Régis Guégan
email: regis.guegan@univ-orleans.fr

Marcelo Giovanela

Fabienne Warmont
email: fabienne.warmont@cnrs-orleans.fr

Mikael Motelica-Heino

Nonionic organoclay: A 'Swiss Army knife' for the adsorption of organic micro-pollutants?

A Na exchanged montmorillonite (Mt) was used as a starting layered material for the preparation of two organoclays synthesized with benzyl decyltrimethyl ammonium (BDTA) cationic surfactant and the tri-ethylene glycol mono n-decyl ether(C 10 E 3 ), a nonconventional nonionic surfactant. The adsorption of the surfactants was performed at an amount of 0.7 times the cation exchange capacity (CEC) for BDTA and below the critical micelle concentration (cmc) where C 10 E 3 is in a monomer state, leading to the intercalation of a lateral monolayer surfactant arrangement within the interlayer space and about 5-7% organic carbon content in organoclays. The environmental properties of both nonionic (C 10 E 3 -Mt) and cationic (BDTA-Mt) organoclays were compared to those of the starting Mt clay with the sorption of three micro-pollutants: benzene, dimethyl-phthalate and paraquat. The adsorption isotherms and the derivative data determined through the fitting procedure by using Langmuir, Freundlich and Dubinin-Radushkevitch equation models explicitly highlighted the importance of the chemical nature of the micropollutants, which play on the adsorbents efficiency. The adsorption data combined with FTIR and XRD supplementary results suggest that C 10 E 3 -Mt nonionic organoclay, although being less efficient for the retention of the different micropollutants, turned out to be the most polyvalent adsorbent since such hybrid material could sequestrate the entire studied organic compounds.

Introduction

The anthropogenic input organic micro-pollutants hydro-systems generates severe human health risks and impacts on aquatic ecosystems [1]. Despite the establishment of strict regulations on industrial discharges and the improvement of specific treatments for both wastewater and drinking water, numerous organic products, incriminated to be responsible of several cancers, are accumulated at significant concentrations in the water [2,3].

Clay minerals have been for a long time known for their outstanding adsorption properties. However, although being used in drinking water treatment, these layered materials turned out to be ineffective for the sequestration of persistent nonpolar hydrophobic contaminants [4][5][6][7][8][9][10]. The intercalation of cationic surfactants, through ion exchange with the inorganic cations, switches the chemical nature of the starting layered material from hydrophilic to hydrophobic. Moreover, the resulting organoclay composite has a wide opening of its interlayer space that significantly improve the adsorption of numerous organic compounds [4][5][6][7][8][9][10]. Nevertheless, the improvement of the sequestration mainly depends on both the chemical nature and the structural organization of the intercalated surfactants. Thus, surfactants showing a long alkyl chains such as hexadecyltrimethylammonium (HDTMA) create an appropriate organic environment within the inorganic frame for the adsorption of alkanes whereas modifiers such as benzyl decyltrimethylammonium (BDTA) show an excellent affinity with aromatic compounds [10].

While numerous works focused on the improvement of the sequestration of organic contaminants by using cationic organoclays, the study on the adsorption properties of organoclays synthesized with unconventional nonionic surfactants showed far much less interest which is even more surprising in regards to the major interests that nonionic surfactants show: (i) a biodegradability and a no toxicity [11]; (ii) a good thermal and chemical stabilities [12,13]; (iii) a preservation of the exchangeable inorganic cations after adsorption onto clay minerals, conferring a dual hydrophilic / hydrophobic character [14][15][16][17][18]; (iv) a still possible cation exchange of the resulting nonionic organoclay with both organic and inorganic cations [13,16,17,19]; and (v) aggregates of which structural arrangement depends on the surfactant state in aqueous solution expanding at wide openings the interlayer space of the layered material [13,17,19].

Thus, this work focuses on the study of the adsorption properties of nonionic organoclay using the triethylene glycol mono n-decyl ether (C 10 E 3 ) and compare them to those of the starting clay mineral (Mt) and a cationic organoclay. The selected BDTA cationic surfactant combines a long alkyl chain and an aromatic ring of which affinity with organic compounds was shown in various works. It should be also noticed that in order to get realistic and economical viable organoclay materials, only a monolayer of surfactant was confined within the interlayer space of Mt. Such organoclay arrangement, where the surfactant undergoes severe effects of confinement, shows a maximal thermal stability and prevents any unwanted surfactant release. The adsorption properties of the three sorbent materials were studied by focusing on three water micro-pollutants: (i) benzene; (ii) dimethyl phthalate; and (iii) paraquat,. If the adsorption of benzene and paraquat on both clay minerals and cationic organoclays has been the subject of numerous studies focusing on the behavior of the adsorption isotherms, details on the evolution of the interlayer space determined by XRD and FTIR analyses are still scarce. Moreover, no study has been focused on the sequestration of these two pollutants by a nonionic organoclay as well as the adsorption of phthalate. Results obtained in this study with the use of a set of complementary techniques (XRD, FTIR, and adsorption isotherms), beyond serving as only data for the effectiveness comparison of the three different sorbents will be useful to identify the interactional mechanisms involved in the adsorption of organic micro-pollutants.

Materials and Methods

Clay mineral

Wyoming sodium montmorillonite (Mt), obtained from the Source Clay Minerals Repositery, University of Missouri (Columbia, MO) was used in this study as a starting material. This Mt clay mineral shows a cation exchange capacity (CEC) of 80 meq per 100 g clay, and a BET specific surface area of 35 m².g -1 (condensation of a non-polar N 2 gas) as well as a 660 m 2 .g -1 accessible surface area determined through the adsorption of ethylene glycol due to the solvation of Na + cations in Mt for polar molecules [21,22]. The Mt clay was fractioned to <2 µm by gravity sedimentation, purified and Na exchanged by well-established procedures [21].

Experimental techniques

The concentration of carbon in solution and on solids were measured using an elemental analyzer (Shimadzu TOC 5050 /SSM 5000-A). Moreover, the amounts of adsorbed organic compound and equilibrium isotherms were determined by using a Hitachi U-1900 UV spectrometer on the basis of the concentration decrease after both clay mineral and organoclays impregnation.

Fourier transform infrared (FTIR) measurements in the range 650-4000 cm -1 , were recorded using a Thermo Nicolet 6700 FT spectrometer equipped with a Deuterated Triglycine Sulfate (DTGS) detector and a Nicolet Continum microscope. The powder samples were spread over a NaCl window of the microscope. The analyzed sample area was a square of side 100 m chosen under the microscope 15X Infinity Reflechromat objective. The analyses were performed in transmission mode and each spectrum was the average of 256 scans collected at 2 cm -1 resolution.

The d 001 spacing's of the starting layered materials and after being in contact with organic pollutants were determined by the first 00l reflection from the X-rays patterns which were recorded in a conventional θ-θ Bragg-Brentano configuration by using a Thermo Electron ARL'XTRA diffractometer equipped with a Cu anode (CuK α1,2 = 1.5418 Å) coupled with a Si(Li) solid detector. The diffractogramms on dry samples (100°C for 24 hours) were performed between 1 and 62° (2θ) with an angular and time steps of 0.04° and 10s respectively.

The morphology of both Na-Mt and organoclays was visualized by transmission electron microscopy (TEM) using a Philips CM20 microscope operating at 200 kV. The samples were initially dispersed in a 50% ethanol solution and then dropped on holey carbon grids and dried at 80°C before TEM observations.

Organoclay preparation

Tri-ethylene glycol mono n-decyl ether (C 10 E 3 ) nonionic surfactant, purchased from Nikko Chemicals, Inc., (Tokyo, Japan), and assumed to have a purity > 99.8% was used without any further purification.

Benzyldimethyltetradecyl ammonium chloride (BDTAC) cationic surfactant was supplied from Sigma Aldrich. Aqueous solutions of BDTAC and C 10 E 3 with Millipore deionized water were prepared at room temperature for concentrations at 0.7 times the cation exchange capacity (CEC) and 4.10 -4 mol.L -1 (below the cmc of C 10 E 3 ) respectively. These surfactant solutions were poured in aqueous Mt dispersions and the solutions were stirred at 250 rpm for 48 hours. After agitation, the solutions were centrifuged at 5500 rpm for 30 min and the supernatants were removed. The resulting organoclays were washed with Millipore deionized water and centrifuged twice and dried at 100°C for 48 hours before FTIR and XRD analyses.

Sorption of the micro-pollutants

Benzene, dimethyl phthalate (dimethyl benzene-1,2-dicarboxylate) and paraquat (1,1'-dimethyl-4,4'bipyridylium chloride) were purchased from Sigma-Aldrich Chemical. Batch sorption isotherms were conducted in duplicate using at least 15 initial aqueous concentrations ranging from 5 mg.L -1 to 1 g.L -1 for the organic micro-pollutants: benzene, paraquat, and dimethyl-phthalate. All of these organic compounds have aqueous solubility better than 1 g.L -1 . The aqueous solutions were added to the two

(1) (2) (3)
organoclays and the starting clay mineral sorbents contained in 60 mL centrifuge tubes. Samples were shaken on a rotary shaker at 50 rpm for 48h in order to achieve the equilibrium of adsorption determined with preliminary tests, and then centrifuged at 5500 rpm for 30 min. Both supernatants and sorbents after contact with the pollutants were removed and analyzed through UV spectroscopy. The amount of adsorbed organic contaminants was calculated by the difference between the initial and equilibrium final concentrations that allowed us to determine the adsorption isotherms for the whole pollutants on the clay mineral and its organoclay derivatives. Before their XRD and FTIR characterizations, the entire resulting organoclays and Mt after adsorption of contaminants were dried at 100°C for 48 hours.

Adsorption isotherm fitting procedure

The resulting adsorption isotherms of the contaminants onto organoclays and Mt were fit using Langmuir model expressed as: 23 Where q e is the equilibrium paraquat amount adsorbed on sorbents (mol.g -1 ), C e the equilibrium micropollutant concentration in the resulting solution (mol.L -1 ), q max the maximum adsorption capacity of the sorbents (mol.g -1 ) and K L is the Langmuir adsorption constant (L.mol -1 ) which is related to the free energy ( ) of adsorption by the relation (2):

Since the adsorption of surfactant contributes to a change of the chemical nature of the surface as well as a heterogeneity in the distribution of the adsorption sites, a Freundlich and a Dubinin-Radushkevish (D-R) model isotherm equations were used. The Freundlich isotherm is an empirical equation to describe heterogeneous systems of which the linear form is: 23, 24 (4)

(5) (6)
where K F (L.g -1 ) and n are Freundlich adsorption isotherm constants, being indicative of the extent of the adsorption and the degree of nonlinerarity between the solute concentration and adsorption respectively. The D-R isotherm represents a more general model for adsorption, since it assumes a heterogeneous surface and a variability for the adsorption potential and was applied successfully for fitting the adsorption of organic species onto Mt. 24, 25 The linear form of this latter model is written by the eq (4):

where is a constant related to the mean free energy of adsorption per mole of the sorbents (mol 2 .J -2 ), q m , the theoretical saturation capacity and corresponds to the Polanyi potential, which is equal to:

( )
Where R (J.mol -1 .K -1 ) is the gas constant and T (K) is the absolute temperature. The constant provides the identification of the mean free energy E (kJ.mol -1 ) of adsorption of organic compounds to sorbents surface and can be determined by using the relationship (6): 25, 26 √ This latter parameter gives information whether the adsorption mechanism involves ion exchange or physical adsorption. Indeed, if the magnitude of E is between 8 and 16 kJ.mol -1 , the adsorption process follows an ion-exchange, while for E< 8kJ.mol -1 physisorption is envisaged. 27

Results and Discussion

Organoclays properties

The C 10 E 3 -Mt and BDTA-Mt organoclays were characterized by XRD, TEM and FTIR analyses. Both 00l diffraction patterns of the prepared organoclays shift to lower angular values underlining the expansion of the interlayer space at 14 and 13.5 Å for dehydrated BDTA-Mt and C 10 E 3 -Mt respectively in contrast to 9.7 Å for the untreated dehydrated Na-Mt (Figure 1). As expecting with the concentrations of the prepared surfactant solutions, where C 10 E 3 , below the cmc is only in monomers form, and for BDTA below 1 times the CEC, the adsorption leads to the intercalation of a lateral surfactant monolayer within the interlayer space [13,18]. Indeed, the intersheet separation in organoclay materials matches the height of both BDTA and C 10 E 3 where surfactants lie parallel to the plane of the silicate layers.

