G M Greenway

Munshi A B,C , Gómez

J D Elliott

Biological responses to contaminants in the Humber Estuary: disentangling complex relationships

Keywords: Trace metals, organic pollutants, GST, sediment toxicity, environmental homeostasis, Humber estuary

Due to the ecological importance of estuaries, it is necessary to understand the biological effects that potentially toxic contaminants induce in bioindicator species. A key aspect is whether effects at lower levels of biological organisation transfer through the system to higher levels. In understanding such processes, characterising multivariate relationships between contaminants, sediment toxicities and detoxification processes are important. Worms (Hediste diversicolor) and sediments were collected along the Humber Estuary, England, and inorganic and organic contaminants were quantified. Sediment toxicities and glutathione S-transferases (GSTs) activity in the ragworm were analysed. Concentrations of metals were highest near urban and industrial areas, whereas organic contaminants appeared at upstream locations. GST activity correlated with heavy metals. The genotoxicity, estrogenicity, dioxin and dioxin-like activity were higher at upstream locations. Estrogenicity correlated with alkylphenols and some organochlorines, whilst genotoxicity correlated with organochlorines and heavy metals. Despite this, higher level biological responses could not be predicted, indicating that homeostasis is operating.

Introduction

Estuaries are highly productive ecosystems, important nursery and recruitment areas for many species and support dense populations of invertebrates, which act as key prey species for higher animals [START_REF] Mclusky | The Estuarine Ecosystem; ecology, threats and management, 3 rd Edn[END_REF][START_REF] Dauvin | Effects of heavy metal contamination on the macrobenthic fauna of estuaries: the case of the Seine[END_REF]. As they are characterised by fine grained and organic rich sediments, they act as sink for contaminants and they are therefore priority areas for identifying potential bioindicator species and biomarkers of pollution [START_REF] Ducrotoy | The use of biotopes in assessing the environmental quality of tidal estuaries in Europe[END_REF].

Direct and diffuse (agricultural and urban run-off, riverine inputs) inputs to catchments have long degraded estuarine ecosystems worldwide. Many key contaminants such as metals, organochlorines and endocrine disrupting chemicals are well-characterised and increasingly regulated and monitored, thus the occurrence of lethal toxic effects and a number of higher level sub-lethal effects, such as severe impacts on reproductive success, are now rare. However, much less is known about combined effects of mixtures of contaminants at the biochemical and molecular level in relation to chronic, lower level exposure, and the link between these effects and those at higher ecological levels (i.e. populations, communities and ecosystems).

Previously it has been assumed that an impact on the lower levels of biological organisation (the cell and individual) would, if the stressor had not been removed or reduced, been transferred through to higher ecological levels (Lawrence and Hemingway, 2003). However, it is now hypothesised that because of the system complexity, changes at lower levels may get absorbed through the system or the organisms could accommodate the changes such that there are no concomitant changes at higher levels -a feature described at environmental homeostasis [START_REF] Elliott | The Estuarine Quality Paradox, Environmental Homeostasis and the difficulty of detecting anthropogenic stress in naturally stressed areas[END_REF]. It was suggested by Elliott and Quintino that estuaries are environmentally variable and their organisms adapted to that variability such that

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4 stress is more difficult to detect in whole estuarine systems. Hence there is a need to disentangle these factors when assessing environmental impacts.

Several metals and metalloids are known to produce oxidative stress with biological effects such as lipid peroxidation and genotoxicity (Bocchetti et al., 2004, Sun andZhou, 2008). Metals, and other persistent contaminants, often bio-accumulate and may bio-magnify through the food chain with the potential to affect human health (McLusky andElliott, 2004, Luoma and[START_REF] Luoma | Metal contamination in aquatic environments: Science and lateral management[END_REF]. Stable synthetic organic compounds are considered persistent, bioaccumulative and toxic contaminants (PBT).

Among them, xeno-estrogens, nonylphenols (NP), octylphenol (OP) and bisphenol A (BPA) and organochlorines (PCBs and some pesticides) may also act as endocrine disruptors (EDCs) at concentrations found in estuarine systems [START_REF] Lutz | Amphibians as model to study endocrine disruptors: I. Environmental pollution and estrogen receptor binding[END_REF]. They are also oxidative stressors and genotoxic to many organisms [START_REF] Nice | Long-term and transgenerational effects of nonylphenol exposure at a key stage in the development of Crassostrea gigas. Possible endocrine disruption?[END_REF].

Species respond specifically to these challenges with different mechanisms of detoxification that can in turn be used as biomarkers. Detoxification enzymes are used widely to detect pollution in aquatic environments, and include the glutathione-stransferases (GSTs) which comprising a superfamily of multifunctional proteins with fundamental roles in the detoxification of a wide range of compounds (Frova, 2006[START_REF] A N U S C R I P T A C C E P T E D Accepted Manuscript 26 Davies | The toxicology and metabolism of chlorothalonil in fish. III. Metabolism, enzymatics and detoxication in Salmo spp. and Galaxias spp[END_REF][START_REF] Bainy | Biochemical responses in farmed mussels Perna perna transplanted to contaminated sites on Santa Catarina Island, SC, Brazil[END_REF]. Additionally, peroxidase and isomerase activities have been described for GST enzymes, enabling their use as biomarkers for oxidative stress (Sheehan et al., 2001). GST has been employed in biomonitoring and impact assessments of heavy metals, organic contaminants including EDCs (Pérez-López, et al., 2002[START_REF] Martín-Díaz | Field validation of a battery of biomarkers to assess sediment quality in Spanish ports[END_REF][START_REF] Moreira | Effects of estuarine sediment contamination on feeding and on key physiological functions of M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 30 polychaete Hediste diversicolor: Laboratory and in situ assays[END_REF][START_REF] A N U S C R I P T A C C E P T E D Accepted Manuscript 25 Berglund | Oxidative stress in pied flycatcher (Ficedula hypoleuca) nestlings from metal contaminated environments in northern Sweden[END_REF], Solé et al., 2009). In polychaetes, the use of GST as a biomarker of oxidative stress has been demonstrated in Hediste (Nereis) diversicolor in Portuguese [START_REF] Moreira | Effects of estuarine sediment contamination on feeding and on key physiological functions of M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 30 polychaete Hediste diversicolor: Laboratory and in situ assays[END_REF] and

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French estuaries (Durou et al., 2007), and in the exposure of Alitta (Nereis) succinea to copper (Rhee et al., 2007) and to the xeno-estrogen nonylphenol [START_REF] Ayoola | Xenoestrogen induces stress in Nereis succinea (Polychaeta) and affect the reproductive fitness[END_REF]. The ragworm H. diversicolor was used as a sentinel species in the present study given its success as an estuarine species as shown by large and widespread populations in European estuaries and its previously reported use as a biomarker [START_REF] Pocklington | Polychaetes: key taxa for marine Environmental quality monitoring[END_REF], Poirier et al., 2006, Sun and Zhou, 2008). As they have a variable feeding mode covering both deposits and suspensions, they are exposed to and can bio-concentrate and bio-accumulate persistent contaminants (Gray and Elliott, 2009).

Inorganic and organic contaminants in sedimentary sinks reflect inputs to the aquatic environment. However, the partitioning in surface waters shows that the concentration of metals and the hydrophobic organic compounds in particulate material exceed the water soluble concentration by orders of magnitude [START_REF] Luoma | Metal contamination in aquatic environments: Science and lateral management[END_REF]. Sediments from both the catchment and the sea accumulate at the mouth of rivers and in estuaries, which become sinks for such toxic chemicals. Fine sediments accumulate and create organic films, which also attract and accumulate contaminants (Gray and Elliott, 2009), thus estuarine organisms are assumed to be excellent sentinels for pollution detection, in particular those that live at the sedimentwater interface such as the infaunal H. diversicolor.

Given the complex nature of these relationships, employing multivariate approaches is essential to link the concentration and composition of contaminants to their individual and synergistic toxicities, detoxification and tolerance in specific environments such as estuaries. This work tests the hypothesis that only an integrative approach can be used to analyse the relationships between the concentration of contaminants, toxicities such as genotoxicity, cytotoxicity, estrogenicity and dioxin-

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like activity in the sediments, and GST activity as a general biomarker of stress; furthermore this approach is required to understand more fully toxic effects and biological responses in complex natural environments. These aspects were analysed in samples obtained from the north shore of the Humber Estuary system in Eastern England.

Materials and Methods

Humber as a case study area

The Humber Estuary, Eastern England (UK) is one of the largest basins in the British Isles with a key role in bird conservation [START_REF] Archer | The Humber Estuary -an internationally important bird area[END_REF]. It has historically received industrial and urban waste from a large catchment area (Jaffe andWalters, 1977, Oguchi et al., 2000) and its sediments have a relatively uniform grain size and chemical composition, including some metals (Grant and Middleton, 1993). .

Sample collection

Sediment samples and worms were collected from 12 sampling points covering 4 different zones along the north bank of the Humber Estuary (Figure 1): Kilnsea (1, 2 and 3) located downstream close to the mouth of the estuary; Paull (4, 5 and 6) and Hessle (7, 8 and 9) close to urban and industrial zones and Blacktoft (10, 11 and 12) at the confluence of River Ouse with the Humber. The 3 sampling points at each site may be regarded as field replicates.

Trace elements analysis

The sediment samples were dried at 60 °C for 72 hours before being ground and sieved (100 µm mesh size) and 1 g of each sample was taken for extraction in 55 ml PFA Teflon digestion vessels with 10 ml nitric acid in a microwave digestion system (MARS Xpress), heated at 200 °C for 15 min. The digests were then analysed using an optical emission spectrometer-mass spectrometer (OES-MS). The concentrations of the elements in the digests were calculated in the original dry solid in µg g -1 of sediment.

Organic contaminants

The organochlorines (PCBs and pesticides) and the xeno-estrogens, nonylphenol, octylphenol and bisphenol A (BPA) were extracted from the sediments using Sohxlet apparatus and analysed by gas chromatography electron capture detector (GC-ECD) and GC-mass spectrometry (GC-MS). Organochlorine standards The determination of PCBs and organochlorines was according to [START_REF] Munshi | Organochlorine concentration in various fish from different locations of Karachi coast[END_REF]. Sediment samples (25 g) were extracted with acetone:hexane (1:1 v/v, 250 ml) during 8 hours. Extracts were analyzed using a GC (Perkin Elmer Clarus 500) equipped with an ECD, using a DB-5 column (30 m x 250 µm x 0.25 µm). The detection limit (LD) for pesticides and PCBs was 5 ng g -1 dry weight.

Determination of xeno-estrogens was according to [START_REF] Ayoola | Xenoestrogen induces stress in Nereis succinea (Polychaeta) and affect the reproductive fitness[END_REF] adapted for sediments. Five grams of sediment was spiked with 100 µl of internal standard ([13C6]-363 NP) and extracted with 90 ml hexane:isopropanol (85:15, v:v) during 6 hours at 60 ºC. Samples were evaporated, re-dissolved in 200 µl of acetone and derivatised with 25 µl of MSTFA at room temperature. A sample volume of 1 µl was injected into the GC-MS for analysis using an Agilent 6890N gas chromatograph directly connected to an Agilent 5973 inert mass selective detector in a selective ion mode (SIM). In most cases, the recoveries of detected organic pollutants ranged between 80 and 98 %. The DL were 3.1 and 2.2 ng g -1 dry weight fro OP and NP respectively.

