Guilhem Boéris

Lorenzo Gori

Marteen D Hoogerland

Avinash Kumar

Eleonora Lucioni

Luca Tanzi

Massimo Inguscio

Thierry Giamarchi

Chiara D 'errico

Giuseppe Carleo

C D'errico

G Modugno

L Sanchez-Palencia

Mott Transition for Strongly-Interacting 1D Bosons in a Shallow Periodic Potential

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction.-The interplay of repulsive interactions and a periodic potential in a quantum fluid triggers a superfluid-insulator transition known as the Mott transition, provided the potential period is commensurate to the inverse fluid density. The most familiar notion of Mott transition takes place in the limit of a deep periodic potential. In this case the lattice Hubbard model microscopically captures the dominant interaction and hopping processes, the strengths of which, U and J, strongly depend on the periodic potential amplitude V . Then, the Mott transition is driven by the competition of these sole two parameters at J ∼ U [1,2]. Quite strikingly in one dimension (1D) a Mott transition can exist even for a vanishingly small periodic potential provided the repulsive interactions are strong enough [3][4][5][6]. In the limit of a shallow potential, its amplitude V becomes sub-relevant and the transition is mostly controlled by the interaction strength g alone [6].

Ultracold atoms provide a remarkable laboratory to study this physics [7,8]. So far the Mott transition has been observed in both deep [9][10][11][12][13] and shallow [14] optical lattices. However, the characterization of the Mott transition in shallow potentials remains a formidable challenge for both theory and experiments, with direct consequences not only in the ultracold atom realm but also in condensed matter for problems such as spin chains for instance [6,15,16].

On the theoretical side, while the Hubbard limit is now well documented [7,8,[START_REF] Georges | Many-Body Physics with Ultracold Gases[END_REF] and its Mott transition has been extensively studied [START_REF] Fisher | [END_REF][19][20][21][22][23][24] its regime of validity beyond the deeplattice limit is still unknown in 1D, with ab-initio results having been reported so-far only for 3 dimensions (3D) [25]. In the limit of a vanishing potential, an estimate of the transition values may be found in the sine-Gordon model whose coefficients are determined perturbatively [14,26]. This, however, ignores the unavoidable renormalization of the field-theoretic coupling parameters by the potential, which may significantly affect the transition. On the experimental side, the Mott transition has been clearly observed in the shallow lattice limit using modulation spectroscopy and transport measurements [14]. However, the experimental uncertainties did not allow to a precise determination of the phase diagram.

In this Letter, we report the first quantitative joint theoretical and experimental investigation of the Mott transition for strongly-interacting 1D Bosons in a shallow periodic potential. Using continuous-space quantum Monte Carlo calculations, we determine the exact quantum phase diagram. Our calculations confirm the field-theoretical universal predictions and provide, in addition, an accurate quantitative values of the critical parameters of the Mott transition. Experimentally, we perform transport measurements on a Bose gas with tunable interactions down to the limit of very shallow lattices and we analyze them with a phase slip based model to determine accurately the Mott transition. The numerical and experimental results are in excellent agreement and show significant deviation from the perturbative sine-Gordon theory using bare Luttinger parameters.

Model and theoretical approach.-We consider zerotemperature interacting 1D Bosons of mass m with a contact interaction of strength g, subjected to a periodic potential V (x) = V sin 2 (kx) of spacing a = π/k and amplitude V . Both the large V and small V limits have the possibility of a Mott transition when the interactions are increased [START_REF] Georges | Many-Body Physics with Ultracold Gases[END_REF]. In spite of their qualitative different natures the two limiting cases are, however, expected to belong to the same universality class for they both lead to the same low-energy sine-Gordon model [4,6,[START_REF] Georges | Many-Body Physics with Ultracold Gases[END_REF]27]. Within the Tomonaga-Luttinger liquid (TLL) approach, the homogeneous superfluid is parameterized by the Luttinger parameter K, which characterizes the interaction strength. For weak interactions, the periodic potential is essentially irrelevant, except in renormalizing the effective value of the Luttinger parameter. For strong interactions, the TLL may be unstable upon introducing a periodic potential, which signals the Mott insulator phase. More precisely, the Mott transition may be triggered either by changing the fluid density to commensurability at sufficiently strong interactions (Mott- The various curves correspond to increasing system sizes, L/a = 30, 50, and 100 (blue, green, and red respectively). The vertical dotted lines show the transition points determined from the criterion K c = 1. (b) Phase diagram in the g-µ plane for the fixed amplitude of the periodic potential V = 2E r . The black points joint by lines are determined from the K c = 1 criterion, while the colored points show the results found from the crossing point of the compressibility (red), the cusp of the compressibility (green), and the crossing point of the Luttinger parameter (blue). The red crosses are the hard-core limits. The inside of the lobe (red region) and the dashed line correspond a density of one particle per potential spacing. Inset: Mott gap versus interaction strength. The black dashed line is a fit to the BKT prediction (see text). The blue dashed line is the prediction of the BH model and the blue cross is the corresponding tip of the lobe.

for the Mott-δ and Mott-U transitions respectively, where p is the commensurability order [5,6,27].

However the TLL theory involves effective parameters that are not easily related to the microscopic Hamiltonian parameters and the critical curve g c (V ) is presently not quantitatively known. To precisely determine the Mott transitions, we use quantum Monte-Carlo (QMC) simulations. This allows us (i) to determine quantitatively the phase diagram in terms of the microscopic parameters and (ii) to compute explicitly the Luttinger parameter K as a function of the microscopic ones and make the link with field theory. We use the same implementation of the continuous-space worm algorithm [28,29] in the grand-canonical ensemble as used in Ref. [30], which is numerically exact for all the physical quantities we study in the following [31].

Incommensurate transition.-We start with the incommensurate (Mott-δ ) transition, which may be triggered by changing the chemical potential µ. In order to accurately determine the critical point, several quantities are examined. The particle density n is computed directly in QMC and the compressibility κ ≡ ∂ n/∂ µ is computed independently from particle number fluctuations. The hydrodynamic superfluid density n s is found from the superfluid stiffness ϒ s , defined as the response of the system to a twist of the boundary conditions, which is computed using the winding-number estimator [31]. We then deduce the superfluid fraction f s = n s /n and the Luttinger parameter K = π √ n s κ.

The QMC results are shown versus the chemical potential in Fig. 1 for V = 2E r where E r = h2 k 2 /2m is the recoil energy, g = 7h 2 /ma, and various system sizes. The density [Fig. 1(a1)] increases monotonically with the chemical poten-tial µ and exhibits a plateau at commensurability, na = 1, where the superfluid density drops to zero. This is the signature of the Mott-δ transition. The critical chemical potentials µ ± c corresponding to the two edges of the plateau are accurately determined from the crossing points of the compressibility for different system sizes [Fig. 1(a2)]. They can also be found from the drop of the superfluid fraction f s , which yields similar values for µ ± c [Fig. 1(a3)]. At the Mott-δ transition, the Luttinger parameter is expected to exhibit the universal discontinuity from K = 1 to K = 0. Our data [Fig. 1(a4)] are perfectly compatible with this prediction of the TLL theory.

Repeating the same calculations for various values of the interaction strength, we find the Mott lobe in the gµ plane shown in Fig. 1(b). The black points and joining lines are determined from the K c = 1 criterion while the colored points are extracted from the crossing point of the compressibility (red), the cusp of the compressibility (green), and the crossing point of the Luttinger parameter (blue). The different methods yield results in excellent agreement all along the lobe within a few percents. The Mott gap ∆ = µ + cµ - c is shown in the Inset of Fig. 1 versus the interaction strength. The black dashed line is a fit with the exponential closing ∆(g) ∝ exp(-b/ √ gg c ) predicted by the Berezinskii-Kosterlitz-Thouless (BKT) theory [32][33][34], with b and g c as fitting parameters. We find that the fit is good for all values of g except very close to the hardcore limit, g → +∞. The fit also yields a fair estimate of the critical value at commensurability, g c /(h 2 /ma) = 2.2 ± 0.1, which corresponds to the tip of the lobe.

Commensurate transition.-In order to characterize the commensurate (Mott-U) transition more precisely, we now vary the interaction strength γ ≡ mg/h 2 n at commensura-

(x) = ψ † (x) ψ(0) for V /E r = 2.
The decay is exponential in the insulator (left inset, γ = 7) and algebraic in the superfluid (right inset, γ = 1.25). Left: Correlation length in the insulator. Right: Luttinger parameter in the superfluid extracted from the decay of the correlation function (black) and from the formula K = π √ n s κ (red). The QMC results are the points, connected by straight lines to guide the eye. Note that the inflection of the correlation length curve close to the transition is due to finite size effects. bility, na = 1. We compute the one-body correlation function g 1 (x) = ψ † (x) ψ(0) , where ψ is the field operator, from the statistics of endpoints on open world lines in the QMC [28,29]. When increasing γ along the line with na = 1 (dashed line in Fig. 1), we observe a clear change of behavior of the g 1 function from algebraic for γ < γ c to exponential for γ > γ c (see Insets of Fig. 2). This is the signature of the Mott-U transition. The finite correlation length l c in the insulating phase in shown in the left panel of Fig. 2. It is of only a few lattice sites long for strong interactions and increases up to a value comparable to the system size for γ c ∼ 2. This is compatible with the expected divergence of the correlation length at the transition. In the superfluid phase, the algebraic decay of the correlation function is compatible with the TLL theory prediction g 1 (x) ∝ 1/x 1/2K . The two values of the Luttinger parameter found from a fit to this prediction and from the thermodynamic prediction K = π √ n s κ are in good agreement (see right panel of Fig. 2). When increasing the interaction towards the insulating phase, the Luttinger parameter decreases down to K 2 as predicted by the TLL theory.

To locate the Mott-U transition point accurately, we resort to the BKT renormalization group equations to perform the finite size scaling of the Luttinger parameter [31]. The results are shown in Fig. 3 (black points). In the strong potential limit, the results are compatible with the prediction of the BH model with the critical value (J/U) c = 0.297 ± 0.01 [21,22] and the hopping J and interaction strength U calculated from the exact Wannier functions [20]. In the vanishing potential limit, the results converge to the critical value γ c (V = 0) 3.5 (red cross in Fig. 3) found from the exact relation K(g) for the integrable Lieb-Liniger model [35]. However, in the inter- ). The black and green points, both with errorbars) are the Monte Carlo and experimental results, respectively. The blue line is the BH prediction using exact Wannier functions and the critical value (J/U) c = 0.297 ± 0.01 (the shaded area corresponds to this error bar). The red cross indicates the critical interaction strength g c such that K(g c ) = 2 in the Lieb-Liniger model. Along the whole transition line the effective Luttinger parameter remains K = 2. The red line is the result of the bare sine-Gordon theory [26].

mediate regime, we find a strong deviation from the pinning transition line (red dashed line) computed in Ref. [26] from the perturbative sine-Gordon theory using the bare Luttinger parameters. This shows that to quantitatively obtain the phase diagram the renormalization of the Luttinger parameters by even relatively weak interactions is significant and cannot be ignored in the perturbative field theory.