Transmission electron microscopy pictures confirm the proper intercalation of surfactant within the interlayer space of starting montmorillonite. In addition, TEM pictures give information on morphology of the organoclay systems where some area show regularly intercalated layers with regular layer-tolayer distance whereas other are display randomly intercalated layers suggesting an irregular intercalation with lower surfactant concentration (Figure 2). FTIR organoclays spectra show typical features characteristics of the organic compounds: the absorption bands at 2840-2920 cm -1 relative to the symmetric and antisymmetric CH 2 stretching vibrations of the surfactant alkyl chains (Figure 3). It is well established that the wavenumber of the CH 2 stretching bands of hydrocarbon chains are extremely sensitive to the conformational ordering and change in the gauche-trans conformer ratio of the chains which can be used as probe, in correlation with the d 001 spacing variation, for the surfactant arrangement within the silicate layers. At such low surfactant concentration for the organoclay preparation, the wavenumber shift to high value (up to 5 and 12 cm -1 for symmetric and antisymmetric CH 2 stretching vibrations) comparatively to bulk surfactant system (not shown) indicating a disordered monolayer structure in the interlayer space of Mt. Despite such small loadings of surfactant, estimated through elementary analysis at 5.2 and 6.1 % of organic content for C 10 E 3 and BDTA respectively, the resulting organoclays exhibit a hydrophobic behavior. Indeed, such behavior is confirmed through the measure of the hydraulic conductivity for both organoclays, which is increased in contrast to raw Mt (results not shown here). Since, the adsorption of C 10 E 3 keeps the whole Na + exchangeable cations, the prepared C 10 E 3 -Mt organoclay is assumed to show a dual hydrophilic/hydrophobic character, suitable for the adsorption of large range of organic contaminants.

Adsorption of an organic cation herbicide: the paraquat

The affinity between paraquat a quaternary ammonium herbicide and the untreated Mt and the prepared organoclays was shown through the lineshape of the adsorption isotherms, which could be fit by using Langmuir equation model (Figure 4). The thermodynamic data and different Langmuir,

Freundlich and D-R parameters point out the better affinity of paraquat for untreated Mt than organoclays which differs considerably from a previous study where cationic organoclays seemed to improve slightly the adsorption of the organic cation (Table 1). Here, both Langmuir and Freundlich constants for the determination of the adsorption of paraquat onto Mt are in good agreement with previous works as well as the maximum adsorbed amount at 5.2 x 10 -4 mol.g -1 , which matches the value of the CEC of Mt underlining electrostatic interaction as the main vector for the adsorption of paraquat [28][29][30][31][32]. This last point is confirmed through D-R fitting procedure with the magnitude of E that reaches a value >8 kJ.mol -1 , stressing out an adsorption through ion exchange onto Mt as well as C 10 E 3 -Mt. Nevertheless, although paraquat shows a particular affinity for the nonionic organoclay, it never reaches the magnitude for the untreated Mt as both Langmuir and Freundlich constants expressed. Moreover, the estimation of the maximum of adsorbed paraquat represents 0.3 times the CEC of the layered materials, which indicates that only a part of the whole accessible Na + cations are exchanged. From isotherm data, we may think that the spatial arrangement of C 10 E 3 in a lateral monolayer within the interlayer space should block the intercalation of paraquat due to steric effect leaving an exchange with the 20% of the Na + cations located on the external surface of clay particle. However, XRD patterns revealed a shift of the 00l reflection to large angular value, suggesting a contraction of the interlayer space at a value of 12.75 Å, equivalent to the d 001 of the Mt after being impregnated into paraquat solutions (Figure 5). Such d 001 spacing conveys a planar arrangement for the intercalation of paraquat within the interlayer space of Mt as well as C 10 E 3 -Mt and suggests that Na + cations involved in the exchange with paraquat are mainly located within the interlayer space for the nonionic organoclay. A control of the supernatant after the impregnation into paraquat solution revealed that C 10 E 3 was not expelled from the layered structure of the organoclay and the density of C 10 E 3 remains strictly identical as FTIR spectra exhibited. Indeed, the integrated intensity of CH 2 stretching bands at 2850-2920 cm -1 related to the alky tails of C 10 E 3 evolved gradually as the density of surfactant increases and was proposed as a probe for adsorption isotherm study [13,17,18]. These two bands did not show any significant difference before and after the impregnation in the pollutant solution whereas new absorption bands attributed to paraquat could be identified (Figure 6). Once intercalated, paraquat as a divalent organic cation, in contrast to Na + may considerably reduce the electrostatic repulsion between the phylosilicate layers and as a result stabilize the composite materials with a shrinking of its interlayer space. Despite a small loadings of C 10 E 3 , only a few part of the accessible Na + cations (30%) are involved in the exchange with paraquat. This can be explained by the possible lining of the linear arrangement of C 10 E 3 around the inorganic cations preventing any cation exchange.

In contrast to raw clay mineral, cationic organoclay usually showed an improvement for the adsorption of numerous organic compounds due to their hydrophobic behavior and a wide opening of the interlayer space [10,14]. However, in the case of paraquat, these last points play minor roles since the adsorption is mainly driving through electrostatic interaction [28][29][30][31][32]. Although keeping some Na + cations, however at small ratio, chemical analysis of the supernatant as well as XRD and FTIR characterizations showed that BDTA-Mt was not suitable for the adsorption of paraquat. The previous study of Seki and Yurdakoç pointed out the capacity of cationic organoclays prepared with nonyl-and dodecylammonium surfactants for the sequestration of paraquat, which was somehow improved compared to the untreated clay minerals [32]. The authors stressed out the ion-exchange as the main adsorption mechanism between paraquat and the cationic surfactant previously adsorbed onto clay mineral. Although using a different chemical nature for the cationic surfactant as modifier and maybe a different protocol for the impregnation of the organoclays into the contaminant solution, the results presented here support that BDTA-Mt as a cationic organoclay is ineffective for the adsorption of inorganic and organic cationic compounds or at least is not the most appropriate sorbent. Indeed, numerous studies revealed such failure for cationic organoclay in which surfactant binds irreversibly to the interlayer of the base clay mineral, excluding any further ion-exchange with both inorganic and organic compounds.

Nevertheless, cationic organoclay could adsorb in a smaller proportion through ion complexation or Van der Waals interaction trace elements and organic cationic compounds respectively [33,34]. Despite the generation of a hydrophobic environment with BDTA and the preparation of the cationic organoclay at 70% of the CEC, the whole data confirms the no paraquat adsorption. Moreover, the phenyl rings of BDTA in contrast to the linear form of C 10 E 3 may block the herbicide cation from the phenyl rings, thereby reducing considerably the sorption, which only solicit the external surface of particles. Thus, it appears that paraquat mainly interact through electrostatic interaction and as a result cannot be properly sequestrated by using a BDTA-Mt.

Adsorption of a nonpolar hydrophobic contaminant: the benzene

The adsorption of benzene has been intensively studied with the use of numerous porous materials such as actived carbon, clay minerals and organoclays [35][36][37][38][39]. If cationic organoclays show a certain capacity to retain such nonpolar compound, clay minerals cannot adsorb it [35][36][37][38][39]. With the use of montmorillonite deploying a BET specific surface of 35 m 2 .g -1 , we obtained similar results to previous studies with the no benzene sorption by the untreated clay mineral. Both chemical analysis of the supernatant, in the limit of the resolution of the analytical methods, and solid characterizations (FTIR and XRD) of Mt after impregnation in solution at high benzene concentration, did not show any traces of its adsorption (Figures 789). As expected, the absence or low content of organic carbon in the starting clay mineral, of which behavior is mainly hydrophilic, cannot retain nonpolar hydrophobic compounds and is consistent with previous works [35,38].

On the other hand, the chemical modification with C 10 E 3 and BDTA surfactants, even at low loadings, provides an adequate media for the adsorption of nonpolar hydrophobic compound.

If the density of surfactant increases a hydrophobic character, the improvement of the adsorption properties of organoclays for non-polar hydrophobic pollutants does not evolve linearly and depends on both the chemical nature of the surfactant and its arrangement. Indeed, for organoclays with cationic phenyl rings surfactants (BDTA or BTEA for instance) at small loadings (5-7% of organic content) showed a maximum uptake for nonpolar compounds while organoclays prepared by linear surfactants with long alkyl tails (HDTMA) demonstrated the opposite trend [38]. The adsorption isotherms of benzene on C 10 E 3 -Mt and BDTA-Mt show similar behavior where the adsorbed benzene amount nonlinearly grows, and the lineshapes are perfectly fit with a Langmuir model (Figure 7). The parameters derived from the fitting procedure clearly show the preferential affinity of benzene for BDTA-Mt with K L > than C 10 E 3 -Mt (Table 1).

Moreover, the maximum benzene adsorbed amount follows the same trend by reaching at the plateau of adsorption 1.09 and 1.64 x 10 -4 mol.g -1 for C 10 E 3 -Mt and BDTA-Mt. Both

Freundlich and D-R models overestimates the maximum of adsorption which may indicates a monolayer covering of benzene onto organoclays as well as none variability for the adsorption potentials. Despite the intercalation of long alkyl surfactants leading to new possible adsorption sites, BET specific surface area of organoclays usually decreases with organic content due to surfactant block in Mt micropores [38]. Nevertheless, for 5.2 and 6.1%

surfactant loadings, C 10 E 3 -Mt and BDTA-Mt showed a BET surface area of 28 and 25 m 2 .g -1 respectively. Benzene displays a spherical geometry with a 2.8 Å radius, yielding to an apparent packing area per molecule of 26 Å 2 . If we assume the entire surface of both organoclays is totally accessible, the calculated amount of benzene covering in monolayer reaches 1.82 and 1.73 x 10 -4 mol.g -1 for C 10 E 3 -Mt and BDTA-Mt respectively. Such adsorbed amount of benzene suggests its complete or partial lining onto the whole organoclay surface including that of the interlayer space. While the d 001 spacing of C 10 E 3 -Mt gradually increases following the equilibrium benzene concentration and reaches for high concentration a value of 14 Å, consistent with the size of the intercalated organic compounds, the BDTA-Mt interlayer spacing remains at its original value of 14.2 Å (Figure 8). Benzene is somehow quite a small molecule, which can be easily blended among the alkyl tails without introducing any significant change of the interlayer space of the cationic organoclay. FTIR spectra of the resulting organoclays after their impregnation into high concentrated benzene solution confirmed its proper sequestration with the identification of absorption bands characteristics of this nonpolar hydrophobic compound (Figure 9).

Figure 9: FTIR spectra -from bottom to topof the benzene, and Na-Mt, C 10 E 3 -Mt as well as BDTA-Mt sorbents after being impregnated in this nonpolar hydrophobic compound at high concentration regime (concentration range once a plateau for the adsorbed amount is reached). Spectra were normalized with respect to the Si-O stretching band at 1100 cm -1 . While Na-Mt after being impregnated in benzene did not show any additional bands attesting the no sorption of this compound, the confirmation of its proper adsorption onto both organoclays is highlighted by representative absorption bands attributed to CH stretching vibrations at 2850-3000 cm -1 and C=C stretching modes at 1450-1500 cm -1 of benzene (dashed lines).

