S Pullanhiotan

A Chatterjee

B Jacquot

A Navin

M Rejmund

Improvement in the reconstruction method for VAMOS spectrometer

Keywords: Magnetic spectrometer, trajectory reconstruction PACS: 07.55.-w, 29.30.-h

The VAMOS spectrometer at GANIL is a large acceptance magnetic spectrometer employing a trajectory reconstruction technique to identify the reaction products. The spectrometer when coupled with the EXOGAM clover array provides a powerful tool for the spectroscopy of very weak reaction channels. Parameters are reconstructed using a numerical procedure with a polynomial relationship between the measured final coordinates and the quantities of interest. The coefficients of this polynomial function are determined from a numerical fit to the data generated from ray-tracing calculations using the ion optics code ZGOUBI. To reconstruct each parameter, a single polynomial is used with a set of coefficients that are valid across the full accepted phase space of the spectrometer. To further improve upon the reconstruction method, we developed an alternate procedure for reconstruction based on selecting an optimum set of trajectories and interpolating the data within these trajectories. For each detected event, a set of coordinates lying close to the measured trajectory are selected from the database and the target parameters of interest are determined by polynomial interpolation. The results for the reconstructed parameters using the two methods are presented.

Introduction

The VAMOS spectrometer is a large acceptance ray tracing spectrometer [1] used for selecting the reaction products from heavy ion reactions using both, the direct beams from the GANIL cyclotrons and the SPIRAL [2] beams. It consists of two large aperture quadrupoles, an E × B Wien filter and a large magnetic dipole configured to operate in different ion optical modes. Depending upon the kinematics of the reaction, the operating mode of VAMOS can be varied to optimize different requirements. In the mass dispersive mode operation, the spectrometer selects and separates the reaction products according to the momentum to charge( p q ) ratio and their unique identification is achieved via event-by-event reconstruction of ion trajectories in magnetic fields. When operated as a velocity filter at zero degrees, the spectrometer physically separates the reaction products from beam backgrounds and transports them to the focal plane. The spectrometer has an angular acceptance ranging from -125 mrad to +100 mrad in hor-izontal plane and a momentum acceptance of ±5%( at 25 msr solid angle). The main operational features of the spectrometer are listed in table 1 and detailed specifications can be found in Ref [1].

The large acceptance of the spectrometer induce significant image aberrations in the focal plane of the spectrometer making it practically difficult to resolve the mass spectrum by direct position measurement alone. It is essential to employ trajectory reconstruction to determine the particle momentum and scattering angles from the measured final coordinates. In VAMOS, event reconstruction is implemented using numerical methods that relate the final coordinates to the initial parameters by polynomial functions. The unique identification of reaction products is achieved by combining the reconstructed parameters (magnetic rigidity Bρ and path length l) with measured quantities such as time of flight (T OF ), energy (E) and energy loss (∆E) obtained from focal plane detectors. VAMOS coupled with a high efficiency γ detector array EXOGAM [3], has been used in a recent experiment to identify the γ ray transitions belonging to very neutron rich nuclei produced in deep inelastic reactions [4]. Using kinematic reconstruction of the velocity and emission angle at the target, the Recently a new approach has been attempted in VAMOS to further improve upon its reconstruction results. The reconstruction methods and its application to identification of reaction products from experimental data is presented in this paper.

Focal plane detector setup for VAMOS

The VAMOS spectrometer has a versatile detection system suitable for both light fast particles having energy around 2-20 MeV/u ( direct reactions) and heavy slow particles having energy less than 2 MeV/u ( fusion reactions). The focal plane detector system consists of a pair of two dimensional position sensitive tracking detectors, followed by multi segmented ionization chamber and an array of Silicon detectors measuring the particle coordinates(x,y), energy loss(∆E) and energy(E) respectively. Two kinds of position sensitive detectors are used in VAMOS. For light (Z < 15) ions with high energy (E > 2MeV/u), low-pressure drift chambers are used. The drift chambers have a x and y position resolution of 0.3 and 1 mm (FWHM) respectively. For heavier and slower (E < 2 MeV/u) ions, position sensitive detectors based on secondary electron detection (SED) [5] are used.

Fig. 1 shows a typical focal plane detection setup used in dispersive mode operation. A pair of identical drift chambers [6] separated by 1 m measure the horizontal (x) and transverse (y) position of the particles passing through it. The measured x, y coordinates in two planes are used to compute the angle of the trajectory and its projection onto an "image plane" located between the two detectors. A large area SED detector mounted between the drift chambers is used to obtain fast timing and trigger signals. The second drift chamber is followed by a 30 cm deep segmented anode ionization chamber which measures the differential energy loss(∆E) of the particles. An array of silicon detectors mounted behind the ion chamber is used to stop the particles, giving residual energy(E) and an additional time measurement. The charge Z identification is achieved by the standard method of combining the differential energy loss and residual energy from the detectors. The time of flight (T OF ) is recorded between the beam pulse radio frequency(RF ) and the SED as well as between the SED and silicon. In experiments where low energy heavy recoils are to be detected, the two drift chambers are replaced by a pair of identical SED providing x, y position and time of flight signals.

Event Reconstruction in VAMOS

VAMOS is a "software spectrometer" employing trajectory reconstruction technique [7,8] to determine the momentum and scattering angles of detected particles. The reconstruction method used in VAMOS is based on two independent steps: trajectories simulation and reconstruction algorithms.

Step 1: Trajectories Simulation and Tracking codes

The modeling of ion trajectories in an optical systems is generally realized by two approaches. The first approach relies on a transfer map calculation which relates the initial coordinates to the final coordinates of a trajectory. The corresponding transfer map may be represented with a Taylor series of given order computed for some predefined optical object (TRANSPORT [9], GIOS [10]). Codes using a differential algebraic method even allow the computation of the transfer map up to an arbitrary order(COSY INFIN-ITY [11]). The second approach uses multi-particle tracking codes based on the numerical integration of Newton-Lorenz equation for individual particles in a set of analytical field models (ZGOUBI [12], RAYTRACE [13]). In the case of VAMOS spectrometer, the complexity of the quadrupoles (a pole shim in Q2 induces field corrections up to fifth order) prevent the usage of the standard and predefined field model proposed in most of the codes. The precise computation of trajectories can only be done with a very accurate field model. The trajectories in VAMOS are reconstructed using the field maps obtained during the magnet design(generated with the 3D electromagnetic computation code TOSCA [START_REF]TOSCA Static Field Analysis OPERA-3D, Vector Fields[END_REF]) in combination with the ZGOUBI multi-particle tracking code. ZGOUBI offer different op-tions for the treatment of field maps including both 2D and 3D lattices of field values. The accurate description of the VAMOS spectrometer in a given mode is simulated by ray-tracing a set of 20000 trajectories covering the full acceptance and storing their initial(δ, x i , θ i , y i , φ i ) and final coordinates (x f , θ f , y f , φ f , l). For a given trajectory the parameters x and y corresponds to the two transverse distance from the reference trajectory, θ and φ refers to the inclination angle in horizontal and vertical planes. The parameter δ = (pp 0 )/p 0 defines the fractional momentum deviation from the reference momentum and l, the difference in path length between the given and the reference path. The subscripts 'i' and 'f ' here refer to the parameters corresponding to initial and final space respectively. In all calculations, the nominal image plane considered is an arbitrary plane normal to the reference trajectory and located midway between the two drift chambers. The four parameters x f , θ f , y f , φ f are determined on this image plane by projecting the position and angle measurement from two drift chamber detectors.

Step 2: The reconstruction algorithm

In a spectrometer, generally the determination of momentum(or mass) is directly obtained from the particle position, recorded with a detector in the dispersive focal plane. In a large acceptance spectrometer such as VAMOS, the non-linearity make the extraction more involved. The particle momentum is to be reconstructed from the measured four final coordinates (two position x f , y f and two angles θ f , φ f ) using mathematical transformations. The reconstruction algorithm employed in VAMOS uses a numerical procedure that calculates a polynomial relationship between the four final coordinates and the quantities to be reconstructed ( δ, l, θ i , φ i ) [15]. For each detected particle, the parameters that are to be be reconstructed are expressed as the following set of polynomial functions:

δ = F N 1 (x f , θ f , y f , φ f ) θ i = F N 2 (x f , θ f , y f , φ f ) φ i = F N 3 (x f , θ f , y f , φ f ) l = F N 4 (x f , θ f , y f , φ f ) ( 1
)
where N is the order of polynomial(7 th order in the present case). As an example, the expression for δ is written as:

δ = i+j+k+l=7 i,j,k,l=0 C ijkl x i f θ j f y k f φ l f (2)
where the coefficients C ijkl are related to the transfer map of the spectrometer. These coefficients are determined off-line, from fitting polynomials to the set of trajectories computed by ZGOUBI. Due to the mid plane symmetry in the magnet, the coefficients in Eq. 2 are null for odd values of k +l and only the remaining relevant coefficients are considered for reconstructing the δ parameter. This method of reconstruction uses a universal set of coefficients to reconstruct all trajectories across the full acceptance window of the spectrometer. It is difficult to get a clear idea, if these polynomials have sufficient precision over the full region of phase space. The merit function χ 2 gives only a feeling of the quality of the fit on an average.

In order to reduce the complexity of the problem it is possible to decompose the full acceptance of the spectrometer into N smaller bins and use a set of piece-wise lower order polynomials locally. We have tested a different algorithm for reconstructing trajectory parameters in VAMOS using a third order polynomial interpolation within a subset of trajectories selected close to the region of interest. The method of reconstruction is implemented through the following steps. a) For each detected particle, the measured data (x m , θ m , y m , φ m ) in the image plane is compared with the set of trajectories from ray-tracing calculation to select an optimum number ( ∼ 125) of nearest trajectories. The criteria for selecting the nearest trajectories is based on the minimum squared distance D to the points where

D 2 = α 1 (x m -x f ) 2 +α 2 (θ m -θ f ) 2 +α 3 (y m -y f ) 2 +α 4 (φ m -φ f ) 2 .
The choice for the number of trajectories and weight factors α i are optimized empirically by determining the best χ 2 fit for the reconstructed parameters with least computation time.

b) Only those selected nearest trajectories are used to obtain the coefficients of local third order polynomial functions for δ, θ i , φ i and l independently.

c) The required reconstructed parameters are then obtained using Eq. 1.

This method has the advantage of avoiding the global fitting procedure involving several hundreds of coefficients. Coefficients of these piece-wise polynomials can be stored, or can be recomputed for each trajectories on event by event basis.

For a complete identification of the reaction products, the following basic relations between Bρ, measured energy (E tot ) and particle characteristics( mass number M , velocity v and atomic charge state q) are used.

M q = Bρ 3.105 × β M = 2E tot 931.5 × β 2 (3)
where E tot is total energy in MeV, Bρ in T-m, and β = v/c. Fig. 2 shows a comparison of the reconstructed m/q spectrum using the two approaches: a global fitting procedure using 7 th order polynomials (a) and the reconstruction procedure using a local third order polynomials whose coefficients have been adjusted for the nearest trajectories. The data shown here correspond to quasi-elastically scattered target like recoils in the reaction 238 U + 58 N i at 1.3 GeV energy [START_REF] Sugathan | [END_REF]. Though the second method is computationally Fig. 2. The M (mass) of the recoils plotted as a function of M q for different charge states. a) using a 7 th order polynomial function with set of coefficients obtained using data across the full phase space; b) using a local third order polynomial for a selected set of trajectories. (See text). more intensive, it yields improved results as seen in Fig. 2. An improvement of the experimental M q resolution( 8% improvement in FWHM) could be achieved in the present data. The advantages of the second method are of operational interest, since it is clear that the first method (a) could be improved by selecting the coefficients more carefully or/and improving the quality of the fit with a bigger set of computed trajectories. However a systematic checking of the quality of large order polynomials over the whole acceptance of the spectrometer could be difficult. The second method, appears to be well adapted in the context of the flexible operation of the spectrometer and can improve the VAMOS resolution when required. Further improvements may be obtained by employing alternative methods for the reconstruction as proposed in the literature [START_REF] Shchepunov | [END_REF][START_REF] Berz | Advances in imaging and electron, modern maps methods[END_REF][START_REF] Thiessen | [END_REF].

Apart from determining the particle momentum, the initial angles of the particles are also determined by reconstruction methods. Fig. 3 shows the results of reconstruction applied to determine the initial angles of well defined trajectories using a sieve slit (1 mm thick collimator) placed in front of the magnetic aperture of first quadrupole(Q1) of VAMOS. The slit contained a pattern of pin holes (11 x11) defining a set of well defined trajectories with known angles (37 mrad between holes) in θ, φ. Elastically scattered 58 N i passing through the slit were detected in the focal plane detectors and their final coordinates at the image plane were determined from the drift chamber measurements. For each scattered particle passing through individual sieve-holes, its initial angles were reconstructed from the measured final positions and angles and compared with the calculated angle of the trajectory. The figure shows the reconstructed angle θ (horizontal plane) for the trajectories through the collimator. The results shows a close agreement between the reconstructed angles and the real angles of the sieveholes suggesting that the reconstruction method can be applied to determine the scattering angles of the particles to reconstruct the kinematics of the reaction.

