Jean Auriol

Florent Di Meglio

Two sided boundary stabilization of heterodirectional linear coupled hyperbolic PDEs

Keywords: Distributed parameter systems, Linear systems, Backstepping, Stability of linear systems, Linear system observers

We solve the problem of stabilizing a general class of linear first-order hyperbolic systems using actuation at both boundaries of the spatial domain. We design a novel Fredholm transformation similarly to backstepping approaches to derive a boundary controller and a boundary observer enabling stabilization by output feedback. This yields an explicit full-state feedback law that achieves the theoretical lower bound for convergence to zero.

I. INTRODUCTION

T HIS article solves the problem of boundary stabilization of a general class of coupled heterodirectional linear first-order hyperbolic systems of Partial Differential Equations (PDEs) in minimum time, with arbitrary numbers m and n of PDEs in each direction. The actuation is applied on both boundaries.

Most physical systems involving a transport phenomenon can be modeled using hyperbolic partial differential equations (PDEs): traffic flow [START_REF] Amin | On stability of switched linear hyperbolic conservation laws with reflecting boundaries[END_REF], heat exchangers [START_REF] Xu | Exponential stability and transfer functions of processes governed by symmetric hyperbolic systems[END_REF], open channel flow [START_REF] Coron | A lyapunov approach to control irrigation canals modeled by saint-venant equations[END_REF], [START_REF] De Halleux | Boundary feedback control in networks of open channels[END_REF] or multiphase flow [START_REF] Di | Dynamics and control of slugging in oil production[END_REF], [START_REF] Dudret | Stability and asymptotic observers of binary distillation processes described by nonlinear convection/diffusion models[END_REF]. The backstepping approach [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic pdes[END_REF], [START_REF] Vazquez | Local exponential h 2 stabilization of a 2× 2 quasilinear hyperbolic system using backstepping[END_REF] has enabled the design of stabilizing full-state feedback laws for these systems. These controllers are explicit, in the sense that they are expressed as a linear functional of the distributed state at each instant. The (distributed) gains can be computed offline. Comparing results obtained via backstepping design with existence results for stabilizing controllers reveals a gap. In [START_REF] Li | Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems[END_REF], an extensive review of controllability results for linear hyperbolic systems is given, along with the theoretical lower bounds for convergence times. These bounds vary according, mainly, to the number and location of available actuators. Backstepping results have, until now, focused on single-boundary actuation, see e.g. [START_REF] Coron | Local exponential hˆ2 stabilization of a 2\times2 quasilinear hyperbolic system using backstepping[END_REF] for the case of two coupled PDEs, [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic pdes[END_REF] for an arbitrary number of PDEs or [START_REF] Auriol | Minimum time control of heterodirectional linear coupled hyperbolic pdes[END_REF] for a minimum-time result in the general (single boundary actuation) case.

When actuation is applied at both boundaries, the literature usually focuses on design of dissipative boundary conditions to stabilize the system. This does not guarantee stabilization in the minimum theoretical time, and is only possible for small coupling terms between PDEs, but can generally be achieved using static boundary output feedback, which is much less computationally intensive. Recently the problem of stabilizing a system of two coupled PDEs with control at both sides has been solved in [START_REF] Vazquez | Bilateral boundary control of one-dimensional first-and second-order pdes using infinite-dimensional backstepping[END_REF] in the case of reaction-diffusion PDEs and for 2-state heterodirectional linear PDEs with equal transport velocities. A generalization of this result for a general system of two hyperbolic PDEs is given in [START_REF]Two sided boundary stabilization of two linear hyperbolic pdes in minimum time[END_REF]. The main contribution of this paper is a generalization of this Manuscript received December 1, 2012; revised August 26, 2015. Corresponding author: J. Auriol (email: jean.auriol@mines-paristech.fr).

result with the design of a minimum time stabilizing controller in the general case of heterodirectional hyperbolic PDEs with actuation at both boundaries. A proposed boundary feedback law ensures finite-time convergence of the states to zero in minimum time. The minimum time defined [START_REF] Li | Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems[END_REF] is the largest time between the different transport times in each direction.

Applications where controls and/or sensors are located at the two boundaries include control of open channel flow [START_REF] Bastin | On boundary feedback stabilization of non-uniform linear 2 × 2 hyperbolic systems over a bounded interval[END_REF] and state and parameter estimation for oil drilling [START_REF] Di Meglio | An adaptive observer for hyperbolic systems with application to underbalanced drilling[END_REF]. The process of drilling for hydrocarbons involves the circulation of a drilling fluid inside the serveral kilometer-long well to pressurize and clean the borehole. Under certain conditions, oil and gas flow from the reservoir into the well, resulting in a multiphase flow that needs to be monitored to ensure safety of operations. Sensors are typically located at the surface and, on more modern drilling facilities, at the bottom of the well. In this paper, we investigate the benefits of using sensors at both boundaries by conducting simulations on a distributed model for two-phase flow [START_REF] Aarsnes | Controloriented drift-flux modeling of single and two-phase flow for drilling[END_REF].

From a theoretical point of view, our approach, similar to the one presented in [START_REF]Two sided boundary stabilization of two linear hyperbolic pdes in minimum time[END_REF] is the following. Using a Fredholm transformation, the system is mapped to a target system with desirable stability properties. This target-system is designed as an exponentially stable cascade. The well-posedness of the Fredholm transformation is a consequence of a clever choice of the domain on which the kernels are defined and of the cascade structure of the target system. A full-state feedback law guaranteeing exponential stability of the zero equilibrium in the L 2 -norm is then designed. This full-state feedback law would require fully distributed measurements in practice, which is not realistic. For this reason we derive a boundary observer relying on measurements of the states at both boundaries.

The main technical difficulty of this paper is to prove wellposedness of the Fredholm transformation and its invertibility. Interestingly, the transformation kernels satisfy a system of equations with a cascade structure akin to the target system one. This structure enables a recursive proof of existence of the transformation kernels.

The paper is organized as follows. In Section II we introduce the model equations and the notations. In Section III we present the stabilization result: the target system and its properties are presented in Section III-A. In Section III-B, we derive the integral transformation and we present the domains on which the kernels are defined. The well-posedness of the kernel equations is proved in Section IV. This proof uses an important theorem which is given in Appendix in order to make the whole paper more readable. In Section V we prove the invertibility of the Fredholm transformation using an operator approach. The control feedback law and its properties are given in Section VI. In Section VII we present the uncollocated observer design. This observer is obtained with the a similar approach than the one developed for the control law. Finally, some simulation results are given in Section VIII. We conclude this article with some remarks presented in Section IX.

II. PROBLEM DESCRIPTION

A. System under consideration and notations

We consider the following general linear hyperbolic system which appear in Saint-Venant equations, heat exchangers equations and other linear hyperbolic balance laws (see [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF]).

u t (t, x) + Λ + u x (t, x) = Σ ++ u(t, x) + Σ +-v(t, x) (1) v t (t, x) -Λ -v x (t, x) = Σ -+ u(t, x) + Σ --v(t, x) (2)
evolving in {(t, x)| t > 0, x ∈ [0, 1]}, with the following linear boundary conditions

u(t, 0) = U (t), v(t, 1) = V (t) (3)
where

u = (u 1 . . . u n ) T , v = (v 1 . . . v m ) T (4)
Λ + =    λ 1 0 . . . 0 λ n    , Λ -=    µ 1 0 . . . 0 µ m    (5)
with constant speeds:

-µ m < • • • < -µ 1 < 0 < λ 1 < • • • < λ n (6)
and constant real coupling matrices

Σ ++ = {σ ++ ij } 1≤i≤n,1≤j≤n Σ +-= {σ +- ij } 1≤i≤n,1≤j≤m (7)
Σ -+ = {σ -+ ij } 1≤i≤m,1≤j≤n Σ --= {σ -- ij } 1≤i≤m,1≤j≤m (8)
The initial conditions denoted u 0 and v 0 are assumed to belong to L 2 ([0, 1]). In the following we denote

∀i, j a ij = λ i λ i + µ j (9)
We also define ā such that |ā -

1 2 | = min 1≤i≤n,1≤j≤m |a ij - 1 2 | ( 10
)
The a ij and ā play an important role in the design of the target system.

Remark 1: ā can be written as a kl with k ∈ [0, n] and l ∈ [0, m] (the uniqueness is not guaranteed). For this particular solution we denote λ and μ the corresponding transport velocities.

Remark 2: The coupling terms are assumed constant here but the results of this paper can be adjusted for spatiallyvarying coupling terms.

Remark 3: Without any loss of generality we can assume that ā ≥ 1 2 (if this is not the case we make the change of variables x = 1 -x). This assumption will be done in the following

B. Well-posedness and operator formulation

We can rewrite the system in the abstract form

d dt u v = A u v + B U V (11)
where the operators A and B can be identified through their adjoints. The operator A is thus defined by

A :D(A) ⊂ (L 2 (0, 1)) n+m → (L 2 (0, 1)) n+m u v -→ -Λ + u x + Σ ++ u + Σ +-v Λ -v x + Σ -+ u + Σ --v (12)
with

D(A) = {(u, v) ∈ (H 1 (0, 1)) n+m |u(0) = v(1) = 0} (13)
A is well defined and its adjoint A * is

A * :D(A * ) ⊂ (L 2 (0, 1)) n+m → (L 2 (0, 1)) n+m u v -→ Λ + u x + (Σ ++ ) T u + (Σ -+ ) T v -Λ -v x + (Σ +-) T u + (Σ ++ ) T v (14)
with

D(A * ) = {(u, v) ∈ (H 1 (0, 1)) n+m |u(1) = v(0) = 0} (15)
The operator B is defined by

< B U V , z 1 z 2 >= z 1 (0) t Λ + U + z 2 (1) T Λ -V (16)
Its adjoint is

B * z 1 z 2 = z 1 (0) T Λ + z 2 (1) T Λ - (17)

C. Control problem

The goal is to design feedback control inputs U (t) = (U 1 (t), . . . , U n (t)) T and V (t) = (V 1 (t), . . . , V m (t)) T such that the zero equilibrium is reached in minimum time t = t F , where

t F = max 1 µ 1 , 1 λ 1 (18)
This "minimum time" is the time needed for the slowest characteristic to travel the entire length of the spatial domain.

The existence of a control law reaching the null equilibrium in time t F is proved in [START_REF] Li | Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems[END_REF] using a method of characteristics.

To the best of our knowledge, no explicit feedback law has been designed to achieve this goal. Previous approaches yield

• exponential stability for small coupling terms when twosided static output feedback is used [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF].

• finite-time stability in time

1 λ1 + 1 µ1 > max 1 µ 1 ,
1 λ 1 when one-sided backstepping design is used, i.e with one controlled boundary only ( [START_REF] Auriol | Minimum time control of heterodirectional linear coupled hyperbolic pdes[END_REF]).

• finite-time stability in time max 1 µ , 1 λ when only two equations are considered ( [START_REF]Two sided boundary stabilization of two linear hyperbolic pdes in minimum time[END_REF]). In the second case, the system is mapped with a Volterra transformation to a target system that has a cascade structure, which is natural for backstepping. In the third case, a Fredholm transformation is used to map the system to a target system with desirable properties of stability. In the following we use a combination of these two approaches.