As a non-polar hydrophobic compound, even though D-R fit process overestimates the maximum of adsorbed amount; this model gives an idea of the energy involved for the benzene adsorption onto organoclays, which are below 8kJ.mol -1 confirming a physisorption mechanism. The alkyl carbon chains of both surfactants contribute to weak short range Van der Waals interaction with benzene as well as phenyl ring-ring interaction for BDTA-Mt. This last contribution, although displaying a shorter accessible surface area and pore space favors an adsorption of benzene in contrast to C 10 E 3 -Mt with 60% of its surface covered by benzene shows its ability to adsorb such kind of chemical product. Among alkyl-phthalate compounds, dimethyl-phthalate (DP) with the smallest partition coefficient K o/w , represents the most hydrophilic compound in the phthalate series which has been identified in diverse environmental media, and more particularly in drinking water [40,41]. The affinity of DP with the two organoclays and Mt was shown with its nonlinear growth of the adsorbed amount as a function of its equilibrium concentration (Figure 10). If the adsorption isotherms are well fit by Langmuir equation model with parameters like the maximum adsorbed amount, which matches the experimental data, the parameters determined through Freundlich or D-R models overestimate the maximum adsorbed amount (Table 1).

Adsorption of the dimethyl-phthalate

Both Freundlich and D-R parameters even though departing from experimental data with a ≈50% error deviation follow the same trend as Langmuir constants stressing out the affinity of DP which decreases in the following order: Mt >C 10 E 3 -Mt > BDTA-Mt with K L of DP for Mt 68% and 57% greater than the cationic and nonionic organoclays respectively. This sorbents affinity rank for DP goes against what could be thought where the hydrophobic character of organoclays, identified as a key parameter in phthalates adsorption [40][41][42][43][44]. appears here to play finally a minor role in the sequestration of DP. Indeed, if hydrophobic interaction was the main driving force for the adsorption of DP onto the studied sorbents, the entire Langmuir, Freundlich and DR constants would have been greater for both organoclays than the untreated Mt which is obviously not the case here. Moreover, FTIR spectra of the sorbents after being impregnated in high concentrated DP solution did not put in evidence absorption bands related to carboxylic groups (Figure 11).

Thus, we assumed that the energies of DP for sorbents determined through D-R model are somehow overestimated and are below 8 kJ.mol -1 underlining as it was reported a physisorption mechanism for the adsorption of nonionic organic compounds [27,40,41]. In contrast to nonpolar hydrophobic pollutant like benzene and even larger phthalate compounds where the adsorption results to hydrophobic interaction and π-π electron donor acceptor mechanism with hydrophobic substrate [41], DP with the highest hydrophilic character of the phthalates series and hydrogen bonding donor and acceptor functions of 0.8 owns the ability to form some H-bonds and as a result could be adsorbed through ion-dipole and H-bonds interaction with its -COOCH 3 group, which is consistent to the hydrophilic behavior of the sorbents. The greater hydrophilic behavior of the sorbents is, the larger of the adsorbed amount is. Since Mt shows a low silanol (Si-OH) density on its surface [45], the hydrophilic behavior of untreated Mt mainly results to the presence of exchangeable cations within the interlayer space. Such character is particularly enhanced with alkyl metal exchangeable cations as Na-Mt shows thus favoring the adsorption of nonionic organic compounds around them associated by ion-dipole interaction [23]. This sorption mechanism also contributes to the sorption of DP onto organaclays with the possible association of BDTA-DP through iondipole and H-bonds between C 10 E 3 and DP as well as ion-dipole interaction modulated around the accessible Na + cations in the nonionic C 10 E 3 -Mt organoclay.

Figure 12: Evolution of the d 001 basal spacing determined by the 00l reflection of the silicates sheets obtained from X-ray diffraction patterns of Na-Mt (black circles), BDTA-Mt cationic organoclay (red squares) and C 10 E 3 -Mt nonionic organoclay (blue diamonds) after being impregnated in dimethyl phthalate solutions as a function of the equilibrium contaminant concentration.

The wide layer expansion of Mt after the impregnation in DP solution with a d 001 spacing of 18.2 Å suggests a lateral bilayer arrangement of the contaminant within the interlayer space whereas for both organoclays, only a monolayer is confined with a d 001 increase of ≈ 4 Å (Figure 12). Such molecular arrangement is consistent with the adsorbed DP amount that matches the total accessible surface area of the entire sorbents which provide the major adsorption sites. For a proper understanding of the sorption of polar molecule, instead of considering BET specific surface area of sorbents which properly describes the adsorption of nonpolar molecule, it is rather preferable to take into account that one determined by ethylene glycol, a polar liquid, estimated at 660 m 2 .g -1 [22]. The molecular volume of DP is approximately 600 Å 3 giving an area per molecule of 86 Å 2 . The adsorption of DP was achieved in solution where the Mt suspension is well swollen and shows a stable hydration state in which the layers are widely expanded suggesting that the specific surface area is totally accessible. In that case, it leads to a calculated amount of DP in monolayer of 1.28 x 10 -3 mol.g -1 that exceeds the experimental data determined at the plateau of adsorption confirming a lateral bilayer organization within the interlayer space. It is possible to repeat this process for both organoclays even through the specific surface area may be reduced in view of their hydrophobic behavior and the block of intercalated surfactants within the interlayer space. The experimental and even the deduced adsorbed amount of DP through Langmuir fit, onto both organoclays are consistent with a lateral monolayer arrangement of the phthalate compound.

Conclusions

The chemical modification of clay minerals via the intercalation of cationic surfactants for the preparation of environmental organoclay obviously improves the efficiency of the adsorption of numerous chemicals [10,14]. Nevertheless, this observation is only correct for a restraint range of hydrophobic chemicals where cationic organoclays mainly excel, due to their hydrophobic character, but show some limits for other organic compounds [35]. Worse, cationic organoclays proved to be an inappropriate sorbent for cationic species where the irreversibility of the first adsorption of quaternary ammonium surfactants prevents any further ion exchange and as a result are permeable to such pollutant class [33,34]. If we neglect the nonpolar hydrophobic contaminant, untreated clay minerals appear to be the most efficient material for the adsorption of paraquat and DP chemicals where their large specific surface area and exchangeable cations provides a large adsorption sites. However, the identified recalcitrant contaminant in drinking water and other environmental samples are usually highly lipophilic large molecules where raw clay minerals show their adsorption limits. C 10 E 3 -Mt nonionic organoclay, although not being the best efficient sorbent in this study provide an interesting behavior since it could adsorb the entire studied contaminants. The large amplitude of this material for the adsorption of various chemical class results to its particular C 10 E 3 coating onto Mt driving to a dual hydrophilic/hydrophobic behavior. While the introduction of amphiphilic molecules generates a hydrophobic media suitable for the adsorption of nonpolar hydrophobic compound through hydrophobic interaction, the accessibility of Na + cations allows the sequestration of cationic compounds by cation exchange and favors as well the adsorption of polar molecule through ion-dipole and H-bonds interaction (Figure 13). Therefore, such nonionic organoclay, by involving several possible adsorption mechanisms for the adsorption of various class contaminants, appears to be a convenient tool for environmental applications as a 'Swiss Army Knife' provides in daily life.

However, since the adsorption efficiency mainly depends on the organic content in organoclay, the effect of nonionic surfactant density as well as the ratio between the hydrophilic to hydrophobic portion of the amphiphilic molecules adsorbed on clay mineral need to be examined along with other environmental factors like pH for instance.

Figure 1 :

 1 Figure 1: X ray diffraction patterns for untreated dehydrated Na-Mt (black solid line), dehydrated C 10 E 3 -Mt nonionic organoclay (blue solid line) and dehydrated BDTA-Mt cationic organoclay (red line). The (00l) and (021) reflections of the layered materials are labeled. (Q) refers to the presence of quartz in the samples.

Figure 2 :

 2 Figure 2: Transmission Electron Microscopy (TEM) images of a) BDTA-Mt orgonclay dispersed with 50% ethanol solution and b) C 10 E 3 -Mt organoclay dispersed with 50% ethanol.

Figure 3 :

 3 Figure 3: FTIR spectra of the Na-Mt, C 10 E 3 -Mt and BDTA-Mt organoclays. The coating of surfactants is highlighted by the presence of additional absorbance bands relative to the CH 2 stretching vibrations of the hydrocarbon chains of C 10 E 3 and BDTA.

Figure 4 :

 4 Figure 4: Adsorption isotherms of paraquat herbicide onto both Na-Mt (black circles), and C 10 E 3 -Mt nonionic organoclay (blue diamonds). The solid lines represent Langmuir model fit.

Figure 5 :

 5 Figure 5: Evolution of the d 001 basal spacing determined by the 00l reflection of the silicates sheets obtained from X-ray diffraction patterns of Na-Mt (black circles), and C 10 E 3 -Mt nonionic organoclay (blue diamonds) after their impregnation in paraquat herbicide solutions at several concentrations for 48h.

Figure 6 :

 6 Figure6: FTIR spectra -from bottom to top -of the paraquat, and Na-Mt, C 10 E 3 -Mt as well as BDTA-Mt sorbents after being impregnated in paraquat at high concentration regime (concentration range once a plateau for the adsorbed amount is reached). Spectra were normalized with respect to the Si-O stretching band at 1100 cm -1 . The adsorption of paraquat onto Na-Mt and C 10 E 3 -Mt is confirmed by the presence of additional absorption bands relative of the alkyl stretching modes at 2850-3000 cm -1 , aromatic C-C stretching vibrations 1570 and 1620 cm -1 which are highlighted by dashed lines whereas BDTA-Mt spectrum does not show any evidence of these particular bands.

Figure 7 :

 7 Figure 7: Adsorption isotherms of nonpolar hydrophobic benzene onto C 10 E 3 -Mt nonionic organoclay (blue diamonds) and BDTA-Mt cationic organoclay (red squares). The solid lines represent Langmuir model fit.

Figure 8 :

 8 Figure 8: Evolution of the d 001 basal spacing determined by the 00l reflection of the silicates sheets obtained from X-ray diffraction patterns of BDTA-Mt cationic organoclay (red squares) and C 10 E 3 -Mt nonionic organoclay (blue diamonds) after being impregnated in benzene solutions as a function of the equilibrium nonpolar hydrophobic compound concentration.

Figure 10 :

 10 Figure 10: Adsorption isotherms of dimethyl phthalate Na-Mt (black circles), BDTA-Mt cationic organoclay (red squares) and C 10 E 3 -Mt nonionic organoclay (blue diamonds). The solid lines represent Langmuir model fit.

Figure 11 :

 11 Figure11: FTIR spectra -from bottom to top -of the dimethyl phthalate, and Na-Mt, C 10 E 3 -Mt as well as BDTA-Mt sorbents after being impregnated in dimethyl phthalate at high concentration regime (concentration range once a plateau for the adsorbed amount is reached). Spectra were normalized with respect to the Si-O stretching band at 1100 cm -1 . The adsorption of dimethyl phthalate onto whole sorbents is confirmed by the presence of additional absorption bands relative of the CH 3 stretching vibration at 2850-3000 cm -1 , C-O and C=O stretching bands of ester groups at 1300 and 1722 cm -1 which are highlighted by dashed lines.

Figure 13 :

 13 Figure 13: Possible adsorption mechanisms in a C 10 E 3 -Mt nonionic organoclay showing a lateral surfactant arrangement involved for the sequestration of paraquat, an organic herbicide cation(ion exchange); benzene, a nonpolar hydrophobic organic compound (hydrophobic interaction), and dimethyl-phthalate compound (ion dipole and H-bonds interaction).

Table 1 .

 1 Adsorption isotherm constants determined with Langmuir, Freundlich, and Dubinin-Radushkevich model fit for the adsorption of organic contaminants onto Mt, BDTA-Mt, and C 10 E 3 -Mt organoclays.

	Pollutant	Adsorbent	Langmuir

Acknowledgements

We are grateful to the Canon Foundation in Europe for their support in this research, which allowed the first author to be hosted in laboratory of the Professor Ogawa at the University of Waseda last year 2013 for 6 months where some experiments were performed.

Associated Content

Additional figures on Freundlich and Dubinin-Radushkevish (D-R) fits of the adsorption isotherms of the micro-pollutants on the three sorbents are included.