Experimental Results

In this section we illustrate the application of trajectory reconstruction and identification method using the first approach (using 7 th order polynomials) to the identification of nuclei formed in deep inelastic transfer reactions. The experimental setup consisted of the VAMOS spectrometer coupled to the high efficiency γ-detector array EXOGAM.

The experimental data shown are taken from Ref [4] using a 238 U beam at 1.3 GeV energy bombarding an isotopically enriched 1 mg/cm 2 thick 48 Ca target. The resulting target-like recoils from deep inelastic transfer processes were detected and identified in the VAMOS spectrometer. The spectrometer was rotated at 35 • (around the calculated grazing angle) with respect to the beam direction. VAMOS was operated in the mass dispersive mode with magnetic fields set for mass M = 52 with charge state q = 19 + and energy E = 430 MeV optimizing the yield for 52 Ca like products. At the image plane the reaction products were characterized by an event by event measurement of the final coordinates(x f , θ f , y f , φ f ), energy loss ∆E, energy E and time of flight T OF For each detected particle, the magnetic rigidity (Bρ), path length (l) and initial angles (θ i , φ i ) were determined by reconstruction method and reaction products were identified using the values of M, M/q, v and energy loss (∆E)

The resulting identification spectra is shown in Fig. 4 which displays the two dimensional image of the energy loss (∆E) plotted as a function mass over charge 'M/q'. As seen in the figure, unambiguous identification of target like residues with Z ≥ 19 could be achieved by gating on charge state q =19. The data shown here are selected by a particular silicon detector covering only a small region of the focal plane. By software gates on a particular Z, the corresponding mass spectra are generated as shown in Fig. 5 which shows the mass spectrum belonging to Vanadium isotope (Z=23). All these measurements were done with a T OF resolution of ∼ 2 ns mainly limited by the time resolution of the beam pulsing. More details on the experimental results are given in Ref [4].

Conclusions

The VAMOS spectrometer is now fully operational at GANIL and uses trajectory reconstruction method for complete identification of the reaction products. Two different algorithms have been developed and compared for trajectory reconstruction. The method using polynomial interpolation using a subset of selected trajectories shows improved resolution compared to that using a single polynomial across the full phase space. Results from recent ex-periments showed that the reconstruction and identification procedures used in VAMOS are capable to unambiguously identify products. High sensitivity and selectivity of the spectrometer coupled with the EXOGAM detector array proved successful in identifying and assigning gamma ray transitions in exotic nuclei.

Fig. 1 .

 1 Fig. 1. Schematic of VAMOS focal plane detection setup used in Deep inelastic transfer reactions.

Fig. 3 .

 3 Fig. 3. The reconstructed horizontal angle θ of holes in sieve slit. The slit contained a pattern of pin holes separated by an angle of 37 mrad.

Fig. 4 .Fig. 5 .

 45 Fig.4. Identification plot displaying ∆E plotted as a function of M/q, showing an unambiguous identification of the products. The spectrum shown here is generated by gating on the charge state q = 19.

Table 1

 1 Operational features of VAMOS for dispersive mode operation

	Horizontal Acceptance -125 mrad to +100 mrad
	Vertical Acceptance	±160 mrad
	Momentum Acceptance	±5 %( at 25 msr)
	M/q resolution	∼ 0.6 %
	Maximum rigidity Bρ	1.6 T-m
	Deflection Angle	0 • -60 • (variable)
	Flight Path length	760 cm
	Linear Translation	40 to 120 cm
	Angular rotation	0 • to 60 •
	coincident γ-rays are Doppler corrected resulting in an im-
	proved resolution of the individual states in final nuclei.

Acknowledgment

The authors are thankful to the GANIL team for their excellent support in operating the facility. They would also like to thank various colleagues for help and insightful discussions. This work has been founded by the European Commission under contract 506065 (EURONS/INTAG).


Martin Gugat
email: martin.gugat@fau.de

AND Vincent Perrollaz
email: vincent.perrollaz@lmpt.univ-tours.fr

Lionel Rosier
email: lionel.rosier@mines-paristech.fr

BOUNDARY STABILIZATION OF QUASILINEAR HYPERBOLIC SYSTEMS OF BALANCE LAWS: EXPONENTIAL DECAY FOR SMALL SOURCE TERMS

Keywords: 2000 Mathematics Subject Classification. 35L50, 35L60, 76B75, 93D15 System of balance laws, shallow water equations, telegraph equation, finite-time stability, dynamical boundary conditions, exponential stability, decay rate

We investigate the long-time behavior of solutions of quasilinear hyperbolic systems with transparent boundary conditions when small source terms are incorporated in the system. Even if the finite-time stability of the system is not preserved, it is shown here that an exponential convergence towards the steady state still holds with a decay rate which is proportional to the logarithm of the amplitude of the source term. The result is stated for a system with dynamical boundary conditions in order to deal with initial data that are free of any compatibility condition.

Introduction

Solutions of certain hyperbolic systems can reach the equilibrium state in finite time. Such a property, called finite-time stability in [START_REF] Alabau-Boussouira | Finite-time stabilization of a network of strings[END_REF][START_REF] Perrollaz | Finite-time stabilization of hyperbolic systems over a bounded interval[END_REF][START_REF] Perrollaz | Finite-time stabilization of 2 × 2 hyperbolic systems on tree-shaped networks[END_REF] or super-stability in [START_REF] Shang | Super-stability and the spectrum of one-dimensional wave equations on general feedback controlled networks[END_REF], was first noticed in [START_REF] Komornik | Rapid boundary stabilization of the wave equation[END_REF][START_REF] Majda | Disappearing solutions for the dissipative wave equation[END_REF] for the (linear) wave equation. The extension of such a property to the wave equation on networks was addressed in [START_REF] Alabau-Boussouira | Finite-time stabilization of a network of strings[END_REF][START_REF] Shang | Super-stability and the spectrum of one-dimensional wave equations on general feedback controlled networks[END_REF].

Fortunately, the finite-time stability still occurs for systems of 2 × 2 quasilinear hyperbolic equations of diagonal form without source terms, as it was noticed in [START_REF] Leugering | On the modelling abd stabilization of flows in networks of open canals[END_REF] with initial data satisfying some compatibility conditions to prevent the emergence of shockwaves, and next in [START_REF] Perrollaz | Finite-time stabilization of hyperbolic systems over a bounded interval[END_REF][START_REF] Perrollaz | Finite-time stabilization of 2 × 2 hyperbolic systems on tree-shaped networks[END_REF] for arbitrary initial data by replacing homogeneous boundary conditions by some dynamical boundary conditions.

The finite-time stabilization of a quasilinear hyperbolic system with source terms seems to be very challenging. In [START_REF] Coron | Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping[END_REF], the authors proved that a 2 × 2 linear hyperbolic system with source terms can be stabilized to the origin in finite-time by using some boundary feedback laws designed with the backstepping approach.

On the other hand, the finite-time stability of a system may be lost when a small, bounded perturbation is added to the system. A famous example is provided by the telegraph equation

∂ 2 t y -∂ 2
x y + ∂ t y = 0, (t, x) ∈ (0, +∞) × (0, L), (1.1) y(t, 0) = 0, y x (t, L) = -y t (t, L).

(1.2) However, as it was noticed in [START_REF] Gugat | Boundary feedbcak stabilization of the telegraph equation: Decay rates for vanishing damping term[END_REF] for (1.1)-(1.2) or more generally for a nonlinear perturbation of the wave equation, the exponential stability of the system is preserved, with a decay rate proportional to ln( -1 ). See also [START_REF] Gugat | Neumann boundary feedback stabilization for a nonlinear wave equation: A strict H 2 -Lyapunov function[END_REF] for the exponential stabilization of the isothermal Euler equations and [START_REF] Datko | An example of the effect of time delays in boundary feedback stabilization of wave equations[END_REF] for the loss of the stability when incorporating an arbitrarily small delay in a transparent boundary condition for the wave equation. The aim of this paper is to show that the robustness property noticed in [START_REF] Gugat | Boundary feedbcak stabilization of the telegraph equation: Decay rates for vanishing damping term[END_REF] is shared by most of the finite-time stable systems. The first result in this paper shows that a linear finite-time stable system with a (small) disturbance is exponentially stable with a decay rate proportional to the logarithm of the amplitude of the perturbation. We refer the reader to [START_REF] Xu | Stabilization of string system with linear bounded feedback[END_REF]Theorem 4.2] for a sufficient condition involving the resolvent for the finite-time stability of a linear system. Theorem 1. Let A be an operator generating a strongly continuous semigroup (e tA ) t≥0 in an Hilbert space H, and let B ∈ L(H) be a bounded operator. Assume that e T A = 0 for some T > 0. Then there exist some positive numbers 0 , M, C such that for any ∈ (0, 0 ), it holds

e t(A+ B) L(H) ≤ M inf(1,
e -(C ln -1 )t ) ∀t ≥ 0.

(1.

3)

The (simple) proof of Theorem 1 is given in Appendix. It rests on the observation that the solution of a Cauchy problem can be obtained as a fixed-point of a map, derived from Duhamel formula, in a weighted space. The weight is related to the decay rate. It is unclear whether such an approach could be extended to quasilinear systems.

It should be noticed that the estimate in (1.3) is essentially sharp. Indeed, for the system

       ∂ t u + c∂ x u = v, x ∈ (0, L), t > 0, ∂ t v -c∂ x v = u,
x ∈ (0, L), t > 0, u(t, 0) = v(t, L) = 0, t > 0, u(0, x) = u 0 (x), v(0, x) = v 0 (x),

x ∈ (0, L),

we shall prove that the decay rate is roughly speaking bounded from below by (c/L) ln -1 .

Theorem 2. Let c > 0, let A(u, v) := (-c∂ x u, c∂ x v) be the operator with domain

D(A) := {(u, v) ∈ [H 1 (0, L)] 2 ; u(0) = v(L) = 0} ⊂ H := [L 2 (0, L)] 2 ,
and let B(u, v) := (v, u). Then for any κ > c/L, there exist some numbers K, 0 > 0 such that for any ∈ (0, 0 ), it holds e t(A+ B) L(H) ≥ Ke -(κ ln -1 )t ∀t ≥ 0.

(1.5)

The main aim of the paper is to investigate the application of transparent boundary conditions, or more generally of dynamical boundary conditions as in [START_REF] Perrollaz | Finite-time stabilization of hyperbolic systems over a bounded interval[END_REF][START_REF] Perrollaz | Finite-time stabilization of 2 × 2 hyperbolic systems on tree-shaped networks[END_REF], to 2×2 quasilinear hyperbolic systems in diagonal form with small source terms

∂ t u + λ(u, v)∂ x u = f (u, v), (1.6) ∂ t v -µ(u, v)∂ x v = g(u, v) (1.7)
where λ(u, v) > c, µ(u, v) > c for some constant c > 0, and 0 < 1. Our results are stated when f and g do not depend on , but there are still valid when f and g depend on but are bounded in W 2,∞ (0, L) for 0 < 1. We shall prove that for small enough and for initial data sufficiently close to a steady state of (1.6)-(1.7), the solution of (1.6)-(1.7) with dynamical boundary conditions converge exponentially to the steady state with a decay rate proportional to ln( -1 ).

Our result can be applied to e.g. the Saint-Venant system with sources terms (see e.g. [START_REF] Bouchut | A new model of Saint Venant and Savage-Hutter type for gravity driven shallow water flows[END_REF][START_REF] Gugat | Global boundary controllability of the Saint-Venant system for sloped canals with friction[END_REF]), which is commonly used as a model for the water flow regulation in a canal with a slowly varying topography and some damping:

∂ t H + ∂ x (HV ) = 0, (1.8)
∂ t V + ∂ x ( V 2 2 + gH) = -g∂ x b - c f 2 V 2 H .
(1.9)

In (1.8)-(1.9), t is time, x is the space variable, H = H(t, x) is the water depth, V = V (t, x) is the flow velocity in the direction parallel to the bottom, g is the gravitation constant, c f is the friction coefficient, and z = b(x) is the equation of the bottom.

Using the Riemann invariants

u := V + 2 gH, v := V -2 gH,
we easily see that system (1.8)-(1.9) can be rewritten as

∂ t u + λ(u, v)∂ x u = F (u, v), (1.10) ∂ t v -µ(u, v)∂ x v = F (u, v) (1.11)
where

λ := V + gH = 1 4 (3u + v), µ := -V + gH = - 1 4 (u + 3v), F (u, v) := -g∂ x b -2c f g u + v u -v 2 .
Thus, our results can be applied when ∂ x b L ∞ (0,L) + c f 1. The above model is valid when the function ∂ x b takes "small values". A more accurate model, the so-called Savage-Hutter system (see [START_REF] Bouchut | A new model of Saint Venant and Savage-Hutter type for gravity driven shallow water flows[END_REF]), reads

∂ t H + ∂ X (HV ) = 0, (1.12) ∂ t V + ∂ X ( V 2 2 + g cos(θ)H) = -g sin(θ)• (1.13)
Here, X denotes a curvilinear coordinate along the bottom, θ = θ(X) is the angle of the bottom tangent with some fixed horizontal axis, H = H(t, X) is the width of fluid in the normal direction at a point X of the bottom, and V (t, X) is the tangential velocity.