Genotoxicity and cytotoxicity assay -GreenScreen GC TM

Genotoxicity was assessed using the GreenScreen GC TM assay from Gentronix Ltd. which uses a DNA repair-competent strain of the yeast Saccharomyces cerevisiae. The reporter consisted of a fusion of the DNA damage-inducible promoter from an endogenous DNA repair gene, RAD54, with a gene encoding a yeast enhanced green fluorescent protein (yEGFP). RAD54 is known to be specifically upregulated by the cells in response to DNA damage, and thus on exposure to a Simultaneously cytotoxicity was assessed by the reduction in cell proliferation during incubation compared to a vehicle-treated control. Cell density was quantified by optical absorbance. Methyl methanesulfonate (MMS, Sigma Aldrich) was used as a genotoxic standard, methanol was used as a cytotoxic standard and 2 % (aq.) DMSO (for molecular biology, Sigma Aldrich) was used as the diluent for all samples and standards. The protocol has been previously reported [START_REF] Cahill | The GreenScreen genotoxicity assay: a screening validation[END_REF]. In brief, sediment samples were dried at 60 ºC and 200 mg was re-suspended with 1 ml 100% dimethyl sulfoxide (DMSO), vigorously vortexed 3 times for 5 min and centrifuged at 8000 g. 40 µl of DMSO solution was transferred to a tube containing 960 µl of double distilled water. Extracts were added to the yeast in proprietary growth media at serial dilution in a 96 well microplate, together with the standard chemicals and vehicletreated controls, incubated overnight (16 -20 hours) and then optical absorbance (450 nm) and fluorescence (excitation 485 nm, emission 510 nm) were monitored using a microplate reader POLARstar OPTIMA (BMG Labtech, UK). Fluorescence and optical absorbance data were processed using Excel based software provided by Gentronix. The LEC (lowest effective concentration) in g ml -1 is the lowest test concentration at which the significance threshold for genotoxicity or cytotoxicity is exceeded.

CALUX assays

Estrogenicity (ER-Calux ® ) and Dioxin-like activity (DR Calux ® ) measurements were performed according to Houtman et al. (2006). Sediments were dried and then 5 g extracted using an accelerated solvent extraction system with 100 ml hexane/acetone. A concentration series of E 2 ( for ER-CALUX) or 2,3,7,

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ACCEPTED MANUSCRIPT 10 8-tetrachlorodibenzo-p-dioxin (TCDD, for DR-CALUX) was employed as standards and the analyses were performed in the BioDetection Systems laboratories (Amsterdam, Netherlands). Sediments were extracted, desulphurised and cleaned up prior to analysis, and then dissolved in DMSO and the activity determined after 24 hours exposure. Measurements were in pg EEQ g -1 of sediment for ER-CALUX and as polychlorinated dibenzofurans (PCDD/F)-PCBs-toxic equivalent quantity (TEQ) g - 1 for DR-CALUX.

GST activity analysis and protein quantification

One gram of worms (10 worms of approximate 100 mg each) per sampling point was homogenized using an ultrasonicator (Soniprobe, 7533A) in cold buffer solution (100 mM potassium phosphate buffer, pH 6.5, containing 2 mM EDTA). The tissue homogenates were immediately centrifuged at 15,000 g for 15 min. at 4 o C. The supernatants were collected and aliquots of the supernatants were stored at -80 o C prior to use. The protein extracts were used to quantify the GST activity and normalised with the total protein concentration calculated using the Bradford method with bovine serum albumin as a standard.

The activity of GST was determined according to the method developed by Habig et al. (1974), adapted to a microplate reader. In brief, this involves measuring the increment in absorbance generated by the conjugation of 1-chloro-2,4dinitrobenzene with reduced glutathione in the presence of GSTs at 340 nm using a microplate reader. Dulbecco phosphate buffer saline, 2 mM reduced glutathione and 1 mM 1-chloro-2, 4-dinitrobenzene were added to the aliquots and the enzyme activity was recorded by following the rate of change in optical absorbance (min -1 ) at 340 nm

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11 for 5 min. GST from equine liver (Sigma-Aldrich, #G6511) was employed as a positive control. The results were calculated in nmol min -1 mg -1 of protein.

Statistical analysis

The variables were regarded as environmental (independent) factors (locality, salinity and organic matter), inorganic contaminants (metals and metalloids), organochlorines (principally pesticides), xeno-estrogens (OC, NP and BPA) and the potential dependent (biological response) variables (GST activity in H. diversicolor, dioxin and dioxin-like activity, estrogenicity, genotoxicity and cytotoxicity). Initially the intra-and inter-group correlations (using two-tailed Pearson test) were obtained between the independent variable groups using the following factors: distance downstream, salinity and organic matter content (after testing for normality). The linear and non-linear relationships were explored to determine which of the variables were correlated with, and thus be responsible for, the sediment chemistry/contaminants/biological response data. Principal component analysis (PCA) was performed to identify relationships between GST activity, sediment toxicity, organic matter content and salinity, related to each specific group of contaminants according to chemical similarity. After obtaining a main factor representing the contaminants with major effects in the total variance, the factor coordinates of cases, based on covariance, were also correlated with biological effects. In general, a linear model was used by default and quadratic relationships were used when these models explained more than 20 % of the variance in comparison to the linear models. Exploratory correlation analyses and multivariate analysis (PCA) were performed using SPSSv17 and PRIMERv6 [START_REF] Clarke | Primer v6: User manual/tutorial[END_REF]. Given that all the variables have had different units, these were standardised

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12 and normalised using the Brodgar formula, according to Zurr et al., (2007) before plotting the PCA. GST activity in H. diversicolor from different locations in the Humber Estuary was compared by one-way analysis of variance (ANOVA) followed by Tukeys test post-hoc comparisons after prior normalisation using logtransformation (significant level p < 0.05).

Results

The organic content in the sediments ranged between 6.31 (location 9) and 10.79 % (location 3) representing relatively little difference between locations. Salinity range were from 1.96 to 19.78 at the most downstream location (table 1S). The abundance and the variability (max. and min. values) of each contaminant analysed are also shown in Table S1. The 14 PCB congeners analysed were below the detection limit (< DL) so were not significantly higher than the blanks. In addition, several organochlorine compounds were below the limit of detection (Quintozene; Heptachlor exo-epoxide; Heptachlor endo-epoxide; alpha-Endosulfan and beta-Endosulfan).

Element traces in sediments

The relative abundance of the elements analysed was:

Al>Zn>Pb>Cr>Cu>Ni>>Co>As>Cd. The relative distributions of almost all elements were positively correlated thus showing similar distributions and accumulation in the sediments (Fig. S1).

Several elements showed their highest values in mid-estuary locations 13 and although they did not correlate with the interstitial salt concentration, in most cases they were positively correlated with organic matter content (Fig. 2).

Organic contaminants

Almost all organochlorines analysed were found being positively correlated, indicating a similar distribution and accumulation process (Fig. S2). Endrin and Aldrin showed the highest concentrations being found in upstream locations at 201 ± 9.5 and 127 ± 4.5 ng g -1 respectively. In relation to environmental parameters, organochlorine distributions differed, some in relation to distance and therefore salinity, whilst other pesticides correlated well with the organic matter content (Fig. 3). For instance, the antifungal hexachlorobenzene was concentrated in downstream sediments, alpha HCH was higher at middle stream locations, whilst all other organochlorines were present at their highest concentrations in the upstream locations (Fig. 3).

As with some organochlorines, xeno-estrogens were more abundant in the sediments at the upstream locations. Concentrations of alkylphenols (NP and OP)

were also higher at the upstream points and were therefore negatively correlated with salinity and interesting, with organic matter content (Fig. 4). NP was detected at a maximum concentration of 1.29 µg g -1 while OP at the same location was at 0.38 µg g -1 . BPA was detected at lower concentrations (highest value was 0.038 µg g -1 ) and close or below the limit of detection in the other sampling locations (Table S1).

The potential toxic elements and the organic contaminants were in general negatively associated with each other or no relationship was observed (Fig. S3).

Glutathioine-S-transferase (GST) activity in H. diversicolor

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The GST activity in worms differed with locations with those at Paull (downstream of the city of Hull) showing the highest values of 56.27 ± 6.23 nmol min -1 mg -1 of protein which were significantly different (One way ANOVA, F (3,11) = 6.029, p < 0.05) from other zones of the estuary. The lowest values were close to the mouth of the estuary (Kilnsea, locations 1 to 3) with 41.21 ± 2.17 nmol min -1 mg -1 of protein. Moreover, the GST activity showed a positive correlation with As and Cu (Fig. 6).

Sediment toxicity

In general, sediment toxicity was higher at upstream locations, except for cytotoxicity which was highest in sediments in downstream areas (Fig. 5).

Genotoxicity clearly increased towards upstream locations with no genotoxicity being detected in the most downstream points (Fig. 5). Upper-Middle locations 7 to 9

(Hessle) showed the highest response and the LEC was 1.3 mg of sediment per ml of culture, and also higher in the upstream locations 10 to 12 (Blacktoft), the LEC value was 2.5 mg ml -1 . However, the stations 4 to 6 (Paull) close to industrial sources showed the lowest genotoxicity (LEC = 5.0 mg ml -1 ). Cytotoxicity was highest in samples from station 3 (Kilnsea) and, in contrast to the genotoxicity results, in downstream sediments (LEC = 1.3 mg ml -1 in both cases). Low cytotoxicity was detected in the upstream location (LEC = 2.5 mg ml -1 ) and Paull (LEC = 5 mg ml -1 ), indicating higher cytotoxicity in the outer estuary, although the absolute values indicate that the cytotoxicity is relatively low throughout the Humber without any correlation with environmental variables (Fig. 5).

The estrogenicity of sediments also varied among the locations. respectively. None of the biological responses were associated with the organic matter content (Fig. 5).

Relationship between contaminants, sediment toxicities and detoxification in

H. diversicolor

Given the spatial complexity in the contamination and biological response variables described above, it is necessary to interrogate the interrelationships. There were no other relationships between toxicities and metals/metalloids other than a positive correlation of As and Cu with the GST activity. It is of note that none of the organic contaminants analysed in this study showed significant correlations with the GST activity in ragworms (Fig. 6). The PCA plot suggests that Alpha-HCH and the expected heavy metals appeared associated with GST activity (Fig. 7). The organochlorines (i.e. aldrin, dieldrin and the factor coordinates of cases of the main factor obtained on the PCA of total organochlorines) correlate with genotoxicity (Fig

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16 6 and 7). In relation to estrogenicity, aldrin, dieldrin, endrin, alkylphenols (NP and OP) and the factor coordinates of cases of the main factor obtained on the PCA of total organochlorines and xeno-estrogens correlated well (Fig. 6) and are very consistent with the multivariate analysis (Fig. 7).

Analysing the combined effect of all the contaminants together with the GST activity (PCA) and sediment toxicity, the main component was found to be determined by two clear groups: trace metals and the organic contaminants, with some exceptions such as HCB and Apha-HCH which are involved in explaining the second component (Fig. 7). The biological responses found in worms and sediments suggest potential synergism or additive response of some contaminants explained with the main component such as genotoxicity, DR-Calux and ER-Calux (Fig 7). These results suggest a possible interaction between the different chemicals and thus specific biological responses depending on the prevailing contaminant regime. Furthermore, the distribution for some chemicals were not related to any environmental parameters measured in this work, indicating the general uniformity of the physico-chemical characteristics of the macrotidal Humber estuary.

