V Choqueuse

A Belouchrani

M Benbouzid

PHASOR ESTIMATION USING CONDITIONAL MAXIMUM LIKELIHOOD: STRENGTHS AND LIMITATIONS

Keywords: Phasor Measurement Units, Smart Grid, Maximum Likelihood, Condition Monitoring

This paper focuses on the estimation of the phasor parameters in three-phase power systems for smart grid monitoring. Specifically, it investigates the use of the Conditional Maximum Likelihood (ML) for phasor parameter estimation. The contribution of this paper is twofold. First, it presents the condition on the signal model for identifiability of the phasor parameters. Then, it shows that the Conditional Maximum Likelihood estimator has a simple closed form expression, which can be determined from simple geometrical properties. Simulation results illustrate the effectiveness of the proposed approach for the estimation of the phasor amplitude and angle shift under dynamic conditions.

INTRODUCTION

For practical and economical reasons, electrical grids use three-phase electric power at the generation, transmission and distribution sides. Under nominal conditions, three-phase systems are composed of three conductors, each carrying a 50 Hz (or 60 Hz) sine wave. In balanced systems, these sine waves have the same amplitude but are phase-rotated from each other by an angle of 2π/3. In practice, many phenomena can introduce deviations from the nominal conditions. For example, a mismatch between energy generation and load introduces frequency deviations [START_REF] Top | Observing the power grid: Working toward a more intelligent, efficient, and reliable smart grid with increasing user visibility[END_REF]. Similarly, abnormal grid events, such as voltage sags, introduce three-phase unbalance [START_REF] Bollen | Signal processing of power quality disturbances[END_REF]. These deviations must be detected at an early stage to avoid severe consequences. For these reasons, electrical signal monitoring is of main importance for improving the power grid reliability and enabling energy management systems.

To monitor the state of the grid, electronic devices called Phasor Measurement Units (PMUs) are dispersed throughout the power grid [START_REF] Phadke | Synchronized phasor measurements and their applications[END_REF][START_REF] Giannakis | Monitoring and optimization for power grids: A signal processing perspective[END_REF]. From a signal processing viewpoint, these devices compute for each phase a complex number d k e jϕ k called the phasor, where d k and ϕ k correspond to the amplitude and phase estimates of the electrical signal evaluated at the fundamental frequency. The IEEE Standard C37.118.1 specifies the performance requirements for phasor estimation [5]. To be compliant with this standard, the phasor estimator must meet certain requirements in terms of accuracy and ability to track fast variations. Several techniques have been proposed in the literature for estimating the phasor. Under stationary conditions, classical techniques include the Root Mean Square and DFT estimators [START_REF] Bollen | Signal processing of power quality disturbances[END_REF]6]. Under non-stationary conditions, more sophisticated algorithms have been proposed. A complete state-of-art can be found in [7]. From an estimation perspective, it should be mentioned that the most challenging requirement is the ability to track fast variations. Recently, a new estimator based on the Maximum Likelihood has been proposed for the estimation of the phasor amplitude [8]. The advantage of this estimator is twofold. First, by considering the modulated signals as unknown parameters, it is well suited for signals with amplitude and or phase modulation. Then, as it is based on the Maximum Likelihood framework, the proposed estimator has the attractive property to be asymptotically optimal under some conditions. Nevertheless, the main limitation of this estimator is the fact that it assumes a perfect 2π/3 angle shift between phases. In practice, this assumption limits its use for phasor estimation.

In this paper, we extend the Conditional Maximum Likelihood Technique in [8] by considering a more general setting, where both the amplitude and phase of the phasors are unknown. The contribution of this paper is twofold. First, we provide the conditions for identifiability of the phasor parameters. Then, we show that the phasor parameters can be simply estimated using basic geometrical properties. This paper is organized as follows. Section 2 presents the signal model and section 3 addresses the phasor estimation problem. Finally, section 4 provides some simulation results.

SIGNAL MODEL

In unbalanced three-phase systems, the electrical waveform on the k th phase can be modeled as (k = 0, 1, 2)

x k [n] = d k a[n] cos(φ[n] + ϕ k ) + b k [n], (1)
where a[n] and φ[n] correspond to the instantaneous scaling factor and instantaneous phase offset, respectively, and b k [n] is the additive noise. Note that this model is quite general and includes all the dynamic scenarios of the IEEE standard C37.118.1 [5, 5.5.6-5.5.8]. The parameters d k e jϕ k corresponds to the phasor on the k th phase. Let us define c = [d 0 e jϕ0 , d 1 e jϕ1 , d 2 e jϕ2 ] T the 3 × 1 column vector containing the phasors. Then, let us use the notation c(θ) to stress that this vector depends on several unknown parameters, θ. Using matrix notations, the three-phase signals can be expressed as

x[n] = A(θ)s[n] + b[n] (2)
where

• x[n] et b[n]
are column matrices which are defined as

x[n]   x 0 [n] x 1 [n] x 2 [n]   b[n] =   b 0 [n] b 1 [n] b 2 [n]   , (3)
• A(θ) is a 3 × 2 matrix defined as

A(θ) e[c(θ)] -m[c(θ)] , (4)
• s[n] is a 2 × 1 column vector containing the direct and quadrature components i.e.

s[n] a[n] cos(φ[n]) a[n] sin(φ[n]) . (5)
In this paper, we assume that the noise is a zero-mean, white Gaussian noise with covariance matrix σ 2 Under this assumption, the goal of this study is to estimate the phasor parameters, θ, from the three-phase signals without knowing

S [s[0], • • • , s[N -1]].

CONDITIONAL MAXIMUM LIKELIHOOD

The Conditional Maximum Likelihood (CML) models the inphase and quadrature components as unknown deterministic parameters. This is a reasonable assumption when no a-priori information is available about the direct and quadrature components. However, as the in-phase and quadrature components are treated as unknown quantities, there is no guarantee that the unique identification of the parameters is enabled. For this reason, we investigate the conditions for parameter identifiability before addressing the estimation problem.

Conditions for parameter identifiability

The parameter identifiability requires that

A(θ)S = A(θ 2 )S 2 ⇒ θ = θ 2 (6)
where S and S 2 are arbitrary 2 × N full rank matrix.

As A(θ) is a full rank 3 × 2 matrix, the orthogonal projector onto the range of A(θ) can be decomposed as [9, p266]

P A (θ) = A(θ)(A T (θ)A(θ)) -1 A T (θ) (7) = I -u(θ)u T (θ) (8)
where u(θ) is the unit-norm eigenvector of A(θ)A T (θ) associated with the zero eigenvalue. If u(θ) = u(θ 2 ), it can be checked that the orthogonal projector onto the range of A(θ) and A(θ 2 ) are equal i.e. P A (θ) = P A (θ 2 ). Therefore, if it exists θ 2 = θ such as u(θ) = u(θ 2 ), then it is possible to find a matrix S 2 = (A T (θ 2 )A(θ 2 )) -1 A T (θ 2 )A(θ)S such as the condition in (6) is not satisfied. Therefore, we obtain the following lemma.

Lemma 1. To enable unique identification of the parameter, it must exist a one-to-one mapping between u(θ) and θ.

By definition, there is a one-to-one mapping between u(θ) and θ if the value of θ satisfying u T A(θ) = 0 is unique. As A(θ) is a 3 × 2 matrix and u a 3 × 1 column vector, the system u T A(θ) = 0 is composed of two equations. Therefore, a unique solution requires that θ contains at most 2 (real) unknown parameters. Note that this condition does not guarantee uniqueness of the solution. Specifically, for particular phasor models c(θ), it is possible to show that u(θ) = u for all θ. For example, let us consider the phasor model c(θ) satisfying [1 1 1]c(θ) = 0 for all θ. This condition implies that [1 1 1]A(θ) = 0. By identification, we conclude that the unit-norm eigenvector is equal to

u = 1/ √ 3[1 1 1] T regardless the value of θ.
Proposition 1. To enable unique identification of the parameters θ, the following conditions are necessary • C 1 : θ contains at most 2 (real) unknown parameters,

• C 2 : the so-called zero sequence component is non-zero for all θ i.e.

[1 1 1]c(θ) = 0 (9)
To illustrate proposition 1, we provide several examples where the parameter θ cannot be identified.

As θ = {α, β} contains two parameters, condition C 1 is satisfied. However, as [1 1 1]c(θ) = 0, the second condition does not hold and θ is not uniquely identifiable.

Example 2. The type-C phasor model of the ABC classification is given by [2, Table 6.2]

c(α) =   1 -1 2 + jα -1 2 -jα   , (11)
As [1 1 1]c(α) = 0, α is not identifiable. In the ABC classification, this problem occurs for sags of type A, C, D, F and G.

Estimation of the phasor parameters.

Assuming that conditions C 1 and C 2 hold, we show in this subsection that the Maximum Likelihood estimator has a simple expression. Using the CML, the estimate of the unbalance parameters is given by [START_REF] Stoica | Music, maximum likelihood, and cramer-rao bound[END_REF] θ = arg min

θ Tr P ⊥ A (θ) R (12)
where • R is the sample covariance matrix, which is defined as

R 1 N N -1 n=0 y[n]y T [n] (13)
• P ⊥ A (θ) is the orthogonal projector onto the kernel of A T (θ), which is defined as

P ⊥ A (θ) = I -A(θ) A T (θ)A(θ) -1 A T (θ) (14)
Using ( 8) and ( 14), P ⊥ A (θ) can be decomposed as

P ⊥ A (θ) = u(θ)u T (θ) (15)
where u(θ) is the unit eigenvector associated with the null eigenvalue of A(θ)A T (θ). Using this decomposition, the cost function in ( 12) can be simplified as follows

θ = arg min θ Tr u(θ)u T (θ) R (16)
= arg min

θ u T (θ) Ru(θ) (17)
To determine θ analytically, we resort to the invariance property of the Maximum Likelihood. First, we estimate u from R, then we estimate θ from u.