Optimality of affine control system of several species in competition on a Sequential Batch Reactor

Keywords: Subject classification: 49J15, 49J20, 97M10 affine control systems, singular trajectories, optimal control problem, sequential batch reactors, pontryagin maximum principle

In this paper we analyze the optimalty of affine control system of several species in competition for a single substrate on a Sequential Batch Reactors (SBR), with the objective being to reach a given (low) level of the substrate. We allow controls to be bounded measurable functions of time plus possible impulses. A suitable modification of the dynamics leads to a slightly different optimal control problem, without impulsive controls, for which we apply different optimality conditions derived from Pontryagin principle and the Hamilton-Jacobi-Bellman equation. We thus characterize the singular trajectories of our problem as the extremal trajectories keeping the substrate at a constant level. We also establish conditions for which a Immediate One impulse (IOI ) strategy is optimal. Some numerical experiences are then included in order to illustrate our study and show that those conditions are also necessary to ensure the optimality of the IOI strategy.

1 Introduction.

In this paper, we focus on optimal control problem for an affine control system where several species compete for a single substrate on a Sequential batch reactor, the objective being to reach a given (low) level of the substrate.

The Sequential batch reactor (SBR) is a device that consists typically in a tank which is filled with biological micro-organisms capable to degrade some undesirable substrate. The method consists then in a sequence of cycles composed by three phases: -Phase 1: filling the reactor with water to be treated, -Phase 2: waiting the concentration of the undesirable substrate to decrease until a given (low) concentration, -Phase 3: emptying the reactor from the clean water, leaving the sludge inside.

The Sequential batch reactor are often used in biotechnological application, notably in waste-water treatments. See [START_REF] Irvine | Sequencing batch reactors for biological wastewater treatment[END_REF], [START_REF] Moreno | Optimal control of bioreactors for the wastewater treatment[END_REF], [START_REF] Gajardo | Minimal time impulsive control of sequential batch reactors with one or more species[END_REF], [START_REF] Smith | The Theory of the Chemostat[END_REF] for more details about fundamental role of SBR in bioengineering.

We consider only increasing growth functions for these species. So, this paper extends some results obtained for one and two species in [START_REF] Moreno | Optimal control of bioreactors for the wastewater treatment[END_REF] and [START_REF] Gajardo | Minimal time impulsive control of sequential batch reactors with one or more species[END_REF]. For a multi-species setting, we fully characterize the existence of singular arc (which existence can lead to complex solutions that are not easily tractable from the mathematical point of view). The extremal trajectories of the singular arc type are characterized as the strategies used to maintain the substrate at a constant level, and we study under which conditions the strategy that consists of filling as fast as possible the tank up to its maximum capacity and then waiting is optimal. This strategy is called on Immediate One Impulsive (IOI ) in the impulsional framework developed here and it will be precisely defined in Section 5. The last issue has been sucessfully solved for one single species in [START_REF] Moreno | Optimal control of bioreactors for the wastewater treatment[END_REF] by only assuming that the growth function is nondecreasing. However, when two species coexist in the SBR, it is shown in [START_REF] Gajardo | Minimal time impulsive control of sequential batch reactors with one or more species[END_REF] that the fact that both growth functions are increasing is no longer enough to ensure the optimality of the IOI strategy. In Section 5, we have only proved that the IOI strategy is optimal when one species is highly performant and the others all are sufficiently close one to each other, which somehow captures the two species case. However, the study of IOI and singular strategies for our problem in which some of the functions of growth could be Haldane type and all remaining being increasing functions must be analyzed to obtain new optimal strategies in the presence of several species, and this will be done in future research by imposing appropriate additional conditions on growth functions or their derivatives, and the use of higher-order optimality conditions of affine control systems, the envelope theory, symplectic geometric methods. See for instance [START_REF] Krener | The high order maximal principle and its application to singular extremals[END_REF], [START_REF] Sussmann | Envelopes, higher order optimality conditions and Lie Brackets[END_REF] and references therein.

Formulation of the optimal control problem

In this section we quickly describe the model and the optimal control problem involved in our analysis.

We consider a SBR with several species in competition for a single substrate. The dynamics of this process can be described as follows (see [START_REF] Smith | The Theory of the Chemostat[END_REF])

           ẋi = µ i (s)x i - F v x i , x i (t 0 ) = x i0 (i = 1 • • • n), ṡ = - n j=1 µ j (s)x j + F v (s in -s), s(t 0 ) = s 0 , v = F, v(t 0 ) = v 0 , (2.1)
where s and v stand, respectively, for the concentration of the substrate and the current volume of water present in the tank. The parameter s in > 0 is a constant which represents the substrate concentration in the input flow. The concentration of the ith species is denoted by x i whose growth functions µ i (•) are non-negative smooth functions such that µ i (0) = 0, and the input flow F is a non-negative control variable.

Given a (desirable) substrate concentration s out ∈]0, s in [ and a volume (of the reactor)

v max > 0, consider the domain D = (IR n + \ {0})×]0, s in ]×]0, v max [ and the target T = IR n + ×]0, s out ] × {v max }.
From any initial condition ξ = (x 10 , . . . , x n0 , s 0 , v 0 ) in D at time t 0 , the objective is to reach T in minimal time. This means to treat as fast as possible the maximum quantity of water, which in the case of the considered SBR is given by v max .

In the context of optimal control theory it leads to the following optimization problem inf

F (•) t -t 0 | s t0,ξ,F (t) ≤ s out , v t0,ξ,F (t) = v max ,
where s t0,ξ,F (•), v t0,ξ,F (•) denote solutions of (2.1) with initial condition ξ ∈ D at time t 0 and control F (•).

Here F (•) is allowed to be a non-negative measurable function plus possible positive impulses. Indeed, instead of an ordinary control F (•), we consider a measure dF (•) that we decompose into a sum of a measure absolutely continuous with respect to the Lebesgue measure u(t)dt and a singular or impulsive part dσ (see [START_REF] Wolenski | A differential solution concept for impulsive systems[END_REF][START_REF] Wolenski | A sampling method and approximation results for impulsive systems[END_REF])

dF (t) = u(t)dt + dσ ,
where, u(•) is a measurable non-negative control that we impose to be bounded from above by u max , because it corresponds to the use of a pump device. At time t, the nonnegative impulse dσ corresponds to an (instantaneous) addition of volume from v -(t) to v + (t).

From [START_REF] Gajardo | Minimal time impulsive control of sequential batch reactors with one or more species[END_REF] we know that a time parameterization τ ≥ t 0 such that dt = r(τ )dτ with r(τ ) = 1 when the pump device is used 0 when an impulse is used, permits to replace dynamics (2.1) by the system

                 dx i dτ = rµ i (s)x i - u v x i (i = 1 • • • n) , ds dτ = -r n j=1 µ j (s)x j + u v (s in -s) , dv dτ = u , (2.2)
where the controls u(•) and r(•) are sought among measurable functions w.r.t. τ , taking values in [0, u max ] and {0, 1}, respectively. Notice that in this formulation u(•) plays the both role of an ordinary control when r = 1 and the control of the amplitude of the jump when r = 0, with the same single constraint u ∈ [0, u max ].

Next, the system (2.2) can be rewritten as the following affine control system

ż = u 1 G 1 (z) + u 2 G 2 (z), (2.3)
where

u 1 = r, u 2 = u, z = (x 1 , • • • , x n , s, v)
, and G 1 (z(τ )), G 2 (z(τ )) be the vector fields defined, respectively, by

G 1 (z(τ )) =              µ 1 (s)x 1 . . . µ n (s)x n - n i=1 µ i (s)x i 0              , G 2 (z(τ )) =              -x1 v . . . -xn v (sin-s) v 1             
The formalism given by system (2.3) will be intensively exploited in the next sections.

Remark 1 Any trajectory of (2.3) with initial condition ξ = (x 10 , . . . , x n0 , s 0 , v 0 ) ∈ D lies in the region defined by:

ρ(ξ) = v   n j=1 x j + s -s in   = w   n j=1 y j + z -s in   . (2.4)
This leads to a reduction of variables in dynamics (2.2), which will be used in Section 5.

Remark 2 Since one can always take r = 0 and u = 0 on a arbitrarily large τ -interval without modifying the total time, the minimal time problem has no unique solution.

Hence, we will be only interested in controls satisfying r(τ ) = 0 or u(τ ) = 0 for all time τ .

Consequently, from now on, we work with V (•) the value function of the reformulated problem (2.3) given by

V (ξ) = inf (u,r)(•) τ t0 r(θ)dθ | s t0,ξ,u,r (τ ) ≤ s out , v t0,ξ,u,r (τ ) = v max , (2.5)
where s t0,ξ,u,r (•), v t0,ξ,u,r (•) denote solutions of (2.3) with initial condition ξ ∈ D at time t 0 and controls u(•) and r(•).

Generalities and Pontryagin's Maximum Principle

In this section we recall the main notions and some results in [START_REF] Gajardo | Minimal time impulsive control of sequential batch reactors with one or more species[END_REF] in order to have a self-contained paper. Thus, in the next subsections we recall the equations and properties obtained from the application of Pontryagin's principle and some known results concerning the optimality of the IOI strategy.

For our purposes it is enough to use the version of the Pontryagin's Maximum Principle (PMP) stated and proved in [START_REF] Pontryagin | The mathematical theory of optimal processes, Translated from the Russian[END_REF]. Recall that the principle gives a first order necessary conditions for continuous-time optimal problems. Its geometric formulation takes place on the cotangent bundle of the state space.

In our particular case, the (PMP) for the affine control system (2.3) on R n+2 reads: If u * = (u 1 * , u 2 * ) is an optimal control and z * is the associated trajectory of (2.3) on R n+2 , there exists a constant λ 0 ≥ 0 and an absolutely continuous function λ :

[0, T ] -→ R n+2 * , τ → λ(τ ) = (λ(τ ), λ n+1 (τ ), λ n+2 (τ )) with λ(τ ) = (λ 1 (τ ), λ 2 (τ ), • • • , λ n (τ ))
such that for almost every τ ∈ Dom(z * ), (λ, λ 0 ) never vanishing and the extremals satisfy

ż = ∂H ∂λ (z, λ, u 1 , u 2 ), (3.1) λ = -λ(u 1 J G 1 + u 2 J G 2 ), (3.2)
Futhermore, the optimal control u * minimizes the Hamiltonian

H = u 1 H G1 + u 2 H G2 + u 1 λ 0 (3.3)
over the set of admissible controls, through the curve (λ(τ ), z(τ )). Here H Gi (z, λ) = λ(τ ), G i (z(τ )) , i = 1, 2 are the Hamiltonian lifts corresponding to each vector field G i respectively. Throughout this paper we assume λ 0 = 1, which corresponds to the normal extremals.

It is well know that the Poisson bracket of the Hamiltonian functions H G1 and H G2 , denote by { •, •}, is associated with the Lie bracket by the relation

{H G1 , H G2 } = λ, [G 1 , G 2 ] (3.4)
where , and the Lie bracket is defined to be

[X, Y ](z) = J Y (z)(X(z)) -J X(z)(Y (z))
for any pair of vector fields X and Y . Here J stand for the Jacobian differential operator with respect to z, and •, • is the Euclidean inner product in R n+2 (see for instance, [START_REF]Méthodes Mathématiques de Mécanique Classique[END_REF]). Now, introducing the auxiliary variables λ i = λ i -λ n+1 , the adjoint system (3.2) become the following dynamical systems

d λ dτ = A(τ ) λ, λ i (T ) = -1, i = 1, • • • , n. (3.5)
where

λ = (λ 1 -λ n+1 , λ 2 -λ n+1 , • • • , λ n -λ n+1
) and A(τ ) is a the n×n time dependent matrix, given by

              u 1 (-µ 1 + µ 1 x 1 ) + u2 v u 1 µ 2 x 2 u 1 µ 3 x 3 • • • • • • • • • • • • u 1 µ n x n u 1 µ 1 x 1 u 1 (-µ 2 + µ 2 x 2 ) + u2 v u 1 µ 3 x 3 • • • • • • • • • • • • u 1 µ n x n . . . . . . . . . . . . . . . . . . u 1 µ n x n u 1 µ 1 x 1 • • • • • • • • • • • • • • • • • • • • • u 1 µ n-1 x n-1 u 1 (-µ n + µ n x n ) + u2 v               .
Consequently, one has

λ(τ ) = 0 for any τ ∈ [τ 0 , T ]. (3.6)
Note that the Hamiltonian (3.3) can be equivalently written as follows

H = u 1 φ u1 (z, λ) + u 2 φ u2 (z, λ) (3.7)
where

       φ u1 (z(τ ), λ(τ )) = 1 + n j=1 λ j (τ )µ j (s(τ ))x j (τ ), φ u2 (z(τ ), λ(τ )) = λ n+2 + λ n+1 (sin-s) v -1 v n j=1 λ j x j (3.8)
For notationally simplicity, we now write φ ui (τ ) for φ u1 (z(τ ), λ(τ )).