Discussion

In this study, most of the inorganic contaminants in sediment samples were in general at low concentrations and the sum of all elements analysed was similar to those found in clean British estuaries and lower compared with historically polluted ones [START_REF] Bryan | A guide to the assessment of heavy-metal contamination in estuaries[END_REF]. The low level of PCBs and the distribution confirmed previous work on PCBs in the Humber estuary (Tyler and Millward, 1996).

Organochlorine pesticides were all at trace levels and low amounts of xeno-estrogens were detected (Table S1).

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ACCEPTED MANUSCRIPT 17 However, significantly higher GST activities in H. diversicolor were found at stations 4 to 6 corresponding to Paull area, indicating the bioavailability of xenobiotics and a potential toxicity on ragworms. Our results (Fig. 6 and7) suggest that metals such as Cu could be directly related to this biological response.

The highest percentages of organic matter in the intertidal mudflat corresponded to mid-and down estuary locations, especially those at Kilnsea at Spurn (near the mouth of the estuary) and Hessle where there is a creek (Haven) coming from an urbanised area. The mudflats are wider here reflecting a less energetic water flux on the upper shore compared to the steeper, narrower mudflats in the upper estuary.

Several elements correlated with the organic matter content of the sediments, although (Fig. 2). In particular, dissolved Cu and Cd have high affinity for sewage-derived particulate matter [START_REF] Comber | Comparison of the partitioning of trace metals in the Humber and Mersey estuaries[END_REF] and are expected to increase its concentration in organic-rich sediments.

The upper limit for copper was 62.6 µg g -1 , corresponding to a moderately-high contaminated range (50-200 µg g -1 ) according to [START_REF] Luoma | Metal contamination in aquatic environments: Science and lateral management[END_REF].

Concentrations of Cd, which is a toxic and genotoxic element, were elevated (up to 9.35 µg g -1 ) when compared with other British estuaries [START_REF] Luoma | Metal contamination in aquatic environments: Science and lateral management[END_REF] after [START_REF] Bryan | A guide to the assessment of heavy-metal contamination in estuaries[END_REF]. Due to its increasing use, it is not unexpected to find elevated concentrations of Cd in estuarine sediments. The concentrations observed here are higher than those measured in British estuaries 20 years ago (see [START_REF] Luoma | Metal contamination in aquatic environments: Science and lateral management[END_REF] and specifically for the Humber, the uniform distribution of metals described previously for intertidal sediments (Grant and Middleton, 1993) has been modified over the past 2 decades. In addition, water samples from the Humber Estuary presented the highest concentration of Cd with an average of 0.08 µg l -1 in

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18 subsurface waters of British estuaries [START_REF] A N U S C R I P T A C C E P T E D Accepted Manuscript 29 Law | Contaminants in seawater around England and Wales: Results from monitoring surveys, 1990-1992[END_REF] and similar concentration of Cd was determined in surface waters [START_REF] Comber | Comparison of the partitioning of trace metals in the Humber and Mersey estuaries[END_REF]. The highest concentration of elements analysed in sediments were found in mid-estuary locations (Paull and Hessle) suggesting the proximity to contaminant inputs and/or settlement and sink areas. The presence of quadratic models rather than linear models indicates, for example, that in relation to the locations, there is either a potential point source within the area or a differential accumulation along the estuary's north bank (Fig. 2).

Concentrations of Zn appeared significantly higher in mid-estuary locations and could reflect discharges from both current and past industrial activity, which is the main source of Zn in estuarine sediments (Joslin, 2000, Luoma and[START_REF] Luoma | Metal contamination in aquatic environments: Science and lateral management[END_REF]. Zn is used worldwide for its corrosion-resistance properties on ferrous metals implicating its anthropogenic distributions (Gordon et al., 2003). However the absolute concentration values do not represent a contaminated system as occurred in other historical polluted estuaries [START_REF] Luoma | Metal contamination in aquatic environments: Science and lateral management[END_REF]. Two potential toxic elements (As and Cu) appeared related to GST activity in ragworms (Fig. 6) suggesting be stressors that can increase the GST activity in polychaetes.

The bioavailability and therefore the toxicity of metals decreases with an increasing salinity and an increase in oxic properties of the sediments [START_REF] Mclusky | The effects of temperature and salinity on the toxicity of heavy metals to marine and estuarine invertebrates[END_REF][START_REF] Laing | Effect of salinity on heavy metal mobility and availability in intertidal sediments of the Scheldt estuary[END_REF] so toxic effects of metal traces are more likely in upstream sediments compared to the marine-influenced zone. Since the distribution of most of the elements analysed occurred in middle locations of the Humber, no correlation were observed with salinity (Fig. 2).

Humber sediments are dominated by silicate clay minerals that readily adsorb part of the organic fraction including many contaminants (Jarvie et al., 1997). Organic contaminants such as organochlorines, have a greater ability to adsorb to the organic

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19 fraction than to the clays, therefore we expected to find higher concentration of the congeners which contain more Cl -ions, since they are more hydrophobic. Many pesticides are organochlorine compounds which are PBT chemicals. Endrin and Aldrin, both now banned substances, were the most abundant organochlorines found in upstream locations suggesting an input coming from past agricultural activity in the catchment (Table S1, Fig. 3).

Xeno-estrogens appeared at moderate levels in comparison to other estuarine sediment analysed [START_REF] Bennett | Distribution of alkylphenol compounds in Great Lakes sediments, United States and Canada[END_REF]Metcalfe, 1998Kawahata et al., 2004). As expected, the strongest estrogenic compounds OP and NP, correlated well with the estrogenicity of the sediments (Fig. 6). Although estrogenic signalling have been clearly observed in nereid polychaetes (García-Alonso et al. 2006, Durou and Mouneyrac, 2007[START_REF] Keay | Hormone-activated estrogen receptors in annelid invertebrates: implications for evolution and endocrine disruption[END_REF] and that xeno-estrogens induced GST in a closely related species (N. succinea, [START_REF] Ayoola | Xenoestrogen induces stress in Nereis succinea (Polychaeta) and affect the reproductive fitness[END_REF] GST activity did not correlated with xeno-estrogen in H. diversicolor on this study.

Genotoxicity appeared higher at mid-estuary (Hessle) and upstream locations (Blacktoft) with an LEC of 0.0013 and 0.0025 g ml -1 respectively, decreasing along the sampling sites until no genotoxicity was observed in downstream points (Paull and Kilnsea). This may be the result of a pool of inorganic and organic contaminants as suggested in Fig. 7; further chemical analyses are required to better understand this sediment toxicity.

Whilst the present study has quantified the concentrations of metal ions, and these include metal ions with known genotoxic and carcinogenic properties (i.e. chromium, as dichromate, nickel and cadmium), there is limited direct correlation between genotoxic activity and the concentration of any particular metal or combinations of metal ions. Since sediments are complex mixtures, simple correlations between

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ACCEPTED MANUSCRIPT 20 mutagenic potential and the presence and concentrations of known sample components are often not possible since genotoxicity arises through the synergistic activity of several contaminants present in low concentration, or through complex interactions between contaminants. The GreenScreen GC assay clearly discriminates between cytotoxic and genotoxic endpoints and has been shown to demonstrate a high specificity in the assessment of genotoxicity, i.e. samples do not produce a positive result for genotoxicity in this assay as a result of cytotoxicity alone [START_REF] Cahill | The GreenScreen genotoxicity assay: a screening validation[END_REF]. Genotoxic contaminants can arise from a number of anthropogenic sources and include diverse organic chemicals such as organochlorine pesticides from agrochemical run off [START_REF] Bull | Evidence for genotoxicity of pesticides in pesticides applicators: a review[END_REF][START_REF] Poletta | Genotoxicity of the herbicide formulation roundup (glyphosate) in broad-snouted caiman (Caiman latirostris) evidenced by the comet assay and the micronucleus test[END_REF] among other chemicals (Claxton et al., 1998, White andRasmussen, 1998). In addition, dredging should also be considered as sediments which are re-located within the estuary may contain historic contamination which is re-liberated and re-distributed around the system (Gray and Elliott, 2009).

Dioxin activity is present in the Humber sediments at very low level. Even though, there is a clear gradient of response -suggesting that this assay could also respond to furans or dioxins, which were not measured in this work. Estrogenicity was positively associated with the concentrations of NP and OP but not with BPA which is a weak estrogenic chemical (Fig. 6 and7). Also, estrogen responses appear associated with pesticides such as aldrin, eldrin and dieldrin (Fig. 6), which may have some effects in activation of the estrogen receptor transcription factor.

The Humber estuary has been shown to be moderately polluted with Cu and As which levels have been reported to induce certain tolerance in H. diversicolor [START_REF] Nedwell | Intraspecific variation in the responses to copper by two estuarine invertebrates[END_REF], Jones et al., 2000). Accordingly, the GST activity was highest in

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT 21
worms at the mid-estuary points (p < 0.05) in contrast to the upstream and downstream sites. This detoxification activity appeared very well associated with the concentration of metals in general, includings Cu and As (Fig. 8). Cu was found at toxic level in Paull, and is a well known oxidative stressor. Moreover,. Rhee et al.

(2007) demonstrated that concentration as low as 12 µg l -1 of CuCl 2 induces the activity of GST in nereids. Cu appeared at a toxic level in mid-estuary sites, reaching an average of 55.5 µg g -1 supporting the hypothesis of metals as an oxidative stressor in H. diversicolor. The multivariate approach advocated here describing the association of contaminants and GST confirms the individual correlation with Cu and

As in the mid-estuary locations, but also a pool of metal and Alpha HCH contaminants could explain an additive effect in the deteoxification response of the worms by GST (Fig. 7).

In particular, the cytotoxicity of sediments is the most intriguing characteristic even though and it is not clearly related to any of the individual contaminants analysed here. Given the distinction between contamination (as enhanced levels of substances) and pollution (implying a biological health response) (McLusky and [START_REF] Mclusky | The Estuarine Ecosystem; ecology, threats and management, 3 rd Edn[END_REF] then cytotoxicity of sediments is of high relevance here despite not being related to any single contaminant analysed here. Biological effects observed in the field (i.e. de facto pollution), the association with their cause (de facto contaminants) and the link with subtle ecological consequences are poorly understood or lacking completely, including in very well studied estuarine systems [START_REF] Langston | Contaminants in fine sediments and their consequences for biota of the Severn Estuary[END_REF]. As shown here, despite the presence of well-studied contaminants, the specific pollutant responses should occur but often do not, as the case of some metals and genotoxicity in this work. Similarly, locations which had specific pollutant responses were not contaminated by potential pollutants such as Dioxin-like

M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT
22 substances and PCBs. Hence this implies that often, the different approaches could give false positive results, that the responses are not sufficiently strong or that the complex system has the ability to absorb the change, i.e. environmental homeostasis [START_REF] Elliott | The Estuarine Quality Paradox, Environmental Homeostasis and the difficulty of detecting anthropogenic stress in naturally stressed areas[END_REF]. The latter suggests the ecosystem has the ability to absorb changes such that changes at the genetic or cellular levels do not follow through the system and do not lead to higher level changes at the individual, population or community levels. If the system can be subject to a stressor but there is no resulting effect at higher biological levels then the system has absorbed the stress ergo environmental homeostasis [START_REF] Elliott | The Estuarine Quality Paradox, Environmental Homeostasis and the difficulty of detecting anthropogenic stress in naturally stressed areas[END_REF].

