Romain Loyer
email: romain.loyer@gipsa-lab.grenoble-inp.fr

Numerical control of the one-dimensional wave-equation with variable coefficients

Keywords: backstepping approach, wave equations, null controllability, numerical methods

The aim of this article is to study, both theoretically and numerically, the one-dimensional wave equation with variable coefficients. The coefficient appearing in the main part of the equation is of the class C 1 . A backstepping transformation is used to obtain the global finite-time stabilisation of the wave equation and thus its zero controllability. The kernel equations and the finite-time stabilisation of the wave equation are solved numerically with first-order convergence under a CFL condition and numerical simulations are presented to validate them.

Introduction

The backstepping approach is a powerful tool for the design of stabilising controllers in the context of finite dimensional systems. The key point of this method is to select a target system and backstep to it using a change of variables given by a Volterra operator. The boundary control is obtained explicitly and more easily from this Volterra transformation (see for example [START_REF] Krstic | Boundary Control of PDEs[END_REF][START_REF] Anfinsen | Adaptive control of linear2×2 hyperbolic systems[END_REF]). The solution of some hyperbolic systems with boundary conditions can reach equilibrium in finite time, this property is called finite time stability (see [START_REF] Alabau-Boussouira | Finite-time stabilization of a network of strings[END_REF][START_REF] Polyakov | Fixed-time stabilization via second order sliding mode control[END_REF]). This property was first observed and pointed out in [START_REF] Majda | Disappearing solutions for the dissipative wave equation[END_REF] in the case of the wave equation. In the case of linear systems, the use of a boundary feedback law based on the backstepping transform allows to obtain the finite-time stability for hyperbolic systems with source terms. The backstepping approach has been used for finite-time stability in [START_REF] Andréa-Novel | Exponential stabilization of an overhead crane with flexible cable via a back-stepping approach[END_REF][START_REF] Wijnand | Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension[END_REF] in the case of a flexible cable, for which the coefficient in the main part of the hyperbolic equation is given explicitly and for which a physical constraint is imposed. Our aim is to generalise these results to general hyperbolic systems with variable coefficients but no constraint.

Moreover, this method allows us to design an efficient numerical scheme without the use of regularisation, filtering or the addition of viscous terms. In fact, it's been known since the first works of [START_REF] Glowinski | A numerical approach to the exact boundary controllability of the wave equation[END_REF] that the design of numerical schemes for the wave equation is delicate. Using the HUM method and a finite difference scheme, it can be shown that the scheme is consistent and stable under the condition ∆ t ≤ h 2 (where ∆ t and h are the time and space steps, respectively). However, the discrete control obtained does not converge to the control because of the appearance of spurious oscillations for the high frequencies (see [START_REF] Zuazua | Propagation, observation, and control of waves approximated by finite difference methods[END_REF]) that don't exist at the continuous level. In other words, the control of the numerical approximation of a controllable system isn't necessary a good way to encode a control approximation for the partial derivatives model.

To sum up, the stability and convergence of a numerical scheme for solving a boundary value problem doesn't guarantee its stability at the level of controllability. A number of techniques have been developed to overcome this difficulty, we can mention the Tychonoff regularisation [START_REF] Glowinski | A numerical approach to the exact boundary controllability of the wave equation[END_REF], a multigrid strategy [START_REF] Glowinski | Ensuring well-posedness by analogy; stokes problem and boundary control for the wave equation[END_REF][START_REF] Mehrenberger | An Ingham type proof for a two-grid observability theorem[END_REF], a mixed finite element method [START_REF] Glowinski | A mixed finite element formulation for the boundary controllability of the wave equation[END_REF] or high frequency filtering [START_REF] Zuazua | Boundary observability for the finite-difference space semi-discretizations of the 2-d wave equation in the square[END_REF]. We refer to [START_REF] Ervedoza | The Wave equation : control and numerics[END_REF] for more details on these methods, but briefly discuss their advantages and disadvantages. Regularisation is an efficient method for moderate-sized problems with good stabilisation of the results, but the choice of the regularisation parameter is tricky. The multigrid strategy provides an efficient approach with the most explicit estimates, but does not provide the optimal control time. (For the wave equation we have T > 2 2 [START_REF] Ervedoza | The Wave equation : control and numerics[END_REF]). As with the Tychonoff regularisation, the choice of parameters for the filtering makes the task more difficult. However, with the right choice of parameters, this method can be used to get closer to the optimum time.

In [START_REF] Castro | Numerical approximation of the boundary control of the 2-d wave equation with mixed finite elements[END_REF], a semi-discrete finite element method is implemented with a discrete multiplier method for the 2D wave equation. The main advantage of this method is that it allows the construction of a convergent sequence of approximate controllers without filtering. The main disadvantage of this approach is a stricter stability condition: ∆ t ≤ h 2 instead of ∆ t ≤ h for the other methods. In [START_REF] Münch | A uniformly controllable and implicit scheme for the 1-d wave equation[END_REF], a step forward has been taken with the addition of viscous terms of order h 2 -∆ 2 t in the spirit of [START_REF] Krenk | Dispersion-corrected explicit integration of the wave equation[END_REF]. Under the condition

∆ t ≤ h T /2
, a uniform control is obtained using discrete Ingham inequalities. In [START_REF] Asch | Uniformly controllable schemes for the wave equation on the unit square[END_REF], a uniform control is obtained under the condition ∆ t ≤ h 2 using a constructive method based on [START_REF] Münch | A uniformly controllable and implicit scheme for the 1-d wave equation[END_REF].

