Pistocchi Chiara
email: chiarapi@ethz.ch

Federica Tamburini

Gerard Gruau

André Ferhi

Dominique Trevisan

Jean-Marcel Dorioz

Tracing the sources and cycling of phosphorus in river sediments using oxygen isotopes: methodological adaptations and first results from a case study in France

Keywords: Phosphorus transfer, Agriculture, In-stream processes, Eutrophication, Phosphate oxygen isotopes

An essential aspect of eutrophication studies is to trace the ultimate origin of phosphate ions (P-PO 4 ) associated with the solid phase of river sediments, as certain processes can make these ions available for algae. However, this is not a straightforward task because of the diversity of allochthonous and autochthonous sources that can supply P-PO 4 to river sediments as well as the existence of in-stream processes that can change the speciation of these inputs and obscure the original sources. Here, we present the results of a study designed to explore the potentials, limitations and conditions for the use of the oxygen isotope composition of phosphate (δ 18 Op) extracted from river sediments for this type of tracing. We first tested if the method commonly applied to soils to purify P-PO 4 and to measure their δ 18 Op concentrations could be adapted to sediments. We then applied this method to a set of sediments collected in a river along a gradient of anthropogenic pressure and compared their isotopic signatures with those from

Introduction

Phosphorus (P) associated with the solid phase and designated as "particulate-P" often accounts for the major part of phosphorus transferred from terrestrial to aquatic ecosystems (Kleinman et [START_REF] Kronvang | Phosphorus losses at the catchment scale within Europe: an overview[END_REF][START_REF] Sharpley | Sources of phosphorus exported from an agricultural watershed in Pennsylvania[END_REF]. The impact of particulate-P on the trophic state and eutrophication of the receiving aquatic ecosystems, especially lentic ecosystems, depends on the size of its potentially bioavailable fraction. The amount of P that occurs as inorganic phosphate ions (P-PO 4 pools), either in solution (dissolved P-PO 4 ), sorbed onto, or precipitated within stream and river sediments, is thought to represent the most readily bioavailable P pool [START_REF] Moorleghem | The bioavailability of colloidal and dissolved organic phosphorus to the alga Pseudokirchneriella subcapitata in relation to analytical phosphorus measurements[END_REF]. Therefore, an essential aspect of studies dedicated to the restoration of eutrophicated aquatic ecosystems is to quantify P-PO 4 pools in river water and sediments and to trace the ultimate origin of these pools [START_REF] Kleinman | Managing agricultural phosphorus for water quality protection: principles for progress[END_REF][START_REF] Mellander | Flow paths and phosphorus transfer pathways in two agricultural streams with contrasting flow controls[END_REF].

However, this quantification is not a straightforward task because of the diversity of the allochthonous and autochthonous sources that may supply P-PO 4 to river sediments. Several allochthonous compartments in the catchment can supply particulate-P to river sediments, including pristine and weathered basement rocks, forested or agricultural soils, riverbank sediments, etc. For example, the autochthonous addition of P-PO 4 to the sediments can occur through sorption onto the sediments of dissolved P-PO 4 ions supplied by sewage effluents or the in-stream mineralization of sewage organic matter [START_REF] Poulenard | Analytical Electron-Microscopy Fractionation of Fine and Colloidal Particulate-Phosphorus in Riverbed and Suspended Sediments[END_REF]. The in-stream retention of dissolved P-PO 4 by the sediments primarily occurs during low-flow periods when the retention time of the sediment is long enough to allow the necessary physical-chemical reactions to develop (Dorioz et al., 1998a[START_REF] Dorioz | Dynamique du phosphore dans les bassins versants: Importance des phenomenes de retention dans les sediments[END_REF][START_REF] Ensign | Nitrogen and phosphorus attenuation within the stream network of a coastal, agricultural watershed[END_REF][START_REF] Pistocchi | A Simple Model to Assess Nitrogen and Phosphorus Contamination in Ungauged Surface Drainage Networks: Application to the Massaciuccoli Lake Catchment, Italy[END_REF] and is particularly intense immediately downstream of the point source pollution inputs.

Autochthonous P-PO 4 pools can also come from the recycling of allochthonous particulate-P by aquatic flora or by microorganisms in the sediment. The biological uptake by biofilm is another important seasonal process that is involved in the retention of P-PO 4 ions by the sediment [START_REF] Wang | Importance of Instream Nutrient Storage to P Export from a Rural, Eutrophic River in Vermont, USA[END_REF]. These in-stream recycling and retention processes, which help the P to turn over within the river ecosystem, are expected to be triggered by the bioavailability of M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 4 allochthonous P and to regulate the internal productivity of the river system. Their quantification can be used to assess the efficiency of a river system to recycle and transform any particulate P into more bioavailable P-PO 4 , but an efficient recycling can eventually obscure the ultimate origin of the P-PO 4 exported at the outlet.

It is often impossible to quantify and trace the different sources of particulate-P that are potentially involved in eutrophication using only the variability in P speciation as determined by classical chemical extraction methods [START_REF] Burrus | Recovery and concentration of suspended solids in the upper rhone river by continuous flow centrifugation[END_REF][START_REF] Pacini | Speciation of riverine particulate phosphorus during rain events[END_REF][START_REF] Poulenard | Infrared spectroscopy tracing of sediment sources in a small rural watershed (French Alps)[END_REF][START_REF] Walling | Tracing suspended sediment and particulate phosphorus sources in catchments[END_REF]. As a result, there a need to develop tools or/and strategies that can be reliably used to trace this important P compartment, particularly its dominant inorganic P-PO 4 fraction, and to understand the extent and rate at which particulate-P is biologically recycled and modified within streams and rivers.

Oxygen isotopes could provide one such possible tool. The oxygen isotopic composition of phosphate (δ 18 Op) has already been successfully used to trace P sources and study the P-PO 4 cycle in soils, freshwater and marine sediments [START_REF] Davies | Phosphate oxygen isotopes within aquatic ecosystems: Global data synthesis and future research priorities[END_REF][START_REF] Goldhammer | Phosphate oxygen isotopes: Insights into sedimentary phosphorus cycling from the Benguela upwelling system[END_REF][START_REF] Gooddy | A multi-stable isotope framework to understand eutrophication in aquatic ecosystems[END_REF][START_REF] Gruau | The oxygen isotope composition of dissolved anthropogenic phosphates: a new tool for eutrophication research[END_REF][START_REF] Melby | Biological decay of 18O-labeled phosphate in soils[END_REF][START_REF] Tamburini | Oxygen Isotopes for Unraveling Phosphorus Transformations in the Soil-Plant System: A Review[END_REF]. This approach is based on: i) the existence of measurable differences in the abundance ratio for 18 O and 16 O atoms in the phosphate group of potential P-PO 4 sources (e.g. primary P-rich minerals, chemical fertilizers, chemical detergents, phosphate from wastewater, etc.); and ii) the fact that biologically mediated processes have the possibility to modify δ 18 Op in a predictable way [START_REF] Blake | Biogeochemical cycling of phosphorus: insights from oxygen isotope effects of phosphoenzymes[END_REF]. The first property allows the oxygen isotope composition of P-PO 4 to be potentially used as a source tracer in environmental systems, whereas the second property allows it to be used to quantify the rate of biological recycling of P-PO 4 . Two enzymatic processes are mainly responsible for changes in the δ 18 Op value. One is the isotope equilibration between oxygen in P-PO 4 and oxygen in ambient water. This process, which is caused by reversible phosphorylation/dephosphorylation reactions that are catalyzed by the [START_REF] Blake | Biogeochemical cycling of phosphorus: insights from oxygen isotope effects of phosphoenzymes[END_REF], is temperature dependent and has been described by an empirical equation [START_REF] Longinelli | Revised phosphate-water isotopic temperature scale[END_REF] that was recently revised by [START_REF] Chang | Precise calibration of equilibrium oxygen isotope fractionations between dissolved phosphate and water from 3 to 37°C[END_REF]. The second is the kinetic isotope fractionation produced by the enzymatic hydrolysis of phosphoesters [START_REF] Liang | Oxygen isotope signature of Pi regeneration from organic compounds by phosphomonoesterases and photooxidation[END_REF][START_REF] Sperber | The oxygen isotope composition of phosphate released from phytic acid by the activity of wheat and Aspergillus niger phytase[END_REF][START_REF] Sperber | The effect of phosphomonoesterases on the oxygen isotope composition of phosphate[END_REF], the magnitude of which depends on the type of enzyme involved (e.g. acid or alkaline phosphatases) and the final effect of which on the isotopic signature of the recycled P-PO 4 also depends on the δ 18 Op value for the original substrate, i.e. inheritance effect [START_REF] Liang | Oxygen isotope signature of Pi regeneration from organic compounds by phosphomonoesterases and photooxidation[END_REF]. Small kinetic isotope effects are associated with the initial stages of the abiotic reaction, such as the sorption of P-PO 4 onto the solid phases (Jaisi et al., 2010;Melby et al., 2013b) or biological uptake [START_REF] Blake | Biogeochemical cycling of phosphorus: insights from oxygen isotope effects of phosphoenzymes[END_REF].

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 5 inorganic intracellular pyrophosphatase enzyme
In freshwater ecosystem studies, the use of δ 18 Op is at an early stage [START_REF] Davies | Phosphate oxygen isotopes within aquatic ecosystems: Global data synthesis and future research priorities[END_REF]. For river sediments, the lack of a standardized method to extract, purify and analyze δ 18 Op is a hinderance. In this study, we first adapted the method that is commonly used for soils to sediments and validated the capability of this method to provide precise and accurate results.

We then applied this method to riverbed sediments from a river in the French Alps with the following objectives: i) to characterize the variability of the δ 18 Op signatures in the riverbed sediments and the potential external sediment sources (soils, riverbank materials) in an exemplar river system; ii) to explore the potential, limitations and conditions of using the isotopic tool to assess the respective role of external inputs and internal recycling processes to build up the P-PO 4 budget of the riverbed sediments in this river.

Materials and methods

Site description and sampling

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The Redon River is a small alpine torrential tributary (i.e. a shallow high-gradient, high-energy stream with an annual average discharge of 0.5 m 3 /s at the outlet) of Lake Geneva (France).