III. CONTROL DESIGN

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

A. Target system design

We map the system (1)-(3) to the following system α t (t, x) + Λ + α x (t, x) = Ω(x)α(t, x) + Γ(x)β(t, x) (19) [START_REF] Li | Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems[END_REF] with the following boundary conditions α(t, 0) = 0 β(t, 1) = 0 [START_REF] Vazquez | Local exponential h 2 stabilization of a 2× 2 quasilinear hyperbolic system using backstepping[END_REF] while Ω and Ω ∈ L ∞ (0, 1) are upper triangular matrices with the following structure

β t (t, x) -Λ -β x (t, x) = Ω(x)β(t, x) + Γ(x)α(t, x)
Ω(x) =       ω 1,1 (x) ω 1,2 (x) . . . ω 1,n (x) 0 . . . . . . . . . . . . . . . ω n-1,n-1 (x) ω n-1,n (x) 0 . . . 0 ω n,n (x)       (22)
Ω(x) =       ω1,1 (x) ω1,2 (x) . . . ω1,m (x) 0 . . . . . . . . . . . . . . . ωm-1,m-1 (x) ωm-1,m (x) 0 . . . 0 ωm,m (x)       (23)
The coefficients of the matrices Γ(x) and Γ(x) are defined by

∀1 ≤ i ≤ n ∀1 ≤ j ≤ m Γ ij (x) = 0 if a ij ≥ ā or x < a ij γ ij (x) otherwise ( 24
)
∀1 ≤ i ≤ m ∀1 ≤ j ≤ n Γij (x) = 0 if a ji < 1 -ā or x > a ji γij (x) ohterwise (25)
Remark 4: The a ij coefficients correspond to the spatial position where the characteristic leaving x = 0 with velocity λ i and the one leaving x = 1 with velocity µ j intersect.

Remark 5: As it will appear in the proof, the coefficients Γ and Γ do not make the system unstable due to their particular cascade structure. Their presence is necessary to prove the well-posedness of the backstepping transformation presented bellow. The following example illustrates this particular structure in a simple case.

Example 1: We consider the following coefficients

n = 3, m = 2 Λ + =   0.5 0 0 0 2 0 0 0 4   Λ -= 1 0 0 3 (26)
We define the matrix A such that where the potential non-null terms are represented by * . These matrices have some structural properties that will be analyzed in Section IV-B. Besides, the following lemma assesses the finite-time convergence of the target system. Lemma 1: The system ( 19)-( 20) reaches its zero equilibrium in finite-time 20) can be rewritten as

A ij = a ij = λi λi+µj A =
t F = max{ 1 λ , 1 µ } = 1 λ . Proof: ∀1 ≤ i ≤ n, ∀1 ≤ j ≤ n, system (19)-(
α i t (t, x) + λ i α i x (t, x) = n p=i ω ip (x)α p (t, x) + m p=1 γ ip (x)h [aip,1] (x)β p (t, x) (29)
β j t (t, x) -µ j β j x (t, x) = m p=j ωjp (x)β p (t, x) + n p=1 γjp (x)h [0,apj ] (x)α p (t, x) (30)
where, for any interval I, h I is defined by

h I (x) = 1 if x ∈ I 0 else (31)
with the convention

γ ij (x) = 0 if a ij > ā (32) γij (x) = 0 if a ji ≤ 1 -ā (33)
By induction, let us consider the following property P (s) defined for all 1 ≤ s ≤ n P (s) :

∀p ≥ n + 1 -s, if t ≥ x λ p then α p (x, t) = 0
Initialization: The initialization can be proved using a similar technique than the one presented bellow in the induction and is not detailed here. Induction: Let us assume that the property P (s -1) (1 < s ≤ n) is true. We denote r = n + 1 -s Integrating the r th line of (29) along its characteristic lines and using the boundary condition α r (t, 0) = 0, yields:

α r (t, x) = x λr 0 n p=r ω rp (x -λ r ν)α p (t -ν, x -λ r ν) + m p=1 γ rp (x -λ r ν)h [arp,1] (x -λ r ν)β p (t -ν, x -λ r ν)dν (34)
with x ∈ [0, 1] and t ≥ x λr . Consequently, ∀p > r:

t ≥ x λ r ⇒ (1 - λ r λ p ) x λ r ≤ t - x λ p ⇒ (1 - λ r λ p )ν ≤ t - x λ p ∀ν ∈ [0, x λ r ] ⇒ t -ν ≥ x -λ r ν λ p ⇒ α p (t -ν, x -λ r ν) = 0 (35)
The last implication uses the fact that P (s -1) is true. Let us now consider the following property P 1 (q) defined for all

1 ≤ q ≤ m P 1 (q) : ∀x > a rq , ∀t ≥ 1 -x µ q , β q (t, x) = 0
Initialization: The initialization can be proved using a similar technique than the one presented bellow in the induction and is not detailed here. Induction: Let us assume that the property P 1 (q -1) (1 < q ≤ m) is true. Integrating the q th line of (30) along its characteristic lines and using the boundary condition β q (t, 1) = 0, yields:

β q (t, x) = 1-x µq 0 n p=q ωqp (µ q ν + x)β p (t -ν, µ q ν + x) + n p>r γqp (µ q ν + x)h [0,apq] (µ q ν + x)α p (t -ν, µ q ν + x)dν (36)
with 1 ≥ x > a rq , and t ≥ 1-x µq (this explains why the last sum starts at p > r). Consequently, ∀p ≥ q and ∀ν ∈ [0, 1-x µq ] such that µ q ν + x ≤ a pq (in order to have h [0,apq] (µ q ν + x) = 0):

t ≥ 1 -x µ q ⇒t - x λ p ≥ ( λ p µ q (λ p + µ q ) - x µ q )(1 + µ q λ p ) ⇒t - x λ p ≥ ( a pq µ q - x µ q )(1 + µ q λ p ) ⇒t - x λ p ≥ (1 + µ q λ p )ν ⇒t -ν ≥ x + µ q ν λ p
Consequently, using the fact that

P (s -1) is true, it yields ∀ν ∈ [0, 1-x µq ] such that µ q ν + x ≥ a pq α p (t -ν, µ q ν + x) = 0 (37)
Consequently the second sum in (36) is always null for t ≥ 1-x µq . Moreover, using the fact that P 1 (q -1) is true, we can simplify the first sum removing most of the terms. We can rewrite (36) as:

β q (t, x) = 1-x µq 0 ωqq (µ q ν + x)β q (t -ν, µ q ν + x)dν (38)
with 1 ≥ x > a rq , and t ≥ 1-x µq . Consequently β q (t, x) = 0 and P 1 (q) is true. This achieves the proof of P 1 (q) for all 1 ≤ q ≤ m. For a given p we now focus on the following term of (34)

γ rp (x -λ r ν)h [arp,1] (x -λ r ν)β p (t -ν, x -λ r ν) (39)
This term can be non null only if

x -λ r ν ≥ a rp ∀ν ∈ [0, x λ r ] (40)
It yields

µ q + λ r λ r x -(µ q + λ r )ν ≥ 1 ⇔ µ q λ r x + x -1 ≥ (µ q + λ r )ν (41) Since t ≥ x λr it yields µ q t + x -1 ≥ (µ q + λ r )ν ⇒ µ q (t -ν) ≥ 1 -x + λ r ν (42)
Using P 1 we can deduce that (39) is always null. Consequently, combining this result with (35), we can rewrite (34) as

α s (t, x) = x λr 0 ω rr (x -λ r ν)α r (t -ν, x -λ r ν) (43) with x ∈ [0, 1] and t ≥ x λr . Consequently it yields ∀x ∈ [0, 1], t ≥ x λ p ⇒ α p (x, t) = 0 (44)
It achieves the recursion. It is then quite straightforward to prove a similar result for β. Consequently we have

∀t ≥ 1 λ 1 , α(x, t) = 0 (45) ∀t ≥ 1 µ 1 , β(x, t) = 0 (46)
This concludes the proof. Using an operator framework, system ( 19)-( 20) rewrites as

d dt α β = A 0 α β (47)
The operator A 0 is defined by

A 0 : D(A 0 ) ⊂ (L 2 (0, 1)) 2 → (L 2 (0, 1)) 2 α β -→ -Λ + α x + Ωα + Γβ Λ -β x + Ωβ + Γα (48)
with

D(A 0 ) = {(α, β) ∈ (H 1 (0, 1)) 2 |α(0) = β(1) = 0} (49)
A 0 is well defined and its adjoint

A * 0 is A * 0 : D(A * 0 ) ⊂ (L 2 (0, 1)) 2 → (L 2 (0, 1)) 2 α β -→ α T x Λ + + α T Ω + β T Γ -β T x Λ -+ β T Ω + α T Γ T ( 50
)
with

D(A * 0 ) = {(α, β) ∈ (H 1 (0, 1)) 2 |α(1) = β(0) = 0} (51)

B. Fredholm transformation 1) Definition of the transformation

In order to map the original system ( 1)-( 3) to the target system ( 19)-( 21), we use the following transformation

α(t, x) = u(t, x) + h [0,ā] (x) -μ λ x+1 x (K(x, ξ)u(t, ξ) + L(x, ξ)v(t, ξ))dξ + h ]ā,1] (x) x λ μ (1-x) (M (x, ξ)u(t, ξ) + N (x, ξ)v(t, ξ))dξ (52) β(t, x) = v(t, x) + h [0,ā] (x) -μ λ x+1 x ( K(x, ξ)u(t, ξ) + L(x, ξ)v(t, ξ))dξ + h ]ā,1] (x) x λ μ (1-x) ( M (x, ξ)u(t, ξ) + N (x, ξ)v(t, ξ))dξ (53)
where we recall that for any interval I, h I (x) is defined by

h I (x) = 1 if x ∈ I 0 else ( 54
)
We define the following triangular domains, depicted in Figure 1:

T 0 = {(x, ξ)| x ∈ [0, ā], x ≤ ξ ≤ -μ λ x + 1} (55)
T1 = {(x, ξ)| x ∈ [ā, 1], λ μ (1 -x) < ξ ≤ x} (56)
The kernels K, L, K and L are defined on T 0 . The kernels M, N, M and N are defined on T1 . They are assumed continuous in their domains of definition. They all have yet to be defined.

Remark 6: One may think that due to the use of the hfunctions, the transformation presents a discontinuity in x = ā. Nevertheless, one can check that the right and left limits are equal since the integrals vanish and that consequently we do not have any discontinuity.

Remark 7: Since α(0) = β(1) = 0 the two control laws U and V can be computed as functions of (u, v).

Remark 8: This transformation is a Fredholm transformation and can be rewritten using integrals between 0 and 1 as follows

α(t, x) = u(t, x) - 1 0 Q 11 (x, ξ)u(t, ξ) + Q 12 (x, ξ)v(t, ξ)dξ (57) β(t, x) = v(t, x) - 1 0 Q 21 (x, ξ)u(t, ξ) + Q 22 (x, ξ)v(t, ξ)dξ (58) with Q 11 (x, ξ) = -K(x, ξ)h [x,-μ λ x+1] (ξ)h [0,ā] (x) -M (x, ξ)h [ λ μ (1-x),x] (ξ)h ]ā,1] (x) (59)
Q 12 (x, ξ) = -L(x, ξ)h [x,-μ λ x+1] (ξ)h [0,ā] (x) -N (x, ξ)h [ λ μ (1-x),x] (ξ)h ]ā,1] (x) (60) Q 21 (x, ξ) = -K(x, ξ)h [x,-μ λ x+1] (ξ)h [0,ā] (x) -M (x, ξ)h [ λ μ (1-x),x] (ξ)h ]ā,1] (x) (61)
Q 22 (x, ξ) = -L(x, ξ)h [x,-μ λ x+1] (ξ)h [0,ā] (x) -N (x, ξ)h [ λ μ (1-x),x] (ξ)h ]ā,1] (x) (62)
2) Kernel equations We now differentiate the Fredholm transformation (52)-(53) with respect to time and space to compute the equations satisfied by the kernels. We start with the α-transformation (52).