Infante and Zuazua [START_REF] Infante | Boundary observability for the space semi-discretizations of the 1d wave equation[END_REF] detail the analysis of the 1D case and Zuazua [START_REF] Zuazua | Propagation, observation, and control of waves approximated by finite difference methods[END_REF] consider a semi-discretisation of the 2D wave equation with a finite difference or finite element method, but in practice this is not efficient because of the need to have good filtering in both spatial directions. High frequency filtering is used and uniform controllability is achieved. In [START_REF] Micu | Uniform boundary controllability of a semi-discrete 1-d wave equation[END_REF], the finite difference problem is also considered and it has been shown that if high frequencies are appropriately filtered, continuous level controls exist for the wave equation. It has also been shown that the discrete norm of the HUM control grows exponentially with the number of discretisation points if no filtering is applied.

More recently, the study of the numerical wave equation in three dimensions related to Huygens' principle has been carried out in [START_REF] Rosier | Numerical control of the wave equation and Huyghens' principle[END_REF]. A domain extension method has been used and the study of numerical control in a box is then carried out. Using Fourier series expansions, the convergence of the sequence of approximate controls is then established, giving us a direct approximation without filtering, damping or regularisation methods. Another numerical study of the multidimensional wave equation has been done in [START_REF] Burman | Spacetime finite element methods for control problems subject to the wave equation[END_REF], using a finite element method that does not reveal the spurious oscillations. Our aim is also to construct an approximation method without using the remedial ones.

We are therefore interested in the controllability of one-dimensional wave equations with variable coefficients. More specifically, we consider the null controllability of the following system :

a(s)y s s + b(s)y s + c(s)y -ρ(s)y t t = 0 s ∈ (0, 1), t ∈ (0, T ) (1)
y(0, t ) = 0 t ∈ (0, T ) (2) y(1, t ) = u(t ) t ∈ (0, T ) (3) y(s, 0) = y 0 (s), y t (s, 0) = y 1 (s) s ∈ (0, 1) (4)
where T > 0, (y 0 , y 1 ) are the initial data and u is the control. The functions a, b, c, ρ are given and satisfy the following assumptions :

a, ρ ∈ C 1 ([0, 1]), b ∈ C 0 ([0, 1]) and c ∈ L 1 (0, 1) (5)
a(s) > 0 and ρ(s) > 0 for all s ∈ [0, 1] (6)

c(s) ≤ 0 for s ∈ [0, 1] (7)
With these assumptions, we have 1 a ∈ L 1 (0, 1) and b a ∈ L 1 (0, 1).

To study this controllability problem, we will use a backstepping approach which is an efficient tool to obtain a stabilising controller (see Definition 2.1). This method consists in transforming the initial problem into another one for which the finite-time stability is easier to study. Thanks to this, we can show that the controllability of the initial problem is also solved thanks to the finite-time stability of the constructed controller.

In Section 2.1, we are reduced to a system of two first-order partial differential equations of the form [START_REF] Coron | Local exponential H 2 stabilization of a 2x2 quasilinear hyperbolic system using backstepping[END_REF] :

w t = Σ(x)w x +C (x)w (8) w 1 (0, t ) = q w 2 (0, t ) (9)
w 2 (1, t ) = W (t ) (10)
where w = w 1 w 2 is expressed in terms of y, W is a control law depending on u, q is real and Σ,C ∈ M 2 (R) will be made explicit (like w, q and W ) in Paragraph 2.1. In Section 2.2, we perform a backstepping transformation to obtain a system of the form:

γ t = Σ(x)γ x (11) α(0, t ) = qβ(0, t ) (12) β(1, t ) = 0 ( 13
)
where γ = α β . We will show, in Section 2.2, that γ is finite-time stable then we'll show that from the construction of γ, we can build a finite-time stable control which allows us to construct the solution of ( 1)-( 4) which is also stable in finite time.

