316L stainless steel by high-power direct laser deposition[END_REF], it is possible to observe delta ferrite from such a process. The DED-LB mixed area is mainly martensitic with a thin austenitic/martensitic strip. Weld metal is also mainly martensitic with few austenitic areas. At the interface between austenite and martensite some unmixed areas remain both in DED-LB transitions and EB welds.

On the Fe-9Cr-1Mo side DED-LB and EB welds are fully martensitic. A smoother chemical gradient is observed in DED-LB samples; as demonstrated by Zuback et al. [START_REF] Zuback | Additive manufacturing of functionally graded transition joints between ferritic and austenitic alloys[END_REF], this can slow down carbon diffusion across the sample and improve ageing resistance. It is also possible to control this gradient in DED-LB by controlling the build parameters.

Applying heat treatment to DED-LB gradients leads to austenite formation in the graded area as it does for EB weld metal (Figure 12).

MECHANICAL PROPERTIES

The hardness changes observed in DED-LB gradient materials are also observed in EB welds (Figure 13). EB weld's base material hardnesses are around 150 HV for 316L and 230 HV for Fe-9Cr-1Mo (Figure 13 (a)). After welding, weld metal hardness is around 350 HV. Soft areas in weld metal correspond to austenitic areas. The higher hardness values (around 400 HV) are reached in the Fe-9Cr-1Mo HAZ due to formation of fresh martensite in the HAZ during welding. Such hardness variations are the usual observations for austenitic/martensitic dissimilar welds, it is for example observed by Albert et al. [START_REF] Albert | Mechanical properties of similar and dissimilar weldments of RAFMS and AISI 316L (N) SS prepared by electron beam welding process[END_REF]. After heat treatment (Figure 13 (b)) hardness in the Fe-9Cr-1Mo HAZ and in the weld metal is reduced due to martensite tempering. As in the DED-LB graded area, weld metal hardness remains above 300 HV after heat treatment but with some soft austenitic areas.

CONCLUSION

This paper characterizes a DED-LB junction between 316L and Fe-9Cr-1Mo steels and compares results obtained with those from EB welds.

Comparison of EDS profiles obtained across DED-LB junctions and EB welds reveal composition changes with DED-LB can be more gradual than in EB welds. Furthermore, it is possible to control the width of the graded area in DED-LB.

Microstructural study before heat treatment reveals that the DED-LB 316L side is fully austenitic whereas the weld's 316L base metal contains few percent of delta ferrite. Some unmixed areas resulting in austenitic and martensitic strips are observed in EB welds at the interface between 316L and the weld metal. Such unmixed strips are also observed in DED-LB junctions.

Weld metal and mixed areas are mainly martensitic. Weld metal in EB welds contains few austenitic areas in contrast to the thin area at the austenitic/martensitic transition of DED-LB graded samples. In DED-LB graded area, the microstructure changes suddenly from austenite to martensite. This result highlights the composition at which Ms becomes lower than room temperature because of the composition change.

Besides, even if the change of microstructure is not very gradual, such chemically graded part could be interesting because its gradual change in chemistry is able to slow down element's diffusion during thermal ageing [START_REF] Zuback | Additive manufacturing of functionally graded transition joints between ferritic and austenitic alloys[END_REF].

DED-LB Fe-9Cr-1Mo is fully martensitic with a uniform microstructure both close to and far from the gradient area. In EB welds Fe-9Cr-1Mo, microstructure is divided into three regions, depending on the distance from the weld: the CGHAZ close to the weld, the FGHAZ and the base metal which is not affected by the welding process.

Microhardness maps reveal 316L and Fe-9Cr-1Mo values far from the junction to be harder in DED-LB materials than in heat treated and laminated plates used for EB welds. Hardness of EB weld metal attains values of 350 HV, with the highest hardness values in the Fe-9Cr-1Mo HAZ. Weld metal contains some softer areas which are austenitic. By contrast, hardness in the DED-LB graded area can reach 430 HV.

Hardness in EB weld metal and Fe-9Cr-1Mo HAZ is reduced after heat treatment, as it is for the DED-LB mixed area and for the Fe-9Cr-1Mo side. SEM microstructure analysis shows a decrease in the amount of delta ferrite formed on the 316L side in EB welds and formation of austenite islands in the martensitic microstructure in the EB weld metal and DED-LB graded area following heat treatment.

Tensile tests show it is possible to obtain mechanical properties from DED-LB comparable to those from EB welding. Samples from both typically exhibit failure in 316L base metal at 20°C and 400°C and failure in Fe-9Cr-1Mo base metal at 550°C.

This study highlights the great interest in DED-LB as a versatile process to induce composition-graded steel with tailored microstructure and local properties in the graded area.