In the rest of the paper we will consider only increasing growth functions: Assumption A1. The functions µ i (•) are non decreasing. Lemma 3.1 Under Assumption A1, the following assertions hold:

(i) The matrix A(τ ) has non-negative off-diagonal terms, i.e., the dynamical system (3.5) is cooperative.

(ii) The vector m 1 (•), defined by

m 1 (τ ) =    µ 1 (s(τ ))x 1 (τ ) . . . µ n (s(τ ))x n (τ )    . (3.9)
lies in IR n + . Proof: The proof is direct and it was already established in [START_REF] Gajardo | Minimal time impulsive control of sequential batch reactors with one or more species[END_REF].

2

Notice that the controls are not obviously determined by the minimum condition at times τ when φ u1 (τ ) = φ u2 (τ ) = 0. Indeed, PMP is not able to distinguish minima from maxima in that periods of time. However, it is well known from the optimal control theory that an optimal trajectory may well be singular; this is, switching functions φ ui (τ ), i = 1, 2 may vanish identically along the trajectory. The characterization of such trajectories, is in general a difficult task, but we provide a context for which this holds for our problem defining the value function in (2.5).

Singular trajectories

In our context, a singular trajectory or singular arc corresponds to an extremal curve for which there exists a nontrivial interval [τ 1 , τ 2 ] ⊂ [0, T ] where both switching functions φ ui (τ ) are identically zero. This can be understood as an extension of the standard definition of singular arcs when only one control is considered (e.g. [7, Part III, Ch. 2]).

Since the switching functions play a crucial role in the definition of a singular arc, we proceed here below to analyze them. In particular, we focus on the computation of their derivatives.

Lemma 4.1 For dynamics (2.2), the first derivative with respect to the time τ of the switching functions φ ui (τ ), i = 1, 2, defined in (3.8), are given by

         dφ u2 dτ = -u 1 (s in -s) v λ, m 1 , dφ u1 dτ = u 2 (s in -s) v λ, m 1 .
(4.1)

where λ = (λ 1 -λ n+1 , λ 2 -λ n+1 , • • • , λ n -λ n+1
) and m 1 = m 1 (τ ) was defined in (3.9).

Proof: This property was shown in [11, Eq. (6.5)].

Denote by

ψ := (s in -s) v λ, m 1 (4.2)
the function appearing in the first derivative of the switching functions. We now establish a formula of the n-order derivative for ψ.

Proposition 4.2 Consider a fixed time τ. For a given integer j, define m j = m j (τ ) by

m j (τ ) =      µ (j) 1 (s(τ ))x 1 (τ ) . . . µ (j) n (s(τ ))x n (τ )      , (4.3)
where µ (j) i denotes the j-th derivative of µ i with i = 1, 2, • • • , n. Then if λ, m k = 0 for every k = 1, • • • , j, we have

d j ψ dτ j = (s in -s) v λ, m j+1 ds dτ j , (4.4)
Proof: We proceed to prove the desired implication by induction on j. The case j = 1 follows directly from the definition of ψ given in (4.2). Now, suppose that the induction hypothesis holds true for a given j, that is, the following assertion is fulfilled: if λ, m k = 0 for every k = 1, • • • , j then (4.4) is verified. In order to establish our induction argument, we also suppose that λ, m k = 0 for every k = 1, • • • , j, and we proceed to prove that equation (4.4) holds true when we replace j by j + 1.

The induction hypothesis implies that (4.4) holds, this yield

d j+1 ψ dτ j+1 = (s in -s) v ds dτ j d dτ λ, m j+1 + d dτ (s in -s) v ds dτ j λ, m j+1 (4.5)
On the other hand, direct computations lead to

λ, A(τ ) m j+1 = u2 v λ, m j+1 + u 1 i µ (j+1) i x i λ, m 1 -u 1 λ, m j+1 (4.6) and λ, d dτ m j+1 = ds dτ λ, m j+2 -u2 v λ, m j+1 + u 1 λ, m j+1 , ( 4.7)
where we have used the auxiliary notation

m j =      µ (j) 1 µ 1 x 1 . . . µ (j) n µ n x n      .
So, equations (4.6) and (4.7) lead to

d dτ λ, m j+1 = λ, A(τ ) m j+1 + λ, dm j+1 dτ = ds dτ λ, m j+2 + u 1 n i=1 µ (j+1) i x i λ, m 1 , (4.8)
Replacing (4.8) into (4.5) we have

d j+1 ψ dτ j+1 = (s in -s) v ds dτ j+1 λ, m j+2 + u 1 ds dτ j n i=1 µ (j+1) i x i λ, m 1 + d dτ (s in -s) v ds dτ j λ, m j+1 , (4.9)
Finally, since by hypothesis the two last terms in the right hand side of (4.8) are equal to zero, it follows that

d j+1 ψ dτ j+1 = (s in -s) v ds dτ j+1 λ, m j+2 ,
which proves our induction argument. 2

Now we are ready to state the main result of this section.

Theorem 4.3 Consider initial condition ξ = (x 10 , . . . , x n0 , s 0 , v 0 ) ∈ D. Suppose that the determinant of the matrix

D =           µ (1) 1 (s) • • • µ (1) n (s) µ (2) 1 (s) • • • µ (2) n (s) . . . . . . . . . µ (n) 1 (s) • • • µ (n) n (s)           (4.10)
is nonsingular for any s ∈ (0, s in ). Here µ (j) i denotes the j-th derivative of function µ i .Then, an extremal curve is a singular arc on (τ 1 , τ 2 ) ⊂ [0, T ] if and only if s(•) is constant on (τ 1 , τ 2 ).

Proof: First note that an initial condition ξ = (x 10 , . . . , x n0 , s 0 , v 0 ) ∈ D implies that s(τ ) ∈ (0, s in ) for all time τ , so we will always suppose that s will remain in this interval.

Suppose that a singular arc defined in an interval of time (τ 1 , τ 2 ) ⊂ [0, T ] is optimal. Since u 1 and u 2 are not simultaneously equal to zero, λ, m 1 = 0 on (τ 1 , τ 2 ) (via equations (4.1)); this in particular implies that the derivatives of ψ with respect to the time τ are null on (τ 1 , τ 2 ) . So, we can apply inductively Proposition 4.2 and obtain that

d j ψ dτ j = (s in -s) v ds dτ j λ, m j+1 = 0 (4.11)
on (τ 1 , τ 2 ), and for all j = 1, 2,

• • • , n -1.
Suppose now that ds dτ = 0. It follows from (4.11) that

λ, m j = 0, j = 1, • • • , n on (τ 1 , τ 2 ). (4.12)
Hence, using the fact that λ(τ ) = 0 for all τ (see (3.6)), we have that m 1 , m 2 , • • • , m n are linearly dependent for any τ ∈ (τ 1 , τ 2 ). However, we have m j = Xη j with

X =       x 1 . . . x n       and η j =       µ (j) 1 (s(τ )) . . . µ (j) n (s(τ ))       ,
And, since X is clearly non-singular and the vectors η j are linearly independent (because they are the rows of the matrix D, which is non-singular), these is a contradiction with the linear dependence of m j , concluding thus ds dτ = 0. Reciprocally, suppose now ds dτ = 0 on (τ 1 , τ 2 ) . Equation (4.8) (with j = 0 for which the formula is still valid) leads to Suppose now that C = 0. One obtains from (4.1) that the switching functions φ u1 and φ u2 are both monotone on (τ 1 , τ 2 ) . This implies that φ u1 and φ u2 are both strictly positive on (τ 1 , τ 2 ) . Consequently, (3.7) implies that controls u 1 and u 2 are both equal to zero. This is a contradiction.

Therefore, the only possibility is C = 0. In this case, (4.1) implies that the switching functions φ u1 and φ u2 are both constant on (τ 1 , τ 2 ). Since u 1 and u 2 are strictly positive, it follows from Lemma 4.1 that φ u1 = φ u2 = 0 on (τ 1 , τ 2 ) . The theorem follows.

2

Assumption A2. The determinant of D given in (4.10) is non null.

Remark 3

The Assumption A2 extends to an arbitrary number of species the also called Assumption A2 in [START_REF] Gajardo | Minimal time impulsive control of sequential batch reactors with one or more species[END_REF] established only for two species.

From an initial state ξ = (x 10 , . . . , x n0 , s 0 , v 0 ) ∈ D at time t 0 . We define the IOI consisting of making:

1. An impulse of volume v max -v 0 at t 0 . This can be achieved by r(τ

) = 0, u(τ ) = u max , for τ ∈ [t 0 , t 0 + (v max -v 0 )/u max ].
2. A null control (no feeding) until the concentration s(τ ) reaches s out .

For convenience, we shall denote by xi0 (ξ), i = 1, • • • , n and s0 (ξ) the concentrations obtained with an impulse of volume v max -v 0 from a state ξ = (x 10 , . . . , x n0 , s 0 , v 0 ) ∈ D :

xi0 (ξ) = x i0 v 0 v max , i = 1, • • • , n , s0 (ξ) = s 0 v 0 v max + s in 1 - v 0 v max . (5.1)
Notice that for the particular case s0 (ξ) ≤ s out , the first step only is used. We consider a family of functions ϕ c (•) defined on (IR n + \{0})×IR + and parameterized by c > 0:

ϕ c (x 10 , . . . , x n0 , s 0 ) = inf t -t 0 | s t0,x10,...,xn0,s0 (t) ≤ c , (5.2)
where (s t0,x10,...,xn0,s0 )(•) is solution of the free dynamics:

         dx i dτ = µ i (s)x i , x i (t 0 ) = x i0 (i = 1 • • • n), ds dτ = - n i=1 µ i (s)x i , s(t 0 ) = s 0 . (5.3)
Standard analysis of minimal time problems shows that ϕ c (•) are Lipschitz-continuous functions and solutions, in the viscosity sense, of the partial differential equation (see for instance [START_REF] Bardi | Capuzzo-Dolcetta: Optimal Control and Viscosity solutions of Hamilton-Jacobi-Bellman Equations[END_REF])

n j=1 (∂ xj0 ϕ c (x 10 , . . . , x n0 , s 0 ) -∂ s0 ϕ c (x 10 , . . . , x n0 , s 0 ))µ j (s 0 )x j0 + 1 = 0 , (5.4)
on the domain (IR n + \ {0}) × (c, +∞) with boundary conditions

ϕ c (. , s 0 ) = 0, ∀s 0 ∈ (0, c]. (5.5)
The time cost of the IOI strategy can then be simply written in terms of above functions, as follows

T IOI (ξ) = ϕ sout (x 10 (ξ), . . . , xn0 (ξ), s0 (ξ)), (5.6)
where (x 10 (ξ), . . . , xn0 (ξ)) and s0 (ξ) are given by (5.1). Now, we assume the following regularity on the function ϕ c (•),

Assumption A3. For any c > 0, the function ϕ c (•) is C 1 on (IR n + \ {0}) × (c, +∞).
Remark 4 Observe that the sub-optimal IOI strategy has a finite time cost T IOI (ξ) for any initial condition ξ in the domain D. Consequently, the optimal value V (ξ) is finite for any ξ in D.

Finally, Proposition 5.3 in [START_REF] Gajardo | Minimal time impulsive control of sequential batch reactors with one or more species[END_REF] provides a direct way to verify when the IOI strategy is optimal for any initial condition ξ = (x 10 , . . . , x n0 , s 0 , v 0 ) ∈ D. We recall here below this key tool.