Consequently, we can indicate the sequence of changes at the lower levels of biological organisation based on the results here to show the relationship between contaminants and toxicities from our results and the potential outcome. Despite this, it

is not yet possible to show the consequences of these at the higher levels of biological organisation. The presence of differential biological responses to the presence of contaminants along the estuary may have some influence on population tolerance, survival or adaptation (e.g. GST activity) and these harmful effects produced by contamination, and thus constituting de facto pollution, should be included in environmental risk assessments. For example, there are small areas of the Humber where the benthic communities are impoverished as a direct result of industrial discharge (e.g. [START_REF] Mazik | The effects of chemical pollution on the bioturbation potential of estuarine intertidal mudflats[END_REF]. Such severe impacts are highly localised and there have been significant efforts (through the Environment Agency's Review of Consents) to improve effluent quality throughout the estuary in order to further minimise or eliminate such impacts [START_REF] Mazik | Environmental Impact Assessment for the Killingholme Refinery discharge[END_REF].

To date, no links have been found between cellular, genetic or biochemical responses and population/community level responses. As such, there is no way of S1. Mean, median, minimum and maximum concentrations of metals and metalloids, PCBs, organochlorines, xeno-estrogens (µg g -1 ), genotoxicity and cytotoxicity (LEC in g ml -1 ), DR-Calux (PCDD/F-PCB-TEQ g -1 ) and ER-Calux (pg EEQ g -1 ) of sediment; GST activity (nmol min -1 mg -1 of protein) and organic matter content (% of dry weight), in sediment and worms samples (n = 12) collected at the Humber estuary. <DL, below the quantitation limit.

  Samples were taken during the low tide in the upper intertidal zone during May 2007. The upper 5 cm layer of brownish oxidised sediment and ragworms were transported in a cool box at 8 ºC to the University of Hull. Sediments were dried and stored at -20 ºC. Worms (H. diversicolor) were dried with absorbent paper, weighed, liquid nitrogen and stored at -80 ºC until analysis. Sediment organic content was measured as percentage loss on ignition (dry sediment kept at 550 ºC for 3 hours). Salinity was obtained from measuring sodium (Na) concentration by optical emission spectroscopy (Optima 5300DV, Perkin Elmer) in dry sediment. Sediment moisture was calculated and concentration of Na per interstitial water volume was estimated.

(

  Mix 71) and individual PCBs standards were purchased from Dr. Ehrenstofer Laboratories, Germany and Accute Standards, USA. Nonylphenol (technical mixture isomers), octylphenol 99 %, Bisphenol A 99+ %, the internal standard 4-(3,6-heptyl)phenol (ring-13 C 6 ) b ([13C6]-363 NP) and derivatising agent MSTFA, were obtained from Sigma-Aldrich, Poole, UK.

  cells become increasingly fluorescent as GFP accumulates.

(

  corresponding to Paull and Hessle) and thus the quadratic model provides a good fit to their data. Hence the elements show a quadratic pattern in relation to the locations, M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT

  Mid and upstream locations (Hessle and Blacktoft) showed the highest values with 320 and 346 pg EEQ sediment respectively, while sediments from Kilnsea and Paull showed 191 and 171 pg EEQ g -1 of sediment respectively. No correlation was observed with organic matter content (Fig.5).The DR-CALUX test showed a very low response at all locations, suggesting no significant dioxin-like pollution in sediments of the Humber Estuary and reflecting the low levels of PCBs measured in the sediments. Upstream sediments (Blacktoft and Hessle) showed the highest dioxin and dioxin-like PCB response, which negatively correlated with salinity (Fig.5). Samples from Hessle had the highest value 71 pg PCDD/F-PCB-TEQ g -1 of sediment and Blacktoft 52 pg PCDD/F-PCB-TEQ g -1 of sediment while Paull and Kilnsea showed 23 and 25 pg PCDD/F-PCB-TEQ g -1

Figure 1 .

 1 Figure Legends

Figure 2 .

 2 Figure 2. Relationships between the concentration of metals and metalloids and

Figure 3 .

 3 Figure 3. Relationships between the concentrations of organochlorines and

Figure 4 .M

 4 Figure 4. Relationships between the concentration of xeno-estrogens (NP, OP and

Figure 5 .

 5 Figure 5. Toxic responses in relation to environmental parameters. Relationships between the Glutathione-S-transferase (GST) activity in nmol min -1 mg -1 of protein

Figure 6 .

 6 Figure 6. Significant relationships (p < 0.05) between metal concentrations,

Figure 7 .

 7 Figure 7. PCA showing the associations between contaminants (green), biological

Figure S1 .Figure S2 .Figure S3 .

 S1S2S3 Figure S1. Correlation analyses within the elements found in the intertidal mudflat of

Acknowledgments

The authors would like to thank H Besselink (BDS B.V. Amsterdam) for ER-Calux and DR-Calux assays, R Knight for metal analysis and Dr Z Velez for helping in field

Electronic Supplementary material

A summary of the concentrations of inorganic and organic pollutants measured (Table S1). The correlations between inorganic pollutants (Fig. S1). The correlations between organochlorines (Fig. S2) and the correlations between inorganic and organic pollutants (Fig. S3).


Christophe Prieur
email: christophe.prieur@gipsa-lab.grenoble-inp.fr

Frédéric Mazenc
email: frederic.mazenc@lss.supelec.fr

ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws

Keywords: Strictification, Lyapunov function, hyperbolic PDE, system of balance laws

A family of time-varying hyperbolic systems of balance laws is considered. The partial differential equations of this family can be stabilized by selecting suitable boundary conditions. For the stabilized systems, the classical technique of construction of Lyapunov functions provides a function which is a weak Lyapunov function in some cases, but is not in others. We transform this function through a strictification approach to obtain a time-varying strict Lyapunov function. It allows us to establish asymptotic stability in the general case and a robustness property with respect to additive disturbances of Input-to-State Stability (ISS) type. Two examples illustrate the results.

Introduction

Lyapunov function based techniques are central in the study of dynamical systems. This is especially true for those having an infinite number of dynamics. These systems are usually modelled by time-delay systems or partial differential equations (PDEs). For the latter family of systems, Lyapunov functions are useful for the analysis of many different types of problems, such as the existence of solutions for the heat equation [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF], or the controllability of the semilinear wave equation [START_REF] Coron | Global steady-state stabilization and controllability of 1D semilinear wave equations[END_REF]. The present paper focuses on a class of onedimensional hyperbolic equations like those written as a system of conservation laws. The study of this class of PDEs is crucial when considering a wide range of physical networks having an engineering relevance. Among the potential applications we have in mind, there are hydraulic networks (for irrigation or navigation), electric line networks, road traffic networks [START_REF] Haut | A second order model of road junctions in fluid models of traffic networks[END_REF] or gas flow in pipeline networks [START_REF] Bastin | Using hyperbolic systems of balance laws for modeling, control and stability analysis of physical networks[END_REF][START_REF] Dick | Classical solutions and feedback stabilization for the gas flow in a sequence of pipes[END_REF]. The importance of these applications motivates a lot of theoretical questions on hyperbolic systems which for instance pertain to optimal control and controllability as considered in [START_REF] Colombo | Optimal control in networks of pipes and canals[END_REF][START_REF] Gugat | Optimal control for traffic flow networks[END_REF][START_REF] Gugat | Global boundary controllability of the Saint-Venant system for sloped canals with friction[END_REF].

The stabilizability of such systems is often proved by means of a Lyapunov function as illustrated by the contributions [START_REF] Dick | Classical solutions and feedback stabilization for the gas flow in a sequence of pipes[END_REF][START_REF] Krstic | Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF][START_REF] Prieur | Stabilization of a 1-D tank containing a fluid modeled by the shallow water equations[END_REF] where different control problems are solved for particular hyperbolic equations. For more general nonlinear hyperbolic equations, the knowledge of Lyapunov functions can be useful for the stability analysis of a system of conservation laws (see [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF]), or even for the design for these systems of stabilizing boundary controls (see the recent work [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF]).

To demonstrate asymptotic stability through the knowledge of a weak Lyapunov function i.e. a Lyapunov function whose derivative, along the trajectories of the system which is considered, is nonpositive, the celebrated LaSalle invariance principle has to be invoked (see e.g. [START_REF] Cazenave | An introduction to semilinear evolution equations[END_REF][START_REF] Luo | Stability and stabilization of infinite dimensional systems and applications[END_REF][START_REF] Slemrod | A note on complete controllability and stabilizability for linear control systems in Hilbert space[END_REF]). It requires to state a precompactness property for the solutions, which may be difficult to prove (and is not even always satisfied, as illustrated by the hyperbolic systems considered in [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF]). This technical step is not needed when is available a strict Lyapunov function i.e. a Lyapunov function whose derivative, along the trajectories of the system which is considered, is negative definite. Thus designing such a strict Lyapunov function is a way to overcome this technical difficulty, as done for example in [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF]. These remarks motivate the present paper which is devoted to new Lyapunov techniques for the study of stability and robustness properties of Input-to-State-Stable (ISS) type for a family of time-varying linear hyperbolic PDEs with disturbances. By first applying the classical technique of construction of Lyapunov functions available in the literature, we will obtain a function which is a weak Lyapunov function for some of the systems we consider, but not for the others. Next, we will transform this function through a strictification approach, which owes a great deal to the one presented in [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF], (see also [START_REF] Mazenc | A simplified design for strict Lyapunov functions under Matrosov conditions[END_REF][START_REF] Mazenc | Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem[END_REF]) and obtain that way a strict Lyapunov function that makes it possible to estimate the robustness of the stability of the systems with respect to the presence of uncertainties and/or external disturbances. This function is given by an explicit expression.

It is worth mentioning that although the ISS notion is very popular in the area of the dynamical systems of finite dimension (see e.g. the recent surveys [START_REF] Jayawardhana | The circle criterion and input-to-state stability[END_REF][START_REF] Sontag | Input to state stability: Basic concepts and results[END_REF]), or for systems with delay (see for instance [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF]), the present work is, to the best of our knowledge, the first one which uses it to characterize a robustness property for hyperbolic PDEs. This work parallels what has been done in [START_REF] Mazenc | Strict Lyapunov functions for semilinear parabolic partial differential equations[END_REF] where ISS-Lyapunov functions for semilinear time-invariant parabolic PDEs are derived using strictification techniques (see also [START_REF] Dashkovskiy | On the uniform input-to-state stability of reactiondiffusion systems[END_REF] where ISS properties are compared for a reaction-diffusion system with its finite dimensional counterpart without diffusion). On the other hand, the construction we shall present is significantly different from the one in [START_REF] Mazenc | Strict Lyapunov functions for semilinear parabolic partial differential equations[END_REF] because the family of systems we will study is very different. In particular the systems studied in [START_REF] Mazenc | Strict Lyapunov functions for semilinear parabolic partial differential equations[END_REF] are time-invariant whereas the systems considered in the present work are time-varying. Moreover an existence result for the stability when the perturbations are vanishing is given in [START_REF] Prieur | Robust boundary control of systems of conservation laws[END_REF] by studying the Riemann coordinates. The present paper gives a constructive estimation even when the perturbations are not vanishing.

The main result of the present paper is applied to the problem of the regulation of the flow in a channel. This problem has been considered for a long time in the literature, as reported in the survey paper [START_REF] Malaterre | Classification of canal control algorithms[END_REF] which involves a comprehensive bibliography, and is still an active field of research, as illustrated for instance in [START_REF] Pham | Predictive control with guaranteed stability for hyperbolic systems of conservation laws[END_REF] where a predictive control is computed using a Lyapunov function. We will consider the linearized Saint-Venant-Exner equations which model the dynamics of the flow in an open channel coupled with the moving sediment bed, see [START_REF] Graf | Hydraulics of sediment transport[END_REF][START_REF] Hudson | Formulations for numerically approximating hyperbolic systems governing sediment transport[END_REF] (see also [START_REF] Diagne | Lyapunov exponential stability of linear hyperbolic systems of balance laws[END_REF]). For this hyperbolic system of balance laws, we will construct a strict Lyapunov function from which an ISS property can be derived.