Figure 1 :

 1 Figure 1: Operating scheme of the DED-LB process applied to the manufacture of gradient materials.

Figure 2 :

 2 Figure 2: Optical micrographs after two-step etching of the as-built DED-LB sample (BD -build direction, SD -scanning direction). In the BD plane, perpendicular to the SD -details of (a) martensitic and (b) austenitic microstructures, (c) overview of the graded sample. In both the BD and SD planes -(d) overview of the graded sample and details of (e) the martensitic and (f) the austenitic microstructures.

Figure 3 and

 3 Figure3and Figure4shows detail of the microstructure in the graded area. In Figure3(a) the 316L side is fully austenitic with large grains crossing several melt pools and elongated in the thermal gradient direction; the Fe-9Cr-1Mo side is fully martensitic. The EDS profile (Figure3(b)) shows that the chemical gradient is actually larger than the unetched area in figure1. According to the forescattered electron (FSE) image and EDS profile (Figure3(a) and (b)), the first 300 µm of the composition change is mainly austenitic whereafter the microstructure becomes mainly martensitic. The EBSD map (Figure4(a) and (b)) reveals a band of almost 100 µm where the composition induces a mix between austenite and BCC phase. The morphology of some of these centred cubic regions suggests that they may be ferritic rather than martensitic. Distinguishing these two phases by EBSD is not easy and the BCC domains are relatively small, so it is not possible to separate them here. However, the intermediate compositions between 316L and Fe-9Cr-1Mo pass through the three-phase F+M+A zone of the Schaeffler diagram, so it is possible to observe ferrite for these unusual compositions. Indeed, it is possible to maintain delta ferrite from the high temperature domain in additive manufacturing, due to the very high cooling rate undergone[START_REF] Villaret | Modelling of delta ferrite to austenite phase transformation kinetics in martensitic steels: application to rapid cooling in additive manufacturing[END_REF].

Figure 3 :

 3 Figure 3: Analysis of the graded area: (a) forescattered electron (FSE) image overview of the microstructure and (b) corresponding EDS profile, (c) FSE image of the Fe-9Cr martensitic microstructure, far from the graded area.

Figure 4 :

 4 Figure 4: (a) EBSD IPF map and (b) EBSD phase map at the interface between 316L and graded area (1 pixel = 0.2 µm). IPF map is projected parallel to the build direction (BD), 10° grain boundaries are shown in black. (color figure)

Figure 5 :

 5 Figure 5: Effect of the heat treatment on microstructure: (a) IPF map and (b) phase map as built and (c) IPF map and (d) phase map after 630°C/8h heat treatment. (color figure)

Figure 6 :

 6 Figure 6: Microhardness measurements for DED-LB sample across the graded area: (a) mean hardness as a function of distance and corresponding maps for the sample (a) as built and (b) after 630°C/8h heat treatment. (color figure)

Figure 7 :

 7 Figure 7: Tensile stress-strain curves for EB welds (dotted) and DED-LB transitions (solid) after 630°C/8h heat treatment at three test temperatures: 20°C (blue), 400°C (yellow) and 550°C (red).

Figure 8 :

 8 Figure 8: For an EB weld (a) large SEM BSE image, (b) EBSD phase map on 316L side showing martensitic strips (bold arrow), (c) EDS profile along the white dashed line in (b), (d) EBSD phase map on Fe-9Cr-1Mo side and (e) the EDS profile along dashed line. For EBSD maps, 1 pixel = 1.13 µm, 10° grain boundaries in black. (color figure)

Figure 9 :

 9 Figure 9: Detail of the 316L HAZ, EBSD phase map (a) as welded and (b) after 630°C/8h heat treatment (1 pixel = 0.1 µm, 10° grain boundaries in black). (color figure)

Figure 10 :

 10 Figure 10: Weld metal in EB welds: (a) phase and (b) IPF EBSD maps as welded and (c) phase and (d) IPF maps after a 630°C/8h heat treatment, emphasizing the formation of austenite in weld metal. (1 pixel = 0.1 µm, 10° grain boundaries in black). (color figure)

Figure 11 :

 11 Figure 11: EBSD IPF maps of the Fe-9Cr-1Mo HAZ (1 pixel = 0.1 µm): (a) large view of the as welded (AW) Fe-9Cr-1Mo HAZ showing the CGHAZ and the FGHAZ. Detail of the FGHAZ (b) as welded and (c) after 630°C/8h heat treatment. (color figure) Figure 12 summarizes in schematic form the differences and similarities induced on microstructures by heat treatment on EB welds (Figure 12 (a) and (b)) and DED-LB transitions (Figure 12 (c) and (d)). In DED-LB, delta ferrite was not observed on the 316L side unlike in EB welds. Depending on build parameters, for example with a lower energy density, as experienced by Guo et al.[START_REF] Guo | Study on microstructure, mechanical properties and machinability of efficiently additive manufactured AISI 316L stainless steel by high-power direct laser deposition[END_REF], it is possible to observe delta ferrite from such a process. The DED-LB mixed area is mainly martensitic with a thin austenitic/martensitic strip. Weld metal is also mainly martensitic with few austenitic areas. At the interface between austenite and martensite some unmixed areas remain both in DED-LB transitions and EB welds.On the Fe-9Cr-1Mo side DED-LB and EB welds are fully martensitic. A smoother chemical gradient is observed in DED-LB samples; as demonstrated by Zuback et al.[START_REF] Zuback | Additive manufacturing of functionally graded transition joints between ferritic and austenitic alloys[END_REF], this can slow down carbon diffusion across the sample and improve ageing resistance. It is also possible to control this gradient in DED-LB by controlling the build parameters.

