Ping Yong

Iryna P Mikheenko

Kevin Deplanche

Mark D Redwood

Lynne E Macaskie

Biorefining of precious metals from wastes -an answer to manufacturing of cheap nanocatalysts for fuel cells and power generation via an integrated biorefinery?

Keywords: Metal recovery, Fuel cell catalyst, Palladium, Desulfovibrio desulfuricans, Escherichia coli

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Biorefining of precious metals from wastes: an answer to manufacturing of cheap nanocatalysts for fuel cells and power generation via an integrated biorefinery?

Introduction

Proton-exchange membrane fuel cells (also described as polymer electrolyte fuel cells: PEM-FC: [START_REF] Larminie | Fuel Cells Explained[END_REF][START_REF] Litster | PEM fuel cell electrodes[END_REF], have been identified as a hydrogenbased future technology to power motor vehicles and generators, and for small stationary applications. The PEM-FC membrane electrode assembly (MEA) comprises precious metalbased electro-catalysts (anode and cathode) and a proton conducting membrane between them.

H 2 is supplied to the anode, and O 2 (i.e. air) to the cathode [START_REF] Litster | PEM fuel cell electrodes[END_REF][START_REF] Schmittinger | A review of the main parameters influencing long-term performance and durability of PEM fuel cells[END_REF]. In the presence of precious metal (PM) catalyst (usually platinum), H 2 is oxidised at the anode; the released protons traverse the proton exchange membrane to the cathode and are reunited with the electrons and oxygen (from air) to form water as the only product. The electrons, via an electrical circuit, power a load; electric current is generated since the membrane is not electrically conductive [START_REF] Larminie | Fuel Cells Explained[END_REF][START_REF] Litster | PEM fuel cell electrodes[END_REF].

Although the most efficient type of fuel cell, PEM-FCs require a clean source of H 2 [START_REF] Larminie | Fuel Cells Explained[END_REF]. For commercial economic operation a fuel cell should operate for 5 years but the current operational lifetime may be as low as ~167 days [START_REF] Aki | Penetration of residential fuel cells and CO 2 mitigationcase studies in Japan by multi-objective models[END_REF].

Commercial H 2 contains traces of CO, a potent catalyst poison. It is feasible to produce COfree bio-H 2 by fermentation, and electricity was produced by a PEM-FC fuelled by a bio-H 2 stream from E. coli [START_REF] Macaskie | Applications of bacterial hydrogenases in waste decontamination, manufacture of novel bionanocatalysts and in sustainable energy[END_REF].

The performance of a PEM-FC is also sensitive to a range of operating parameters in the MEA including the active catalyst surface area and catalyst particle size, with metallic nanoparticles (NP) commonly used [START_REF] Ivanchev | Fluorinated proton-conduction nafion-type membranes, the past and the future[END_REF][START_REF] Schmittinger | A review of the main parameters influencing long-term performance and durability of PEM fuel cells[END_REF].

However even with avoidance of catalyst poisoning and effective NP production, the cost and availability of precious metals (PM) will ultimately limit the application of PEM-FC into the hydrogen economy as sustainable energy production replaces fossil fuels.

Bio-derived PM nanoparticles are alternative fuel cell quality catalysts [START_REF] Dimitriadis | Pt-based electro-catalytic materials derived from biosorption processes and their exploitation in fuel cell technology[END_REF][START_REF] Yong | From bio-mineralisation to fuel cells: biomanufacture of Pt and Pd nanocrystals for fuel cell electrode catalyst[END_REF]), the dual biosynthesis and nanoparticle-supporting functions overcoming the agglomeration problems that can beset conventional, non-biological methods of nanoparticle catalyst manufacture (see [START_REF] Creamer | Novel supported Pd hydrogenation bionanocatalyst for hybrid homogeneous/heterogeneous catalysis[END_REF]. Catalytically active Pd(0) nanoparticles were manufactured by Desulfovibrio desulfuricans and also by E. coli [START_REF] Deplanche | New nanocatalysts made by bacteria from metal solutions and recycling of metal wastes[END_REF][START_REF] Orozco | Towards an integrated system for bio-energy: Hydrogen production by Escherichia coli and use of palladium-coated waste cells for electricity generation in a fuel cell. Biotechnol Letts in submission Redwood MD, Paterson-Beedle M, Macaskie LE[END_REF], prompting the possibility to utilise waste E. coli biomass from biohydrogen production as the precursor to a fuel cell catalyst [START_REF] Orozco | Towards an integrated system for bio-energy: Hydrogen production by Escherichia coli and use of palladium-coated waste cells for electricity generation in a fuel cell. Biotechnol Letts in submission Redwood MD, Paterson-Beedle M, Macaskie LE[END_REF]. Metallic NP synthesis is achieved by using hydrogenases [START_REF] Mikheenko | Bioaccumulation of palladium by Desulfovibrio fructosovorans wild-type and hydrogenase-deficient strains[END_REF][START_REF] Deplanche | Role of hydrogenases in the formation of highly catalytic Pd(0) nanoparticles by bioreduction of Pd(II) using Escherichia coli mutant strains. Microbiology in press Deplanche K, Macaskie LE, Biorecovery of gold by Escherichia coli and Desulfovibrio desulfuricans[END_REF], expressed maximally under fermentative conditions (H 2 synthetic direction), to split H 2 (catabolic direction) and reduce, as electron acceptors, Pd(II) and Pt(IV) ions to Pd(0) and Pt(0) NPs.

In addition to D. desulfuricans and E. coli [START_REF] Mikheenko | Bioaccumulation of palladium by Desulfovibrio fructosovorans wild-type and hydrogenase-deficient strains[END_REF][START_REF] Deplanche | Role of hydrogenases in the formation of highly catalytic Pd(0) nanoparticles by bioreduction of Pd(II) using Escherichia coli mutant strains. Microbiology in press Deplanche K, Macaskie LE, Biorecovery of gold by Escherichia coli and Desulfovibrio desulfuricans[END_REF][START_REF] Deplanche | Role of hydrogenases in the formation of highly catalytic Pd(0) nanoparticles by bioreduction of Pd(II) using Escherichia coli mutant strains. Microbiology in press Deplanche K, Macaskie LE, Biorecovery of gold by Escherichia coli and Desulfovibrio desulfuricans[END_REF], other organisms with high hydrogenase activity, like Shewanella (De [START_REF] Windt | Bioreductive deposition of palladium (0) nanoparticles on Shewanella oneidensis with catalytic activity towards reductive dechlorination of polychlorinated biphenyls[END_REF] and Ralstonia [START_REF] Ludwig | Oxygen-tolerant oxidation by membrane-bound [NiFe] hydrogenase of Ralstonia species. Coping with low level H 2 in air[END_REF], can also reduce Pd(II) [START_REF] Deplanche | New nanocatalysts made by bacteria from metal solutions and recycling of metal wastes[END_REF]. Cupriavidus (formerly Ralstonia) metallidurans can biomineralise gold by various mechanisms including its reduction to Au(0) [START_REF] Reith | 12 Mechanism of gold biomineralization by the bacterium Cupriavidis metallidurans[END_REF]. The first objective of this study was to compare the activity of a PEM-FC anode made with Pd(0) manufactured by C. metallidurans (Bio-Pd C. metallidurans ) with that of Bio-Pd D. desulfuricans and Bio-Pd E. coli.

Use of bio-manufacturing, unlike many physico-chemical manufacturing methods, is intrinsically scalable technology but is limited ultimately by the high cost of precious metals (PMs). Previous studies showed that PMs can be bio-recovered from wastes and secondary sources with the resulting mixed-metal deposits on the bacterial cells being catalytically active [START_REF] Mabbett | Biorecovered precious metals from industrial wastes: Single-step conversion of a mixed metal liquid waste to a bioinorganic catalyst with environmental application[END_REF][START_REF] Murray | Biorecovery of platinum group metals from secondary sources[END_REF]. Therefore the second objective was to evaluate the activity of a PEM-FC made using PMs recovered from an industrial processing waste by a strain of E. coli previously shown to have potential for both H 2 production and for use as a FC catalyst [START_REF] Orozco | Towards an integrated system for bio-energy: Hydrogen production by Escherichia coli and use of palladium-coated waste cells for electricity generation in a fuel cell. Biotechnol Letts in submission Redwood MD, Paterson-Beedle M, Macaskie LE[END_REF].

Materials and Methods

Preparation of bacteria E. coli IC007, a derivative of E.coli MC4100, provided by Prof F. Sargent (University of Dundee, UK) was as described previously [START_REF] Orozco | Towards an integrated system for bio-energy: Hydrogen production by Escherichia coli and use of palladium-coated waste cells for electricity generation in a fuel cell. Biotechnol Letts in submission Redwood MD, Paterson-Beedle M, Macaskie LE[END_REF]). E. coli was cultured anaerobically at 37 °C in nutrient broth no. 2 supplemented with 0.4% (w/v) sodium fumarate and 0.5% (v/v) glycerol. Desulfovibrio desulfuricans NCIMB 8307 was cultured anaerobically in Postgate C medium at 30 o C [START_REF] Yong | Bioaccumulation of palladium by Desulfovibrio desulfuricans[END_REF]2007).