The sediment dynamic of this river has been already intensively studied [START_REF] Al-Charideh | Contribution à l'étude hydrologique du bassin du Redon, Haute-Savoie : relation pluie-débit et quelques aspects isotopiques[END_REF]Dorioz et al., 1998b;[START_REF] Feuillade | Enzymatic release of phosphate in sediments of various origins[END_REF][START_REF] Hamid | Etude des transferts d'eau à l'intérieur d'une formation morainique dans le bassin du Léman-Transferts d'eau dans la zone non saturée[END_REF][START_REF] Poulenard | Analytical Electron-Microscopy Fractionation of Fine and Colloidal Particulate-Phosphorus in Riverbed and Suspended Sediments[END_REF] making it a particularly appropriate site for the present exploratory study. The total catchment surface area is 3,300 ha with a land use that consists mainly of forests (1,100 ha) in its upland part and grasslands (1,200 ha) and cultivated farmlands (300 ha) in its lowland part. Several wetlands (20 ha) are present, while urbanized areas are concentrated near the outlet, close to the shore of Lake Geneva. The Redon River is impacted by both point and diffuse sources of P.

The main point source of P is located in the lower part of the river and consists of inputs of predominantly dissolved P from a wastewater treatment plant (WWTP), whereas the main diffuse sources occur in the upper part of the catchment, consisting mainly of particulate-P from agricultural and forested lands (Fig. 1).

The soils are mainly Cambisols, formed from the weathering products of quaternary moraine material consisting of a large range of mainly calcareous sedimentary, metamorphic and igneous rocks from the Rhône glacier [START_REF] Nicoud | Les complexes détritiques de marge glaciaire active du Bas Chablais oriental : implications hydrogéologiques[END_REF]. The soil mineralogy is comprised of a mixture of minerals inherited from the weathering of this heterogeneous parent material and their secondary weathering products. Previous studies (Dorioz et al., 1998a(Dorioz et al., , 1998b) ) have shown that the Redon River sediments are CaCO 3 -rich (10 to 20% CaCO 3 , pH > 7). Their fine (< 200 µ) fraction is dominated by loamy and sandy materials, with an organic matter content between 2 to 4.5%. The clay fractions (7-11%) consist mostly of soil-derived illite, biotite and chlorite [START_REF] Poulenard | Analytical Electron-Microscopy Fractionation of Fine and Colloidal Particulate-Phosphorus in Riverbed and Suspended Sediments[END_REF]. The water in the Redon River is bicarbonated-calcic.

[Figure 1]

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Sampling strategy and sample preparation

Details regarding the sampling sites and sampling conditions are given in Tables 1 and2. Three types of samples were collected: i) eight riverbed sediments collected at six locations (A, B, C, D, E and F in Fig. 1) along a downstream gradient of land use that changes from forested to mixed agricultural/urban areas, thus representing a gradient of increasing anthropogenic pressure; this includes two samples collected in streamlets draining a small agricultural subcatchment (locations A and B in Fig. 1) that is representative of the type of agriculture practiced in the Redon River catchment [START_REF] Dorioz | Pollution diffuse et gestion du milieu agricole: Transferts compares de phosphore et d'azote dans un petit bassin versant agricole[END_REF]; ii) four samples representative of the potential sources of river sediments, including cultivated topsoil and deep horizon, riverbank material, and soil from a 2 ha wetland, hydraulically connected to the river; and iii) two samples from sediments from Lake Geneva representing an average reference for the local superficial lacustrine sediments (collected in 2012 and provided by the EDYTEM laboratory).

Given that it was difficult to access the river bed, it was not possible to visit all six riverbed sediment sampling locations at the same time. Three locations were first sampled at the end of March 2012 (B, C and F), and five (A, B, D, E and F) were sampled (again) at the beginning of September 2012. Both periods were low-flow periods, where the September period corresponds to the lowest weekly discharge period of the year. Samples were taken in areas of preferential accumulation of fine sediments. At each location, single samples of the entire sediment matrix were taken manually at 3-5 different points and then mixed to obtain an average sample (0.5-1 kg dry weight). Simultaneously, the electric conductivity (EC) was measured and a water sample was collected to determine the SRP (soluble reactive phosphorus) concentration and water oxygen isotope composition (δ 18 O w ). Hourly water temperatures were obtained from automatic recorders at the outlet. The riverbank sample was collected in the upstream, forested part of the catchment (see Fig. 1). The cultivated soil samples were collected in the same small

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8 sub-catchment used to sample the streamlet sediments (Fig. 1), as this sub-catchment is known to represent zones of maximum soil erosion risk in the Redon catchment [START_REF] Dorioz | Pollution diffuse et gestion du milieu agricole: Transferts compares de phosphore et d'azote dans un petit bassin versant agricole[END_REF]. In addition to taking a topsoil sample, a deeper soil sample was taken at a depth of approximately -70 cm to account for the possible erosion of deep soil materials along ditches or/and soil embankments. These two samples were obtained by pooling five subsamples (200-300 g) collected in different locations within a single field or its embankment. The pooled samples were then air-dried and sieved at 200 µm before analysis. No river water samples were collected for the analysis of the δ 18 Op values for the dissolved P-PO 4 , although these could have been useful in the interpretation of the riverbed sediment data. The decision not to sample the river water was due to the very low concentrations of dissolved P-PO 4 recorded in this river (Table 1), which makes it extremely difficult to analyze the δ 18 Op in the current state of the analytical protocols.

[Table 1]

[Table 2]

Sediment and soil P chemical analyses

The total phosphorus (Pt) and Olsen P (P Olsen ) concentrations were determined in all of the samples. P Olsen [START_REF] Olsen | Residual phosphorus availability in long-term rotations on calcareous soils[END_REF], which is considered as being representative of the bioavailable fraction of P in soils, has been shown to correlate linearly with P extracted from the sediment in bio-assays with algae (Jordan-Meille, 1998). The Pt concentrations were measured after a hot sodium hydroxide digestion. The total inorganic and organic P in the samples were determined according to [START_REF] Saunders | Observations on the Determination of Total Organic Phosphorus in Soils[END_REF] (1974). The spectrophotometric determination of P-PO 4 in all of the extracts/digestates was carried out using the molybdenum blue method (Murphy and Riley, 1962) at 882.0 nm on a Cary 50 UV-Vis spectrometer (Agilent).

Isotopic analyses and data treatment

Extraction and purification protocol

We used the extraction and purification protocol initially developed by [START_REF] Tamburini | A method for the analysis of the δ18O of inorganic phosphate extracted from soils with HCl[END_REF] for soils (hereafter referred as to Tb10), which we adapted for sediments. The Tb10 protocol consists of the following six steps: i) extraction of phosphate from the soil/sediment with 1 M HCl, which in sequential extraction mainly targets the P-PO 4 pool (HCl-P) associated with Ca, but in our case (singular extraction) also P sorbed onto Fe and Al oxides;

ii) dissolved organic matter (DOM) removal from the solution using an Amberlite resin, iii) ammonium phospho-molybdate (APM) precipitation and subsequent dissolution, iv) magnesium ammonium phosphate (MAP) precipitation and dissolution, v) cation removal by a cation exchange resin (BioRad, AG50 x8), vi) silver phosphate (Ag 3 PO 4 ) precipitation through the addition of an Ag-ammine solution to the purified phosphate solution.

The goal of these steps, after HCl extraction, is to avoid sources of oxygen (i.e. DOM, nitrate, etc.) other than P-PO 4 and other contaminants, e.g. chloride, in the final silver phosphate precipitate.

We modified the Tb10 protocol in several ways as follows (Fig. 2 in the supplementary material). First, the samples were sieved at 200 µm instead of 2 mm, since this finer fraction contains the bulk of the bio-geochemically active P in river sediments [START_REF] Poulenard | Analytical Electron-Microscopy Fractionation of Fine and Colloidal Particulate-Phosphorus in Riverbed and Suspended Sediments[END_REF].

The liquid-to-solid ratio in the extraction step was modified to 100:1 instead of 50:1, since the

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10 pre-tests showed that, with our highly carbonated sediments, the solution did not reach the targeted pH (1 ± 0.5). The quantity of sediment was adjusted to have at least 10 µmoles of P for the purification, corresponding to 10 to 20 g of sieved sediments. The DOM removal step using an Amberlite resin was not performed as the subsequent steps successfully eliminated the organic matter (see below). Finally, for samples with a high calcium carbonate content, an additional step had to be introduced to account for the formation of a calcium molybdate precipitate during the APM precipitation. To eliminate the precipitate, we repeated the subsequent MAP precipitation step twice.

The 1 M HCl extraction may hydrolyze some of the organic or condensed P [START_REF] Turner | Extraction of soil organic phosphorus[END_REF] and the released Pi may bias the isotopic composition of the inorganic P pool targeted here. To assess the existence of such a possible artifact, extractions were run in duplicate using 18 O-labeled and unlabeled reagents. An 18 O-labelled 1M HCl solution was prepared by adding approximately 0.05 ml of 18 O-enriched water (97‰, Aldrich) to 1 liter of 1 M HCl solution.

Labeled and unlabeled solutions were analyzed for δ 18 O w . If hydrolysis occurred, it should result in different isotopic compositions depending on whether labeled or unlabeled reagents were used, i.e. a higher δ 18 O p for the labeled sample compared to the unlabeled one. In this case, we calculated the "real" isotopic value by a revised version of the mass balance equation used by McLaughlin et al. (2006b):

ܱ81ߜ = (ഃభఴೀ • ഃభఴೀ ೢ ష ഃభఴೀ • ഃభఴೀ ೢ ) (ഃభఴೀ ష ഃభఴೀ ష ഃభఴೀ ೢ శ ഃభఴೀ ೢ ) (1)
where δ 18 O pl and δ 18 O pnl are the isotopic composition of P-PO 4 for the labeled and unlabeled samples, respectively, and δ 18 O wl and δ 18 O wnl are the isotopic composition of water for the labeled and unlabeled reagents, respectively.