Experiment.-Having quantitatively characterized the Mott transition from theory, we now turn to the experimental investigation. The experiment starts with a Bose-Einstein condensate of 35 000 39 K atoms with tunable scattering length at a broad Feshbach resonance [36]. The BEC is split into about 1000 vertical 1D tubes by adiabatically loading a strong horizontal 2D optical lattice. Each tube contains on average 36 atoms and the transverse trapping frequency, ω ⊥ = 2π × 40 kHz corresponds to an energy higher than all other energy scales, realizing an effective 1D geometry. In the longitudinal direction we then adiabatically rise a weak optical lattice with spacing a = λ /2 =532 nm and normalized amplitude V /E r ranging from 1.0(1) to 4.0(4). The temperature of the system, T 30 nK, is below the 1D degeneracy temperature T D 50 nK [37]. A magnetic field along the vertical direction holds the system against gravity and a residual harmonic trap potential, with frequency ω z = 2π × 160 Hz, makes it inhomogeneous. By varying the 3D scattering length a 3D , we can tune the Lieb-Liniger parameter γ in the range 0.07 -7.4. The system parameters are chosen to obtain a mean atom number per potential period na =1. This implies that in most of the tubes there are one or two regions with local commensurate density, na=1, which can undergo a Mott-U transition. There is however a fraction of the tubes for which na < 1 which can- V /E r =1 (red), 2 (orange), 2.8 (green), 4 (blue). A piecewise second-order polynomial fit determines the critical values for the SF-MI transition: respectively a c /a 0 =392 ± 12, 214 ± 6, 160 ± 30, 122 ± 8. Inset: time evolution of the momentum distribution peak p for a 3D =109 a 0 and V /E r =2. The solid line is the theoretical damped oscillation fitting the data p < p c before the dynamical instability sets in. The error bars represent the root mean square of the imaging resolution and statistical uncertainties.

not become insulating [31].

To detect the Mott transition we excite a sloshing motion of the system through a shift of the trapping potential, obtained by suddenly switching off the magnetic field gradient [14,38]. We let the atoms evolve in the trap for a variable time t, after which all optical potentials are switched off and time-of-flight absorption images are recorded. An example of the time evolution of the momentum distribution peak p(t) is shown in the inset of Fig. 4: One can notice an initial increase of p followed by a subsequent decrease. We analyze this behavior in the frame of a phase slip based model [38,39]. Phase slips, i.e. the dominant excitations in 1D, make the system dynamics dissipative already at small momenta: p(t) can be fit with a damped oscillation function of the form p(t) = p max e -Gt sin(ω t), where the frequency ω is renormalized by the damping rate G and by the lattice [31]. At larger momenta one observes a clear deviation from such behavior, with a sudden increase of the damping that significantly reduces the growth of p. We identify the momentum where the experimental data points deviate with respect to the theoretical curve as the critical momentum p c for the occurrence of a dynamical instability, driven by a divergence of the phase slip rate [38]. The critical momentum p c is expected to vanish at the superfluid-insulator transition [39].

The behavior of p c as a function of the scattering length a 3D is reported on Fig. 4 for several values of the lattice depth. The measured p c initially decreases for increasing a 3D , and then reaches a finite constant value. We interpret the onset of the plateau as the Mott transition for the commensurate regions of the system, which drives all the corresponding tubes into an effective insulating regime (i.e. transport along individual tubes is globally suppressed). The fraction of tubes that does not reach the critical density na=1 keeps instead moving also beyond this point, originating the observed plateau for p c . This interpretation is corroborated by the observed increase of p c at the plateau for decreasing V , since the increase of the interaction strength that is necessary to reach the insulating regime produces an overall decrease of the density of the 1D systems, hence an increase of the fraction of tubes that does not reach na=1. For each set of measurements with a given value of V , we therefore identify the critical scattering length value a c to enter the insulator regime by determining the beginning of the plateau with a piecewise fit. We use a second-order polynomial fit, which is justified by the phase slip based model [31,39,40]. We clearly see that as V decreases, a c -and thus also γ c -increases.

For each periodic potential depth we get the Mott-U transition point converting a c into γ c for na = 1. The experimental results are shown as green points in Fig. 3. Within our uncertainties [41], the experiment is in very good agreement with the numerical simulations, except for the case V /E r =1, where the finite size of the system might start to play a role. Our results are also consistent with those reported in Ref. [14] within their uncertainties. Our experiment confirms the clear deviation of the transition line from the bare sine-Gordon prediction observed in the QMC results. Note that, surprisingly enough, while the BH model is justified only for V E r , both numerics and experiments show that the BH prediction for the Mott-U transition is quite accurate down to the limit V → 0. This agreement is however rather accidental and the breakdown of the BH model is manifest in other quantities. For instance, the BH prediction for the Mott gap deviates significantly from the exact QMC result [see Inset of Fig. 1(b)].

Conclusions.-We have studied, both theoretically and experimentally, the Mott transition of strongly-interacting 1D Bosons in a periodic potential from deep to shallow potentials. Our ab-initio QMC calculations validate the field-theoretic universal predictions and in addition provide a quantitative determination of the phase diagram. It shows that the renormalization of the Luttinger parameter is significant even for weak periodic potentials. The numerical analysis give excellent agreement with experiments for the Mott-U transition. The experimental observation of our numerical phase diagram for the Mott-δ transition is still beyond reach for ultracold atomic systems due to the requirement of a fine control of atom number in box-shaped potentials. In spite of recent progress in that direction [42,43], it remains a great challenge for future studies.

This research was supported by the EU FET-Proactive QUIC (H2020 grant No. 641122), the ERC ALoGlaDis (FP7/2007-2013 grant No. 256294), Marie Curie IEF (FP7/2007-2013 grant No. 327143), MIUR (grant No. RBFR12NLNA), and the Swiss NSF under Division II. Numerical calculations were performed using HPC resources from GENCI-CCRT/CINES (grant No. c2015056853) and the GMPCS cluster of the LUMAT federation (FR LUMAT 2764), and make use of the ALPS scheduler library and statistical analysis tools [44][45][46].

Note added.-During the completion of this manuscript, a preprint appeared reporting the numerical study of the Mott-U transition with results consistent with ours [47].

-Supplemental Material-

Mott Transition for Strongly-Interacting 1D Bosons in a Shallow Periodic Potential

In this Supplemental Material we give additional details on the methods and the analysis we have performed in the main Letter. In particular we provide a detailed description of i) the quantum Monte Carlo calculations, ii) the determination of the superfluid density, iii) the finite-size scaling analysis, and iv) the experimental methods.

Quantum Monte Carlo calculations

We consider 1D interacting Bosons at zero temperature governed by the continous-space Hamiltonian,

Ĥ = dx h2 2m ∇ ψ † ∇ ψ + g 2 ψ † ψ † ψ ψ +V (x) ψ † ψ , (S1)
where ψ(x) is the Bose field, m is the particle mass, g is the contact interaction strength, and V (x) = V sin 2 (kx) is a periodic potential of spacing a = π/k and amplitude V .

We study the interacting quantum system from first principles, by means of quantum Monte Carlo (QMC) numerical simulations. Path-integral QMC approaches allow us to compute the exact thermodynamic properties of a generic, interacting bosonic quantum system in any dimensions. This is achieved by means of the path-integral representation of the grand-canonical partition function Z = Tr[e -β ( Ĥ-µ N) ], where β = 1/k B T with k B the Boltzman constant and T the temperature, µ is the chemical potential, and N = dx ψ † ψ is the total particle number operator. The quantum partition function is expressed as an equivalent classical partition function of interacting polymers living in dimension D + 1, where the additional dimension is the imaginary-time direction [1]. The equivalent classical partition function can be treated stochastically by means of Monte Carlo sampling. An efficient way of sampling the associated partition function is given by the worm algorithm [2,3].

In our implementation, the imaginary-time propagator entering the path-integral representation is written as

{r i }|e -ε Ĥ |{r i } = N ∏ i=1 ρ (0) ε (r i , r i ) N ∏ i=1 e -U (1) ε (r i ,r i ) × × ∏ i< j e -U (2) ε (r i ,r j ,r i ,r j ) , (S2)
where

ρ (0) ε (r i , r i ) = m/2π h2 ε × e -m|r i -r i | 2 /2h
2 ε is the freeparticle propagator. The one-body contribution e -U (1) ε (r i ,r i ) = r i |e -ε[ p2 /2m+V (r)] |r i /ρ (0) ε (r i , r i ) is computed exactly by matrix squaring [1]. The two-body interaction is taken into account at the pair-product level, thanks to the explicit expression for the two-body propagator e -U (2) ε (r i ,r j ,r i ,r j ) = r i , r j |e -ε[( p2

i + p2 j )/2m)+gδ (r i -r j )] |r i , r j /ρ (0) ε (r i , r i )ρ (0)
ε (r j , r j ) which is known [4]. The systematic error coming from the discretization of the path-integral along the imaginary-time direction is smaller than the statistical errorbars reported in our results.

Superfluid density

The stiffness ϒ s = -mL

β h2 ∂ 2 ln Z ∂ θ 2 0 0
measures the response of the system when the periodic boundary conditions are twisted by an angle θ 0 . It is directly computed by QMC using the usual winding number estimator [1]. However, this quantity strongly depends on the ratio β /L [5]. The relevant quantity to study is the superfluid density n s appearing in the hydrodynamic action

S[θ ] = L 0 dx hβ 0 dτ h2 n s 2m (∂ x θ ) 2 + h2 κ 2 (∂ τ θ ) 2 . (S3)
Computing the partition function Z = Dθ e -S[θ ]/h , correctly taking into account winding configurations of the form θ (x, τ) = 2πkx/L with k ∈ Z, one deduces the relation

ϒ s = n s 1 -4π 2 n s h2 β mL ∑ +∞ k=-∞ k 2 e -2π 2 n s k 2 h2 β /mL ∑ +∞ k=-∞ e -2π 2 n s k 2 h2 β /mL . (S4)
In our calculations, we invert this relation and extract n s from the stiffness ϒ s computed by QMC.