Cupriavidus metallidurans CH34 (NCIMB14178) was cultured anaerobically at 30 °C in tryptone soy broth (TSB). Cultures were grown in sealed 2 l Durham bottles with anaerobic conditions promoted by degassing under vacuum (30 min) and bubbling O 2 -free N 2 (OFN 30 min. Growing cultures (OD 600 = 0.5-0.7) were harvested by centrifugation, washed three times in 100 ml degassed MOPS/NaOH buffer (20 mM, pH 7.2), resuspended in the same buffer (1/20 of the original volume) and stored at 4 o C under N 2 for up to 2 days before use.

Bio-deposition of Pd(0) and precious metals (PM)

The concentrated cells were resuspended anaerobically (room temperature) in dilute HNO 3 (pH 2, pre-sparged with OFN) containing 2 mM Na 2 PdCl 4 or 1/500 dilution of an industrial reprocessing waste solution (to pH 2.3 and Pt: 2.8 mM; Pd: 1.9 mM and Rh: 0.14 mM; Degussa Ltd, Germany) to give ~25% Pd/cell dry weight (w/w), respectively. The suspension was left for 2 h to allow biosorption of metal (much of the hydrogenase activity was retained for this duration: I. Mikheenko, unpublished) then H 2 was bubbled through the solution for ~30 min. Complete Pd(II) reduction was confirmed as previously described [START_REF] Yong | Bioaccumulation of palladium by Desulfovibrio desulfuricans[END_REF] after the resulting black precipitate settled under gravity.

Examination of metal crystals by transmission electron microscopy (TEM) and X-ray powder diffraction (XRD)

Bacterial cells without or with Pd(0)-precipitates ('Bio-Pd') were washed twice in deionised water, fixed with 2.5% (w/v) glutaraldehyde (4 o C, 1 h), sectioned, stained with uranyl acetate and lead citrate and examined under a TEM (Jeol JEM-1200 EX electron microscope accelerating voltage 80 kV). The identity of the deposited metals as Pd(0) was confirmed using X-ray powder diffraction as previously described [START_REF] Yong | Bioaccumulation of palladium by Desulfovibrio desulfuricans[END_REF].

Pd and PM catalyst preparation, electrode preparation and fuel cell testing

Bio-Pd or Bio-PM (made from the precious metal waste as above) samples were centrifuged, washed with water three times and then with acetone, dried at room temperature and placed into 10 ml alumina ceramic crucibles. The crucibles were put in a program-controlled furnace, with the temperature raised from room temperature to 700 o C within 7 h gradually and held at 700 o C for a further hour. The sintered samples were cooled in the furnace to room temperature. Sintered Bio-Pd/PM samples (20 mg, as metal) were each mixed with 80 mg pure activated carbon powder (BDH Chemicals Ltd, England). Nafion perfluorinated ionexchange resin (0.2 ml; 10 % (w/v) in water, Sigma-Aldrich) and 0.2 ml water were added into each sample. The sample was mixed well and applied evenly by hand on to 16 cm 2 Teflon treated carbon paper (Fuel Cell Scientific, USA), and dried at room temperature.

Power generation was tested using the laboratory made Bio-Pd and Bio-PM electrodes and parallel electrodes made using commercial Pd powder (Aldrich, UK) as anodes using a commercial fuel cell testing unit as described previously [START_REF] Yong | From bio-mineralisation to fuel cells: biomanufacture of Pt and Pd nanocrystals for fuel cell electrode catalyst[END_REF]). Current (A) and voltage (V) were measured and recorded for variable resistance (R) as described by [START_REF] Voigt | Fuel Cell Technology for Classroom Instruction[END_REF].

Results

Formation of bio-Pd by bacterial cells

Electron micrographs of the bacteria without added Pd(II) are shown in Fig 1 (A,C,E,G).

After introduction of H 2 and soluble Pd(II), the metal was reduced to black deposits on the cells (Fig. 1B,D,F,H). The identity as Pd(0) was confirmed by X-ray powder diffraction. Pd(0) on cells of D. desulfuricans was visible as homogeneous small nano-particles (Fig. 1B) while that on the two E. coli strains was similar (fewer, larger particles). Cells of C. metallidurans appeared to show Pd(0) deposits (Fig. 1H) of size intermediate between those on the cells of D. desulfuricans (Fig. 1B) and E. coli (Fig. 1 D,F).

Activity of FC anodes made with Bio-Pd

Preliminary tests using electrodes made with Bio-Pd using native dried palladised biomasses without sintering gave negligible electricity output, probably due to the poor conductivity of biological materials. However, after sintering Bio-Pd (and also Bio-Pt; P. Yong, unpublished work) catalysts performed better than commercial Pd(0) and comparably (Bio-Pt) to fuel cell grade Pt black [START_REF] Yong | From bio-mineralisation to fuel cells: biomanufacture of Pt and Pd nanocrystals for fuel cell electrode catalyst[END_REF]. Note that the electrode area was only 16 cm 2 ; commercial PEMS are much larger and commercially are employed as 'stacks' for power output in the region of ~1kW [START_REF] Wang | Development and performance of 5 KW proton exchange membrane fuel cell stationary power system[END_REF].

Preliminary tests showed that Bio-anodes made using a 5% mass loading of Pd(0) [START_REF] Orozco | Towards an integrated system for bio-energy: Hydrogen production by Escherichia coli and use of palladium-coated waste cells for electricity generation in a fuel cell. Biotechnol Letts in submission Redwood MD, Paterson-Beedle M, Macaskie LE[END_REF]) gave approximately half the power output (54 10%; pooled data for all the strains) as anodes made with 25% mass loading of Pd(0) and the latter was adopted for the fuel cell studies (Fig. 2). A previous study [START_REF] Orozco | Towards an integrated system for bio-energy: Hydrogen production by Escherichia coli and use of palladium-coated waste cells for electricity generation in a fuel cell. Biotechnol Letts in submission Redwood MD, Paterson-Beedle M, Macaskie LE[END_REF] showed high activity of Bio-Pd D. desulfuricans (comparable to commercial materials) and this was taken as the positive control in this study. Conversely, Bio-Pd E. coli MC4100 had only about ~ 10% of this activity [START_REF] Orozco | Towards an integrated system for bio-energy: Hydrogen production by Escherichia coli and use of palladium-coated waste cells for electricity generation in a fuel cell. Biotechnol Letts in submission Redwood MD, Paterson-Beedle M, Macaskie LE[END_REF]) and was taken as the negative control for the evaluation of Bio-Pd C. metallidurans . The maximum power output of the latter was intermediate between the controls (~140 and ~29 mW respectively: Fig. 2) at ~61 mW (Fig 2C ; summarised in Table 1).

For Bio-Pd C. metallidurans the activity was comparable to that of bio-Pd Ecoli MC4100 at 5% Pdloading (not shown) and was also enhanced by ~4-fold at 25% Pd(0) loading whereas the bio-Pd E. coli MC4100 was only increased by 2-fold (18 mW at 5% loading [START_REF] Orozco | Towards an integrated system for bio-energy: Hydrogen production by Escherichia coli and use of palladium-coated waste cells for electricity generation in a fuel cell. Biotechnol Letts in submission Redwood MD, Paterson-Beedle M, Macaskie LE[END_REF]; 29 mW at 25% Table 1). It is apparent that use of C. metallidurans confers no advantage in terms of the catalytic activity of its Bio-Pd and this strain was not considered further.

A previous study using 5% mass loading of Pd(0) [START_REF] Orozco | Towards an integrated system for bio-energy: Hydrogen production by Escherichia coli and use of palladium-coated waste cells for electricity generation in a fuel cell. Biotechnol Letts in submission Redwood MD, Paterson-Beedle M, Macaskie LE[END_REF] showed the mutant E. coli IC007 produced Bio-Pd giving a maximum power output of ~56 mW as compared to ~18 mW for the parent strain MC4100, i.e. an increase of ~ 3-fold for reasons which are still unclear. In comparison, this study (25% mass loading of Pd(0)) showed the respective maximum power outputs of Bio-Pd E. coli MC4100 and Bio-Pd E. coli IC007 to be 29 and 114 mW (Fig. 2 D,E), i.e. an increase of ~ 4-fold by use of the mutant strain. Hence, increasing the mass loading of Pd(0) from 5% to 25% enhances the activity of the mutant strain Bio-Pd more than that of the parent strain MC4100.

A notable feature of the low-activity Bio-Pd E. coli MC4100 and for Bio-Pd C. metallidurans (Fig. 2C,D) the maximum power output was sustained over a wide voltage range whereas for Bio-Pd D. desulfuricans and Bio-Pd E. coli IC007 (Fig 2B ,E) the maximum power output was observed as a sharp peak at V = ~0.4 (Fig. 2). Hence these two active electrodes gave a higher current than the 'low active' electrodes when the V was decreased. Apparently the mutation(s) imposed in strain IC007 (Fig. 2E) yield a pattern of Bio-Pd behaviour more consistent with that of D. desulfuricans (Fig. 2B) than that of E. coli MC4100 (Fig. 2C) and the profile and power output was almost identical to that of the commercial Pd(0) (Fig. 2A,E). A comparison of the electron micrographs of strains MC4100 and IC007 (Fig. 1D,F) revealed no apparent morphological differences in the appearance of their respective Pd(0) deposits. A high resolution TEM study is clearly warranted, in conjunction with examination of possible pleiotrophic effects of the mutations imposed on strain IC007, as noted by [START_REF] Orozco | Towards an integrated system for bio-energy: Hydrogen production by Escherichia coli and use of palladium-coated waste cells for electricity generation in a fuel cell. Biotechnol Letts in submission Redwood MD, Paterson-Beedle M, Macaskie LE[END_REF].