Figure 12 :

 12 Figure 12: Schematic comparison of microstructure evolution for EB welds (a) before and (b) after 630°C/8 h heat treatment and for the DED-LB transition zone (c) before and (d) after heat treatment. (color figure)

Figure 13 :

 13 Figure 13: Microhardness maps for EB welds (a) before heat treatment and (b) after 630°C/8h heat treatment. 316L is on left side of the maps shown. (color figure)

Table 1 : Chemical Composition, suppliers and size distribution of 316L and Fe-9Cr-1Mo powders.

 1 . Samples were gas atomised powders with spherical particles.

	Composition (wt%)	supplier	C	Cr	Ni	Mo	Mn	Si	N	D10 (µm) D50 (µm) D90(µm)
	316L	Erasteel 0.016 17.7	12.6	2.33	0.29	0.58	0.08	57	78	108
	Fe-9Cr-1Mo	Nanoval 0.10	9.3	0.22	1.04	0.48	0.28	0.004	10	30	65

Carrying/shielding gas Argon Spot diameter (Dspot) 1.4 mm 316L powder feed rate 4 g/min Fe-9Cr-1Mo powder feed rate 2 g/min Energy density (P/v.Dspot)

	Setup	Optomec LENS 850 R
	Laser power (P)	400 W
	Scanning speed (v)	5 mm/s
	Layer height	0.2 mm
		57 J/mm²

Table 2 : DED-LB building parameters

 2

Table 3 : DED-LB gradient and EB weld mechanical properties compared to base materials 4. DISCUSSION: COMPARISON WITH EB WELD

 3

			)			UTS (MPa)		Total elongation %
		20°C	400°C	550°C	20°C 400°C	550°C	20°C	400°C	550°C
	316L	303	176	153	614	447	412	91.8	52,7	58.7
	Fe-9Cr-1Mo	562	453	397	698	558	419	25.9	18,5	24.8
	DED-LB gradient	420	346	287	610	478	357	23	18	23
	EB weld	330	204	189	647	459	402	43.6	19.1	24.7

ACKNOWLEDGEMENTS

Authors would like to thank Patrice Peyre and Thierry Malot (PIMM Laboratory, Arts et Métiers ParisTech) for their expertise and their help in the production of the DED-LB samples used for this study, Danielle Ayrault and Arnaud Courcelle (CEA-Saclay) for their precious advice in carrying out this study and Owen Johnson for his very valuable advice.

DATA AVAILABILITY

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.


Norbert Mercier

Hélène Valladas

Thierry Aubry

João Zilhão

Jean-Louis Joron

Jean-Louis Reyss

Farid Sellami

Fariseu: first confirmed open-air Palaeolithic parietal art site in the Côa Valley (Portugal)

, since the animals represented and pictorial compositions showed great affinities with upper Palaeolithic cave art of Spain and France. Objections were raised to attribut ing so many well-preserved petroglyphs in an open-air site to the Palaeolithic. Miscellaneous attempts to get direct ages for the engravings themselves failed to establish the true antiquity of this site because the dating methods employed were ill-suited to the task. We wish to report below some thermoluminescence (TL) dates for burnt lithic remains from the archaeological levels of the Fariseu site which should close the debate, since some of the levels dated rise above the lower portions of numerous designs. In this letter we show that most if not all of the compositions are undoubtedly more than 14 500 years old.