Introducing the Riemann invariants

u := V + 2 g cos(θ)H, v := V -2 g cos(θ)H,
we derive again a system of the form (1.10)-(1.11), with x = X and

λ := V + g cos(θ)H = 1 4 (3u + v), µ := -V + g cos(θ)H = - 1 4 (u + 3v), F (u, v) := -g sin(θ).
Again, our results can be applied when |θ| 1. The paper is outlined as follows. The main result (Theorem 3) is stated in Section 2. Its proof is displayed in Section 3. It is divided in three parts. The first one is a rephrasing of the problem. The second part establishes the existence and uniqueness of global solutions for small initial data using Schauder's fixed-point theorem. The last one introduces some Lyapunov functions with exponential weights needed to prove the exponential convergence towards the steady state. The paper ends with an Appendix which contains the proofs of Theorem 1 and of Theorem 2 and which provides some background about linear transport equations.

Stationary states and Main result

We are interested in the following system of balance laws

∂ t u + λ(u , v )∂ x u = f (u , v ) ∂ t v -µ(u , v )∂ x v = g(u , v ) t > 0, x ∈ (0, L). (2.1) System (2.1) is supplemented with the initial conditions u (0, x) = u 0 (x), v (0, x) = v 0 (x) (2.2)
and the boundary conditions

u (t, 0) = y l (t), v (t, L) = y r (t). (2.3)
In (2.3), the boundary data y l and y r are defined as the solutions of the initial value problem (2.4) where γ ∈ (0, 1) and K ∈ (0, +∞) are any given numbers.

             dy l dt = -K y l - ū |y l -ū| γ , dy r dt = -K y r - v |y r -v| γ , y l (0) = u 0 (0), y r (0) = v 0 (1),
Here and in what follows, we assume that the functions λ, µ, f and g are of class C 2 . We fix a pair (ū, v) ∈ R 2 such that

λ(ū, v) > 0 µ(ū, v) > 0, (2.5)
and we introduce two real numbers c > 0 and R > 0 such that

∀(u, v) ∈ R 2 , ||(u, v) -(ū, v)|| ≤ 2R ⇒ inf(λ(u, v), µ(u, v)) ≥ c. (2.6)
Let us construct for each 0 < 1 a pair (ū , v ) of stationary states for (2.1).

We denote by X the Banach space of continuous functions from [0, L] to R 2 equipped with the uniform norm (u, v) = sup x∈[0,L] ||(u(x), v(x))||, and by Ω the subset of X consisting of the functions taking their values in the open ball B((ū, v), R); that is,

Ω = {(u, v) ∈ X; (u(x) -ū, v(x) -v) < R ∀x ∈ [0, L]}.
We can now define a functional F : (u, v,

) ∈ Ω × R → F (u, v, ) = (U, V ) ∈ X by U (x) := u(x) -ū - x 0 f (u(y), v(y)) λ(u(y), v(y))
dy, (2.7)

V (x) := v(x) -v - L x g(u(y), v(y)) µ(u(y), v(y)) dy. (2.8) It is easy to see that (i) the set Ω is open; (ii) F (ū, v, 0) = 0; (iii) the functional F is of class C 1 on Ω; (iv) the differential with respect to (u, v) of F at (ū, v, 0) is given by D (u,v) F (ū, v, 0) = Id X .
We infer from the implicit function theorem the local existence and uniqueness of a map

∈ [-0 , 0 ] → (ū , v ) ∈ Ω which is of class C 1 and which satisfies ∀ ∈ [-0 , 0 ], F (ū , v ) = 0.
(2.9)

It follows then from (2.7)-(2.8) that the functions ū and v are of class C 1 in [0, L] and that they satisfy

         λ(ū , v )∂ x ū = f (ū , v ), ∀x ∈ [0, L], -µ(ū , v )∂ x v = g(ū , v ), ∀x ∈ [0, L], ū (0) = ū, v (L) = v.
(2.10)

We are now in a position to state the main result in this paper.

Theorem 3. There exist 0 > 0 and δ > 0 such that for any ∈ [0, 0 ], any (γ, K) ∈ (0, 1) × (0, +∞), and any initial data (u 0 , v 0 ) ∈ Lip([0, L]) 2 , with

||u 0 -ū|| W 1,∞ (0,L) ≤ δ and ||v 0 -v|| W 1,∞ (0,L) ≤ δ, (2.11)
the system (2.1)-(2.4) has a unique solution

(u , v ) ∈ Lip([0, +∞) × [0, L]) satisfying (2.1)
almost everywhere, and it holds

∀t ≥ 0, ||(u -ū , v -v )(t)|| 2 L 2 (0,L) ≤ M inf(1, e -C t 1+κ ) ||(u 0 -ū , v 0 -ū )|| 2 L ∞ (0,L) , (2.12
)

∀t ≥ 0, ||(u -ū , v -v )(t)|| L ∞ (0,L) ≤ M inf(1, e -1 3 C t 1+κ 3 ) ||(u 0 -ū , v 0 -ū )|| 2 3 L ∞ (0,L) , (2.13)
where κ := (cδ γ )/(KLγ), M = M (δ) > 0 and C = C (δ) ∼ -c L ln( ) as → 0 + . We shall use some Lyapunov function to prove (2.12) (see Section 3.3).

Remark 1. We notice that for any fixed

t > (1 + κ) L c = L c + δ γ Kγ , (2.14)
we have that e -C t 1+κ = e (ln 1 )[1+κ-t( c L +o( 1))]

and therefore that the r.h.s. of (2.12) tends to 0 as → 0 + . This result, combined with the boundedness of {(u , v )} 0< < 0 in Lip([0, T ] × [0, L]) for all T > 0, yields again the finite-time stability around (ū, v) of the limit system (without source term)

               ∂ t u + λ(u, v)∂ x u = 0, t > 0, x ∈ (0, L), ∂ t v -µ(u, v)∂ x v = 0, t > 0, x ∈ (0, L), u(t, 0) = y l (t), t > 0, v(t, 1) = y r (t), t > 0, u(0, x) = u 0 (x), x ∈ (0, L), v(0, x) = v 0 (x)
x ∈ (0, L).

that was established in [START_REF] Perrollaz | Finite-time stabilization of 2 × 2 hyperbolic systems on tree-shaped networks[END_REF] with an extinction time very similar to (2.14) (δ γ in the r.h.s. of (2.14) being replaced by

||(u 0 -ū, v 0 -v)|| γ L ∞ (0,L) ).
3. Proof of Theorem 3

3.1. Reduction of the problem. We aim to show that if (2.11) holds with δ small enough, then the solutions of (2.1)-(2.4) tend to (ū , v ) as t → +∞.

To this end, we introduce the functions

U := u -ū , V := v -v .
The original system (2.1) can be written

         ∂ t U + λ(ū + U, v + V )∂ x U = [f (ū + U, v + V ) -f (ū , v )] -∂ x ū [λ(ū + U, v + V ) -λ(ū , v )], ∂ t V -µ(ū + U, v + V )∂ x V = [g(ū + U, v + V ) -g(ū , v )] +∂ x v [µ(ū + U, v + V ) -µ(ū , v )]. (3.1)
The boundary conditions become

U (t, 0) = y l (t) -ū, V (t, L) = y r (t) -v, (3.2)
where y l and y r still solve (2.4), and the initial condition read

U (0, x) = U 0 (x) := u 0 (x) -ū (x), V (0, x) = V 0 (x) := v 0 (x) -v (x). (3.3) Note that y l (t) = ū (resp. y r (t) = v) for t ≥ (γK) -1 |U 0 (0)| γ (resp. t ≥ (γK) -1 |V 0 (1)| γ ). Therefore, if U 0 L ∞ (0,L) ≤ δ and V 0 L ∞ (0,L) ≤ δ, then we have U (t, 0) = V (t, 1) = 0 ∀t ≥ (γK) -1 δ γ . (3.4)
Thanks to the definition of Ω, we notice that for any

∈ [-0 , 0 ], it holds ∀x ∈ [0, L], ||(ū (x), v (x)) -(ū, v)|| < R.
For any given r > 0 and any f : R 2 → R, let

f r := sup{ |f (u, v)|; (u, v) ∈ B((ū, v); r)}.
In the following, for any function f = f (u, v), the quantity f will always denote f R where R is as in (2.6). (Note that R does not depend on .) For h :

[0, L] → R, we denote h [0,L] = sup{|h(x)|; x ∈ [0, L]}.
Using (2.10), we see that

||∂ x ū || [0,L] ≤ ||f || c , ||∂ x v || [0,L] ≤ ||g|| c •
Differentiating in (2.10), we infer that

||∂ 2 xx ū || [0,L] ≤ 2 ||f ||.||∂ 1 f || + ||g||.||∂ 2 f || c 2 + ||∂ 1 λ||.||f || 2 c 3 + ||∂ 2 λ||.||f ||.||g|| c 3 , ||∂ 2 xx v || [0,L] ≤ 2 ||f ||.||∂ 1 g|| + ||g||.||∂ 2 g|| c 2 + ||∂ 2 µ||.||g|| 2 c 3 + ||∂ 1 µ||.||f ||.||g|| c 3 .
We now simplify the notations by setting

                             α(x, a, b) := λ(ū (x) + a, v (x) + b), β(x, a, b) := µ(ū (x) + a, v (x) + b), F (x, a, b) := [f (ū (x) + a, v (x) + b) -f (ū (x), v (x))] -∂ x ū (x)[λ(ū (x) + a, v (x) + b) -λ(ū (x), v (x))], G(x, a, b) := [g(ū (x) + a, v (x) + b) -g(ū (x), v (x))] +∂ x v (x)[µ(ū (x) + a, v (x) + b) -µ(ū (x), v (x))], Y l (t) := y l (t) -ū, Y r (t) := y r (t) -v. (3.5)
The system (3.1) can be written as

∂ t U + α(x, U, V )∂ x U = F (x, U, V ), ∂ t V -β(x, U, V )∂ x V = G(x, U, V ). (3.6)
It is supplemented with the initial conditions

U (0, x) = U 0 (x), V (0, x) = V 0 (x), (3.7)
and the boundary conditions

U (t, 0) = Y l (t), V (t, L) = Y r (t), (3.8)
where the functions Y l and Y r solve the system

           dY l dt = -K Y l |Y l | γ , dY r dt = -K Y r |Y r | γ , Y l (0) = u 0 (0) -ū, Y r (0) = v 0 (1) -v. (3.9)
The functions α, β, F and G enjoy the following properties:

∀(a, b) ∈ B((0, 0), R), inf(α(x, a, b), β(x, a, b)) ≥ c; (3.10) ∀x ∈ [0, L], F (x, 0, 0) = 0 = G(x, 0, 0); (3.11) ∀x ∈ [0, L], ∀(a, b) ∈ B((0, 0), R) |F (x, a, b)| ||(a, b)|| ≤ ||Df || B((ū,v),2R) + ||f || B((ū,v),R) c ||Dλ|| B((ū,v),2R) ; |G(x, a, b)| ||(a, b)|| ≤ ||Dg|| B((ū,v),2R) + ||g|| B((ū,v),R) c ||Dµ|| B((ū,v),2R) ; ∀x ∈ [0, L], ∀(a, b) ∈ B((0, 0), R)                                        |∂ x F (x, a, b)| ≤ 2 2 c 2 1 + ||λ|| c c||f ||.||∂ 1 f || + c||g||.||∂ 2 f || + ||∂ 1 λ||.||f || 2 + ||∂ 2 λ||.||f ||.||g|| ; |∂ x G(x, a, b)| ≤ 2 2 c 2 1 + ||µ|| c c||∂ 1 g||.||f || + c||∂ 2 g||.||g|| + ||g|| 2 .||∂ 2 µ|| + ||g||.||f ||.||∂ 1 µ|| ; |∂ a F (x, a, b)| ≤ ||∂ 1 f || 2R + ||∂ 1 λ|| 2R .||f || c ; |∂ b F (x, a, b)| ≤ ||∂ 2 f || 2R + ||∂ 2 λ|| 2R .||f || c ; |∂ a G(x, a, b)| ≤ ||∂ 1 g|| 2R + ||∂ 1 µ|| 2R .||g|| c ; |∂ b G(x, a, b)| ≤ ||∂ 2 g|| 2R + ||∂ 2 µ|| 2R .||g|| c .
(3.12)

3.2. The fixed-point argument. We show in this section the existence and uniqueness of the solution of (3.1)-(3.3) for small enough. Introduce some positive constants I 0 , I 1 , P , and M such that

||U 0 || L ∞ (0,L) ≤ I 0 , ||V 0 || L ∞ (0,L) ≤ I 0 , ||U 0 || L ∞ (0,L) ≤ I 1 , ||V 0 || L ∞ (0,L) ≤ I 1 , sup(||∂ x F || ∞ , ||∂ a F || ∞ , ||∂ b F || ∞ , ||∂ x G|| ∞ , ||∂ a G|| ∞ , ||∂ b G|| ∞ ) ≤ P, and sup(||α|| ∞ , ||∂ x α|| ∞ , ||∂ a α|| ∞ , ||∂ b α|| ∞ , ||β|| ∞ , ||∂ x β|| ∞ , ||∂ a β|| ∞ , ||∂ b β|| ∞ ) ≤ M.
We notice that we can assume that (i) the constants I 0 , I 1 are small by picking the numbers δ and 0 in Theorem 3 small enough; (ii) the constant P is small by adjusting thanks to (3.12); (iii) the constant M is bounded for ∈ [0, 0 ]. To display our fixed-point argument based on Schauder fixed-point theorem, we have to define (i) a class of functions containing the desired solution; (ii) a map whose fixed-point is the desired solution of (3.1)- (3.3). This is done in the following two definitions. Definition 1. Given two positive constants A and B, we define D as the set of functions

(U, V ) ∈ Lip(R + × [0, L]) 2 such that ∀(t, x) ∈ R + × [0, L], |U (t, x)| ≤ A, |V (t, x)| ≤ A; (3.13) ∀(t, x, y) ∈ R + × [0, L] 2 , |U (t, x) -U (t, y)| ≤ B|x -y|, |V (t, x) -V (t, y)| ≤ B|x -y|; (3.14)
and

∀(t, s, y) ∈ R + × R + × [0, L], |U (t, x) -U (s, x)| ≤ B|t -s|, |V (t, x) -V (s, x)| ≤ B|t -s|. (3.15)
Definition 2. Given (U, V ) ∈ D, we introduce the unique solution ( Ũ , Ṽ ) (thanks to Theorem 7 in Appendix) of the system

                   ∂ t Ũ + α(x, U, V )∂ x Ũ = F (x, U, V ), ∂ t Ṽ -β(x, U, V )∂ x Ṽ = G(x, U, V ), Ũ (t, 0) = Y l (t), Ṽ (t, L) = Y r (t), Ũ (0, x) = U 0 (x), Ṽ (0, x) = V 0 (x).
(3.16)

We set F(U, V ) := ( Ũ , Ṽ ).