Activity of electrode made using waste precious metals solution

Since E. coli strain IC007 gave a power output profile from its Bio-Pd similar to that of Bio-Pd D. desulfuricans (Fig. 2) and was only compromised by ~20% in terms of the maximum power output as compared to the positive control (P max of 114 and 140 mW respectively, Table 1) and since E. coli biomass residual from hydrogen production is desirable in terms of process efficiency [START_REF] Orozco | Towards an integrated system for bio-energy: Hydrogen production by Escherichia coli and use of palladium-coated waste cells for electricity generation in a fuel cell. Biotechnol Letts in submission Redwood MD, Paterson-Beedle M, Macaskie LE[END_REF]) strain IC007 was used for estimation of power output using metallised cells that had recovered Pd, Pt and Rh from a commercial processing waste. All three metals were removed from solution and the composition of the material was therefore assumed to be Pd (6.6%), Pt (17.9%) and Rh (0.5%) but this was not tested. The power output profile of the Bio-PM material in the FC is shown in Fig 2 E; significant power was generated, at ~68% of commercial finely-divided Pd(0) (Table 1). It is noteworthy that the sharp peak of power output at ~ 0.4V was absent and in this respect the Bio-PM E.coliIC007 (Fig. 2F) resembled the poorly perfoming Bio-Pd E.coliMC4100 . (Fig. 2D).

Discussion

In the PEM FC the power output relies on the catalyst activity but can also be affected by the electrode preparation method, e.g. by the formation of catalyst layer, gas diffusion layer and electrode design [START_REF] Litster | PEM fuel cell electrodes[END_REF]. Such commercial methods are proprietary; hence, a positive and negative control for this study was fabricated using our standard method (see Materials and Methods) together with a commercial catalyst material for comparison. A fixed composition of metal powder, activated carbon, Nafion, and water was applied by hand in 16 cm 2 electrode preparations. In this way a previous study showed that bio-Pt D. desulfuricans gave a power output identical to commercial fuel cell Pt catalyst and the outputs of both were ~15% less than the proprietary intact FC, indicating the extent to which our method differs from the proprietary method [START_REF] Yong | From bio-mineralisation to fuel cells: biomanufacture of Pt and Pd nanocrystals for fuel cell electrode catalyst[END_REF]). However our method was reproducible; for all except Bio-Pd C. metallidurans standard deviations of within 2 mW were observed between three independent fuel cells with each strain (Fig. 2) Consistent with tests in other, chemical, catalyses [START_REF] Creamer | Novel supported Pd hydrogenation bionanocatalyst for hybrid homogeneous/heterogeneous catalysis[END_REF][START_REF] Deplanche | New nanocatalysts made by bacteria from metal solutions and recycling of metal wastes[END_REF][START_REF] Creamer | Novel supported Pd hydrogenation bionanocatalyst for hybrid homogeneous/heterogeneous catalysis[END_REF];) Bio-Pd D. desulfuricans gave good power generation [START_REF] Yong | From bio-mineralisation to fuel cells: biomanufacture of Pt and Pd nanocrystals for fuel cell electrode catalyst[END_REF].

However the requirement for obligately anaerobic growth, the need to grow biomass specially, and the need to remove all traces of H 2 S (a powerful catalyst poison) from the biomass product would make the use of D. desulfuricans unattractive in large scale manufacturing of fuel cell catalyst. Previous attempts to use Bio-Pd E.coli were unsuccessful, with only a low power output reported [START_REF] Orozco | Towards an integrated system for bio-energy: Hydrogen production by Escherichia coli and use of palladium-coated waste cells for electricity generation in a fuel cell. Biotechnol Letts in submission Redwood MD, Paterson-Beedle M, Macaskie LE[END_REF]) despite that the chemical catalytic activity of bio-Pd D. desulfuricans and Bio-Pd E. coli was comparable [START_REF] Deplanche | New nanocatalysts made by bacteria from metal solutions and recycling of metal wastes[END_REF]. A strain of E. coli (IC007) developed with the initial aim of increasing hydrogen production was unexpectedly found to make a Bio-Pd with enhanced FC anode activity [START_REF] Orozco | Towards an integrated system for bio-energy: Hydrogen production by Escherichia coli and use of palladium-coated waste cells for electricity generation in a fuel cell. Biotechnol Letts in submission Redwood MD, Paterson-Beedle M, Macaskie LE[END_REF]. As noted previously [START_REF] Orozco | Towards an integrated system for bio-energy: Hydrogen production by Escherichia coli and use of palladium-coated waste cells for electricity generation in a fuel cell. Biotechnol Letts in submission Redwood MD, Paterson-Beedle M, Macaskie LE[END_REF]) the activity using Bio-Pd E. coli IC007 was enhanced by several-fold (Fig. 2; Table 1). A previous study [START_REF] Creamer | Novel supported Pd hydrogenation bionanocatalyst for hybrid homogeneous/heterogeneous catalysis[END_REF] showed that the optimal Pd loading onto biomass for chemical catalysis was 5% by mass However, regardless of the bacterial strain used, the activity of electrodes made from 5% of Pd/biomass (w/w) samples was ~ half than these made from 25% of Pd/biomass (w/w). This may be attributable to the proportional residual (cell ashes) from the samples with 5% Pd after sintering being much greater than these with 25% Pd, possibly giving a greater resistance. However this would not account for the different activities of the two E. coli strains Bio-Pd MC4100 and Bio-Pd IC007 , at the same Pd(0) loading by mass; an in-depth TEM study of the progression of the bio-materials during the sintering process is in progress.

In conclusion, bio-manufacture of fuel cell-grade precious metal catalysts is achieved using resting cells of D. desulfuricans or the E. coli mutant IC007. Electrodes from the latter gave up to 81% of maximum power generation compared to these from bio-Pd of D. desulfuricans, and ~ 115% of maximum power generation compared to those from a commercial Pd catalyst (maximum power: ~99 mW). A Bio-PM (precious metal) catalyst made directly from an industrial reprocessing solution by E. coli IC007 made into a FC anode gave ~ 68% of the power generation as pure Pd(0). This ground-breaking study paves the way to use FC catalysts made directly from precious metal waste solutions using waste biomass from biohydrogen production, with the consumption of the OAs downstream into further hydrogen (Redwood et al. 2008) 'closing the loop'.

The concept of an 'integrated biorefinery' to convert agri-food wastes and precious metal wastes into power is shown in Fig. 3. Light-driven photofermentation and waste supply may be intermittent whereas power demand may be constant; for this reason a hydrogen store is inserted before the PEM FC to hold the hydrogen and deliver it as required, decoupling the rates of hydrogen production and consumption. In this context, preliminary studies have shown the effective uptake of bio-hydrogen into a chemical store from a fermenter outflow at ~ 1 bar, and later discharge of H 2 for power production (M.D. Redwood and A. Bevan, unpublished. This will be reported in full in a later publication.

Further study is needed to optimise the power generation by Bio-PM directly from a greater variety of precious metal wastes. In this context, PMs recovered from acidic leachate from metallic scraps [START_REF] Murray | Biorecovery of platinum group metals from secondary sources[END_REF] illustrates the possibility to bypass commercial metal reprocessing. The integrated biorefinery shown in Fig. 3 is currently under construction. systems. Electrodialytic (ED) separation of organic acids is employed to generate biohydrogen streams from the two bioreactors. These streams are pooled with the H 2 generated via electrolysis of water within the ED cell, into a hydrogen store. The waste biomasses from fermentation can be used [START_REF] Orozco | Towards an integrated system for bio-energy: Hydrogen production by Escherichia coli and use of palladium-coated waste cells for electricity generation in a fuel cell. Biotechnol Letts in submission Redwood MD, Paterson-Beedle M, Macaskie LE[END_REF] to recover catalytically-active precious metals from waste streams (i.e. this study) and also leachates made from PM scrap [START_REF] Creamer | Palladium and gold removal and recovery from precious metal solutions and electronic scrap leachates by Desulfovibrio desulfuricans[END_REF][START_REF] Mabbett | Biorecovered precious metals from industrial wastes: Single-step conversion of a mixed metal liquid waste to a bioinorganic catalyst with environmental application[END_REF][START_REF] Murray | Biorecovery of platinum group metals from secondary sources[END_REF]). The PEM fuel cell is made using a Bio-Pd anode (this study) and biohydrogen is fed [START_REF] Macaskie | Applications of bacterial hydrogenases in waste decontamination, manufacture of novel bionanocatalysts and in sustainable energy[END_REF] to generate 68 mW of power per 16 cm 2 anode (Table 1). This compares with 170 mW obtained with commercial Pt fuel cell catalyst and also Bio-Pt D. desulfuricans [START_REF] Yong | From bio-mineralisation to fuel cells: biomanufacture of Pt and Pd nanocrystals for fuel cell electrode catalyst[END_REF]) and suggests that an improvement of 2.5-fold is needed for the Bio-Pd IC007 to become competitive with a commercial PEMFC. Note however that optimisation studies have not yet been done and also that bio-H 2 does not contain CO, which reduces FC operational life using commercial hydrogen.

Legends to Figures

Fig 1 .

 1 Fig 1. Cells of D. desulfuricans (A, B), E. coli MC4100 (C, D), E. coli IC007. (E, F) and C.

  Fig 1. Cells of D. desulfuricans (A, B), E. coli MC4100 (C, D), E. coli IC007. (E, F) and C. metallidurans (G, H) viewed under TEM. A, C, E, and G: Pd-unchallenged cells. B, D, F and H: Pd(II) (25% of Pd to Biomass, w/w) was reduced to Pd(0) after reduction under H 2 . Bars show 500 nm.