Theorem 5.1 Under Assumption A3, the one impulse strategy is optimal for any ξ ∈ D if and only if

∆ u1 T IOI (ξ) ≥ 0, ∀ ξ ∈ D s.t. T IOI (ξ) > 0.
where ∆ u1 T IOI (ξ) is computed as follows

∆ u1 T IOI (ξ) = n j=1 ∂ xj0 ϕ sout (x 10 (ξ), • • • , xn0 (ξ), s0 (ξ)) -∂ s0 ϕ sout (x 10 (ξ), • • • , xn0 (ξ), s0 (ξ)) × x j0 (µ j (s 0 ) -µ j ( s 0 (ξ))) (5.7)
In order to evaluate whenever the IOI strategy is optimal, the following assumption will be crucial in the sequel.

Assumption A4 For any s 1 , s 2 ∈ [s out , s in ] and j = 1, • • • , n -1, one has s 2 ≥ s 1 ⇒ µ n (s 2 )µ j (s 1 ) ≥ µ j (s 2 )µ n (s 1 ).

(5.8)

Remark 5 Notice that for Monod laws of the form (4.14), implications (5.8) are exactly fulfilled when

K n ≥ K j , j = 1, • • • , n -1. Indeed, direct computations lead to µ n (s 2 )µ j (s 1 ) -µ j (s 2 )µ n (s 1 ) = µ max,n µ max,1 s 1 s 2 (s 2 -s 1 ) (K n -K j ) (K j + s 2 ) (K j + s 1 ) (K n + s 2 ) (K n + s 1 )
.

In [START_REF] Gajardo | Minimal time impulsive control of sequential batch reactors with one or more species[END_REF] it is shown that, when only two species coexist and one is clearly more performant than other one (which is reflected via the condition µ 2 (s) ≥ µ 1 (s), for all s ∈ (0, s sin ], on the growth functions), then the IOI strategy is optimal. Moreover, numerical experiences proved that this condition is also necessary. In other words, the fact that growth functions are increasing is not enough to ensure the optimality of the IOI strategy. For instance, see example below. Following this idea of our analysis, one would like to investigate if similar behaviors occur in this case when more than two species coexist (n > 2). For this, in what follows, we assume the existence of a species which is clearly more performant than the other ones. This is stated via the following assumption.

Assumption A5 µ n (s) ≥ µ j (s), ∀ j = 1, • • • , n -1, ∀ s ∈ (0, s in ].
In this framework, next numerical example shows that Assumptions A1, A3, A4 and A5 do not ensure the optimality of the strategy IOI. In other words, we establish the surprising result that in the case of more than two species, even the existence of a "most performant species"is not enough for this purpose. Next example shows that, even under the Assumptions A1, A3, A4 and A5', strategy IOI is not necessarily optimal.

Example: We consider the following three growth functions (see Figure 1):

µ 1 (s) = s 1 + s , µ 2 (s) = 2s 1.5 + s , µ 3 (s) = 4s 2 + s .
They clearly fulfill Assumptions A1, A2, A3, A4 and A5.

Consider also the parametric values v max = 10, s out = 0.1 and s in = 5, and the initial conditions x 10 = 1, x 20 = 0.001, s 0 = 3 and v 0 = 1. In table 1 we compare, for different values of the initial conditions x 30 , the time achieved by the strategy IOI with the time achieved by an alternative strategy consisting of to reach, as fast as possible, a given level s * in (s out , s in ), then to keep s constant and equal to s * until v reaches v max , and finally, put u = 0 and r = 1 (which means to close SA(s * ), and the level s * is numerically computed in order to minimize the cost of this type of strategies among all possible values of s * in (s out,sin ). Recall that in this case of two species, it was shown in [START_REF] Gajardo | Minimal time impulsive control of sequential batch reactors with one or more species[END_REF] that this strategy besides IOI when Assumptions A1, A2, A3, A4 and A5 were fulfilled.

So, reported results in Table 1 establish that SA(s * ) has an improvement close to 25% with respect to IOI for some values of x 30 . We thus conclude that IOI is not optimal for this particular setting.

A5' µ n (s) ≥ µ n-1 (s) ≥ • • • ≥ µ 1 (s), ∀ s ∈ (0, s in ].
This means that the performance of all the species are perfectly ordered. Thus Assumption A5' can be interpreted as a stronger extension of the corresponding condition in [START_REF] Gajardo | Minimal time impulsive control of sequential batch reactors with one or more species[END_REF]. However, the example above states that even under this stronger condition it is not possible to ensure the optimality of the IOI strategy. We proceed to investigate other hypotheses (stronger than A5) for which one can prove that IOI strategy is optimal. For this, at this stage of our analysis, we need to state a technical lemma for the family of functions ϕ c (•). This is direct extensions of the corresponding results in [START_REF] Gajardo | Minimal time impulsive control of sequential batch reactors with one or more species[END_REF]. Since its proof is very similar to those appearing in [START_REF] Gajardo | Minimal time impulsive control of sequential batch reactors with one or more species[END_REF], it is omitted here. Lemma 5.2 Under Assumptions A1, A3, A4 and A5, functions ϕ c (•) posses the following properties for all c ∈ (0, s in ) , (y, z) := (x 10 , . . . , x n0 , s 0 ) ∈ IR 2 + \ {0} × (c, s in ] and i

= 1, • • • , n -1 : i. ∂ z ϕ c (y, z) ≥ ∂ xn0 ϕ c (y, z), ii. µn(z)-µn(z) µn(z)
≤ µi(z)-µi(z) µi(z) ≤ 0. Now, we are in position to establish an additional condition for which IOI is optimal in this framework.

Theorem 5.3 Suppose that the Assumptions A1, A3, A4 and A5 are fulfilled. Suppose also that growth functions µ i , for i = 1, . . . , n -1, are all close enough one each other on [0, s in ], that is, there exists a small enough > 0 such that for all i, j = 1, . . . , n -1, s ∈ [0, s in ] one has µ i (s) -µ j (s) ≤ .

(5.9)

Then IOI strategy is optimal for any initial condition in D.

Proof: Consider ξ = (x 10 , . . . , x n0 , s 0 , v 0 ) := (y, z, v 0 ) ∈ D such that T IOI (ξ) > 0. Let us denote for all i = 1, . . . , n -1 :

γ i = µ i (z) -µ i (z) µ i (z) / n-1 j=1
µ j (z) -µ j (z) µ j (z) .

Part iv of Lemma 5.2 ensures that γ i ≥ 0 for all i. Moreover, Hypothesis (5.9) implies that

γ i ∈ 1 n -1 -˜ , 1 n -1 + ˜ , ∀i = 1, ..., n -1, (5.10)
for some ˜ > 0. Note that the magnitude of ˜ is driven by , that is, ˜ can be set as small as necessary provided that is choosen small enough. Indeed, it suffices to note that µ i (z) ≥ µ i (z) and the fact that the µ i are bounded on [0, s in ].

Notice that the term ∆ u1 T IOI (ξ), given in (5.7), can be written as follows

∆ u1 T IOI (ξ) = n j=1
∂ yj ϕ sout (ỹ, z) -∂ z ϕ sout (ỹ, z) µ j (z) ỹj µ j (z) -µ j (z) µ j (z) .

So, from Part iv of Lemma 5.

  d dτ λ, m = ϕ(τ ) λ, m , for some given function ϕ(•). Hence, λ, m = C exp τ 0 ϕ(τ )dτ on (τ 1 , τ 2 ), for a real constant C.

Figure 1 :

 1 Figure 1: Graphs of the three growth functions considered in the example.

∂∂∂∂

  2 and (5.10) we obtain that γ n ≥ γ i ≥ 1 n-1 -˜ , for all i = 1, ..., n -1. Then, Part i of Lemma 5.2 implies∆ u1 T IOI (ξ) yj ϕ sout (ỹ, z) -∂ z ϕ sout (ỹ, z) µ j (z) ỹj γ j + (∂ yn ϕ sout (ỹ, z) -∂ z ϕ sout (ỹ, z)) µ n (zyj ϕ sout (ỹ, z) -∂ z ϕ sout (ỹ, z) µ j (z) ỹj + ˜ n j=1 ∂ yj ϕ sout (ỹ, z) -∂ z ϕ sout (ỹ, z) µ j (zyj ϕ sout (ỹ, z) -∂ z ϕ sout (ỹ, z) µ j (z) ỹj ,where this last equality is due to (5.4). Consequently∆ u1 T IOI (ξ) ≥yj ϕ sout (ỹ, z) -∂ z ϕ sout (ỹ, z) µ j (z) ỹj  Therefore, since y, ỹ, z, z remain in a compact set and all the function involved are C 1 , the right hand side term in the above expression is positive provided that ˜ is small enough. We thus conclude by Theorem 5.1.2

The n-species case with a Monod growth function

In section we analyze the case when the growth function of each species of microorganism considered in the SBR follows a Monod law growth function [START_REF] Monod | Recherches sur la croissance des cultures bactriennes[END_REF], that is

where, for the i-species, µ max,i and K i stand for the maximal growth rate and for the half saturation constant or Monod constant. As a first step, we establish a formula for the derivatives of a general Monod function.

Lemma 4.4 For any Monod function described by (4.14), its derivatives with respect to s are given by

Proof: The proof follows directly from induction on j. 2

As an application of the results obtained in this section, a characterization of singular arcs, for the case in which several species compete for the same substrate and all of them have Monod growth function, is given in the following proposition. Proposition 4.5 Assume that any growth function in our setting follows a Monod law, which is different from the growth function of any other species. This is, if

and K i = K j , for all i = j. Then, an extremal curve is a singular arc on (τ 1 , τ 2 ) ⊂ [0, T ] if and only if s(•) is constant on (τ 1 , τ 2 ) .

Proof: Lemma 4.4 and some additional computations show that the determinant of the matrix D, defined in (4.10), is given by

Now, since K i = K j , for all i, j = 1, • • • , n it follows that the determinant of D is non null, and consequently D is nonsingular. Then, the desired result follows from Theorem 4.3. 2

The Immediate One Impulse strategy

This section is devoted to evaluate whenever the Immediate One Impulse strategy (denoted from now on by IOI ) is optimal for our minimal time control problem, when one species is highly performant and the others all are sufficiently close one to each other. We begin by first describing the IOI strategy.


Paul Bosch
email: paul.bosch@udp.cl

Julio López
email: julio.lopez@udp.cl

Héctor Ramírez
email: hramirez@dim.uchile.cl

Hugo Robotham
email: hugo.robotham@udp.cl

Support vector machine under uncertainty: An application for hydroacoustic classification

Keywords: Support vector machines, multi-class classification, robust chance constraints, second-order cone programming, species identification

come   L'archive ouverte pluridisciplinaire

Introduction

Acoustics surveys are used to estimate the abundance and to study the behavior of many species. The last decades shows a great progress in the development of acoustics devices, and of new methods and techniques for automatically identifying species (see [START_REF] Horne | Acoustic approaches to remote species identification: a review[END_REF], [START_REF] Horne | Acoustic approaches to remote species identification: a review[END_REF] and [START_REF] Fernandes | The SIMFAMI Project: Species Identification Methods for Acoustic Multifrequency Information[END_REF], [START_REF] Fernandes | The SIMFAMI Project: Species Identification Methods for Acoustic Multifrequency Information[END_REF]). The echograms obtained with acoustics devices provide information of size, location and echo intensity of fish schools. However, its classification is still not detectable by modern devices. Indeed, according to [START_REF] Horne | Acoustic approaches to remote species identification: a review[END_REF], the species identification is the big challenge of fisheries and plancton acoustics.

A wide range of statistical techniques have been used to classify mono-specific fish-school. [START_REF] Fernandes | Classification trees for species identification of fish-school echotraces[END_REF](see [START_REF] Fernandes | Classification trees for species identification of fish-school echotraces[END_REF]) presents a complete list of studies including principal component analysis, discriminant-function analysis, artificial neural networks, nearest-neighbor analysis, k-means clustering and mixture models. [START_REF] Buelens | Kernel methods for the detection and classification of fish schools in single beam and multibeam acoustic data[END_REF] (see [START_REF] Buelens | Kernel methods for the detection and classification of fish schools in single beam and multibeam acoustic data[END_REF]) use kernel methods to classify fish schools in single beam and multibeam acoustic data. In [START_REF] Robotham | Acoustic identification of small pelagic fish species in Chile using support vector machines and neural networks[END_REF] (see [START_REF] Robotham | Acoustic identification of small pelagic fish species in Chile using support vector machines and neural networks[END_REF]), support vector machine methods (tool for discriminating between two groups, see [START_REF] Cortes | Support-vector networks[END_REF]) are used for the automatic classification of small pelagic fish species from acoustic surveys data. The results obtained by this approach was compared with an artificial neural networks methods.