The following result is classical (see e.g. [?]).

Proposition 1. The function d : [C 0 ([0, +∞[×[0, L]) 2 ] 2 → R + defined by d((U, V ), (u, v)) := +∞ n=1 1 2 n ||U -u|| L ∞ ([0,n]×[0,L]) + ||V -v|| L ∞ ([0,n]×[0,L]) 1 + ||U -u|| L ∞ ([0,n]×[0,L]) + ||V -v|| L ∞ ([0,n]×[0,L]) is a distance and [C 0 ([0, +∞[×[0, L])] 2 endowed with this distance is a Fréchet Space. On the other hand, D is a convex compact subset of [C 0 ([0, +∞[×[0, L])] 2 .
We first have to show that F maps D into itself for an appropriate choice of the constants. Proposition 2. For any A, B, c, K, L, M and γ in (0, +∞), there exist some numbers I 0 , I 1 and P in (0, +∞) such that F(D) ⊂ D.

Proof. Using Corollary 1 in Appendix, we obtain the following estimates:

|| Ũ || L ∞ ≤ 2L c P A + I 0 ; (3.17) || Ṽ || L ∞ ≤ 2L c P A + I 0 ; (3.18) ||∂ x Ũ || L ∞ ≤ L c exp( L c M (1 + 2B)) P (1 + 2B) + sup(2P A + KI 1-γ 0 , cI 1 ) L ; (3.19) ||∂ x Ṽ || L ∞ ≤ L c exp( L c M (1 + 2B)) P (1 + 2B) + sup(2P A + KI 1-γ 0 , cI 1 ) L ; (3.20) ||∂ t Ũ || L ∞ ≤ M L c exp( L c M (1 + 2B)) P (1 + 2B) + sup(2P A + KI 1-γ 0 , cI 1 ) L + 2P A; (3.21) ||∂ t Ṽ || L ∞ ≤ M L c exp( L c M (1 + 2B)) P (1 + 2B) + sup(2P A + KI 1-γ 0 , cI 1 ) L + 2P A. (3.22)
It is thus sufficient to show that for I 0 and I 1 small enough, one can choose P sufficiently small so that there exist some numbers A > 0 and B > 0 with

2L c P A + I 0 ≤ A, (3.23) L c exp( L c M (1 + 2B)) P (1 + 2B) + sup(2P A + KI 1-γ 0 , cI 1 ) L ≤ B, (3.24) M L c exp( L c M (1 + 2B)) P (1 + 2B) + sup(2P A + KI 1-γ 0 , cI 1 ) L + 2P A ≤ B. (3.25)
For given A, B, M, I 0 and I 1 in (0, +∞), it is thus sufficient to impose that as P → 0 + , the limit of the leftsided terms in (3.23)-(3.25) are strictly less than those of the corresponding rightsided terms; that is,

I 0 < A, (3.26)
L c exp( L c M (1 + 2B)) sup(KI 1-γ 0 , cI 1 ) L < B, (3.27)
M L c exp( L c M (1 + 2B)) sup(KI 1-γ 0 , cI 1 ) L < B. (3.28)
Now, it is clear that given A, B, c, K, L, M and γ in (0, +∞), the conditions (3.26)-(3.28) are fulfilled by choosing I 0 and I 1 small enough.

Next, we have to show the continuity of the map F.

Proposition 3. The map F : D → D is continuous. Proof. Let {(U n , V n )} ⊂ D be any sequence such that d((U n , V n ), (U, V )) → n→+∞ 0
for some (U, V ) ∈ D. This is equivalent to saying that (U n ) n≥0 tends to U (and (V n ) n≥0 tends to V ) uniformly on all the sets [0, T ] × [0, L], T > 0. This yields that α(., U n , V n ) → α(., U, V ) and β(., U n , V n ) → β(., U, V ) uniformly on all the sets [0, T ] × [0, L], T > 0. Using Theorem 6 in Appendix, we deduce that F(U n , V n ) converges uniformly on all the sets [0, T ] × [0, L], T > 0, to a pair ( Ũ , Ṽ ) ∈ D which is the unique solution of the system

                   ∂ t Ũ + α(x, U, V )∂ x Ũ = F (x, U, V ), ∂ t Ṽ -β(x, U, V )∂ x Ṽ = G(x, U, V ), Ũ (t, 0) = Y l (t), Ṽ (t, L) = Y r (t), Ũ (0, x) = U 0 (x), Ṽ (0, x) = V 0 (x).
(3.29)

By Definition 2, we conclude that ( Ũ , Ṽ ) = F(U, V ). It follows that F is continuous.

We are now in a position to prove that F has a fixed-point in D. Proof. Clearly, it is sufficient to prove the uniqueness of the solution (U, V ) to the system (3.6)-

(3.8). Assume given U 0 , V 0 ∈ Lip([0, L]) and two pairs (U i , V i ) ∈ Lip([0, T ] × [0, L]) 2 (i = 1, 2) of solutions of (3.6)-(3.8); that is, we have for i = 1, 2        ∂ t U i + α(x, U i , V i )∂ x U i = F (x, U i , V i ), ∂ t V i -β(x, U i , V i )∂ x V i = G(x, U i , V i ), U i (t, 0) = Y l (t), V i (t, L) = Y r (t), U i (0, x) = U 0 (x), V i (0, x) = V 0 (x). Set Û := U 1 -U 2 , V := V 1 -V 2 , and α := α(x, U 1 , V 1 ) -α(x, U 2 , V 2 ), β := β(x, U 1 , V 1 ) -β(x, U 2 , V 2 ), F := F (x, U 1 , V 1 ) -F (x, U 2 , V 2 ), Ĝ := G(x, U 1 , V 1 ) -G(x, U 2 , V 2 ).
Then we see that Û , V ∈ Lip([0, T ] × [0, L]) and that the pair ( Û , V ) solves the system

∂ t Û + α(x, U 1 , V 1 )∂ x Û + α∂ x U 2 = F , (3.30) ∂ t V -β(x, U 1 , V 1 )∂ x V -β∂ x V 2 = Ĝ, (3.31) Û (t, 0) = V (t, L) = 0, (3.32) Û (0, x) = V (0, x) = 0.
(3.33)

After multiplying (3.30) (resp. (3.31)) by Û (resp. by V ), and integrating over (0, t) × (0, L), we obtain

1 2 ( Û 2 L 2 (0,L) + V 2 L 2 (0,L) ) = t 0 L 0 [ F Û + Ĝ V ]dxdτ - t 0 L 0 [α(x, U 1 , V 1 ) Û ∂ x Û -β(x, U 1 , V 1 ) V ∂ x V ]dxdτ - t 0 L 0 [α Û ∂ x U 2 -β V ∂ x V 2 ]dxdτ =: I 1 -I 2 -I 3 . (3.34) Then |I 1 | ≤ t 0 L 0 [P (| Û | + | V |)| Û | + P (| Û | + | V |)| V |]dxdτ ≤ 2P t 0 L 0 | Û | 2 + | V | 2 dxdτ.
Similarly,

|I 3 | ≤ M B t 0 L 0 (| Û | + | V |) 2 dxdτ ≤ 2M B t 0 L 0 (| Û | 2 + | V | 2 )dxdτ.
On the other hand, integrating by part in I 2 yields

-I 2 = t 0 L 0 (∂ x [α(x, U 1 , V 1 )]| Û | 2 -∂ x [β(x, U 1 , V 1 )]| V | 2 )dxdτ - t 0 α(x, U 1 , V 1 )| Û | 2 -β(x, U 1 , V 1 )| V | 2 L 0 dτ ≤ t 0 L 0 (∂ x [α(x, U 1 , V 1 )]| Û | 2 -∂ x [β(x, U 1 , V 1 )]| V | 2 )dxdτ ≤ M (1 + 2B) t 0 L 0 (| Û | 2 + | V | 2 )dxdτ
where we used the definitions of α, β, M, B, (2.6) and (3.32). Gathering together all the estimates, we arrive to

L 0 [| Û (t, x)| 2 + | V (t, x)| 2 ]dx ≤ [M (1 + 4B) + 2P ] t 0 L 0 (| Û | 2 + | V | 2 )dxdτ, ∀t ≥ 0.
An application of Gronwall's lemma yields Û ≡ 0 and V ≡ 0.

Lyapunov Functions and Decay

Rates. Let (U, V, Y r , Y l ) denote the solution of system (3.6)-(3.9).

To simplify the computations, we denote by C a positive constant such that for all x ∈ [0, L] and all (a, b) ∈ B((0, 0), R), we have

|F (x, a, b)| ≤ C 2 (|a| + |b|), |G(x, a, b)| ≤ C 2 (|a| + |b|)
and such that for all (t, x) ∈ R + × [0, L], we have sup α(x, U (t, x), V (t, x)), β(x, U (t, x), V (t, x)) ≤ C and

| d dx α(x, U (t, x), V (t, x)) | ≤ C, | d dx β(x, U (t, x), V (t, x)) | ≤ C. (3.35)
We first introduce a Lyapunov function to investigate the long-time behavior of (U, V ).

Definition 3. Given any θ > 0, let the function L θ be defined by

L θ (a, b) := L 0 [a 2 (x)e -θx + b 2 (x)e -θ(L-x) ]dx ∀(a, b) ∈ [L 2 (0, L)] 2 .
Then we have the following result.

Proposition 6. For almost every t ∈ R + , it holds

d dt L θ (U (t, .), V (t, .)) ≤ ( C 3 + e θL 2 + C -cθ)L θ (U (t, .), V (t, .)) + C(Y 2 r (t) + Y 2 l (t)).
Proof. Since U and V are uniformly bounded and Lipschitz continuous, and hence differentiable almost everywhere by Rademacher theorem, the derivative of L θ (U (t, .), V (t, .)) exists almost everywhere and it is obtained by differentiating the integrand with respect to t. Thus we obtain

d dt L θ (U (t, .), V (t, .)) = L 0 2F (x, U (t, x), V (t, x))U (t, x)e -θx dx + L 0 2G(x, U (t, x), V (t, x))V (t, x)e -θ(L-x) dx - L 0 α(x, U (t, x), V (t, x))∂ x (U 2 )(t, x)e -θx dx + L 0 β(x, U (t, x), V (t, x))∂ x (V 2 )(t, x)e -θ(L-x) dx =: I 1 + I 2 + I 3 + I 4 .
Let us begin with I 3 .

I 3 = -[e -θx α(x, U (t, x), V (t, x))U 2 (t, x)] L 0 -θ L 0 α(x, U (t, x), V (t, x))U 2 (t, x)e -θx dx + L 0 d dx (α(x, U (t, x), V (t, x)))U 2 (t, x)e -θx dx.
As far I 4 is concerned, we have that

I 4 = [e -θ(L-x) β(x, U (t, x), V (t, x))V 2 (t, x)] L 0 -θ L 0 β(x, U (t, x), V (t, x))V 2 (t, x)e -θ(L-x) dx - L 0 d dx (β(x, U (t, x), V (t, x)))V 2 (t,
x)e -θ(L-x) dx.