The paper is organized as follows. Basic definitions and notions are introduced in Section 2. In Section 3 the analysis of the robustness of a linear time-varying hyperbolic PDE with uncertainties is carried out by means of the design of an ISS-Lyapunov function. In Section 4, the main result is illustrated by an academic example. Section 5 is devoted to an other application, that is the design of stabilizing boundary controls for the Saint-Venant-Exner equations. Concluding remarks in Section 6 end the work.

Notation. Throughout the paper, the argument of the functions will be omitted or simplified when no confusion can arise from the context. A continuous function α : [0, ∞) → [0, ∞) belongs to class K provided it is increasing and α(0) = 0. It belongs to class K ∞ if, in addition, α(r) → ∞ as r → ∞. For any integer n, we let Id denote the identity matrix of dimension n. Given a vector ξ in R n , the components of ξ are denoted ξ i for each i = 1 . . . , n. Given a continuously differentiable function A : R n → R, ∂A ∂Ξ (Ξ) stands for the vector ( ∂A ∂ξ1 (Ξ), . . . , ∂A ∂ξn (Ξ)) ∈ R n . The norm induced from the Euclidean inner product of two vectors will be denoted by | • |. Given a matrix A, its induced matrix norm will be denoted by ||A||, and

Sym(A) = 1 2 (A + A ⊤ )
stands for the symmetric part of A. Moreover, given a vector (a 1 , . . . , a n ) in R n , Diag(a 1 , . . . , a n ) is the diagonal matrix with the vector (a 1 , . . . , a n ) on its diagonal. We denote the set of diagonal positive definite matrices in R n×n by

D n,+ . The norm | • | L 2 (0,L) is defined by: |φ| L 2 (0,L) = L 0 |φ(z)| 2 dz, for any function φ : (0, L) → R n such that L 0 |φ(z)| 2 dz < +∞.
Finally, given two topological spaces X and Y , we denote by C 0 (X; Y ) (resp. C 1 (X; Y )) the set of the continuous (resp. continuously differentiable) functions from X to Y . Following [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF], we introduce the notation, for all matrices M ∈ R n×n ,

ρ 1 (M ) = inf{ ∆M ∆ -1 , ∆ ∈ D n,+ } .
(1)

Basic definitions and notions

Throughout our work, we will consider linear partial differential equations of the form

∂X ∂t (z, t) + Λ(z, t) ∂X ∂z (z, t) = F (z, t)X(z, t) + δ(z, t) , (2)
where z ∈ [0, L], t ∈ [0, +∞), and Λ(z, t) = Diag(λ 1 (z, t), . . . , λ n (z, t)) is a diagonal matrix in R n×n whose m first diagonal terms are nonnegative and the n-m last diagonal terms are nonpositive (we will say that the hyperbolicity assumption holds, when additionally the λ i 's are never vanishing). We assume that the function δ is a disturbance of class C 1 , F is a periodic function with respect to t of period T and of class C 1 , Λ is a function of class C 1 , periodic with respect to t of period T . The boundary conditions are written as

X + (0, t) X -(L, t) = K X + (L, t) X -(0, t) , (3)
where

X = X + X - , X + ∈ R m , X -∈ R n-m , and K ∈ R n×n is a constant matrix.
The initial condition is

X(z, 0) = X 0 (z) , ∀z ∈ (0, L) , (4)
where X 0 is a function [0, L] :→ R n of class C 1 satisfying the following zeroorder compatibility condition:

X 0 + (0) X 0 -(L) = K X 0 + (L) X 0 -(0) (5)
and the following first-order compatibility condition:

            -λ 1 (0, 0) dX 0 1 dz (0) + (F (0, 0)X 0 (0) + δ(0, 0)) 1 . . .
-λ m (0, 0)

dX 0 m dz (0) + (F (0, 0)X 0 (0) + δ(0, 0)) m -λ m+1 (L, 0) dX 0 m+1 dz (L) + (F (L, 0)X 0 (L) + δ(L, 0)) m+1 . . . -λ n (L, 0) dX 0 n dz (L) + (F (L, 0)X 0 (L) + δ(L, 0)) n             = K             -λ 1 (L, 0) dX 0 1 dz (L) + (F (L, 0)X 0 (L) + δ(L, 0)) 1 . . . -λ m (L, 0) dX 0 m dz (L) + (F (L, 0)X 0 (L) + δ(L, 0)) m -λ m+1 (0, 0) dX 0 m+1 dz (0) + (F (0, 0)X 0 (0) + δ(0, 0)) m+1
. . .

-λ n (0, 0)

dX 0 n dz (0) + (F (0, 0)X 0 (0) + δ(0, 0)) n             . (6)
As proved in [START_REF] Kmit | Classical solvability of nonlinear initial-boundary problems for first-order hyperbolic systems[END_REF], if the function Λ is of class C 1 and satisfies the hyperbolicity assumption, if the function δ is of class C 1 , and if the initial condition is of class C 1 and satisfies the compatibility conditions ( 5) and ( 6), there exists an unique classical solution of the system (2), with the boundary conditions (3) and the initial condition (4), defined for all t ≥ 0. Now we introduce the notions of weak, strict and ISS-Lyapunov functions that we consider in this paper (see for instance [START_REF] Luo | Stability and stabilization of infinite dimensional systems and applications[END_REF]Def. 3.62] for the notion of Lyapunov function and [START_REF] Mazenc | Strict Lyapunov functions for semilinear parabolic partial differential equations[END_REF] for the notion of ISS-Lyapunov function in an infinite dimensional context).

Definition 1 Let ν : L 2 (0, L) × R → R be a continuously differentiable function, periodic with respect to its second argument. The function ν is said to be a weak Lyapunov function for (2) with (3), if there are two functions κ S and κ M of class K ∞ such that, for all functions φ ∈ L 2 (0, L) and for all t ∈ [0, +∞),

κ S |φ| L 2 (0,L) ≤ ν(φ, t) ≤ L 0 κ M (|φ(z)|) dz (7)
and, in the absence of δ, for all solutions of (2) satisfying ( 3), and all t ≥ 0, dν(X(., t), t) dt ≤ 0 .

The function ν is said to be a strict Lyapunov function for ( 2) with (3) if, in the absence of δ, there exists a real number λ 1 > 0 such that, for all solutions of (2) satisfying (3), and for all t ≥ 0,

dν(X(., t), t) dt ≤ -λ 1 ν(X(., t), t) .
The function ν is said to be an ISS-Lyapunov function for ( 2) with (3) if there exist a positive real number λ 1 > 0 and a function λ 2 of class K such that, for all continuous functions δ, for all solutions of (2) satisfying (3), and for all t ≥ 0, dν(X(., t), t) dt ≤ -λ 1 ν(X(., t), t)

+ L 0 λ 2 (|δ(z, t)|)dz .
Remark 1 1. For conciseness, we will often use the notation ν instead of dν(X(.,t),t) dt .

2.

Let us recall that, when is known a weak Lyapunov function, asymptotic stability can be often established via the celebrated LaSalle invariance principle (see [START_REF] Luo | Stability and stabilization of infinite dimensional systems and applications[END_REF]Theorem 3.64] among other references).

3. When a strict Lyapunov function ν exists for ( 2) with (3) and δ is not present, then the value of ν along the solutions of ( 2) satisfying (3) exponentially decays to zero and therefore each solution X(z, t) satisfies lim t→+∞ |X(., t)| L 2 (0,L) = 0. When in addition, there exists a function κ L of class K ∞ , such that, for all functions φ ∈ L 2 (0, L) and for all t ≥ 0,

ν(φ, t) ≤ κ L |φ| L 2 (0,L) , (8)
then the system (2) is globally asymptotically stable for the topology of the norm L 2 . 4. When the system (2) with ( 3) admits an ISS-Lyapunov function ν, then, one can check through elementary calculations 1 that, for all solutions of (2) satisfying ( 3) and for all instants t ≥ t 0 , the inequality

|X(., t)| L 2 (0,L) ≤ κ -1 S (ϕ 1 (t, t 0 , X)) + κ -1 S (ϕ 2 (t, t 0 )) with ϕ 1 (t, t 0 , X) = 2e -λ1(t-t0) L 0 κ M (|X(z, t 0 )|)dz and ϕ 2 (t, t 0 ) = 2 λ 1 sup τ ∈[t0,t] L 0 λ 2 (|δ(z, τ )|)dz
holds. This inequality is the analogue for the PDEs (2) with (3) of the ISS inequalities for ordinary differential equations. It gives an estimation of the influence of the disturbance δ on the solutions of the system (2) with (3). •

1 For instance this inequality follows from the fact that we have, for all κ of class K (that is for all continuous, zero at zero, and increasing functions κ : [0, +∞) → [0, +∞)), and for all positive values a and b, κ(a

+ b) ≤ κ(2a) + κ(2b) ,
and from the fact that the function κ -1 S is zero at zero and increasing.

ISS-Lyapunov functions for hyperbolic systems

Before stating the main theoretical result of the work, some comments are needed. Since, in the case where the system (2) is such that m < n, we can replace X(z, t) by

X + (z, t) X -(L -z, t)
and we may assume without loss of generality that Λ is diagonal and positive semidefinite. Then the boundary conditions (3) become

X(0, t) = KX(L, t) . (9)
Next, we recall an important result given in [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF] because it sheds light on the problem we consider and, more precisely, on the assumptions we introduce below. If Λ is constant, diagonal and positive definite and if ρ 1 (K) < 1, where the ρ 1 is the function defined in (1), then the system (2), when F (z, t) = 0 and δ(z, t) = 0 for all z ∈ [0, L], t ≥ 0, with the boundary conditions ( 9) is exponentially stable in H 2 -norm. Moreover, there exist a diagonal positive definite matrix 2 Q ∈ R n×n and a positive constant ε > 0 such that

Sym(QΛ -K ⊤ QΛK) ≥ εId . (10)
Furthermore, following what has been assumed for parabolic equations in [START_REF] Mazenc | Strict Lyapunov functions for semilinear parabolic partial differential equations[END_REF], it might seem natural to consider the case where F (z, t) possesses some stability properties. On the other hand, there is no reason to believe that this property is always needed. These remarks lead us to introduce the following assumption:

Assumption 1 For all t ≥ 0 and for all z ∈ [0, L], all the entries of the diagonal matrix Λ(z, t) are nonnegative. There exist a symmetric positive definite matrix Q, a real number α ∈ (0, 1), a continuous real-valued function r, periodic of period T > 0 such that the constant

B = T 0 max{r(m), 0} ||Q|| + min{r(m), 0} λ Q dm , (11)
where λ Q is the smallest eigenvalue of Q, is positive. Moreover, for all t ≥ 0 and for all z ∈ [0, L], the following inequalities:

Sym αQΛ(L, t) -K ⊤ QΛ(0, t)K ≥ 0 , (12)
Sym (QΛ(z, t)) ≥ r(t)Id , (13)
Sym Q ∂Λ ∂z (z, t) + 2QF (z, t) ≤ 0 ( 14
)
are satisfied.