Finite size scaling

To precisely determine the Mott-U critical interaction strength g c , we study the evolution of the Luttinger parameter K for the increasing system sizes L/a = 30, 50, and 100. When L is increased, the temperature T is lowered by keeping the ratio β /L constant in order to consistently compute ground-state properties. We resort to the known Berezinkii-Kosterlitz-Thouless (BKT) renormalization group (RG) equations to perform the finite size scaling and determine the critical interaction strength in the thermodynamic limit. In particular, the finite-size behaviour of the Luttinger parameter K and of renormalized potential strength V are governed by the renormalization equations [6][7][8][9][10],

dK d = - π 6 16 V E r 2 K 2 ; dV d = (2 -K)V. (S5)
Dividing these two equations one by another, one finds that the quantity

ζ ≡ 2 K + ln K 2 - π 2 16 V E r 2 (S6)
is conserved along the RG trajectories. The separatrix between the superfluid and the insulating phases then corresponds to K = 2 and V = 0, i.e. ζ = 1. In practice, we use the values the Luttinger parameter K 1 and K 2 for two system sizes L 1 and L 2 , and compute ζ using the integral relation

K 1 K 2 dK K 2 (ln K/2 -ζ ) + 2K = ln(L 2 /L 1 ). (S7)
The Mott-U critical point is then determined using the criterion ζ = 1.

Experimental methods

In the experiment, a Bose-Einstein condensate of 39 K atoms is split in about 10 3 1D subsystems (potential tubes) by means of a 2D optical lattice. The overall Thomas-Fermi distribution of atoms in the tube labelled by i and j along the two horizontal directions is given by N i, j = N 0,0 [1 -2πN 0,0 (i 2 + j 2 )/5N T ] 3/2 , where N T = 3.5 × 10 4 is the total atom number and N 0,0 50 is the atom number in the central tube. On N T there is a 6% statistical error while a 30% systematic uncertainty is due to the imaging calibration. Along the vertical direction of the tube, where we load the weak periodic potential, the trapping frequency is ω z = 2π × 160 Hz. A good estimate of the mean atomic density for each tube is provided by the largest of the Thomas-Fermi and the Tonks value [11]. The mean site occupation na is then calculated by averaging over all the tubes. During the lattice loading we employ an optimal value of the 3D scattering length to get commensurability on average, i.e. na = 1. Given ω z and N T , the optimal value is a 3D = 220 a 0 .

After the lattice loading, by varying the 3D scattering length a 3D , we tune the 1D scattering length a 1D = 2 ⊥ (1 -1.03a 3D / ⊥ )/2a 3D , and thus the Lieb-Liniger parameter γ, where ⊥ = h/mω ⊥ is the harmonic transverse size of a tube, fixed by the 2D lattice. The value of γ is estimated by averaging over all the tubes 1/(n max aa 1D ), where the peak occupation n max a is the largest of the Thomas-Fermi and the Tonks value. The mean peak occupation, i.e. the peak occupation averaged over all the tubes, is nmax a 1.2 and the peak density in the central tube is less than two, thus preventing localization mechanisms at occupations other than one. Despite the inhomogeneity of our system, at sufficiently strong interactions we clearly observe a suppression of the system dynamics. We interpret the latter as due to the fact that within each tube, a part of the atoms reaches the localization condition na = 1 stopping also the remaining adjacent parts with different occupation. We estimate that about one quarter of the atoms resides in tubes where the occupation is always na < 1, justifying the plateau in p c shown in Fig. 4 of the main paper.

The complete model we use to fit the time dependence of the momentum, p(t), as shown in Fig. 4 of the main paper, is p(t) = p max e -Gt sin(ω t) with amplitude p max = m * ω * 2 z 0 /ω , frequency ω = √ ω * 2 -G 2 and damping rate G. Here m * is the effective mass due to the shallow lattice, z 0 3 µm is the trap displacement, and ω * = ω z m/m * is the lattice renormalized frequency.

In the absence of an exact theoretical model for the critical momentum p c to enter the dynamical instability regime at finite interaction, we use a quantum phase slips based model to predict the interaction dependence of p c . Along the lines of Ref. [12] we use the equation for the quantum phase slip nucleation rate [13,14] Γ =B L(U) (S8)

Here L(U) 2U 1/3 is the mean length of the tubes and B is a phenomenological constant. Assuming that the system enters the strongly dissipative regime when the nucleation rate Γ exceeds a constant critical value, from Eq. (S8) we can obtain p c for each given value of the interaction U. Since this relation is valid in the Bose-Hubbard regime, we can employ it to estimate the critical momentum p c (U) for the experimental case with V /E R = 4. The phenomenological parameter B is arbitrarly adjusted to reproduce the measured p c at one interaction value, U = 2.4J. In Fig. S1 we compare the measured and the predicted p c as a function of the scattering length a 3D ∝ U. The quantum phase slip-based model well reproduces the experimental behaviour of p c with the interaction, showing a quadratic dependence of p c on a 3D . This result justifies the choice of a quadratic polynomial fit in Fig. 4 of the main paper, at least for the measurements with larger V . For shallower lattices the Tomonaga-Luttinger Liquid model for the nucleation rate could in principle be used [13], but this would require the Luttinger parameter K as a function of the interaction γ in the presence of the lattice, which is still unknown.

2 µFigure 1 .

 21 Figure1. (Color online) (a) QMC results for the number of particles per site, na, the compressibility κ, the superfluid fraction f s , and the Luttinger parameter K, versus the chemical potential µ, for a periodic potential amplitude V = 2E r and an interaction strength g = 7h 2 /ma. The various curves correspond to increasing system sizes, L/a = 30, 50, and 100 (blue, green, and red respectively). The vertical dotted lines show the transition points determined from the criterion K c = 1. (b) Phase diagram in the g-µ plane for the fixed amplitude of the periodic potential V = 2E r . The black points joint by lines are determined from the K c = 1 criterion, while the colored points show the results found from the crossing point of the compressibility (red), the cusp of the compressibility (green), and the crossing point of the Luttinger parameter (blue). The red crosses are the hard-core limits. The inside of the lobe (red region) and the dashed line correspond a density of one particle per potential spacing. Inset: Mott gap versus interaction strength. The black dashed line is a fit to the BKT prediction (see text). The blue dashed line is the prediction of the BH model and the blue cross is the corresponding tip of the lobe.

1 Figure 2 .

 12 Figure 2. (Color online) Analysis of the correlation function g 1(x) = ψ † (x) ψ(0) for V /E r = 2.The decay is exponential in the insulator (left inset, γ = 7) and algebraic in the superfluid (right inset, γ = 1.25). Left: Correlation length in the insulator. Right: Luttinger parameter in the superfluid extracted from the decay of the correlation function (black) and from the formula K = π √ n s κ (red). The QMC results are the points, connected by straight lines to guide the eye. Note that the inflection of the correlation length curve close to the transition is due to finite size effects.

Figure 3 .

 3 Figure 3. (Color online) Phase diagram in the g-V plane at commensurability (na = 1). The black and green points, both with errorbars) are the Monte Carlo and experimental results, respectively. The blue line is the BH prediction using exact Wannier functions and the critical value (J/U) c = 0.297 ± 0.01 (the shaded area corresponds to this error bar). The red cross indicates the critical interaction strength g c such that K(g c ) = 2 in the Lieb-Liniger model. Along the whole transition line the effective Luttinger parameter remains K = 2. The red line is the result of the bare sine-Gordon theory[26].

Figure 4 .

 4 Figure 4. (Color online) Critical momentum p c as a function of the scattering length a 3D for four periodic potential depth values:V /E r =1 (red), 2 (orange), 2.8 (green), 4 (blue). A piecewise second-order polynomial fit determines the critical values for the SF-MI transition: respectively a c /a 0 =392 ± 12, 214 ± 6, 160 ± 30, 122 ± 8. Inset: time evolution of the momentum distribution peak p for a 3D =109 a 0 and V /E r =2. The solid line is the theoretical damped oscillation fitting the data p < p c before the dynamical instability sets in. The error bars represent the root mean square of the imaging resolution and statistical uncertainties.

√ naJU π/ 2 -

 2 pλ /2h× 7.1(π/2pλ /2h) 5/2 2π U/naJ × exp -7.1 naJ/U(π/2pλ /2h) 5/2 .

a 3 D

 3 Figure S1. For V /E R = 4 the experimental critical momentum (blue dots) is compared to the one predicted by the quantum phase slip based model, which applies up to the critical scattering length a c to enter the MI regime (gray dashed line). For a > a c , p c has instead a plateau due to tubes with occupation na < 1 (black dotted line)..


Mouhamadou Diaby

Abderrahman Iggidr
email: abderrahman.iggidr@inria.fr

Mamadou Sy

Observer design for a schistosomiasis model

Keywords: high-gain, Interval observer, Observer, Schistosomiasis model, State estimation

This paper deals with the state estimation for a schistosomiasis infection dynamical model described by a continuous nonlinear system when only the infected human population is measured. The central idea is studied following two major angles. On the one hand, when all the parameters of the model are supposed to be well known, we construct a simple observer and a high-gain Luenberger observer based on a canonical controller form and conceived for the nonlinear dynamics where it is implemented.

On the other hand, when the nonlinear uncertain continuous-time system is in a bounded-error context, we introduce a method for designing a guaranteed interval observer. Numerical simulations are included in order to test the behavior and the performance of the given observers.

Introduction

Human schistosomiasis is a behavioral and occupational disease associated with poor human hygiene, insanitary animal husbandry and economic activities. Among human parasitic diseases, schistosomiasis ranks second behind malaria as far as the socio-economic and public health importance in tropical 5 and subtropical areas are concerned. Urinary schistosomiasis, caused by the species Schistosoma haematobium, is common in Africa and the Middle East. The main clinical sign of schistosomiasis infection is haematuria itself caused by the depositions of eggs by an adult female's worms through the bladder by urinary intermediary [START_REF] Organization | Prevention and Control of Schistosomiasis and Soil-Transmitted Helminthiasis: Report of a WHO Expert Committee[END_REF].

The most effective form of treatment for infected individuals is the use of the drug praziquantel a drug that kills the worms with high efficiency. Control programs often consist on mass chemotherapy possibly supplemented by snail (intermediate host) control. Since school-age children are the heaviest infected group that suffer the most from morbidity and by that are major sources of infection for the community, school targeted chemotherapy can be then an adequate effective approach to control that morbidity [START_REF] Organization | Prevention and Control of Schistosomiasis and Soil-Transmitted Helminthiasis: Report of a WHO Expert Committee[END_REF][START_REF] Tang | Development of larval Schistosoma japonicum blocked in Oncomelania hupensis by pre-infection with larval Exorchis sp[END_REF].

Schistosomiasis have one of the most complex host-parasite process to model mathematically because of the different steps of growth of larval assumed by the parasite and the requirement of two host elements (definitive human host and intermediate snail hosts) during their life cycle.

Current world-wide interest in the control of schistosomiasis has focused attention upon the intermediate hosts of the causative parasite, since there is general agreement that the most promising method of controlling the disease is to eliminate or greatly reduce the numbers of these vector snails. It is necessary to obtain information about snail populations, whether the information is used for snail-control evaluation, for ecological research, or for the study of transmission potential.

In epidemiology, mathematical models are very often used to describe the dynamic evolution of the diseases. Deterministic Ordinary differential Equations (ODEs) are one of the major modeling tools and are used in our case.