If x ≤ ā: Differentiating (52) with respect to space and using the Leibniz rule yields

α x (t, x) = u x (t, x) -K(x, x)u(t, x) -L(x, x)v(t, x) -μ λ K(x, -μ λ x + 1)u(-μ λ x + 1) -μ λ L(x, -μ λ x + 1)v(-μ λ x + 1) + -μ λ x+1 x (K x (x, ξ)u(t, ξ) + L x (x, ξ)v(t, ξ))dξ (63)
Differentiating (52) with respect to time, using (1), (2) and integrating by parts yields

α t (t, x) = -Λ + u x (t, x) + Σ ++ u(t, x) + Σ +-v(t, x) + K(x, x)Λ + u(t, x) -L(x, x)Λ -v(t, x) -K(x, -μ λ x + 1)Λ + u(t, -μ λ x + 1) + L(x, -μ λ x + 1)Λ -v(t, -μ λ x + 1) + -μ λ x+1 x (K ξ (x, ξ)Λ + u(t, ξ) + L ξ (x, ξ)Λ -v(t, ξ))dξ (64)
Plugging these expressions into the target system ( 19)-( 20) yields the following system of kernel equations

0 =Σ ++ -Λ + K(x, x) + K(x, x)Λ + -Ω(x) (65) 0 =Σ +--Λ + L(x, x) -L(x, x)Λ --Γ(x) (66) 0 = -μ λ Λ + L(x, -μ λ x + 1) + L(x, -μ λ x + 1)Λ - (67) 0 = μ λ Λ + K(x, -μ λ x + 1) + K(x, -μ λ x + 1)Λ + (68) 0 =Λ + K x (x, ξ) + K ξ (x, ξ)Λ + + K(x, ξ)Σ ++ + L(x, ξ)Σ -+ -Ω(x)K(x, ξ) -Γ(x) K(x, ξ) (69) 0 =Λ + L x (x, ξ) -L ξ (x, ξ)Λ -+ K(x, ξ)Σ +- + L(x, ξ)Σ ---Ω(x)L(x, ξ) -Γ(x) L(x, ξ) (70)
if x > ā: Similarly we get

0 =Σ ++ + Λ + M (x, x) -M (x, x)Λ + -Ω(x) (71) 0 =Σ +-+ Λ + N (x, x) + N (x, x)Λ --Γ(x) (72) 0 = - λ μ Λ + N (x, - λ μ (x -1)) + N (x, - λ μ (x -1))Λ -(73) 0 = λ μ Λ + M (x, - λ μ (x -1)) + M (x, - λ μ (x -1))Λ + (74) 0 =Λ + M x (x, ξ) + M ξ (x, ξ)Λ + + M (x, ξ)Σ ++ + N (x, ξ)Σ -+ -Ω(x)M (x, ξ) -Γ(x) M (x, ξ) (75) 0 =Λ + N x (x, ξ) -N ξ (x, ξ)Λ -+ M (x, ξ)Σ +- + N (x, ξ)Σ ---Ω(x)N (x, ξ) -Γ(x) N (x, ξ) (76)
We now compute the kernels for the β-transformation if x ≤ ā: Differentiating (53) with respect to space and time and then plugging into the target system ( 19)-( 20) yields the following system of kernel equations

0 =Σ -+ + Λ -K(x, x) + K(x, x)Λ + -Γ(x) (77) 0 =Σ ---Λ -L(x, x) + L(x, x)Λ --Ω(x) (78) 0 = -K(x, - μ λ m x + 1)Λ + + μ λ Λ -K(x, -μ λ x + 1) (79) 0 = μ λ Λ -L(x, -μ λ x + 1) + L(x, - μ λ m x + 1)Λ - (80) 0 = -Λ - Kx (x, ξ) + Kξ (x, ξ)Λ + + K(x, ξ)Σ ++ + L(x, ξ)Σ -+ -Ω(x) K(x, ξ) -Γ(x)K(x, ξ) (81) 0 = -Λ - Lx (x, ξ) -Lξ (x, ξ)Λ -+ K(x, ξ)Σ +- + L(x, ξ)Σ ---Ω(x) L(x, ξ) -Γ(x)L(x, ξ) (82)
if x > ā: Similarly, we get

0 =Σ -+ -Λ -M (x, x) -M (x, x)Λ + -Γ(x) (83) 0 =Σ --+ Λ -N (x, x) -N (x, x)Λ --Ω(x) (84) 0 = λ μ Λ -N (x, - λ μ (x -1)) + N (x, - λ μ (x -1))Λ -(85) 0 = - λ μ Λ -M (x, - λ μ (x -1)) + M (x, - λ μ (x -1))Λ + (86) 0 = -Λ - Mx (x, ξ) + Mξ (x, ξ)Λ + + M (x, ξ)Σ ++ + N (x, ξ)Σ -+ -Ω(x) M (x, ξ) -Γ(x)M (x, ξ) (87) 0 = -Λ - Nx (x, ξ) -Λ - Nξ (x, ξ) + M (x, ξ)Σ +- + N (x, ξ)Σ ---Ω(x) N (x, ξ) -Γ(x)N (x, ξ) (88)
The well-posedness of all these kernel equations is assessed in the following theorems Theorem 1: Consider system (65)-( 70) and ( 77)-( 82). There exists a unique solution K, L, K and L in L ∞ (T 0 )

Theorem 2: Consider system (71)-( 76) and ( 83)-( 88). There exists a unique solution M , N , M and N in L ∞ (T 1 ) The proof of these theorems is described in the following section and uses the cascade structure of the kernel equations (which is due to the particular shapes of the matrices Ω, Ω, Γ and Γ)

IV. WELL-POSEDNESS OF THE KERNEL EQUATIONS

In this section we prove Theorem 1. The proof of Theorem 2 is quite similar and is not detailed here. The proof follows three steps:

• First, we develop the kernel equations and the associated boundary conditions. • Then we define a particular sequence in which to solve the equations using the properties of the matrices Γ and Γ.

• Finally, this sequence is used in a recursive approach to complete the proof. For every iteration of the recursion we prove that the i th line of the system of kernel PDEs K ij , L ij (resp. Kij , Lij ) is well-posed. This proof involves the following well-posedness theorem.

Theorem 3: Consider the following system of hyperbolic equations

j ∂ x F j (x, ξ) + ν j ∂ ξ F j (x, ξ) = Σ j (x, ξ)F (x, ξ) (89)
where F = (F 1 . . . F n+m ) is defined on the triangle D:

D = {(x, ξ)|x ≤ ξ c 1 ξ ≤ c 1 -c 2 x d 1 ξ ≥ d 1 -d 2 x} (90)
where the coefficients c 1 , c 2 , d 1 , d 2 are all positive. The corresponding boundary conditions are defined on a closed subset R j included on the boundary of the domain ∂D

F j|Rj = f j (91)

Assume

• that the homogeneous system, obtained by taking Σ(x, ξ) = 0 in (89) along with boundary conditions (91) is well-posed; • that there exists α j > 0 such that the following inequalities holds for all j = 1, . . . , n + m ∀(x, ξ) ∈ D α j j (x) + ν j (ξ) < -δ < 0 (92)

Then the system (89) with boundary conditions (91) has an unique solution F ∈ L ∞ (D).

Proof: The proof of this theorem is given in Appendix.

Remark 9: A necessary and sufficient condition for the first assumption to be satisfied is that, for every j = 1 . . . n + m the characteristics defined by the j , ν j uniquely connect each point of T 0 to D j .

A. Development of the kernel equations

We only focus on the kernels K, L, K and L defined on T 0 since the proof is similar for the remaining kernels. Developing (65)-( 70) and ( 77)-(82) we get the following set of kernel PDEs:

for

1 ≤ i ≤ n, 1 ≤ j ≤ n λ i ∂ x K ij (x, ξ) + λ j ∂ ξ K ij (x, ξ) = - n k=1 σ ++ kj K ik (x, ξ) - m p=1 σ -+ pj L ip (x, ξ) + i≤p≤n K pj (x, ξ)ω ip (x) + 1≤p≤m Kpj (x, ξ)Γ ip (x) (93)
for

1 ≤ i ≤ n, 1 ≤ j ≤ m λ i ∂ x L ij (x, ξ) -µ j ∂ ξ L ij (x, ξ) = - m k=1 σ -- kj L ik (x, ξ) - n p=1 σ +- pj K ip (x, ξ) + i≤p≤n L pj (x, ξ)ω ip (x) + 1≤p≤m Lpj (x, ξ)Γ ip (x) (94) for 1 ≤ i ≤ m, 1 ≤ j ≤ n µ i ∂ x Kij (x, ξ) -λ j ∂ ξ Kij (x, ξ) = n k=1 σ ++ kj Kik (x, ξ) + m p=1 σ -+ pj Lip (x, ξ) - i≤p≤m Kpj (x, ξ)ω ip (x) - 1≤p≤n K pj (x, ξ) Γip (x) (95)
for

1 ≤ i ≤ m, 1 ≤ j ≤ m µ i ∂ x Lij (x, ξ) + µ j ∂ ξ Lij (x, ξ) = m k=1 σ -- kj Lik (x, ξ) + n p=1 σ +- pj Lip (x, ξ) - i≤p≤m Lpj (x, ξ)ω ip (x) - 1≤p≤n L pj (x, ξ) Γip (x) (96)
with the following set of boundary conditions (in order to make the whole content more readable we have removed the domains of definition of the indices)

K ij (x, 1 -μ λ x) = 0 (97) K ij (x, x) = σ ++ λ i -λ j i > j (98) Lij (x, -μ λ x + 1) = 0 (99) Lij (x, x) = σ -- µ i -µ j i > j (100) ( µ j λ i -μ λ )L ij (x, 1 -μ λ x) = 0 (101)
(-λ i + µ i μ λ ) Kij (x, - μ λ m x + 1) = 0 (102) ∀x < a ij L ij (x, x) = - σ +- ij λ i + µ j (103) if 1 -a ji ≥ ā Kij (x, x) = - σ -+ λ j + µ i (104)
We add the following arbitrary boundary conditions (in order to have a well-posed system)

if 1 -a ji < ā Kij (0, ξ) = 0 (105)
Besides, (65) imposes

∀i ≤ j ω ij (x) = (λ j -λ i )K ij (x, x) + σ ++ ij ( 106
)
and (66) imposes

∀a ij < x < 1 γ ij (x) = -(λ i + µ j )L ij (x, x) + σ +- ij (107)
Similarly (78) imposes

∀i ≤ j ωij (x) = (µ j -µ i ) Lij (x, x) + σ -- ij (108)
and (77) imposes

∀0 ≤ x ≤ a ji γij (x) = (µ i + λ j ) Kij (x, x) + σ -+ ij ( 109
)
This induces a coupling between the kernels through equations ( 93), (94), ( 95) and (96) that could appear as non linear at first sight. However, the coupling has a linear cascade structure due to the particular shapes of the matrices Ω, Ω, Γ and Γ. We now define two sequences r i and ri that will be used in a recursive proof of the well-posedness. Some of these equations with the corresponding characterisitc lines are represented on Figure 1 and Figure 2.

Remark 10: The artificial boundary condition we add for the kernel K is not a degree of freedom since it has no impact on the control law and on the stability of the target system.