Once we've established this, we are interested in obtaining direct numerical control without the use of regularisation, filtering or damping processes. To do this, we approximate the Goursat system (38), which is satisfied by the direct kernels of the backstepping transform, using a finite difference scheme with a clever choice of the step described by [START_REF] Anfinsen | Adaptive control of linear2×2 hyperbolic systems[END_REF] and used next in [START_REF] Wijnand | Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension[END_REF]. However, the numerical analysis of this scheme has not yet been done, so we propose to study it. The system (11)-( 13) is approximated by classical upwind-downwind schemes (see [START_REF] Sainsaulieu | Calcul scientifique: cours et exercices corrigés pour le 2e cycle et les écoles d'ingénieurs[END_REF][START_REF] Ciarlet | Introduction à l'analyse numérique matricielle et à l'optimisation[END_REF]), for which the Courant-Friedrichs-Lewy condition (or CFL for short), which originally appeared in [START_REF] Courant | On the partial difference equations of mathematical physics[END_REF], is necessary. Using these approximations, we return to the previous construction and, using trapezoidal methods, we can approximate the systems ( 8)- [START_REF] Andréa-Novel | Exponential stabilization of an overhead crane with flexible cable via a back-stepping approach[END_REF] as well as the control. Finally, we obtain an approximation of the initial system (1)-( 4). The approximate control sequence converges to order one in time, and the approximation to the solution of the system (1)-( 4) converges to order one in time and space.

Finite-time stabilisation of the system

In this section, we focus on the theoretical aspects of backstepping. We recall the transformations performed in [START_REF] Wijnand | Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension[END_REF][START_REF] Martin | Null controllability of one-dimensional parabolic equations by the flatness approach[END_REF] to put on the system (1)-( 4) under the form ( 11)-( 13) in Paragraph 2.1. Next, we will show that the system (11)-( 13) is stabilisable (see Paragraph 2.2.1), and this property can then be traced back to the initial system in Paragraph 2.3.

Transformations

In this section, we're going to reduce our system to a system of two PDEs of order one for the purpose of applying a backstepping type transformation (see Section 2.2.1) which will be useful to show the finite-time stabilisation of the solution of the system (1)-( 4).

We do a first transformation to get an equation of the form z t t -λ 2 z xx = 0. With this in hand, we can perform a second transformation as in [START_REF] Wijnand | Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension[END_REF] to obtain the equation (8).

Première transformation

Following [START_REF] Martin | Null controllability of one-dimensional parabolic equations by the flatness approach[END_REF], we introduce: s) .

B (s) := s 0 b(σ) a(σ) d σ, a(s) := a(s)e B (s) , c(s) := -c(s)e B (
So, we see that

B ∈ C 1 ([0, 1]), c ∈ L 1 (0, 1
) and that a(s) > 0, c(s) > 0 for all s ∈ (0, 1). We then introduce v the solution to the elliptic problem:

-av s s + c v = 0 for s ∈ [0, 1], ( 14
)
v(0) = v(1) = 1, (15)
such a function v exists, it is unique and we have the following proposition.

Proposition 2.1. The function v ∈ W 1,1 (0, 1), av ′ ∈ W 1,1 (0, 1) and 0 < v(s) ≤ 1 for all s ∈ [0, 1].
Proof. The positivity of v and the fact that v ∈ W 1,1 (0, 1) are treated in the same way as in [START_REF] Martin | Null controllability of one-dimensional parabolic equations by the flatness approach[END_REF]. We set w(x)

= x 0 c(σ)v(σ)d σ. So w ∈ W 1,1 (0, 1
) and the derivative in the sense of distributions of av ′w equals 0. We deduce that av ′ ∈ W 1,1 (0, 1). Let y(s, t ) := y(s,t ) v(s) and

L := 1 0 a(σ)v(σ) 2 e B (σ) -1 d σ , x(s) := 1 L s 0 a(σ)v(σ) 2 e B (σ) -1 d σ. ( 16
)
Finally, we set: z(x, t ) := z(x(s), t ) := y(s, t ), ( 17)

ρ(x) := ρ(x(s)) := L 2 a(s)v(s) 4 e 2B (s) ρ(s), ( 18
) λ(x) := ρ(x) -1 2 . ( 19
)
With all this in hand, we can then make the following statement:

Proposition 2.2.
i) The change of variables x : [0, 1] → [0, 1] is an increasing bijection with x, x -1 ∈ W 1,1 (0, 1).

ii) We have ρ(s) > 0, λ(s) > 0 for all s ∈ [0, 1] and ρ, λ ∈ C 1 ([0, 1]).

iii) The unknown z is a solution of the wave system:

z t t -λ(x) 2 z xx = 0 x ∈ (0, 1) t ∈ (0, T ) (20) z(0, t ) = 0 t ∈ (0, T ) (21) z(1, t ) = u(t ) t ∈ (0, T ) (22) z(x, 0) = z 0 (x), z t (x, 0) = z 1 (x) x ∈ (0, 1) (23)
with z 0 (x(s)) = y 0 (s) v(s) and z 1 (x(s)) = y 1 (s) v(s) for s ∈ (0, 1).