In 1991 over 17km of the cliff faces bordering the river Côa (a tributary of the Douro, in north-eastern Portugal, Figure 1) were found to be covered with carvings of horses, aurochs, deer, and other wild animals. Cave art was believed by many specialists to be associated with magic practices, because it was generally located deep inside caves, far from human occupation sites. Though the animal forms and composition resembled those of French and Spanish cave art [START_REF] Bahn | Cave art without the caves[END_REF][START_REF] Clottes | Are the Foz Côa engravings actually Holocene?[END_REF][START_REF] Zilhão | The Age of the Côa Valley (Portugal) Rock-Art: Validation of Archaeological Dating to the Palaeolithic and Refutation of 'Scientific' Dating to Historic or Proto-Historic Times[END_REF], not all prehistorians were willing to accept that this astounding open-air site was contemporary with similar cave art, and dates as recent as a few centuries have been proposed [START_REF] Bednarik | The Côa petroglyths: an obituary to the stylistic dating of Palaeolithic rock-art[END_REF][START_REF] Watchman | Recent petroglyths, Foz Côa, Portugal[END_REF]. While there is no reliable method of determining absolute dates of petroglyphs, a number of attempts were made to narrow down the date of execution. Radiocarbon dates obtained on thin mineral accretions formed on petroglyphs gave ages ranging from a few centuries to several millennia [START_REF] Dorn | Constraining the age of the Côa Valley (Portugal) engravings with radiocarbon dating[END_REF]. Of course, these results give no information on the age of the engravings themselves, since there is no way to tell where the carbon dated came from. Rock-erosion or varnish-deposit techniques [START_REF] Bednarik | The Côa petroglyths: an obituary to the stylistic dating of Palaeolithic rock-art[END_REF] were no more helpful than 14 C, since they are strongly dependent on environmental factors [START_REF] Zilhão | The Age of the Côa Valley (Portugal) Rock-Art: Validation of Archaeological Dating to the Palaeolithic and Refutation of 'Scientific' Dating to Historic or Proto-Historic Times[END_REF]. Fortunately, archaeological excavations within the valley [START_REF] Zilhão | The rock art of the Côa valley (Portugal) and its archaeological context: first results of current research[END_REF][START_REF] Aubry | Open-air rock-art, territories and modes of exploitation during the Upper Palaeolithic in the Côa Valley (Portugal)[END_REF] quickly revealed evidence of human occupation at several sites with levels dated from Gravettian (Olga-Grande 4 and Cardina) to Magdalenian (Quinta da Barca). The initial cultural attributions were confirmed by TL ages of burnt lithics which ranged from c. 28 000 to 12 000 years [START_REF] Valladas | TL Dating of Upper Palaeolithic sites in the Côa Valley (Portugal)[END_REF]. Of course, those favouring a more recent date pointed out that the upper Palaeolithic humans who inhabited the valley were not necessarily the authors of these petroglyphs. The work reported below establishes a terminus ante quem for the lower levels of these magnificent works of art.

When the Douro river dam, located downstream from Côa, was emptied in December 1999 the lowered water level at Fariseu exposed the lower section of an engraved rock panel (No. 1) partly hidden by a thick layer of sediment (Figure 2a). Excavation carried out soon after to a depth of 2.5m exposed a succession of deposits rich in schist fragments and containing remains of human activity (lithic artefacts) separated by sterile layers of alluvium [START_REF] Aubry | Open-air rock-art, territories and modes of exploitation during the Upper Palaeolithic in the Côa Valley (Portugal)[END_REF]. Of key importance was the fact that these archaeological deposits were in direct contact with and concealed some of the 82 animal figures (horses, deer, aurochs and goats) engraved on a 3m 2 rock panel. Moreover, a careful examination of the superposition of the design lines suggested that all the figures of this particular panel were traced within a short span of time [START_REF] Baptista | The Quaternary Rock Art of the Côa Valley[END_REF].

The uppermost excavated layer one consisted of modern sediments deposited since the Douro dam was built and layer two contained fragments of ceramics (medieval, modern?). Micromorphological analysis of the lower layers (three to eight), antedating the dam and which are in contact with the decorated rock-face, indicated that the sediments were only affected by pedological processes which are not able to move significantly the soil and disturb the organization of artefact assemblages [START_REF] Sellami | Dynamique du sol et fossilisation des ensembles archéologiques sur les sites de plein air. Données expérimentales sur l'organisation des micro-artefacts et les traits pédo-sédimentaires[END_REF]). In the upper part of archaeological level four, the flint tools showed all the characteristics typical of Late Magdalenian industries, whereas those of the lower part could be attributed to Early Magdalenian or Late Gravettian periods. It is important to point out that these archaeological deposits rose well above the bottom of the engraved panel to the extent that the top of layer seven was at the level of the lowest petroglyphs, which were concealed by layer six (Figure 2b). Moreover, two thirds of the upper part of this engraved panel were hidden by archaeological levels three and four. 1. The site formation process was greatly influenced by the fact that the site is located along a glacis slopping towards the river bed. Layers C3 to C8 are characterized by a succession of pebbly layers, particularly at the lowest strata, and layers rich in silty alluvion in the upper strata: this succession most likely reflects climatic evolution. The lower layers contain numerous schist fragments produced by the erosion of schist formations found higher up the slope. The angular nature of these fragments suggests that they did not travel far before being deposited [START_REF] Bullock | Hand-book for soil thin description[END_REF][START_REF] Fedoroff | Organisation du sol aux échelles microscopiques[END_REF]. The fact that these rocks fragments show no significant physicochemical changes suggests that layers seven and eight were deposited primarily during a cold period. The dotted and coloured lines show boundaries between the different layers, which in some case are in direct contact with engravings: note that layers six to three concealed some of the 82 animal figures of this panel, whereas the top of layer seven is located at the base of the lowest petroglyphs. b) Position of some engraved animals of panel No. 1. relative to layer boundaries. The superposition of the figures and the intersection of the drawing lines allow to establish that some of the engravings had already been done when the lowest part of this decorated panel was covered by layer 6, dated to 15 200±1600 years. Click to enlarge.