ξ 1 0 1 x = ξ x a λ ξ = -µ x + λ µ ξ = - λ ( x - 1 ) Τ 1 Τ 0 Κ L Fig. 1. Representation of the K-kernels and the L-kernels (a ij ≥ ā) ξ 1 0 1 x = ξ x a λ ξ = -µ x + λ µ ξ = - λ ( x - 1 ) Τ 1 Τ 0 L Fig. 2. Representation of the L-kernels (a ij < ā)

B. Definition of the sequences r i and ri

In this subsection we define two sequences r i and ri that we are going to use in the recursive proof. Let us consider the matrices ∆ and ∆ defined by the following relations

∀1 ≤ i ≤ n ∀1 ≤ j ≤ m ∆ ij = 0 if a ij ≥ ā 1 else (110) ∀1 ≤ i ≤ m ∀1 ≤ j ≤ n ∆ij = 0 if a ji < 1 - ā 1 else (111)
These matrices have exactly the same structure as the matrices Γ and Γ. We have the following results (some of the proofs are quite straightforward and are omitted there). Lemma 2: ∆ = 1 -∆ T Proof: The proof relies on the fact that due to the definition of ā (Equation ( 10

)), if a ij > ā then a ij > 1 -ā. Suppose that ∆ ij = 0, then a ij ≥ ā ⇒ a ij ≥ 1 -ā (since ā > 1 2 ). It yields ∆ji = 1. Suppose now that ∆ ij = 1, then a ij < ā ⇒ a ij < 1 -ā (due to the definition of ā). It yields ∆ji = 0. Lemma 3: If ∆ ij = 0 then ∀k > i ∆ kj = 0 Lemma 4: If ∆ ij = 0 then ∀k < j ∆ ik = 0
The two previous lemmas use the fact that a ij < a i+1,j and a ij > a i,j+1 . Same results hold for ∆.

Lemma 5: Either the last line of ∆ or the last line of ∆ is null.

Proof: Let us assume that the last line of ∆ is non-null. Consequently, ∀j ∈ [1, n] a jm ≤ 1-ā and particularly a nm ≤ 1 -ā. This implies a nm ≤ ā (due to the definition of ā) and it yields ∆ mn = 0. Using the previous Lemma concludes the proof.

In the following we denote s i (resp si ) the number of coefficients which are equal to 1 in the i th line of ∆ (resp ∆). For 0 ≤ i ≤ n + m, we define the sequences r i and ri

if r0 -ri ≥ s ri-1 then r i+1 = r i -1 ri+1 = ri if r 0 -r i ≥ sri-1 then ri+1 = ri -1 r i+1 = r i where r 0 = n + 1, r0 = m + 1. We use the convention s 0 = s0 = ∞.
Theorem 4: The sequences r i , ri are well defined. Moreover r n+m = rn+m = 1

Proof: To prove that the sequences are well defined we need to prove that for any 0 ≤ i ≤ n + m one and one only of the following assumptions is true

r0 -ri ≥ s ri-1 (112) r 0 -r i ≥ sri-1 (113)
We start by proving that at least one of the two assumptions is true. By contradiction let us assume that none of them hold. Consequently we have, for some i

r0 -ri < s ri-1 (114) r 0 -r i < sri-1 (115)
• By definition of s ri-1 , we have exactly s ri-1 coefficients that are equal to 1 on the (r i -1) th line of the matrix ∆. Using Lemma 2 yields that we have exactly m -s ri-1 coefficients equal to 1 on the (r i -1) th column of ∆ • By definition of sri-1 , we have exactly sri-1 coefficients that are equal to 1 on the (r i -1) th line of ∆. Consequently ∆ri-1,sr i -1 = 1. Adjusting Lemma 3 to ∆ yields that on the (n + 1 -sri-1 ) th column of ∆ we have at least ri -1 coefficients equal to 1.

• Since n + 1 -sri < r i we get n + 1 -sri ≤ r i -1. It means that the column n + 1 -sri is located lefter than the column r i -1. Consequently, using Lemma 3, we must have a larger number of coefficients equal to 1 on the column r i -1 than on the column n + 1 -sri . This implies

m -s ri-1 ≥ ri -1 (116)
which is a contradiction with (114). To achieve the proof of the well posedness of the two sequences r i and ri , we need to prove that the two assumptions (112)-(113) cannot both be true. This is quite straightforward using the same ideas. If we assume that (113) holds, then n + 1 -sri ≥ r i implies that the column n + 1 -sri of ∆ is located strictly righter than the column r i -1 and that consequently (Lemma 4), the number of coefficients equal to 1 on the former column is larger than the number on the later. This implies

m -s ri-1 < ri -1 (117)
and consequently (112) is false. The same holds when (112) is true.

The following lemma makes the link between the matrices ∆, ∆, and Γ, Γ. Lemma 6: The matrix Γ(x) has at least m -s i nullcoefficients on its i th line. Similarly, the matrix Γ(x) has at least n -si null-coefficients on its i th line.

Corollary 1: ∀i ≤ n, ∀j ≤ m -s i , Γ(x) ij = 0 and ∀i ≤ m, ∀j ≤ n -si , Γ(x) ij = 0
Proof: The proofs of this lemma and of this corollary are quite straightforward noticing that the matrices Γ (resp. Γ) and ∆ (resp. ∆) have exactly the same structure and that consequently the properties described above can be easily extended to Γ and Γ.

C. Induction hypothesis

By induction, let us consider the following property P (q) defined for all 1 ≤ q ≤ m + n:

∀r q ≤ i ≤ n, ∀r q ≤ ī ≤ m, ∀1 ≤ j ≤ n, ∀1 ≤ d ≤ m, ∀1 ≤ j ≤ n, ∀1 ≤ d ≤ m, the problem (93)-(104)
where Ω, Ω, Γ and Γ are defined by ( 106)-( 108) has an unique solution

K ij (•, •), L id (•, •), Kīj(•, •), Lī d(•, •) ∈ L ∞ (T 0 ).
Initialization : For q = 1, we have either (r 1 = n and r1 = m + 1) or (r 1 = n + 1 and r1 = m) due to Theorem 3. Assuming that r 1 = n and r1 = m + 1, system (93)-(104) rewrites as follow for 1 ≤ j ≤ n

λ n ∂ x K nj (x, ξ) + λ j ∂ ξ K nj (x, ξ) = - n k=1 σ ++ kj K nk (x, ξ) - m p=1 σ -+ pj L np (x, ξ) + K nj (x, ξ)σ ++ nn (118) for 1 ≤ j ≤ m λ n ∂ x L nj (x, ξ) -µ j ∂ ξ L nj (x, ξ) = - m k=1 σ -- kj L nk (x, ξ) - n p=1 σ +- pj K np (x, ξ) + L nj (x, ξ)σ ++ nn ( 119
)
with the corresponding set of boundary conditions. The wellposedness of such system is quite straightforward using [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic pdes[END_REF] or Theorem 3. The initialization still holds for r 1 = n + 1 and r1 = m.

Induction : Let us assume that the property P (q -1)

(1 < q ≤ n + m -1) is true. We consequently have that ∀r q-1 ≤ i ≤ n, ∀r q-1 ≤ ī ≤ n, ∀1 ≤ j ≤ n, ∀1 ≤ d ≤ m, ∀1 ≤ j ≤ n, ∀1 ≤ d ≤ m, K ij (•, •), L id (•, •), Kīj(•, •), and Lī d(•, •) are bounded.
In the following we assume that rq = rq-1 (and that consequently r q = r q-1 -1). The result still holds if r q = r q-1 and the proof is similar. We denote i = r q .

Using the induction hypothesis yields that

∀1 ≤ j ≤ n, ∀1 ≤ d ≤ m, ∀r q = rq-1 ≤ ī ≤ n Kīj(•, •), and Lī d(•, •) are well-posed.
Rewriting equation ( 93) yields

-λ i ∂ x K ij (x, ξ) -λ j ∂ ξ K ij (x, ξ) = n k=1 σ ++ kj K ik (x, ξ) + m p=1 σ -+ pj L ip (x, ξ) - i≤p≤m K pj (x, ξ) • ((λ p -λ i )K ip (x, x) + σ ++ ip ) - 1≤p≤m Kpj (x, ξ)h [aip,1] (x) • ((λ i + µ p )L ip (x, x) + σ +- ip ) (120)
with the boundary conditions

K ij (x, 1 -μ λ x) = 0 K ij (x, x) = σ ++ λ i -λ j i > j
The one-but-last sum uses the expression of K pj for i ≤ p ≤ m. This term is known and bounded for p > i (induction assumption). For p = i, λ i = λ p and the term

(λ p -λ i )K ip (x ij (x, ξ, s), x ij (x, ξ, s)) cancels.
Using Corollary 1 and relation (107) it is possible to rewrite the last sum as

m+1-si≤p≤m Kpj (x, ξ)(σ +- ip + (λ i + µ p )L ip (x, ξ))h [aij ,1] (x) (121)
Using the definition of r i , we have

s i ≤ s i-1 = s rq-1 ≤ m + 1 -rq ⇒ m -s i ≥ rq -1 (122)
Consequently, the last sum uses the expression of Kpj for rq ≤ p ≤ m which is known according to the induction assumption. Therefore, all the non-linearites that could appear at first sight on the kernel equations actually involve terms that have been computed in the previous iterations and that are bounded. We can rewrite (120) as

-λ i ∂ x K ij (x, ξ) -λ j ∂ ξ K ij (x, ξ) = n k=1 C ++ kj (x, ξ)K ik (x, ξ) + m k=1 C -+ kj (x, ξ)L ik (x, ξ)ds (123)
where the coefficients C ++ kj and C -+ kj are known and bounded (since they are either constants or computed during the previous iteration of the recursion).

Similarly, we can rewrite (94) as

-λ i ∂ x L ij (x, ξ) + µ j ∂ ξ L ij (x, ξ) = n k=1 C -+ kj (x, ξ)K ik (x, ξ) + m k=1 C -- kj (x, ξ)L ik (x, ξ)ds ( 124
)
where the coefficients C -+ kj and C -- kj are known and bounded. Moreover we have the boundary condition

( µ j λ i -μ λ )L ij x, 1 -μ λ x = 0 (125)
and

∀x < a ij = λ i λ i + µ j L ij (x, x) = σ +- ij λ i + µ j (126)
Each L ij has a discontinuity line defined by ξ = 1 -µj λi x. The characteristics are integrated in opposite directions on each side of the discontinuity: away from the ξ = x boundary for ξ ≤ 1-µj λi x and away from the line ξ = 1-μ λ for ξ ≥ 1-µj λi x. Therefore the parameters α and δ of Theorem 3, which have to satisfy (92) for all L ij on the domain on which the equations are considered, vary on each side of the discontinuity of all the kernels. Therefore, in what follows, we define a sequence of triangular domains, depicted on Figure 3, on which there exists α and δ satisfying (92). More precisely, assuming that a i1 ≥ ā (this specific case will be presented in Remark 10), for all k ≤ m + 1 such that a ik < ā (with the convention a i(m+1) = 0), consider the domains T k :

T k = {(x, ξ)|x ≤ ξ ξ ≤ 1 - µ k-1 λ i x ξ ≥ 1 - µ k λ i x} T m+1 = {(x, ξ)|0 ≤ x ≤ ξ ξ ≤ 1 - µ m λ i x} (127)
The equations can be solved successively on these triangles, starting from the rightmost one. Their are represented on Figure 3.

The trace of the solution on the boundary of a given T k provides boundary conditions of the system considered on T k+1 . By induction, let us now consider the property Q(k) defined for all k such that a ik < ā by: The system (123)-( 126) is well-posed on T k ∩ T 0 .

Initialization: Let k 0 such that a ik0 < ā ≤ a i(k0-1) . ξ 1 0 1 x = ξ x a λ ξ = -µ x + λ µ ξ = -λ ( x -1 ) Τ 1 Τ k a ik a ik+1 Τ k+1 Fig. 3.