Using (3.10) and (3.35), we infer that

I 3 + I 4 ≤ α(0, U (t, 0), V (t, 0))U 2 (t, 0) + β(L, U (t, L), V (t, L))V 2 (t, L) +( C -θc)L θ (U (t, .), V (t, .)). (3.36)
Let us evaluate the remaining terms I 1 and I 2 . We have that

I 1 + I 2 ≤ C L 0 [U 2 (t, x)e -θx + V 2 (t, x)e -θ(L-x) + U (t, x)V (t, x) e -θ(L-x) + e -θx ]dx ≤ C L θ (U (t, .), V (t, .)) + C L 0 U 2 (t, x) + V 2 (t, x) 2 (e -θ(L-x) + e -θx )dx ≤ C 3 + e θL 2 L θ (U (t, .), V (t, .)), (3.37)
where we used twice in the last inequality the following estimate ∀x ∈ [0, L], e -θ(L-x) + e -θx ≤ (e θL + 1)e -θx .

The proof of Proposition 6 is completed by gathering together (3.36) and (3.37).

Next, we introduce another Lyapunov function for the dynamics of the boundary conditions. Then the following result holds.

Proposition 7. We have for all t ≥ 0

d dt Lθ (Y l (t), Y r (t)) ≤ -cθ Lθ (Y l (t), Y r (t)) -C(Y 2 l (t) + Y 2 r (t)). Proof. From ∂ a Lθ (a, b) = C K(γ + 2) ((γ + 2)a|a| γ + θc K a|a| 2γ )e θ c Kγ |a| γ , ∂ b Lθ (a, b) = C K(γ + 2) ((γ + 2)b|b| γ + θc K b|b| 2γ )e θ c Kγ |b| γ , and
Ẏg (t) = -K Y l (t) |Y l (t)| γ , Ẏd (t) = -K Y r (t) |Y r (t)| γ , we infer that d dt Lθ (Y l (t), Y r (t)) = Ẏg (t)∂ a Lθ (Y l (t), Y r (t)) + Ẏd (t)∂ b Lθ (Y l (t), Y r (t)) = -CY 2 l (t)e θ c Kγ |Y l (t)| γ -CY 2 r (t)e θ c Kγ |Yr(t)| γ -cθ Lθ (Y l (t), Y r (t)) ≤ -CY 2 l (t) -CY 2 r (t) -cθ Lθ (Y l (t), Y r (t)).
We are in a position to define the Lyapunov function for the full state (U, V, Y l , Y r ). Let the functional L be defined by

∀t ≥ 0, L(U (t, .), V (t, .), Y l (t), Y r (t)) := L θ (U (t, .), V (t, .)) + Lθ (Y l (t), Y r (t)).
Then the following holds. On the other hand, the best decay rate obtained by taking the minimum of the parenthesis over θ reads The estimate (3.38) will give the exponential decay of the L 2 norm of (U, V ). To derive an exponential decay for the L ∞ norm of (U, V ), we need the following result.

-C := min θ∈R C 3 + e θL 2 + C -cθ = - c L ln( -1 )+[ c L + C - c L ln 2c CL ]+ 3 C 2 ∼ →0 + - c L ln( -1 ). ( 3
Lemma 1. Let u ∈ Lip([0, L]). Then ||u|| L ∞ (0,L) ≤ 2 √ L ||u|| L 2 (0,L) or ||u|| 3 L ∞ (0,L) ≤ 8||u|| 2 L 2 (0,L) ||∂ x u|| L ∞ (0,L) . (3.40)
If, in addition u(0) = 0, then

||u|| 3 L ∞ (0,L) ≤ 16||u|| 2 L 2 (0,L) ||∂ x u|| L ∞ (0,L) . (3.41) Proof. If ∂ x u L ∞ (0,L) = 0, then the function u is constant and the first inequality in (3.40) is obvious. Assume that ∂ x u L ∞ (0,L) > 0 and pick any x ∈ [0, L] such that u(x) = ||u|| L ∞ (0,L) .
Then we have

∀y ∈ [0, L], |u(y)| ≥ |u(x)| -||∂ x u|| L ∞ (0,L) |y -x|, Letting D := |u(x)| 2||∂ x u|| L ∞ (0,L) = ||u|| L ∞ (0,L) 2||∂ x u|| L ∞ (0,L) ,
we see that

∀y ∈ [0, L], |x -y| ≤ D ⇒ |u(y)| ≥ |u(x)| 2 = ||u|| L ∞ (0,L) 2 • Now if I := [sup(0, x -D), inf(L, x + D)], we have ||u|| 2 L 2 (0,L) ≥ I |u(y)| 2 dy ≥ |I| ||u|| 2 L ∞ (0,L) 4 ,
and hence

||u|| 2 L ∞ (0,L) ≤ 4||u|| 2 L 2 (0,L) |I| •
From the definition of I (using x ∈ [0, L]), we have that |I| ≥ inf(L, D), so that

||u|| 2 L ∞ (0,L) ≤ 4 sup 1 D , 1 L ||u|| 2 L 2 (0,L) •
Using the definition of D, we obtain (3.40). Assume in addition that u(0) = 0. We claim that

u 2 L 2 (0,L) ≤ 2L 3 2 u L 2 (0,L) ∂ x u L ∞ (0,L) . (3.42)
Indeed, we have by Cauchy-Schwarz inequality

u(x) 2 -0 = x 0 2u(y)∂ x u(y) dy ≤ 2 u L 2 (0,L) ∂ x u L 2 (0,L) ≤ 2 √ L u L 2 (0,L) ∂ x u L ∞ (0,L)
and an integration w.r. Let us complete the proof of Theorem 3. Picking any θ ∈ (0, +∞), we infer from (3.38) that (2.12) holds for t ∈ [0, 1 + (1 + κ)L/c] for some constant M > 0. Increasing M if needed, we see that (2.13) holds as well for t ∈ [0, 1 + (1 + κ)L/c], by using Lemma 1. Assume now that t > 1 + (1 + κ)L/c. We pick θ := L -1 ln[(2c)/( C L)], so that, for 0 < < 0 , we have

h(θ) = -C ∼ -c L ln -1 . It follows that L 0 [U 2 (t, x) + V 2 (t, x)]dx ≤ L 0 [e θ(L-x) U 2 (t, x) + e θx V 2 (t, x)]dx ≤ e θL L(t) ≤ e θL e -C t L(0) ≤ 2c C L e -C t L 0 [U 2 0 (x) + V 2 0 (x)] dx + Lθ (Y l (0), Y r (0)) .
But we have

|Y l (0)| = |u 0 (0) -ū| = |u 0 (0) -u (0)| ≤ δ and similarly |Y r (0)| ≤ δ, so that Lθ (Y l (0), Y r (0)) ≤ C K(γ + 2) δ γ exp(θ cδ γ Kγ ) (u 0 -ū , v 0 -v ) 2 L ∞ (0,L) ≤ C K(γ + 2) δ γ exp( cδ γ KLγ ln( 2c C L )) (u 0 -ū , v 0 -v ) 2 L ∞ (0,L) ≤ Const κ (u 0 -ū , v 0 -v ) 2 L ∞ (0,L) • It follows that (u -u , v -v )(t) 2 L 2 (0,L) ≤ Const e -C t 1+κ (u 0 -ū , v 0 -v ) 2 L ∞ (0,L) •
Thus the estimate (2.12) holds for t > 1 + (1 + κ)L/c. Finally, (2.13) follows from (2.12) and (3.40). The proof of Theorem 3 is complete.

Conclusion

In this paper, we have shown the exponential decay of the L 2 -norm and of the L ∞ -norm of solutions of a quasilinear hyperbolic system of balance laws with sufficiently small source terms and appropriately chosen boundary controls. In fact, with vanishing source terms the decay rates become arbitrarily large, and thus in the limit, the case of finite-time stability is recovered. Since we work with solutions that are at each time t ≥ 0 in the space W 1,∞ (0, L) with respect to the space variable, the question arises whether also the L ∞ -norm of the space derivative of the solution decays exponentially. This question will be the subject of future investigations.

Appendix

Proof of Theorem 1.

We can find some constants N ≥ 0 and ω ∈ R such that e tA L(H) ≤ N e ωt for all t ≥ 0. On the other hand, it is well known that A + B generates also a strongly continuous semigroup in H. For any u 0 ∈ H, the (mild) solution of the Cauchy problem

∂ t u = Au + Bu, u(0) = u 0 (5.1)
is denoted by u(t) = e t(A+ B) u 0 , and it is the solution in C 0 (R + , H) of the Duhamel integral equation

u(t) = e tA u 0 + t 0 e (t-s)A [ Bu(s)]ds, ∀t ≥ 0. (5.2)
Pick any u 0 ∈ H. For given λ ∈ (0, +∞), introduce the Banach space

E = {u ∈ C 0 (R + , H); u E := sup t≥0 e λt u(t) H < +∞}.
For any u ∈ E, we define a function v : R + → H by

v(t) = e tA u 0 + t 0 e (t-s)A [ Bu(s)]ds, ∀t ≥ 0. ( 5.3)
Let Γ(u) = v. We aim to show that for ∈ (0, 0 ) (with 0 small enough) and λ > 0 conveniently chosen, Γ has a unique fixed-point in E. First, we note that Γ maps the space E into itself. Indeed, for t ∈ [0, T ] we have

e λt v(t) ≤ e λT N sup(1, e ωT ) u 0 + 0 T B L(H) u E < ∞,
and for t ≥ T , we have (using the fact that e sA w = 0 for all s ≥ T and all w ∈ H)

e λt v(t) ≤ t t-T
e λt e (t-s)A Bu(s) ds

≤ N sup(1, e ωT ) B L(H) u E t t-T e λ(t-s) ds ≤ N sup(1, e ωT ) B L(H) u E e λT -1 λ < ∞.
Let us show now that Γ contracts in E. Pick any u 1 , u 2 ∈ E, and let us denote v 1 = Γ(u 1 ), v 2 = Γ(u 2 ). Then we have for all t ≥ 0

e λt v 1 (t) -v 2 (t) ≤ t sup(0,t-T )
e λt e (t-s)A B(u

1 (s) -u 2 (s)) ds ≤ N sup(1, e ωT ) B L(H) e λT -1 λ u 1 -u 2 E
so that Γ contracts for any given λ > 0 if 1. Then by the contraction mapping theorem, Γ has a unique fixed-point in E which is nothing but the mild solution of (5.1). Now, pick λ > 0 of the form λ = C ln -1 . Then Γ contracts in

E if k := N sup(1, e ωT ) B L(H) e λT λ < 1, which becomes k = 1-CT N sup(1, e ωT ) B L(H) C ln -1 < 1.
This holds if C < 1/T and ∈ (0, 0 ) with 0 > 0 small enough. On the other hand, we have that

u(t) -e tA u 0 E ≤ (1 -k) -1 e tA u 0 -Γ(e tA u 0 ) E ≤ (1 -k) -1 k u 0 ant that e tA u 0 E ≤ N sup(1, e ωT )e λT ≤ Const u 0 • It follows that u E ≤ Const u 0 / , i.e. u(t) ≤ Const e -(C ln -1 )t u 0 / for all t ≥ 0. On the other hand, sup t≥0 u(t) ≤ sup t≥0 u(t) -e tA u 0 + sup t≥0 e tA u 0 ≤ [(1 -k) -1 k + N sup(1, e ωT )] u 0 .
The proof of Theorem 1 is complete.

Proof of Theorem 2.

To prove (1.5), it is sufficient to find an eigenvalue

λ of A + B of the form λ ∼ - c L ln -1 .
Thus, we investigate the following spectral problem

λu + c∂ x u = v, (5.4) λv -c∂ x v = u,
(5.5) u(0) = v(L) = 0.

(5.6)

Differentiating with respect to x in (5.4), replacing ∂ x v by its expression in (5.5) and next v by its expression in (5.4), we arrive to the following ODE for u

∂ 2 x u = λ 2 -2 c 2 u. (5.7)
Similarly, we see that v solves the ODE

∂ 2 x v = λ 2 -2 c 2 v. (5.8)
Let α ∈ C be such that

α 2 = λ 2 -2 c 2 • (5.9)
Then u and v can be written as (5.14) Now, using (5.9), we see that (5.12) and (5.13) follow respectively from (5.14) and (5.11), so that (5.11)-(5.14) is equivalent to

u(x) = Ae αx + Be -αx , u(x) = Ce αx + De -αx (5.
C = λ + cα A, B = λ + cα D. Setting µ = λ + cα , (5.15)
we wee that the solutions of (5.4)-(5.5) are the functions of the form

u(x) = Ae αx + µDe -αx , v(x) = µAe αx + De -αx ,
where A, D ∈ C are arbitrary. The boundary conditions (5.6) yield the following constraints for A and D: A + µD = 0, µAe αL + De -αL = 0.

The above system admits some nontrivial solutions if and only if the determinant of the associated matrix is null:

e -αL -µ 2 e αL = 0.

(5.16)

Gathering together (5.9), (5.15) and (5.16), we conclude that λ is an eigenvalue of the operator A + B if there is some α ∈ C such that

λ 2 -c 2 α 2 = 2 , λ+cα 2 = e -2αL
. ,

or equivalently if there are some numbers α ∈ C and s ∈ {-1, 1} such that

λ + cα • λ -cα = 1, (5.17)
λ + cα = se -αL .