Estimation of u

In the first step, we find the value of u that minimizes the cost function in (17). This problem can be formalized as

u = arg min u u T Ru, subject to u T u = 1. (18)
As the scalar u T Ru is a Rayleigh quotient, the minimum is reached at u = g, where g is the eigenvector associated with the smallest eigenvalue of the sample covariance matrix R [11, Theorem 4.2.2].

Estimation of θ

In the second step, we extract the value of θ from g. As u(θ) is the eigenvector of A(θ)A T (θ) associated with the null eigenvalue, then u T (θ)A(θ) = 0. Replacing θ and u T (θ) by their estimates, we obtain It should be noted that this system is the basis of subspace algorithms such as MUSIC. This system is composed of two equations involving real numbers. By introducing the complex number j, this equation can be compacted as

g T A( θ) = 0. (19)
g T A( θ) 1 -j = 0. ( 20
)
Using the definition of A(θ), we obtain Theorem 1.

Theorem 1. The ML estimate of θ, denoted θ, is obtained by finding the value of θ for which the phasors are orthogonal to the eigenvector g i.e.

g T c( θ) = 0. (21)
The orthogonality condition in Theorem 1 has a simple geometric interpretation. Let us decompose g and c(θ) as g = [g 0 , g 1 , g 2 ] T and c(θ) = [d 0 e jϕ0 , d 1 e jϕ1 , d 2 e jϕ2 ] T . Note that for convenience, we write d k and ϕ k instead of d k (θ) and ϕ k (θ). Using this decomposition, it follows that

2 k=0 g k d k e jϕ k = 0. ( 22
)
In the complex plane, this equality corresponds to a triangle with side lengths d k g k and inner angles given by

β 02 = ϕ 2 -ϕ 0 -π (23a) β 01 = π -ϕ 1 + ϕ 0 (23b) β 12 = π -ϕ 2 + ϕ 1 . ( 23c
)
This triangle is presented in Fig. 1. Using this representation, the estimation problem can be treated as a geometrical problem.

Closed form estimators

In this subsection, we address the two following estimation problems: i) the estimation of the amplitude parameters d 1 and d 2 and ii) the estimation of the phase-shift parameters ϕ 1 , and ϕ 2 . In each scenario, the first phase is taken as the reference i.e. d 0 = 1 and ϕ 0 = 0

Estimation of the amplitude parameters

In the first estimation problem, the unknown parameters are θ = {d 1 , d 2 }. To estimate d 1 and d 2 , we use the sine rule in the triangle of Fig. 1. After some simplifications, we obtain

d 1 = - g 0 sin(ϕ 2 ) g 1 sin(ϕ 2 -ϕ 1 ) (24a)
d 2 = g 0 sin(ϕ 1 ) g 2 sin(ϕ 2 -ϕ 1 ) . ( 24b
)
Note that when the angle shift between phases is equal to 2π/3, the triangle in Fig. 1 is equilateral. It follows that d k = g 0 /g k , which corresponds to the particular solution in [8].

Estimation of the phase parameters

In the second estimation problem, the unknown parameters are θ = {ϕ 1 , ϕ 2 }. Using the cosine law, we obtain

g 2 2 d 2 2 = g 2 0 + g 2 1 d 2 1 -2g 0 g 1 d 1 cos(β 01 ) (25a)
g 2 1 d 2 1 = g 2 0 + g 2 2 d 2 2 -2g 0 g 2 d 2 cos(β 02 ). (25b)
Therefore, the angle-shift estimators are given by

ϕ 1 = arccos g 2 2 d 2 2 -g 2 0 -g 2 1 d 2 1 2g 0 g 1 d 1 (26a)
ϕ 2 = arccos g 2 2 d 2 2 + g 2 0 -g 2 1 d 2 1 2g 0 g 2 d 2 + π. (26b)

SIMULATION RESULTS

In this section, the performances of the amplitude and angle shift estimators are accessed through simulations using Python/Numpy. The phasor parameters are set to

d 0 = 1, d 1 = 1.2, d 2 = 0.2, φ 0 = 0, φ 1 = 2.
a[n] = 1 + 0.1 cos(2πf m n/F e ) (27a) φ[n] = 2πf 0 n/F e + 0.1 cos(2πf m n/F e -π) (27b)
where f m = 5Hz, f 0 = 50Hz, F e = 1000Hz. For each estimator, the Mean Square Error (MSE) is evaluated through 10000 Monte Carlo simulations. The MSE is analyzed and compared (asymptotically) with the Cramér Rao bounds for different data lengths and signal-to-noise ratios (SNRs), where the SNR is defined as

SN R = 10 log trace(A(θ)SS T A T (θ)) 3N σ 2 . ( 28
)
In the following simulations, the SNR ranges from -20 to 50dB. It should be mentioned that low values of the SNR are unlikely in power system applications. Figure 2 presents the performance of the amplitude estimator described in (24). We observe that the MSE and CRB of d 2 are lower than that of d 1 . As reported in [8], this behavior comes from the fact that d 2 < d 1 . A constant offset between the MSE also suggests that the ratio M SE[ d 1 ]/M SE[ d 2 ] is constant for large values of SNR or N. Furthermore, we note that the proposed estimator seems to reach the CRB for SNR→ ∞ but not for N → ∞, which is compliant with the properties of the Conditional ML [START_REF] Renaux | On the high-snr conditional maximum-likelihood estimator full statistical characterization[END_REF]. Figure 3 reports on the performance of the angle shift estimator described in (26). We note that the MSE of φ 1 is lower than that of φ 2 , except for low N or SNR. Additional simulations suggest that the performances of φ 1 and φ 2 are equal when φ 1 = 2π/3 and φ 2 = 4π/3. Concerning the CRB, the proposed estimator is biased for low SNRs, making the comparison non-relevant in this asymptotic region. For moderate and large SNRs, we note that the angle shift estimator has the same behavior than the amplitude estimator (efficient for SNR→ ∞ but not efficient for N → ∞).

CONCLUSION

This paper has presented a technique based on the Conditional Maximum Likelihood (ML) for the estimation of the phasor parameters in three-phase systems. When the identification is enabled, it has been demonstrated that the ML estimation of the phasor parameters can be determined by finding a phasor vector orthogonal to an eigenvector of the sample covariance matrix. Furthermore, this problem is transformed into the determination of the geometrical properties, such as side lengths and inner angles, of a triangle. This paper has also demonstrated that the Conditional ML can estimate at most two parameters. This limitation suggests that additional information must be incorporated in the signal model for the identification of all the phasor parameters. nal Processing Magazine, IEEE, vol. 30, no. 5, pp. 107-128, Sept 2013.

[5] IEEE standards, ""ieee standard for synchrophasor measurements for power systems"," IEEE Std C37.118.

Example 1 .

 1 Let us consider the following phasor model c(α, β) = αe jβ

12 Fig. 1 :

 121 Fig. 1: Geometrical interpretation of (22) .

d 2 (

 2 a) MSE versus SNR (N = 200).

2 (

 2 b) MSE versus sample length (SN R = 10dB).

Fig. 2 :

 2 Fig. 2: Estimation of the amplitude parameters d 1 and d 2 .

[ 8 ]φ 2 (

 82 V. Choqueuse, A. Belouchrani, E.-H. El Bouchikhi, and M. El Hachemi Benbouzid, "Estimation of amplitude, phase and unbalance parameters in three-phase systems: Analytical solutions, efficient implementation a) MSE versus SNR (N = 200).

2 (

 2 b) MSE versus N (SN R = 10dB).

Fig. 3 :

 3 Fig. 3: Estimation of the angle shift parameters ϕ 1 and ϕ 2 .

  (Revision of IEEE Std C37.118-2005), pp. 1-61, Dec 2011. [6] M. Wang and Y. Sun, "A practical, precise method for frequency tracking and phasor estimation," IEEE Transactions on Power Delivery, vol. 19, no. 4, pp. 1547-1552, Oct 2004. [7] F. Aminifar, M. Fotuhi-Firuzabad, A. Safdarian, A. Davoudi, and M. Shahidehpour, "Synchrophasor measurement technology in power systems: Panorama and state-of-the-art," IEEE Access, vol. 2, pp. 1607-1628, 2014.


Christian Darne
email: christian.darne@inrs.fr

Catherine Coulais

Francine Terzetti

Caroline Fontana

Stéphane Binet

Laurent Gaté

Yves Guichard

In vitro comet and micronucleus assays do not predict morphological transforming effects of silica particles in Syrian Hamster Embryo cells

Keywords: silica, genotoxicity, SHE cell transformation, in vitro micronucleus assay, in vitro comet assay, asbestos

Crystalline silica particles and asbestos have both been classified as carcinogenic by the International Agency for Research on Cancer (IARC). However, because of the limited data available, amorphous silica was not classifiable.

In vitro, the carcinogenic potential of natural crystalline and amorphous silica particles has been revealed by the Syrian Hamster Embryo (SHE) cell transformation assay. On the other hand, the genotoxic potential of those substances has not been investigated in SHE cells. And yet, genotoxicity assays are commonly used for hazard evaluation and they are often used as in vitro assays of reference to predict a possible carcinogenic potential. The main objective of this study was to compare the genotoxic potential and the carcinogenic potential of different crystalline and amorphous silica particles in SHE cells. Three silica samples of different crystallinity were used: natural amorphous silica, partially crystallized silica and quartz silica particles. Their genotoxicity were tested through the in vitro micronucleus assay and the comet assay in SHE, and their carcinogenic potential through the SHE transformation assay.