Representation of the triangles T k

On T k0 ∩ T 0 , equations ( 123) and ( 124) can be simply rewritten, for l = 1 . . . n + m as

l ∂ x F l (x, ξ) + ν l ∂ ξ F l (x, ξ) = Σ l (x, ξ)F (x, ξ) (128)
where

F = (K i,1 . . . K i,n , L i,1 , . . . L i,m ) T .
The functions l and ν l are defined according to the location of the boundary condition by

l =    -λ i if l ≤ n -λ i if l > n and l -n < k 0 +λ i else (129)
ν l =    -λ l if l ≤ n +µ l if l > n and l -n < k 0 -µ l else (130)
The homogeneous system, obtained by taking Σ l (x, ξ) = 0 along with the corresponding boundary conditions is wellposed. If we choose α k0 such that

µ k0 -1 λ i < α k0 = µ k0-1 + µ k0 2λ i < µ k0 λ i (131)
we easily get

α k0 l + ν l =      -µ k 0 -1 -µ k 0 2 -λ l if l ≤ n -µ k 0 -1 -µ k 0 2 + µ l if l > n and l -n < k 0 +µ k 0 -1 +µ k 0 2 -µ l else (132)
In the first case, the result is always negative.

If l -n < k 0 , µ l < µ k0 < µ k 0 +µ k 0 -1 2
. Consequently, for the second case the result is still negative. The same holds for the third case. Consequently, the two hypothesis of Theorem 3 are verified and we can conclude to the well-posedness of the kernel equations on T k0 . This concludes the initialization.

Recursion If we assume that Q(k) holds (for k 0 ≥ k < m) we can easily prove using Theorem 3 that Q(k + 1) holds. The well-posedness of the homogeneous system is direct using Q(k) and one can easily check that the second hypothesis of the theorem holds choosing α k+1 :

α k+1 = µ k+1 + µ k 2 (133)
with the convention µ m+1 = µ m + 1. Moreover this iteration provides us the boundary condition for the next triangle. This concludes the proof. Remark 11: If a i1 < ā, the previous result still holds taking a i0 = ā. This proves the well-posedness of the i th line of the kernels K and J on T 0 . Consequently P (q) is true and the well-posedness of the kernels K, L, K, L on T 0 is proved.

V. INVERTIBILITY OF THE FREDHOLM TRANSFORMATION

Unlike the Volterra transformation, the Fredholm transformation is not always invertible. In [START_REF] Coron | Stabilization and controllability of first-order integro-differential hyperbolic equations[END_REF], the authors prove the invertibility of such a transformation in the case of a first-order integro-differential hyperbolic equation. In [START_REF]Two sided boundary stabilization of two linear hyperbolic pdes in minimum time[END_REF] the invertibility of the transformation is proved in the scalar case. In this section we use similar arguments (in particular we rely on the Fredholm alternative) to prove the invertibility of our transformation.

1) Operator formulation of the Fredholm transformation and properties

In this subsection we rewrite the previous Fredholm transformation using operators. This will lead to some relations verified by the adjoint operators. The Fredholm transformation (57)-(58) can be written as an operator P acting on u v . More precisely we have

P = Id -Q (134) α β = P u v (135)
where Q : (L 2 (0, 1)) n+m → (L 2 (0, 1)) n+m is the integral operator defined by

Q u v = 1 0 Q 11 (x, ξ)u(t, ξ) + Q 12 (x, ξ)v(t, ξ) Q 21 (x, ξ)u(t, ξ) + Q 22 (x, ξ)v(t, ξ) dξ (136)
Its adjoint is:

Q * u v = 1 0 Q 11 (ξ, x)u(t, ξ) + Q 21 (ξ, x)v(t, ξ) Q 12 (ξ, x)u(t, ξ) + Q 22 (ξ, x)v(t, ξ) dξ (137)
One can easily check that:

Q * (D(A * )) ⊂ D(A * ) (138)
The control U V can also be rewritten using operators

U V = Γ u v (139)
with

Γ u v = 1 0 Q 11 (0, ξ)u(t, ξ) + Q 12 (0, ξ)v(t, ξ) Q 21 (1, ξ)u(t, ξ) + Q 22 (1, ξ)v(t, ξ) dξ (140)
Using ( 47) and (135) yields

d dt α β = A 0 α β = A 0 P u v (141)
Moreover using [START_REF] Coron | Stabilization and controllability of first-order integro-differential hyperbolic equations[END_REF] and (135) we get

d dt α β = d dt (P u v ) = P A u v + P BΓ u v (142)
Consequently P and Γ satisfy the following relation:

A 0 P = P A + P BΓ (143)
Taking the adjoints, this is equivalent to

P * A * 0 = A * P * + Γ * B * P * (144)
2) The Fredholm alternative We give first the following useful lemmas: Lemma 7: ker P * ⊂ D(A * 0 ) = D(A * ) Proof: Let us consider z ∈ ker P * . Consequently we have P * z = 0. We can rewrite it

z 1 z 2 = 1 0 Q 11 (ξ, x)z 1 (t, ξ) + Q 21 (ξ, x)z 2 (t, ξ) Q 12 (ξ, x)z 1 (t, ξ) + Q 22 (ξ, x)z 2 (t, ξ) dξ (145)
If we evaluate the first line for x = 1 and the second one for x = 0, using the fact that Q 11 (ξ, 0) = Q 21 (ξ, 0) = Q 12 (ξ, 1) = Q 22 (ξ, 1) = 0, we get

z 1 (1) = z 2 (0) = 0 (146)
Consequently z ∈ D(A * 0 ) and we can write

ker P * ⊂ D(A * 0 ) (147)
Lemma 8: ker P * ⊂ ker B * Proof: Let us consider z ∈ ker P * . Consequently we have P * z = 0. We can rewrite it

z 1 z 2 = 1 0 Q 11 (ξ, x)z 1 (t, ξ) + Q 21 (ξ, x)z 2 (t, ξ) Q 12 (ξ, x)z 1 (t, ξ) + Q 22 (ξ, x)z 2 (t, ξ) dξ (148)
If we evaluate the first line for x = 0 and the second one for x = 1, using the fact that

Q 11 (ξ, 1) = Q 21 (ξ, 1) = Q 12 (ξ, 0) = Q 22 (ξ, 0) = 0, we get z 1 (0) = z 2 (1) = 0 (
= z 1 (t, x) T x Λ + + z 1 (t, x) T Ω + z 2 (t, x) T Γ -νz 1 (t, x) T -z 2 (t, x) T x Λ -+ z 2 (t, x) T Ω + z 1 (t, x) T Γ -νz 2 (t, x) T (151)
with the boundary conditions

z 1 (0) = z 2 (0) = 0 (152)
Consequently (using the Cauchy-Lipschitz' theorem) we have z = 0 0 We can now state the following theorem Theorem 5: The map P * = Id -Q * is invertible Proof: Since Q * is a compact operator we can use the Fredholm alternative (e.g [START_REF] Brezis | Functional analysis, sobolev spaces and partial differential equations[END_REF]): Id 2 -Q * is either non-injective or surjective. Consequently it suffices to prove that P * is injective. In addition, the Fredholm alternative also gives [START_REF] Brezis | Functional analysis, sobolev spaces and partial differential equations[END_REF] dim ker(Id -Q * ) < +∞ (153)

By contradiction we assume that ker P * = {0}. We first prove that ker P * is stable by A * 0 . We have ker P * ⊂ D(A * 0 ). Let then consider z ∈ ker P * . Using (144) we can obtain

P * A * 0 z = 0 (154)
We thus have A * 0 z ∈ ker P * . Consequently the restriction A * 0 |kerP * of A * 0 to ker P * is a linear operator from ker P * to ker P * . Since the dimension of ker P * is finite we can find at least one eigenvalue ν. Let e ∈ ker P * be a corresponding eigenvector (by definition e = 0). We have e ∈ ker P * and so e ∈ ker B * . Moreover we have A * 0 e = νe. Consequently e ∈ ker(νId -A * 0 ) ∩ kerB * (155) which contradicts Lemma 9 and concludes the proof.

VI. CONTROL LAW AND MAIN RESULTS

We now state the main stabilization result as follows: Theorem 6: System (1)-( 2) with the following feedback control laws

U (t) = - 1 0 (K(0, ξ)u(t, ξ) + L(0, ξ)v(t, ξ))dξ (156)
V (t) = - 1 0 ( M (1, ξ)u(t, ξ) + N (1, ξ)v(t, ξ))dξ (157)
where K, L and M , N are defined by (65)-( 70) and ( 83)-(88), reaches its zero equilibrium in finite time t F , where t F is given by [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic pdes[END_REF]. The zero equilibrium is exponentially stable in the L 2 -sense.

Proof: Notice that evaluating (52) at x = 0 yields (156) and evaluating (53) at x = 1 yields (157). Since the kernels are invertible, there exists a unique operator S such that

u v = S α β (158)
Applying Lemma 2 implies that (α, β) go to zero in finite time t F , therefore (u, v) converge to zero in finite time t F VII. OBSERVER DESIGN AND OUTPUT FEEDBACK

CONTROLLER

In this section we design an observer that relies on the measurements of u at the right boundary and v at the left boundary, i.e we measure

y 1 (t) = u(t, 1) (159) y 2 (t) = v(t, 0) (160)
Then, using the estimates given by our observer and the control law ( 156)-(157), we derive an output feedback controller.

A. Observer design

The observer equations reads as follows

ût (t, x) + Λ + ûx (t, x) = Σ ++ û(t, x) + Σ +-v(t, x) (161)
-P 11 (x)(û(t, 1) -u(t, 1)) -P 12 (v(t, 0) -v(t, 0)) (162)

vt (t, x) -Λ -vx (t, x) = Σ -+ û(t, x) + Σ --v(t, x) -P 21 (x)(û(t, 1) -u(t, 1)) -P 22 (v(t, 0) -v(t, 0)) (163)
with the boundary conditions

û(t, 0) = U (t), v(t, 1) = V (t) (164)
where P 11 (•), P 21 (•), P 12 (•) and P 22 (•) have yet to be designed. This yield the following error system ũt (t, x) + Λ + ũx (t, x) = Σ ++ ũ(t, x) + Σ +-ṽ(t, x) -P 11 (x)ũ(t, 1) -P 12 (x)ṽ(t, 0) (165) ṽt (t, x) -Λ -ṽx (t, x) = Σ -+ ũ(t, x) + Σ --ṽ(t, x) -P 21 (x)ũ(t, 1) -P 22 (x)ṽ(t, 0)

with the boundary conditions ũ(t, 0) = 0, ṽ(t, 1) = 0 (167)

This system evolves in [0, T ] × [0, x] and its initial condition (ũ(0, x), ṽ(0, x)) = (ũ 0 (x), ṽ0 (x)) belongs to L 2 ([0, 1]). We define the following system (which is the same as above changing t in T -t).

-

d dt ū v = Ā ū v (168) evolving in [0, T ] × [0, x] with the L 2 ([0, 1]) arbitrary initial condition ū(0, x) v(0, x) = ū0 (x) v0 (x) (169)
The operator Ā is thus defined by

Ā :D( Ā) ⊂ (L 2 (0, 1)) n+m → (L 2 (0, 1)) n+m ū v -→ -Λ + ūx + Σ ++ ū + Σ +-v Λ -vx + Σ -+ ū + Σ --v + -P 11 (x)ū(t, 1) -P 12 (x)v(t, 0) -P 21 (x)ū(t, 1) -P 22 (x)v(t, 0) (170)
with

D( Ā) = {(u, v) ∈ (H 1 (0, 1)) n+m |ū(0) = v(1) = 0} (171) B. A dual system
In order to find the observer gains we define a control problem that is the dual of the observer problem. The observer gains will then be defined by the gains of the dual controller.