2 such a matrix Q may be obtained by selecting a diagonal positive definite matrix ∆ such that ∆K∆ -1 < Id. Then selecting Q = ∆ 2 Λ -1 and ε > 0 sufficiently small we have [START_REF] Diagne | Lyapunov exponential stability of linear hyperbolic systems of balance laws[END_REF].

Let us introduce the function

q(t) = µ max{r(t), 0} ||Q|| + min{r(t), 0} λ Q - µB 2T , (15)
where Q and r are the matrix and the function in Assumption 1

We are ready to state and prove the main result of the paper.

Theorem 1 Assume the system (2) with the boundary conditions ( 9) satisfies Assumption 1. Let µ be any real number such that

0 < µ ≤ - 1 L ln(α) . ( 16
)
Then the function U :

L 2 (0, L) × R → R defined, for all φ ∈ L 2 (0, L) and t ∈ R, by U (φ, t) = exp 1 T t t-T t ℓ q(m)dmdℓ L 0 φ(z) ⊤ Qφ(z)e -µz dz , ( 17
)
where q is the function defined in [START_REF] Gugat | Optimal control for traffic flow networks[END_REF], is an ISS-Lyapunov function for the system (2) with the boundary conditions (9).

Remark 2 1. Assumption 1 does not imply that for all fixed z ∈ [0, L], the ordinary differential equation Ẋ = F (z, t)X is stable. In Section 4, we will study an example where this system is unstable.

2. The fact that Q is symmetric positive definite and all the entries of Λ(z, t) are nonnegative does not imply that Sym (QΛ(z, t)) is positive definite. That is the reason why we do not assume that r is a nonnegative function.

3. Assumption 1 holds when, in the system (2), Λ(z, t) is constant, F (z, t) is constant, δ(z, t) = 0 for all z ∈ [0, L] and t ≥ 0 and the boundary condition (3) satisfies Sym QΛ -K ⊤ QΛK ≥ 0 , Sym (QF ) ≤ 0 for a suitable symmetric positive definite matrix Q. Therefore Theorem 1 generalizes the sufficient conditions of [START_REF] Diagne | Lyapunov exponential stability of linear hyperbolic systems of balance laws[END_REF] for the exponential stability of linear hyperbolic systems of balance laws (when F is diagonally marginally stable), to the time-varying case and to the semilinear perturbed case (without assuming that F is diagonally marginally stable).

4. The Lyapunov function U defined in ( 17) is a time-varying function, periodic of period T . In the case where the system is time-invariant, one can choose a constant function q, and then it is obtained a time-invariant function [START_REF] Haut | A second order model of road junctions in fluid models of traffic networks[END_REF] which is a quite usual Lyapunov function candidate in the context of the stability analysis of PDEs (see e.g., [START_REF] Coron | On the null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domain[END_REF][START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF][START_REF] Xu | Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems[END_REF]).

•

Proof. We begin the proof by showing that the function V defined by, for all φ ∈ L 2 (0, L),

V (φ) = L 0 φ(z) ⊤ Qφ(z)e -µz dz
is a weak Lyapunov function for the system (2) with the boundary conditions (9) when Assumption 1 and ( 16) are satisfied, r is a nonnegative function, and δ is identically equal to zero.

We note for later use that, for all φ ∈ L 2 (0, L),

λ Q L 0 |φ(z)| 2 e -µz dz ≤ V (φ) ≤ ||Q|| L 0 |φ(z)| 2 e -µz dz . (18)
Now, we compute the time-derivative of V along the solutions of ( 2) with ( 9):

V = 2 L 0 X(z, t) ⊤ Q ∂X ∂t (z, t)e -µz dz = 2 L 0 X(z, t) ⊤ Q -Λ(z, t) ∂X ∂z (z, t) + F (z, t)X(z, t) + δ(z, t) e -µz dz = -R 1 (X(., t), t) + R 2 (X(., t), t) + R 3 (X(., t), t) , (19)
with

R 1 (φ, t) = 2 L 0 φ(z) ⊤ QΛ(z, t) ∂φ ∂z (z)e -µz dz , R 2 (φ, t) = 2 L 0 φ(z) ⊤ QF (z, t)φ(z)e -µz dz , (20)
R 3 (φ, t) = 2 L 0 φ(z) ⊤ Qδ(z, t)e -µz dz . (21)
Now, observe that

R 1 (φ, t) = L 0 ∂(φ(z) ⊤ QΛ(z, t)φ(z)) ∂z e -µz dz - L 0 φ(z) ⊤ Q ∂Λ ∂z (z, t)φ(z)e -µz dz .
Performing an integration by parts on the first integral, we obtain

R 1 (φ, t) = φ(L) ⊤ QΛ(L, t)φ(L)e -µL -φ(0) ⊤ QΛ(0, t)φ(0) +µ L 0 φ(z) ⊤ QΛ(z, t)φ(z)e -µz dz - L 0 φ(z) ⊤ Q ∂Λ ∂z (z, t)φ(z)e -µz dz . (22)
Combining ( 19) and ( 22), we obtain V = -X(L, t) ⊤ QΛ(L, t)X(L, t)e -µL + X(0, t) ⊤ QΛ(0, t)X(0, t)

+R 4 (X(., t), t) + R 3 (X(., t), t) , with R 4 (φ, t) = -µ L 0 φ(z) ⊤ QΛ(z, t)φ(z)e -µz dz + L 0 φ(z) ⊤ Q ∂Λ ∂z (z, t)φ(z)e -µz dz + R 2 (φ, t) ,
where R 2 is the function defined in [START_REF] Kmit | Classical solvability of nonlinear initial-boundary problems for first-order hyperbolic systems[END_REF]. Using [START_REF] Dashkovskiy | On the uniform input-to-state stability of reactiondiffusion systems[END_REF], we obtain V = -X(L, t) ⊤ QΛ(L, t)X(L, t)e -µL + X(L, t) ⊤ K ⊤ QΛ(0, t)KX(L, t) +R 4 (X(., t), t) + R 3 (X(., t), t) .

By grouping the terms and using the notation

N (t) = K ⊤ QΛ(0, t)K
we obtain V = X(L, t) ⊤ N (t)e -µL QΛ(L, t) X(L, t) + R 4 (X(., t), t) + R 3 (X(., t), t) .

Grouping the terms in R 4 (X(., t), t), we obtain

V = X(L, t) ⊤ N (t) -e -µL QΛ(L, t) X(L, t) - L 0 X(z, t) ⊤ QM (z, t)X(z, t)e -µz dz + R 3 (X(., t), t) , with M (z, t) = µΛ(z, t) - ∂Λ ∂z (z, t) -2F (z, t) .
The inequalities ( 12) and ( 16)

imply that V ≤ - L 0 X(z, t) ⊤ QM (z, t)X(z, t)e -µz dz + R 3 (X(., t), t) .
It follows from ( 14) that

V ≤ -µ L 0 X(z, t) ⊤ QΛ(z, t)X(z, t)e -µz dz + R 3 (X(., t), t) .
Using [START_REF] Graf | Hydraulics of sediment transport[END_REF], we deduce that V ≤ -µr(t) L 0 |X(z, t)| 2 e -µz dz + R 3 (X(., t), t) .

From [START_REF] Hudson | Formulations for numerically approximating hyperbolic systems governing sediment transport[END_REF] and the definition of R 3 in (21), we deduce that

V ≤ -µ max{r(t),0} ||Q|| + min{r(t),0} λ Q V (X(., t)) + 2||Q|| L 0 |X(z, t)||δ(z, t)|e -µz dz .
From Cauchy-Schwarz inequality, it follows that, for all κ > 0,

V ≤ -µ max{r(t), 0} ||Q|| + min{r(t), 0} λ Q V (X(., t)) +2||Q||κ L 0 |X(z, t)| 2 e -µz dz + ||Q|| 2κ L 0 |δ(z, t)| 2 e -µz dz (23)
≤ -q κ (t)V (X(., t)) + ||Q|| 2κ L 0 |δ(z, t)| 2 dz , (24)
with q κ (t) = µ max{r(t),0}

||Q|| [START_REF] Löfberg | Yalmip : A toolbox for modeling and optimization in MATLAB[END_REF] implies that when r(t) is nonnegative, δ is identically equal to zero and κ small enough, the function V is a weak Lyapunov function for the system (2) with the initial conditions [START_REF] Dashkovskiy | On the uniform input-to-state stability of reactiondiffusion systems[END_REF]. However, we did not assume that the function r is nonnegative and we aim at establishing that the system is ISS with respect to δ. This leads us to apply a strictification technique which transforms V into a strict Lyapunov function. The technique of [START_REF] Malisoff | Constructions of Strict Lyapunov Functions[END_REF]Chapter 11] leads us to consider the time-varying candidate Lyapunov function

+ min{r(t),0} λ Q -2||Q||κ λ Q . The inequality
U κ (φ, t) = e sκ(t) V (φ) , with s κ (t) = 1 T t t-T t ℓ
q κ (m)dmdℓ. Through elementary calculations, one can prove the following:

Claim 2 For all t in R, we have d dt s κ (t) = q κ (t) -1 T t t-T q κ (m)dm.
With [START_REF] Luo | Stability and stabilization of infinite dimensional systems and applications[END_REF] and Claim 2, we get that the time-derivative of U κ along the solutions of ( 2) with the initial conditions (9) satisfies:

Uκ ≤ -e sκ(t) q κ (t)V (X(., t))

+ ||Q|| 2κ e sκ(t) L 0 |δ(z, t)| 2 dz +e sκ(t) q κ (t) -1 T t t-T q κ (m)dm V (X(., t)) ≤ ||Q|| 2κ e sκ(t) L 0 |δ(z, t)| 2 dz -e sκ(t) 1 T t t-T q κ (m)dm V (X(., t)) .
Since r is periodic of period T , we have

t t-T q κ (m)dm = µ t t-T max{r(m), 0} ||Q|| + min{r(m), 0} λ Q dm - 2T ||Q||κ λ Q = µB -2T ||Q||κ λ Q ,
where B is the constant defined in [START_REF] Dick | Classical solutions and feedback stabilization for the gas flow in a sequence of pipes[END_REF]. We deduce that the value

κ = µBλ Q 4T ||Q|| , (25)
which is positive because B is positive, gives

U ≤ -µ 2T BU (X(., t), t) + ||Q|| 2κ e sκ(t) L 0 |δ(z, t)| 2 dz ≤ -µ 2T BU (X(., t), t) + c 1 L 0 |δ(z, t)| 2 dz , with c 1 = 2T ||Q|| 2 µBλ Q e T µr M ||Q|| , r M = sup {m∈[0,T ]}
{r(m)} and U = U κ for κ defined in [START_REF] Malaterre | Classification of canal control algorithms[END_REF]. Moreover, [START_REF] Hudson | Formulations for numerically approximating hyperbolic systems governing sediment transport[END_REF] ensures that there are two positive constants c 2 and c 3 such that, for all t ∈ R and φ ∈ L 2 (0, L),

c 2 L 0 |φ(z)| 2 dz ≤ U (φ, t) ≤ c 3 L 0 |φ(z)| 2 dz .
Therefore inequalities of the type (7) are satisfied. We deduce that U is an ISS-Lyapunov function, as introduced in Definition 1. This concludes the proof of Theorem 1. •

Benchmark example

In this section we consider the system (2) and the boundary conditions ( 3) with the following data, for all z in [0, L] and for all t ≥ 0,

X(z, t) ∈ R , L = 1 , Λ(z, t) = sin 2 (t) cos 2 (t) + 1 -1 2 z , F (z, t) = sin 2 (t) 5 , K = 1 2 .
A remarkable feature of this system if that the system ξ = F (z, t)ξ, which rewrites as ξ = sin 2 (t) 5 ξ, is exponentially unstable. Now, we show that Theorem 1 applies to the PDE we consider. Let Q = 1 and α = 1 2 . Then we have, for all t ≥ 0,

αQΛ(1, t) -K ⊤ QΛ(0, t)K = 1 4 sin 2 (t) cos 2 (t)
and thus ( 12) is satisfied. It is clear that [START_REF] Graf | Hydraulics of sediment transport[END_REF] holds with the function r(t) = sin 2 (t) cos 2 (t) + 1 2 , which is periodic of period 2π. Then the corresponding value of B is positive, and we compute B = 3π 4 . Finally, for all z ∈ [0, 1], t ≥ 0,

Sym Q ∂Λ ∂z (z, t) + 2QF (z, t) = -sin 2 (t) 2 + 2 sin 2 (t) 5 ≤ -sin 2 (t) 10
.