Fortunately, the levels containing flint tools also yielded a number of heated quartzite fragments for which age-estimates could be obtained by the thermoluminescence dating method (TL). Among the burnt lithics collected in the course of the excavation, five were suitable for TL dating. Three came from archaeological level four and two from level seven. To get chronological information about some other levels, sediments collected from layers six and eight were dated by optical luminescence (OSL) since they have been exposed to sunlight before being covered by overlaying layers. Dosimetric measurements were carried out on site during the excavation to determine the gamma and cosmic dose-rates the samples experienced during burial. The radioisotopic contents of quartzites and sediment were determined by Neutron Activation Analysis and gamma-ray spectrometry, respectively. The sample preparation and the experimental procedures which were used to get all the relevant radiometric data for the age calculations have been published earlier [START_REF] Mercier | Application de la méthode de la thermoluminescence à la datation des occupations paléolithiques de la Vallée du Côa[END_REF].

The TL and OSL age-estimates, which fall within the Tardiglacial period, range from c. 18 500 for layer eight at the bottom, and c. 14 500 for layers seven and six, to c. 11 000 for layer four, near the top of the archaeological sequence (Table 1). For layers four and seven, the TL age-estimates yield a weighted mean of 11 400±700 and 13 900±800 years, respectively. Moreover, the two methods, though applied to materials with distinctly different luminescence and radiometric characteristics, such as quartzite pebbles for TL and silt-sized quartz grains for OSL, give coherent results that dovetail the stratigraphy. Thus, this set of dates confirms the conclusions derived from the study of the lithic industries which attributed the Fariseu human occupation levels to the Upper Palaeolithic. According to the luminescence results, layers seven and six were deposited in a relatively short time around 14 500 years ago and they antedate by at least 1000 years the deposition of layer four, which falls within the same time interval as the Quinta da Barca Late Magdalenian settlement site (12 100±600 years) at the end of the Pleistocene period. Finally, as the archaeological level six concealed the lowest petroglyphs, the luminescence dates allow us to conclude that some of the designs exposed during excavation must be more than 14 500 years old.

Table 1. Luminescence ages were obtained by the thermoluminescence (TL) method on the burnt quartzite pebbles Far-17, 2, 8 and 9, recovered in the archaeological levels and by the optically stimulated luminescence (OSL) method on Far-24 and silt-sized quartz grains extracted from sediments (C6 and C8). The radioisotopic contents of the dated specimens (columns 3 to 5) were measured by Neutron Activation Analysis at the Pierre Süe Laboratory (Saclay). The α-sensitivity was determined by comparison of the TL-α and TL-β signals induced by α and β particles coming from a Pu-238 source and a Sr-90 source, respectively. Gamma and cosmic dose-rates were deduced from measurements taken in the field with a portable gamma spectrometer. Measurements performed on sediment samples in the laboratory with a high purity Ge detector showed no significant disequilibrium in the uranium series. Paleodoses of the burnt lithics (except Far-24) were computed by measuring the TL blue-UV signal emitted between 340 and 400 °C. For some of them (Far-17, 8, 9 and 24), paleodoses were also measured by optical stimulation (OSL) using a single-aliquot regenerative-dose protocol [START_REF] Murray | Luminescence dating of quartz using an improved single-aliquot regenertative dose protocol[END_REF] and the results were concordant. This protocol was used to determine the paleodoses of silt-sized quartz grains extracted from sediments C6 and C8. The quoted uncertainties on the TL and OSL ages, at one sigma level, include statistic and systematic errors. Click to enlarge.

Thus our luminescence dating results should definitively close the controversy raised ten years ago about the antiquity of the Côa petroglyphs by demonstrating that they were carved during the Pleistocene period and not during the last millennium.

The stylistic and morphological similarities between these particular petroglyphs and the thousands that adorn a long stretch of the cliffs enclosing the Côa river valley [START_REF] Baptista | The Quaternary Rock Art of the Côa Valley[END_REF] leave little doubt that this amazing creative impulse can definitely be attributed to Upper Palaeolithic humans. It is hard to believe, in view of its sheer magnitude, that this open-air site is unique, so other similar sites executed by the hunter-gatherer groups of the last ice age probably remain to be discovered. The age estimates reported here suggest that, whatever their motivations, Palaeolithic parietal artists were allowed to express their talents elsewhere than in secluded sites hidden from the profane eyes.

Figure 1 .

 1 Figure 1. Location of rock-art and Upper Palaeolithic sites in the Douro and Côa valleys. Click to enlarge.

Figure 2 .