1) A new control problem Let us consider the following system

φ t (t, x) -Λ + φ x (t, x) = (Σ -+ ) T ψ(t, x) + (Σ ++ ) T φ(t, x) (172)
ψ t (t, x) + Λ -ψ x (t, x) = (Σ --) T ψ(t, x) + (Σ +-) T φ(t, x) (173)
evolving in {(t, x)| t > 0, x ∈ [0, 1]}, with the following linear boundary conditions

ψ(t, 0) = U (t), φ(t, 1) = V (t) (174)
and the arbitrary initial conditions (belonging to L 2 ([0, 1]))

φ(0, x) = φ 0 (x), ψ(0, x) = ψ 0 (x) (175)
Using Theorem 6, we can explicitly compute kernels K 1 , L 1 , M1 and N1 such that system (172)-( 173) with the following feedback law

U (t) = - 1 0 (K 1 (0, ξ)φ(t, ξ) + L 1 (0, ξ)ψ(t, ξ))dξ (176) V (t) = - 1 0 ( M1 (1, ξ)φ(t, ξ) + N1 (1, ξ)ψ(t, ξ))dξ (177)
reaches its zero equilibrium in time t F where t F is given by [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic pdes[END_REF]. As seen in Section II-B, this system can be rewritten in the abstract form

d dt φ ψ = A φ ψ + B U V (178)
with the initial condition

φ(0, x) ψ(0, x) = φ 0 (x) ψ 0 (x) (179)
where A and B are defined in a similar form as the one presented in equations ( 12)-( 17).

2) Definition of the observer gains

We now define the gains of the observer as

P 11 (x) = M1 (1, x) T Λ + P 21 (x) = N1 (1, x) T Λ + (180) P 12 (x) = K 1 (0, x) T Λ -P 22 (x) = L 1 (0, x) T Λ -(181)
Theorem 7: System (168)-(169) where P 11 , P 12 , P 21 and P 22 are defined by (180)-( 181) is the adjoint of system (178)-(179). Moreover, both system reach their zero equilibrium in time t F where t F is defined by [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic pdes[END_REF].

Proof: We define < •, • > as the scalar product associated to the L 2 -norm. For every solution (ũ, ṽ) of ( 168)-( 169) and every solution (φ, ψ) of ( 178)-(179) (with any initial conditions) we have

< d dt φ ψ -A φ ψ -B V U , ũ ṽ >= 0 (182) It yields 0 = τ 0 1 0 (-ū T t (t, x) + ūT x (t, x)Λ + -ūT (t, x)(Σ ++ ) T -v(t, x) T (Σ +-) T )φ(t, x) + (-v T t (t, x) -vT x (t, x)Λ - -ū(t, x)(Σ -+ ) T -v(t, x)(Σ --) T )ψ(t, x)dxdt + 1 0 (v T (τ, x)ψ(τ, x) -vT (0, x)ψ(0, x) + ūT (τ, x)φ(τ, x) -ūT (0, x)φ(0, x))dx + τ 0 1 0 (ū(t, 1) T Λ + ( M1 (1, x)φ(t, x) + N1 (1, x)ψ(t, x)) + v(t, 0) T Λ -(K 1 (0, x)φ(t, x) + L 1 (0, x)ψ(t, x)))dt (183)
Using this expression, it can easily be seen that system (168)-( 169) is the adjoint of system (178)-(179). Moreover, taking

τ = t F yields 0 =< φ ψ , - d dt ū v -Ā ū v > (184) + 1 0 -ū T (0, x)φ(0, x) -vT (0, x)ψ(0, x)dx ⇒ 0 = ūT (0, x)φ(0, x) + vT (0, x)ψ(0, x) (185)
Since this has to be true for any initial condition, it implies that

v(0, x) = 0, ū(0, x) = 0 (186)
Consequently using the change of variable t = t F -t yields ṽ(t F , x) = 0, ũ(t F , x) = 0 (187)

C. Output feedback controller

The estimates can be used in a observer-controller to derive an output feedback law yielding finite-time stability of the zero equilibrium Lemma 10: Consider the system composed of (1)-(3) and target system (162)-(164) with the following control law

U (t) = - 1 0 (K(0, ξ)û(t, ξ) + L(0, ξ)v(t, ξ))dξ (188) V (t) = - 1 0 ( M (1, ξ)û(t, ξ) + N (1, ξ)v(t, ξ))dξ (189)
where K, L and M , N are defined by (65)-( 70) and ( 83)-(88). Its solutions (u, v, û, v) converge in finite time to zero Proof: The convergence of the observer error states ũ, ṽ to zero for t F ≤ t is ensured by Theorem 7, along with the existence of the backstepping transformation. Thus, once t F ≤ t, v(t, 0) = v(t, 0) and one can use Theorem 6. Therefore for 2t F ≤ t, one has (ũ, ṽ, û, v) ≡ 0 which yields (u, v) ≡ 0.

VIII. APPLICATION: STATE ESTIMATION DURING UNDERBALANCED DRILLING

In this section, we illustrate the benefits of our approach by applying it to an industrial problem. around its main axis, through which is injected a drilling fluid, typically water-based mud. At the end of the pipe, the fluid exits through the drillbit (which chatters and cuts the rock) and circulates back to the surface inside the annulus, carrying rock cuttings. For numerous reasons [START_REF] Aarsnes | Modeling of two-phase flow for estimation and control of drilling operations[END_REF], it is desirable to produce oil and gas from the reservoir as the drilling process goes on when possible, a technique referred to as UnderBalanced Drilling (UBD). The term underbalanced refers to the value of the pressure at the bottom of the annulus, which must be lower than the value of the pressure of hydrocarbons inside the reservoir (called the balance point) for the oil and gas to flow in. To ensure safety and efficiency of operations, it is desirable to monitor at all time the amount of gas inside the well. This is a difficult task since sensors cannot be placed all along the drillpipe, and the dynamics of multiphase flow are known to be complex [START_REF] Aarsnes | A methodology for classifying operating regimes in underbalanced-drilling operations[END_REF]. The presence of gas, in particular, makes the distributed, delay-like nature of the dynamics predominant and may generate instabilities, such as severe slugging.

B. Observer design

The simulation model is a Drift-Flux Model (DFM) described in [START_REF] Aarsnes | Controloriented drift-flux modeling of single and two-phase flow for drilling[END_REF]. It models the flow of liquid (oil, water and drilling fluid being considered as one liquid phase) and gas along the drillstring using two mas conservation laws and one combined momentum conservation law. Along with closure relations, this yields a set of three nonlinear transport PDEs with appropriate boundary conditions. The model can be linearized around a given equilibrium profile, which yields a system of the form (1)-( 3) with n = 2 and m = 1 (see, e.g., [START_REF] Aarsnes | A methodology for classifying operating regimes in underbalanced-drilling operations[END_REF]).

The model used to design the observer is the same model, starting from a different initial condition. This choice is debatable, since perfect knowledge of the model is unrealistic, however assessing the robustness of the model with respect to model uncertainty is out of the scope of this paper and will be the topic of future contributions. Notice that the nonlinearity of the model, combined with the offset in the initial condition already yields differences in the linearized model parameters.

Similarly to [START_REF] Di Meglio | A backstepping boundary observer for a class of linear first-order hyperbolic systems[END_REF], the observer is designed by copying the nonlinear equations and adding the linear output error correction terms such that linearizing the observer equations yields (161)-(164).

C. Simulation results

Figure 5 depicts snapshots of the volume fraction of liquid (holdup), as a function of well depth at different time instants of a transient simulation. The three curves respectively correspond to the 'plant', the observer and an open-loop simulation starting from the same initial condition as the observer, without any correction terms. The holdup profile of the observer converges in approximately 25 minutes to the plant profile, while the open-loop profile would take around two hours (the settling time of this particular system) to converge. This plot is not shown here due to lack of space.

IX. CONCLUDING REMARK

Using a backstepping approach we have presented a stabilizating boundary feedback law for a general class of linear first-order system controlled in both boundaries. Moreover, the zero equilibrium of the system is reached in minimum time t F which is the largest time between the two transport times in each direction. Besides, the presented result narrows the gap with the theoretical controllability results of [START_REF] Li | Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems[END_REF]. These results, although they do not provide explicit control law, ensure exact minimumtime controllability with less control inputs than what is currently achievable using backstepping. More generally, this raises the question of the links between controllability and stabilization by backstepping. In particular future works will consider first-order systems with a smaller number of controls in each boundary.

  149) Consequently z ∈ ker B * and we can write kerP * ⊂ kerB * (150) Lemma 9: ∀λ ∈ ker(λId -A * 0 )∩ ker B * = {0} Proof: Let us consider ν ∈ and z ∈ ker(νId -A * 0 )∩ ker B * = {0}. Consequently we have 0

A

  Fig. 4. Schematic view of drilling facilities. In UnderBalanced Drilling (UBD), oil and gas enter the annulus as the drilling process goes on.

Fig. 5 .

 5 Fig. 5. Snapshots of the spatial profile of the liquid holdup as a function of depth. The profile of the observer converges to the true profile in approximately 25 minutes.

APPENDIX

In this appendix, we prove Theorem 3. Assume that the assumptions of Theorem 3 hold then the system (89) with boundary conditions (91) has an unique solution F ∈ L ∞ (D). Classically (see [START_REF] John | Continuous dependence on data for solutions of partial differential equations with a prescribed bound[END_REF] and [START_REF] Beresford | Linear and nonlinear waves[END_REF]), the proof follows three steps:

• First, we compute the characteristic lines • In each domain the equations are transformed into integral equations. • Finally, a method of successive approximations is used to find a solution to the integral equations.

A. Transformation into integral equations

The first assumption of Theorem 3 yields the existence and uniqueness of characteristic curves defined as follows.

For each 1 ≤ j ≤ n + m and (x, ξ) ∈ D, we now define the following characteristic lines (x j (x, ξ, •), ξ j (x, ξ, •)) corresponding to equation (89)

These lines originate at the point (x 0 j (x, ξ), ξ 0 j (x, ξ)) located on the boundary of the domain and terminate on (x, ξ). Integrating (89) along these characteristics and using boundary conditions (91) yields

where we denote M j (x, ξ, s) = (x j (x, ξ, s), ξ j (x, ξ, s)) and M 0 j (x, ξ, s) = (x 0 j (x, ξ, s), ξ 0 j (x, ξ, s))

B. Method of successive approximations

In order to solve (192) we use the method of successive approximations. We define

We now construct the sequence F p defined by

Consequently, if the sequence F p has a limit, then this limit is a solution of the integral equation and therefore of the original system.

We define the increment ∆F p = F p -F p-1 . Provided the limit exists one has

We now prove the convergence of the series.

C. Convergence of the successive approximations series

The proof of convergence is based on the following Lemmas.