Proof. For the first point, we refer to [START_REF] Martin | Null controllability of one-dimensional parabolic equations by the flatness approach[END_REF].

ii) From the strict positivity of v, from (6) and from [START_REF] Lax | Survey of the stability of linear finite difference equations[END_REF], we deduce that ρ(s) > 0 for all s ∈ [0, 1] and so that the function λ is well defined. Moreover, λ(s) > 0 for all s ∈ [0, 1] because it's the root of a non-zero quantity. Furthermore, from av ′ ∈ W 1,1 (0, 1) ⊂ C 0 ([0, 1]) and v ′ = av ′ a , we deduce with (5) that v ′ ∈ C 0 ([0, 1]). This, combined with (5), we obtain ρ ∈ C 1 ([0, 1]). And so that λ ∈ C 1 ([0, 1]).

iii) First, we derive the partial differential equation verified by y, we have (see [START_REF] Martin | Null controllability of one-dimensional parabolic equations by the flatness approach[END_REF]): e -B av 2 e B y s s = ρv 2 y t t .

(

) 24
With ( 16), we have ∂ x = Lav 2 e B ∂ s and so by ( 17)-( 19) and ( 24), we deduce that (20) is satisfied.

For the boundary conditions ( 21)-( 22), it comes from (2)-( 3) and ( 17) by simple computations.

Finally, for the initial data [START_REF] Münch | A uniformly controllable and implicit scheme for the 1-d wave equation[END_REF], it comes from ( 4) and [START_REF] Krstic | Boundary Control of PDEs[END_REF].

Second transformation

Secondly, we put the equation ( 20) into the form of two first-order equations. By noting that

∂ t -λ∂ x ∂ t + λ∂ x z = -λλ ′ z x , ( 25
)
∂ t + λ∂ x ∂ t -λ∂ x z = -λλ ′ z x , (26)
we see that the Riemann invariants S := z t + λz x and D := z t -λz x satisfy the system:

S t -λ(x)S x = - λ ′ (x) 2 S -D , (27)
D t + λ(x)D x = - λ ′ (x) 2 S -D . ( 28
)
By setting

w(x, t ) := w 1 (x, t ) w 2 (x, t ) = 1 λ(x) D(x, t ) S(x, t ), (29)
we obtain the following first-order system:

w t = -λ(x) 0 0 λ(x) w x + 0 -λ ′ (x) 2 λ ′ (x) 2 0 w, ( 30
)
with the boundary conditions:

w 1 (0, t ) = -w 2 (0, t ) (31)
w 2 (1, t ) = W (t ). ( 32
)
The equation ( 31) is derived from [START_REF] Mehrenberger | An Ingham type proof for a two-grid observability theorem[END_REF] which implies that z t (0, t ) = 0 for all t > 0. W denotes a feedback law which can be expressed as a function of the control u and the derivative z x (1, t ):

W (t ) = 1 λ(1) u ′ (t ) + λ(1)z x (1, t ) . ( 33
)
The system (30)-( 32) is now in the necessary form to apply the backstepping transformation [8, 29] which we will introduce below.

Backstepping et stabilisation

This section explains how to obtain the (11)-( 13) using the backstepping transformation (see Definition 2.2) as mentioned above.

According to [START_REF] Coron | Local exponential H 2 stabilization of a 2x2 quasilinear hyperbolic system using backstepping[END_REF], the system (11)-( 13) is finite-time stable in the following sense:

Definition 2.1. Let Φ := Φ(x, z, t ) with x ∈ H , z ∈ [0, 1] and t ∈ R + the flow associated with an evolution system in a Hilbert space H . We suppose that:

• Φ satisfies the semi-group property:

∀x ∈ H , ∀z ∈ [0, 1], ∀(s, t ) ∈ R 2 + , Φ(Φ(x, z, s), z, t ) = Φ(x, z, s + t ),
• x = 0 is an equilibrium point:

∀z ∈ [0, 1], ∀t ∈ R + , Φ(0, z, t ) = 0.
We then say that the flow Φ(x, z, t ) (x,z,t )∈H ×[0,1]×R + is globally finite-time stable if:

• we have the finite-time convergence property: there is a non-decreasing function T : (0, +∞) → (0, +∞) called the stabilisation time function, such that:

Φ(x, z, t ) = 0, ∀x ∈ H \{0}, ∀z ∈ [0, 1], ∀t ≥ T (∥x∥ H ),
• the equilibrium point x = 0 is Lyapunov stable:

∀ε > 0, ∃δ > 0, tel que ∥x∥ H < δ =⇒ ∥Φ(x, z, t )∥ H < ε, ∀z ∈ [0, 1], ∀t ∈ R + .
Once the stability for intermediate systems is satisfied, we construct y the solution of ( 1)-( 4) whose finite-time stability can be established using the inverse backstepping transformation and a finitetime stable control.