 2 Figure 2. a) Picture of the Fariseu Palaeolithic sequence in relation to the engraved rock panel No. 1. The site formation process was greatly influenced by the fact

Acknowledgements

We thank the Parque Arqueológico do Vale do Côa, the Portuguese Ministry of Culture, CNRS and CEA for financial support.


Eurequa Hippolyte D'albis

R Amir

J M Bonnisseau

B Cornet

S Demichelis

Existence of Solutions in Continuous-time Optimal Growth Models *

Keywords: Existence of solutions, optimization, continuous time, optimal growth. JEL Classification C61, D91, E13

This paper studies the existence of solutions in continuous time optimization problems. It provides a theorem whose conditions can be easily checked in most models of the optimal growth theory including those with increasing return and multi-sector economies.

Introduction

The modern theory of economic growth is based on a behaviorist tradition that analyses both optimal and equilibrium allocations of scarce resources over time. To discuss this issue, infinite dimensional optimization techniques in continuous time have been extensively used over the last forty years. In most cases the problem may be re-written as the following variational problem:

max +∞ 0 G (x t , ẋt ) e -rt dt s.t. ∀t, ẋt ∈ Γ t (x t ) x 0 ∈ R l + given
This problem has a solution provided that +∞ 0

G (x t , ẋt ) e -rt dt exhibits some upper semi-continuity and Γ some compactness. However, in many papers, those properties are not checked and the existence of the discussed solution is simply assumed. The reason is technical and specific to continuous time: to be a feasible path, x t should belong to a ball of L 1 which is not compact for the L 1 topology. General results have nevertheless been established. Notably, [START_REF] Magill | Infinite horizon programs[END_REF] provides an existence theorem for problems with a G concave in its arguments. However, his theorem does not apply to the large body of the literature that consider increasing returns. It is true that if the problem is not concave, the difficulty increases since traditional optimality conditions for the Social Planner problem are necessary but not anymore sufficient. [START_REF] Chichilnisky | Existence and characterization of optimal growth paths including models with non-convexities in utilities and technologies[END_REF] and Romer (1986a) hence make an important step by providing an existence theorem for optimal solution that apply to models with increasing returns. While proved in a different way, the conditions they proposed are based on the concavity of G with respect to the one of its arguments which have the highest derivative. Their demonstrations are very nice but the precise set of conditions they propose is rather technical and it is not simple to know if a given economic problem satisfies it. We hence propose an original existence theorem whose conditions are easy to verify in a given economic model. We show this by confronting some well-known models to our existence conditions.

The demonstration we propose extends an initial result of Askenazy and Le [START_REF] Askenazy | A model of optimal growth strategy[END_REF] about the regularity of the objective function. Then it uses the Dunford-Pettis Criterion to get some sequential compactness for some weak topology and a multi-dimensional version of the Fatou Lemma to deduce properties for the limit with respect to this topology.

The paper is organized as follows. In section 2, the model is developed while our theorem is presented in section 3. In section 4, we show how our theorem can be easily applied to models of neoclassical growth or endogenous growth.

The Model

x t ∈ R C + , y t ∈ R K + , x 0 ∈ R C + , r > 0 are given.
In this setting, x t is the control variable, y t is the state variable, and

y t ∈ R K +
is the externality. The sets C and K are finite (for simplicity, (y 1 , ..., y K ) denote elements of R K and (x 1 , ..., x C ) elements of R C .

Assume:

A1. The functions F and G are continuous. A2. For any j ∈ K, the function G j is concave with respect to x. A3. There exist

(b i ≥ 0, A i > 0, i = 1, ..C), (a i ≥ 0, A i > 0, i = 1, ..K) such that: ∀t, F (y t ) ≤ . y t ≤ G(y t , y t , x t ), ∀i ∈ K, y i t ≤ A i e a i t =⇒ ∀t, ∀i ∈ K, ∀j ∈ C, y i t ≤ A i e a i t , . y i t ≤ A i e a i t , x j t ≤ A i e b j t . A4. r > sup {b i , i ∈ C} . A5. The function u is concave, non decreasing, continuous on R C + . Lemma 1 The function x ∈ L 1 + (e -rt dt) → +∞ 0
u(x t )e -rt dt is finite valued and upper semi-continuous for the topology σ(L 1 (e -rt dt), L ∞ ). [START_REF] Askenazy | A model of optimal growth strategy[END_REF]; note that the assumption u(0) = 0 made in their article is not necessary for the proof of this lemma. Moreover, their proof is still true when the function u satisfies the following assumption A5bis.

Proof. See Askenazy and Le

Remark 1 Assumption A5 could be replaced by: A5bis. The function u is concave and upper semi-continuous from R C + into R∪ {-∞} and finite valued on R C ++ .

Assumption A5bis allows for utility functions unbounded from below, such as u(c) = log(c) or u(c) = c σ /σ with σ < 0 that are extensively used in economic models.

Existence of Solutions

This section provides the proof of the existence of a solution for Problem (P).