In this paper, we are interested in the estimation problem of the unknown snails population state of a schistosomiasis model whose dynamics are modeled by a continuous time system.

Symbolically, we can write a dynamical system as :

   Ẋ(t) = F (X(t)), Y (t) = h(X(t)), (1)
with

X(t) ∈ R n , Y (t) ∈ R p , p < n.
If it is possible to have the value of the state at some time t 0 then it is possible to compute X(t) for all t ≥ t 0 by integrating the differential equation with the initial condition X(t 0 ). Unfortunately, it is not often possible to measure the whole state at a given time and by the same way to integrate the differential equation because one does not know the initial condition. One can only have a partial information on the state and this partial information is precisely given by Y (t) the output of the system. Therefore we shall show how to use this partial information Y (t) together with the given model in order to have a reliable estimation of the unmeasurable state variables. A state observer is usually employed, in order to accurately reconstruct the state variables of the dynamical system. In the case of linear systems, the observer design theory developed by Luenberger [START_REF] Luenberger | An introduction to observers[END_REF], offers a complete and comprehensive answer to the problem. In the field of nonlinear systems, the nonlinear observer design problem is much more challenging and has received a considerable amount of attention in the literature.

An observer for (1) is a dynamical system

   Ż(t) = F (Z(t), Y (t)), X(t) = L(Z(t), Y (t)) ( 2
)
whose task is state estimation. It is expected to provide a dynamical estimate X(t) of the state X(t) of the original system. The output is in general a function of the state variable, that is, Y (t) = h(X(t)).

One usually requires at least that X(t) -X(t) goes to zero as t → ∞.

When the convergence of X(t) towards X(t) is exponential, the system (1) is an " exponential observer ". More precisely, system (2) is an exponential observer for system (1) if there exists λ > 0 and c 0 ≥ 0 such that, for all t ≥ 0 and for all initial conditions X(0), X(0) , the corresponding solutions of ( 1)-( 2) satisfy

X(t) -X(t) ≤ e -λ t X(0) -X(0) + c 0 .
The best situation corresponds to the case where c 0 = 0. In this situation a good estimate of the real unmeasured state is rapidly obtained. One must notice that we do not need to care about the initial condition of the observer since the convergence of X(t) towards the real state X(t) does not depend on this choice. There are numerous means to deal with the synthesis of nonlinear observers.

The most general method to tackle it is to use a "high-gain method observer " when the functions of the variables are perfectly known in the dynamical model. This means is much more general than "the output injection model " developed in [START_REF] Krener | Linearization by output injection and nonlinear observers[END_REF][START_REF] Krener | Nonlinear Observers with Linearizable Error Dynamics[END_REF][START_REF] Hammouri | Bilinearization up to output injection[END_REF][START_REF] Hammouri | Global time-varying linearization up to output injection[END_REF], which is applied to a very special class of systems only.

If it happens that some functions of the variables are partially known in the dynamical model but bounded with a priori known bounds, we can define a bounded error observer giving X(t) with X(t) -X(t) bounded by a "reasonable" positive real constant (depending on the uncertainty), "reasonable" meaning that this constant is small with respect to the measurement errors as developed in [START_REF] Gouzé | A simple improvement of interval asymptotic observers for biotechnological processes[END_REF].

This paper shows out first a high-gain observer for a reduced nonlinear model of schistosomiasis as proposed by Allen [9]. This high-gain observer method, has been initiated in [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF][START_REF] Gauthier | Observers for nonlinear systems[END_REF][START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF]. However, the convergence of this kind of observers is difficult to prove (because of the global Lipshitz condition). So, we propose a simpler observer whose convergence analysis is studied. This nonlinear observer design does not require Lipschitz extension of functions and change of coordinates for the system contrary to the high-gain observer.

In the second part, we will present an interval observer design to handle the already mentioned uncertainties of the model parameters. The methodology of interval observers has already been studied using a theoretical framework [START_REF] Rapaport | Parallelotopic and practical observers for nonlinear uncertain systems[END_REF][START_REF] Meslem | Interval Observers for Uncertain Nonlinear Systems. Application to Bioreactors[END_REF], and interval observers have been developed for particular models [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF][START_REF] Rocha | Design of estimators for specific growth rate control in a fed-batch E. coli fermentation[END_REF], and have been validated experimentally [START_REF] Meslem | Interval Observers for Uncertain Nonlinear Systems. Application to Bioreactors[END_REF]. In these works the authors address conditions for stability of the interval observer.

The construction of an observer requires some properties of observability and requires essentially the existence of globally defined and globally Lipschitzian change of coordinates.

The paper is organized as follows: In Section 2 we present the biological assumptions that guided the model's structure and the model's equations. In Section 3 we perform a high-gain observer design. Section 4 will point out a simple observer design. Section 5 tackles the guaranteed interval observer construction. Finally Sections 6 and Section 7 will respectively be constituted of the different estimator simulations and the conclusion.

Model and assumptions

In this section, the model proposed is a modified version of Allen's model [START_REF] Allen | Modelling and simulation of a schistosomiasis infection with biological control[END_REF]. The main point in the model presented in Allen in a relatively isolated community, based on the model presented in Allen [START_REF] Allen | Modelling and simulation of a schistosomiasis infection with biological control[END_REF] is to take into account an additional mammalian host as well as a competitor snails. The model assumes that hosts population and infection rates are independent of environmental factors. The totality of simplifying assumptions lead one to question the quantitative predictions of the model. However, the qualitative features of the results are in themselves of considerable interest [START_REF] Allen | Modelling and simulation of a schistosomiasis infection with biological control[END_REF]. Here, we ignore competitor snails population. Thus, the total human population size, denoted by N H (t), is split into susceptible individuals (X 1 (t)) and infected individuals (X 2 (t)) so that N H (t) = X 1 (t) + X 2 (t), and the total mammal population size, denoted by N M (t), is also subdivided into susceptible mammals (X 6 (t)) and infected mammals (X 7 (t)) so that N M (t) = X 6 (t) + X 7 (t). Whereas, the total snails population, denoted by N S (t), is subdivided into susceptible snail host (X 3 (t)), infected snails which are not yet shedding cercariae (X 4 (t)) and infected and shedding snail (X 5 (t)). Thus N S (t) = X 3 (t) + X 4 (t) + X 5 (t). We assume that the total time interval considered, T , is not too large so that the infection in the definitive hosts (e.g. human) does not result death. Further, it is assumed that infected snails and infected mammals do not recover from schistosomiasis as their life spans are short in comparison to that for humans. For simplicity, assume that births in each population group (human, snail and alternate host) were allowed to enter only the uninfected populations. Another assumption made is that the latent periods can be ignored in both definitive and intermediate hosts. This means that we disregard the time period between the moment when a cercaria penetrates a definitive host and the moment when the cercaria has grown to a sexually mature parasite. Other assumptions are that the recovery rate of infected intermediate hosts is independent of the length of the infectious period, and that the rate of output of cercariae from an infected intermediate host is constant throughout the period when it remains infected. We furthermore assume that births and deaths were considered to be proportionate to population size . Thus, the differential equations which govern the disease are :

                                                 dX 1 dt = -t 15 X 5 X 1 + r 12 X 2 , dX 2 dt = t 15 X 5 X 1 -r 12 X 2 , dX 3 dt = b 3 (X 3 + X 4 + X 5 ) -t 32 X 2 X 3 -d 3 X 3 -t 37 X 3 X 7 , dX 4 dt = t 32 X 2 X 3 + t 37 X 3 X 7 -d 4 X 4 -r 54 X 4 , dX 5 dt = r 54 X 4 -d 5 X 5 , dX 6 dt = b 6 (X 6 + X 7 ) -t 65 X 5 X 6 -d 6 X 6 , dX 7 dt = t 65 X 5 X 6 -d 7 X 7 . (3)
Where :

• t 15 = transmission rate from infected snails to uninfected humans,

• t 32 = transmission rate from infected humans to uninfected snails,

• t 37 = transmission rate from infected mammals to susceptible snail,

• t 65 = transmission rate from infected snails to susceptible mammals.

Birth and death rates for the various sub-populations are denoted by b i et d i , respectively, for i = 1, 2, ..., 7. Also, r 12 is the treatment rate of infected human population and r 54 denotes the rate that the latent snail population X 4 becomes shedding X 5 .

It is assumed for simplicity that b

1 = d 1 = d 2 , b 3 = d 3 = d 4 = d 5
, and b 6 = d 6 = d 7 . Then all total populations are constant thanks to the assumptions on b i and d i . This is a technical assumption aimed to reduce the complexity of the mathematical analysis, but is not always a plausible approximation. This assumption of a constant population size is also realistic when modeling a disease over many years if the births are approximately balanced by the natural deaths. More general models can be constructed and analyzed, but our goal here is to show what may be deduced from simple model. Furthermore, this simple model has additional value as it is based on models that include more detailed structure.

We introduce the proportions of the snails and mammals sub-populations:

x i = X i N S
for i = 3, 4, 5, and

x i = X i N M for i = 6, 7.
Using the number of infected humans and the proportions of the other subpopulations and the fact that X 1 = N H -X 2 , x 3 + x 4 + x 5 = 1, and x 6 + x 7 = 1, system (3) reduces to one of the following equivalent systems (4) and ( 5). The first one using X 2 , x 3 , x 5 , and x 7 is given by:

                         d X 2 dt = t 15 (N H -X 2 ) N S x 5 -r 12 X 2 , d x 3 dt = b 3 -(t 32 X 2 + t 37 N M x 7 + b 3 ) x 3 , d x 5 dt = r 54 (1 -x 3 -x 5 ) -b 3 x 5 , d x 7 dt = t 65 N S x 5 (1 -x 7 ) -b 6 x 7 . (4)
The second one using X 2 , x 4 , x 5 , and x 7 is given by:

                         d X 2 dt = t 15 (N H -X 2 ) N S x 5 -r 12 X 2 , d x 4 dt = (t 32 X 2 + t 37 N M x 7 ) (1 -x 4 -x 5 ) -(b 3 + r 54 ) x 4 , d x 5 dt = r 54 x 4 -b 3 x 5 , d x 7 dt = t 65 N S x 5 (1 -x 7 ) -b 6 x 7 . (5)
To compute X(t) = (X 2 (t), x 4 (t), x 5 (t), x 7 (t)) at time t using equations 5, we need to know the value of the state X(t 0 ) at a given time t 0 < t. However, this is not possible in general. To make the model useful, we have to find how to estimate the unknown value X(t). This is the main concern we will address in this paper. To achieve this goal, we will use a tool from control theory called observer. What is noticeable in this model is that the state of snails and mammalians are not available for the measurement in so far as the only available information at time t is the value of the infected human population. This means that it is possible to detect through clinical signs the number of infected people at a given time t (it is provided by health department). Then the measurable output is y(t) = X 2 (t).