Backstepping and first stabilisation results

The aim of a backstepping transformation is to define a transformation that links the variable w to a target variable solution of a target system for which finite-time stability is easier to establish. Definition 2.2. The backstepping transformation between the initial state variable w and the target state variable γ := α β satisfying the target system:

γ t = -λ(x) 0 0 λ(x) γ x , ( 34
) α(0, t ) = -β(0, t ), ( 35
)
β(1, t ) = 0, ( 36
)
is defined by a Volterra integral:

γ(x, t ) := w(x, t ) - x 0 K (x, ξ)w(ξ, t )d ξ (37)
with the direct kernel K (x, ξ) decomposed as K (x, ξ)

:= K uu (x, ξ) K uv (x, ξ) K vu (x, ξ) K v v (x, ξ) ,
where K uu (x, ξ), K uv (x, ξ), K vu (x, ξ) and K v v (x, ξ) are given by (38). By simple computations we see that (see [START_REF] Coron | Local exponential H 2 stabilization of a 2x2 quasilinear hyperbolic system using backstepping[END_REF]) the direct kernels are the solution of a Goursat system of two pairs of coupled first-order hyperbolic partial differential equations on the triangular domain. T := {(x, ξ) : 0 ≤ ξ ≤ x ≤ 1}. More precisely, we have:

             λ(x)K uu x + λ(ξ)K uu ξ + λ ′ (ξ)K uu + λ ′ (ξ) 2 K uv = 0 (x, ξ) ∈ T , λ(x)K uv x -λ(ξ)K uv ξ -λ ′ (ξ)K uv - λ ′ (ξ) 2 K uu = 0 (x, ξ) ∈ T , λ(x)K vu x -λ(ξ)K vu ξ -λ ′ (ξ)K vu - λ ′ (ξ) 2 K v v = 0 (x, ξ) ∈ T , λ(x)K v v x + λ(ξ)K v v ξ + λ ′ (ξ)K v v + λ ′ (ξ) 2 K vu = 0 (x, ξ) ∈ T , (38)
with the following boundary conditions:

           K uu (x, 0) = -K uv (x, 0), K uv (x, x) = -λ ′ (x) 4λ(x) , K vu (x, x) = -λ ′ (x) 4λ(x) , K v v (x, 0) = -K vu (x, 0). (39)
Using the third point of Proposition 2.2, we can apply the result of [START_REF] Coron | Local exponential H 2 stabilization of a 2x2 quasilinear hyperbolic system using backstepping[END_REF] which asserts that there is a unique solution to this system which is C 0 (T ).

In this way, we are able to transform the ( 1)-( 4) into the target system (34)-( 36). Let's discuss some properties of this system. We have a first finite-time stabilisation result for the system (34)-(36) (see [START_REF] Coron | Local exponential H 2 stabilization of a 2x2 quasilinear hyperbolic system using backstepping[END_REF]):

Proposition 2.3. The system (34)-(36) admits a unique solution γ globally stable in finite time in the sense of Definition 2.1 by setting the time t

F := 1 0 2 λ(x) d x. In addition, by choosing initial data α 0 , β 0 ∈ H 1 (0, 1), its components verify α, β ∈ C 0 [0, T ], H 1 (0, 1) ∩ C 0 [0, 1], H 1 (0, T ) for all T > 0.
Proof. We recall the main steps of the proof of [START_REF] Coron | Local exponential H 2 stabilization of a 2x2 quasilinear hyperbolic system using backstepping[END_REF]. We define φ(x) :=

x 0 1 λ(s) d s. The equations satisfied by α and β (34) can be rewritten as

∂ ∂t α φ -1 (x), t + ∂ ∂x α φ -1 (x), t = 0, ∂ ∂t β φ -1 (x), t + ∂ ∂x β φ -1 (x), t = 0,
whose solutions are given by α(x, t ) = F α φ(x)t and β(x, t ) = F β φ(x) + t with F α , F β some arbitrary functions. Let α 0 and β 0 the initial data. Hence, we obtain that for 0 < x < φ(1):

F α (x) = α 0 φ -1 (x) and F β (x) = β 0 φ -1 (x) .
Using the boundary conditions ( 35)-(36), we deduce the expressions for α and β:

α(x, t ) = α 0 φ -1 (φ(x) -t ) if t ≤ φ(x), -β(0, t -φ(x)) if t ≥ φ(x), ( 40
) β(x, t ) = β 0 φ -1 (φ(x) + t ) if t ≤ φ(1) -φ(x), 0 if t ≥ φ(1) -φ(x). ( 41
)
Let's show the regularity of the solution. Remember that we have λ > 0 on [0, 1] and λ ∈ C 1 ([0, 1]) so φ ∈ C 1 ([0, 1]) and is strictly positive. The same applies to φ -1 according to the Bijection Theorem. Since β 0 ∈ H 1 (0, 1), thanks to the expression (41), we deduce that

β ∈ C 0 [0, T ], H 1 (0, 1) ∩C 0 [0, 1], H 1 (0, T ) .
In particular, β(0, •) ∈ H 1 (0, T ) and thus with (40) and

α 0 ∈ H 1 (0, 1), we deduce that α ∈ C 0 [0, T ], H 1 (0, 1) ∩ C 0 [0, 1], H 1 (0, T ) .
We will now show global stability in finite time by following the demonstration of [START_REF] Coron | Local exponential H 2 stabilization of a 2x2 quasilinear hyperbolic system using backstepping[END_REF]. Let's start by looking at the stability in the Lyapunov sense. We define

D(x) := A e -µx λ(x) 0 0 B e µx λ(x)
, where B := A + λ 2 , A := λ 2 e µ and µ := λ 1 ε where ε := max

x∈[0,1] 1 λ(x) and λ 1 , λ 2 > 0 are arbitrary. We choose V 1 (t ) := 1 0 γ ⊺ (x, t )D(x)γ(x, t )d x.
Then, using integration by parts and (34), we find that

V ′ 1 ≤ -λ 1 V 1 -λ 2 α 2 (1, t ) + β 2 (0, t ) .
This shows the stability we were looking for. For convergence in finite time according to the expressions ( 40)-( 41), it's clear that after the time t F , we have α ≡ β ≡ 0.