Lemma 11: Assume that (92) holds. then for all j = 1 . . . n + 1, (x, ξ) ∈ D, the following function is strictly increasing

Proof: The proof is trivial. Recalling (92) yields

This concludes the proof. Lemma 12: For all j = 1 . . . n+m the following inequalities hold

Proof: Consider the following change of variables

It yields

Using ( 92) and the definition of δ, this yields

Since (x 0 j (x, ξ), ξ 0 j (x, ξ)) ∈ D and since α > 0, one has -αx 0 j (x, ξ) -ξ 0 j (x, ξ) + α + 1 ≥ 0 which yields the result. Lemma 13: Let M > 0 be such that

where Σ is defined as

If for some 1 ≤ q and some f > 0 one has, for all (x, ξ)

Then one has ∀j = 1 . . . m + n

Proof: Assume that (205) holds for some fixed

Using (205) this yields


A Ristori

M Orain

L Vingert

C Brossard

N Dorval

A Nicole

A Guy

V Rialland

S Godel

Caractérisation expérimentale de la couche de mélange dans un jet de post-combustion et comparaison aux simulations LES

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Lorsqu'un missile balistique vole à 20 km d'altitude, la pression en sortie de tuyère est d'environ 20 fois la pression atmosphérique locale. Dans ces conditions, le jet supersonique présente une structure caractéristique comprenant une succession de chocs droits qui forment des "disques de Mach". Cette situation peut être partiellement reproduite sur le banc MASCOTTE de l'ONERA à Palaiseau [START_REF] Vingert | Mascotte, a Test Facility for Cryogenic Combustion Research at High Pressure[END_REF][START_REF] Vingert | Upgrading of the Mascotte Cryogenic Test Bench to the LOX/Methane Combustion Studies[END_REF]. Dans le cadre du Projet de Recherche Fédérateur (PRF) SIMBA (SIgnature de jet de Missile Balistique), une expérimentation a été menée en 2015 sur ce banc, afin de constituer une base de données de mesures de référence sur une flamme de post-combustion de propulseur. Cette base permettra de valider la simulation LES réactive mise en oeuvre pour simuler cette flamme. Différentes mesures ont été obtenues dans le jet avec de hautes cadences d'acquisition afin de suivre les fluctuations liées aux instabilités dans la couche de mélange, détecter la transition vers la turbulence et étudier l'évolution des structures cohérentes, les oscillations de la flamme de diffusion turbulente, ainsi que la dynamique temporelle de la signature infrarouge : mesures de signature infrarouge (imagerie et spectroradiométrie), champs de vitesse obtenus par PIV stéréoscopique, champs de concentrations de OH par PLIF et émission de OH excité (OH*).

Cet article présente l'ensemble de l'expérimentation mise en place, et les principaux résultats obtenus par PIV, PLIF-OH, et par visualisation de l'émission de OH*. Ces résultats permettent de mettre en évidence l'aérodynamique du jet, ainsi que l'évolution temporelle des structures tourbillonnaires dans la couche de mélange et de la structure du front de flamme. Les résultats sont ensuite comparés aux simulations RANS et LES du jet expérimental calculées par le code CEDRE de l'ONERA.

Montage expérimental 2.1 Chambre de combustion et tuyère

Le montage est visible sur la figure 1. La chambre de combustion BHP (Boîtier Haute Pression) a déjà été mise en oeuvre avec succès à l'ONERA dans un PRF précédent pour une combustion cryotechnique H 2 /O 2 [START_REF] Roblin | Comparison between computations and measurements of a H2/LOX rocket motor plume[END_REF][START_REF] Bresson | Experimental investigation of a supersonic rocket engine plume using OH*-emission, OH-PLIF and CARS thermometry[END_REF]. La tuyère a été définie de sorte à avoir un rapport de pression Ps/Pa de 5 pour une pression de 50 bar dans la chambre de combustion. Elle a été réalisée en cuivre massif. Son rapport de section As/Ac (Section Sortie / Section Col) est de 2,48. Pour la mise en oeuvre de la technique PLIF-OH, la tuyère en cuivre massif déjà mise en oeuvre dans [START_REF] Roblin | Comparison between computations and measurements of a H2/LOX rocket motor plume[END_REF][START_REF] Bresson | Experimental investigation of a supersonic rocket engine plume using OH*-emission, OH-PLIF and CARS thermometry[END_REF] a été reprise. Pour la mise en oeuvre de la technique PIV, une tuyère de même profil mais avec fente annulaire d'injection au culot a été spécifiquement conçue et utilisée afin d'assurer l'ensemencement de la couche de mélange en traceurs. L'utilisation d'un mélange réactif gazeux CH 4 /O 2 permet d'améliorer la représentativité des émissions d'un jet de post-combustion de propulseur à kérosène en générant des produits de combustion carbonés en sortie de chambre de combustion. Le choix du point de fonctionnement a résulté d'un compromis entre différentes contraintes contradictoires liées d'une part à la tenue thermique de la chambre de combustion (chambre et tuyère massives refroidies uniquement en puits de chaleur), et d'autre part à la détectabilité du signal de PLIF-OH. Les conditions nominales de ce point sont un rapport de mélange (RM, rapport des débits massiques d'oxygène et de méthane) de 1,8, et un débit total de 81 g/s, correspondant à une pression chambre de 8,5 bar. L'ensemble des données de PLIF-OH et PIV ont été obtenues sur deux campagnes d'essais successives. Pour chacune, une caméra intensifiée (modèle Photron APX-I2), munie d'un filtre passe-bande (10 nm) centré à 310 nm, a été synchronisée avec chaque technique laser, afin de visualiser en champ large l'émission de OH*. Cette émission permet de tracer la zone de réaction, et ainsi de disposer d'une référence de comparaison sur l'ensemble des tirs. L'intensité enregistrée est aussi, en première approximation, proportionnelle au taux de dégagement de chaleur, qui est l'une des sorties du calcul LES. Le signal enregistré est intégré sur toute l'épaisseur optique du jet, mais on peut se ramener dans un plan par une inversion d'Abel, moyennant une hypothèse d'axisymétrie, bien vérifiée sur une image moyenne.

Technique PLIF-OH haute cadence

Le système PLIF-OH à haute cadence est composé :

-d'un laser Nd:YAG pompé par diodes laser (Modèle IS2011-E, EdgeWave) de forte puissance et opérant à haute cadence doublé en fréquence (532 nm). Il délivre des impulsions laser d'une durée comprise entre 6 et 10 ns selon la cadence de 4 à 10 kHz. L'énergie par impulsion est au maximum égale à 9,7 mJ à 10 kHz (P moyenne = 97 W) et 12 mJ à 4 kHz (P moyenne = 47 W). Ce laser sert à pomper le laser à colorant ; -d'un laser à colorant (Modèle Credo, Sirah Lasertechnic) qui opère avec de la rhodamine 6G diluée dans l'éthanol. Le laser est accordé à 566 nm et doublé en fréquence par un cristal BBO pour générer un faisceau laser à 283 nm. La largeur spectrale de la raie laser est de 0,12 cm -1 à 283 nm. L'énergie par impulsion est égale à 420 µJ à 10 kHz et 650 µJ à 4 kHz ; -d'une caméra de type CMOS (Modèle HSS6, LaVision), munie d'un intensificateur (Modèle IRO, LaVision). Ce système est équipé d'un objectif UV 94 mm (F/4,1, Cerco) et d'un jeu de filtres interférentiel (centré à 310 nm, de largeur à mi-hauteur égale à 10 nm, transmission maximale égale à 70 %) et passe-haut (transmission égale à 95 % au-dessus de 300 nm). La taille du champ observé est de 64×64 mm 2 , ce qui conduit à une résolution spatiale égale à 62,5 µm/pixel. L'ensemble est piloté par le logiciel DaVis 8 (LaVision). Pour les présentes expériences, la cadence de répétition du système PLIF-OH est fixée à 4 kHz, de manière synchrone avec la caméra détectant l'émission OH*, opérée à cette même cadence. 10000 images sont enregistrées par acquisition, sur une durée de 2,5 s. L'implémentation du système PLIF-OH sur le banc MASCOTTE est visible sur la figure 1. Le faisceau laser UV est renvoyé depuis la salle de contrôle à la position du jet par l'intermédiaire de miroirs. Il est transformé en une nappe laser focalisée et collimatée à l'aide d'une lentille sphérique (de focale égale à 1000 mm) et de deux lentilles cylindriques rectangulaires (de focales respectivement égales à -19 mm et 150 mm). L'épaisseur de la nappe est de 150 μm. La nappe laser est située sur l'axe de la veine d'essais et vient exciter les radicaux OH présents dans le jet réactif. La propagation de la nappe laser se fait verticalement, de bas en haut. Le système de détection, une partie des optiques de génération de nappe laser, et les deux derniers miroirs de renvoi, sont solidaires d'une table de déplacement motorisée, afin de pouvoir explorer le jet sur sa longueur. 12 secteurs adjacents localisés à différentes positions longitudinales Z du plan de sortie de la tuyère comprises entre X=16 mm et X=256 mm ont été sondés. Lorsque l'on s'éloigne de la sortie de la tuyère, la couche de mélange réactive s'épaissit, avec une augmentation de la concentration du radical OH, ce qui suggère un taux de dégagement de chaleur plus élevé. Ces observations sont en accord avec les visualisations synchronisées d'émission de OH*.

La faible étendue spatiale du radical OH dans les 2 ème (36 < X < 56 mm) et 3 ème (56 < X < 76 mm) secteurs est expliquée par le très faible niveau de signal de fluorescence, qui ne peut pas alors être corrigé efficacement pour l'inhomogénéité de la nappe laser (le signal est présent uniquement au centre de la nappe où le laser est suffisamment intense).

L'observation de séquences d'images instantanées successives permet de mettre en évidence l'intermittence de la flamme. Par exemple, la figure 4 montre une séquence de dix images enregistrées dans le 5 ème secteur (96 < X < 116 mm). On constate que la flamme peut être présente simultanément dans les parties basse et haute de l'image, seulement dans la partie basse, uniquement dans la partie haute, ou enfin totalement absente. L'observation de séquences d'images instantanées montre également que la cadence d'acquisition, bien qu'élevée (4 kHz), reste insuffisante pour permettre un suivi temporel de structures turbulentes dans le jet supersonique réactif, ce qui n'est pas étonnant compte tenu des hauts niveaux de vitesse (1000 à 2000 m/s mesurés par PIV dans le coeur du jet, cf. figure 9). La figure 5 montre les résultats déduits des mesures PIV. Sur l'image de gauche, le champ de vitesse moyenne a été reconstitué sur une distance de 290 mm depuis le plan de sortie de la tuyère. Les valeurs de X reportées sur la figure correspondent aux coordonnées axiales amont de chacun des 4 secteurs sondés. Cette image montre également les zones de recouvrement entre les différents champs mesurés. L'augmentation de l'étendue verticale de la zone de vecteurs vitesses lorsqu'on s'éloigne du plan de sortie de la tuyère traduit l'épaississement de la couche de mélange : en effet, aucune mesure hors de la couche de mélange sur l'extérieur du jet n'est possible faute de traceurs. Sur l'image de droite, le champ moyen de vitesse (avec un vecteur sur 10 pour plus de visibilité) obtenu en sortie de tuyère est superposé au champ moyen de vorticité. La zone où la vorticité change de signe correspond à la zone de cisaillement. Figure 5 : Champs moyens (~5500 champs instantanés) obtenus par PIV dans le jet réactif supersonique : de vitesse (gauche), reconstitué sur une distance de 256 mm en aval du plan de sortie de la tuyère (X=0) ; de vitesse et de vorticité, de 0 à 90 mm du plan de sortie de la tuyère La figure 6 montre 12 champs instantanés successifs de lignes de courant déduites des champs de vitesse, obtenus entre 60 et 150 mm du plan de sortie de la tuyère. Ces champs font apparaître des structures en forme de « vagues », qui semblent correspondre à la partie inférieure de larges structures tourbillonnaires, telles que celle visible sur le champ 5. Il est parfois possible de suivre visuellement le déplacement de ces structures entre deux images successives (par exemples sur les champs 1 à 5), ce qui suggère que le suivi temporel est possible lorsque l'échelle spatiale de ces structures est suffisamment grande.