Theorem 2 Under assumptions A1, ..., A5, Problem (P) has a solution.

Proof. First note that the variables (x t , y t ) will be said feasible if they satisfy the constraints of Problem (P). Observe that assumptions A3 and A4 imply that +∞ 0 u(x t )e -rt dt is uniformly bounded above on the set of feasible controls x t . Let the sequence (x n ) of elements in

L 1 + (e -rt dt) satisfy lim n→∞ +∞ 0 u (x n t ) e -rt dt = M = sup +∞ 0
u(x t )e -rt dt over the set of feasible controls. Assumption A3 implies that the sequence satisfies the Dunford-Pettis criterion1 and has a subsequence, denoted also by (x n t ) for simplicity, which converges to x * ∈ L 1 + for the topology σ(L 1 (e -rt dt), L ∞ ). From Lemma 1, +∞ 0 u (x * t ) e -rt dt ≥ M. To end the proof, it remains to show that x * is feasible.

Observe first that for a defined ρ > max {a i : i ∈ K} then the associated sequences (y n , . y n ) also satisfy the Dunford-Pettis criterion for L 1 (e -ρt dt). It can therefore be assumed that they converge respectively to y * , z * for the topology σ(L 1 (e -ρt dt), L ∞ ). Then for all t:

t 0 . y n (s)ds → ϕ(t) = t 0 z * (s)ds.
In other words, y n (t) → ϕ(t) + y 0 , for every almost t. Since ∀t, ∀i, 0 ≤ y n,i (t) ≤ A i e a i t ∈ L 1 (e -rt dt), use Lebesgue Theorem to obtain that y n converges to y * for the strong topology (and consequently for the weak topology) of L 1 (e -ρt dt). Thus y * = ϕ + y 0 and . y * = z * .

Using the multidimensional Fatou Lemma (see Appendix), given s, there exist (θ sj ) j=1,...,K+C+1 such that K+C+1 j=1 θ sj = 1, θ sj ≥ 0, ∀j, and

( . y * s , x * s ) = K+C+1 j=1 θ sj (ς 1 sj , ς 2 sj ),
where (ς ). Given (j, s), there will be some increasing function φ js : N → N such that (

.

y φ js (n) s , x φ js (n) s ) -→ n→∞(ς 1 sj , ς 2 sj ). Then, . y * i s = K+C+1 j=1 θ sj ς 1i sj = K+C+1 j=1 θ sj lim n→∞ . y i,φ is (n) s ≤ K+C+1 j=1 θ sj lim n→∞ G i (y s , y φ js (n) s , x φ js (n) s
).

Since G is continuous and y n converges pointwise, it yields:

.

y * i s ≤ K+C+1 j=1 θ sj G i (y s , lim n→∞ y φ js (n) s , lim n→∞ x φ js (n) s ) = K+C+1 j=1 θ sj G i (y s , y * s , ς 2 sj ).
From A2,

.

y * i s ≤ G i (y s , y * s , K+C+1 j=1 θ sj ς 2 sj ) = G i (y s , y * s , x * s ).

Recall that

.

y * i s = K+C+1 j=1 θ sj ς 1i sj = K+C+1 j=1 θ sj lim n→∞ . y i,φ is (n) s ≥ K+C+1 j=1 θ sj lim n→∞ F i (y φ js (n) s
).

Since F is continuous and y n converges pointwise, it yields:

.

y * i s ≥ K+C+1 j=1 θ sj F i ( lim n→∞ y φ js (n) s ) = K+C+1 j=1 θ sj F i (y s ) = F i (y s ).
Thus, (y * , x * ) have been proved to be feasible. Hence M = sup

+∞ 0 u(x t )e -rt dt = +∞ 0 u (x * t ) e -rt dt.
Remark 2 This proof can be easily adapted in order to study problems with finite horizon:

max xt,yt T 0 u(x t )e -rt dt, T < +∞ subject to: ∀t ≥ 0, F (y t ) ≤ . y t ≤ G(y t , y t , x t ), x t ∈ R C + , y t ∈ R K + , x 0 ∈ R C + , r > 0, T ∈ ]0, +∞[ are given.
In this setting, Assumption A4 is not necessary.

Remark 3

The Social Planner Problem is a particular case of Problem (P): just consider y t =1 K , ∀t ≥ 0.

4 Some Examples in the Literature

The Ramsey model with discounting

Consider the model developed by [START_REF] Cass | Optimum growth in an aggregative model of capital accumulation[END_REF] and [START_REF] Koopmans | On the concept of optimal economic growth[END_REF]:

max ct,kt +∞ 0 u(c t )e -rt dt, subject to ∀t ≥ 0, -δk t ≤ . kt ≤ f (k t ) -δk t -c t c t ≥ 0, k t ≥ 0, k 0 > 0, r > 0 are given.
Variables c t and k t denote respectively the consumption and the capital stock at time t. It is supposed that the instantaneous utility function u satisfies assumption A5, and that the production function f is concave, continuous, increasing, differentiable, and satisfies f (0) = 0, f (∞) = 0. Define x t = c t , y t = k t , y t = 1, G(y t , y t , x t ) = y t (f (y t ) -δy t -x t ) , F (y t ) = -δy t . Now, check that Assumptions A1,...,A4 are satisfied: A1 : It is obvious that F and G are continuous.