Inverse backstepping and second stabilisation result

Now that we know how to construct a solution of the system (34)-( 36) which is stable in finite time, we show that we can then construct a solution of the system (30)-(32) which is also stable in finite time thanks to an inverse backstepping transformation.

Definition 2.3. The inverse backstepping transformation between the target state variable γ and the initial state variable w is giving by

: w(x, t ) := γ(x, t ) + x 0 L(x, ξ)γ(ξ, t )d ξ (42) with the inverse kernel decomposed as L(x, ξ) = L αα (x, ξ) L αβ (x, ξ) L βα (x, ξ) L ββ (x, ξ) .
The inverse kernels are solution of a similar system to (38) on the same triangular domain T :

             λ(x)L αα x + λ(ξ)L αα ξ = -λ ′ (ξ)L αα - λ ′ (ξ) 2 L βα , λ(x)L αβ x -λ(ξ)L αβ ξ = λ ′ (ξ)L αβ - λ ′ (ξ) 2 L ββ , λ(x)L βα x -λ(ξ)L βα ξ = λ ′ (ξ)L βα - λ ′ (ξ) 2 L αα , λ(x)L ββ x + λ(ξ)L ββ ξ = -λ ′ (ξ)L ββ - λ ′ (ξ) 2 L αβ , ( 43
)
with the following boundary conditions:

           L αα (x, 0) = -L αβ (x, 0), L αβ (x, x) = -λ ′ (x) 4λ(x) , L βα (x, x) = -λ ′ (x) 4λ(x) , L ββ (x, 0) = -L βα (x, 0). ( 44
)
According to the third point of Proposition 2.2, there is a unique solution to this system, which is C 0 (T ) (see [START_REF] Coron | Local exponential H 2 stabilization of a 2x2 quasilinear hyperbolic system using backstepping[END_REF]). Let's start with a result from [START_REF] Wijnand | Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension[END_REF]: Proposition 2.4. The inverse kernels satisfy the following equalities:

L αα (x, ξ) = L ββ (x, ξ), ∀(x, ξ) ∈ T and L αβ (x, ξ) = L βα (x, ξ), ∀(x, ξ) ∈ T
A second property that can be obtained is a relationship between direct and indirect kernels. (see [START_REF] Anfinsen | Adaptive control of linear2×2 hyperbolic systems[END_REF]). This relationship will be useful for the numerical approximation of inverse kernels:

L(x, ξ) = K (x, ξ) + x ξ K (x, σ)L(σ, ξ)d σ. ( 45
)
We have a second finite-time stabilisation result for the system (30)-(32) (see [START_REF] Coron | Local exponential H 2 stabilization of a 2x2 quasilinear hyperbolic system using backstepping[END_REF][START_REF] Wijnand | Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension[END_REF]):

Proposition 2.5. The system (30)-( 32) is sent to (34)-( 36) with finite-time stability through backstepping transformation (37) with the kernels given by the solution of the problems (38)-( 39) and (43)-( 44) and by the feedback law W (t )

= 1 0 K vu (1, ξ)w 2 (ξ, t )d ξ + 1 0 K v v (1, ξ)w 1 (ξ, t )d ξ by setting the time t F = 1 0 2 λ(x) d x.
In addition, by carefully selecting the initial data w 0 1 and w 0 2 in H 1 (0, 1), we have the regularity w 1 ,

w 2 ∈ C 0 [0, T ], H 1 (0, 1) ∩ C 0 [0, 1], H 1 (0, T ) for all T > 0.
Proof. We follow the framework of the proof presented in [START_REF] Coron | Local exponential H 2 stabilization of a 2x2 quasilinear hyperbolic system using backstepping[END_REF]. We choose the initial data w 0 1 , w 0 2 in H 1 (0, 1). We denote by γ * the solution of (34)-( 36) with the initial data

γ 0 (x) := w 0 (x) - x 0 K (x, ξ)w 0 (ξ)d ξ.
The solution of ( 30)-( 32) is given explicitly by the formula:

w(x, t ) = γ * (x, t ) + x 0 L(x, ξ)γ * (ξ, t )d ξ. ( 46
)
In particular, thanks to the Proposition 2.3, we know that γ * ≡ 0 for t ≥ t F and so does w. Since the origin is exponentially stable for the system verified by γ * , with the formula (46), this property is also satisfied for the system verified by w, proving Lyapunov stability.