(5.18)

This yields (using

s -1 = s) λ -cα = se αL . (5.19)
Eliminating λ in (5.18)-(5.19)), we conclude that system (5.17 To complete the proof, it is sufficient to pick s = -1 and to limit ourselves to the solutions α ∈ R + of the equation sinh(Lα)

Lα = c L • (5.22)
It is easily seen that the map x → sinh x/x is (strictly) increasing on (0, +∞) and onto (1, +∞), so that for any ∈ (0, c/L) there is a unique α = α( ) ∈ (0, +∞) satisfying (5.22). Moreover, the map → α is decreasing and α → +∞ as → 0 + . Finally, e Lα /(2Lα) ∼ c/(L ) yields by taking the logarithm Lα ∼ ln -1 . This gives by using again (5.22) We infer from the implicit function theorem (proceeding as in Section 2) that φ is of class C 1 on (R + ) 2 × R. On the other hand, differentiating in (5.24) yields

Lα = ln -1 + O(ln(ln -1 )) so that λ = -cα + o( ) = - c L ln -1 + O(ln(ln -1 )) > -κ ln -1 if κ > c/L
∀(s, t, x) ∈ (R + ) 2 × R, ∂ 2 φ(s, t, x) = -a(t, φ(t, t, x)) + s t ∂ 2 a(r, φ(r, t, x))∂ 2 φ(r, t, x)dr.
Noticing that the last equation can be viewed as a linear ODE and using the fact that a(t, φ(t, t, x)) = a(t, x), we obtain the desired formula for ∂ 2 φ. The other one for ∂ 3 φ is proven in the same way.

Remark 2. Is should be noted that ∀(s, t, x) ∈ (R + ) 2 × R, ∂ t φ(s, t, x) + a(t, x)∂ x φ(s, t, x) = 0. Definition 6. Given a positive real number L, we introduce for any (t, x) ∈ R + × [0, L] the sets

F -:= {r ∈ [0, t]; ∀s ∈ [r, t], φ(s, t, x) ∈ [0, L]}, F + := {r ∈ [t, +∞); ∀s ∈ [t, r], φ(s, t, x) ∈ [0, L]}.
We also set e(t, x) := inf F -.

Proposition 9. Assume that a is a function of class C 1 and that there exists c > 0 such that

∀(t, x) ∈ [0, +∞) × [0, L], a(t, x) ≥ c. ( 5
G := {(t, x) ∈ (0, T ) × (0, L); x < φ(t, 0, 0)}, I := {(t, x) ∈ (0, T ) × (0, L); x > φ(t, 0, 0)}.
Furthermore, it holds

∀(t, x) ∈ G,    ∂ t e(t, x) = -a(t,x)
a(e(t,x),0) exp -t e(t,x) ∂ 2 a(r, φ(r, t, x))dr , ∂ x e(t, x) = -1 a(e(t,x),0) exp -t e(t,x) ∂ 2 a(r, φ(r, t, x))dr , while ∀(t, x) ∈ I, e(t, x) = 0. The set G coincide with the set of pairs (t, x) ∈ (0, T ) × (0, L) such that e(t, x) > 0.

Remark 3. The following holds

∀(t, x) ∈ (0, T ) × (0, L), x = φ(t, 0, 0) ⇒ ∂ t e(t, x) + a(t, x)∂ x e(t, x) = 0.
Since the propagation speed a(t, x) is greater than c, it should be clear that

∀(t, x) ∈ R + × [0, L], 0 ≤ t -e(t,
    ∂ t y + a(t, x)∂ x y = b(t, x), y(t, 0) = y l (t), y(0, x) = y 0 (x) (t, x) ∈ (0, T ) × (0, L). (5.27)
We say that a function y ∈ Lip([0, T ] × [0, L]) is a strong solution of (5.27) if the first equation in (5.27) holds almost everywhere and if the second and third equations in (5.27) hold everywhere. We shall need also to introduce the concept of weak solution of (5.27), following [START_REF] Perrollaz | Initial boundary value problem and asymptotic stabilization of the Camassa-Holm equation on an interval[END_REF]. ). Then y is a strong solution of (5.27) if and only if y is a weak solution of (5.27).

Definition 7. We say that y ∈ C 0 ([0, T ] × [0, L]) is a weak solution of (5.27) if for any function ψ ∈ C 1 ([0, T ] × [0, L]) satisfying ∀t ∈ [0, T ], ψ(t, L) = 0, ∀x ∈ [0, L], ψ(T, x) = 0, it holds T 0 L 0 [y(t, x) ∂ t ψ(t, x) + a(t, x)∂ x ψ(t, x) + ∂ x a(t, x)ψ(t, x) + b(t, x)ψ(t, x)]dxdt
2. If a, b, y l and y 0 are Lipschitz continuous functions, then there is at most one weak solution of (5.27).

Proof. The first assertion follows from classical arguments. The second one is proven in the Appendix of [START_REF] Perrollaz | Initial boundary value problem and asymptotic stabilization of the Camassa-Holm equation on an interval[END_REF].

Theorem 5. Let a, b, y l and y 0 be functions of class C 1 such that y l (0) = y 0 (0).

(5.29)

Then the system (5.27) admits exactly one (strong or weak) solution, and it is given explicitly by the formula

∀(t, x) ∈ [0, T ] × [0, L], y(t, x) = y l (e(t, x)) + t e(t,x) b(r, φ(r, t, x))dr if x < φ(t, 0, 0), y 0 (φ(0, t, x)) + t 0 b(r, φ(r, t, x))dr otherwise.
(5.30)

Proof. Using Propositions 8 and 9, it is straightforward to check that the function y given by formula (5.30) is in Lip([0, T ]×[0, L]) and is of class C 1 , except possibly on the curve x = φ(t, 0, 0) (where it is likely merely continuous), and that it is a strong solution of (5.27). On the other hand, the uniqueness of a weak solution of (5.27) follows from Proposition 10.

Corollary 1. The solution y of (5.27) satisfies the estimates: Proof. See the Appendix in [START_REF] Perrollaz | Initial boundary value problem and asymptotic stabilization of the Camassa-Holm equation on an interval[END_REF]. (t, x) ∈ (0, T ) × (0, L).

||y|| ∞ ≤ L c
(5.35)

Proof. Since y n is the unique solution of (5.34), it follows from Corollary 1 that the sequence (y n ) n≥0 is bounded in Lip([0, T ] × [0, L]). From Ascoli-Arzela theorem, we can extract a subsequence (y n k ) k≥0 which converges uniformly towards a function y. On the other hand, since (∂ x a n ) n≥0 is bounded in L ∞ ((0, T ) × (0, L)) and since the only possible weak- * limit of a convergent subsequence is ∂ x a, we have by weak- * compactness that ∂ x a n → ∂ x a weakly- * in L ∞ . But this is enough to pass to the limit as k → +∞ in the weak formulation that is, y is a weak solution of the transport equation (5.35). As the weak solution of (5.35) is unique by Proposition 10, we infer that there is only one possible limit for any convergent subsequence of (y n ) n≥0 , so that the whole sequence (y n ) n≥0 converges uniformly towards y.

Theorem 7. Theorem 5 and Corollary 1 are still valid when we assume merely that the functions a, b, y l and y 0 are Lipschitz continuous.

Proposition 4 .Proposition 5 .

 45 The map F has a fixed-point in D, and therefore the system (2.1)-(2.3) has at least one solution.Proof. As the map F :D → D is continuous and D is a convex, compact subset of the Fréchet space [C 0 ([0, +∞[×[0, L])]2 , it has at least one fixed-point by Tihonov fixed-point theorem (see e.g. [20, Corollary 9.6].) The pair (U, V ) solves (3.6)-(3.8), while the pair (u , v ) := (U + ū , V + v ) solves (2.1)-(2.3). We are now concerned with the uniqueness of the solution of the system (2.1)-(2.3). There is at most one solution to (2.1)-(2.3) in the class Lip([0, T ] × [0, L]) 2 .

Definition 4 .

 4 Given any θ > 0, let the function Lθ be defined by∀(a, b) ∈ R 2 ,Lθ (a, b)

Theorem 4 .

 4 The functional satisfies d dt L ≤ ( C 3 + e θL 2 + C -cθ)L for a.e. t ≥ 0. (3.38)

2 +

 2 .39)Proof. The estimate (3.38) follows at once from Propositions 6 and 7. On the other hand, the minimum of the function h(θ) := C 3+e θL C -cθ is achieved when e θL = (2c)/( C L), which yields (3.39).

  t. x ∈ (0, L) yields at once (3.42). Next, if the first estimate in (3.40) holds, taking the cube of each term and using (3.42), we arrive to (3.41). If the second estimate in (3.40) holds, then (3.41) is obvious.

  10) for some constants A, B, C, D ∈ C. Plugging the expressions of u and v in (5.4)-(5.5) yields the system λ(Ae αx + Be -αx ) + c(Aαe αx -Bαe -αx ) = (Ce αx + De -αx ), λ(Ce αx + De -αx ) -c(Cαe αx -Dαe -αx ) = (Ae αx + Be -αx ), which is satisfied if (λ + cα)A = C, (5.11) (λ -cα)B = D, (5.12) (λ -cα)C = A, (5.13) (λ + cα)D = B.

  )-(5.18) is equivalent to the system λ + cα = se -αL ,

. 25 )

 25 Then the function e is of class C 1 on the two open sets

+ T 0 a 0 y 0

 000 (t, 0)y l (t)ψ(t, 0)dt + L (x)ψ(0, x)dx = 0. (5.28) Then we have the following result. Proposition 10. 1. Let y ∈ Lip([0, T ] × [0, L]

Proposition 11 .

 11 Let (t, x) ∈ [0, T ]×[0, L] and let {a n } ⊂ C 0 ([0, T ]×[0, L])∩L ∞ (0, T, Lip([0, 1])) be a sequence such that ||a n || L ∞ (0,T,Lip([0,L])) is bounded and ||a n -a|| C 0 ([0,T ]×[0,L]) → 0 as n → +∞,and let {(t n , x n )} ⊂ [0, T ] × [0, L] be a sequence such that (t n , x n ) → (t, x). Then e n (t n , x n ) → e(t, x).

Theorem 6 .∂∂

 6 Let y n , a n , b n , y 0,n and y l,n be Lipschitz continuous functions such that (i) for any n ≥ 0, the function y n is a strong solution of t y n + a n (t,x)∂ x y n = b n (t, x), y n (t, 0) = y l,n (t), y n (0, x) = y 0,n (x); (t, x) ∈ (0, T ) × (0, L) (5.34) (ii) the sequence (a n ) n≥0 is bounded in Lip([0, T ] × [0, L]) and it converges uniformly on [0, T ] × [0, L] towards a function a; (iii) the sequence (b n ) n≥0 is bounded in Lip([0, T ] × [0, L])and it converges uniformly on [0, T ] × [0, L] towards a function b; (iv) the sequence (y l,n ) n≥0 is bounded in Lip([0, T ]) and it converges uniformly on [0, T ] towards a function y l ; (v) the sequence (y 0,n ) n≥0 is bounded in Lip([0, T ]) and it converges uniformly on [0, T ] towards a function y 0 .Then there exists a Lipschitz continuous function y on [0, T ] × [0, L] such thaty n → y in C 0 ([0, T ] × [0, L]),and y is the unique solution of t y + a(t, x)∂ x y = b(t, x), y(t, 0) = y l (t), y(0, x) = y 0 (x).

+ T 0 a 0 y 0

 000 (t, 0)y l (t)ψ(t, 0)dt + L (x)ψ(0, x)dx = 0,(5.37)

  Transport equation. This part follows closely the Appendix of[START_REF] Perrollaz | Initial boundary value problem and asymptotic stabilization of the Camassa-Holm equation on an interval[END_REF].Definition 5. Let a = a(t, x) be a Lipschitz continuous function on R + × R such that Assume that a = a(t, x) satisfies (5.23) and is of class C 1 on R + × R. Then the function φ a is of class C 1 on (R + ) 2 × R, and we have

	5.3. sup (t,x)∈R + ×R	|a(t, x)| 1 + |x|	< +∞.	(5.23)
	For any pair (t, x) ∈ R + × R, we denote by s → φ a (s, t, x) the maximal solution to the following
	Cauchy problem
			θ(s) = a(s, θ(s)),
			θ(t) = x,
	which is defined on R + thanks to (5.23).
	Proposition 8. ∀(s, t, x) ∈ (R + ) 2 × R,	∂ 2 φ(s, t, x) = -a(t, x) exp ∂ 3 φ(s, t, x) = exp s t ∂ 2 a(r, φ(r, t, x))dr . s t ∂ 2 a(r, φ(r, t, x))dr ,
	Proof. Using the integral form of the differential equation, we obtain
					s
	∀(s, t, x) ∈ (R + ) 2 × R,	φ(s, t, x) = x +	a(r, φ(r, t, x))dr.	(5.24)
					t
	and 0 <	1.