A high-gain observer for a schistosomiasis model

We construct a high-gain observer for our system using the techniques developed in [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF][START_REF] Gauthier | Observability and observers for nonlinear systems[END_REF]. The high-gain observer construction involves some complicated computations [START_REF] Gauthier | Observers for nonlinear systems[END_REF][START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF][START_REF] Iggidr | Controllability, observability and stability of mathematical models, Encyclopedia of life support systems (EOLSS). Developed under the auspices of the UNESCO[END_REF]. Since systems (4) and ( 5) are equivalent, we shall use system (4) which is more adapted to the high-gain observer construction.

Let us denote by x(t) = (X 2 (t), x 3 (t), x 5 (t), x 7 (t)) the state vector of system (4), f the vector field defining the dynamics of the system (4), and h the output function, that is y(t) = h(x(t)) = X 2 (t), and

f =     t 15 (N H -X 2 ) N S x 5 -r 12 X 2 -(t 32 X 2 + t 37 N M x 7 + b 3 ) x 3 + b 3 r 54 (1 -x 3 -x 5 ) -b 3 x 5 t 65 N S x 5 (1 -x 7 ) -b 6 x 7     .
To construct a high-gain observer for ( 4), one has to perform a change of coordinates in order to write the system in a simpler form. This usually done

The Jacobian of Φ can be written:

d Φ d x =         1 0 0 0 -r 12 -N S t 15 x 5 0 N S t 15 (N H -X 2 ) 0 α 0 -N S r 54 t 15 (N H -X 2 ) α 1 0 α 2 α 3 α 4 N M N S r 54 t 15 t 37 (N H -X 2 ) x 3        
, where:

α 0 = (r 12 + N S t 15 x 5 ) 2 + N S t 15 (b 3 x 5 + r 54 (-1 + x 3 + x 5 )) α 1 = -N S t 15 (b 3 (N H -X 2 ) + N H (r 12 + r 54 + 2 N S t 15 x 5 ) -X 2 (2 r 12 + r 54 + N S t 15 x 5 )) α 2 = N S r 54 t 15 t 32 (N H -X 2 ) x 3 + N S t 15 (b 3 + 2 r 12 + r 54 + 2 N S t 15 x 5 ) + (-r 12 -N S t 15 x 5 ) (-b 3 x 5 -r 54 (-1 + x 3 + x 5 )) (r 12 + N S t 15 x 5 ) 2 + N S t 15 (b 3 x 5 + r 54 (-1 + x + x 5 )) +N S r 54 t 15 (b 3 -x 3 (b 3 + t 32 X 2 + N M t 37 x 7 )) α 3 = N S r 54 t 15 (2 b 3 (N H -X 2 ) + N H (r 12 + r 54 + t 32 X 2 + 3N S t 15 x 5 + N M t x 7 )) -N S r 54 t 15 (X 2 (3r 12 + r 54 + t 32 X 2 + 3 N S t 15 x 5 + N M t 37 x 7 )) α 4 = b 2 3 N H N S t 15 + b 3 N H N S r 12 t 15 + N H N S r 2 12 t 15 + 2 b 3 N H N S r 54 t 15 + N H N S r 12 r 54 t 15 +N H N S r 2 54 t 15 -3 N H N 2 S r 54 t 2 15 -b 2 3 N S t 15 X 2 -3 b 3 N S r 12 t 15 X 2 -3 N S r 2 12 t 15 X 2 -2 b 3 N S r 54 t 15 X 2 -3 N S r 12 r 54 t 15 X 2 -N S r 2 54 t 15 X 2 + 3 N 2 S r 54 t 2 15 + 3 N H N 2 S r 54 t 2 15 x 3 -3 N 2 S r 54 t 2 15 X 2 x 3 + 6 b 3 N H N 2 S t 2 15 x 5 + 4 N H N 2 S r 12 t 2 15 x 5 + 6 N H N 2 S r 54 t 2 15 x 5 -6 b 3 N 2 S t 2 15 X 2 x 5 -6 N 2 S r 12 t 2 15 X 2 x 5 -6 N 2 S r 54 t 2 15 X 2 x 5 + 3 N H N 3 S t 3 15 2 5 -3 N 3 S t 3 15 X 2 x 2 5 .
The determinant of d Φ d x can be expressed by:

Γ(X 2 , x 3 , x 5 ) = N M N 3 S r 2 54 t 3 15 t 37 (N H -X 2 ) 3 x 3 .
The Jacobian d Φ d x is non-singular in the region D and moreover Φ(x) is one-toone from D to Φ( D). So the map Φ is a diffeomorphism from D to Φ( D). This implies that the system (4) with the output y(t) = X 2 (t) is observable. In the

180 news coordinates defined by (z 1 , z 2 , z 3 , z 4 ) T = z = Φ(x) = (h(x), L f h(x), L 2 f h(x), L 3 f h(x)
) T , our system can be written in the canonical form as follows:

                   ż(t) =     0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0     A z(t) +     0 0 0 Ψ(z(t))     , y(t) = z 1 (t) = (1, 0, 0, 0) C z(t), (6)
where:

Ψ(z) = L 4 f h(Φ -1 (z)) = L 4 f h(x) = ψ(x).
The function ψ is smooth (it is a polynomial function of x = (X 2 , x 3 , x 5 , x 7 )) on the compact set D. Hence, it is globally Lipschitz on D. Therefore it can be extended by ψ, a Lipschitz function on R 4 which satisfies ψ(x) = ψ(x), for all x ∈ D. In the same manner, we define Ψ the Lipschitz prolongation of the function Ψ. So we have the following system (7) defined on the whole space R 4 . The restriction of [START_REF] Hammouri | Global time-varying linearization up to output injection[END_REF] to the domain D is the system (6):

           ż = A z +     0 0 0 Ψ(z)     , y = C z. (7)
According to [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF], an exponential (high-gain) observer for system [START_REF] Hammouri | Global time-varying linearization up to output injection[END_REF] is given by

ż = A z +     0 0 0 Ψ(z)     + S -1 (θ) C T (y -C z), (8)
where

S(θ) is the solution of 0 = -θ S(θ) -A T S(θ) -S(θ) A T + C T C and θ is large enough. Here, S(θ) =              1 θ - 1 θ 2 1 θ 3 - 1 θ 4 - 1 θ 2 2 θ 3 - 3 θ 4 4 θ 5 1 θ 3 - 3 θ 4 6 θ 5 - 10 θ 6 - 1 θ 4 4 θ 5 - 10 θ 6 20 θ 7              .
This observer is particularly simple since it is only a copy of system ( 7), together with a corrective term depending on θ. For the proof one can see [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF].

An observer for the original system (4) can then be given by:

               ż = A z +     0 0 0 Ψ(z)     + S -1 (θ) C T (y -C z), x(t) = Φ -1 (z(t)). (9)
Or more simply a high-gain observer for the original system (4) can be given by:

ẋ = f (x) + d Φ d x -1 x=x × S(θ) -1 C T (y -h(x)). (10)
However, the set D which is positively invariant for system (4) is not necessary positively invariant for the observer [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF], and Φ(D) is not positively invariant for the observer [START_REF] Gouzé | A simple improvement of interval asymptotic observers for biotechnological processes[END_REF]. Therefore the expressions

d Φ d x -1
x=x and Φ -1 (z(t)) are not well defined in general.

If there exists Φ a prolongation of the diffeomorphism Φ to the whole space R 4 , that is Φ is a diffeomorphism from R 4 to R 4 whose restriction to D is Φ, then it will be sufficient to replace Φ by Φ in ( 9) and ( 10) and so all the expressions will be well defined. However, for our system such a prolongation does not exist since d Φ d x is singular on the set {X 2 = N H } ∪ {x 3 = 0}. So instead of working on D, we have to consider first a set D ⊂ D given by

D = D ∩ {X 2 < N H -, x 3 > }.
The positive number has to be chosen in such away that D is positively invariant for system (4).

On

{X 2 = N H -}, we have d X 2 dt = t 15 N S x 5 -r 12 (N H -) ≤ (t 15 N S + r 12 ) -r 12 N H (since x 5 ≤ 1).
On {x 3 = }, we have

d x 3 dt = -(t 32 X 2 + t 37 N M x 7 + b 3 ) + b 3 ≥ b 3 -(t 32 N H + t 37 N M + b 3 ) (since X 2 < N H and x 7 ≤ 1).
So we take ≤ min

r 12 N H t 15 N S + r 12 , b 3 t 32 N H + t 37 N M + b 3 .
With this choice of , the map Φ is a diffeomorphism from D to Φ(D ). It is then theoretically possible to find a diffeomorphism Φ from R 4 to R 4 whose restriction to D is Φ. However, it is generally difficult to give an explicit analytical expression of the extension Φ . Therefore, the simulations will be done without extending the diffeomorphism Φ.

A " simple observer" for a schistosomiasis model

Following the high-gain implementation difficulties related in the previous section, we provide a simple observer design. In this section we consider system [START_REF] Krener | Nonlinear Observers with Linearizable Error Dynamics[END_REF]

on the set D = [0, N H ] × [0, 1] × [0, 1] × [0, 1].
We shall prove that a simple candidate observer for system (5) on the set D is given by:

                             d X2 dt = t 15 (N H -X2 ) N S x5 -r 12 X2 + L 1 (y -X2 ) d x4 dt = t 32 X2 + t 37 N M x7 (1 -x4 -x5 ) -(b 3 + r 54 ) x4 , d x5 dt = r 54 x4 -b 3 x5 , d x7 dt = t 65 N S x5 (1 -x7 ) -b 6 x7 . (11)
It is remarkable that the set

D = [0, N H ] × [0, 1] × [0, 1] × [0, 1
] is a positively invariant compact set for system [START_REF] Gauthier | Observers for nonlinear systems[END_REF].

This observer is simply a copy of system (5) plus a corrective term given by L 1 (y -X2 ). The number L 1 is a constant positive real number that will be chosen in order to ensure the convergence of the estimation error.

We will denote x(t) = (X 2 (t), x 4 (t), x 5 (t), x 7 (t)) the state vector of system (5), and x(t) = X2 (t), x4 (t), x5 (t), x7 (t) the state vector of the candidate observer [START_REF] Gauthier | Observers for nonlinear systems[END_REF]. The estimation error is e(t) = (e 2 (t), e 4 (t), e 5 (t), e 7 (t)) = x(t)x(t).