The efficiency and reliability of the modified Tb10 protocol used here were evaluated by considering the repeatability of the extraction/purification process, the final Ag 3 PO 4 yield and its purity. The purity was assessed by i) dissolving the obtained Ag 3 PO 4 and comparing the

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11 actual concentration of phosphate in the solution with the theoretical concentration, i.e. the amount expected if all of the solid mass was Ag 3 PO 4 , and ii) determining the C and N concentration of the obtained Ag 3 PO 4 on the elemental analyzer (Flash 2000, Termo Scientific); these two concentrations are used as indicators of possible contamination by organic matter and/or nitrate. The purity of the obtained Ag 3 PO 4 crystals was also assessed through quantification of the oxygen yield during mass spectrometry analyses. This was done by graphically comparing the weight of the Ag 3 PO 4 standards and samples introduced into the machine against the peak area of the produced CO ("oxygen yield"), which is proportional to the oxygen content in the sample. If other sources of oxygen or contaminants were present in the sample, the oxygen yield should be different from the standard Ag 3 PO 4 . P-PO 4 losses at each step of the purification were also quantified (see [START_REF] Pistocchi | Développement d'une méthode d'extraction et purification des phosphates à partir de matrices sédimentaires pour l'analyse isotopique de leur oxygène[END_REF] for more details).

Isotopic analyses and calculation of the theoretical equilibrium values

The oxygen isotope composition of the purified Ag 3 PO 4 was measured using a thermal conversion elemental analyzer (Variopyro Cube Elementar), coupled to an IsoPrime100 isotopic ratio mass spectrometer (IRMS) at the Plant Nutrition Group laboratory (ETH Zurich).

Two to three replicates of each sample were weighed (300 to 600 µg) in silver capsules with a small added amount of glassy carbon powder. The temperature of the TC/EA furnace was kept at 1450°C for pyrolysis.

During each run, repeated measurements of an internal standard (Acros Organics Ag 3 PO 4 > 97.5%, δ 18 O p = +14.2‰) and two international benzoic acid standards (IAEA 601 and IAEA 602, δ 18 O = +23.3‰ and +71.4‰, respectively, IAEA, Vienna, Austria) were included and used to calibrate and correct the instrumental drift. The δ 18 O p values presented in this paper are reported in the standard δ notation with respect to VSMOW (Vienna Standard Mean Oceanic Water), i.e. as ‰ deviation values. Analytical uncertainties, as determined from the replicate analysis of the standards, were estimated to be below 0.3‰.

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 12

In order to assess the extent to which the sedimentary P-PO 4 pools investigated in this study could be secondary recycled P-PO 4 (i.e. P-PO 4 that has been cycled by the river biota), a theoretical equilibrium δ 18 O p value -i.e. the oxygen isotopic composition the phosphate should have if it is in equilibrium with the ambient water -was calculated for each sample using the following equation [START_REF] Chang | Precise calibration of equilibrium oxygen isotope fractionations between dissolved phosphate and water from 3 to 37°C[END_REF]:

ߜ18ܱ = (ߜ18ܱ ௪ + 10 ଷ ) ⋅ ݁ ቈቆభర.రయ⋅ భబ య ቇషమల.ఱర భబ య -10 ଷ (2)
where δ 18 Op is the calculated oxygen isotopic composition of the P-PO 4 sample at equilibrium in ‰, δ 18 O w is the measured oxygen isotopic composition of the water in ‰, and T is the temperature in degrees Kelvin.

Considering the stability of the δ 18 O w values during low-flow periods in the Redon River due to the dominance of the ground water contribution to the discharge during these periods [START_REF] Al-Charideh | Contribution à l'étude hydrologique du bassin du Redon, Haute-Savoie : relation pluie-débit et quelques aspects isotopiques[END_REF], we used the δ 18 O w of the river water measured at each sampling date as the reference isotopic value for water in equation 2. We considered both the minimum and maximum temperatures measured over the period running between the sampling date and the closest big storm flow event (around 10-20 days; Table 1 in the Supplementary Material), since the sediments are (partially) renewed at each storm flow. The temperatures measured at the time of sampling fell within these ranges. For samples 9 to 12 (sources), we used the δ 18 O w values measured in the local water sources, including rain, groundwater and soil water data, and the extremes for the soil temperature measured within the previous 10 days at an agrometeorological station close to the studied sites. This allowed us to have a conservative and time-integrated estimate of the equilibrium interval. For these samples, we also calculated the deviation from equilibrium using the average stream water temperature and isotopic composition, to verify whether or not their δ 18 O p values would fall within the equilibrium range once transferred into the river. For each sample, the deviation from equilibrium was calculated

Method validations

Repeated extractions and purifications (n = 3 or 4) on the same sediment sample yielded δ 18 Op values with a standard deviation < 0.4 ‰. This repeatability can be considered as acceptable, since it is within the range of the precision obtained on the repeated standards measurements on the TC/EA-IRMS. sediments sampled immediately downstream from the WWTP discharge. These variations are the consequence of differences in the source, erosion and deposition conditions of the sediments as well as differences in the P inputs that accumulate along the river (Table 1). The downstream sediments (samples 5 to 8) and streamlet sediments (samples 1 and 2) were enriched in Pt, Pi and P Olsen compared to the sediments coming from the head of the catchment (samples 3) or to the riverbank material (sample 12). The same relative enrichment is observed for the cultivated topsoil (sample 10) compared to its lower horizon (sample 11).

Pt and P Olsen were weakly and positively correlated (r² = 0.49, p < 0.01) as well as the P Olsen /Pi ratio with Pt concentrations of r ²= 0.76 (p < 0.01), providing evidence that the P enrichment process recorded by the downstream sediments is accompanied by an increase in the most reactive fraction of the total P-PO 4 pool. The highest Po concentrations were found in the topsoil samples (9 and 10).

Measured isotopic signatures and comparison with the equilibrium values

The 3). This sample is representative of both the principal source of the upstream, unpolluted sediments and the parent material of most of the soils in this catchment [START_REF] Cipel | Rapport de l'étude des pollutions diffuses dans le bassin lémanique[END_REF] Interestingly, the variability in the δ 18 O p values for the allochthonous sources that may possibly contribute toward the river particulate-P load was higher than the variability in the riverbed sediments. Whereas while the soil, riverbank and riverbed sediments collected in the head of the catchment (samples 1 to 3 and 9 to 12) are representatives of these sources [START_REF] Dorioz | Pollution diffuse et gestion du milieu agricole: Transferts compares de phosphore et d'azote dans un petit bassin versant agricole[END_REF] and exhibit δ 18 O p values varying by > 6 ‰ (from +11.8 to +18.3‰), this variation is reduced to approximately 2‰ for the sediments immediately upstream/downstream the WWTP and to 1‰ for the outlet sediments (+12.8 to +13.8‰; Fig. 2a), the δ 18 O p signatures of which were comparable to those recorded by the lake sediments (samples 13 and 14; Table 3).

[Figure 2]

We compared the measured δ 18 O p values with the theoretical equilibrium calculated with Equation 2 (see Table 2). The deviation between the calculated and measured δ 18 O p values is presented in Fig. 2b. The δ 18 O p value for the riverbank moraine sample (sample 12) was above the calculated equilibrium by ca. 2‰, as was the wetland soil sample (9). This was not the case for the cultivated soil samples that showed δ 18 O p values that were either very close to (-0.1‰, The riverbed sediment samples collected in the streamlets of the agricultural sub-catchment (samples 1, 2) exhibited δ 18 O p values that were lower than the calculated equilibrium values by ca. 2‰. For the sediments located within the main hydrogeological network, the differences between the measured and calculated equilibrium values generally tended to be lower (see Table 3 and Fig. 2b): ca. ± 1.5‰ in the middle part, upstream and downstream from the WWTP discharge point (samples 4 to 6) and < 0.5‰ for the sediments collected at the outlet (samples 7 and 8) and the lacustrine sediments (samples 13 and 14).

Discussion

Significance and variability of the measured and equilibrium δ 18 O p values

The present study shows that the method originally established and optimized by [START_REF] Tamburini | A method for the analysis of the δ18O of inorganic phosphate extracted from soils with HCl[END_REF] for soils is suitable for CaCO 3 -rich riverbed sediment samples, following some modifications to the methodology. However, as for soils, it should be kept in mind that the extraction is not entirely specific with regards to the nature of the extracted P-PO 4 . First, 1 M HCl can be used to extract the fraction that is primarily composed of Ca-bound P, which can contain a broad spectrum of P compounds (more or less crystallized apatite forms, CaCO 3bound P, etc.). This reagent can also extract some P-PO 4 sorbed onto Fe and Al oxides, if used as a single extractant. Thus, the δ 18 O p signatures obtained with this method correspond to a pooled P-PO 4 isotope signature of a mixture of different P fractions with a different degree of stability, exchangeability and bioavailability [START_REF] Jarvie | Phosphorus sources, speciation and dynamics in the lowland eutrophic River Kennet, UK. Science of The Total Environment, Water quality functioning of lowland permeable catchments:inferences from an intensive study of the River Kennet and upper River[END_REF] rather than to the signature of a geochemically specific P-PO 4 compartment. Despite this limitation, the oxygen isotope composition of P-PO 4 extracted from sediment using HCl can provide information that is not

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17 available with other methods, including the extent to which a P pool has been biologically processed or derived from primary inputs or secondary anthropogenic sources [START_REF] Tamburini | Oxygen Isotopes for Unraveling Phosphorus Transformations in the Soil-Plant System: A Review[END_REF]Davis et al., 2014).