Fig. 2 .Fig. 3 .

 23 Fig.2. The output current and power by fuel cells tested with anodes made (as described in Materials and Methods) from commercial finely divided Pd (A) and Bio-Pd from 25% Pd/dry cells (w/w) of D. desulfuricans (B), C. metallidurans (C), E. coli MC4100 (D), E. coli IC007 (E) and 25% PM/dry cells of E-coli IC007 (F). PM: an industrial PM reprocessing (Degussa Ltd, Germany) solution diluted 1/500 for use and containing a mixture of Pt (2.8 mM), Pd (1.9 mM) and Rh (0.14 mM). : Power (Watts);  Current (Amps). Data are from three experiments; standard deviations generally (< 2 mW) where not shown were within the dimensions of the symbols

  Fig. 1

Acknowledgments

This work was supported by the Royal Society (Industrial Fellowship and a Brian Mercer Senior Award for Innovation to LEM) and by BBSRC (Grant nos BB/C516195/2 and BB/E003788/1) and EPSRC (Grant nos EP/D05768X/1 and EP/E034888/1). The authors wish to thank Professor K. Kendall and Dr B.G. Pollet (Department of Chemical Engineering, University of Birmingham) for helpful discussions.


Higher order microphones

The method of array processing involves the use of multiple microphones to receive a signal carried by propagating waves. Microphone arrays have a variety of applications such as sonars, radars and acoustic tomography. The main objective of this work is the implementation of higher order microphone arrays in DirAC. A review on previous research followed by simulations of various microphone array designs has been performed. Prototype arrays will be constructed and evaluated within a DirAC system. Simulating the various microphone arrays revealed the advantages and possible drawbacks of each design. Various higher order designs have been implemented in the past and are used as guidelines. First order microphones are well implemented into DirAC system by using the omnidirectional component to estimate pressure and the X,Y and Z to estimate the velocity. In principle DirAC input can be extended to higher order microphones. Any microphone array can be used for the estimation of the incoming direction of the wavefront and the diffuseness of the soundfield. One way of achieving that is by combining different microphone orders and create virtual microphones of higher order. There are various models that will be tested and applied in order to compute direction of the wavefront and diffuseness of the soundfield. Some of these are single and multiple source reverberant field models, cross correlation models or beamforming techniques. The use of these models has the advantage that the analysis and synthesis part will no longer be a separate process within the DirAC analysis but one single process using methods such as the cross-spectrum analysis. 1

Microphone array simulations

A microphone simulator is used to determine optimal design of microphone arrays and check their performance before construction. If the simulator is accurate it can be a very cost effective method. The initial design consisted of four microphone capsules placed at the edges of cube with inner distance of 0.5cm . By using this arrangement we can calculate components of 0 th , 1 s t and 2 n d order. These components are extracted by simple additions and subtractions between the channel. The equation system below explain how these are calculated:

W = A 1 (P 1 + P 2 + P 3 + P 4 )
(1)

X = A 2 (P 1 -P 3 ) (2) U = A 3 (P 1 -P 2 + P 3 -P 4 ) (3)
where A 1 , A 2 and A 3 are normalization coefficients. The rotated components of 1 st and 2 nd order can be obtained as Y = -X and V = -U . Figure 1 shows the calculated components if we use four omnidirectional capsules.

The same simulation is also performed by using cardioid capsules. Results are shown in Figure 1. The advantage in this case is the lack of boost in low frequencies especially in the 0 t h order component. In addition to that the components X and U are closer to the ideal ones. According to this analysis it is possible to apply beamforming. Beamforming is a technique that combines a number of microphones in order to perform spatial filtering by creating a narrow directivity pattern. The purpose of a beamformer is to capture sound from a preferred location. There are various ways to implement beamformers. The cross-spectrum analysis described in this report can be also used a a beamforming technique.

Directional Audio Coding

Various spatial sound systems have been developed during the last 3 decades as a way for representing a sound field as accurately as possible. Some of them are based on the accurate reconstruction of the soundfield i.e. ambisonics, WFS while others are perception based. A good example of a latter systems is DirAC which is developed in the Laboratory of Acoustics and Audio Signal Processing, Helsinki University of Technology. DirAC is based on the following assumptions and makes use that the human auditory system can process one directin of arrival for each time-frequency bin.

• Direction of arrival of sound will transform into interaural time difference (ITD), interaural level difference (ILD), and monaural localization cues. This has been proven and discussed extensively in Blauert's book, Spatial Hearing. • Timbre depends on the ITD, ILD, monaural spectrum and interaural coherence of sound [2].

• The direction of arrival, diffuseness, and spectrum of sound measured in a point with the temporal and spectral resolution of human hearing determines the auditory spatial image the listener is perceiving [3,4].

Higher order systems

The main idea behind a higher order implementation in DirAC is to use higher order microphone directivities instead of the standard b-format. Higher order microphones are able to be narrower and hence provide more directional information for the arrival of sound and the calculation of diffuseness.

The simulations that follow are based on the analysis of higher order ideal microphones and the use of a cross-spectrum function.

Theory

Spectral analysis shows how the energy is distributed in frequency. Autocorrelation is the correlation of a function with a shifted version of itself Hence the auto-spectrum is the Fourier transform in a simplified way is:

R xx [m] = n x * [n]x[n + m] (4)
G xx = f f t(x(t) * x(t)) (5)
The cross-spectrum is similar but instead of using the autocorrelation, it makes use of the cross-correlation between two signals. In this work the cross-spectrum is described between two microphone inputs. For example:

g simeon = (M x × M y ) M x (n) 2 + M y (n) 2 (6)
where M x and M y are the time-frequency analysis of two different microphone inputs. The quantity M x (n) 2 + M y (n) 2 is a normalization coefficient.

indicates the real part and M * x the complex conjugate. The cross-spectrum describes the correlation between two inputs. It estimates the coherence between two input signals. If we take the real part of this quantity we get the in phase fluctuation between the inputs. A sample diagram of the process described in this chapter is shown in Figure 2 The input of this system is a microphone array from which higher order directivities are extracted by using either the analysis in the spherical harmonic domain or a simple addition and multiplication between the array's signals. In this diagram ideal microphones are shown. Specifically the microphones that are used in this example are up to 3 rd : an omnidirectional W , the dipoles X and Y and a quadrapole U . By taking the cross-spectrum of the X and U components a narrower directivity pattern can be obtained. The resulting directivity provided specific gain values for each angle. It is also possible to use the remaining components of 1 st 2 nd in order to construct directive microphones. A standard DirAC analysis is performed in order to calculate the active intensity and the azimuth. These quantities provide

  Research visit in the Department of Acoustics and Signal Processing, Espoo Finland: Simulations of second order microphones in audio coding order microphones . . . . . . . . . . . . . . . . . . . . 1 0.2 Microphone array simulations . . . . . . . . . . . . . . . . . . 2 0.3 Directional Audio Coding . . . . . . . . . . . . . . . . . . . . 2 0.4 Higher order systems . . . . . . . . . . . . . . . . . . . . . . . 3 0.5 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Figure 1 :

 1 Figure 1: Unequalized components y using four omnidirectional microphones

Figure 2 :

 2 Figure 2: Analysis of Directional Audio Coding

information on the direction and diffuseness of the soundfield by using the 0 th and 1 st order components. In order to synthesize this information for reproduction a loudspeakers gain needs to be computed: the g DirAC . This is done by mapping the values of the azimuth to the values of calculated cross-spectrum for the specific angles. Hence we map

where g XY is the cross-spectrum between the microphone inputs from X and U . Now the proposed analysis is performed where the cross spectrum between X and U is calculated as:

Virtual microphones are created by using the calculated components (X + U ). It should be noted that any combination of components can be used in order to create virtual microphones. The combination of X and U provides a relative directional pattern which is usefull for this analysis. The calculated virtual microphones that contain the signal information are filtered through g DirAC and g Simeon . The analysis is based on the comparison between the existing DirAC system and proposed HODirAC by comparing:

(9)

Summary

My research goals in the department of acoustics and signals processing in Aalto university can be summarized as follows:

1. A model to compare higher order with first order input in DirAC analysis is created. Coincident microphones were used and the theory was extended to microphone arrays.

2. A slight advantage is gained by using higher higher order microphones instead of first order.

3. Future work will support these findings with listening tests. Subjective evaluation will check the validity of a higher order microphone array system analysis in DirAC. A study on the subjective effects of the physical limitations of microphone arrays (i.e. spatial aliasing) will also be performed. The results will be compared with the original DirAC analysis to check the potential advantages of the implemented system.


LDL from obese patients with the metabolic syndrome show increased lipid peroxidation and activate platelets.

  Fig.3


T Horsin

PRESCRIBING THE MOTION OF A SET OF PARTICLES IN A 3D PERFECT FLUID

We establish a result concerning the so-called Lagrangian controllability of the Euler equation for incompressible perfect fluids in dimension 3. More precisely we consider a connected bounded domain of R 3 and two smooth contractible sets of fluid particles, surrounding the same volume. We prove that given any initial velocity field, one can find a boundary control and a time interval such that the corresponding solution of the Euler equation makes the first of the two sets approximately reach the second one.

1. Introduction 1.1. Presentation of the problem. In this paper, we are concerned with the Lagrangian controllability of the three dimensional Euler equation for perfect incompressible fluids by means of a boundary control. The problem under view is the following.