Finally, from the explicit formula (46), we have:

w 1 , w 2 ∈ C 0 [0, T ], H 1 (0, 1) ∩ C 0 [0, 1], H 1 (0, T ) for all T > 0.

Finite-time stabilisation

In this paragraph we show that we can reconstruct the initial variable y from the target state γ. We've already shown that w can be reconstructed from γ with finite time stabilisation thanks to Proposition 2.5.

To do this, we use [START_REF] Wijnand | Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension[END_REF] to construct S and D, then construct the solution z of the system (54)-( 56). Once all this is done, we find our starting variable y by using [START_REF] Krstic | Boundary Control of PDEs[END_REF].

First step: Regularity and stabilisation of the unknown z.

Firstly, we begin by defining the initial data space. Let:

H = (z 0 , z 1 ) ∈ H 2 (0, 1) × H 1 (0, 1)/z 0 (0) = 0 . ( 47
)
We provide this space with the norm ∥(z 0 , z 1 )∥ 2

H := ∥z 0 ∥ 2 H 2 (0,1) + ∥z 1 ∥ 2 H 1 (0,1)
. We choose (z 0 , z 1 ) ∈ H and

w 0 1 w 0 2 (x) = 1 λ(x) z 1 (x) -λ(x)z 0 x (x) z 1 (x) + λ(x)z 0 x (x) , ∀x ∈ [0, 1] (48)
So we have w 0 1 , w 0 2 ∈ H 1 (0, 1). We define the initial data for the target state α 0 , β 0 by the formula (37) of the definition of the backstepping transform. Hence, we also have α 0 1 , α 0 2 ∈ H 1 (0, 1).

Next, we define the control and the Riemann invariants:

W (t ) := 1 0 L βα (1, ξ)α(ξ, t ) + L ββ (1, ξ)β(ξ, t )d ξ (49) u(t ) := - λ(1) 2 t F t w 1 (1, τ) + W (τ)d τ (50) D S (x, t ) = λ(x)w(x, t ) (51)
Lemma 2.1. We have the following regularities:

• For all T > 0, W ∈ H 1 ([0, T ]), u ∈ H 2 ([0, T ]) • For all T > 0, D, S ∈ C 0 [0, T ], H 1 (0, 1) ∩ C 0 [0, 1], H 1 (0, T )
In addition, D and S are globally stable in finite time, setting time t F = 1 0 2 λ(x) d x and after this time t F , u ≡ 0.

Proof. From the above definitions, kernel regularity and Propositions 2.3 and 2.5 we see that for all T > 0, W ∈ C 0 ([0, T ]) and therefore u ∈ C 1 ([0, T ]) using Proposition 2.5. Indeed, u is clearly derivable and we have:

u ′ (t ) = λ(1) 2 (w 1 (1, t ) + W (t )) (52)
As γ and w satisfy respectively (34) and ( 30), we obtain the fact that D ans S satisfy ( 27)- [START_REF] Sainsaulieu | Calcul scientifique: cours et exercices corrigés pour le 2e cycle et les écoles d'ingénieurs[END_REF].

As in [START_REF] Wijnand | Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension[END_REF], using the explicit formula (51) and the Proposition 2.5, we have:

D, S ∈ C 0 [0, T ], H 1 (0, 1) ∩ C 0 [0, 1], H 1 (0, T ) for all T > 0. ( 53
)
Now let's check the finite-time stabilisation of D and S. According to the Propositions 2.3 and 2.5, for t ≥ t F , α ≡ β ≡ w 1 ≡ w 2 ≡ 0. So according to (49) and (51), after t = t F , it holds W ≡ D ≡ S ≡ 0. And so from (50), after the time t = t F , we also have u ≡ 0. The convergence in finite time is then shown for these functions, and it remains to study stability in the Lyapunov sense. Using the Proposition 2.5 and (51), it's easy to see that S and D are Lyapunov stable and so are globally stable in finite time.

Thanks to the above, we can now construct z. Let f := S+D 2 and g := S-D 2λ . Note that the Schwarz condition f x = g t is checked by having ( 27)- [START_REF] Sainsaulieu | Calcul scientifique: cours et exercices corrigés pour le 2e cycle et les écoles d'ingénieurs[END_REF]. It follows that there is a unique solution z := z(x, t ) to the system: (this for all T > 0)

z t = S + D 2 x ∈ (0, 1), t ∈ (0, T ), ( 54
)
z x = S -D 2λ x ∈ (0, 1), t ∈ (0, T ), ( 55
) z(1, 0) = z 0 (1), ( 56
)
where we recall that we choose (z 0 , z 1 ) ∈ H . This solution is explicitly given by:

z(x, t ) = z 0 (1) + t 0 f (1, s)d s + x 1 g (s, t )d s (57) = z 0 (1) + x 1 g (s, 0)d s + t 0 f (x, s)d s (58)
This combined with (53) shows that (for all T > 0)