The calculation of the equilibrium δ 18 O p values is another critical point for the interpretation of the δ 18 O p isotope data. Information regarding the kinetics of the equilibrium is only derived from cell-free or soil incubation experiments [START_REF] Blake | Biogeochemical cycling of phosphorus: insights from oxygen isotope effects of phosphoenzymes[END_REF][START_REF] Melby | Biological decay of 18O-labeled phosphate in soils[END_REF], but not from conditions that are close to the natural state. Our strategy was to take samples during lowflow periods because of the stability of the δ 18 O w value and the temperature, but there is still the question of how to choose appropriate time points when calculating the theoretical equilibrium values with equation 2. This is also relevant for soils where sharp variations in the δ 18 O w value in the soil surface layers due to the progressive evaporation of the soil water [START_REF] Hsieh | Oxygen isotopic composition of soil water: Quantifying evaporation and transpiration[END_REF] Except for a small number of studies [START_REF] Elsbury | Using Oxygen Isotopes of Phosphate To Trace Phosphorus Sources and Cycling in Lake Erie[END_REF][START_REF] Gooddy | A multi-stable isotope framework to understand eutrophication in aquatic ecosystems[END_REF][START_REF] Markel | Phosphorus cycling and phosphorus sources in Lake Kinneret: tracing by oxygen isotopes in phosphate[END_REF]McLaughlin et al., 2006a), freshwater ecosystems have been poorly explored in terms of δ 18 O p . Our results expand the global δ 18 O p library for these systems, comparing, for the first time, the oxygen isotope signatures of HCl-P extracted from river sediment samples from upstream to downstream with those of the main potential allochthonous sources in the catchment. The isotope signatures of the analyzed river sediments, ranging from +12.2 to +15.8‰, lie in the interval reported in previous studies, i.e. +8.6 to 22.5 [START_REF] Markel | Phosphorus cycling and phosphorus sources in Lake Kinneret: tracing by oxygen isotopes in phosphate[END_REF],

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+12.4 to 23.5‰ [START_REF] Davies | Phosphate oxygen isotopes within aquatic ecosystems: Global data synthesis and future research priorities[END_REF] and +9.7 to +20.0‰ [START_REF] Tamburini | Oxygen Isotopes for Unraveling Phosphorus Transformations in the Soil-Plant System: A Review[END_REF]. The same is true for our soil samples [START_REF] Amelung | The δ 18 O signatures of HCl-extractable soil phosphates: methodological challenges and evidence of the cycling of biological P in arable soil: The δ 18 O signatures of HCl-extractable soil phosphates[END_REF][START_REF] Tamburini | Oxygen Isotopes for Unraveling Phosphorus Transformations in the Soil-Plant System: A Review[END_REF][START_REF] Zohar | Phosphorus dynamics in soils irrigated with reclaimed waste water or fresh water -A study using oxygen isotopic composition of phosphate[END_REF]. 3), as in other cultivated soils, is the long-term result of the accumulation and transformation of P favoring a high P bioavailability, which supports the assumption of an intense biological activity.

Biological

The most common fertilizers used in the area have δ 18 O p values that are higher (16.2-22.5‰) or lower (9.5‰; Guyard 2010) than the values for the studied agricultural soils. Even if we cannot exclude that the observed equilibrium values were only the result of a mixing between the parent material (close to +15.9‰) and a lighter fertilizer input, we assume that the fertilizer has little influence because the fertilizer inputs are almost totally cycled over the long-term (> 1 year) and the amount applied is too small (2-3% of the HCl-P soil stock) over the short-term (one season).

The high and above-equilibrium δ 18 O p value in the wetland surface soil (sample 9) can also reflect the impact of biota, e.g. plants. [START_REF] Pfahler | 18 O enrichment in phosphorus pools extracted from soybean leaves[END_REF][START_REF] Tamburini | Oxygen Isotopes for Unraveling Phosphorus Transformations in the Soil-Plant System: A Review[END_REF]. This suggests that the extracted fraction might partially originate from the release of P from plant litter, either directly or via decomposition. A significant fraction of this enriched P pool that is not taken up by microbes [START_REF] Liang | Oxygen isotope signature of Pi regeneration from organic compounds by phosphomonoesterases and photooxidation[END_REF][START_REF] Oberson | Microbial turnover of phosphorus in soil[END_REF] might have precipitated or been sorbed into the soil mineral pool. Interestingly, this sample had a particularly high Po concentration. Preferential sorption/desorption reactions with soil minerals could also have helped increase the δ 18 O p value for this sample. However, previous studies have reported both negative and positive fractionation associated with sorption (Jaisi et al., 2010;Melby et al., 2013b). In any case, this effect is probably too small (<1 ‰) to justify the observed value by itself.

When compared to the equilibrium values calculated using the average river conditions (the values in brackets in Table 3, Fig. 2b), the subsoil/embankment sample and the wetland topsoil are no longer found within the equilibrium range. The situation is different for the agricultural topsoil sample, the actual δ 18 O p value of which falls within the equilibrium range, meaning that under these conditions the oxygen isotopic composition alone would not help to distinguish between a sedimentary P-PO 4 pool inherited from this soil and in-stream recycled P-PO 4 .

River sediments upstream and downstream from the wastewater treatment plant discharge

The September sampling showed that the input of P from wastewater was marked by a threefold increase in the SRP concentration in the river water sampled upstream and

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ACCEPTED MANUSCRIPT 20 downstream from the discharge point (Table 1), accompanied by a drastic change in the Pt, Pi and P Olsen concentrations in the corresponding sediments (Table 3). No changes were detected in the Po content, meaning that the impact of the point discharge on this pool in the bed sediments was negligible (Dorioz et al., 1998a). Interestingly, despite the small distance (< 100 m) separating the upstream and downstream sites, the δ 18 Op values were also distinct, with a difference of 3‰. Upstream from the WWTP discharge point, the P pools in the sediment are mainly impacted by sources comprising a mixture of riverbanks and agricultural soil erosion (materials similar to samples 12 and 1, 2). The δ 18 Op value (12.2‰) and P speciation for sample 4 were intermediate between those of these source materials.

The two September samples (4 and 5, immediately upstream and downstream from the WWTP) deviate from the isotopic equilibrium values (by -1.5 and +1.4‰, respectively; Fig. 2b),

suggesting the existence of conditions that prevent the complete biological turnover of P at both sites (also see [START_REF] Gooddy | A multi-stable isotope framework to understand eutrophication in aquatic ecosystems[END_REF]. Under the assumption that biological equilibration is the dominant process during sewage treatment [START_REF] Gruau | The oxygen isotope composition of dissolved anthropogenic phosphates: a new tool for eutrophication research[END_REF], the δ 18 Op value for dissolved Pi from the WWTP would vary between 13.0 and 14.8‰ using equation 2 (confirmed by the value of 14.0‰ measured by [START_REF] Guyard | Contribution des méthodes isotopiques à la détermination des origines du phosphore apporté au Lac Léman[END_REF], meaning that the higherδ 18 Op value downstream (15.8‰) cannot be interpreted as the result of a mixing between wastewater derived P and P from the upstream sediments. One possible explanation for this could be that part of the Pi injected into the river by the WWTP is rapidly (within a few meters from the injection point)

and continuously sorbed onto the sediment particles, as also hypothesized by Dorioz et al. (1998a), and that the heavier isotopologues are preferentially sorbed. However, there is no consensus regarding the associated fractionation (Jaisi et al., 2010;Melby et al., 2013b). An alternative explanation would be that the upstream sample 4 represents a local anomaly, and the signature of the downstream sediments, sample 5, derives from local riverbank erosion. Sample

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21 to that of sample 5. However, this hypothesis is unlikely as it does not explain the drastic change in P speciation observed in the sediments downstream from the WWTP (see Table 3).

Only the locality situated immediately downstream from the WWTP discharge point could be sampled in March 2012 (sample 6). The SRP, Pt, Pi and P Olsen concentrations were as high as those found in sample 5, which was collected in September, and much higher than the concentrations measured further upstream, at location 3, once again providing evidence for the strong impact of the WWTP discharge on the river and sediment P pool. The δ 18 Op value for sample 6 was similar to the September sample (sample 5), although it was close to the calculated isotopic equilibrium. The lack of a reference value equivalent to sample 4 in March means that it is not possible to conclude whether the observed value resulted from the complete equilibration of the sedimentary P-PO 4 pool or if it was instead produced by selective sorption processes as we suggest for sample 5. A repeated seasonal sampling of the sediments immediately upstream and downstream from the WWTP could possibly clarify this point.

Head and outlet sediments

The possible mixing of isotopically different sediment source materials coupled with differences in the sediment retention time complicates the interpretation of the δ 18 Op values for the riverbed sediments. Under conditions of low biological activity and a short residence time, the isotopic signature of the sediments is expected to represent primary source signatures. In this context, an isotopic disequilibrium would indicate the existence of factors that limit the instream P cycling, intended as uptake by the biota, intracellular equilibration and subsequent release. As judged from the present data, a limiting situation for in-stream P recycling could have been prevalent in the head of the Redon River catchment, where strong out-of-equilibrium δ 18 Op signatures have been recorded (see Table 3 and Fig. 2b). This could be first related to the physical conditions in this part of the catchment -high-energy stream (total renewal of the

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22 sediments with each storm flow) and a short distance from the sources, i.e. riverbanks -which both determined the short residence times. The oligotrophic state of the running water (SRP < 10 µg P/l, NO 3 < 0.5 mg N/l) and the low P bioavailability of the riverbed sediments (as indicated by low P Olsen concentrations) could contribute to limit the development of benthic biota and thus P uptake and subsequent release. For the same reasons the contribution of extracellular mineralization of organic P is expected to be negligible. The lack of an isotopic difference and the small differences in the Pt, Pi and Po content between the riverbank sample ( 12) and the sediment sample (3) collected immediately downstream further confirms the direct filiation between the source and the sediment.

The hypothesis that a rapid transfer does not significantly modify the δ 18 Op signature is further supported by the disequilibrium signature recorded in the streamlet samples 1 and 2 and their Pi, Po and P Olsen concentrations, as all of these parameters are intermediate between the values found in the analyzed deep and topsoil samples. This suggests i) a mixed origin for these two sediment samples derived from both superficial soils and ditch banks or field embankments;

and ii) that the biological activity was not intense or not long enough to equilibrate the original P oxygen isotope signatures, consistently with the expected very short residence time of the sediments in the streamlets (stream order of 1).

The situation was different for the sediments that were collected closer to the river outlet or in the lake (stream order higher than 3). Close-to-equilibrium values were found in these samples.

A corresponding increase in the SRP concentration in running water (a factor of 8 between upstream and the outlet; see Table 1), as the result of the cumulative anthropogenic inputs in the river, was also observed.