In addition, silica samples were also tested with the same genotoxicity assays in V79 hamsterlung cells, a common in vitro model for particle exposure.

Results obtained in the micronucleus and the comet assays show that none of the silica was capable of inducing genotoxic effects in SHE cells and only the amorphous silica induced genotoxic effects in V79 cells. However in the SHE cell transformation assays, the partially crystallized and quartz silica were able to induce morphological cell transformation. Together, these data suggest that, in vitro, the short-term genotoxic assays alone are not sufficient to predict the hazard and the carcinogenic potential of this type of particles; SHE transformation assay appears a more reliable tool for this purpose and should be included in the "in vitro battery assays" for hazard assessment.

-INTRODUCTION

Occupational exposure to some forms of silica is associated with silicosis and other respiratory pathologies. The International Agency for Research on Cancer [START_REF]Silica, Some Silicates, Coal Dust and para-Aramid Fibrils[END_REF] has classified quartz and cristobalite as carcinogenic to humans (Group 1) on the basis of sufficient evidence for carcinogenic effects in experimental animals and in humans. Because of the limited epidemiological and experimental data available, amorphous diatomaceous earth silica has not been classifiable for its carcinogenicity (Group 3).

The mechanisms of silica-induced carcinogenesis remain only partially understood, even if the IARC classification was based mainly on the crystallinity. Dimension, crystallinity, solubility and biopersistence are all parameters that could influence the toxicity of silica [START_REF] Fubini | Variability of biological responses to silicas: effect of origin, crystallinity, and state of surface on generation of reactive oxygen species and morphological transformation of mammalian cells[END_REF]. Another parameter thought to be involved is the production of reactive oxygen or nitrogen species (ROS, RNS) [START_REF]Silica, Some Silicates, Coal Dust and para-Aramid Fibrils[END_REF], [START_REF] Fubini | Reactive oxygen species (ROS) and reactive nitrogen species (RNS) generation by silica in inflammation and fibrosis[END_REF] and [START_REF] Shi | Reactive oxygen species and silicainduced carcinogenesis[END_REF].

Several in vitro studies on genotoxic effects have focused on quartz particles [START_REF] Zhong | Detection of mineral-dust-induced DNA damage in two mammalian cell lines using the alkaline single cell gel/comet assay[END_REF] and [START_REF] Zhong | Studies on the relationship between treatment condition and micronucleus induction in V79 cells exposed to silica and glass fibers[END_REF] but little data is available regarding the effects of natural amorphous silica such as diatomaceous earth. In previous studies [START_REF] Fubini | Variability of biological responses to silicas: effect of origin, crystallinity, and state of surface on generation of reactive oxygen species and morphological transformation of mammalian cells[END_REF] and [START_REF] Elias | Cytotoxic and transforming effects of silica particles with different surface properties in Syrian hamster embryo (SHE) cells[END_REF], crystalline silica was found to be cytotoxic and it also induced morphological transformation of Syrian hamster embryo (SHE) cells, which further acquired tumorigenic properties, whereas amorphous silica did not.

The assessment of silica hazard and of their carcinogenic potential is often based on long-term in vivo studies as for other chemicals. But in order to limit animal use, analyses are now often conducted with in vitro tests. Generally, these assays look at the cytotoxicity, the mutagenicity, the inflammatory response and the genotoxicity [START_REF] Msiska | DNA double-strand breaks by asbestos, silica, and titanium dioxide: possible biomarker of carcinogenic potential?[END_REF], [START_REF] Barrett | Antioxidant treatment attenuates cytokine and chemokine levels in murine macrophages following silica exposure[END_REF] and [START_REF] Nagalakshmi | Silica-induced micronuclei and chromosomal aberrations in Chinese hamster lung (V79) and human lung (Hel 299) cells[END_REF]. Few studies use the in vitro cell transformation assays (CTA) for determine this carcinogenic potential, whereas these assays are useful to evaluate all carcinogens (genotoxic and non genotoxic) as demonstrated by the OECD detail review paper [START_REF] Oecd | Detailed review paper on Cell transformation assays for detection of chemical carcinogens[END_REF]. For the SHE CTA, more than 200 chemicals have been tested and meta-analysis of the results show a sensitivity of 92%, a specificity of 66%, a positive predictivity of 88%, a negative predictivity of 75% and a concordance of 85%. The SHE CTA is a useful alternative to animal experimentation for the detection of carcinogenic chemicals and could be a useful in vitro predictive tool since the only carcinogenicity test accepted by authorities so far is the in vivo two-year bioassay.

A recent study, conducted by Benigni, et al. [START_REF] Benigni | The Syrian hamster embryo cells transformation assay identifies efficiently nongenotoxic carcinogens, and can contribute to alternative, integrated testing strategies, Mutation research[END_REF], has shown the benefit of the Syrian hamster embryo cells transformation assay to efficiently identify carcinogens and especially those that are nongenotoxic. Similarly, no study has determined in the same cellular model the genotoxic and transforming effects of silica particles.

The purpose of this work was to determine whether the capacity of natural silica materials to transform SHE cells were linked to their ability to induce micronucleus formation and DNA damage in SHE and V79 cells. By this mean, we have also looked at the benefit that the SHE CTA could present for the evaluation of silica hazard. Silica materials of different crystallinity were used to test this hypothesis: a quartz silica (Min-U-Sil 5), an amorphous diatomaceous earth (DE) and its calcination product (Chd), partially crystallized into cristobalite form as a consequence of flux calcinations. Occupational exposure to these particles may occur during extraction or processing of the native material (DE, Min-U-Sil), and commercial use (Chd) in industry as a filtration agent, abrasive or adsorbent material [START_REF]Silica, Some Silicates, Coal Dust and para-Aramid Fibrils[END_REF].

An asbestos sample (chrysotile) with transforming potential [START_REF] Darne | In vitro cytotoxicity and transforming potential of industrial carbon dust (fibers and particles) in syrian hamster embryo (SHE) cells[END_REF] was also selected as a reference particulate sample for this study.

-MATERIAL AND METHODS

-Materials

The materials investigated (described in Table 1) comprised: (i) an amorphous biogenic diatomaceous earth (DE), which is the source of (ii) a commercial dust, industrially prepared by flux calcination of the DE at high temperature to yield a partially crystallized cristobalite (Chd); (iii) Min-U-Sil 5 quartz, the most widely investigated silica in in vitro and in vivo studies [START_REF]Silica, Some Silicates, Coal Dust and para-Aramid Fibrils[END_REF]; and (iv) a chrysotile asbestos sample provided by J. Fournier (Laboratoire de Réactivité de surface, Jussieux, France), originally obtained from UICC A (International Union against Cancer, South Africa ), which was used as a positive control in the SHE cell transformation assay

-Physicochemical characterisation of the materials

The degree of crystallinity was determined by X-ray diffractometry (Philips diffractometer, Philips Electronics, The Netherlands). Impurities were analysed by inductively coupled plasma (ICP) spectrometry (Spectro Ciros CCD, Germany) (Table 1). Particle size distribution was measured by SEM (JEOL 840A, Japan) (Table 2).

-Cell culture

Syrian hamster embryo (SHE) cell cultures were established from individual 13-day gestation foetuses (inbred colony, INRS, France). All of the experiments involving animals were performed in the INRS laboratory animal facility approved by the French Ministry of Agriculture, according to the French regulations regarding the protection of animals used for experimental and other scientific purposes and the INRS ethical policy. The culture medium used was Dulbecco's MEM (DMEM; Invitrogen, France), pH 7, supplemented with 20% preselected fetal calf serum (Dutscher, France) and 2 mM L-glutamine (Invitrogen, France) without antibiotics.

Cells were incubated at 37°C and 10% CO 2 . Cryopreserved primary cultures were selected for cell growth, cloning efficiency, and spontaneous and induced morphological transformation. The primary and secondary cultures used in this study were from a batch that yielded results consistent with the historical laboratory data.

V79 cells (lung fibroblast from Chinese hamster, ATCC, USA, reference CCL-93) were selected for this study as they are one of the cell models recommended in OCDE guideline n° 487 for use in the in vitro micronucleus assay. Cells were grown in Dulbecco's MEM (DMEM; Invitrogen, France), supplemented with 10% fetal calf serum (Dutscher, France) and 0.5% Penicillin/Streptomycin (5000 U-5000 µg/mL, Invitrogen, France). Cells were incubated at 37°C and 10% CO 2 , as recommended by the supplier for optimal culture with our medium.

-Cell proliferation

Cells (30,000 SHE cells/ml or 20,000 V79 cells/mL) were cultured for 24 h at 37°C and 10% CO 2 in culture medium. The cell cultures were then treated for 24 h with culture medium (control) or with particle/fibre suspensions in final concentrations between 3.3 and 13.2 µg/cm 2 for the asbestos and between 20 and 80 µg/cm 2 for silica. After treatment, cells were trypsinized and counted (Coulter Z1, Beckman Coulter, France). Each assay was repeated at least three times. The mean number of cells ± SD for each treatment concentration was calculated and compared using an ANOVA-LSD test (Fisher's Least Significant Difference) (Statgraphics Centurion, Statpoint Technologies, USA).

-SHE cloning efficiency and transformation assay.

The assay was performed as described previously [START_REF] Elias | Cytotoxic and transforming effects of silica particles with different surface properties in Syrian hamster embryo (SHE) cells[END_REF] and [START_REF] Elias | Cytotoxic and neoplastic transforming effects of industrial hexavalent chromium pigments in Syrian hamster embryo cells[END_REF]. X-irradiated SHE feeder cells were seeded at 3 x 10 4 cells/ml in a 60-mm dish. After 24 h of incubation (37°C, 10% CO 2 ), 300 SHE target cells/dish were seeded onto the feeder cells. Cells were incubated for 24 h at 37°C and 10% CO 2 , and were then exposed for 7 days to at least three different concentrations: between 0.48 µg/cm 2 and 1.91 µg/cm 2 for asbestos and between 3.81 µg/cm 2 and 30.48 µg/cm 2 for silica. Control cells received culture medium alone.