We shall make the following assumptions on the model parameters:

Assumption 4.1. N H t 32 + N M t 37 b 3 + r 54 < 1, r 54 b 3 ≤ 1 2 , N S t 65 b 6 ≤ 1
Proposition 4.1. Under the assumption 4.1, the system governed by ( 11) is an exponential observer for the system (5) for L 1 satisfying

L 1 ≥ max N H N S t 15 t 32 b 3 + r 54 -r 12 , 0 ,
i.e, there exists a positive real number λ such that for all initial conditions (x(0),

x(0)) ∈ D × D, one has |x(t) -x(t)| ≤ e -λ t |x(0) -x(0)| .
Proof. The estimation error e(t) = (e 2 (t), e 4 (t), e 5 (t), e 7 (t)) = x(t)-x(t) obeys the following differential equation:

ė = A d e + f (x) -f (x) (12)
where

A d =       -L 1 -r 12 0 0 0 0 -b 3 -r 54 0 0 0 0 -b 3 0 0 0 0 -b 6       , f (x) =     t 15 N S x 5 (N H -X 2 ) (t 32 X 2 + t 37 N M x 7 ) (1 -x 4 -x 5 ) r 54 x 4 t 65 N S x 5 (1 -x 7 )     . Let P =           1 2 L 1 + 2r 12 0 0 0 0 1 2b 3 + 2r 54 0 0 0 0 1 2b 3 0 0 0 0 1 2b 6          
and consider the following candidate Lyapunov function for the error equation ( 12):

V (e) = e T P e

We can write:

250 f (x) -f (x) = 1 0 ∂f ∂x (sx + (1 -s)x)ds e = R(e, x) e
The explicit expression of the matrix R(e, x) is given in appendix A. Therefore ė = (A d + R(e, x)) e and then the derivative of V (e) with respect to time along the solutions of the estimation error equation is V (e) = e T P A d + A T d P + P R(e, x) + R(e, x) T P e.

Simple computations give

V (e) = -e The expression of V can be written:

V (e) = -(a 2 + 1) e

b 57 = (1 -x7 ) N S t 65 b 7 ; b 25 = -X2 + N H N S t 15 L 1 + r 12 .
The derivative of V (e) can be seen as a quadratic form in e i . Applying the Gauss-Lagrange reduction to this quadratic form leads to:

-V (e) = (1 + a 2 ) (e 2 +F 2 (e 4 , e 5 )) 2 +l 1 (e 4 + F 4 (e 5 , e 7 ))

2 +l 2 (e 5 + F 5 (e 7 ))

2 +l 3 e 2 7
where l 1 , l 2 , l 3 are functions of the model parameters, and the F i are linear forms in their arguments. The exact expressions of the l i and the F i are given in Appendix A.

In Appendix A we show that if the parameters satisfy Assumption 4.1, then it is possible to choose a gain L 1 in such a way that all the l i are positive. This proves that V is negative definite which ends the proof.

Design of the interval observers for a schistosomiasis model

The logic of interval observers is to generate estimated bounds that are caused by a lack of reliability in the models of measurements [START_REF] Rapaport | Parallelotopic and practical observers for nonlinear uncertain systems[END_REF][START_REF] Meslem | Interval Observers for Uncertain Nonlinear Systems. Application to Bioreactors[END_REF]. Here, we intend to explore the possibility of designing interval observers in the case where transmission rate of the model (3) that are t 15 , t 32 , t 37 and t 65 remain partially known. Moreover, we seek to obtain an estimation even during the transients of the system that is to choose bounds that have been valid since the beginning.

Given the uncertain bounds in the model, we are looking for dynamic ones to estimate the variables. This situation resembles many epidemiological models either by the lack of confidence in model parameters calibrated from experimental data, or because of models dynamical simplicity taking into account their complexity.

We will consider in this case that such parameters are bounded by a given positive number rather than given by a single value. The design is based on two points observers which help estimate in real time the lower and upper bound of the vectors state, given the following type of uncertain nonlinear systems: (S) :

       ẋ(t) = Ax(t) + ψ(x, p), y(t) = Cx(t), x(t 0 ) ∈ [x 0 ] ∧ p ∈ [p],
where A is a matrix of dimension n × n, t

∈ [t 0 , t n T ], [p] = [p -, p + ] is a real interval vector of R np , ψ ∈ C k-1 (D × [p]), D × [p] ⊆ R n+np is an open set; n, m
and n p are the dimension of respectively the state vector x, the output vector y and the uncertain parameter vector p. We assume that measurements y m (t) are subject to an unknown but bounded, with known bound, additive error. Thus the feasible domain for measurements are given by the following boxes

Y = [y m (t) -b, y m (t) + b],
where b is the vector of maximal measurement error.

We suppose that the input is uncertain with known bounds ψ -, ψ + such that:

ψ -≤ ψ ≤ ψ + , ∀t ∈ R + .
Remark 5.1. The inequalities applied to vectors must be considered term by term.

Under this assumption, we build two asymptotic observers.

Definition 5.1. Let us consider the system (S). The pair of systems (S -, S + ) with

(S -) : ẋ-(t) = Ax -(t) + B -(ψ -(t), ψ + (t)),
x -(t 0 ) = x - 0 , (S + ) :

ẋ+ (t) = Ax + (t) + B + (ψ -(t), ψ + (t)),
x + (t 0 ) = x + 0 , where x - 0 ≤ x 0 ≤ x + 0 is an interval estimator for the system (S) if for any compact set D 0 ⊂ D, the coupled system (S, S -, S + ) verifies for any initial conditions x(t 0 ) ∈ D 0 :

∀t ≥ t 0 , x -(t) ≤ x(t) ≤ x + (t). Function B + (respectively B -) is such that: B -(ψ -(t), ψ + (t)) ≤ B(ψ(t)) ≤ B + (ψ -(t), ψ + (t)).
We assume that the imprecisely known function ψ can be bounded by a lower and upper Lipschitz function. Moreover, there exists two known functions ψ -(•) and ψ + (•) built according to the bounds of [p] and a known number M < +∞ such that: If there exists a gain K, a positive matrix, such as the non-diagonal elements of the matrix (A -KC) are non-negative and Hurwitz with the condition (13) fulfilled, we can propose the interval observer for system (S) the same spirit as for the classical Luenberger approach, provided x - 0 ≤ x 0 ≤ x + 0 :

       ∀p ∈ [p], ∀x ∈ D, ψ -(x, p) ≤ ψ(x, p) ≤ ψ + (x, p), ψ -(x, p) -ψ + (x, p) ≤ M.
   ẋ+ (t) = (A -KC) x + (t) + ψ + (x -, x + , p -, p + , u(t)) + K y + m (t), ẋ-(t) = (A -KC) x -(t) + ψ -(x -, x + , p -, p + , u(t)) + K y - m (t). ( 14
)
Remark: It is shown in [START_REF] Meslem | Interval Observers for Uncertain Nonlinear Systems. Application to Bioreactors[END_REF] that the observation error remains positive and with A -K C a Hurwitz matrix this error converges.

We consider our model whose dynamics are expressed in (4) and rewriting as follows for some convenience.

                         d X 2 dt = t 15 (N H -X 2 ) N S x 5 -r 12 X 2 , d x 4 dt = (t 32 X 2 + t 37 N M x 7 ) (1 -x 4 -x 5 ) -(b 3 + r 54 ) x 4 , d x 5 dt = r 54 x 4 -b 3 x 5 , d x 7 dt = t 65 N S x 5 (1 -x 7 ) -b 6 x 7 . (15)
We write system [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF] in the typical form of (S) where

A =     -r 12 0 0 0 0 -(b 3 + r 54 ) 0 0 0 r 54 -b 3 0 0 0 0 -b 6     and ψ =     t 15 (N H -X 2 ) N S x 5 (t 32 X 2 + t 37 N M x 7 ) (1 -x 4 -x 5 ) 0 t 65 N S x 5 (1 -x 7 )     and C = (1, 0, 0, 0).
We assume that the transmission rates t All the components ψ(•) of the vector function are Lipschitz on D with respect to the state vector for any p ∈ [p]. Then we can consider a Lipschitz extension of ψ(•) on R 4 since D is positively invariant by the system [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF], see for example [START_REF] Mcshane | Extension of range of functions[END_REF], that we also denote by ψ(•). Moreover, there exists two known functions ψ -(•) and ψ + (•) built according to the bounds of [p] such as

ψ -=     t - 15 (N H -X 2 ) N S x 5 t - 32 X 2 + t - 37 N M x 7 (1 -x 4 -x 5 ) 0 t - 65 N S x 5 (1 -x 7 )     and ψ + =     t + 15 (N H -X 2 ) N S x 5 t + 32 X 2 + t + 37 N M x 7 (1 -x 4 -x 5 ) 0 t + 65 N S x 5 (1 -x 7 )     .
It is not hard to show that there exists a real positive number M < +∞ such that the condition ( 13) fulfilled (see Appendix B). For the observer gain K = [l, 0, 0, 0] T , where l is a positive real number, the matrix

A -KC =     -r 12 -l 0 0 0 0 -b 3 -r 54 0 0 0 r 54 -b 3 0 0 0 0 -b 6     is Hurwitz. Therefore, given an interval estimate ([z - 1 , z + 1 ], [z - 2 , z + 2 ], [z - 3 , z + 3 ], [z - 4 , z + 4 ]
) T , the estimation in the basis (X 2 , x 4 , x 5 , x 7 ) is given by

320                                      ż+ 1 = -(r 12 + l) z + 1 + t + 15 N H -z + 1 z + 3 N S + l y + m (t), ż+ 2 = -(b 3 + r 54 ) z + 2 + t + 32 z + 1 + t + 37 N M z + 4 1 -z + 2 -z + 3 , ż+ 3 = r 54 z + 2 -b 3 z + 3 ż+ 4 = -b 6 z + 4 + t + 65 N S z + 3 1 -z + 4 , ż- 1 = -(r 12 + l) z - 1 + t - 15 N H -z - 1 z - 3 N S + l y - m (t), ż- 2 = -(b 3 + r 54 ) z - 2 + t - 32 z - 1 + t - 37 N M z - 4 1 -z - 2 -z - 3 , ż- 3 = r 54 z - 2 -b 3 z - 3 , ż- 4 = -b 6 z - 4 + t - 65 N S z - 3 1 -z - 4 .
(16)

Simulations

This part consists in showing out some simulation exercises to stress on the efficiency of the proposed observers of system (4) and system [START_REF] Krener | Nonlinear Observers with Linearizable Error Dynamics[END_REF]. The different snails and mammals populations are estimated via infected humans population measurements. In the first place the "simple observer" design was implemented.

325

The population sizes estimation behave satisfactory along the simulations. For the simulation of the high-gain observer we extend the function f that defines the system (4) by continuity in order to make it globally Lipschitz on R 4 in the following way: We denote f the prolongation of f to R 4 and the function π the projection on the domain D and we construct f = f • π. The extend function f has the same Lipschitz coefficient as f . The projection π is defined as follows: for x ∈ R 4 , π(x) = x where x ∈ D is such that dist(x, D) = x -x , i.e., x satisfies x -x = min u∈D u -x . The initial values of state variables in all simulations concerning the high-gain observer are X 2 (0) = 1600, x 3 (0) = 0.4,

x 5 = 0.3, x 7 = 0.5, X2 (0) = 2000, x3 (0) = 0.7, x5 (0) = 0.1, x7 (0) = 0.6.
The parameter values used in the simulation are those estimated and reported in [START_REF] Allen | Modelling and simulation of a schistosomiasis infection with biological control[END_REF]. More specifically, t 15 = 2, 23 × 10 -7 , t 37 = 5, 21 × 10 -6 , t 32 = 5, 23 × 10 -6 , t 65 = 2, 78 × 10 We choose the following N H = 5000, N S = 95000 and N M = 2500.