The equilibrium values observed in the lowest part of the Redon River suggest the occurrence of processes that are able to efficiently recycle the sedimentary P-PO 4 . This hypothesis would be consistent with the decrease in the δ 18 Op values observed during the September campaign decline in the δ 18 Op value between sample 5, which was collected immediately downstream from the WWTP discharge point, and sample 8, which was collected approximately 2 km further downstream. This decrease was accompanied by a strong narrowing of the gap between the measured and calculated equilibrium δ 18 Op values. Whereas this gap was +1.4‰ for sample 5, it was reduced to < 0.4‰ for sample 8 (see Table 3). The isotopic equilibration inferred by these data would likely occur in the sediments, given that in this kind of torrential rivers most of the biological activity essentially takes place in the benthic system. Equilibration could be further promoted by the longer residence time of the sediments in the downstream part of the river [START_REF] Dorioz | Dynamique du phosphore dans les bassins versants: Importance des phenomenes de retention dans les sediments[END_REF]. It may have been also driven by the enhanced bio-availability of P in the lower part of the river, notably downstream from the WWTP, as evidenced by the sharp increase in the P Olsen concentration starting from this point (see Table 3, Figs. 2a and2b).

Most likely, the overall process should be viewed more as the integration of a succession of partial equilibrium occurring over different storage stages under similar equilibrium conditions, as it is unlikely that the bulk of the P-PO 4 could be completely cycled by the benthic biota during a single low-flow period.

However, we lack information about the real residence time of the sediments and the P recycling rates in the Redon River. Considering that the measured δ 18 O P values for the sediments collected near the river outlet (samples 7 and 8) are identical within error to those measured in the upstream streamlet sediments (samples 1 and 2) and are within the range of the values found in the possible sediment sources (see Fig. 2a), we cannot entirely exclude that the

Potential and limits of the oxygen isotopes tool

We hypothesize that is possible to trace the potential sources of sediment particulate-P-PO 4 using oxygen isotopes, assuming a limited number of potential sources and under conditions of low biological activity and a short residence time in the river network. In the studied torrential system, it appeared possible to trace these sources in the head of the catchment (only two sources, rapid and high-energy transfer, limiting conditions for the biological activity). It would be also possible at the catchment scale, during winter high-flow periods, when the sediments that originate from banks and agricultural soil erosion are rapidly transferred as suspended matter and without significant interaction with point source discharge and biological activity. In this case, the δ 18 Op value would primarily be the result of a two end-members mixing between the cultivated soil (values close to 11-14‰) and moraine material (values close to 15-16‰).

However, the isotopic tracing is expected to be less straightforward during the summer and early fall periods, as the storm flows that occur at that time primarily mobilize riverbed sediments. Therefore, it could become more difficult to perform a quantitative assessment of the original particulate-P sources using δ 18 Op due to an overprinting of the sediment isotopic signature induced by the high biological activity. During summer low flows, an even stronger overprinting of the original isotopic signature is expected to occur in the downstream part of the river, due to the input of anthropogenic, bioavailable P, a longer sediment residence time and increased biological activity. This being the case, δ 18 Op could then instead be used as a tool to assess the in-stream biological turnover of the P.

Conclusion

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• In this study, we successfully adapted Tamburini et al.'s method (2010) for soil for use with river sediments to provide the first assessment of the variability in δ 18 Op from HClextracted P in river sediments along a gradient of increasing anthropogenic impact.

•

We observed a contrast between the δ 18 Op signature and the speciation of the sediments collected in the head and at the outlet of the catchment. The sediments that were collected upstream have isotopic signatures that are far from isotopic equilibrium and within the range of the signatures found in the possible sediment sources (i.e. riverbank material), while those collected at the outlet showed δ 18 Op values close to isotopic equilibrium. We related this difference to the contrast between the highly dynamic character (short residence time) of the transfer system in the head of the catchment (preservation of the isotopic signature of the source material in the sediments) and the "quieter" sediment transport system in the lower catchment, which, along with the input of the bioavailable anthropogenic P, likely fostered the recycling of P by benthic biota, leading to an equilibration in the δ 18 O signatures. However, with the available data, we cannot exclude that the signatures at the outlet were the result of the mixing of sediments with higher (upstream/wetland sediments) and lower signatures (agricultural soil erosion).

• Uncertainties remain about the calculation of the equilibrium window. For aquatic environments, the duration of the in-stream equilibration processes should be taken into account, together with the perturbation caused by changes in the hydrological regime. This would require an experimental approach for a proper estimation.

•

In terms of the quantitative tracing of potential particulate-P sources, future investigations should focus on winter high flows, when rapid sediment transfer maximally reduces the possibility of biological interactions. On the other hand, the sensitivity of δ 18 Op to biological processes opens up new promising perspectives for studying in-stream processes. In • We observe a contrast in the oxygen isotope composition between upstream and downstream sediments

• This contrast can be related to anthropogenic impacts on the river system and to the sediment residence time

• We have shown that in some situations it is possible to trace particulate-P sources of sediments with this isotopic tool

  as described byFares et

  interval (in ‰) between the measured δ 18 O p values and the closest values for the equilibrium range calculated from equation 23. Results

(

  2011) for C and N, respectively. The efficiency of the purification procedure was confirmed by the oxygen yield given by the mass spectrometer (Fig. 1 in the Supplementary Material). All of the samples were close to the regression line defined by the Ag 3 PO 4 standards, indicating that the amount of oxygen in the samples was close to the stoichiometry of the pure compound. Most of the labeled/unlabeled pairs did not show significantly different δ 18 O p values. Only four samples (2, 5, 8 and 9) showed different labeled and unlabeled values, meaning that the organic P or condensed phosphate was hydrolyzed during the HCl extraction. These values have been corrected with equation 1, as all of the other values are the average of the labeled and unlabeled results.

  measured δ 18 O p values for the HCl extracts varied between +11.8‰ (deep soil horizon, sample 11) and +18.3‰ (wetland soil, sample 9; Fig.2a). No significant statistical relationships were found between the δ 18 O p values for the samples and their Pt, P Olsen or Pi content or ratios, nor with the water characteristics, notably the SRP concentrations. The in the head of the catchment (sample 12) corresponds to a weakly weathered moraine, as confirmed by field observations, i.e. no visible weathering and no vegetation or biofilm on the banks, since they are eroded at each storm flow, and by the P speciation data (very low P Olsen and Po fractions; see Table

  10) or slightly lower than (-0.7‰, deep horizon, sample 11) the calculated equilibrium values.

  might be a major source of uncertainty.[START_REF] Amelung | The δ 18 O signatures of HCl-extractable soil phosphates: methodological challenges and evidence of the cycling of biological P in arable soil: The δ 18 O signatures of HCl-extractable soil phosphates[END_REF] suggest calculating the equilibrium δ 18 O p value with the annual average δ 18 O w for rainwater, arguing that the biological recycling of Ca-bound P in soil is a long-term process. Using an annual rainwater δ 18 O w value and temperature values, the calculated equilibrium value for the agricultural soil samples (10 and 11) would increase by approximately +2‰, resulting in measured signatures below the equilibrium.

  recycling versus tracing the sources using δ 18 O p values 4.2.1 Soils[START_REF] Tamburini | Oxygen Isotopes Unravel the Role of Microorganisms in Phosphate Cycling in Soils[END_REF][START_REF] Tamburini | Oxygen Isotopes for Unraveling Phosphorus Transformations in the Soil-Plant System: A Review[END_REF] have shown that the HCl-P fraction in the upper and most biologically active part of the oldest soil of a chronosequence was formed by secondary minerals precipitated from Pi that were previously cycled by biota. The close-to-equilibrium δ 18 Op values measured in the agricultural soil samples (samples 10 and 11) can be viewed as evidence that biological cycling had a large impact on the HCl-P fraction (which represents roughly half of the Pt) in these samples. The soil P is supplied by the parent material and fertilizations and is redistributed throughout the soil profile via biological activities. As the parent material is low in Pt, P Olsen and Po, the high concentrations observed in the topsoil sample (see Table

  WWTP discharge point, i.e. between samples 5 and 8. There was a 3‰

δ

  18 O P values in these downstream sediments are simply source-inherited isotopic values. More direct evidence for the real residence time of the sediments and the ability of aquatic biota to efficiently recycle P-PO 4 in the downstream part of the river is needed to decide between these two hypotheses.

  could help develop these studies by providing unique information about the biological uptake, turnover and/or organic P mineralization.•More insights about the way and extent the biological activity controls P speciation in the soil and sediments could be possibly provided by the development of refined extraction methods that are able to selectively and quantitatively extract other P pools like the one associated with Al and Fe hydroxides, P Olsen and Po or finer fractions (e.g. < 50 µm, < 2 µm).

Figure 2 .

 2 Figure 2. Oxygen isotope composition (δ 18 O p ) and deviation from isotopic equilibrium. The

Figure 3 .

 3 Figure 3. Conceptual model describing the evolution of δ 18 Op for the riverbed sediments in the

  conductivity SRP = soluble reactive phosphorus concentration in water, measured on filtered samples (0.45 µm) by the molybdate-blue method (Murphy and Riley, 1962) NPS = Non-point source, PS = Point source WWTP = waste-water treatment plant

a

  samples collected within an agricultural sub-catchment of 33 ha (Figure1)

Figure 1 .Figure 2 .•

 12 Figure 1. Redon River catchment and sampling sites, capital letters refer to sampling locations, numbers to sample code

Table 3 .

 3 P speciation and isotopic dataThe P content and speciation varied strongly from one sample to another, with the Pt value ranging from 459 to 1256 mgP/kg, Pi from 179 to 631 mgP/kg and Po from 36 to 450 mgP/kg. P Olsen accounted for 0.7 to 10.6 % of Pt depending on the samples. The highest P Olsen concentrations (65-67 mgP/kg, representing 8.4 to 10.6% of Pt) were found in the river

  . Its δ 18 O p value (+15.9‰) indicates that it has a mixed igneous/sedimentary origin, as this value is intermediate between the δ 18 O p values reported for igneous or metamorphic apatite (approximately +7‰; Tamburini et al., 2012) and for sedimentary apatite (between +17 and +23‰; Tamburini et al., 2014).