Therefore [START_REF] Graf | Fluvial hydraulics[END_REF] holds and Assumption 1 is satisfied. We conclude that Theorem 1 applies. It follows that the system defined by

∂X ∂t (z, t) + sin 2 (t) cos 2 (t) + 1 - 1 2 z ∂X ∂z (z, t) = sin 2 (t) 5 X(z, t) + δ(z, t) (26
) for all z in (0, 1) and for all t ≥ 0, with the boundary condition

X(0, t) = 1 2 X(1, t)
is asymptotically stable and, for this system, the function ( 17) is an ISS-Lyapunov function (with respect to δ). Let us compute this ISS-Lyapunov function. With [START_REF] Gugat | Optimal control for traffic flow networks[END_REF] and the choice Q = 1 and µ = 1 2 , we compute

1 2π t t-2π t ℓ q(m)dmdℓ = 1 2π t t-2π t ℓ 1 2 sin 2 (m)[cos 2 (m) + 1 2 ] -3 32 dmdℓ , = 1 2π t t-2π t ℓ 1-cos(2m) 8 + 1-cos(4m) 16 -3 32 dmdℓ , = 1 2π t t-2π 3 32 (t -l) -sin(2t) 16 -sin(4t) 64 dℓ , = 3π
32 -sin(2t)

16

sin(4t)

64

.

Therefore the ISS-Lyapunov function is given by the expression, for all φ in L 2 (0, 1) and for all t ≥ 0,

U (φ, t) = exp 3π 32 - 1 16 sin(2t) - 1 64 sin(4t) 1 0 |φ(z)| 2 e -1 2 z dz . ( 27
)
To numerically check the stability and the ISS property of the system, let us discretize the hyperbolic equation ( 26) using a two-step variant of the Lax-Friedrichs (LxF) method [START_REF] Shampine | Two-step Lax-Friedrichs method[END_REF] and the solver [START_REF] Shampine | Solving hyperbolic PDEs in Matlab[END_REF] on Matlab c3 . We select the parameters of the numerical scheme so that the CFL condition for the stability holds. More precisely, we divide the space domain [0, 1] into 100 intervals of identical length, and, choosing 10 as final time, we set a time discretization of 5 × 10 -3 . For the initial condition, we select the function X(z, 0) = z + 1, for all z ∈ [0, 1]. For the perturbation, we choose, for all z ∈ [0, 1],

δ(z, t) = sin 2 (πt) when t < 5 and δ(z, t) = 0 when t ≥ 5 .

The time evolutions of the solution X and of the Lyapunov function U given by ( 27) are in Figures 1 and2 respectively. We can observe that the solution converges as expected to the equilibrium.

5 Application on the design of boundary control for the Saint-Venant-Exner equations

Dynamics using the physical and Riemann variables

In this section, we apply the main result of Section 3 to the Saint-Venant-Exner model which is an example of nonlinear hyperbolic system of balance laws.

We consider a prismatic open channel with a rectangular cross-section, and a unit width. The dynamics of the height and of the velocity of the water in the pool are usually described by the shallow water equation (also called the Saint-Venant equation) as considered in [START_REF] Graf | Fluvial hydraulics[END_REF]. To take into account the effect of the sediment on the flow, this system of equations should be modified as described in [START_REF] Graf | Hydraulics of sediment transport[END_REF][START_REF] Hudson | Formulations for numerically approximating hyperbolic systems governing sediment transport[END_REF] (see also [START_REF] Diagne | Lyapunov exponential stability of linear hyperbolic systems of balance laws[END_REF]). This yields the following model

∂H ∂t + V ∂H ∂z + H ∂V ∂z = δ 1 , ∂V ∂t + V ∂V ∂z + g ∂H ∂z + g ∂B ∂z = gS b -C f V 2 H + δ 2 , ∂B ∂t + aV 2 ∂V ∂z = δ 3 , (28)
where

-H = H(z, t)
is the water height at z in [0, L] (L is the length of the pool), and at time t ≥ 0; -V = V(z, t) is the water velocity; -B = B(z, t) is the bathymetry, i.e. the sediment layer above the channel bottom; -g is the gravity constant; -S b is the slope (which is assumed to be constant); -C f is the friction coefficient (also assumed to be constant); -a is a physical parameter to take into account the (constant) effects of the porosity and of the viscosity; -δ = δ(z, t) = (δ 1 (z, t), δ 2 (z, t), δ 3 (z, t)) ⊤ is a disturbance, e.g. it can be a supply of water or an evaporation along the channel (see [START_REF] Graf | Fluvial hydraulics[END_REF]). In the previous model the disturbance may come from many different phenomena. The system (28) admits a steady-state H ⋆ , V ⋆ and B ⋆ (i.e. a solution which does not depend on the time) which is constant with respect to the z-variable. Of course H ⋆ , V ⋆ are such that the equality gS b H ⋆ = C f V ⋆2 is satisfied. The linearization of (28) at this equilibrium is carried out in [START_REF] Diagne | Lyapunov exponential stability of linear hyperbolic systems of balance laws[END_REF] and is:

∂h ∂t + V ⋆ ∂h ∂z + H ⋆ ∂v ∂z = δ 1 , ∂v ∂t + V ⋆ ∂v ∂z + g ∂h ∂z + g ∂b ∂z = C f V ⋆2 H ⋆2 h -2C f V ⋆ H ⋆ v + δ 2 , ∂b ∂t + aV ⋆2 ∂v ∂z = δ 3 . (29)
Now we perform a change of coordinates for system [START_REF] Mazenc | Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem[END_REF]. More precisely, we consider the classical characteristic Riemann coordinates (see e.g. [START_REF] Li | Global classical solutions for quasilinear hyperbolic systems[END_REF]) defined for each k = 1, 2, 3 by4

X k = 1 θ k (λ k -λ i )(λ k -λ j ) [((V ⋆ -λ i )(V ⋆ -λ j ) + gH ⋆ ) h + H ⋆ λ k v + gH ⋆ b] (30)
for suitable constant characteristic velocities λ k (which are quite complicated but explicitly computed in [START_REF] Hudson | Formulations for numerically approximating hyperbolic systems governing sediment transport[END_REF]), and denote

θ k = C f V ⋆ H ⋆ λ k (λ k -λ i )(λ k -λ j ) . (31)
Using this change of coordinates, we rewrite the system (29) as, for each k = 1, 2, 3,

∂X k ∂t + λ k ∂X k ∂z + 3 s=1 (2λ s -3V ⋆ )θ s X s = δ k . ( 32
)
This system belongs to the family of systems [START_REF] Bastin | On Lyapunov stability of linearised Saint-Venant equations for a sloping channel[END_REF], where X = (X 1 , X 2 , X 3 ) ⊤ , Λ(z, t) = Diag(λ 1 , λ 2 , λ 3 ), and, for all z ∈ [0, L], t ≥ 0,

F (z, t) =   α 1 α 2 α 3 α 1 α 2 α 3 α 1 α 2 α 3   , (33)
with α k = (3V ⋆ -2λ k )θ k for all k = 1, 2, 3. It is proved in [START_REF] Hudson | Formulations for numerically approximating hyperbolic systems governing sediment transport[END_REF] that, for all k = 1, 2, 3, α k < 0 (and thus the matrix in ( 33) is stable) and that the three eigenvalues of Λ satisfy

λ 1 < 0 < λ 2 < λ 3 . (34)
Let us explain how Theorem 1 can be applied to design a stabilizing boundary feedback control for the system (32).

Boundary conditions

The boundary conditions of (28) (and thus of ( 29)) are defined by hydraulic control devices such as pumps and valves. Here it is assumed that the water levels are measured at both ends of the open channel, and that the control action can be directly prescribed by the control devices. More precisely, in the present paper, we consider the following set of boundary conditions (see [START_REF] Bastin | On Lyapunov stability of linearised Saint-Venant equations for a sloping channel[END_REF] for the two first boundary conditions and [START_REF] Diagne | Lyapunov exponential stability of linear hyperbolic systems of balance laws[END_REF] for the last one): 1) the first boundary condition is made up of the equation that describes the operation of the gate at outflow of the reach:

H(L, t)V(L, t) = k g [H(L, t) -u 1 (t)] 3 , (35)
where k g is a positive constant coefficient and, at each time instant t, u 1 (t) denotes the weir elevation which is a control input; 2) the second boundary condition imposes the value of the channel inflow rate that is controlled. It is denoted u 2 (t):

H(0, t)V(0, t) = u 2 (t) ; (36)
3) the last boundary condition is a physical constraint on the bathymetry:

B(0, t) = B , (37)
where B is a constant value. By linearizing these boundary conditions, we derive the following boundary conditions for [START_REF] Mazenc | Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem[END_REF]:

H ⋆ v(L, t) + h(L, t)V ⋆ = 3 2 k g (h(L, t) -u 1 (t) + u ⋆ 1 ) H ⋆ -u ⋆ 1 , (38)
H ⋆ v(0, t) + h(0, t)V ⋆ = u 2 (t) -u ⋆ 2 , (39)
b(0, t) = 0 ,

where u ⋆ 1 and u ⋆ 2 are the constant control actions at the equilibrium (H ⋆ , V ⋆ , B ⋆ ). Due to [START_REF] Mazenc | Strict Lyapunov functions for semilinear parabolic partial differential equations[END_REF], for all constant values k 12 , k 13 and k 21 in R, the following conditions

X 1 (L, t) = k 12 X 2 (L, t) + k 13 X 3 (L, t) , (41)
X 2 (0, t) = k 21 X 1 (0, t) (42)
are equivalent to [START_REF] Sturm | Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones[END_REF] and ( 39) for a suitable choice of the control actions u 1 (t) and u 2 (t). More precisely, assuming [START_REF] Sturm | Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones[END_REF] gives the following

H ⋆ = u ⋆ 1 , (
u 1 (t) = u ⋆ 1 - 2(H ⋆ v(L, t) + h(L, t)V ⋆ ) 3k g H ⋆ -u ⋆ 1 + h(L, t) . (43)
Before computing v(L, t), note that, by letting

A = -C 2 f V ⋆2 H ⋆2 λ 1 λ 2 λ 3 (λ 1 -λ 2 ) 2 (λ 1 -λ 3 ) 2 (λ 3 -λ 3 ) 2
and, for each triplet of different index (i, j, k) in {1, 2, 3}, Ψ k = θiθj (λ k -λi)(λ k -λj ) , due to [START_REF] Pham | Predictive control with guaranteed stability for hyperbolic systems of conservation laws[END_REF], we have