Typically, the methods used to classify species of fishes require a scrutiny process of the echograms, the guide of expert criteria and additional information from trawl sampling [START_REF] Simmonds | Fisheries Acoustics: Theory and Practice[END_REF], see [START_REF] Simmonds | Fisheries Acoustics: Theory and Practice[END_REF]). This procedure incorporates some grade of subjectivity. Thus, other approaches have been developed based only on the information provided by the echosounders. For instance, the identification of species based on school "descriptors parameters" from single-frequency acoustic data represents one type of approach. A second type of approach, which will not be exploded in this work, makes use of multifrequency acoustic data [START_REF] Korneliussen | Acoustic species identification of schooling fish[END_REF], see [START_REF] Korneliussen | Acoustic species identification of schooling fish[END_REF]).

The purpose of this article is to compare the results of support vector machines (deterministic SVM) and classification when the uncertainty of data is incorporated for automatic acoustic identification of small pelagic fish species; anchovy (Engraulis ringens), common sardine (Strangomera bentincki) and jack mackerel (Trachurus murphyi) in southern-central Chile. The latter makes no precise assumption on the class-conditional densities with given mean and covariance matrix, that is, it is a "worst-case" setting.

Data collection and descriptors

School data were obtained from 11 acoustic assessment surveys performed with the R/V Abate Molina in northern and south-central Chile (18 • 25'S-43 • 50'S) between 1991 and 2006. The data were collected using a scientific echosounder (SIMRAD EK-500) with a split-beam transducer (ES38 38 kHz) with a nominal -3 dB beam with of 7 • , calibrated according to standard procedures (Foote et al., 1987). The ping rate of the echosounder in the surveys was 1 vs-1, the pulse duration was 1 ms and a minimum threshold of -65 dB. An Engel pelagic trawl with a 14-m vertical opening and 14-mm mesh size in the codend as used to identify the species in the acoustic survey. The flotation line of this net was adapted for fishing near the surface.

The acoustic records of the fish schools detected by the echosounder were processed with Echoview 3.0 software (Sonardata). Schools were identified by fishing hauls carry out using a midwater trawl. The parameters of the fish schools were determined automatically by the algorithm SHAPES programmed into the software Echoview and described in [START_REF] Barange | Acoustic identification, classification and structure of biological patchiness on the edge of the Agulhas Bank and its relation to frontal features[END_REF], and [START_REF] Coetzee | Use of a shoal analysis and patch estimatiom system (SHAPES) to characterize sardine schools[END_REF], (see [START_REF] Barange | Acoustic identification, classification and structure of biological patchiness on the edge of the Agulhas Bank and its relation to frontal features[END_REF] and [START_REF] Coetzee | Use of a shoal analysis and patch estimatiom system (SHAPES) to characterize sardine schools[END_REF], respectively). Each aggregation was manually marked in a region on the image of the echogram, and each case was individually analyzed. The parameters used were minimum candidate heigth=1m, minimum candidate length =1m, maximum vertical linking distance =1m, and maximum horizontal linking distance =15m.

The input data for a classification of fish-school is a collection of acoustics records. Each acoustics record, is characterized by a duple (x, y), where x ∈ R n is the n-dimensional descriptors vector and y is a category (class) species which belong this descriptor. We used 12 descriptors for each school detected, which were grouped in four categories (see Scalabrín and Massé (1993), [START_REF] Scalabrin | Acoustic detection of the spatial and temporal distribution of fish shoals in the Bay of Biscay[END_REF]):

1. Morphological: mean height (H), length (L) and perimeter (P ) with units in meter (m); area (m 2 ) and unidimensional descriptors:

-Elongation = length mean height . -F ractal dimension = 2 ln 0.25P ln area .

2. Bathymetric: Bottom depth (D) and mean school depth (D m ) with units in meter and unidimensional descriptor:

-School altitude index = 100 D-Dm D .

3. Energetic: Acoustic energy (m 2 /mn 2 ) and acoustic density (dB).

4. Space position: School-shore distance (mn).

Some bathymetric and morphological descriptors are presented in Figure 1. 2.1 l 2 -Support vector machines Support vector machines (SVMs) is a statistical classification method originally designed for binary classification. Given a set T = {(x 1 , y 1 ), . . . , (x m , y m )} of m training data, where x i ∈ R n representing the ith training data and y i ∈ {-1, 1} the class label of x i , SVM provides the optimal hyperplane f (x) = w x -b that separates two classes. When the training data are linearly separable, this hyperplane separates two classes with no training error, and maximizes minimum distance from the training data to the hyperplane. In order to maximize this minimum distance, we need to classify correctly the vectors x i of the training set into two different classes y i , using the smallest norm of coefficients w. The maximum hyperplane problem can be formulated as the following Quadratic Programming (QP) optimization problem (see [START_REF] Cortes | Support-vector networks[END_REF]):

min w,b 1 2 w 2 y i (w x i -b) ≥ 1, i = 1, . . . , m.
(2.1)

Support vector machines under uncertainty

Let X 1 and X 2 be two random vector variables associated to the positive and negative classes, respectively. In order to construct a maximum margin linear classifier such that the false-negative and false-positive error rates do not exceed η 1 ∈ (0, 1] and η 2 ∈ (0, 1], respectively, we consider the following Quadratic Chance-Constrained Programming (QCCP) problem:

min w,b 1 2 w 2 Pr{w X 1 -b ≤ -1} ≤ η 1 , Pr{w X 2 -b ≥ 1} ≤ η 2 .
(

In other words, we require that the random variable X i lies on the correct side of the hyperplane with probability greater than 1 -η i for i = 1, 2. Assume that for i = 1, 2 we only know the mean µ i ∈ R n and covariance matrix Σ i ∈ R n×n of the random vector X i . In this case, for each i = 1, 2 we want to be able to classify correctly, up to the rate η i , even for the worst distribution in the class of distributions which have common mean and covariance X i ∼ (µ i , Σ i ), replacing the probability constraints in (2.2) with their robust counterparts sup

X 1 ∼(µ 1 ,Σ 1 ) Prob{w X 1 -b ≤ -1} ≤ η 1 , sup X 2 ∼(µ 2 ,Σ 2 ) Prob{w X 2 -b ≥ 1} ≤ η 2 .
Thanks to an appropriate application of the multivariate Chebyshev inequality, this worst distribution approach leads to the following quadratic second-order cone programming (QSOCP) problem (see [START_REF] Alizadeh | Second-order cone programming[END_REF]), which is a deterministic formulation of (2.2) (see [START_REF] Saketha Nath | Maximum margin classifiers with specified false positive and false negative error rates[END_REF][START_REF] Alvarez | Interior proximal algorithm with variable metric for second-order cone programming: applications to structural optimization and support vector machines[END_REF] for all details):

min w,b 1 2 w 2 w µ 1 -b ≥ 1 + κ 1 S 1 w b -w µ 2 ≥ 1 + κ 2 S 2 w , (2.3)
where Σ i = S i S i (for instance, Cholesky factorization) and η i and κ i are related via the

formula κ i = 1-η i η i , for i = 1, 2.
Note that any feasible hyperplane must separate the means, hence the natural condition µ 1 = µ 2 is necessary for (2.3) to be feasible. Since κ i → 0 when η i → 1, the problem (2.3) can be made feasible whenever µ 1 = µ 2 by choosing appropriate values for η 1 and η 2 . The election of different values of η 1 and η 2 leads to classify with preferential bias towards a particular class. For instance, in many binary medical classification problems, the cost of misclassifying one category is higher than the other, and in these applications it is desirable to employ a classifier with selective sensitivity or specificity (Cismondi et al [START_REF] Cismondi | Multi-stage modeling using fuzzy multi-criteria feature selection to improve survival prediction of ICU septic shock patients[END_REF]). This is the case of medical diagnosis of cancer (e.g. [START_REF] Kononenco | Machine learning for medical diagnosis: History, State of Art and Perspective[END_REF][START_REF] Nahar | Computational intelligence for microarray data and biomedical image analysis for the early diagnosis of breast cancer[END_REF]), where the cost of misclassifying a cancer patient is higher than the cost of misclassifying of a healthily patient.

Finally, since κ i → ∞ when η i → 0, we note that problem (2.3) becomes unfeasible for small values of η 1 and/or η 2 .

Practical implementation

So far we have assumed that the mean-covariance pairs (µ i , Σ i ) are known. However, in many practical situations we only have the training data set T = {(x 1 , y 1 ), . . . , (x m , y m )}. Assuming that T consists of two samples of independent observations of the random vectors X 1 for y = 1 and X 2 for y = -1, the idea is to replace (µ i , Σ i ) with a statistical estimator ( µ i , Σ i ); this can be done by computing the sample mean and covariance for each class from the available observations. We denote by

T 1 = [x 1 1 , . . . , x 1 m 1 ] ∈ R n×m
µ i = μi = 1 m i T i 1 m i , Σ i = Σi = S i S i with S i = 1 √ m i (T i -µ i 1 m i ), for i = 1, 2
, where 1 m i denotes a vector of ones of dimension m i . Since w ∈ R n , it can be written as w = [T 1 , T 2 ]s + Mr, where M is a matrix with its columns as vectors orthogonal to training data points and s, r are vectors of combining coefficients. The columns of T 1 , T 2 and M together span entire R n . Now, the terms involving w in the constraints of (2.3) can be written as

w µ i = s g i , w Σ i w = s G i s, i = 1, 2,
where

g 1 = 1 m 1 [K 11 1 m 1 ; K 21 1 m 1 ], g 2 = 1 m 2 [K 12 1 m 2 ; K 22 1 m 2 ], G 1 = 1 m 1 [K 11 ; K 21 ](I m 1 - 1 m 1 1 m 1 1 m 1 )[K 11 , K 12 ]
and

G 2 = 1 m 2 [K 12 ; K 22 ](I m 2 - 1 m 2 1 m 2 1 m 2 )[K 21 , K 22 ]
with I m i denoting the identity matrix of dimension

m i × m i , K 11 = T 1 T 1 , K 12 = K 21 = T 1 T 2 , K 22 = T 2
T 2 matrices whose elements are inner products of data points. For instance, the entry (i, j) for the matrix K 12 is

(K 12 ) ij = x 1 i , x 2 j .
(2.4) Thus, the formulation (2.3) can be written as:

min s,b 1 2 s Ks s g 1 -b ≥ 1 + κ 1 s G 1 s b -s g 2 ≥ 1 + κ 2 s G 2 s, (2.5)
where

K = [K 11 , K 12 ; K 21 , K 22 ]
. Note that in order to solve the above problem, we need to know only the dot products of training data points. Thus, one can solve the above problem in any feature space as long as the inner products in that space are available.