After 7 days of incubation at 37°C, 10% CO 2 , dishes were washed (HBSS, Invitrogen) and colonies were fixed (absolute methanol) and stained (10% Giemsa). Colonies were counted and examined for morphological transformation with a stereomicroscope (Wild, Germany) (for photograph examples see [START_REF] Maire | Photo catalogue for the classification of cell colonies in the Syrian hamster embryo (SHE) cell transformation assay at pH 7.0[END_REF]). Ten cell cultures were used per treatment concentration and control. For each treatment concentration and control in an individual assay, the following were scored: (i) total colony number; (ii) cloning efficiency (CE) = (total colony number/total target cell number seeded) x 100; (iii) relative plating efficiency (RPE) = (CE of treated cells/CE of the control) x 100; (iv) number of morphologically transformed colonies; and (v) transformation frequency (TF) = (the number of transformed colonies/total number of colonies) x 100.

The mean CE of the control cultures was 23.71% (n = 16). Only one spontaneous transformed colony was recorded in one of a total of 16 experiments. For each treatment concentration, data reported the pooled results from a minimum of three individual assays. TF was compared to the control using the Chi-squared test.

-Comet assay

Cell membrane integrity was verified just before comet experiments using the trypan blue exclusion method (data not shown).

The Fpg-modified comet assay was used to evaluate DNA damage. This test uses the Fpg enzyme (formamido pyrimidine DNA glycoylase), a glycosylase that recognizes and cuts the modified bases, in particular at 8oxoguanine sites, producing apurinic that are converted into breaks by the associated AP-endonuclease activity.

These breaks can therefore indicate oxidative DNA damage [START_REF] Azqueta | Enhancing the sensitivity of the comet assay as a genotoxicity test, by combining it with bacterial repair enzyme FPG[END_REF]. We followed the procedure of Collins et al. [START_REF] Collins | Serum carotenoids and oxidative DNA damage in human lymphocytes[END_REF], with minor modifications. The concentrations tested ranged from 2.9 to 11.4 µg/cm 2 for the asbestos and from 11.4 to 45.7 µg/cm 2 for silica. Methylmethane sulfonate (MMS, Sigma-Aldrich, France) was used as a positive control at 0.125mM. About 20,000 cells (SHE or V79) were mixed in 1% low-melting agarose (Sigma Aldrich, France) in complete medium at 37°C and transferred onto a slide pre-coated with 1% normal melting agarose (Sigma Aldrich, France). The slides were then immersed in lysis solution and kept in the dark for 1 h at 4°C. The slides were washed, drained and incubated in the dark with 50 µl of either buffer or Fpg (10 U/mL, Sigma, France) in enzyme buffer, for 30 min at 37°C. The slides were then immersed in cold alkaline solution for 20 min. Electrophoresis was performed at 25 V and 300 mA for 40 min. The slides were then washed with Tris-HCl 0.4 M for 15 min and stained with propidium iodide.

Images of 100 randomly selected comets were analysed from each sample, using specific image analyser software (Comet assay IV, Perceptive Instruments, United Kingdom). Experiments were repeated three times.

Data were expressed as the mean of the medians of the tail DNA ± SEM, and statistical analyses were performed on the mean values using a multiple comparison method one-way ANOVA (Fisher least significant difference (LSD)) with Statgraphics Centurion software (Statpoint Technologies, USA). The p < 0.05 level was considered to be statistically significant.

-Micronucleus assay.

For the micronucleus assay, 2.5 x 10 4 cells (SHE or V79) were seeded in Labtek ® slides (Nunc A/S, Denmark) with 1 mL of culture medium. After 24h, they were treated with 1 mL of sample preparation for 24h.

Concentrations ranged from 2.9 to 11.4 µg/cm 2 for the asbestos and from 11.4 to 54.4 µg/cm 2 for silica. MMS was used as a positive control at 0.25 mM. Cells were washed with PBS (phosphate buffer saline, Invitrogen, France) and fixed for 15 min in methanol. Slides were washed in PBS, drained, and then received one drop of Pro Long Gold antifade reagent ® with DAPI (Molecular Probe, Invitrogen, France). About 1000 cells/point were analyzed for the presence of MN (micronucleus) and mitotic cells. Each assay was repeated three times.

Statistical analysis of MN induction was performed using an ANOVA-LSD test (Fisher's Least Significant Difference) (Statgraphics Centurion, Statpoint Technologies, USA). The p < 0.05 level was considered to be statistically significant.

3-RESULTS

-Physicochemical characterisation of the materials:

Chemical compositions and sample impurities are presented in Table 1. Amorphous silica DE contained 1.3 % iron. The heated DE (Chd) was partially crystallized (47% cristobalite) and contained 1.9 % iron by mass. The Min-U-Sil quartz (100% crystallized) contained 0.05 % iron and 0.16 % aluminium.

Both DE and Min-U-Sil samples both presented homogeneous size distributions (Table 2). The Chd silica was heterogeneous, with 68.9 % of particles having a diameter of less than 5 µm and 31.1 % with a diameter greater than 5 µm.

Our chrysotile sample was composed of very small fibres (96.1 % L/D >3, L<5 µm).

-Cell proliferation:

The cell proliferation data for SHE and V79 cells are presented in Figures 1A and1B, respectively. Cells were counted 24 hours after treatment.

In SHE cells, the following concentrations yielded 50% reductions in cell count: 8.2 µg/cm 2 for chrysotile, 65.7 µg/cm 2 for Chd and 68.7 µg/cm 2 for DE. At the highest Min-U-Sil concentration (80 µg/cm 2 ) the cell count was 58% of the control. As expected, the asbestos was more cytotoxic than any of the silica. Min-U-Sil was the least cytotoxic silica tested.

Sample toxicity can be ranked as follows: Chrysotile >> Chd ≥ DE > Min-U-Sil.

In V79 cells, a 50% decrease in cell count was recorded at a concentration of 60.1 µg/cm 2 for silica DE. At the highest dose (90.4 µg/cm 2 ) the cell counts were at 72 % of the control for Chd silica and 64.6 % for Min-U-Sil. 11.4 µg/cm 2 of chrysotile gave a cell count that was 55 % of the control. The sample toxicity in V79 cells can be ranked as follows:

Chrysotile >> DE >> Min-U-Sil > Chd.

-Relative plating efficiency (RPE) and transformation frequency (TF) of SHE cells:

The relative plating efficiency and TF induced by the different silica and chrysotile are reported in Table 3.

The silica samples significantly reduced the RPE but the cytotoxic potency of each type of silica varied: DE > Chd> Min-U-Sil. Amorphous DE induced no morphological transformation (except for 3 colonies in one experiment at 15.24 µg/cm 2 ), while a concentration-dependent increase in TF was induced by the two other silica samples. As reported previously [START_REF] Fubini | Variability of biological responses to silicas: effect of origin, crystallinity, and state of surface on generation of reactive oxygen species and morphological transformation of mammalian cells[END_REF]and [START_REF] Elias | Cytotoxic and transforming effects of silica particles with different surface properties in Syrian hamster embryo (SHE) cells[END_REF], Min-U-Sil did not express any cytotoxicity in concentrations up to 30.48 µg/cm 2 but did induce morphological transformation in a dose-dependent manner.

The partially crystallised Chd was slightly cytotoxic and also induced transformation in a dose-dependent manner.

The chrysotile was more cytotoxic and more transforming than silica and the effects were dose-dependent.

-Micronuclei (MN) assessment:

In order to detect the potential genotoxic effects of samples, induction of MN in the SHE and V79 cells was assessed after 24 h of treatment (~1.3 cell-division cycles for SHE and ~1.7 cell-division cycles for V79).

After 24 h of treatment, none of the silica samples was able to induce an increase in MN frequency in SHE cells. Even though no significant difference was observed between the control and the highest concentrations, there was a general tendency for the number of cells with micronuclei to decrease with concentration (Figure 2A). This cannot be related to a decrease in cell division induced by the treatment because no significant change in the number of mitotic cells was observed in the assays (data not shown). The only positive sample was the asbestos, for which we observed a positive relationship between MN frequency and concentration. A significant response was obtained with 11.34 µg/cm 2 of chrysotile.

In V79 cells (Figure 2B), the DE and chrysotile induced a significant increase in the number of micronucleated cells at the two highest concentrations and a concentrationdependent relationship is apparent. In contrast, the Chd and Min-U-Sil had no effect.

-Comet assay:

Cell membrane integrity, as evaluated by the trypan blue exclusion method (data not shown), was not significantly affected (over 80% survival) in either cell type after 3 or 24 hours of treatment.

The results of the comet assay are expressed as percentage of tail DNA. In the SHE cells, the mean values obtained for the positive control (methyl methanesulfonate 0.125 mM) were, after 3 h of treatment, 23.58 % ± 3.50 and 62.62 % ± 4.71 in the absence or presence of Fpg (which cuts DNA strands where bases are modified), respectively, and after 24 h of treatment, 29.51 % ± 9.63 and 67.79% ± 5.73 in the absence or presence of Fpg, respectively.

In V79 cells, values obtained were 15.4 ± 2.88 and 62.37 ± 5.44 after 3h of MMS treatment and 20.17 ± 1.84 and 67.44% ± 1.81 after 24h of treatment without or with Fpg, respectively. 3A and4A) did not induce any DNA damage, as evaluated by the alkaline comet assay with or without Fpg in both cell types. Treatment with Fpg led to an increase in the tail DNA values, but without any significant difference from the control, which also presented a higher tail DNA value.