A2 : It is also obviously satisfied. A3: Since f (∞) = 0, for any ε > 0, it exists B such that f (y) ≤ B+εy, ∀y ≥ 0. It is easy to see that it exists A > 0, such that y t ≤ A e εt , ∀t ≥ 0. This implies that ∀t ≥ 0, c t ≤ Ae εt , with some A > 0. Since -δy t ≤ . y t , then . y t ≤ A = max δk 0 , A .

A4 : Choose ε < r.

The Ramsey model with leisure

Consider now max ct,lt,kt +∞ 0 u(c t , l t )e -rt dt, subject to ∀t ≥ 0, -δk t ≤ . kt ≤ f (k t , 1 -l t ) -δk t -c t , c t ≥ 0, k t ≥ 0, 0 ≤ l t ≤ 1, k 0 > 0, r > 0 are given.
The variable l t denotes the leisure at time t. It is supposed that the utility function u satisfies A5 and that the production function f is concave, continuously differentiable, and satisfies f (0, l) = f (k, 0) = 0, and f (∞, l) = 0. As in the previous example, one can find some ε in ]0, r[ such that for any t ≥ 0, c t ≤ Ae εt , k t ≤ A e εt , . kt ≤ A e εt . Put x t = (c t , l t ), y t = k t , y t = 1, G(y t , y t , x t ) = y t (f (y t ) -δy t -x t ) , F (y t ) = -δy t . It is then easy to check that Assumptions A1,..., A4 are satisfied.

The Romer Model (1986b)

For a given path k t ≥ 0, solve:

max ct,kt +∞ 0 u(c t )e -rt dt, subject to ∀t ≥ 0, c t + i t ≤ F (k t , k t ), . kt k t = g( i t k t ), c t ≥ 0, i t ≥ 0, k t ≥ 0, k 0 > 0, r > 0 are given.
Variables i t , k t , c t respectively denote the investment, the stock of knowledge and the consumption at time t. It is supposed that the function g is concave, increasing, and satisfies g(0) = 0, g(∞) = λ > 0. Assume, moreover, that F is concave, nonnegative, and satisfies F (y, k) ≤ y α k β for every (y, k) ≥ 0, and α > 0, β > 0. The function u, as previously, satisfies Assumption A5. Define G(y, y, x) = yg( F (y,y)-x y ) when y > 0, F (y, y) -x ≥ 0, and G(y, 0, 0) = 0.

When x t = c t , y t = k t , the problem is then equivalent to: kt ≤ G(k t , k t , c t ), k t ≥ 0, k 0 > 0, r > 0 are given. Now, check that Assumptions A1,...,A4 are fulfilled. A1 : It is easy to see that G is continuous. A2 : It is obvious that G is concave in x. A3 : Since 0 ≤ . y t ≤ λy t , it implies that ∀t, y t ≤ k 0 e λt , . y t ≤ λk 0 e λt and x t ≤ k 0 k 0 e λ(α+β)t . A4 : It is satisfied if r > λ(α + β).

The Lucas Model (1988)

For a given human capital path (h t ), solve: max ct,θt,kt,ht +∞ 0 u(c t )e -rt dt,

  t )e -rt dt, subject to ∀t ≥ 0, 0 ≤ .

See Dunford and Schwartz, 1967, corollary 11, p.

subject to:

Variables c t , θ t , k t and h t denote respectively the consumption, the working time, the physical capital and the human capital at time t. Moreover, it is supposed that function φ is increasing, concave and satisfies φ(0) = 0, φ(1) = µ > 0 while function u is concave and increasing. Define

Obviously, Assumption A5 is satisfied. Now, check A1,...,A4. A1 : It is obviously true. A2 : It is immediate since 0 < α < 1 A3 : It follows from the dynamic constraint on human capital that . ht ≤ µh 0 e µt , h t ≤ h 0 e µt . From the constraints on physical capital, one obtains that

(1-α) . A4 : It is satisfied under the additional condition r > µ(1-α+γ)

(1-α) .

Appendix

The original Fatou's lemma has been generalized to this proposition (See [START_REF] Hildenbrand | On Fatou's lemma in several dimensions[END_REF].

Proposition 3 Let (h n ) be an integrably bounded sequence of integrable mappings from Ω ⊂ R to R k , which converges weakly to a integrable mapping f : Ω to R k . Then, we have for a.e. where Ls n {x n } denotes the set of limit points of converging subsequences of (x n ), and co(Z) the convex hull of the set Z.