Let Ω be a smooth bounded domain of R 3 and let Γ be a nonempty open part of its boundary ∂Ω. Given T > 0, we consider the classical Euler system for perfect incompressible fluids in Ω:

∂ t u + (u • ∇)u + ∇p = 0 in (0, T ) × Ω, (1)
div u = 0 in (0, T ) × Ω, (2)

u |t=0 = u 0 in Ω. ( 3
)
As boundary conditions, one usually considers the impermeability condition on ∂Ω: u • n = 0 on (0, T ) × ∂Ω, where n stands for the unit outward normal vector field on ∂Ω. However, in the problem under view, the impermeability condition is merely imposed on ∂Ω \ Γ: [START_REF] Coron | On the controllability of 2-D incompressible perfect fluids[END_REF] u • n = 0 on (0, T ) × (∂Ω \ Γ).

Consequently, since we do not prescribe boundary values for u on (0, T ) × Γ, the problem stated as above is underdetermined. It is in fact an implicit control problem; in other words, we consider the boundary values on Γ as a control on this system, i.e. a way to influence the fluid in a prescribed way. If one wants to close the system, as shown by Khazhikov [START_REF] Kazhikov | Note on the formulation of the problem of flow through a bounded region using equations of perfect fluid[END_REF], one may consider as boundary conditions on Γ the following ones: u • n on (0, T ) × Γ and curl u ∧ n on {(t, x) ∈ (0, T ) × Γ such that u • n < 0}, that is, the normal part of the velocity field on Γ and the tangential entering vorticity. We refer to [START_REF] Kazhikov | Note on the formulation of the problem of flow through a bounded region using equations of perfect fluid[END_REF] for more details on this notion of boundary values. Due to the complexity of this formulation of boundary values, and as in the literature concerning the controllability of the Euler equation (see e.g. [START_REF] Coron | Contrôlabilité exacte frontière de l'équation d'Euler des fluides parfaits incompressibles bidimensionnels[END_REF], [START_REF] Coron | On the controllability of 2-D incompressible perfect fluids[END_REF], [START_REF] Glass | Exact boundary controllability of 3-D Euler equation[END_REF], [START_REF] Glass | An addendum to a J.-M. Coron theorem concerning the controllability of the Euler system for 2D incompressible inviscid fluids[END_REF]), we prefer not to refer to them explicitely and to look for the solution (u, p) itself. Once a solution of (1)-( 4) is given, its suitable trace on (0, T ) × Γ can be thought as the control.

Date: 17/08/2011. 2000 Mathematics Subject Classification. Primary .

The question that we raise is the possibility of prescribing the motion of a set of particles driven by a fluid governed by ( 1)- [START_REF] Coron | On the controllability of 2-D incompressible perfect fluids[END_REF]. It is classical in fluid mechanics that one can describe the motion of a fluid from two different points of view. One possibility, referred as the Eulerian description of the fluid, is to describe the velocity of fluid particles that pass "through" a given point. The other possibility, referred as the Lagrangian description of the fluid consists in following fluid particles along the flow. To that purpose one has to solve the differential equation associated to the velocity field.

The notion of controllability that we consider in this paper is different from the usual notion of controllability and, in some sense, closer to the Lagrangian description of the fluid than the usual notion of controllability. The standard sense of controllability would refer to the possibility of driving the velocity field (the state of the system), from one prescribed value to a fixed target. The study of the controllability of the Euler equation has been initiated by J.-M. Coron in [START_REF] Coron | Contrôlabilité exacte frontière de l'équation d'Euler des fluides parfaits incompressibles bidimensionnels[END_REF] and [START_REF] Coron | On the controllability of 2-D incompressible perfect fluids[END_REF], and then studied by the first author in [START_REF] Glass | Exact boundary controllability of 3-D Euler equation[END_REF] and [START_REF] Glass | An addendum to a J.-M. Coron theorem concerning the controllability of the Euler system for 2D incompressible inviscid fluids[END_REF]. Also related to the boundary control of the velocity field, the question of asymptotic stabilization of this system around zero was studied by J.-M. Coron in 2-D simply connected domains (see [START_REF] Coron | On the null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domain[END_REF]) and by the first author for what concerns more general 2-D domains (see [START_REF] Glass | Asymptotic stabilizability by stationary feedback of the two-dimensional Euler equation: the multiconnected case[END_REF]).

In this paper, we consider the problem of prescribing the displacement of a set of particles, rather than the velocity field in final time. In [START_REF] Glass | Approximate Lagrangian controllability for the 2-d Euler equations. application to the control of the shape of vortex patch[END_REF], the authors showed that in 2-D, one can indeed prescribe approximately the motion of some specific sets of fluids. The goal of the present paper is to consider the case of the dimension 3.

1.2. Notations and definitions. In this section, we fix the notations and give the basic definitions.

In the sequel, smooth curves, surfaces or maps will mean C ∞ ones. We will denote by C ω the class of real analytic curves/surfaces. The volume of a borelian set A in R 3 will be denoted by |A|. Given a suitably regular vector field u, we will denote by φ u the flow of u, defined (when possible) by [START_REF] Coron | On the null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domain[END_REF] ∂ t φ u (t, s, x) = u(t, φ u (t, s, x)) and u(s, s, x) = x.

For A a subset of R 3 and η > 0, we will denote:

V η (A) := {x ∈ R 3 / d(x, A) < η}.
We will also use the notation L p (0, T ; X(Ω(t))) or C k ([0, T ]; X(Ω(t))) of time-dependent functions with values in a space X of functions on a domain depending on time Ω(t) ⊂ R 3 . This refers to the space functions that can be extended to L p (0, T ;

X(R 3 )) or C k ([0, T ]; X(R 3 )).
We now recall some definitions from differential geometry (see e.g. [START_REF] Hirsch | Differential topology[END_REF]).

Definition 1. A Jordan surface γ in R 3 is the image of S 2 by some homeomorphism h.

According to the Jordan-Brouwer Theorem (see [START_REF] Brouwer | Über Jordansche Mannigfaltigkeiten[END_REF] or [START_REF] Guillemin | Differential topology[END_REF]), R 3 \ γ has two connected components, one of which is bounded and will be denoted by int(γ). Moreover when the homeomorphism h is a diffeomorphism, we have a unit outward normal vector field on γ, which we will denote by ν.

Definition 2. Two Jordan surfaces γ 0 and γ 1 embedded in R 3 are said to be isotopic in Ω, if there exists a continuous map I : [0, 1] × S 2 → Ω such that I(0) = γ 0 , I(1) = γ 1 and for each t ∈ [0, 1], I(t, •) is an homeomorphism of S 2 into its image. When, for some k ∈ N \ {0}, this homeomorphism is a C k -diffeomorphism with respect the space variable, I will be said to be a C k -isotopy, or, when k = ∞, a smooth isotopy. A one-parameter continuous family of diffeomorphisms of Ω is called a diffeotopy of Ω.

When their regularity is not specified, the geometrical objects are considered to be smooth in the sequel. We are now in position to define the Lagrangian controllability (the corresponding two-dimensional notion was introduced in [START_REF] Glass | Approximate Lagrangian controllability for the 2-d Euler equations. application to the control of the shape of vortex patch[END_REF]). Definition 3. Let T > 0 be a given time. We will say that the exact lagrangian controllability of (1)-( 4) holds if given two Jordan surfaces γ 0 and γ 1 included in Ω such that γ 0 and γ 1 are isotopic in Ω, [START_REF] Gardiner | Harmonic approximation[END_REF] |int(γ 0 )| = |int(γ 1 )|, [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] and given a regular initial data u 0 satisfying div u 0 = 0 in Ω, (8)

u 0 • n = 0 on ∂Ω \ Γ, (9)
there exists a solution (u, p) of ( 1)-( 4) such that one has ∀t ∈ [0, T ], φ u (t, 0, γ 0 ) ⊂ Ω, (10)

φ u (T, 0, γ 0 ) = γ 1 . (11) up to reparameterization.
Let us say a few words concerning the regularity of u. In general, strong solutions of the Euler equation are considered in a Hölder or Sobolev space with respect to x, which is included in the space of Lipschitz functions. In the sequel, the solutions will be taken in the Hölder space C k,α (Ω), k ≥ 1.

If the exact Lagrangian controllability does not occur, one may try to weaken this definition as follows.

Definition 4. We will say that the approximate Lagrangian controllability of ( 1)-( 4) holds in time T and in norm C k if given γ 0 , γ 1 and u 0 as above, and given any ε > 0, there exists a solution (u, p) of ( 1)-( 4) such that (10) holds and that

(12) φ u (T, 0, γ 0 ) -γ 1 C k (S 2 ) < ε, up to reparameterization.
Remark 1. It is easy to see that the conditions that γ 0 and γ 1 should be isotopic and enclose the same volume are necessary in order that a smooth volume-preserving flow drives γ 0 to γ 1 . In particular |int(γ 0 )| = |int(γ 1 )| comes from the incompressibility of the fluid.

Remark 2. Condition (10) allows to make sure that one "controls" the fluid zone for all time by making it stay in the domain; Conditions [START_REF] Glass | Approximate Lagrangian controllability for the 2-d Euler equations. application to the control of the shape of vortex patch[END_REF] or [START_REF] Guillemin | Differential topology[END_REF] would not have a clear meaning without it.

1.3. Main result. The main result of this paper is the following, establishing an approximate Lagrangian controllability property.