λ 1 Ψ 1 -k 12 λ 2 Ψ 2 -k 13 λ 3 Ψ 3 = A(1 -k 12 -k 13 ) .
Moreover, due to [START_REF] Shampine | Solving hyperbolic PDEs in Matlab[END_REF], A is positive, thus λ 1 Ψ 1 -k 12 λ 2 Ψ 2 -k 13 λ 3 Ψ 3 = 0, as soon as k 12 + k 13 = 1. Therefore, with a suitable choice of the tuning parameters k 12 and k 13 , the condition5 λ 1 Ψ 1k 12 λ 2 Ψ 2k 13 λ 3 Ψ 3 = 0 holds. Therefore, under the condition k 12 + k 13 = 1, we may compute v(L, t) with ( 30) and (41), as a function of h(L, t) and b(L, t) as follows

v(L, t) = h(L, t) gH ⋆ (-Ψ 1 + k 12 Ψ 2 + k 13 Ψ 3 ) -Ψ 1 (V ⋆ -λ 2 )(V ⋆ -λ 3 ) H ⋆ (λ 1 Ψ 1 -k 12 λ 2 Ψ 2 -k 13 λ 3 Ψ 3 ) + k 12 Ψ 2 (V ⋆ -λ 1 )(V ⋆ -λ 3 ) + k 13 Ψ 3 (V ⋆ -λ 1 )(V ⋆ -λ 2 ) H ⋆ (λ 1 Ψ 1 -k 12 λ 2 Ψ 2 -k 13 λ 3 Ψ 3 ) +b(L, t) g(-Ψ 1 + k 12 Ψ 2 + k 13 Ψ 3 ) λ 1 Ψ 1 -k 12 λ 2 Ψ 2 -k 13 λ 3 Ψ 3 . (44)
Combining ( 43) and (44), the control action depends only on h(L, t) and b(L, t), that is

u 1 (t) = u ⋆ 1 + h(L, t) 1 - 2H ⋆ 3k g H ⋆ -u ⋆ 1 × gH ⋆ (-Ψ 1 + k 12 Ψ 2 + k 13 Ψ 3 ) -Ψ 1 (V ⋆ -λ 2 )(V ⋆ -λ 3 ) H ⋆ (λ 1 Ψ 1 -k 12 λ 2 Ψ 2 -k 13 λ 3 Ψ 3 ) + k 12 Ψ 2 (V ⋆ -λ 1 )(V ⋆ -λ 3 ) + k 13 Ψ 3 (V ⋆ -λ 1 )(V ⋆ -λ 2 ) H ⋆ (λ 1 Ψ 1 -k 12 λ 2 Ψ 2 -k 13 λ 3 Ψ 3 ) - 2V ⋆ 3k g H ⋆ -u ⋆ 1 -b(L, t) 2H ⋆ 3k g H ⋆ -u ⋆ 1 gH ⋆ (-Ψ 1 + k 12 Ψ 2 + k 13 Ψ 3 ) H ⋆ (λ 1 Ψ 1 -k 12 λ 2 Ψ 2 -k 13 λ 3 Ψ 3 ) . (45)
Similarly ( 30), ( 39), (40), and (42) give the following control

u 2 (t) = u ⋆ 2 + h(0, t)V ⋆ + H ⋆ v(0, t) = u ⋆ 2 + h(0, t) [V ⋆ (46) + gH ⋆ (-Ψ 2 + k 21 Ψ 1 ) -Ψ 2 (V ⋆ -λ 1 )(V ⋆ -λ 3 ) + k 21 Ψ 1 (V ⋆ -λ 2 )(V ⋆ -λ 3 ) H ⋆ (λ 2 Ψ 2 -k 21 λ 1 Ψ 1 )
.

By defining the output of the system (28) as the height at both ends of the channel and the bathymetry of the water at the outflow, the control (u 1 , u 2 ) defined in (45) and ( 46) is an output feedback law. The last boundary condition (40) is equivalent, for all t ≥ 0, to

i [(λ i -V ⋆ ) 2 -gH ⋆ ]X i (0, t) = 0
and thus, assuming6 (λ 3 -V ⋆ ) 2 -gH ⋆ = 0, and using (42), we have

X 3 (0, t) = - [(λ 1 -V ⋆ ) 2 -gH ⋆ ]X 1 (0, t) + [(λ 2 -V ⋆ ) 2 -gH ⋆ ]X 2 (0, t) (λ 3 -V ⋆ ) 2 -gH ⋆ = - [(λ 1 -V ⋆ ) 2 -gH ⋆ ] + k 21 [(λ 2 -V ⋆ ) 2 -gH ⋆ ] (λ 3 -V ⋆ ) 2 -gH ⋆ X 1 (0, t) .
By performing the same change of spatial variables as in the beginning of Section 3, we may assume that, for all i = 1, 2, 3, λ i > 0.

To summarize the boundary conditions in the characteristic Riemann coordinates are [START_REF] Dashkovskiy | On the uniform input-to-state stability of reactiondiffusion systems[END_REF] with

K =   0 k 12 k 13 k 21 0 0 η(k 21 ) 0 0   , where η(k 21 ) = - [(λ 1 -V ⋆ ) 2 -gH ⋆ ] + k 21 [(λ 2 -V ⋆ ) 2 -gH ⋆ ] (λ 3 -V ⋆ ) 2 -gH ⋆ .

Numerical computation of a stabilizing boundary control and of an ISS-Lyapunov function

In this section, we show that computing a symmetric positive definite matrix Q such that Assumption 1 holds can be done by solving a convex optimization problem. This allows us to compute an output feedback law that is a stabilizing boundary control for [START_REF] Mazenc | Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem[END_REF], and to make explicit an ISS-Lyapunov for the linearized Saint-Venant-Exner equations [START_REF] Mazenc | Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem[END_REF]. Note that [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF] cannot be directly used since ( 29) is a quasilinear hyperbolic system (due to the presence of the non-zero left-hand side), and that [2, 10] cannot be applied since the effect of the perturbations δ 1 , δ 2 and δ 3 is additionally considered in this paper, and since the boundary conditions are different. Finally, in the paper [START_REF] Santos | Boundary control of open channels with numerical and experimental validations[END_REF], only the perturbed Saint-Venant equation is considered (without the dynamics of the sediment).

With the change of variables explained at the beginning of Section 3, we obtain a diagonal positive definite matrix Λ. Therefore, Assumption 1 holds as soon as there exists a symmetric positive definite matrix Q such that Sym(QΛ -K ⊤ QΛK) ≥ 0 , Sym(QF ) ≤ 0 .

(47)

Note that conditions (47) are Linear Matrix Inequalities (LMIs) with respect to the unknown symmetric positive definite matrix Q. Computing this matrix can be done using standard optimization softwares (see e.g. [START_REF] Sturm | Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones[END_REF]). Let us solve this optimization problem and apply Theorem 1 using numerical values of [START_REF] Hudson | Formulations for numerically approximating hyperbolic systems governing sediment transport[END_REF] and of [START_REF] Santos | Boundary control of open channels with numerical and experimental validations[END_REF] for the system [START_REF] Prieur | Stabilization of a 1-D tank containing a fluid modeled by the shallow water equations[END_REF] with the boundary conditions [START_REF] Dashkovskiy | On the uniform input-to-state stability of reactiondiffusion systems[END_REF] with K in (48). Therefore we have proved the following result:

Proposition 3 The boundary control (45) and ( 46), with the tuning parameters k 12 , k 13 and k 21 given by (49), is an asymptotically stabilizing boundary control for [START_REF] Mazenc | Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem[END_REF] when the δ i are not present. With the height at both ends of the channel and the bathymetry of the water at the outflow as outputs of the system [START_REF] Mazenc | Further results on input-to-state stability for nonlinear systems with delayed feedbacks[END_REF], it is an output feedback law. Moreover, performing the inverse of the change of variables [START_REF] Mazenc | Strict Lyapunov functions for semilinear parabolic partial differential equations[END_REF], the function U given by ( 50) is an ISS-Lyapunov function for the linearized Saint-Venant-Exner equations [START_REF] Mazenc | Lyapunov functions for time-varying systems satisfying generalized conditions of Matrosov theorem[END_REF] relative to δ(z, t).

Conclusions

For time-varying hyperbolic PDEs, we designed ISS-Lyapunov functions. These functions are time-varying and periodic. They make it possible to derive robustness properties of ISS type. We applied this result to two problems. The first one pertains to a benchmark hyperbolic equation, for which some simulations were performed to check the computation of an ISS-Lyapunov function.

The second one is the explicit computation of an ISS-Lyapunov function for the linear approximation around a steady state of the Saint-Venant-Exner equations that model the dynamics of the flow and of the sediment in an open channel.

This work leaves many questions open. The problem of designing ISS-Lyapunov functions for nonlinear hyperbolic equations will be considered in future works, possibly with the help of Lyapunov functions considered in [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF]. In addition, it would be of interest to use an experimental channel to validate experimentally the prediction of the offset that is inferred from the proposed ISS-Lyapunov function, in a similar way as what is done in [START_REF] Santos | Boundary control of open channels with numerical and experimental validations[END_REF].

Fig. 1 Fig. 2

 12 Fig. 1 Solution of (26) for t ∈ [0, 10] and for z ∈ [0, 1]

7 . 6 .544 × 10 7 

 767 The equilibrium is selected as H ⋆ = 0.1365 [m], V ⋆ = 14.65 [ms -1 ], andB ⋆ = 0 [m]. Next, consider system (32) with λ 1 = -10 [ms -1 ], λ 2 = 7.72 × 10 -4 [ms -1 ], λ 3 = 13 [ms -1 ]and the matrix F defined in[START_REF] Prieur | Robust boundary control of systems of conservation laws[END_REF]. We take g = 9.81 [ms -1 ], and we compute k 21 such that the following tuning parametersk 12 = 0 , k 13 = 0 , k 21 = -0.09517 .(49)Now employing parser YALMIP[START_REF] Löfberg | Yalmip : A toolbox for modeling and optimization in MATLAB[END_REF] on Matlab c , we may check that 8 × 10 7 -2.668 × 10 3 -7.215 × 10 7 ⋆ 2.927 × 10 2 2.083 × 10 3 ⋆ ⋆ ensures that Assumption 1 is satisfied. Thus selecting µ = 1.5 × 10 -2 , we compute the following ISS-Lyapunov function, defined by, for all φ in L 2 (0, 1), Qφ(z)e -µz dz (50)

The simulation codes can be downloaded from www.gipsa-lab.fr/∼christophe.prieur/Codes/2012-Prieur-Mazenc-Ex1.zip

In[START_REF] Mazenc | Strict Lyapunov functions for semilinear parabolic partial differential equations[END_REF] (and similarly in other equations of this section), given k = 1, 2, or 3, the index i and j are in {1, 2, 3} and such that i, j and k are different. In particular the product (λ kλ i )(λ kλ j ) does not vanish.

The condition k 12 + k 13 = 1 will hold with the numerical values that will be chosen in Section 5.3 below.

This will be the case with the numerical values that will be chosen in Section 5.3 below.

The simulation codes can be downloaded from www.gipsa-lab.fr/∼christophe.prieur/Codes/2012-Prieur-Mazenc-Ex2.zip

In the following equation, the symmetric terms are denoted by ⋆.

Work supported by HYCON2 Network of Excellence "Highly-Complex and Networked Control Systems", grant agreement 257462.