  An application of the implicit function theorem gives that the function e is of class C 1 . Taking partial derivatives in (5.26) yields

							x) ≤	L c	•
	Proof. Using (5.25), we see that for (t, x) ∈ G, e(t, x) is the only solution (at least locally) of
	the equation
			φ(e(t, x), t, x) = 0.	(5.26)
	∂ t e(t, x) = -	∂ 2 φ(e(t, x), t, x) ∂ 1 φ(e(t, x), t, x)	= -	-a(t, x) exp	e(t,x) t a(e(t, x), 0) ∂ 2 a(r, φ(r, t, x))dr
	and
	∂ x e(t, x) = -	∂ 3 φ(e(t, x), t, x) ∂ 1 φ(e(t, x), t, x)	= -	exp	e(t,x) t a(e(t, x), 0) ∂ 2 a(r, φ(r, t, x))dr	•
	Let us now consider the system
	

  ||b|| ∞ + sup (||y 0 || ∞ , ||y l || ∞ ) , ||∂ 2 a|| ∞ sup ||b|| ∞ + ||y l || ∞ + L||∂ 2 b|| ∞ , c||y 0 || ∞ + L||∂ 2 b|| ∞ , (5.32) ||∂ 2 a|| ∞ sup ||b|| ∞ + ||y l || ∞ + L||∂ 2 b|| ∞ , c||y 0 || ∞ + L||∂ 2 b|| ∞ +||b|| ∞ .

					(5.31)
	||∂ x y|| ∞ ≤	1 c	exp	L c
	and
	||∂ t y|| ∞ ≤	||a|| ∞ c	exp	L c
					(5.33)
	Proof. Straightforward from (5.30).

  [y n k (t, x) (∂ t ψ(t, x) + a n k (t, x)∂ x ψ(t, x) + ∂ x a n k (t, x)ψ(t, x)) + b n k (t, x)ψ(t, x)]dxdt [y(t, x) (∂ t ψ(t, x) + a(t, x)∂ x ψ(t, x) + ∂ x a(t, x)ψ(t, x)) + b(t, x)ψ(t, x)]dxdt

	where ψ is any function as in Definition 7. We arrive to
	T	L
	0	0
	T	L
	0	0
		T	L
		+	a n k (t, 0)y l,n k (t)ψ(t, 0)dt +	y 0,n k (x)ψ(0, x)dx = 0, (5.36)
		0	0

Acknowledgments

VP and LR were partially supported by the ANR project Finite4SoS (ANR-15-CE23-0007).


Erik Vanhoutte

Franck Ruffier

Julien Serres
email: julien.serres@univ-amu.fr

A honeybee's navigational toolkit on Board a Bio-inspired Micro Flying Robot

In this paper, a 395-gram micro flying robot equipped with an insect-inspired visual system is presented. The robot's visual system was designed to make it avoid both ground and lateral obstacles, using optic flow-based regulation principles. The quadrotor is an open-hardware X4-MaG drone with an active gimbal system based on a pair of serial servo motors, which stabilizes 8 retinas dedicated to optic flow measurements in the 25 • /s to 1000 • /s range, each of which comprises 12 auto-adaptive pixels working in a 7-decade lighting range. The X4-MaG drone is tested in front of a slanted wall, its quasipanoramic bio-inspired eye on board is able to estimate the angle of incidence in the 0 • to 50 • range with an error of less than 2.5 • when flying. These experimental results are a first step towards a fully autonomous micro quadrotor requiring no magnetometers, which will be able in the near future to "sense and avoid" obstacles in GPS-denied environments.

INTRODUCTION

Mimicking the flight of a tiny honeybee is still an arduous task [START_REF] Srinivasan | Honeybees as a model for the study of visually guided flight, navigation, and biologically inspired robotics[END_REF][START_REF] Serres | Optic flow-based collisionfree strategies: From insects to robots[END_REF]. Since insect-sized micro air vehicles are increasingly becoming reality, however, [START_REF] Duhamel | Altitude feedback control of a flapping-wing microrobot using an on-board biologically inspired optical flow sensor[END_REF][START_REF] Kushleyev | Towards a swarm of agile micro quadrotors[END_REF][START_REF] Ma | Controlled flight of a biologically inspired, insect-scale robot[END_REF][START_REF] Dunkley | Visual-inertial navigation for a camera-equipped 25g nano-quadrotor[END_REF] it will be necessary to endow them in the future with sensors and flight control devices enabling them to perform all kinds of aerial maneuvers, including ground and obstacle avoidance, terrainfollowing and landing, in the same way as honeybees. This is a difficult challenge in the field of robotics, although drones can now be miniaturized thanks to recent advances in embedded electronics.

Stereovision-based strategies on board micro flying robots have recently provided means of exploring and avoiding obstacles indoors, but only under slow flight conditions [START_REF] Mcguire | Efficient optical flow and stereo vision for velocity estimation and obstacle avoidance on an autonomous pocket drone[END_REF][START_REF] Centeye | Centeye nano unmanned aircraft system with 360-degree stereo vision[END_REF]. Micro LiDAR ( Light Detection And Ranging devices) are also available for micro air vehicle applications (e.g., Centeye Inc. in the USA or LeddarTech Inc. in Canada), but these devices require a scanning system and have a high energy consumption. The optic flow (OF) is also a relevant visual cue which can be used for quickly sensing the configuration of the environment using either micro cameras or custom-made OF sensors.

Unfortunately, micro cameras sample tens of thousands of pixels with a refresh rate of only about 30 frames per second [START_REF] De Croon | Optic-flow based slope estimation for autonomous landing[END_REF][START_REF] Sabo | An inexpensive flying robot design for embodied robotics research[END_REF], which is 10 times lower than the temporal resolution of the bees eye (∼ 300 Hz, see [START_REF] Srinivasan | Honeybees as a model for the study of visually guided flight, navigation, and biologically inspired robotics[END_REF]), and they also require much more computational resources than custommade OF sensors. In addition, micro cameras are blinded by the changes in the light which occur when moving from one room to another, from indoor to outdoor conditions, or when encountering strong sunny to shadow contrasts. However, custom-made OF sensors lighten both the weight and the CPU load of micro flying robots , and enable them to make quasi panoramic OF measurements [START_REF] Keshavan | Autonomous vision-based navigation of a quadrotor in corridor-like environments[END_REF][START_REF] Moore | Autonomous mav guidance with a lightweight omnidirectional vision sensor[END_REF][START_REF] Briod | A method for ego-motion estimation in micro-hovering platforms flying in very cluttered environments[END_REF] as well as to work under various unpredictable lighting conditions. Because of their lack of visual stabilization, micro flying robots have to fly at low speeds despite the use of OF regulation principles [START_REF] Sabo | Bio-Inspired Visual Navigation for a Quadcopter using Optic Flow[END_REF][START_REF] Sabo | An inexpensive flying robot design for embodied robotics research[END_REF]. Up to now, no OF sensing strategies based on a stabilization system preventing the rotational effects have yet been implemented on board a micro flying robot.

In [START_REF] De Croon | Optic-flow based slope estimation for autonomous landing[END_REF], the slope of the ground was assessed on board a quadrotor. In [START_REF] Expert | Flying over uneven moving terrain based on optic-flow cues without any need for reference frames or accelerometers[END_REF], 10 local OF measurements coupled to a least mean squares method were used to stabilize a minimalistic quasi-panoramic compound eye on the Beerotor robot with respect to the local downward slope, thus enabling the robot to avoid any very steep relief encountered. In [START_REF] Vanhoutte | A quasipanoramic bio-inspired eye for flying parallel to walls[END_REF], an innovative OF-based algorithm was tested in the horizontal plane as a means of measuring the robot's incidence angle when perceiving a slanting wall showing a moving texture. The results of our experiments showed that our minimalistic algorithm based on 20 local OF measurements could be used to determine the local visual heading with respect to the moving slanting wall with an error of less than 3 • and a mean accuracy of 3 • [START_REF] Vanhoutte | A quasipanoramic bio-inspired eye for flying parallel to walls[END_REF].

In the present paper, a bio-inspired micro flying robot, which will be able to perform both wall-following and ground-avoidance tasks in GPS-denied (Global Positioning System-denied) environments. This robot is fitted with a visual stabilization system involving 8 custom-made OF sensors which can respond appropriately in a 7-decade lighting range [START_REF] Mafrica | A bio-inspired analog silicon retina with Michaelis-Menten auto-adaptive pixels sensitive to small and large changes in light[END_REF] and can measure the OF magnitude from 25 • /s to 1000 • /s [START_REF] Vanhoutte | Time-of-Travel Methods for Measuring Optical Flow on Board a Micro Flying Robot[END_REF]. It was established in real flight tests that this micro flying robot is able to estimate its incidence angle with respect to a slanting wall in order to restore its flight parallel to the wall. The results obtained in the present study show that our quasi-panoramic bio-inspired eye coupled to a computational unit can estimate this incidence angle in the 0 • to 50 • range when flying.

MICRO FLYING ROBOT

The micro flying robot presented here, a 320-gram quadrotor that can carry a maximum payload of 90 g, is based on the open-hardware X4-MaG model [START_REF] Manecy | X4-MaG : A Low-Cost Open-Source Micro-Quadrotor and Its Linux-Based Controller[END_REF]. Thanks to an open-source Matlab/Simulink toolbox [START_REF] Manecy | RT-MaG: An open-source SIMULINK toolbox for Linux-based realtime robotic applications[END_REF], it is possible to monitor this quadrotor in real time by wifi and change the relevant parameters directly during its flight. We have added to this quadrotor a custom-made 3D print gimbal case endowed with a magnet-based system of fixation facilitating its removal. This custom-made gimbal stabilizes the 8 OF sensors on the pitch and roll axes via two 10-gram serial servo motors (Fig. 1). The first group of 3 OF sensors oriented to the left measure a left OF, the second group of 3 sensors oriented to the right measure a right OF: each of these groups has a field of view of 92.8 • in the horizontal plane and 13.4 • in the vertical plane. The third group of two downward-oriented sensors measure the ventral OF with a field of view of 62.8 • in the forward axis and 13.4 • in the orthogonal axis. The gimbal system, which is stabilised in attitude during the robot's flight, has a total mass of 75 grams enabling the robot to fly for 6 minutes.

The quadrotor is fitted with a low-level autopilot based on the Nanowii (ATmega32u4, MultiWii) with a 6 degrees of freedom Inertial Measurement Unit (IMU) (Fig. 2a). This low-level control board makes it possible for the robot to be piloted manually and takes over from the Gumstix if failure of the latter occurs. All attitude and trajectory control processes based on the OF measurements are handled by the highlevel autopilot based on an Overo AirSTORM Computer-On-Module (COM) (Gumstix) featuring a 1-GHz CPU DM3703 processor (Texas Instruments) comprising an ARM Cortex-A8 architecture. A Teensy 3.2 featuring a 72 MHz Cortex-M4 (PJRC) controls the gimbal servo motors and also reads the 8 retinas (dedicated to OF measurements) and transmits the pixel data to the Overo AirSTORM (Fig. 2a, for details).

OPTIC FLOW SENSORS

The M 2 APix sensor is a bio-inspired OF sensor based on a Michaelis-Menten Auto-adaptive Pixel analog silicon retina that can auto-adapt in a 7-decade lighting range and responds appropriately to stepwise changes of up to ± 3 decades [START_REF] Mafrica | A bio-inspired analog silicon retina with Michaelis-Menten auto-adaptive pixels sensitive to small and large changes in light[END_REF]. Each M 2 APix sensor is composed of 12 pixels distributed in two rows of six pixels offset by half the interpixel distance [START_REF] Mafrica | A bio-inspired analog silicon retina with Michaelis-Menten auto-adaptive pixels sensitive to small and large changes in light[END_REF], and is able to make 10 local OF measurements with a high output refresh rate [START_REF] Vanhoutte | Time-of-Travel Methods for Measuring Optical Flow on Board a Micro Flying Robot[END_REF]. The local OF is measured using a time-of-travel algorithm based on a contrast detection matching method (called the thresholding method) running at only 1 kHz along the pixels rows axis.

The CPU load of the Overo AirSTORM is less than 3 percent per M 2 APix sensor and the OF refresh rate is up to 99 Hz per M 2 APix sensor with an OF ranging from 25 • /s to 1000 • /s [START_REF] Vanhoutte | Time-of-Travel Methods for Measuring Optical Flow on Board a Micro Flying Robot[END_REF].

A GIMBAL SYSTEM NAMED OCTOM 2 APIX

A custom-made PCB was designed for the gimbal system named OctoM 2 APixto connect the following components (Fig. 2a): a Teensy 3.2 featuring a 72 MHz Cortex-M4 (PJRC), an 8-bit SPI GPIO EXPANDER called XRA1404 (Exar c ), an Inertial Measurement Unit (IMU) MPU 6050, and connections to 8 M 2 APix sensors and 2 servo motors belonging to the gimbal system, in order to stabilize the bioinspired eye on the pitch and roll axes and thus remove any rotational OF components.