Many practical situations cannot directly solved with the former approach. Consequently, one usually introduces a kernel formulation. This consists of considering a kernel function k : R n × R n → R satisfying the Mercer conditions (see [START_REF] Mercer | Functions of positive and negative type and their connection with the theory of integral equations[END_REF]). Thus, for instance, the inner product (2.4) is replacing by (K 12 ) ij = k(x 1 i , x 2 j ). The typical kernel functions include the Gaussian kernel defined by k(u, v) = exp(-γ u -v 2 ) with γ ∈ R + , and the polynomial function k(u, v) = (u v + 1) d with d ∈ N. After choosing this kernel function, the quantities g 1 , g 2 , G 1 , G 2 and K can be computed. In the case when K is positive definite, we can use the Cholesky factorization K = L L to obtain a full rank matrix L ∈ R m×m . Thus, introducing a new variable v = Ls, the formulation (2.5) is rewritten as follows

min v,b 1 2 v 2 v h 1 -b ≥ 1 + κ 1 v H 1 v b -v h 2 ≥ 1 + κ 2 v H 2 v, (2.6)
where

h i = L -g i and H i = L -G i L -1 , for i = 1, 2.
Again, H i being positive semidefinite can be written as H i = D i D i . Then, (2.6) can be written in the standard QSOCP form

min v,b 1 2 v 2 v h 1 -b ≥ 1 + κ 1 D 1 v , b -v h 2 ≥ 1 + κ 2 D 2 v .
(2.7)

Clearly, by introducing a new variable t and a constraint v ≤ t, (2.7) can be casted as the following linear SOCP with three blocks (cf. [START_REF] Alizadeh | Second-order cone programming[END_REF])

min t,v,b t t ≥ v , v h 1 -b ≥ 1 + κ 1 D 1 v , b -v h 2 ≥ 1 + κ 2 D 2 v .
(2.8)

Hence, one can classify a new data point x using the following decision function

f (x) = sign(w x -b) = sign(s h(x) -b), (2.9)
where s = L -1 v and h(x) = h 1 (x) h 2 (x) with (h i (x)) l = k(x i l , x) for l = 1, . . . , m i and i = 1, 2.

Numerical results and discussion

Support vector machines (SVM) were originally designed for binary classification. However, they can be adapted for multi-class problems. According to (Weston and Watkins, 1999 and Hsu and Lin, 2002) two strategies are used to approach the multi-class SVM problem. A first strategy consist of solving a series of binary classifications using two possible approaches: one-species-against-one (1-vs-1) and one-species-against-the-Rest (1-vs-R). A second strategy directly considers all the data in a single optimization formulation. The latter leads to an optimization problem which is much more difficult to solve numerically [START_REF] Weston | Support vector machines for multi-class pattern recognition[END_REF]. In this paper, we use the first strategy together with criterion 1-vs-R. This constructs M binary SVM classifiers, each of which separates one class from all the rest. Throughout this article this set of machines is called multi-class 1-vs-R classifier, in contrast with each binary 1-vs-R classifier used to construct it. Thus, the deterministic ith binary 1-vs-R classifier is a SVM trained with all the training examples of the ith class with positive labels, and all the others with negative labels. That is, the ith SVM solves problem (2.1) where the decision function is given by

f i (x) = w i x -b i . (3.1)
So, it solves min

w i ,b i 1 2 w i 2 ; ỹj (w i x j -b i ) ≥ 1, j = 1, . . . , m, (3.2)
where ỹj = 1 if y j = i and ỹj = -1 otherwise. Then, at the classification phase, a sample x is classified in the class which attains the largest value of f i (x), that is, x is in the i * th class when f i * (x) = max{f i (x) : i = 1, . . . , M }. In the (exceptional) case when this maximum is attained in more than one class sample x is (by convention) classified in the class associated with the lowest index i * .

Based on this idea and on the stochastic approach described in Section 2, we can similarly construct stochastic binary 1-vs-R classifiers and stochastic multi-class 1-vs-R classifiers. More precisely, in order to construct a stochastic binary 1-vs-R classifier, we solve (2.7) instead of (3.2), where now the variable v is replaced by w i , and function f , defined in (2.9), is replaced by f i , defined in (3.1).

In the next section we construct several multi-class 1-vs-R classifiers and we use the average rate of correct classifications in order to compare their performance. Each multiclass 1-vs-R classifier is composed by three binary 1-vs-R classifiers, one for each fish species. All our numerical experiment were solved using different codes in Matlab. Indeed, we have used the SVM-Light software (http://svmlight.joachims.org/) for solving deterministic SVM and the SeDuMi software for solving SOCP problems of the form (2.8) (http://sedumi.ie.lehigh.edu/).

Deterministic and stochastic formulation with kernel

A well-known variant of the deterministic binary 1-vs-R classifier is to consider margins to classify each class. Roughly speaking, the relevance of these margins in the SVM is driven by a penalty positive parameter denoted by C (the penalization is stronger when C is larger). In (Robotham et al [START_REF] Robotham | Acoustic identification of small pelagic fish species in Chile using support vector machines and neural networks[END_REF]) the estimation of the parameters C and γ was realized for divers machines for classifying fish-schools using acoustic data (all of them with Gaussian kernel). In the following table, these parameters appear together with the classification rate of each one of the three deterministic binary 1-vs-R classifier. Next, we present the confusion matrix1 obtained for a deterministic multi-class classification based on the last three deterministic binary 1-vs-R classifiers. Our first idea is to use some attributes of this confusion matrix (for instance, classification rate, average rate) to compare the performance of the deterministic multi-class 1-vs.-R classifier presented above with the performances of several stochastic multi-class 1-vs.-R classifiers. They are constructed for different sets of parameters η 1 and η 2 (each classifier contains six parameters, one pair for each of the three binary 1-vs.-R classifiers) For this, we first set the values of parameters η 1 and η 2 as the proportion of each class among the total. For instance, in the machine anchovy vs. the rest, the values of these parameters are given by: The next tables illustrate the performances of each stochastic binary 1-vs-R classifiers and of the stochastic multi-class 1-vs-R classifier. The latter via its confusion matrix. Notice that this choice of parameter leads to a worst performance than the deterministic classifier. This is reflected in lower classification rates for the three binary 1-vs-R classifiers (see Table 1 and compare with 3). In particular, in the stochastic binary jack mackerel vs. the rest classifier, the decision function seems to became more important than in the deterministic case. Indeed, there are three anchovy and one common sardine misclassified as jack mackerel (see Table 2 and4). However, in the deterministic SVM neither anchovies nor common sardine were classified as jack mackerel.

Sensitivity analysis of stochastic multi-class 1-vs-R classifiers

In this section, we evaluate the average classification rate of divers stochastic multi-class 1-vs-R classifiers when η 2 (false-positive rate) varies. We start this analysis from the results established in Table 4, it shows the relative importance of the stochastic binary 1-vs-R jack mackerel vs. the rest classifier in the performance of the stochastic multi-class 1-vs-R classifiers. Then, we think it is interesting to start our sensitivity analysis by decreasing the value of η 2 from

η J 2 =
Total Rest Total fish = 201 228 = 0, 88.

Since η 2 represents the false-positive rate, lower values of this value should improve the accuracy of this particular stochastic binary 1-vs-R jack mackerel vs. the rest classifier. However, the performance of the whole stochastic multi-class 1-vs-R classifiers could be affect if the rest of the stochastic binary 1-vs-R classifiers behaves badly. The remaining parameters remain unaltered (see Table 3) as well as the number of species used for testing: 98 anchovy, 27 jack mackerel and 103 sardines. 5 above presents the confusion matrices for the stochastic multi-class 1-vs-R classifiers obtained for different values η J 2 in the stochastic binary 1-vs-R jack mackerel vs. the rest classifier. As expected, lower values of η J 2 imply better classification rates for this particular classifier. This has also led to a improvement of the average classification rate of the stochastic multi-class 1-vs-R classifiers, even when this positive behavior is not ensured by this choice. Indeed, this average rate could be worse due to a poorer performance of one of the two other stochastic binary 1-vs-R classifiers. This effect can be seen for instance in the variation of the performance of the stochastic binary anchovy vs. the rest classifier with respect to η J 2 , which is non-monotonic and seems to attains its best performance for η J 2 between 0.38 and 0.48. Notice that during these experiments the classification of the stochastic common sardine vs. the rest remains unaltered. This situations suggest us to proceed our experiment by varying the values η A 2 corresponding to the false-positive rate of the stochastic binary anchovy vs. the rest classifier.

For this, we set η J 2 = 0.18 and keep constant the rest of values of all parameters except for η A 2 . We have inspected several values of η J 2 but only two of of them give meaningful results: η A 2 = 0.47 and η A 2 = 0.17. They are reported in following Table 6 via their confusion matrices.

Figure 1 :

 1 Figure 1: Representation of bathymetric and morphological descriptors

1 a n × m 1

 1 data matrix for positive class and by T 2 = [x 2 1 , . . . , x 2 m 2 ] ∈ R n×m 2 a n × m 2 data matrix for negative class. The empirical estimates of the mean and covariance are given by

Table 1 :

 1 Classification rates (%) using deterministic binary 1-vs-R classifiers.

	Species	Classification Parameters
		rate (%)	(C, γ)
	Anchovy vs rest	86.7	88.5	(150, 0.14)
	Jack mackerel vs rest	81.5	99.5	(110, 0.12)
	Common sard. vs rest 90.3	94.4	(117, 0.15)

Table 2 :

 2 Confusion matrix for deterministic multi-class 1-vs.-R classifier.

	Species	Anchovy	Jack	Common Testing Classification Average
			mackerel	sardine	data	rate	rate
	Anchovy	89	0	9	98	90.8%
	Jack mackerel	4	23	0	27	85.2%	88.4%
	Common sard.	11	0	92	103	89.3%

Table 3 :

 3 Classification rates (%) using stochastic binary 1-vs-R classifiers.

	Species	Classification		Parameters
		rate (%)	γ	(η 1 , η 2 )
	Anchovy vs rest	87.8	80.0	0.14 (0.43, 0.57)
	Jack mackerel vs rest	74.1	80.6	0.12 (0.12, 0.88)
	Common sard. vs rest 91.3	86.4	0.15 (0.45, 0.55)

Table 4 :

 4 Classification rates (%) using stochastic multi-class 1-vs-R classifier.

	Species	Anchovy	Jack	Common Testing Classification Average
			mackerel	sardine	data	rate	rate
	Anchovy	86	3	9	98	87.76%
	Jack mackerel	4	22	1	27	81.48%	84.57%
	Common sard.	15	1	87	103	84.47%

Table 5 :

 5 Confusion matrices for different values of η J 2 .

	Parameter Species	Anchovy	Jack	Common Classification Average
			mackerel	sardine	rate	rate
	Anchovy	87	2	9	88.78 %
	η J 2 = 0.78 Jack mackerel	4	22	1	81.48 %	84.91 %
	Common sard.	15	1	87	84.47 %
	Anchovy	88	1	9	89.80 %
	η J 2 = 0.68 Jack mackerel	4	22	1	81.48 %	85.25 %
	Common sard.	15	1	87	84.47 %
	Anchovy	88	1	9	89.80 %
	η J 2 = 0.58 Jack mackerel	4	22	1	81.48 %	85.25 %
	Common sard.	15	1	87	84.47 %
	Anchovy	89	0	9	90.82 %
	η J 2 = 0.48 Jack mackerel	4	22	1	81.48 %	85.59 %
	Common sard.	15	1	87	84.47 %
	Anchovy	89	0	9	90.82 %
	η J 2 = 0.38 Jack mackerel	4	22	1	81.48 %	85.59 %
	Common sard.	15	1	87	84.47 %
	Anchovy	88	1	9	89.80 %
	η J 2 = 0.28 Jack mackerel	4	23	0	85.19 %	86.49 %
	Common sard.	15	1	87	84.47 %
	Anchovy	86	3	9	87.76 %
	η J 2 = 0.18 Jack mackerel	3	24	0	88.89 %	87.04 %
	Common sard.	15	1	87	84.47 %
	Table

This matrix represents the count of classifier's class predictions with respect to the actual outcome on some labeled learning set.

Acknowledgements

This research was supported by CONICYT-Chile, under Anillo project 10336: "Analysis of control problems & applications" (Julio López and Héctor Ramírez) and FONDECYT Grants 1090063 (Paul Bosch), 11110188 (Julio López), and 1110888 (Héctor Ramírez). The third author is also supported by BASAL Project (Centro de Modelamiento Matemático, Universidad de Chile), and project BIONATURE of CIRIC, INRIA-Chile. The field data was recollected with the support of projects financed by the Chilean Fisheries Research Fund (FIP in Spanish) of the Undersecretary of Fisheries.

 7shows that the average rate obtained by our last stochastic multi-class 1-vs-R classifier is 89.55%. This value is better than the one obtained by the deterministic multi-class 1-vs-R classifier 88.4% (see Table 2). So, despite of the uncertainties on the data, we have construct a more performant machine via an inspection procedure regarding false-positive rates η 2 .