Comparaison entre résultats expérimentaux et simulations numériques

La simulation de la flamme de recombustion dans le panache d'un moteur-fusée est ardue à mettre en oeuvre car des phénomènes physiques complexes entrent en jeu : jet supersonique, combustion turbulente, écoulement instationnaire. Les tirs expérimentaux réalisés sur le banc MASCOTTE servent alors de base de données de référence pour la validation des modèles utilisés pour les simulations numériques. Tout d'abord, un calcul stationnaire de type RANS (Reynolds-averaged Navier-Stokes equations) a permis une première estimation des caractéristiques globales des jets. Cependant, avec de telles équations moyennes, il n'est pas toujours possible de prédire correctement les caractéristiques du jet et de la flamme, les structures instationnaires restant inaccessibles. Afin d'obtenir une description plus fine des jets, une simulation LES (Large Eddy Simulation) s'est imposée. Elle permet d'améliorer la description de la turbulence dans le jet et la flamme et elle donne accès à des champs instantanés d'écoulements et à leurs moyennes pour une exploitation plus complète de ces résultats numériques.

Conclusion

Un jet supersonique avec flamme de post-combustion a été étudié expérimentalement sur le banc MASCOTTE de l'ONERA. La distance d'apparition de la flamme de post-combustion (par autoallumage) depuis le plan de sortie de la tuyère, et l'épaississement de la couche de mélange réactive ont été mises en évidence par PLIF-OH. La flamme est décrochée de la tuyère d'une distance environ égale à 40 mm. L'aérodynamique de la couche de mélange a été caractérisée par PIV. L'emploi de ces deux techniques laser à haute cadence (~4 kHz) dans un jet supersonique avec post-combustion constitue, à la connaissance des auteurs, une première mondiale. Sur l'ensemble des essais menés en conditions réactives, des visualisations de l'émission de OH* ont été synchronisées avec chacune de ces deux techniques laser, afin de tracer la zone de réaction et de disposer d'une référence de comparaison sur l'ensemble des essais. Si elle permet de mettre en évidence l'intermittence de la flamme, l'observation de séquences d'images instantanées successives obtenues par PLIF-OH montre qu'une cadence d'acquisition de 4 kHz reste insuffisante pour le suivi temporel des fines structures turbulentes dans un tel jet supersonique réactif. En revanche, à une cadence d'acquisition proche, le suivi temporel est possible sur des structures déduites des champs de vitesse obtenus par PIV dans la couche de mélange lorsque l'échelle spatiale de ces structures est suffisamment grande. Les résultats obtenus par PLIF-OH sont en accord avec les visualisations en champ large de l'émission de OH* réalisées de manière synchrone avec l'imagerie LIF, lors de chaque essai.

Les résultats obtenus dans cette étude ont permis de constituer une base de données de mesures de référence sur une flamme de post-combustion de propulseur. Cette base permettra de valider la simulation LES réactive mise en oeuvre pour simuler cette flamme. Les premiers résultats de simulation LES présentés dans cet article utilisent des modèles simples dans un premier temps, notamment en ce qui concerne la combustion turbulente. Les comparaisons entre ces simulations et les résultats expérimentaux montrent une amélioration par rapport aux simulations RANS, à la fois sur les niveaux de vitesse et sur la longueur de la flamme. Des perfectionnements des méthodes utilisées pour le calcul LES sont envisagés, comme le raffinement du maillage et la mise en place d'un modèle de combustion plus complexe.

La structure d'un jet supersonique non réactif a également été caractérisée sur la base de douze essais à l'hélium, non présentés dans cet article, afin d'enrichir la base de données déjà constituée par des données purement aérodynamiques.

Figure 1 :

 1 Figure 1 : Montage expérimental avec montage PLIF-OH sur le banc MASCOTTE : montage optique (gauche) et jet supersonique réactif (droite) Les images brutes enregistrées sont traitées en plusieurs étapes. Elles sont d'abord corrigées pour l'inhomogénéité spatiale d'énergie dans la nappe laser, qui est importante ici car le profil spatial d'énergie du faisceau laser d'origine n'est pas gaussien : la distribution en énergie est très piquée (point chaud au centre de la tache laser). Le faisceau est elliptique, et de petites dimensions (hauteur 2 mm, largeur 1 mm). En conséquence, la taille de la nappe laser efficace pour générer du signal LIF (largeur à 50 % de l'intensité laser maximale) est de 10 mm. Ce défaut d'homogénéité de la nappe laser rend le traitement des images plus difficile et est source d'incertitudes sur l'amplitude réelle du signal aux bords des images. Les images sont ensuite corrigées de l'absorption de la nappe laser le long de sa propagation dans le jet réactif (du bas vers le haut), et des pertes de signal par quenching. Cette correction utilise les cartographies bidimensionnelles de concentrations des trois molécules H 2 O, CO et CO 2 (produits de combustion majoritaires) et de température obtenues par le calcul LES du jet réactif. Ces données sont ensuite utilisées comme données d'entrée pour simuler le signal de fluorescence du radical OH dans les conditions thermochimiques du jet, à l'aide d'un code de simulations spectroscopiques développé à l'ONERA, qui tient également compte des corrections instrumentales (laser et caméra) [5]. Les images sont ensuite lissées à l'aide d'un filtre gaussien (5×5 pixels) permettant de gommer partiellement les imperfections liées aux corrections précédentes, avant d'être converties en images de concentration du radical OH à l'aide de l'étalonnage de la technique PLIF-OH sur une flamme de référence, extrapolé à partir des résultats des simulations spectroscopiques.

Figure 2 :

 2 Figure 2 : Montage expérimental et montage PIV sur le banc MASCOTTE (Mi : miroir de renvoi ; C1 et C2 : miroirs cylindriques de focales respectives -60 et +500 mm) ; S : miroir sphérique de focale +981 mm)

Figure 3 -Figure 4 -

 34 Figure 3 -Image moyenne (10000 images instantanées) de la concentration de OH (en cm -3 ) dans le jet réactif supersonique, reconstituée sur une distance de 256 mm en aval du plan de sortie de la tuyère (X=0)

Figure 6 :

 6 Figure 6 : Evolution temporelle des champs instantanés de lignes de courant obtenus entre 60 et 150 mm du plan de sortie de la tuyère

Figure 7 :

 7 Figure 7 : Zone simulée Figure 8 : Schéma de la configuration numérique

Figure 9 :

 9 Figure 9 : Comparaison de la vitesse sur l'axe

Figure 10 :

 10 Figure 10 : Comparaison OH* et LES entre X/D=0 et X/D=20

2.3 Technique PIV stéréoscopique haute cadence

  Le système PIV stéréoscopique à haute cadence est composé :-d'un système laser Quantronix Darwin-Dual à deux oscillateurs Nd:YLF pompés par diodes, doublés en fréquence à 527 nm et pouvant chacun fonctionner à une cadence comprise entre 0,1 et 10 kHz. Un ensemble optique composé de lames demi-onde, de miroirs de renvoi et d'un polariseur par séparation de faisceau permet de superposer spatialement les deux faisceaux. ) de chaque caméra est de 7525 Hz, correspondant à une cadence PIV moitié de 3762 Hz, valeur utilisée pour ces essais. 5528 doublets d'images sont enregistrés par acquisition, sur une durée de 1,5 s. L'acquisition est synchronisée avec la caméra détectant l'émission OH*, opérée à la même cadence 3762 Hz. Le champ observé a une longueur de 90 mm, correspondant à la taille de la nappe laser, et une hauteur de 56 mm. Seule la partie supérieure du jet est étudiée, le bord inférieur des images se situant à 8 mm en dessous de l'axe du jet. Le point de fonctionnement a été légèrement modifié par la présence de ce débit d'hélium, afin de conserver la même température de gaz en sortie de tuyère : le RM est de 1,9, pour un débit total de 79 g/s (dont 4,8% d'hélium d'ensemencement), correspondant à une pression chambre de 8,3 bar. L'ensemencement du jet permet également de mesurer la vitesse dans le coeur du jet. Pour ensemencer au mieux la couche de mélange, et compte tenu de l'impossibilité technique d'ensemencer l'environnement autour du jet, une tuyère de même profil mais avec fente annulaire d'injection au culot, donc autour du jet supersonique, a été spécifiquement conçue. Par cette fente est injecté un débit d'environ 1,4 g/s d'azote chargé en particules d'alumine (Al 2 O 3 ) après passage dans un générateur de particules TOPAS, modèle SAG-410/H.

	Pour
	chaque cavité, l'énergie maximale est de 21 mJ/pulse à 1 kHz. Deux atténuateurs intra-cavité
	permettent d'ajuster le niveau d'énergie pour chacun des faisceaux (6,4 mJ/pulse à 3,762 kHz
	pour ces essais) ;
	-de 2 caméras de type CMOS (modèle Vision Research Phantom V711), chacune munie d'un
	objectif motorisé Canon 135 mm et d'un filtre passe-bande (10 nm) centré à 527 nm, et montée sur
	un adaptateur Scheimpflug (modèle V4, LaVision). La cadence maximale à pleine résolution
	(1280x800 pixels

L'ensemble est piloté par le logiciel DaVis 8 (LaVision). L'implémentation du système PIV sur le banc MASCOTTE est visible sur la figure

2

. Les faisceaux laser superposés sont renvoyés depuis la salle de contrôle à la position du jet par l'intermédiaire de miroirs. Ils sont transformés en 2 nappes laser situées sur l'axe de la veine d'essais, superposées, focalisées et collimatées, de largeur 90 mm et d'épaisseur estimées à environ 2 mm, par un jeu de lentilles et de miroirs décrits sur la figure

2

. Les caméras visent les nappes laser en diffusion avant, dans le plan horizontal, de part et d'autres des nappes laser, et avec un angle de visée de 45°. Comme pour le montage de PLIF-OH, la propagation des nappes laser se fait verticalement, de bas en haut, et la dernière lentille cylindrique C2, ainsi que les deux derniers miroirs de renvoi M3 et M4, sont solidaires d'une table de déplacement motorisée, afin de pouvoir explorer le jet sur sa longueur. Quatre secteurs localisés à différentes positions longitudinales Z du plan de sortie de la tuyère comprises entre Z=0 et Z=290 mm ont été sondés, avec un recouvrement d'environ 50% (au moins 20 mm sur chaque côté des champs de vitesse). Les deux derniers éléments optiques (lentille C2 et miroir M4) sont protégés par un capotage et ventilés au moyen d'un écoulement d'air afin de les protéger du rayonnement du jet supersonique et de la pollution par les traceurs PIV.

La qualité de l'ensemencement de la couche de mélange en traceurs est un point clé de la réussite de la mesure. C'est aussi un problème techniquement difficile : il exige a priori d'ensemencer à la fois l'écoulement internet du jet et l'environnement externe. L'écoulement interne est ensemencé dans la chambre BHP, en y introduisant un écoulement d'hélium chargé en particules d'oxyde de magnésium (MgO) après passage dans un générateur de particules de type "cyclone" pouvant fonctionner à 30 bar. L'étalonnage spatial des caméras est réalisé à partir d'une mire crénelée sur chaque face (2 plans de mire par face) LaVision (référence type 11). Les calculs de fonctions d'étalonnage spatial des caméras, ainsi que les traitements PIV par algorithme d'inter-corrélations des doublets d'images enregistrés, sont réalisés par le logiciel LaVision DaVis 8.2.

Remerciements

Les auteurs tiennent à remercier la Direction Scientifique Générale de l'ONERA pour le financement de cette étude. Des remerciements tout particuliers sont également adressés à David Carru, Eric Paux et Frank Vannier pour leur soutien technique.