Theorem 1. Let α ∈ (0, 1) and k ∈ N\{0}. Consider u 0 ∈ C k,α (Ω; R 3 ) satisfying (8)-( 9) and γ 0 and γ 1 two contractible C ∞ embeddings of S 2 in Ω satisfying (6)- [START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF]. Then for any ε > 0, there exist a time T > 0 and a solution (u, p) in L ∞ (0, T ; C k,α (Ω; R 4 )) to (1)-( 4) on [0, T ] such that (10) and ( 12) hold (up to reparameterization).

Remark 3. Let us check that, as in the two-dimensional case (see [START_REF] Glass | Approximate Lagrangian controllability for the 2-d Euler equations. application to the control of the shape of vortex patch[END_REF]), the exact lagragian controllability does not hold. As is classical, in 3-D the vorticity of the fluid ( 13) ω := curl u, satisfies the equation ( 14)

∂ t ω + (u • ∇)ω = (ω • ∇)u.
Denoting w(t, x) = ω(t, φ u (t, 0, x)), we see that

∂ t w = (w • ∇)u(t, φ u (t, 0, x)).
Thus if initially curl u 0 = 0 in a neighborhood of γ 0 then curl u = 0 in a neighborhood of φ u (t, 0, γ 0 ) as long as ( 10) is true. Now, since u satisfies (2), for each time t, in such a neighborhood of φ u (t, 0, γ 0 ) and away from ∂Ω, u(t, •) is locally the gradient of a harmonic function, and hence is real analytic . Therefore, if γ 0 is a real-analytic submanifold, so will be φ u (t, 0, γ 0 ) for each t ∈ [0, T ]. It follows that the exact Lagrangian controllability does not hold in general, since one may take γ 1 smooth but non analytic.

Remark 4. As will be clear from the proof, the time T whose existence is granted by Theorem 1 can be made arbitrarily small. The result does not require a time long enough in order for the controllability property to take place, but a time small enough to make sure that no blow up phenomenon occurs.

Let us now briefly describe the strategy to prove Theorem 1. First, we use a construction due to A. B. Krygin [START_REF] Krygin | Extension of diffeomorphisms that preserve volume[END_REF] to obtain a solenoidal vector field

X ∈ C ∞ 0 ((0, 1) × Ω; R 3 ), such that ∀t ∈ [0, 1], φ X (t, 0, γ 0 ) ⊂ Ω, φ X (1, 0, γ 0 ) = γ 1 .
Then we will prove that there exist potential flows (that is, time-dependent gradients of harmonic functions in Ω), which approximate suitably the action of X on γ 0 . This gives us a solution (ū, p) of ( 1)-( 4) with u 0 = 0 and T = 1 such that

φ ū(0, t, γ 0 ) -φ X (0, t, γ 0 ) C k (S 2 ) ≤ ε.
Then we will use a time-rescaling argument due to J.-M. Coron [START_REF] Coron | On the controllability of 2-D incompressible perfect fluids[END_REF] (see also [START_REF] Glass | Exact boundary controllability of 3-D Euler equation[END_REF]), to get the result for non trivial u 0 and T small enough.

The rest of the paper is divided as follows. In Section 2, we consider the existence of the above mentionned vector field X, and then of the potential flows. In Section 3, we prove Theorem 1 by constructing the solution (u, p) taking the initial condition u 0 into account.

Acknowledgements. The authors are partially supported by the Agence Nationale de la Recherche (ANR-09-BLAN-0213-02). They thank Jean-Pierre Puel for many useful discussions leading to this paper.

Potential flows

2.1. A solenoidal vector field mapping γ 0 onto γ 1 . The vector field X is obtained as a direct consequence of the following result due to A. B. Krygin.

Theorem 2 ([16]

). If γ 0 and γ 1 are as in Theorem 1, then there exists a volume-preserving

diffeotopy h ∈ C ∞ ([0, 1] × Ω; Ω) such that ∂ t h is compactly supported in (0, 1) × Ω, h(0, γ 0 ) = γ 0 and h(1, γ 0 ) = γ 1 .
A direct consequence is that the smooth vector field ( 15)

X(t, x) := ∂ t h(t, h -1 (x)),
is compactly supported in (0, 1) × Ω and satisfies

φ X (1, 0, γ 0 ) = γ 1 ,
and div X = 0 in (0, 1) × Ω.

2.2.

Moving the fluid zone by potential flows. In this subsection we prove the following.

Proposition 1. Let γ 0 be a C ∞ contractible two-sphere embedded in Ω and consider

X ∈ C 0 ([0, 1]; C ∞ (Ω; R 3
)) a solenoidal vector field such that

(16) ∀t ∈ [0, 1], φ X (t, 0, γ 0 ) ⊂ Ω,
and let

γ 1 = φ X (1, 0, γ 0 ). For any ε > 0 and k ∈ N there exists θ ∈ C ∞ 0 ((0, 1) × Ω; R) such that ∀t ∈ [0, 1], ∆ x θ = 0 in Ω, ( 17
) ∂θ ∂n = 0 on [0, 1] × (∂Ω \ Γ), ( 18
) ∀t ∈ [0, 1], φ ∇θ (t, 0, γ 0 ) ⊂ Ω, (19) φ ∇θ (1, 0, γ 0 ) -γ 1 C k (S 2 ) ≤ ε, (20)
up to reparameterization.

As in [START_REF] Glass | Approximate Lagrangian controllability for the 2-d Euler equations. application to the control of the shape of vortex patch[END_REF] we prove this proposition by assuming first that γ 0 is an embedded real analytic 2-sphere and that X is real analytic in x. Then we progressively reduce the assumptions to the framework of Proposition 1.

2.2.1.

Case of an analytic 2-sphere moved by an analytic isotopy. The goal of this paragraph is to prove the following proposition.

Proposition 2. The conclusions of Proposition 1 are satisfied if we assume that γ 0 is an analytic 2-sphere (i.e. γ 0 is the image of S 2 by a real-analytic embedding f 0 ) and that X is moreover in

C 0 ([0, 1]; C ω (Ω; R 3 )).
It is clear that in that case, φ X is volume preserving real analytic isotopy between γ 0 and γ 1 . The first step to establish Proposition 2 is the following.

Lemma 1. Let t → γ(t) be a C 0 ([0, 1]; C ω (S 2 )) family of contractible 2-spheres in Ω and X ∈ C([0, 1]; C ω (Ω)
) be a family of vector field such that

(21) γ(t) X • ν dσ = 0, then there exists η > 0 and ψ ∈ C 0 ([0, 1]; C ∞ (V η [int(γ(t))]; R)) such that ∀t ∈ [0, 1], ∆ x ψ = 0 in V η [int(γ(t))], (22) ∀t ∈ [0, 1], ∂ψ ∂ν = X • ν on γ(t). (23)
In other words, Lemma 1 expresses that the solution of the Neumann system (24) ∆ x ψ = 0 in int(γ(t)), ∂ψ ∂ν = X • ν on γ(t), can be extended across the boundary γ(t) uniformly in t.

Proof. An equivalent lemma in dimension 2 is given in [START_REF] Glass | Approximate Lagrangian controllability for the 2-d Euler equations. application to the control of the shape of vortex patch[END_REF]. By means of real analytical local charts (see e.g. the analytic inverse theorem in [START_REF] Krantz | A Primer of Real Analytic Functions[END_REF]), γ(t) is locally mapped to the plan {x 3 = 0} by some φ with φ(int(γ(t))) ⊂ {x 3 > 0}. Moreover, we can require that dφ(ν) is normal to x 3 = 0 on {x 3 = 0}.

To obtain this property, consider

φ : (x 1 , x 2 , x 3 ) → (x 1 , x 2 , 0) + x 3 dφ φ -1 (x 1 ,x 2 ,0) (ν).
Then φ is analytic and invertible (locally) in a neighborhood of {x 3 = 0}. Now, replace φ by φ-1 • φ to obtain the requirement. Now call Assume that for some constant A and L we have

g := ∂ x 3 (ψ • φ -1 ); it satisfies a • ∇ 2 g + b • ∇g + cg = 0,
(25) |∇ p a(x)|, |∇ p b(x)|, |∇ p c(x)|, |∇ p f (x)| ≤ LA |p| ,
for any multi-index p. There exists then R ′ < R depending only on N, A, L and R such that y may be extended analytically on B R N (0, r) for any r < R ′ .

Thus for each x ∈ γ(t), ∇ x ψ • ν can be analytically extended on a neighborhood U x of x across γ(t). Integrating ∇ψ • ν along ν we deduce that ψ is real analytic and by unique continuation its extension is also harmonic on U x . Moreover, using the continuity of γ in the variable t from [0, 1] to C ω (S 2 )1 (so that the coefficients satisfy (25) uniformly in time), we can find U x so that ψ is analytically extended on in U x for each s in some neighborhood of t. Using the compactness of ∪ t {(t, γ(t))} we see that we can choose η > 0 uniform in t, such that for all t, ψ can be analytically (and hence harmonically) extended in V η (γ(t)). Lemma 1 follows.

Back to the proof of Proposition 2. We take γ(t) = φ X (t, 0, γ 0 ). Due to the regularity of X, γ(t) is analytic for any t, and applying Lemma 1 we deduce a function ψ. Reducing η > 0 given by Lemma 1 if necessary, we may assume that V η (γ(t)) does not meet ∂Ω.