Treatment of the SHE cells with chrysotile or silica samples over 3 h (Figures

After 24 h of treatment (Figure 3B), a significant increase in DNA strand breaks was observed at 11.4 µg/cm 2 chrysotile without Fpg treatment in SHE cells. At concentrations up to 45.7 µg/cm 2 , none of the silica samples were able to induce a significant increase in DNA strand breaks, with or without Fpg.

In V79 cells (Figure 4), only the amorphous silica DE induced a significant increase in DNA breaks at the highest concentration (45.7 µg/cm 2 ) without Fpg, after 24h of treatment (Figure 4B).

-DISCUSSION

Crystallized silica and asbestos have both been classified as carcinogenic to humans (Group 1) by the International Agency for Research on Cancer. This classification was based on evidence obtained from both animal models and epidemiological studies [START_REF]Silica, Some Silicates, Coal Dust and para-Aramid Fibrils[END_REF]. Due to limited data, amorphous silica could not be classified. One of hypothesis put forward to explain the carcinogenic potential of fibres and particles is the involvement of reactive oxygen species (ROS). We have previously shown that both crystallized silica (Chd and Min-U-Sil 5) and chrysotile asbestos are able to generate ROS. The reactivity of these samples in releasing  OH radicals correlated to their transforming ability, whereas amorphous silica DE was unable to generate ROS and did not show any transforming capacity [START_REF] Darne | In vitro cytotoxicity and transforming potential of industrial carbon dust (fibers and particles) in syrian hamster embryo (SHE) cells[END_REF] and [START_REF] Elias | Surface reactivity, cytotoxic, and morphological transforming effects of diatomaceous Earth products in Syrian hamster embryo cells[END_REF]. Some other works have shown genotoxic effects, but no link to the carcinogenicity of these products has been made [START_REF] Zhong | Detection of mineral-dust-induced DNA damage in two mammalian cell lines using the alkaline single cell gel/comet assay[END_REF], [START_REF] Zhong | Studies on the relationship between treatment condition and micronucleus induction in V79 cells exposed to silica and glass fibers[END_REF] and [START_REF] Zhang | Involvement of oxidative stress in crystalline silica-induced cytotoxicity and genotoxicity in rat alveolar macrophages[END_REF].

The SHE cell transformation assay, which uses primary cell cultures of individual Syrian hamster embryos, is an in vitro assay that predicts carcinogenicity of chemicals using morphological transformation as an end-point. This transformation is characterised by disorganised growth patterns (criss-cross, high nuclear to cytoplasmic ratio, randomly oriented colony growth). Strong concordance has been reported between morphological changes in SHE colonies upon chemical treatment and results from carcinogenicity studies in animals [START_REF] Oecd | Detailed review paper on Cell transformation assays for detection of chemical carcinogens[END_REF] and [START_REF] Vasseur | OECD Detailed Review Paper (DRP) number 31 on "Cell Transformation Assays for Detection of Chemical Carcinogens": main results and conclusions[END_REF]. This test present the advantage to be able to detect not only genotoxic carcinogens but also non-genotoxic carcinogens that often escape to in vitro tests conventionally used. As meaning, SHE CTA is very useful for hazard assessment of chemicals.

In the present work, we have shown that crystallized silica (Chd and Min-U-Sil) was able to induce transformed SHE colonies in a dose-dependent manner after seven days of treatment. Our results are in agreement with previous transformation studies on quartz [START_REF] Elias | Surface reactivity, cytotoxic, and morphological transforming effects of diatomaceous Earth products in Syrian hamster embryo cells[END_REF], [START_REF] Hesterberg | Dependence of asbestos-and mineral dust-induced transformation of mammalian cells in culture on fiber dimension[END_REF], [START_REF] Williams | Transforming growth factor beta expression and transformation of rat lung epithelial cells by crystalline silica (quartz)[END_REF], [START_REF] Gao | Induction of cell transformation and oncogene activation by silica dust in BALB/c-3T3 cells[END_REF] and [START_REF] Saffiotti | Neoplastic transformation by quartz in the BALB/3T3/A31-1-1 cell line and the effects of associated minerals[END_REF]. Hesterberg and Barrett [START_REF] Hesterberg | Dependence of asbestos-and mineral dust-induced transformation of mammalian cells in culture on fiber dimension[END_REF] showed that two quartz samples induced a concentration-dependent increase in TF, and both of these samples were less potent than asbestos fibres. In our study, concentrations of 3.8 to 15.2 µg/cm 2 induced similar TF (0.24-0.71%) to that shown by Hesterberg and Barrett [START_REF] Hesterberg | Dependence of asbestos-and mineral dust-induced transformation of mammalian cells in culture on fiber dimension[END_REF], and the crystallized silica was again found to be less potent than the asbestos (0.28 % TF with 1.91 µg/cm 2 of chrysotile vs. 0.4 % TF with 30.4 µg/ cm 2 of Chd or 0.24% TF with 3.81 µg/ cm 2 of Min-U-Sil). All of these data are in agreement with the previously reported data [START_REF] Elias | Surface reactivity, cytotoxic, and morphological transforming effects of diatomaceous Earth products in Syrian hamster embryo cells[END_REF].

At the same time, and in order to evaluate a possible link between transforming potential and genotoxic effects, two types of genotoxicity assays were used for the present study: the in vitro comet assay and the in vitro micronucleus (MN) assay. The genotoxicity assays are often used to determine the potential hazard of chemicals and to alert about possible consequences in terms of health impact.

Numerous studies have validated the comet assay as a sensitive method for quantifying DNA breakage and for evaluating the genotoxic potential of xenobiotics [START_REF] Lee | The use of the single cell gel electrophoresis assay in detecting DNA single strand breaks in lung cells in vitro[END_REF] and [START_REF] Leroy | Evaluation of three methods for the detection of DNA single-strand breaks in human lymphocytes: alkaline elution, nick translation, and single-cell gel electrophoresis[END_REF]. The comet assay was first used to detect the ability of chemicals to generate singlestrand and/or double-strand breaks in the DNA [START_REF] Fairbairn | The comet assay: a comprehensive review[END_REF], but DNA breaks can also result from repair of some DNA lesions [START_REF] Fortini | Analysis of DNA alkylation damage and repair in mammalian cells by the comet assay[END_REF] and [START_REF] Alapetite | The comet assay as a repair test for prenatal diagnosis of Xeroderma pigmentosum and trichothiodystrophy[END_REF]. The comet assay can also be modified, as we made in this study, in order to allow the specific detection of damaged DNA resulting from base modification, and mainly from base oxidation, by including an enzymatic treatment step [START_REF] Karlsson | Can the comet assay be used reliably to detect nanoparticle-induced genotoxicity?[END_REF] and [START_REF] Collins | The use of bacterial repair endonucleases in the comet assay[END_REF].

The in vitro micronucleus assay visualizes, in interphase cells, micronuclei induced by chromosome breakage events (clastogenic effects) or chromosome losses (aneugenic effects) that occur in anaphase during nuclear division [START_REF] Fenech | The in vitro micronucleus technique[END_REF]. The comet assay and the micronucleus assay are thus complementary tests.

With crystalline silica, no effect was observed in the comet or in the micronucleus assay in either cell type. Our results differ from those published before where the sample concentrations used were fairly high. Indeed, Nagalakshmi et al. [START_REF] Nagalakshmi | Silica-induced micronuclei and chromosomal aberrations in Chinese hamster lung (V79) and human lung (Hel 299) cells[END_REF] showed that 24h treatment with Min-U-Sil followed by 18-20h treatment with cytochalasin B (CyB) induced an increase in MN frequency from concentrations of 40 µg/cm 2 in V79 cells and from 160 µg/cm 2 in Hel299 cells. In a later study made by the same group, Min-U-Sil at 137.9µg/cm 2 induced a significant increase in MN frequency in V79 in the absence of CyB treatment but with a high cytotoxicity [START_REF] Zhong | Studies on the relationship between treatment condition and micronucleus induction in V79 cells exposed to silica and glass fibers[END_REF].

We chose not to use concentrations above 55µg/cm 2 , firstly to avoid high cytotoxicity that could interfere with the genotoxicity assay, and secondly because concentrations below 30.4µg/cm 2 were able to induce transformed colonies in the SHE cell transformation assay.

Taken together, our micronucleus results and data published before suggest that crystalline silica genotoxicity is probably discreet.

With regards to natural amorphous silica materials, these have been evaluated as nonfibrogenic and unlikely to be carcinogenic [START_REF]Silica, Some Silicates, Coal Dust and para-Aramid Fibrils[END_REF]. In the present work, and as expected [START_REF] Elias | Surface reactivity, cytotoxic, and morphological transforming effects of diatomaceous Earth products in Syrian hamster embryo cells[END_REF], the DE was unable to induce transformed colonies (except for three colonies in one experiment).

However and surprisingly, the DE silica was the only sample able to induce DNA strand breaks and micronucleus in the V79 cells. But no explanation can be given at this stage.