A M 2 APix sensor works with a SPI bus running at a maximum rate of 1 MHz, which transmits a 256-bit data frame, and the maximum theoretical frame rate is therefore 3906 Hz. SPI devices communicate in the full duplex mode using a master-slave architecture with a single master. Only one SPI bus is usually available on dedicated electronic boards. The maximum frame rate possible to obtain data from the 8 slave M 2 APix sensors is therefore less than 488 Hz, and the data will not be processed synchronously but sequentially (Fig. 2b). To give from the 8 M 2 APix sensors higher frame rates and make the processing synchronous, a XRA1404 GPIO EXPANDER was used. The XRA1404 can read the logic state on 8 digital ports and send this data as a byte to an SPI bus working at a frequency of 26 MHz (Fig. 2b). Each of these 8 digital ports is used to read the bit to bit data frame of each M 2 APix sensor (step 1 in the Fig. 2b). As a result, the first bit in all the M 2 APix data frames are arranged in one byte, which is sent on to the SPI bus. Each bit in each M 2 APix sensor is processed in this way.

The Teensy electronic board reads the SPI bus at a frequency of 26 MHz from the XRA1404 and puts each data frame of each M 2 APix sensor back into order (step 2 in Fig. 2b). The M 2 APix sensor sends a mean light value and 12 pixel values, each of which is coded in 10-bit values. The al-gorithm then selects only the pixel data. This leaves us with 12x10 bits per M 2 APix sensor . Lastly, the Teensy removes the offset of the range used and deletes the last bit which is equivalent to the amplitude of the noise, to express the data in the 8-bit format. The 12 pixels coded with the 8-bit value of the 8 M 2 APix sensors are then sent to the serial bus at a speed of 3 Mbps to a computational unit to compute the OF (step 3 shown in Fig. 2b).

FLYING STRATEGIES

The OctoM 2 APix is designed to make the X4-MaG drone both follow walls and adjust its height at relatively high speeds (up to 2.5 m • s -1 ) by removing the rotational OF component of the sensing performed by the 8 M 2 APix sensors. Wall-following and ground-following behaviour will be obtained by merging OF regulation principles [START_REF] Portelli | Modelling honeybee visual guidance in a 3-D environment[END_REF]. The aim of the flying strategies we propose to pursue in the near future will be to maintain the visual contact with the walls in order to make the robot fly in parallel along one of them. These strategies will involve adding another visual feedback loop controlling the yaw component by estimating the relative local angle between the nearest wall (e.g., the red rectangle in Fig. 4a) and the flying robot. Figure 3 and Eq. 1 explain how to obtain this incidence angle α from OF measurements with a very light solution in terms of the computational resources required, where c is cos, s is sin, and t is the tan function. This flight experiment consisted in observing the dynamic responses of the OctoM 2 APix sensor with all the components integrated during real flight. The X4-MaG flying at the Mediterranean Flying Arena1 (6x8x6 m): its trajectory was controlled in the closed loop mode with a motion capture system (VICON TM ). A 4-m long wall covered with a natural pattern was hanging in the arena to generate OF when the X4-MaG approached it. The X4-MaG repeated various straight trajectories 6 times at the same height (0.8 m ± 0.01 m) with the pattern on the right-hand side and with an angle of incidence α between its trajectory and the pattern ranging from 0 • to 50 • in 10 • steps (Fig. 4). The three M 2 APix sensors could therefore potentially detect the pattern and measure the OF during the experiments.

ωB = V OB • s(ϕB) ωD = V OD • s(ϕD) t(α) = OD • s(ϕD) -OB • s(ϕB) OB • c(ϕD) -OD • c(ϕB) t(α) = 1 ω D • s(ϕD) 2 -1 ω B • s(ϕB) 2 1 ω B • s(ϕB) • c(ϕB) -1 ω D • s(ϕD) • c(ϕD) α = t -1 ωB • s(ϕD) 2 -ωD • s(ϕB) 2 ωD • s(ϕB) • c(ϕB) -ωB • s(ϕD) • c(ϕD) (1

FLYING EXPERIMENT

To test the case of future aggressive maneuvers in an indoor environment, the flights were then performed at high speed (up to 2.5 m • s -1 , see Fig. 5) near obstacles (with a clearance of up to 0.5 m, see Fig. 4). The figure 5 gives the speed profiles of all the trajectories including one high acceleration (up to 6 m • s -2 ) phase and one high deceleration phase.

The figure 4b presents all the trajectories tracked. Due to the flying robots proximity with the patterned wall, aerodynamic perturbations were generated at the end of the trajectories, during the last metre (Fig. 4b). During a part of the trajectories from 10 • to 50 • shown by a dashed line, the pattern was not visible to the M 2 APix #3 (oriented at an angle of 120 • in Fig. 3) , and the dotted lines indicate the part of the trajectories where the pattern was not visible to the M 2 APix #2 (oriented at an angle of 90 • in Fig. 3). None of the OF measurements obtained during these dashed and dotted parts of the trajectories were included in the following statistical analyses.

A median filter was applied to the 10 local OF measurements obtained by each M 2 APix sensor, and the M 2 APix out- andc) Errors distribution of the angles of incidence α per pair of M2 APix sensors when all the optic flow measurements were greater than 50 • /s in c), and less than 50 • /s in d), where the red bar is the median, the black whiskers correspond to min and max values, and the blue box is the interquartile range (IQR) at 50 %. b) median ± IQR obtained for pair #1&#2 : 2.4 • ± 11.4 • ; pair #2&#3 : -1.2 • ± 10.1 • ; pair #1&#3 : 0.0 • ± 6.8 • . c) median ± IQR for pair #1&#2 : 8.8 • ± 24.5 • ; pair #2&#3 : -4.0 • ± 11.7 • ; pair #1&#3 : 2.6 • ± 13.4 • . d) Angle of incidence α computed for each pair of M 2 APix sensors. µ is the mean value and σ is the standard deviation.

puts were then used in pairs to compute Eq. 1 in order to estimate the local angle of incidence α (Fig. 3). We thus obtained 3 measurements of the α angle from these pairs (#1&#2, #2&#3 and #1&#3). Statistical results of the angular measurements obtained from each set of trajectories (from 0 • to 50 • ) are presented in Figs. 6 to 11 in 10 • steps.

The optical parameters of the M 2 APix sensors were chosen so as to be able to measure OF values greater than 50 • /s (up to 1000 • /s, see [START_REF] Vanhoutte | Time-of-Travel Methods for Measuring Optical Flow on Board a Micro Flying Robot[END_REF]) accurately. The angle of incidence α was therefore estimated more accurately with OF values greater than 50 • /s, as shown in Figs. 6b-11b than in Figs 6c-11c. The lowest α errors were consistently obtained with the pair of M 2 APix sensors #1 and #2 (Figs. 6b,c), where the median error was less than 2.5 • . This pair of sensors was that which was the most oriented towards the frontal part of the OF field.

The visual strategy developed in this study should therefore be consistent with the confidence range of the angle of incidence α estimated when the OF values measured on the micro aerial vehicle were high, i.e. at high speeds, as well as when the micro aerial vehicle was travelling very near the walls.

CONCLUSION

The 395-gram X4-MaG quadrotor fitted with a gimbal system and a set of 8 custom-made optic flow sensors provides an appropriate flying platform for testing optic-flow regulation principles during real flight with a view to mimicking honeybees' flight performances. The gimbal system makes it possible to minimize the effects of the rotational component of the optic flow measured by the quasipanoramic bio-inspired eye. The results obtained in these ex- periments show that our optic flow based algorithm can estimate the drone's local angle of incidence with respect to a slanting wall in the 0 • to 50 • range during flight with an error of less than 2.5 • , using the optic flow generated on either side (left or right). To measure larger angles of incidence (from 50 • to 90 • ), the fronto-bilateral optic flow could be used in the similar way to that simulated in [START_REF] Serres | Biomimetic autopilot based on minimalistic motion vision for navigating along corridors comprising u-shaped and s-shaped turns[END_REF], but this possibility is beyond the scope of the present study and would require measuring the low optic flow values occurring close to the focus of expansion.

In the near future, flying at high speed (up to 1 m/s) near obstacles (at a distance of less than 0.5 m) may be possible thanks to the large range of optic flow measurements which can now be made, namely up to 1000 • /s. The advantages of the custom-made optic flow sensors used here include low power consumption, low computational requirements, and robustness to high dynamic range lighting conditions (7 decades). In conclusion, the present X4-MaG drone fitted with smart visual sensors is the first step towards designing airborne vehicles capable of autonomous navigation requiring no magnetometers in GPS-denied environments.

Figure 1 :

 1 Figure 1: Photograph of the X4-MaG quadrotor with the gimbal case attached below (total mass: 395 g, span: 30 cm, autonomy: 6 min).

Figure 2 :

 2 Figure 2: a) Component architecture of the X4-MaG quadrotor with the gimbal elements surrounded by a dash-dot line. The development boards and the data transmission units are shown in gray, the actuators in blue, and the sensors in yellow. b) Data diagram of how visual signals originating from the OctoM 2 APix eye are processed: 1 high speed synchronous data capture, 2 data put in order, and 3 signal processing for optic flow computations.

Figure 3 :

 3 Figure 3: Geometry of the OctoM 2 APix eye (centring in O) on the horizontal plane with a tapered wall on its left-hand side. Each green rectangle represents one M 2 APix sensor. In the left-hand side, the M 2 APix are numbered to make the experiments easier to understand.

Figure 4 :

 4 Figure 4: a) Chronophotograph of the X4-MaG drone at the Mediterranean Flying Arena. The flying robot followed a textured "wall" sloping at an angle of 20 • for about 4.5 s (corresponding to one of the blue trajectories in b). b) Top view of all the trajectories taken in the Mediterranean Flying Arena 1 . 6 different trajectories were tested at the same height (0.8 m ± 0.01 m) with a clockwise angle between the trajectories and the patterned wall. The X4-MaG drone followed each trajectory several times. Depending on the angle of the trajectory, either M 2 APix #3 or both M 2 APix #2 & #3 did not perceive the pattern on the wall due to their orientation (their positions are given by a dashed line and a dotted line, respectively).

Figure 5 :

 5 Figure 5: Speed profile (in XY plane) of the aerial robot for all the trajectories presented in Fig. 4b. µ is the mean value and σ is the standard deviation.

Figure 6 :

 6 Figure 6: a) Optic flow measurements of M 2 APix #1 (red), #2 (blue) & #3 (green), during all the trajectories performed at 0 • . b) and c) Errors distribution of the angles of incidence α per pair of M 2 APix sensors when all the optic flow measurements were greater than 50 • /s in c), and less than 50 • /s in d), where the red bar is the median, the black whiskers correspond to min and max values, and the blue box is the interquartile range (IQR) at 50 %. b) median ± IQR obtained for pair #1&#2 : 2.4 • ± 11.4 • ; pair #2&#3 : -1.2 • ± 10.1 • ; pair #1&#3 : 0.0 • ± 6.8 • . c) median ± IQR for pair #1&#2 : 8.8 • ± 24.5 • ; pair #2&#3 : -4.0 • ± 11.7 • ; pair #1&#3 : 2.6 • ± 13.4 • . d) Angle of incidence α computed for each pair of M 2 APix sensors. µ is the mean value and σ is the standard deviation.

Figure 7 :Figure 8 :

 78 Figure 7: a) Optic flow measurements made by M 2 APix #1 (red), #2 (blue) & #3 (green) during all the trajectories performed at 10 • . b) c) Errors distribution of the angles of incidence α recorded per pair of M 2 APix when all the optic flow measurements were greater than 50 • /s in b) and less than 50 • /s in c), where the red bar is the median, the whiskers correspond to min and max values and the blue box is the IQR at 50 %. OF measurements plotted in dotted lines were not generated by the pattern on the wall and were therefore not included in the results presented in b) and c) in this figure and the next ones. b) median ± IQR for pair #1&#2 : 0.5 • ± 9.6 • ; pair #2&#3 : -8.7 • ± 9.8 • ; for pair #1&#3 : -5.6 • ± 6.7 • . c) median ± IQR for pair #1&#2 : 12.3 • ±21.8 • ; pair #2&#3 : -1.2 • ±11.7 • ; pair #1&#3 : 5.7 • ± 11.5 • .

Figure 9 :Figure 10 :

 910 Figure 9: Results obtained on the trajectories 30 • . The M 2 APix 3 did not measure optic flows greater than 50 • /s, which explains the lack of results on the pair of M 2 APix #2&#3 and #1&#3 in b). b) median ± IQR for pair #1&#2 : -2.3 • ± 6.0 • . c) median ± IQR for pair #1&#2 : 6.9 • ± 13.4 • ; pair #2&#3 : -9.1 • ± 23.0 • ; pair #1&#3 : -3.5 • ± 8.5 • .

Figure 11 :

 11 Figure 11: Results obtained on the trajectories at 50 • . The pattern on the wall was not visible to M 2 APix #3 during all these trajectories, which explains the lack of data on the pair of M 2 APix #2&#3 and #1&#3 in c). c) median ± IQR for pair #1&#2 : -3.1 • ± 15.8 • .

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ACKNOWLEDGEMENTS

We thank J. Diperi for the mechanical design of the flying robot and the test bench, M. Boyron for his involvement in the overall electronic design of the flying robot, and K. Contamain, S. Mafrica, and F. Colonnier for their help with both the electronic development and the programming of the M 2 APix sensors. This research was supported by the French Direction Générale de l'Armement (DGA), CNRS, Aix-Marseille Université, the Provence-Alpes-Côte d'Azur region and the French National Research Agency for Research (ANR) in the framework of the Equipex/Robotex project.