By compactness of [0, 1], for a given ε > 0 we can choose 0

≤ t 1 < • • • < t N ≤ 1 and δ 1 , . . . , δ N such that [0, 1] ⊂ ∪ N i=1 (t i -δ i , t i + δ i ), ∀t ∈ [t i -δ i , t i + δ i ], γ(t) ⊂ V η/2 (γ(t i )), ∀s, t ∈ [t i -δ i , t i + δ i ], ψ(s, •) -ψ(t, •) C k (Vη(γ(t i ))) ≤ ε. ( 26
)
For each i = 1, . . . , N, we consider

K i := V η/2 [int(γ(t i ))] ∪ V η/2 (∂Ω \ Γ), ψ i := ψ(t i , •).
Reducing η again if necessary, we may assume that the connected component of

Ω \ K i in R 3 \ K i meets R 3 \ Ω. This is possible since the connected component of Ω \ [int(γ(t i )) ∪ (∂Ω \ Γ)] in R 3 \ K i does meet R 3 \ Ω, since Γ = ∅. It follows that each connected component of R 3 \ K i meets R 3 \ Ω.
Now we use the following harmonic approximation theorem (see [START_REF] Gardiner | Harmonic approximation[END_REF]Theorem 1.7]): We choose points Y 1 , ..., Y P in each connected component of R 3 \ K i , outside Ω. We apply the preceding result with O = R 3 \ {Y 1 , . . . , Y P } and K = K i . In that case, O * can be thought as the quotient of S 3 = R 3 ∪ {∞} by the identification of Y 1 , . . . , Y P and ∞.

Therefore, for any ν > 0, we get a map ψi in C ∞ (R 3 \ {Y 1 , ..., Y P }; R) such that ψi is harmonic on R 3 \ {Y 1 , ..., Y P } and such that ψi -

ψ i C k+2 (V η/2 [int(γ(t i ))]) < ν, (27) ψi C k+2 (V η/2 (∂Ω\Γ)) < ν. ( 28
)
Since ψi is harmonic in Ω, there holds

(29) ∂Ω ∇ ψi • n dσ = 0.
In order for (4) to be satisfied we consider d i in C ∞ (∂Ω; R) such that

d i = ∇ ψi • n on ∂Ω \ Γ, ( 30
)
d i C k+1 (∂Ω) ≤ C ∇ ψi • n C k+1 (∂Ω) , ( 31
) ∂Ω d i dσ = 0. ( 32
)
and introduce the harmonic function h i in Ω by the Neumann problem

∆h i = 0 in Ω, (33) ∂h i ∂n = d i in Ω, (34) Ω h i = 0.
Note that in particular that by standard elliptic estimates, (35)

h i C k+1 ( Ω) ≤ Cν.
We introduce

(36) ψi := ψi -h i .
Taking a partition of unity χ i associated to the covering of [0, 1] by the intervals (t i -

δ i , t i + δ i ), we define (37) θ(t, x) := N n=1 χ i (t) ψi (x).
Due to (30) and (34), θ satisfies [START_REF] Temam | Navier-Stokes Equations and numerical analysis[END_REF]. Moreover according to (26), ( 27), ( 28) and ( 35) we have for ν small enough with respect to ε and for some C > 0 (38) sup

t∈[0,1] ∇θ -∇ψ C k (V η/3 [γ(t)]) ≤ Cε.
In particular by supposing ε small enough, we have a uniform estimate (39) ∇θ C k (φ ∇θ (0,t,γ 0 )) ≤ ∇ψ C k (φ ∇ψ (0,t,γ 0 )) + 1.

As long as φ ∇θ (t, 0, γ 0 ) remains in V η/3 (γ(t)) one has, using Gronwall's lemma,

φ ∇θ (t, 0, γ 0 ) -φ ∇φ (t, 0, γ 0 ) ∞ ≤ ∇θ(t, •) -∇φ(t, •) C 0 ([0,1];C 0 (V η/3 [γ(t)]) exp( ∇ψ L 1 (0,1;Lip(V η/3 [γ(t)]) ).
Then by reducing ε if necessary, we get thanks to (38) that this is valid for all time in [0, 1]. Differentiating φ with respect to x up to the order k, we obtain in the same way that for all t ∈ [0, 1],

φ ∇θ (t, 0, γ 0 ) -φ ∇φ (t, 0, γ 0 ) C k ([0,1]) ≤ ∇θ -∇φ C 0 ([0,1];C k (V γ/3 (γ(t))) exp(C ∇ψ L 1 ((0,1);W k+1,∞ (V η/3 [γ(t)])) ).
This ends the proof Proposition 2.

2.2.2.

Case of a smooth 2-sphere moved by a special analytic isotopy. In this paragraph, Proposition 2 is extended to the following.

Proposition 3. The conclusions of Proposition 1 are satisfied if we assume that γ 0 is a smooth 2-sphere (i.e. γ 0 is the image of S 2 by f 0 a C ∞ embedding) and that X is moreover in C 0 ([0, 1]; C ω (Ω; R 3 )).

Proof. Due to a result of H. Whitney (see [START_REF] Whitney | The imbedding of manifolds in families of analytic manifolds[END_REF]), γ 0 is imbedded in a smooth family of surfaces γ ν , ν ∈ (-ν 0 , ν 0 ), with γ 0 = γ ν for ν = 0, γ ν ∩ γ ν ′ = ∅ for ν = ν ′ and γ ν real analytic for ν = 0. It follows that either for ν > 0 or for ν < 0, one has (40)

γ 0 ⊂ int[γ ν ].
Without loss of generality, we assume that (40) holds for ν > 0. The family γ ν being smooth with respect its parameters, one has (41)

γ ν → γ 0 in C ∞ (S 2 ) as ν → 0 + .
We can then apply Proposition 2 to X on γ ν for ν > 0 small. We construct a θ ε such that ( 17), ( 18), ( 19) and (20) apply for γ ν instead of γ 0 . The construction also generates a family ψ satisfying (24) in int[φ X (t, 0, γ ν )]. Now we have uniform bounds on the ∇ψ C k (φ ∇ψ (t,0,γν )) with respect to ν, because the constants of the elliptic estimates (see e. g. [7, Theorem 6.30 and Lemma 6.5]) are uniform with respect to ν thanks to (41). We deduce from (39) that we have uniform bounds on ∇θ ε C k (φ ∇θε (t,0,γν )) as ν → 0 + . By Gronwall's lemma one gets

φ ∇θε (t, 0, γ0 ) -φ ∇θε (t, 0, γ 0 ) C k (S 2 ) ≤ C k γ 0 -γ ν C k (S 2 ) exp 1 0 ∇θ ε C k+1 (int(φ ∇θε (t,0,γν )) dt .
Hence we deduce the claim by taking ν small enough. Of course by reparameterizing X and thus θ in time we can always assume that θ is compactly supported in time.

2.2.3.

Case of smooth embedded contractible two-sphere moved by a smooth isotopy. We now prove Proposition 1.

Proof of Proposition 1. In the general case, we assume that X is merely C ∞ (Ω; R 3 ). Let λ > 0 such that max

t∈[0,1]
dist(φ X (t, 0, γ 0 ), ∂Ω) > 2λ.

We define U t := V λ (int(γ(t))). Reducing λ if necessary, we can obtain that for all t, U t is diffeomorphic to a ball. Now we can use Whitney's approximation theorem (see e.g. [15, Proposition 3.3.9]), for any µ > 0 and any k ∈ N there exists

X µ ∈ C([0, 1]; C ω (R 3 )) such that X µ -X C([0,1];C k+1 (Ut)) ≤ µ.
Moreover, we can ask that div

X µ = 0 in [0, 1] × R 3 .
To see this, we use the fact that U t is a topological ball; hence its second de Rham cohomology space is trivial, and any divergence-free vector field (in particular X) on U t is of the form curl A. Hence for each time t one can apply Whitney's approximation theorem (at order k + 2) on ϕA where X = curl A and ϕ(t, x) is a smooth cutoff function equal to 1 on U t and to 0 for d(x, U t ) ≥ 2λ. We obtain an approximating vector field B µ and define X µ := curl B µ . Using as before the compactness of the time interval [0, 1] and a partition of unity (as for the proof of Proposition 2), we can obtain a smooth approximation uniformly in time. Now we apply Proposition 3 with Xµ , apply Gronwall's lemma and get the result for µ small enough.

Proof of Theorem 1

This section is devoted to the proof of Theorem 1. As in [START_REF] Coron | On the controllability of 2-D incompressible perfect fluids[END_REF][START_REF] Glass | Exact boundary controllability of 3-D Euler equation[END_REF][START_REF] Glass | Approximate Lagrangian controllability for the 2-d Euler equations. application to the control of the shape of vortex patch[END_REF], we start with the case where u 0 is small, and then treat the general case.

3.1.

Preliminaries. First we introduce some functions on Ω in order to take its topology into account, more precisely to describe its first de Rham cohomology space. We recall the following construction.