With these parameters, the gain chosen is θ = 1 in all the simulations. Based on Figures 1 to 4, one can see that the high-gain observer performs as expected and that the convergence of the estimates delivered by the high-gain observer is quite fast. 4) with its high-gain observer (10): X 2 (t)(red curve) and its estimate X2 (t) (blue curve) delivered by the high-gain observer [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] 4) with its high-gain observer (10): x 7 (t)(red curve) and its estimate x7 (t) (blue curve) delivered by the high-gain observer (10) when f is extended

In order to test the robustness of the high-gain observer to noisy measurements, the measurements are vitiated by an additive Gaussian noise (5% of a normal gaussian noise with mean zero and standard deviation one). In Figures 5 to 8 the states variables and their estimates obtained using noisy data of X 2 are shown. Simulations corresponding to free-noise and noisy data have been carried out with the same θ value. The chosen value is the one which provided the best compromise between fast convergence and well noise rejection. In the case of noisy measurements, the choice of relatively high values of θ are to be avoided since they amplify the noise and the obtained estimates may be unusable.

As it is known, the high-gain observer is very sensitive to data noise (one can see [START_REF] Ball | High-gain observers in the presence of measurement noise: A nonlinear gain approach[END_REF] and references therein) while our simple observer is less sensitive to data noise. Thus if the output measurements X 2 (t) are not good enough, it is appropriate to use the simple observer [START_REF] Gauthier | Observers for nonlinear systems[END_REF] (see Figures 13 to 16) because we obtain a much less good estimation with the high-gain observer [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] as it is illustrated in Figures 6 to 8. 4) and its high-gain observer [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] when the output measurements are corrupted by noise when f is extended: 4) and its high-gain observer [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] when the output measurements are corrupted by noise when f is extended: x 7 (t)(red curve) and its estimate x7 (t) (blue curve) delivered by the high-gain observer [START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] In the following illustrative figures (Figure 9 to Figure 11), we verify the effectiveness of the proposed simple observer defined by system [START_REF] Gauthier | Observers for nonlinear systems[END_REF]. given by ( 5) and its estimate X2 (t) (blue curve) delivered by the simple observer [START_REF] Gauthier | Observers for nonlinear systems[END_REF]. given by ( 5) and its estimate x3 (t) (blue curve) delivered by the simple observer [START_REF] Gauthier | Observers for nonlinear systems[END_REF]. given by ( 5) and its estimate x5 (t) (blue curve) delivered by the simple observer [START_REF] Gauthier | Observers for nonlinear systems[END_REF]. given by ( 5) and its estimate x7 (t) (blue curve) delivered by the simple observer [START_REF] Gauthier | Observers for nonlinear systems[END_REF]. The time histories of the estimates of the bounds on X = (X 2 , x 3 , x 5 , x 7 ), the states of the system (3), under the constraint that the transmission rates t 32 , t 37 , t 65 are uncertain, are given in Figure 17, Figure 18, and Figure 19 Model output is taken as y(t) = X 2 (t) and the maximal measurement error is b = ∓20% y m (∞) where y m (∞) is a nominal value. A gain vector is taken as K T = (3, 0, 0, 0). These last figures demonstrate that even with loose initial estimations on each bounds of unmeasured variables, we obtain estimates of the uncertainty intervals with a reasonable accuracy, the results keep on showing that the true states are always inside the estimated bounds and the interval estimation converge to a box whose width depends itself on those measurements error bounds.

x 3 (t)(

Conclusion

In this paper some nonlinear observers are designed for a mathematical model describing the propagation of schistosomiasis infection among humains, snails and mammalian populations. These observers allow to dynamically estimate the total population size of snails and mammalians (susceptibles as well as infected) using the size of infected humains which is the only available information. More precisely a high-gain observer, a simple nonlinear observer and an interval observer are developed. To test the effectiveness of these observers was tested by performing some numerical simulations that show that the estimates x(t) delivered by the various observers converge quite fast to the true states x(t). The convergence of the high-gain observer is the fastest one when the measurements are not corrupted by noise. On the other hand, in the case of noisy measurements the estimates delivered by the high-gain observer are not accurate anymore, while the simple observer gives good estimates with a good convergence rate. Indeed, the high-gain observer is very sensitive to data noise while our simple observer is more robust.

Both the high-gain and the simple observers assume that all the model parameters are precisely known. For the case where the parameters are only known to belong to some bounded intervals, we designed an interval observer to cope with these uncertainties in the model. This interval observer has very good convergence properties and correctly predicts the dynamic bounds for the unmeasured variables even when measurements are corrupted by noise. 4(a 2 + 1) + 1

e 7 = 1 l 2     - b 47 b 24 b 25 4 (a 2 + 1) - b 45 2 2 l 1 - b 57 2     e 7 .
Let us prove that l 1 , l 2 and l 3 are positives with assumption 4.1:

We recall the following quantities:

a 2 =

( 13 )

 13 Proposition 5.1. ([14])

- 7 ,

 7 r 12 = 4, 47 × 10 -4 , r 54 = 2, 50 × 10 -2 , b 3 = 6, 00 × 10 -2 , d 6 = 2, 74 × 10 -4 , d 3 = 8, 86 × 10 -3 , b 6 = 5.56 × 10 -4 .

Figure 1 :

 1 Figure 1: Simulation of system (4) with its high-gain observer (10): X 2 (t)(red curve) and its estimate X2 (t) (blue curve) delivered by the high-gain observer[START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] when f is extended

Figure 4 :

 4 Figure 4: Simulation of system (4) with its high-gain observer (10): x 7 (t)(red curve) and its estimate x7 (t) (blue curve) delivered by the high-gain observer[START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] when f is extended

Figure 5 :

 5 Figure 5: Simulation of system (4) and its high-gain observer[START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] when the output measurements are corrupted by noise when f is extended: X 2 (t)(red curve) and its estimate X2 (t) (blue curve) delivered by the high-gain observer[START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF]

Figure 6 :

 6 Figure 6: Simulation of system (4) and its high-gain observer[START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] when the output measurements are corrupted by noise when f is extended: x 3 (t)(red curve) and its estimate x3 (t) (blue curve) delivered by the high-gain observer[START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF]

Figure 7 :Figure 8 :

 78 Figure 7: Simulation of system (4) and its high-gain observer[START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF] when the output measurements are corrupted by noise when f is extended: x 5 (t)(red curve) and its estimate x5 (t) (blue curve) delivered by the high-gain observer[START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF]

Figure 9 :

 9 Figure9: Time evolution of the number of infected humans X 2 (t) (red curve) given by (5) and its estimate X2 (t) (blue curve) delivered by the simple observer[START_REF] Gauthier | Observers for nonlinear systems[END_REF].

Figure 10 :

 10 Figure 10: Time evolution of the number of suceptible snails x 3 (t) (red curve)given by (5) and its estimate x3 (t) (blue curve) delivered by the simple observer[START_REF] Gauthier | Observers for nonlinear systems[END_REF].

Figure 11 :

 11 Figure 11: Time evolution of the number of infected snails x 5 (t) (red curve) given by (5) and its estimate x5 (t) (blue curve) delivered by the simple observer[START_REF] Gauthier | Observers for nonlinear systems[END_REF].

Figure 12 :

 12 Figure12: Time evolution of the number of infected mammals x 7 (t) (red curve) given by (5) and its estimate x7 (t) (blue curve) delivered by the simple observer[START_REF] Gauthier | Observers for nonlinear systems[END_REF].

Figure 15 :Figure 16 :

 1516 Figure 15: Infected snails population x 5 (t) (red curve) and its estimate x5 (t) (blue curve) delivered by the simple observer (11) when the output measurements are corrupted by noise.400

Figure 17 :

 17 Figure17: Measures of infected humans population size X 2 (t) (green curve) and its lower (red) and upper (blue) bounds estimated with interval observer[START_REF] Rocha | Design of estimators for specific growth rate control in a fed-batch E. coli fermentation[END_REF] in presence of uncertainty on the measurements with b = ±20% y m (∞).

Figure 18 :

 18 Figure 18: Measures of latent snails population size x 4 (t) (green curve) and its lower (red) and upper (blue) bounds estimated with interval observer (16) in presence of uncertainty on the measurements with b = ±20% y m (∞).

Figure 19 :

 19 Figure19: Measures of infected snails population size x 5 (t) (green curve) and its lower (red) and upper (blue) bounds estimated with interval observer[START_REF] Rocha | Design of estimators for specific growth rate control in a fed-batch E. coli fermentation[END_REF] in presence of uncertainty on the measurements with b = ±20% y m (∞).

Figure 20 :

 20 Figure 20: Measures of infected mammals population size x 7 (t) (green curve) and its lower (red) and upper (blue) bounds estimated with interval observer (16) in presence of uncertainty on the measurements with b = ±20% y m (∞).

F 4 (e 5 , e 7

 457

  N S t 15 L 1 + r 12 ; a 4 = X 2 t 32 + x 7 N M t 37 b 3 + r 54 ; a 7 = x 5 N S t 65 b 6 ;

	with: a 2 = x 5 b 24 = (-1 + x4 + x5 ) t 32 b 3 + r 54	; b 45 =	r 54 b 3	-	X 2 t 32 + x 7 N M t 37 b 3 + r 54	; b 47 =	(-1 + x4 + x5 )N M t 37 b 3 + r 54	;
		2 2 -(a 4 + 1) e 2 4 -e 2 5 -(a 7 + 1) e 2

  [START_REF] Gouzé | Interval observers for uncertain biological systems[END_REF] , t 32 , t 37 and t 65 are unknown but belong to the following intervals [t - We denote p = (t 15 , t 32 , t 37 , t 65 ). The initial state variables are also unknown but within the interval vector.