These results show the better sensitivity of V79 cells compared to SHE cells for determining the genotoxic effects. However, as they are immortalized (derived from a spontaneous cell transformation), defective for p53, the V79 cells present altered functions that may lead to "false positive" effects compared to non-immortalized cells. Indeed, the p53 protein, which mediates the cellular response to DNA damage, is involved in cell-cycle regulation, apoptosis and DNA repair [START_REF] Negrini | Genomic instability-an evolving hallmark of cancer, Nature reviews[END_REF]. As shown by Chaung, et al. [START_REF] Chaung | The p53 status of Chinese hamster V79 cells frequently used for studies on DNA damage and DNA repair[END_REF], the V79 p53 sequence contains two mutation points that result in a nonfunctional protein. Conversely, SHE cells are normal diploid cells with no alterations in the cell-cycle pathway [START_REF] Huberman | The in vitro induction of an increase in cell multiplication and cellular life span by the water-soluble carcinogen dimethylnitrosamine[END_REF] and [START_REF] Isfort | Comparison of the standard and reduced pH Syrian hamster embryo (SHE) cell in vitro transformation assays in predicting the carcinogenic potential of chemicals[END_REF], and SHE cells also contain a normal p53 protein [START_REF] Alexandre | Detection of apoptosis induced by topoisomerase inhibitors and serum deprivation in syrian hamster embryo cells[END_REF] and [START_REF] Rao | Malachite green induced malignant transformation of Syrian hamster embryo (SHE) cells in primary culture: transformation is associated with enhanced expression of altered p53, bcl-2 and decreased sensitivity to apoptosis[END_REF]. The absence of cell transformation and genotoxic effects with DE silica in SHE cells could be explain by effective repair mechanisms of these cells, involving the p53 gene.

As expected, chrysotile asbestos induced micronucleus formation and cell transformation in our experiments. The micronucleus results in SHE and V79 cells were in agreement with those previously published on human mesothelial, SHE or V79 cells [START_REF] Poser | Modulation of genotoxic effects in asbestos-exposed primary human mesothelial cells by radical scavengers, metal chelators and a glutathione precursor[END_REF], [START_REF] Keane | A study of the effect of chrysotile fiber surface composition on genotoxicity in vitro[END_REF] and [START_REF] Dopp | Mitotic disturbances and micronucleus induction in Syrian hamster embryo fibroblast cells caused by asbestos fibers[END_REF]. In the present study, the chrysotile was able to induce significant DNA damage in SHE cells, but comet assays performed with the Fpg showed that this damage was not of the oxidative type. DNA damage induced by asbestos seemed to be strongly dependent on the cell type used, as it was previously demonstrated by Burmeister et al. [START_REF] Burmeister | Effects of asbestos on initiation of DNA damage, induction of DNA-strand breaks, P53-expression and apoptosis in primary, SV40-transformed and malignant human mesothelial cells[END_REF], who reported DNA damage in HMC cells but not in MeT-5A with 3 µg/cm 2 of chrysotile. In the same work, the authors were unable to detect any Fpg sensitive sites regardless of the cellular type, thus corroborating our findings. A genotoxic effect from chrysotile has also been previously observed in RPMC cells (Rat pleural mesothelial cells) from 0.5 to 10 µg/cm 2 [START_REF] Levresse | DNA breakage in asbestostreated normal and transformed (TSV40) rat pleural mesothelial cells[END_REF]. Our results with those published before, suggest that these fibres could probably interfere with mitotic spindle and induce a cell cycle arrest as it was shown by Cortez and Machado-Santelli [START_REF] Cortez | Chrysotile effects on human lung cell carcinoma in culture: 3-D reconstruction and DNA quantification by image analysis[END_REF].

Taken together, these results highlight the interest to implement, in a chemical hazard assessment, a cell transformation assay, such as that carried on SHE cells. The in vitro genotoxicity assay, as comet or micronucleus assays, can reveal genotoxic effects in a simple way. Gene mutation assays also provide information about the effects of chemicals and mechanisms that may be involved. But some substances, by their mode of action, may not meet these tests in vitro. This difficulty can be avoided by using an in vitro carcinogenicity assay, as CTA SHE alternative to long and costly animal testing. The screening of molecules could be improved as well.

-CONCLUSION

In conclusion, amorphous silica DE was able to generate both DNA strand breaks and micronuclei but it did not induce cell transformation, whereas crystalline silica induced cell transformation but without genotoxic effect; and chrysotile induced both micronuclei and cell transformation. Taking into account all these results, the mechanisms involved in the carcinogenicity of silica materials and asbestos are probably different.

In this study we have shown that if comet and micronucleus assays are useful tools for evaluating genotoxic potential of fibres and particles, they do not allow us to predict the transforming potential of silica materials and their carcinogenic potential.

To our knowledge, this is the first time that different forms of micrometric natural silica have been compared for their transforming and genotoxic potential. Further studies will be required to explore the mechanisms involved in the induction of the in vitro carcinogenic effects of these samples.

In the meantime, our results point out that the in vitro genotoxicity assays currently used for hazard assessment cannot alone predict the carcinogenic potential of every substance, as our study has shown for silica materials, and they could be advantageously completed by a cell transformation assay.

Figure 1 .

 1 Figure 1. SHE cell proliferation

Figure 2 .

 2 Figure 2. Micronucleated SHE cells

Figure 3 .

 3 Figure 3. Comet assays after 3 h or 24 h of treatment in SHE cells

Figure 4 .

 4 Figure 4. Comet assays after 3 h or 24 h of treatment in V79 cells

  Figure 1

  Figure 3

TABLE 1 . Physico -chemical characteristics of the silica samples

 1

	Sample	Origin	Structure a	Chemical impurities (%) b
	Diatomaceous earth (DE)	Natural, precursor of the	Amorphous (100%)	2.2 Ca; 1.4 Al; 1.3 Fe; 0.3 Ti; 0.2 Mg
		commercial dust
	Commercial (Chd)	Heated DE	Partially crystallized:	1.9 Fe; 1.6 Al; 0.8 Ca; 0.3 Ti; 0.2 Mg
			Amorphous (53%), cristobalite (47%)
	Quartz Min-U-Sil 5	Natural; US Silica co: (Berkeley)	Crystalline: quartz (100%)	0.16 Al; 0.05 Fe; 0.04 Ca; 0.01 Mg
	Sample	Origin	Structure a	Chemical composition (%)
	Chrysotile	Zimbabwe, UICC A	Serpentine asbestos	26 Mg; 20 Si; 1.8 Fe

a Determined by X-ray diffractometry.

b Determined by inductively coupled plasma (ICP) spectrometry.

TABLE 2 . A) Average particle diameter and size distribution of silica particles in the different samples a

 2

			Percentage of number of particles with diameter (µm)
		Geometric mean
	Sample
		diameter µm (GSD b )	<5	5-10	10-20	> 20
	Diatomaceous earth (DE)	1.35 (1.48)	96.9	3.0	0.1	0
	Commercial (Chd)	4.85 (4.55)	68.9	21.6	7.2	2.3
	Quartz Min-U-Sil 5	1.33 (0.85)	99.5	0.5	0	0

a Measured by scanning electron microscopy (SEM). b Geometric standard deviation B)

Diameter and length distribution of asbestos sample a Sample Percentage of number of fibres with length (µm)

		<1	1-5	5-20	> 20
	Chrysotile (D< 1µm)	54.9	41.2	3.9	0

a Measured by scanning electron microscopy (SEM). D : diameter

TABLE 3 .

 3 Morphological transformation into SHE cells following 7 days of treatment with references. *: statistically significant (p < 0.05) compared to control (culture medium).

	Sample	Dose (µg/cm 2 )	Morphological transformation frequency (TF) (%)	Number of transformed colonies	Total number of colonies	Relative plating efficiency (RPE) (%)
	Control	0	0	0	3247	100
	Chrysotile	0.48	0.09	1	1107	83
	Chrysotile	0.95	0.26*	3	1154	58
	Chrysotile	1.91	0.28*	4	1454	73
	Control	0	0.01	1	11379	100
	DE	3.81	0.02	1	5990	105
	DE	7.62	0.02	1	5397	95
	DE	15.24	0.11*	3	2739	57
	DE	30.48	0	0	811	38
	Chd	3.81	0.16*	12	7398	120
	Chd	7.62	0.19*	18	9543	112
	Chd	15.24	0.23*	15	6604	88
	Chd	30.48	0.4*	16	4005	72
	Min-U-Sil	3.81	0.24*	3	1249	88
	Min-U-Sil	7.62	0.17*	2	1197	84
	Min-U-Sil	15.24	0.71*	33	4678	132
	Min-U-Sil	30.48	0.77*	21	2738	120

ACKNOWLEDGMENTS

The authors are grateful to Mrs C. Eypert-Blaison, INRS, for the SEM analyses; Mr. D. Rousset, INRS, for ICP-MS analyses; Mr. T. Lecaque, INRS, for X-ray diffractometry analyses. This work was funded by INRS.


Jean Auriol

Florent Di Meglio

Minimum time control of heterodirectional linear coupled hyperbolic PDEs with controls in both sides

I. INTRODUCTION

This article solves the problem of boundary stabilization of two coupled heterodirectional linear first-order hyperbolic Partial Differential Equations (PDEs) in minimum case with one PDE in each direction and with actuation applied on both boundaries.

The main contribution of this paper is a minimum time stabilizing controller. More precisely a proposed boundary feedback law ensures finite-time convergence of the two states to zero in minimum time. The minimum time defined [START_REF] Li | Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems[END_REF] in is the largest time between the two transport times in each direction.

Our approach is the following. Using a Fredholm transformation, the system is mapped to a target system with desirable stability properties. This target system is a copy of the original dynamics from which the coupling terms are removed. The well-posedness of the Fredholm transformation is a consequence of a clever choice of the domain on which the kernels are defined. The proof of the invertibility of this transformation is non-trivial and uses an operator-approach inspired by the one developed in [START_REF] Coron | Stabilization and controllability of first-order integro-differential hyperbolic equations[END_REF].