  The δ 18 O p value for this sample (18.3‰) cannot be mixing of any of the potential input materials present in the area (i.e. bank material or cultivated soil), as +18.3‰ is too far outside the range of the values (11.8 and 15.8‰) recorded by these input materials. The only source with a sufficiently high δ 18 O p value would be plant-derived phosphate. Phosphate in plants is enriched in 18 O, with the δ 18 O p values ranging from 28 to 31‰

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	explained by the
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	19

Table and Figure captionsTable 1 .

 1 Characteristics of the sediment sampling sites.

Table 2 .

 2 Characteristics of the soil sampling sites. Table3. P speciation and oxygen isotope composition.Figure1. Redon River catchment and sampling sites; the capital letters refer to the sampling locations, the numbers are the sample code.

Table 1

 1 Characteristics of sediment sampling sites

	Site SO	Sample	Sampling date	Q	EC	SRP Altitude	Drainage	Land use of the	Main sources of
		code						area	drainage basin	stream P
				L/s mS/cm mg P/L m a.s.l	ha
	A 1	1	September 3,	<1 543 0.0012	576	4	Cultivated land	NPS (erosion of
			2012							cultivated soils
										and tiches)
	B 1	2	September 3,	1	557	0.009	577	17	Cultivated land	NPS (erosion of
			2012							cultivated soils
										and tiches)
	C 2	3	March 26, 2012 35 598	0.008	696	510	Steep forest	NPS (natural:
										mainly river bank
										erosion)
	D 3	4	September 3,	55 7203 0.027	510	2310	Forest and	NPS (natural,
			2012						cultivated land	agricultural)
	E 3	5	March 26, 2012 105 654	0.056	508	2311	Forest and	NPS (natural,
									cultivated land	agricultural) + PS
		6	September 3,	55 753	0.092				(WWTP)
			2012
	F 4	7	March 26, 2012 340 597	0.062	405	3300	Forest and	NPS (natural,
									cultivated land	agricultural) + PS
		8	September 3,							(WWTP)
			2012

Table 2

 2 Characteristics of soil sampling sites

	Sample	Date of sampling Location	Soil	Altitude Sampling conditions
	code		depth
			cm	m a.s.l
	9	September 3, 2012 Wetland topsoil	-1 to -5 552	Organic soil, saturated
	10	September 3, 2012 Cultivated topsoil a	-10 to -	558	Cambisol on moraine deposit, dry
			30
	11	September 3, 2012 Cultivated soil	-60 to -	558	Cambisol on moraine deposit,
		(embankment) a	70		saturated
	12	September 3, 2012 Riverbank in a	-	980	Würm moraine deposits, saturated
		forested upland area

Table 3 .

 3 P speciation and oxygen isotopes composition

	Sample code	Sample type	Pt	P Olsen	Pi	Po	P Olsen /Pt a δ 18 O p St. Dev. n Deviation from equilibrium (‰) b
			mgP/kg mgP/kg mgP/kg mgP/kg %	‰	‰	-‰
	1	River bed	791	14	401	180	2.4	12.5 0.3	3 -2.4
	2	sediments d	751	20	323	208	3.8	13.3 $ 0.3	2 -1.6
	3		528	8	411	78	1.7	15.5 0.6	4 + 1.3
	4		609	18	398	202	2.9	12.2 $ 0.4	2 -1.5
	5		1018	65	586	194	8.4	15.8 $ 0.7	2 + 1.4
	6		962	67	470	166	10.6	15.2 0.3	3 < 0.4
	7		734	26	439	138	4.5	13.8 0.4	2 < 0.4
	8		736	21	324	90	5.2	12.8 0.1	3 -0.6
	9	Wetland soil	1163	32	179	383	5.7	18.3 $ 0.2	2 + 2.1 (+ 3.6) c
	10	Agric. topsoil 1256	43	631	450	6.2	14.0 0.4	2 < 0.4 (< 0.4) c
	11	Embankment/	755	5	363	76	1.1	11.8 0.4	2 -0.7 (-2.0) c
		deep soil
	12	Riverbank	511	3	305	36	0.7	15.9 0.2	2 +1.9 (+1.2) c
		(moraine)
	13	Lake	459	15	365	121	3.5	12.9 0.3	2	< 0.4
	14	sediments	539	17	420	70	3.1	13.9 0.5	2 + 0.8

a As a sum of Pi and Po fractions b See section 2.3.2 and equation 2 c Values not in brackets are calculated according to the temperature and water isotopes conditions at the sampling site, the values in brackets are calculated according to the "within-river" conditions d Sediment samples are ranked according to the corresponding drained area

Acknowledgments

This work was funded by the EFPA department of INRA. The authors thank Philippe Quetin, Pascal Perney and Danielle Lacroix (INRA, UMR CARRTEL) for their precious help and Jerome Lazzarotto and Liliane Savoye for their advice and fruitful discussions. The authors acknowledge the EDYTEM Laboratory and in particular J.P. Jenny for providing the sediment samples from Lake Geneva.


Vincent Léchappé
email: lechappe.vincent@gmail.com

Emmanuel Moulay
email: emmanuel.moulay@univ-poitiers.fr

Franck Plestan
email: franck.plestan@irccyn.ec-nantes.fr.

Dynamic observation-prediction for LTI systems with a time-varying delay in the input

Keywords: Prediction-based control, time-varying delay, dynamic prediction, Lyapunov-Krasovskii functional

A predictive-based controller is proposed to control LTI systems in presence of time-varying delay in the input. The control method is based on the computation of an approximated (or asymptotic) prediction thanks to a dynamic system. Then, this prediction is "plugged" into any Lipschitz controller that stabilizes the delay-free system. Explicit conditions that guarantee the closed-loop stability are given thanks to a Lyapunov-Krasovskii analysis. A qualitative analysis of these conditions is performed and the results are illustrated in simulation.

I. INTRODUCTION

Input delay systems are a subclass of time delay systems (TDS). The reader can refer to survey papers [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF] and [START_REF] Gu | Survey on recent results in the stability and control of time-delay systems[END_REF] for a general review on TDS. An input can be delayed because of a large computation time or because of latencies during communications. The latter example is particularly common for remote controlled devices such as UAVs, satellites or in Networked Control Systems (NCS).

There exist two different approaches to control input delay systems: memoryless (or memory free) and memory controllers. The advantage of memory free controllers is that they do not require the computation of an integral. See for example [START_REF] Mazenc | Global asymptotic stabilization for chains of integrators with a delay in the input[END_REF] for bounded control, [START_REF] Choi | Stabilization of a chain of integrators with an unknown delay in the input by adaptive output feedback[END_REF] for adaptive control, [START_REF] Yoon | Truncated predictor feedback control for exponentially unstable linear systems with time-varying input delay[END_REF] for a truncated predictor and [START_REF] Richard | Sliding mode control in the presence of delay[END_REF] for sliding mode techniques. The drawback is that they usually cannot guarantee a good level of performance for unstable systems with large delays.

In this case, memory controllers can be designed. For systems with a single delay in the input (as those considered in this work), a memory controller is often a controller based on the computation of a prediction. It has been highlighted in [START_REF] Mirkin | Every stabilizing dead-time controller has an observer-predictor-based structure[END_REF] that state prediction is a fundamental concept for delay systems, much like state observation is for systems with incomplete state measurements. The most well-known method to control dead time systems is probably the Smith predictor. This frequency approach was introduced by Smith at the end of the 1950s in [START_REF] Smith | Closer control of loops with dead time[END_REF]. At the end of the 1970s and the beginning of the 1980s, the result of Smith has been extended to state-space representation and unstable systems by [START_REF] Olbrot | Stabilizability, detectability, and spectrum assignment for linear autonomous systems with general time delays[END_REF], [START_REF] Kwon | Feedback stabilization of linear systems with delayed control[END_REF] and [START_REF] Artstein | Linear systems with delayed controls: A reduction[END_REF].

The inconvenient of these methods is that they usually require the discretization of an integral. For open-loop stable systems, the integral term can be computed without discretizing the integral [START_REF] Watanabe | A process-model control for linear systems with delay[END_REF]. However, for open-loop unstable systems, the integral has to be discretized in a finite number of points. This step has to be done very carefully since it can destabilize the system as pointed out in [START_REF] Van Assche | Some problems arising in the implementation of distributed-delay control laws[END_REF]. A prediction approach based on an approximated prediction computed thanks to a dynamic system has been proposed for the first time in [START_REF] Besançon | Asymptotic state prediction for continuous-time systems with delayed input and application to control[END_REF] for the control of input delay systems. The advantage is that no discretization is required to compute the prediction. The idea has also been used for the observation of systems with delayed output [START_REF] Germani | A new approach to state observation of nonlinear systems with delayed output[END_REF], [START_REF] Kazantzis | Nonlinear observer design in the presence of delayed output measurements[END_REF], [START_REF] Ahmed-Ali | Cascade high gain predictors for a class of nonlinear systems[END_REF]. Recently, some works have used the same idea of dynamic prediction for control purposes: [START_REF] Najafi | Closed-loop control of dead time systems via sequential subpredictors[END_REF] with full state knowledge, [START_REF] Ahmed-Ali | Robust stabilization of nonlinear globally lipschitz delay systems[END_REF] and [START_REF] Karafyllis | Global stabilisation of nonlinear delay systems with a compact absorbing set[END_REF] considering sample and hold phenomena.

However, the works mentioned above consider a constant delay. In this article, the time-varying delay-case is treated by adding an extra term in the Lyapunov-Krasovskii functionals. The exponential stability is proven for a large class of controllers and observers. In addition, explicit conditions for stability are given and analyzed.

The paper is organized as follows. The different assumptions and the problem are stated in Section II. In Section III, the dynamic observer-predictor is presented along with a convergence proof and a qualitative analysis of the stability conditions. This result is extended to the case of various sub observers-predictors in Section IV. In Section V, simulations support previous theoretical results. Finally, some perspectives are given in Section VI.