	15 , t + 15 ], [t -32 , t + 32 ], [t -37 , t + 37 ], [t -65 , t + 65 ] re-
	spectively.

  when f is extended

			0.82 0.8										Estimate of x_3 by GG Estimate of x_7
													Susceptible Snails x_3 Infected Mammals
			0.8 0.75
			0.78 0.7
			0.76
			0.65
		Populations Populations	0.7 0.72 0.74 0.6 0.55
			0.68 0.5
			0.66 0.45
			0.64
			0.4
			0.62
			50 50		100 100	150	150 200	250	200	300	250 350	400	300 450	350 500	550	400
									time t time t
													Estimate of x_5 by GG
			0.22										Infected Snails x_5
			0.2
			0.18
			0.16
		Populations	0.12 0.14
			0.1
			0.08
			0.06
			0.04
			10	20	30	40	50		60		70	80	90	100	110
									time t
	350	Figure 3: Simulation of system (4) with its high-gain observer (10): x 5 (t)(red
		curve) and its estimate x5 (t) (blue curve) delivered by the high-gain observer
		(10) when f is extended

Figure 2: Simulation of system (

4

) with its high-gain observer (10): x 3 (t)(red curve) and its estimate x3 (t) (blue curve) delivered by the high-gain observer

[START_REF] Gauthier | A simple observer for nonlinear systems applications to bioreactors[END_REF]

when f is extended

  To show the effectiveness of the observer in noise compensation, simulation results are included in Figures13 to 15. We have added to the output mea-

	390
	surements X 2 (t) of the continuous system, 5% of a normal gaussian noise with 0.7
	mean zero and standard deviation one. These results show that estimation
	0.65 populations are weakly affected by the noise. Estimate of x_3 by S-obs
								Susceptible Snails x_3
			0.6
	5 500	Populations	0.5 0.55							Estimate of X_2 by S-obs
										Infected Humans X_2
	5 000		0.45
	4 500		0.4
	4 000		0.35
	3 500 Figure 14: Susceptible snails population x 3 (t) (red curve) and its estimate x3 (t) 0 200 400 100 300 50 150 250 350 time t Populations
	3 000 (blue curve) delivered by the simple observer (11) when the output measure-
	ments are corrupted by noise.
	2 500
	2 000
			0.3
									Estimate of x_5 by S-obs
	1 500		0.28						Infected Snails x_5
		0	0.26	50	100	150	200		250	300	350	400
							time t
			0.24
			0.22
	Figure 13: Infected humans population X 2 (t) (red curve) and its estimate X2 (t) 0.2
	(blue curve) delivered by the simple observer (11) when the output measure-
	ments are corrupted by noise. 0.18
			0.16
			0.14
			0.12
			0.1
			0.08
			0	50	100	150	200	250	300	350	400

  , 2.23 10 -7 ] × [0.05 10 -7 , 1.05 10 -7 ] ×[1.00 10 -7 , 2.00 10 -7 ] × [0.02 10 -7 , 1.02 10 -7 ].

		. The
	partially unknown parameters vector is
	p = [t 15 , t 32 , t 37 , t 65 ]	T ∈ [t -15 , t + 15 ] × [t -32 , t + 32 ] × [t -38 , t + 38 ] × [t -75 , t + 75 ]
	= [1.23 10 -7

  x 5 N S t 15 L 1 + r 12 ; a 4 = X 2 t 32 + x 7 N M t 37 b 3 + r 54 ; a 7 = x 5 N S t 65 b 6 ;-X2 + N H N S t 15 L 1 + r 12 . N H t 32 + N M t 37 b 3 + r 54 1 ≤ 1 + a 4 ≤ 1 + N H t 32 + N M t 37 b 3 + r 54 N H N S t 15 t 32 (L 1 + r 12 ) (b 3 + r 54 ) ≤ b 24 b 25 4(1 + a 2 ) ≤ 1 4 N H N S t 15 t 32 (L 1 + r 12 ) (b 3 + r 54 ) N H N S t 15 t 32 L 1 (b 3 + r 54 ) + r 12 (b 3 + r 54 ) N H t 32 + N M t 37 b 3 + r 54 N H N S t 15 t 32 4 (b 3 + r 54 ) (L 1 + r 12 )-N H N S t15 L 1 + r 12 + N H t 32 + N M t 37 b 3 + r 54 We have p = (t 15 , t 32 , t 37 , t 65 ) where t 15 ∈ [t - 15 , t + 15 ], t 32 ∈ [t - 32 , t + 32 ], t 37 ∈ [t - 37 , t + 37 ], t 65 ∈ [t - 65 , t + 65 ], and x = (X 2 , x 4 , x 5 , x 7) ∈ D = [0, N H ] × [0, 1] × [0, 1] × H -X 2 ) N S x 5 (t 32 X 2 + t 37 N M x 7 ) (1 -x 4 -x 5 ) 0 t 65 N S x 5 (1 -x 7 ) H -X 2 ) N S x 5 t - 32 X 2 + t - 37 N M x 7 (1 -x 4 -x 5 ) 0 t - 65 N S x 5 (1 -x 7 ) H -X 2 ) N S x 5 t + 32 X 2 + t + 37 N M x 7 (1 -x 4 -x 5 ) 0 t + 65 N S x 5 (1 -x 7 )

				2
		-0 ≤ 1 2 V 2 2 ≤ r 54 b 3 4(1 + a 2 ) ≤ -b 45 2 b 24 2 ≤ ≤ -1 4 ⇒ -1 2 r 54 b 3 1 4   V 1 b 47 3 + b 57 2   ≤ 2 4 So l 3 ≥ 1 + a 7 -1 3 N M t 37 b 3 + r 54	+ ≤ -1 2 2 3 2 -4 + 4 a 2 b 24 2 N M t 37 ≤ 0 b 3 + r 54 V 2 2 l 2 l 3 ≥ 1 + a 7 -X 4 +	Y 2 1 12	2 > 0. N S t 65 2 b 7 400 + -1107
		3 4 And finally we give here the full expression of R(e, x) = ≤ 1 + a 4 -b 24 2 4(1 + a 2 ) N H t 32 + N M t 37 ≤ 1 + b 3 + r 54 where:	R1,1 R1,2 R2,1 R2,2
	475	-R 1,1 = 1 4	-1 2 (e5 + 2x5)NSt15 2 (e4 + e5 + 2(x4 + x5 -1))t32 1 -1 2 (-e2 -2 X2)t32 -(e7 + 2x7)NM t37 0
		b 47 4 1 + a 4 -4(1 + a 2 ) 2 b 24 2 -1 2 e2 + 2 X2 -2NH NSt15 ≤ 1 3 N M t 37 b 3 + r 54 R 1,2 = 1 2 (-e2 -2 X2)t32 -(e7 + 2x7)NM t37 2	0 2 (e4 + e5 + 2(x4 + x5 -1))NM t37 -1
		-R 2,1 = 1 2 r 54 b 3		≤ 0 r54 b 24 b 25 4(1 + a 2 ) 0 0 ; R 2,2 = -1 2 b 45 ≤ -1 1 4 2 (e7 + 2x7 -2)NSt65 -1 0 2 (e5 + 2x5)NSt65 -1 2 r 54 b 3 0	+	.
		l 3 ≥ 1 + a 7 -Appendix B Existence of a finite positive real number M satisfying 1 3 N M t 37 b 3 + r 54 2 -V 2 2 l 2 . condition (13)
	480 515	b 24 = b 47 = -1 + (-1 + x4 + x5 ) t 32 b 3 + r 54 (-1 + x4 + x5 )N M t 37 ; b 45 = b 3 + r 54 ; b 57 = r 54 b 3 1 2 r 54 b 3 N H t 32 + N M t 37 b 3 + r 54 ≤ b 24 b 25 -X 2 t 32 + x 7 N M t 37 b 3 + r 54 (1 -x7 ) N S t 65 ; b 25 = ; 4(1 + a 2 ) b 45 -2 b 24 2 ≤ 4 3 Since -r 54 b 3 ≤ Y ⇒ N M t 37 b 3 + r 54 ≤ 2 r 54 b 3 , we obtain l 3 ≥ 1 + a -[0, 1]. 1 + a 4 -4(1 + a 2 ) r 54 b 3 + N H t 32 + N M t 37 If |Y | ≥ r 54 We recall the following vectors : then b 3  t 15 (N  2(b 3 + r 54 ) b 7 We have a 7 ≤ N S t 65 b 7 ; -N M t 37 b 3 + r 54 ≤ b 47 ≤ N M t 37 b 3 + r 54 ; -N S t 65 b 7 ≤ b 57 ≤ N S t 65 1 3 -36 57 b 7 ; r 54 2 b 3 -1 2 N H t 32 + N M t 37 b 3 + r 54 ≤ 1 2 b 45 ≤ 1 2 r 54 b 3 ; 0 ≤ b 25 2 4(1 + a 2 ) ≤ 1 4 N H N S t 15 > 0. L 1 + r 12 4 ≤ 4(1 + a 2 ) ≤ 4(1 + N S t 15 L 1 + r 12 ) ⇒ 1 4 1 + N S t 15 L 1 + r 12 -1 ≤ 4 (1 + a 2 ) -1 ≤ 1 4 ; b 24 b 25 4(1 + a 2 ) -b 45 2 2 1 + a 4 -b 24 2 4(1 + a 2 ) ≤ 1 3 1 2 -r 54 b 3 |V 1 | ≤ | b 24 b 25 4(1 + a 2 ) | + | -1 2 b 45 | ≤ X 4 + Y ψ =       , 2 2 Since N H t 32 + N M t 37 b 3 + r 54 < 1, we have t 32 r 54 + b 3 It follows that 0 ≤ V 2 1 ≤ X 4 2 Y + 2  t -15 (N  So l 2 ≥ 1 -1 4 N H N S t 15 L 1 + r 12 2 -1 3 r 54 2 ψ -=       b 3 < 1, we get l 1 = 1 + a 4 -b 24 2 4(1 + a 2 ) ≥ 1 -1 4 2 l 2 ≥ 41 48 > 0 and  t + 15 (N  t 32 b 3 + r 54 > 0 first case: Iff b 45 ≥ 0 |V 2 | ≤ | V 1 b 47 4 2 3 | + | b 57 2 | ≤ 2 3 N M t 37 b 3 + r 54 X 4 + Y 2 + N S t 2 b 7 ψ + =       .	2

-b 24 e 2 e 4 + b 45 e 4 e 5 -b 47 e 4 e 7 + b 57 e 5 e 7 + b 25 e 2 e 5 ,

Appendix A Proof of Proposition 4.1: the positivity of l i , i = 1, ..., 3

Here we shall prove that the derivative of V (e) is negative definite. Thanks to Gauss-Lagrange algorithm, V (e) can be written as follows:

450

V (e) = -(1 + a 2 ) (e 2 + F 2 (e 4 , e 5 )) 2 + l 1 (e 4 + F 4 (e 5 , e 7 ))

2 + l 2 (e 5 + F 5 (e 7 ))

where:

and

.

And:

We define

and

We choose L 1 ≥ 0 so that X ≤ 1/2. we get

and since b 47 ≤ 0

we get

We have the following inequalities :

as we have (N H -X 2 ) ≥ 0.

In this manner ψ -≤ ψ ≤ ψ + for all x ∈ D.

Let us prove now the last item of the condition [START_REF] Rapaport | Parallelotopic and practical observers for nonlinear uncertain systems[END_REF]. We have