The paper is organized as follow. In Section II we introduce the model equations and the notations. In Section III we present the stabilization result: the target system and its properties are presented in Section III-A. In Section III-B we derive the integral transformation and we present the domains on which the kernels are defined. Some arguments about the well-posedness of the kernels are given in Section III-C. Section IV contains the proof of the invertibility of the Fredholm transformation. In Section V we present the control feedback law and its properties. Finally in Section VI we give some simulation results.

florent.di meglio@mines-paristech.fr

II. PROBLEM DESCRIPTION

A. System under consideration

We consider the following general linear hyperbolic system

u t (t, x) + λu x (t, x) = σ +-v(t, x) (1) v t (t, x) -µv x (t, x) = σ -+ u(t, x) (2)
with the following linear boundary conditions

u(t, 0) = U (t), v(t, 1) = V (t) (3)
with constant coupling terms and constant speeds

0 < λ ≤ µ (4)
The initial conditions are defined by

u(0, x) = u 0 (x), v(0, x) = v 0 (x) (5)
Remark 1: The coupling terms are assumed constant here but the results of this paper can be adjusted for spatiallyvarying coupling terms.

Remark 2: System (1)-( 2) is equivalent to the following system

u t (t, x) + λu x (t, x) = σ ++ u(t, x) + σ +-v(t, x) (6) v t (t, x) -µv x (t, x) = σ -+ u(t, x) + σ --v(t, x) (7)
This can straightforwardly be proved using an exponential transformation.

Remark 3:

The assumption λ ≤ µ can be done without any loss of generality

B. Well-posedness

Taking the scalar product of (1)-(2) with a smooth test function Φ T = (φ 1 , φ 2 ) and integrating by parts leads to the following definition of a solution.

Definition 1: Consider system (1)-( 2) with initial conditions u 0 , v 0 ∈ L2 and control laws U (t) and V (t). We say that u v is a (weak) solution if for every τ ≥ 0 and every

function Φ = (φ 1 , φ 2 ) T ∈ (C 1 ([0, τ ] × [0, 1])) 2 such that φ 1 (•, 1) = φ 1 (•, 0) = 0 we have 0 = τ 0 1 0 -(φ 1t (t, x) + λφ 1x (t, x)+ σ -+ φ 2 (t, x))u(t, x) -(φ 2t (t, x) -µφ 2x (t, x) + σ +-φ 1 (t, x))v(t, x)dxdt + 1 0 (u(τ, x)φ 1 (τ, x) -u(0, x)φ 1 (0, x) + v(τ, x)φ 2 (τ, x) -v(0, x)φ 2 (0, x))dx - τ 0 [λU (t)φ 1 (t, 0) + µV (t)φ 2 (t, 1)] dt (8)
We can consequently rewrite the system in the abstract form

d dt u v = A u v + B U V (9)
where the operators A and B can be identified through their adjoints. The operator A is thus defined by

A :D(A) ⊂ (L 2 (0, 1)) 2 → (L 2 (0, 1)) 2 u v -→ -λu x + σ +-v µv x + σ -+ u (10)
with

D(A) = {(u, v) ∈ (H 1 (0, 1)) 2 |u(0) = v(1) = 0} (11)
A is well defined and its adjoint A * is

A * :D(A * ) ⊂ (L 2 (0, 1)) 2 → (L 2 (0, 1)) 2 u v -→ λu x + σ -+ v -µv x + σ +-u (12)
with

D(A * ) = {(u, v) ∈ (H 1 (0, 1)) 2 |u(1) = v(0) = 0} (13)
The operator B is defined by

< B U V , z 1 z 2 >= U z 1 (0) + V z 2 (1) (14)
Its adjoint is

B * z 1 z 2 = z 1 (0) z 2 (1) (15)

C. Control problem

The goal is to design feedback control inputs U (t) and V (t) such that the zero equilibrium is reached in minimum time t = t F , where

t F = min 1 µ , 1 λ (16)

III. CONTROL DESIGN

The control design is based on a modified backstepping approach: using a specific transformation, we map the system (1)-( 3) to a target system with desirable properties of stability. However, unlike the classical backstepping approach where a Volterra transformation is used, we use a Fredholm transformation here.

A. Target system design

We map the system (1)-(3) to the following system

α t (t, x) + λα x (t, x) = 0 (17) β t (t, x) -µβ x (t, x) = 0 (18)
with the following boundary conditions

α(t, 0) = 0 β(t, 1) = 0 ( 19
)
This system is designed as a copy of the original dynamics, from which the coupling terms of ( 1)-( 2) are completely removed.

Lemma 1: The zero equilibrium of ( 17),( 18) with boundary conditions (19) and initial conditions (α 0 , β 0 ) ∈ L 2 ([0, 1]) is exponentially stable in the L 2 sense.

Proof: The proof, using a Lyapunov function, is quite classical and is omitted here. Besides, the following lemma assesses the finite-time convergence of the target system.

Lemma 2: The system (17),(18) reaches its zero equilibrium in finite-time t F = max{ 1 λ , 1 µ } Proof: The proof of this lemma is quite straightforward and uses the same arguments than the proof of [3, Lemma 3.1] Using operators, we can rewrite system (17)-(18) as

d dt α β = A 0 α β (20)
The operator A 0 is defined by

A 0 : D(A 0 ) ⊂ (L 2 (0, 1)) 2 → (L 2 (0, 1)) 2 α β -→ -λα x µβ x (21)
with

D(A 0 ) = {(α, β) ∈ (H 1 (0, 1)) 2 |α(0) = β(1) = 0} (22)
A 0 is well defined and its adjoint A * 0 is

A * 0 : D(A * 0 ) ⊂ (L 2 (0, 1)) 2 → (L 2 (0, 1)) 2 α β -→ λα x -µβ x (23)
with

D(A * 0 ) = {(α, β) ∈ (H 1 (0, 1)) 2 |α(1) = β(0) = 0} (24)
B. Fredholm transformation 1) Definition of the transformation: Without any loss of generality we recall that λ ≤ µ. In order to map the original system (1)-(3) to the target system (17)-( 19), we use the following transformation α(t, x) = u(t, x)

+ D i=1 h Ji (x) -µ λ x+bi x (K i (x, ξ)u(t, ξ) + L i (x, ξ)v(t, ξ))dξ + h [ µ µ+λ ,1] (x) x µ λ (1-x) (M (x, ξ)u(t, ξ) + N (x, ξ)v(t, ξ))dξ (25) β(t, x) = v(t, x) + h [0, λ µ+λ [ (x) λ µ (1-x) x ( K(x, ξ)u(t, ξ) + L(x, ξ)v(t, ξ))dξ + h [ λ µ+λ ,1] (x) x λ µ (1-x) ( M (x, ξ)u(t, ξ) + N (x, ξ)v(t, ξ))dξ (26)
where h I (x) (I is an interval) is defined by

h I (x) = 1 if x ∈ I 0 else (27)
The sequence b i with i ∈ N is defined by

b n+1 = (1 + µ λ )a n (28)
a n = λ µ (b n -1) (29)
b 1 = µ λ (30)
The parameter D is defined as follow

D = min{n|a n < 0} (31)
One can readily prove that D is well defined. Finally the sequence J i is defined by

J i = [max{a i , 0}, a i-1 [ (32)
with the convention a 0 = µ λ+µ . All the J i are disjoint and are a partition of [0, µ λ+µ [. We define the following domains:

T 0 = {(x, ξ)|x ∈ [ µ λ + µ , 1], µ λ (1 -x) < ξ ≤ x} (33) T i = {(x, ξ)|x ∈ J i , x ≤ ξ < µ λ x + b i } (34) T0 = {(x, ξ)|x ∈ [0, λ λ + µ [, x ≤ ξ < λ µ (1 -x)} (35) T1 = {(x, ξ)|x ∈ [ λ λ + µ , 1], λ µ (1 -x) < ξ ≤ x} (36)
The kernels K i , L i are defined on T i , M, N are defined on T 0 . The kernels K, L are defined on T0 and M , N defined on T1 . They all have yet to be defined.

Remark 4: This transformation is a Fredholm transformation and can be rewritten using integrals between 0 and 1:

α(t, x) = u(t, x) + 1 0 Q 11 (x, ξ)u(t, ξ) + Q 12 (x, ξ)v(t, ξ)dξ (37) β(t, x) = v(t, x) + 1 0 Q 21 (x, ξ)u(t, ξ) + Q 22 (x, ξ)v(t, ξ)dξ (38) with Q 11 (x, ξ) = D i=1 K i (x, ξ)h [x,-µ λ x+bi] (ξ)h Ji (x) + M (x, ξ)h [ µ λ (1-x),x] (ξ)h [ µ λ+µ ,1] (x) (39)
Q 12 (x, ξ) = D i=1 L i (x, ξ)h [x,-µ λ x+bi] (ξ)h Ji (x) + N (x, ξ)h [ µ λ (1-x),x] (ξ)h [ µ λ+µ ,1] (x) (40)
Q 21 (x, ξ) = K(x, ξ)h [x, λ µ (1-x)] (ξ)h [0, λ λ+µ [ (x) + M (x, ξ)h [ λ µ (1-x),x] (ξ)h [ λ λ+µ ,1] (x) (41)
Q 22 (x, ξ) = L(x, ξ)h [x, λ µ (1-x)] (ξ)h [0, λ λ+µ [ (x) + N (x, ξ)h [ λ µ (1-x),x] (ξ)h [ λ λ+µ ,1] (x) (42
) Remark 5: Since α(0) = β(1) = 0 the two control laws U and V can be computed as functions of (u, v).