Let Σ 1 , ..., Σ g be g smooth manifolds with boundaries of dimension n -1 inside Ω such that:

• for all i in {1, . . . , g}, ∂Σ i ⊂ ∂Ω and Σ i is transverse to ∂Ω,

• for all i, j in {1, . . . , g} 2 with i = j, Σ i and Σ j intersect transversally (which, by definition, includes the case of an empty intersection),

• Ω \ ∪ g i=1 Σ i is simply connected. For i = 1, . . . , g, we consider

X i := {p ∈ H 1 (Ω \ ∪ g k=1 Σ k ), [p] i = constant, [p] j = 0, j = i}, where [p] k = p |Σ + k -p |Σ - k
is the jump of p on each arbitrarily fixed side of Σ k in Ω. Then by Lax-Milgram's Theorem there exists a unique q i ∈ X i such that for all p ∈ X i

Ω ∇q i • ∇p dx = [p] i ,
which leads to the existence of a unique p i such that

∆p i = 0 in Ω \ ∪ g k=1 Σ k , ∂ n p i = 0 on ∂Ω, [p i ] i = 1 and [p i ] j = 0 for j = i, [∂ n q i ] i = 0,
and we take Q i := ∇p i , which is regular in Ω. Then we have the following result. Proposition 4 (see e.g. [START_REF] Temam | Navier-Stokes Equations and numerical analysis[END_REF], Appendix I, Proposition 1.1). For any X ∈ L 2 (Ω; R 3 ) such that curl X = 0 in Ω, there exist χ in H 1 (Ω; R) and α 1 , . . . , α g in R such that

(42) X = ∇χ + g i=1 α i Q i .

3.2.

A fixed point operator. Now we introduce an operator, whose fixed point will give a solution of our problem when taking u 0 (suitably small) into account.

We introduce R > 0 such that

Ω ⊂ B R := B R 3 (0, R).
We introduce a linear continuous extension operator π : C(Ω; R 3 ) → C 0 (B R ; R 3 ) such that ∀k ∈ N and all β ∈ (0, 1), π is continuous from C k,β (Ω; R 3 ) to C k,β 0 (B R ; R 3 ). Let δ ∈ (0, 1), and consider µ ∈ C ∞ 0 ([0, +∞); R) with support in [0, δ] and with value 1 on a neighborhood of 0.

Given ε > 0 we denote ȳ := ∇θ, Thus, by combining (49), (50), and the elliptic estimates given by ( 45)-( 46), we infer that (51)

F (u) -ȳ L ∞ (0,1;C k,α (Ω)) ≤ C( y k,α ) u 0 C k,α (Ω) ,
for some constant C depending on k, α and y. It follows that for u 0 C k,α (Ω) small enough, F sends X ν into itself.

• That F (X ν ) is relatively compact is seen easily: given (u n ) ∈ X N ν , the sequence (F (u n )) belongs to X ν and, following the construction, it is easy to see that (∂ t F (u n )) is bounded in L ∞ (0, 1; C k-1 (Ω)). The conclusion follows then from Ascoli's theorem.

• Finally, that F is continuous for the uniform topology can be seen as follows. Let us be given (u n ) ∈ X N ν converging uniformly to u ∈ X ν . The flows Φ n associated to ũn converge uniformly towards the flow Φ associated to ũ. Hence one can see that ω un converges uniformly to ω u . Due to the bounds on ω u , this convergence also takes place in L ∞ (0, 1; C k-1,β (B R )) for any β < α. We deduce in a straightforward manner the convergence of (v n ) and (λ n i ) corresponding to u n towards v and λ i corresponding to u, and the conclusion follows.

It follows that F admits a fixed point u in X ν . This concludes the proof of Proposition 5. 2. Now we prove that ū fulfills the requirements of Theorem 1 for u 0 C k,α small enough. Given x ∈ B R , we have φū (t, 0, x)-φȳ (t, 0, x) = ū(t, φ ū(t, 0, x))-ȳ(t, φ ū(t, 0, x))+ ȳ(t, φ ū(t, 0, x))-ȳ(t, φ ȳ(t, 0, x)).

We deduce easily that for a constant C > 0 depending on y only, | φū (t, 0, x) -φȳ (t, 0, x)| ≤ ν + C|φ ū(t, 0, x)φ ȳ(t, 0, x)|.

Thus by Gronwall's lemma we have |x(t)y(t)| ≤ Cν where C depends only on ȳ. Reasoning in the same way for the derivatives (up to order k) with respect to x of the flows, we obtain φ ũ(t, 0, •)φ y (t, 0, •) C k (B R ) ≤ Cν.

Hence taking ν small enough (and hence u 0 C k,α even smaller), we can obtain [START_REF] Glass | Asymptotic stabilizability by stationary feedback of the two-dimensional Euler equation: the multiconnected case[END_REF] and [START_REF] Guillemin | Differential topology[END_REF] for T = 1. In order words, there exists c > 0, such that for any u 0 in C k,α with u 0 C k,α ≤ c, one can find a solution of the Euler equation for T = 1, satisfying [START_REF] Glass | Asymptotic stabilizability by stationary feedback of the two-dimensional Euler equation: the multiconnected case[END_REF] and [START_REF] Guillemin | Differential topology[END_REF].

Theorem 3 .

 3 with a real-analytic Dirichlet boundary condition given on x 3 = 0 and analytic coefficients a, b and c. We use the following result of Cauchy-Kowalevsky type (see e.g. Morrey [17, Theorem 5.7.1']) Let f , a, b and c be real analytic functions on G R := B R N (0, R) ∩ {(x 1 , ..., x N ), x N ≥ 0}, and y ∈ H 2 (G R ) satisfying a • ∇ 2 y + b • ∇y + c = f in G R and y = 0 on x N = 0.

Theorem 4 .Remark 5 .

 45 Let O be an open set in R N and let K be a compact set in R N such that that O * \ K is connected, where O * is the Alexandroff compactification of O. Then, for each function u which is harmonic on an open set containing K and each ε > 0, there isa harmonic function v in O such that vu ∞ < ε on K.Recall that the Alexandroff compactification of O is obtained by adding a point, say {∞} to O and to consider the topology generated by the open sets of O, and the sets of the form {∞} ∪ (O \ K), with K a subset of O. One may state the same result with the C k norm instead of the uniform one. It suffices to consider a compact K whose interior contains K, apply the above result on K and use standard properties of harmonic functions.

3. 4 . 1 .

 41 Relevance of the fixed point. Call u the fixed point obtained above. Let us first check that u is a solution of the Euler equation in [0, 1] × Ω. From (44), and since ũ = u in [0, 1] × Ω, we deduce that curl (∂ t u + (u • ∇)u) = 0 in [0, 1] × Ω. From (48) and (u • ∇)u = ∇ |u| 22 + (curl u) ∧ u, we see thatΩ (∂ t u + (u • ∇)u) • Q i dx = 0,which together with Proposition 4 proves the claim.

We refer to[START_REF] Krantz | A Primer of Real Analytic Functions[END_REF] for a definition of the topology on this space.

the potential flow obtained by Proposition 1 with X obtained from Theorem 2. For some ν ∈ (0, 1) which will be small in the sequel we define with α ∈ (0, 1):

X ν := u ∈ L ∞ ((0, 1); C k,α (Ω; R 3 )), div u = 0 in (0, 1) × Ω, uȳ L ∞ (0,1;C k,α (Ω)) ≤ ν .

It is straightforward to check that X ν is a closed convex subset of L ∞ ((0, 1) × Ω).

We extend ȳ by π and still denote ȳ the extended function. Now, given u ∈ X ν , we associate F (u) as follows. First, we introduce

)) as the solution of the following transport equation:

Due the support of ũ, one sees by using characteristics that the system (43)-( 44) is wellposed and that indeed ω u has the claimed regularity (details for obtaining the regularity can be found in [START_REF] Glass | Exact boundary controllability of 3-D Euler equation[END_REF]). Now observing that

we easily deduce that div (ω u ) = 0 in B R , so that ω u can be written in the form curl v in B R . Hence it is classical (since

According to Proposition 4, we can determine g time-dependent functions λ 1 , . . . , λ g such that, if we define (46)

we have for all j = 1, . . . , g

and for all t ∈ [0, 1],

(48)

This is possible in a unique way since the matrix ( Ω Q i • Q j dx) i,j is invertible, as a Gram matrix of independent functions. Now we finally define F (u) := V.

3.3.

Finding a fixed point. Our goal is to prove hereafter:

Proposition 5. Given ν > 0, if u 0 C j,α (Ω) is small enough, F admits a fixed point in X ν .

We will use the following lemma (see e.g. [1, Theorem 3.14]):

Lemma 2. Let j ∈ N, a ∈ (0, 1). Let f , v, and g be elements of L ∞ (0, 1;

Then for some C > 0 depending on j and α only, there holds

Proof of Proposition 5. We establish Proposition 5 and find a fixed point of F in X ν via Schauder's fixed point theorem. Accordingly, we prove that, ν being fixed and for u 0 C k,α small enough, F sends X ν into itself, that F is continuous and F (X ν ) is relatively compact for the uniform topology on X ν .

• Using Lemma 2, we see that

with as before

We apply Gronwall's lemma to

and deduce

Thus with the definition of X ν and the continuity of π, we obtain that

Now by (48) we have

and thus, with (47), (49) and the definition of X µ , we deduce (50)

3.

Let us now explain how we can obtain the result without the condition of smallness of u 0 (but for T small enough). Given u 0 ∈ C k,α (Ω), we rescale it by considering v 0 = ρu 0 , with ρ > 0 small enough so that v 0 satisfies v 0 C k,α ≤ c . Applying the above construction to v 0 gives us a solution (u, p) of ( 1)-( 3) defined on t ∈ [0, 1] with u(0,

If we define u ρ by

then u ρ is defined on t ∈ [0, ρ] and φ uρ (ρ, 0, γ 0 )γ 1 < ε.

This concludes the proof of Theorem 1.

Ceremade, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France E-mail address: glass@ceremade.dauphine.fr Université de Versailles, 45 avenue des Etats-Unis, F78030 Versailles, France E-mail address: horsin@math.uvsq.fr