II. PROBLEM STATEMENT

The class of systems considered in this work is the following

ẋ(t) = Ax(t) + Bu(t -h(t)) y(t) = Cx(t) (1)
where h(t) is a known and time-varying delay. The objective is to design prediction-based controllers without the need of computing the exact prediction. Indeed, the discretization is delicate process that often requires a large computing power that may restrict the application range of predictive methods in practice. In addition, for a time-varying delay, future values of the delay have to be known in advance to compute the exact prediction [START_REF] Krstic | Lyapunov tools for predictor feedbacks for delay systems: Inverse optimality and robustness to delay mismatch[END_REF], [START_REF] Nihtila | Adaptive control of a continuous-time system with time-varying input delay[END_REF]. In this article, only the value of h has to be known at instant t. The controller is designed for the delay-free system and then the prediction is "plugged" in it. As a consequence, throughout this paper it is assumed that Assumption 1: There exists a time differentiable and globally Lipschitz controller u : x → u(x), that guarantees the existence a function

V u : R n → R V u (x) = x T P x (2)
with P a n×n symmetric matrix that satisfies the inequalities

   c||x|| 2 ≤ V u (x) ≤ c||x|| 2 , Vu (x(t)) ≤ -c u ||x(t)|| 2 , dVu dx ≤ c||x||, (3)
with c, c, c u , c > 0. Remark 2.1: Note that the argument "t" in u(x(t)) will be omitted for clarity. Furthermore, the notations u(t) or u(x) will be used indifferently when no confusion is possible. Assumption 1 implies that the controller u(x) globally exponentially stabilizes delay-free system. Note that Theorem 4.14 in [START_REF] Khalil | Nonlinear Systems[END_REF] guarantees the existence of a Lyapunov function that verifies (3) for a general class of exponentially stable nonlinear systems. However, here a special form (but classic) of the Lyapunov function (V u = x T P x) is assumed. Remark that differentiating V u along the trajectories of the delay-free system and using inequalities (3), one gets

x T [A T P + P A]x + u T B T P x + x T P Bu ≤ -c u ||x|| 2 . (4)
In addition, since u : x → u(x) is globally Lipschitz, there exists l u > 0 such that

||u(x)|| ≤ l u ||x|| (5)
for all x ∈ R n . It is also assumed that an estimation x of the state x can be computed thanks to an observer:

ẋ(t) = Ax(t) + Bu(t -h(t)) + g(C x(t) -y(t)) (6)
where the correction term g is such that Assumption 2: There exists a globally Lipschitz function g that guarantees the existence of a function

V g : R n → R V g (e) = e T Qe (7)
with e = x -x and Q a n× n symmetric matrix that satisfies the inequalities

     m||e|| 2 ≤ V g (e) ≤ m||e|| 2 , Vg (e(t)) ≤ -m g ||e(t)|| 2 , dVg de ≤ m||e||, (8)
with m, m, m u , m > 0 . Remark 2.2: By a slight abuse of notation, g(Ce) will be sometimes denoted g(e) to underline that it depends on the observation error e. Assumption 2 implies that the function g globally exponentially stabilizes the observation error

ė(t) = Ae(t) + g(Ce(t)). (9)
Similarly to (4), one gets

e T [A T Q + QA]e + g T (Ce)Qe + e T Qg(Ce) ≤ -m g ||e|| 2 . ( 10
)
In addition, since g is globally Lipschitz, there exists

l g > 0 such that ||g(Ce)|| ≤ l g ||Ce|| (11)
for all e ∈ R n . Some assumptions are also made to characterize the time-varying delay h(t): Assumption 3: The time-varying delay h(t) is upper and lower bounded by strictly positive constants.

Assumption 4: The delay is differentiable and its dynamics is bounded. In the next section, the dynamic observation-prediction method is presented.

III. DYNAMIC OBSERVATION AND PREDICTION METHOD

The observation and the prediction are computed by a single dynamic system

ż(t) = Az(t) + Bu(t) + g(Cz(t -h(t)) -y(t)]) (12)
with g a correction term verifying Assumption 2.

Remark 3.1: Note that to compute z(t), the initial condition z(t) = φ z (t) for t ∈ [-h max , 0] with φ z a continuous function, is required. The prediction z will be continuous in t = h(t). Also remark that the variable z(t) has to be stored on [t-h max , t] which imposes constraints on practical implementations. Then, a predictive feedback u(z) with u that verifies Assumption 1 can be applied to stabilize [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF]. Notice that the non-delayed input u(t) enters system [START_REF] Watanabe | A process-model control for linear systems with delay[END_REF] and that the correction term involves the delayed estimation state z(t-h(t)), that is why z(t) is an approximation of x(t + h(t)).

In order to state the next result, the prediction error is denoted by

e p (t) = z(t -h(t)) -x(t). (13)
It is now possible to introduce the following theorem. Theorem 1: Consider system (1), where h(t) is known and complies with Assumptions 3 and 4, and assume that there exist a controller u and the correction term g satisfying Assumptions 1 and 2 respectively. Suppose that system (1) is controlled by u(z) with z computed by observer-predictor [START_REF] Watanabe | A process-model control for linear systems with delay[END_REF] and define

Υ(t) = ||x(t)|| 2 + sup s∈[t-hmax,t] ||z(s)|| 2 + sup s∈[t-hmax,t] || ėp (s)|| 2 .
(14) Then, there exist ς, ̺, h * , δ * > 0 such that, provided

h(t) < h * and | ḣ(t)| < δ * , one has Υ(t) ≤ ςΥ(0)e -̺t ∀t ≥ 0 (15) therefore lim t→+∞ ||x(t)|| = 0.
Proof: Note that the delay h(t) (respectively ḣ(t)) will be denoted h (respectively ḣ) in the proof to facilitate the reading. First, denoting e p (t) = z(t -h(t)) -x(t), equation [START_REF] Watanabe | A process-model control for linear systems with delay[END_REF] becomes

ż(t) = Az(t) + Bu(t) + g(Ce p (t)). (16)
The dynamics of e p (t) reads as

ėp (t) = Ae p (t) + g(Ce p (t -h)) -ḣ[Az(t -h) + Bu(t -h) + g(Ce p (t -h))] (17)
and can be rewritten as follows

ėp (t) = Ae p (t) + g(Ce p (t)) - Cep(t) Cep(t-h) dg ds (s)ds -ḣ[Az(t -h) + Bu(t -h) + g(Ce p (t -h))].
(18) Since the delay is bounded from Assumption 3, there exist h min , h max > 0 such that

h min ≤ h(t) ≤ h max . ( 19
)
The following Lyapunov-Krasovskii functional candidate is proposed

V (t) = γV 1 (t) + V 2 (t) + V 3 (t) + V 4 (t) (20)
where

V 1 (t) = z T (t)P z(t), (21)
with P defined in (2),

V 2 (t) = e T p (t)Qe p (t), (22)
with Q defined in [START_REF] Mirkin | Every stabilizing dead-time controller has an observer-predictor-based structure[END_REF],

V 3 (t) = t t-hmax (h max + s -t)|| ėp (s)|| 2 ds, (23)
V 4 (t) =

V3 (t) ≤ h max || ėp (t)|| 2 -1 2hmax ||m(t)|| 2 -1 2 t t-hmax || ėp (s)|| 2 ds. (30)
In addition, since u and g are globally Lipschitz and using (26) and Hölder's inequality, it can be deduced from ( 18) that

|| ėp || 2 ≤ c 1 ||e p || 2 + c 2 ||m|| 2 + c 3 | ḣ| 2 ||z(t -h)|| 2 +c 2 | ḣ| 2 ||e p (t -h)|| 2 (31) with c 1 = 4(||A|| + l g ||C||) 2 , c 2 = 4l 2 g ||C|| 2 , c 3 = 4(||A|| + l u ||B||) 2 so V3 (t) ≤ h max [c 1 ||e p || 2 + c 2 ||m|| 2 + c 3 | ḣ| 2 ||z(t -h)|| 2 +c 2 | ḣ| 2 ||e p (t -h)|| 2 ] -1 2hmax ||m(t)|| 2 -1 2 t t-hmax || ėp (s)|| 2 ds. (32) Finally, V4 (t) ≤ (h max + h min )(||z(t)|| 2 + ||e p (t)|| 2 ) -h min (1 -| ḣ|)(||z(t -h)|| 2 + ||e p (t -h)|| 2 ) - t t-h (||z(s)|| 2 + ||e(s)|| 2 )ds.
(33) Thus, using Young's inequality, the "completing the square" method and Assumptions 1 and 2, the following inequality is obtained

V +εV ≤ -γcu 2 -2h max -γεc ||z(t)|| 2 - 1 2hmax -R ′2 2mg -h max c 2 ||m(t)|| 2 - mg 2 -γR 2 2cu -h max (2 + c 1 ) -| ḣ|(M/2 + R ′ /2) -ε m ||e p (t)|| 2 -h min -| ḣ|(h min +| ḣ|h max c 3 +M/2) ||z(t-h)|| 2 -h min -| ḣ|(h min +| ḣ|h max c 2 +R ′ /2) ||e p (t-h)|| 2 -1 2 -εh max t t-hmax || ėp (s)|| 2 ds -[1 -2εh max ] t t-h (||z(s)|| 2 + ||e p (s)|| 2 )ds.
(34) with ε > 0. To make V (t) + εV (t) ≤ 0, it is sufficient to have the coefficients pre-multiplying the quadratic terms negative. It leads to the following conditions

                                             γc u 2 -2h max -γεc > 0, 1 2h max - R ′2 2m g -h max c 2 > 0, m g 2 - γR 2 2c u -h max (2 + c 1 ) -| ḣ|(M/2 + R ′ /2) -ε m > 0, h min -| ḣ|(h min + | ḣ|h max c 3 + M/2) > 0, h min -| ḣ|(h min + | ḣ|h max c 2 + R ′ /2) > 0, 1 2 -εh max > 0, 1 -2εh max > 0. (35)
(36)