2) Operator formulations and properties: In this subsection we rewrite the previous Fredholm transformation using operators. This will lead to some relations verified by the adjoint operators. The Fredholm transformation (37)-(38) can be written as an operator P acting on u v . More precisely we have

P = Id 2 -Q (43) α β = P u v (44)
where Q : (L 2 (0, 1)) 2 → (L 2 (0, 1)) 2 is the integral operator defined by

Q u v = 1 0 Q 11 (x, ξ)u(t, ξ) + Q 12 (x, ξ)v(t, ξ) Q 21 (x, ξ)u(t, ξ) + Q 22 (x, ξ)v(t, ξ) dξ (45)
Its adjoint is:

Q * u v = 1 0 Q 11 (ξ, x)u(t, ξ) + Q 21 (ξ, x)v(t, ξ) Q 12 (ξ, x)u(t, ξ) + Q 22 (ξ, x)v(t, ξ) dξ (46)
One can easily check that:

K * (D(A * )) ⊂ D(A * ) (47)
The corresponding domains, characteristic lines and boundary conditions in Figure 1 We now focus on the

λ/µ ξ 1 0 1 λ/(λ+µ) x = ξ x µ ξ = λ ( 1 -x ) _ N = 0 _ L = 0 _ N = 0 _ K = -σ -+ / ( λ + µ ) _ M = σ -+ / ( λ + µ )
Fig. 1. Representation of the beta-kernels alpha-transformation if x ≥ µ µ+λ : As above, differentiating (25) with respect to space and time and then plugging into the target system (17)-(18) yields the following system of kernel equations

0 = λN x (x, ξ) -µN ξ (x, ξ) + σ +-M (x, ξ) (64) 0 = λM x (x, ξ) + λM ξ (x, ξ) + σ -+ N (x, ξ) (65) 0 = N (x, x) + σ +- λ + µ (66) 0 = M (x, µ λ (1 -x)) (67)
if x ∈ J i As above we get the following kernel equations 59), ( 60)-( 63), ( 64)-( 67), ( 68)-(71). There exists a unique solution K i , L i (defined on L ∞ (T i )), M i , N i (defined on L ∞ (T 0 )), K, L (defined on L ∞ ( T0 )), M , N (defined on L ∞ ( T1 )) Classically (see [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic pdes[END_REF], [START_REF] John | Continuous dependence on data for solutions of partial differential equations with a prescribed bound[END_REF] and [START_REF] Beresford | Linear and nonlinear waves[END_REF]) the proof of this theorem consists in transforming the kernel equations into integral equations using the method of the characteristics. These integral equations are then solved using the method of successive approximations.

0 = λL i x (x, ξ) -µL i ξ (x, ξ) + σ +-K i (x, ξ) (68) 0 = λK i x (x, ξ) + λK i ξ (x, ξ) + σ -+ L i (x, ξ) (69) 0 = L i (x, x) - σ +- λ + µ (70) 0 = K i (x, - µ λ x + b i ) (71)

IV. INVERTIBILITY OF THE FREDHOLM TRANSFORMATION

Unlike the Volterra transformation, the Fredholm transformation is not always invertible. In [START_REF] Coron | Stabilization and controllability of first-order integro-differential hyperbolic equations[END_REF], the authors prove the invertibility of such a transformation in the case of a firstorder integro-differential hyperbolic equation. In this section we use similar arguments to prove the invertibility of our transformation. We give first the following useful lemmas:

Lemma 3: ker P * ⊂ D(A * 0 ) = D(A * ) Proof: Let us consider z ∈ ker P * . Consequently we have P * z = 0. We can rewrite it

z 1 z 2 = 1 0 Q 11 (ξ, x)z 1 (t, ξ) + Q 21 (ξ, x)z 2 (t, ξ) Q 12 (ξ, x)z 1 (t, ξ) + Q 22 (ξ, x)z 2 (t, ξ) dξ (72)
If we evaluate the first line for x = 1 and the second one for x = 0, using the fact that

Q 11 (ξ, 0) = Q 21 (ξ, 0) = Q 12 (ξ, 1) = Q 22 (ξ, 1) = 0, we get z 1 (1) = z 2 (0) = 0 (73)
Consequently z ∈ D(A * 0 ) and we can write

kerP * ⊂ D(A * 0 ) (74)
Lemma 4: ker P * ⊂ ker B * Proof: Let us consider z ∈ ker P * . Consequently we have P * z = 0. We can rewrite it

z 1 z 2 = 1 0 Q 11 (ξ, x)z 1 (t, ξ) + Q 21 (ξ, x)z 2 (t, ξ) Q 12 (ξ, x)z 1 (t, ξ) + Q 22 (ξ, x)z 2 (t, ξ) dξ (75)
If we evaluate the first line for x = 0 and the second one for x = 1, using the fact that Q 11 (ξ, 1) = Q 21 (ξ, 1) = Q 12 (ξ, 0) = Q 22 (ξ, 0) = 0, we get Proof: Since Q * is a compact operator we can use the Fredholm alternative (e.g [START_REF] Brezis | Functional analysis, sobolev spaces and partial differential equations[END_REF]): Id 2 -Q * is either noninjective or surjective. Consequently it suffices to prove that P * is injective. In addition, the Fredholm alternative also gives [START_REF] Brezis | Functional analysis, sobolev spaces and partial differential equations[END_REF] dim ker(Id -Q * ) < +∞ (80)

Fig. 2 . 1 :

 21 Fig. 2. Representation of the alpha-kernels

z 1 (

 1 0) = z 2 (1) = 0 (76) Consequently z ∈ ker B * and we can write kerP * ⊂ kerB * (77) Lemma 5: ∀λ ∈ ker(λId 2 -A * 0 )∩ ker B * = {0} Proof: Let us consider λ ∈ and z ∈ ker(λId 2 -A * 0 )∩ ker B * = {0}. Consequently we haveλz 1x (t, x) + σ ++ z 1 (t, x) -λz 1 (t, x) -µz 2x (t, x) + σ --z 2 (t, x) -λz 2 (t, x)We can now state the following theorem Theorem 2: The map P * = Id 2 -Q * is invertible

Jean Auriol is with MINES ParisTech, PSL Research University, CAS -Centre automatique et systèmes, 60 bd St Michel 75006 Paris, France.jean.auriol@mines-paristech.fr

Florent Di Meglio is with MINES ParisTech, PSL Research University, CAS -Centre automatique et systèmes, 60 bd St Michel 75006 Paris, France.

The control

can also be rewritten using operators

with

Using ( 20) and (44) yields

Moreover using ( 9) and (44) we get

Consequently P and Γ satisfy the following relation:

Taking the adjoints, this is equivalent to

3) Kernel equations: We now differentiate the Fredholm transformation (25)-(26) with respect to time and space to compute the equations satisfied by the kernels. if x ≥ λ µ+λ : Differentiating (26) with respect to space and using the Leibniz rule yields

Differentiating (26) with respect to time, using (1), ( 2) and integrating by parts yields

Plugging these expressions into the target system (17)-(18) yields the following system of kernel equations

if x < λ µ+λ : Similarly we get

By contradiction we assume that ker P * = {0}. We first prove that ker P * is stable by A * 0 . We have ker P * ⊂ A * 0 . Let then consider z ∈ ker P * . Using (53) we can obtain

We thus have A * 0 Z ∈ ker P * . Consequently the restriction A * 0 |kerP * of A * 0 to ker P * is a linear operator from ker P * to ker P * . Since the dimension of ker P * is finite we can find at least one eigenvalue λ. Let e ∈ ker P * be a corresponding eigenvector (by definition e = 0). We have e ∈ ker P * and so e ∈ ker B * . Moreover we have A * 0 e = λe. Consequently

which contradicts Lemma 5 and concludes the proof.

V. CONTROL LAW AND MAIN RESULTS

We now state the main stabilization result as follows: Theorem 3: System (1)-( 2) with the following feedback control laws

reaches its zero equilibrium in finite time t F , where t F is given by ( 16). The zero equilibrium is exponentially stable in the L 2 -sense.

Proof: Notice that evaluating (25) at x = 0 yields (83) and evaluating (26) at x = 1 yields (84). Since the kernels are invertible, there exists a unique function S such that

Applying Lemma 2 implies that (α, β) go to zero in finite time t F , therefore (u, v) converge to zero in finite time t F Remark 6: One can notice that the control laws use only four kernels. Consequently from a practical point of view it is not necessary to compute all the kernels.

VI. SIMULATION RESULTS

In this section we illustrate our results with simulations on a tou problem. The numerical values of the parameters are as follow. In this section we design an observer that relies on the measurements of u at the right boundary and of v at the left boundary, i.e we measure

Then using the estimates given by our observer and the control law ( 83)-( 84), we derive an output feedback controller.

A. Observer design

The observer equations read as follows ût (t, x) + λû x (t, x) =σ +-v(t, x) -p 11 (x)(û(t, 1) -u(t, 1))

with the boundary conditions

where p 11 , p 12 , p 21 and p 22 are the observer gains and have yet to be designed. This yields the following error system

with the boundary conditions ũ(t, 0) = 0, ṽ(t, 1) = 0 (93)

B. Target system

We map the system (91)-(92) to the following system αt (t, x) + λα x (t, x) = 0 (94) βt (t, x) -µ βx (t, x) = 0 (95)

with the following boundary conditions α(t, 0) = 0, β(t, 1) = 0 (96)

Lemma 6: The system (94)-(95) reaches its zero equilibrium in a finite time t F where t F is defined by (16).

Proof: The proof of this lemma is straightforward and is omitted here.

C. Fredholm transformation

We use the following transformation ũ(t, x) = α(t, x)

Remark 7: We do have α(t, 0) = ũ(t, 0) = ṽ(t, 1) = β(t, 1). Moreover α(t, 1) = ũ(t, 1) and ṽ(t, 0) = β(t, 0) if x ≤ 1 2 Differentiating the equations ( 97)-( 98) with respect to time and space yields

All those kernels equations seem well-posed. Question : Is the transformation invertible ?