(37) (38) (39) (40) (41)
First, (35), ( 40) and ( 41) can be reformulated as follows: h max < h 1 and h max < h 2 and h max < h 3 with

h 1 = γ 2 c u 2 -εc and h 2 = 1 2ε . (42)
Besides, (36) can be rewritten in the following form

h max < R ′2 2mg 2 + 2c 2 -R ′2 2mg 2c 2 < h 3 (43)
with

h 3 = 1 √ 2c 2 . (44)
Conditions ( 38)-(39) depend simultaneously on h max and | ḣ|. From Assumption 4, there exists δ > 0 such that | ḣ(t)| < δ. Thus, rearranging the terms gives h max < h 4 and h max < h 5 with

h 4 = 1 c 3 δ 2 [h min -δ(h min + M/2)] (45)
and

h 5 = 1 c 2 δ 2 [h min -δ(h min + R ′ /2)] . (46)
Similarly condition (37) depends simultaneously on h max and δ and can be reformulated as follows h max < h 6 with

h 6 = 1 2 + c 1 m g 2 - γR 2 2c u -δ(M/2 + R ′ /2) -ε m .
(47) In order to have h 4 , h 5 and h 6 positive, δ has to comply with the condition below

δ < δ * = min(δ 1 , δ 2 , δ 3 ) (48) with δ 1 = h min h min + M/2 , δ 2 = h min h min + R ′ /2 , ( 49
)
and

δ 3 = 2 (M + R ′ ) m g 2 - γR 2 2c u -ε m . ( 50
)
Choosing ε and γ sufficiently small guarantees that h 1 and δ 3 are positive. In that case, taking

h(t) < h max < h * = min(h 1 , h 2 , h 3 , h 4 , h 5 , h 6 ) (51) and | ḣ(t)| < δ < δ * ensures that V (t) ≤ -εV (t). (52)
Finally, equation ( 15) can be deduced from (52) by lengthy but straightforward computations that are omitted here. Remark 3.2: From ( 15), it can be said that the closedloop system is exponentially stable in terms of the norm Υ and that the convergence of ||x|| to zero is exponential. This theorem shows that a predictive feedback computed from observer-predictor (12) can stabilize system (1) provided that the delay and its variation are sufficiently small. The Lyapunov analysis that has been done to prove the stability allows to find explicit bounds for h and ḣ. A qualitative analysis of these bounds h * and δ * is given below. It is reminded that l u (respectively l g ) is the Lipschitz constant associated to the function u (respectively g) and is defined in (5) (respectively [START_REF] Artstein | Linear systems with delayed controls: A reduction[END_REF]).

• For a small value of ε, then h 1 , h 2 and h 6 increase which means that a slow convergence rate of the closedloop system allows larger value of the delay. • Similarly, for a small value of ε, then δ 3 increases which means that the delay dynamics can be faster for a slow convergence rate of the close-loop system. • For a large value of l g (that appears through c 1 and c 2 ), then h 3 , h 5 and h 6 decrease which means that the admissible delay is smaller for a fast observer. • For a large value of l u (that appears through c 3 ), then h 4 decreases which means that the admissible delay is smaller for a fast controller. • For a large value of δ, then δ 1 and δ 2 increase which means that the delay dynamics can be faster for a small delay interval. • For a small value of δ, then h 4 , h 5 and h 6 increase which means that a slow-varying delay can have a larger amplitude.

Simulations show that for stable systems (A Hurwitz), the real values h * and δ * can be large. On the contrary, for unstable systems, these bounds become very small. However, the use of sequential sub observers-predictors is a solution to relax this constraint. This method is presented in the next section.

IV. SUB OBSERVERS-PREDICTORS

The idea is similar to the one presented in [START_REF] Besançon | Asymptotic state prediction for continuous-time systems with delayed input and application to control[END_REF] and [START_REF] Najafi | Closed-loop control of dead time systems via sequential subpredictors[END_REF]. However, it is extended to the time-varying delay case. In addition, the exponential stability (only asymptotic stability was proved in [START_REF] Najafi | Closed-loop control of dead time systems via sequential subpredictors[END_REF]) is proved for a larger class of observers and controllers. The technique is to design various cascaded observers-predictors; Each observer-predictor will predict the state for a fraction of the delay denoted h. This time-varying prediction horizon is equal to

h(t) = h(t) r , r ∈ N * . ( 53
)
The equations of the cascaded sub observers-predictors are given below

                           ż1 (t) = Az 1 (t) + Bu(t -(r -1) h(t)) +g 1 ([Cz 1 (t -h(t)) -y(t)]) . . . żi (t) = Az i (t) + Bu(t -(r -i) h(t)) +g i (C[z i (t -h(t)) -z i-1 (t)]) . . . żr (t) = Az r (t) + Bu(t) +g r (C[z r (t -h(t)) -z r-1 (t)]) (54)
where z i ∈ R n , i = 1, ..., r. A predictive output feedback u(z r ) that verifies Assumption 1 can be applied to stabilize system [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF].

In order to the state the next theorem, we define the sub prediction errors

e p1 = z 1 (t -h(t)) -x(t) (55)
and

e pi (t) = z i (t -h(t)) -z i-1 (t) (56)
for all i = 2, ..., r. The advantage of the sequential structure is to relax the condition on the maximum delay value that is mentioned in Theorem 1. Theorem 2: Consider system (1), where h(t) is known and complies with Assumptions 3 and 4, and assume that there exist a controller u and gains g i satisfying Assumptions 1 and 2 respectively. Suppose that system (1) is controlled by u(z r ) with z r computed by sequential observers-predictors (54). Then, there exist r * ∈ N * and δ * > 0 such that, provided r > r * and | ḣ| < δ * , ||x(t)|| exponentially converges to zero. The proof, omitted for space limitations, consists of a recursive analysis combined with similar computation as in Theorem 1.

Remark 4.1: In previous theorem, the exponential stability in the sense of the norm Υ was shown. Here, only the exponential convergence of ||x(t)|| is proven because of the induction proof.

Theorem 2 shows that the condition on the delay size h * in Theorem 1 can be overcame by the use of sub observerspredictors. There is still a condition on the delay rate but this condition cannot be removed without any knowledge of the future values of h. The minimum number of sub observerpredictor has to be chosen such that h(t) < h * where h * is defined in Theorem 1.

The result of Theorem 2 (and 1) holds for any given controller u and a given correction term g. In practice, choosing a "slower" controller and a "slower" correction term allows to reduce the number of sub observers-predictors.

Above results are illustrated by simulation in the next section.

V. SIMULATION

An unstable second order system is used to illustrate above results

ẋ(t) = Ax(t) + Bu(t -h(t)) (57)
where A = 0 1 0.1 0.1 and B = 0 1 .

Remark 5.1: Since system (57) is unstable, memoryless predictive technique involving saturated input [START_REF] Mazenc | Global asymptotic stabilization for chains of integrators with a delay in the input[END_REF] can not be applied to stabilize this system. The controller

u(t) = Kz(t) (58)
where the gain K = [-2.1, -3.1] is tuned in order to place the eigenvalues of the delay-free system to -1 and -2. A Luenberger observer has been chosen to observe and predict the state of the system so

g(Ce) = LCe with L = [-1.6, -0.76] T . (59)
The eigenvalues of A + LC are -0.5 and -1. Unlike for a simple observer, the gain L can not be chosen arbitrarily large to impose a faster dynamics of the observer-predictor than of the close-loop system. Remark that controller (58) and the correction term LCe comply with Assumptions 1 and 2 respectively. Different delays will be introduced and the results 1, 2 or 3 sub observers-predictors will be compared. First, the performances of the method are checked with constant delays. On Figure 1, the effect on stability of an increasing constant delay h is illustrated. Note that the delays applied in simulation have the same order of magnitude as the dynamics of the closed loop system. It can be observed on Figure 1a for h = 0.5 s that the system can be stabilized for 1, 2 or 3 sub observers-predictors. However, when the delay is increased to h = 1 s on Figure 1b, the single observerpredictor does not guarantee the stability anymore. This is in accordance with Theorem 1 that states that there exists a maximum bound on the delay to preserve the stability. Finally, increasing again the delay to h = 1.7 s makes the 2 sub observers-predictors become unstable. However, it can be seen that the stability is maintained using 3 sub observerspredictors. This is illustrates Theorem 2 that states that there exists a minimum number of sub observers-predictors to guarantee stability.

On Figure 2, a time-varying delay affects the input of system (57). One can see that the system is stabilized using 2 sub observers-predictors.

VI. CONCLUSION

It has been shown that a single dynamic system can estimate and predict (approximately) the future state of the system for a sufficiently slow time-varying delay. The advantage of this method is that it does not require an integral discretization. Using sequential observers-predictors allows to stabilize the system for an arbitrarily long delay provided it is sufficiently slow-varying. The exponential stability in terms of a particular norm is proven and explicit bounds are provided. The extension to nonlinear systems is considered for future developments.

Fig. 1 :Fig. 2 :

 12 Fig. 1: Influence of the number of sub-predictors with constant delays

  Since g is globally Lipschitz and from (4), the time derivatives of V 1 satisfiesV1 (t) ≤ -c u ||z(t)|| 2 + R||z(t)|| ||e p (t)||(25)with R = 2l g ||P || ||C||. Moreover, by substitution one gets

						so, reminding that
						-	t t-hmax	|| ėp (s)|| 2 ds ≤ -	t t-h(t)	|| ėp (s)|| 2 ds
						≤ -1 hmax	t t-h(t)	|| ėp (s)||ds	2	,
						(29)
						it follows that
						(24)
	and γ > 0. Cep(t)	dg ds	(s)ds ≤ l g ||C||	t	|| ėp (s)||ds, (26)
	Cep(t-h(t))		t-h(t)
	so using (10), it follows that
	V2 (t) ≤ -m g ||e p (t)|| 2 + R ′ ||e p (t)|| ||m(t)|| +| ḣ|R ′ ||e p (t)|| ||e p (t-h)|| +| ḣ|M ||e p (t)|| ||z(t-h)||	(27)
	with R ′ = 2l t
				|| ėp (s)|| 2 ds	(28)
			t-hmax

t t-h (h max + h min + s -t)(||z(s)|| 2 + ||e p (s)|| 2 )ds g ||Q|| ||C||, M = 2||P || (||A|| + l u ||B||) and ||m(t)|| = t t-h || ėp (s)||ds. Furthermore, using Leibniz differentiation rule, one gets V3 (t) = h max || ėp (t)|| 2 -