z ∈ C 0 [0, T ], H 2 (0, 1) ∩ C 1 [0, T ], H 1 (0, 1) ∩ C 0 [0, 1], H 2 (0, T ) . ( 59
)
Now let's check that the unknown z thus defined verifies the system ( 20)-( 23). -As in [START_REF] Wijnand | Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension[END_REF], we can show that for any x ∈ [0, 1], z(x, 0) = z 0 (x) any z t (x, 0) = z 1 (x). Indeed, from the definition of H (47), ( 48) and ( 51), on the one hand we have

z(x, 0) = z 0 (1) + x 1 g (s, 0)d s = z 0 (1) + x 1 z 0 x (s)d s = z 0 (x), ∀x ∈ [0, 1],
and on the other we have

z t (x, 0) = f (x, 0) = λ(x) 2 u 0 (x) + v 0 (x) = z 1 (x), ∀x ∈ [0, 1].
So the initial condition ( 23) is satisfied.

-In the same way as for [START_REF] Wijnand | Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension[END_REF], we can show that (20) is satisfied using the expressions ( 27)-( 28) and ( 54)-( 55):

z t t -λ 2 z xx = S t + D t 2 -λ 2 S x + D x 2λ -λ ′ S -D 2λ 2 = 0.
-Let us now show that for all t ≥ 0, z(1, t ) = u(t ). From ( 51) and (58):

z t (1, t ) = S(1, t ) + D(1, t ) 2 = λ(1) 2 (w 2 (1, t ) + w 1 (1, t ))
Or, according to (32), w 2 (1, t ) = W (t ). Hence, with (50), we get

z t (1, t ) = λ(1) 2 (W (t ) + w 1 (1, t )) ,
and so we find [START_REF] Micu | Uniform boundary controllability of a semi-discrete 1-d wave equation[END_REF].

-It remains to show that z(0, t ) = 0 for all t ≥ 0. Using (58), [START_REF] Micu | Uniform boundary controllability of a semi-discrete 1-d wave equation[END_REF] and the fact that g (s, 0) = z 0 x (s) (see the definition of H (47)), we get

z(0, t ) = u(0) + 0 1 g (s, 0)d s + t 0 f (0, s)d s = u(0) + 0 1 z 0 x (s)d s + t 0 f (0, s)d s = u(0) + z 0 (0) -z 0 (1) + t 0 f (0, s)d s = z 0 (0) + t 0 f (0, s)d s
Moreover, according to (31), we have

f (0, s) = S(0, s) + D(0, s) 2 = λ(0) 2 (w 2 (0, s) + w 1 (0, s)) = 0.
We then deduce that for all t ≥ 0, z(0, t ) = z 0 (0). From the definition of H , we have z 0 (0) = 0. We find [START_REF] Mehrenberger | An Ingham type proof for a two-grid observability theorem[END_REF].

The following property has now been partially demonstrated. We can also show that z is stable in finite time.

Proposition 2.6. For (z 0 , z 1 ) ∈ H , the unknown z defined by (57)-( 58) is solution of (20)-( 23) and has the regularity

C 0 [0, T ], H 2 (0, 1) ∩ C 1 [0, T ], H 1 (0, 1) ∩ C 0 [0, 1], H 2 (0, T ) .

Moreover, z is globally finite-time stable by positing the time t

F = 1 0 2 λ(x) d x.
Proof. Let's now show that z is finite-time stable. According to Lemma 2.1, for t ≥ t F , D ≡ S ≡ 0 and u ≡ 0. We then deduce from (55) that after the time t = t F , z x ≡ 0. Since we have [START_REF] Micu | Uniform boundary controllability of a semi-discrete 1-d wave equation[END_REF], we can write:

z(x, t ) = x 1 z x (s, t )d s + u(t ). ( 60
)
This implies that after t = t F , we have z ≡ 0. Convergence in finite time is then shown. It remains to establish the stability in the Lyapunov sense We follow the reasoning of [START_REF] Wijnand | Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension[END_REF]. We suppose that ∥(z 0 , z 1 )∥ H is small and at least lower than one. Thus, ∥w 0 1 ∥ L 2 (0,1) and ∥w 0 2 ∥ L 2 (0,1) are small from (48) and from Lemma 2.1, we also have ∥w 1 (., t )∥ L 2 (0,1) and ∥w 2 (., t )∥ L 2 (0,1) small. On the one hand, combining this with (51) and (55), ∥z x (., t )∥ H 1 (0,1) is also small. On the other hand, by combining them with (42) and (33), ∥α(., t )∥ H 1 (0,1) , ∥β(., t )∥ H 1 (0,1) are small and so that W (t ) and therefore, with (50), we deduce that u(t )is small too. Finally, by combining the fact that ∥z x (., t )∥ H 1 (0,1) and u(t ) are small, from (60), we infer that ∥z(., t )∥ H 2 (0,1) is small. We have thus demonstrated stability in the Lyapunov sense and hence the global finite-time stability of z.

Conclusion: Regularity and stabilisation of the starting unknown y

We can then reconstruct the initial unknown from the previous constructions. Let y(s, t ) := v(s)z(x(s), t ) where we recall that v is the solution of the elliptic problem ( 14)- [START_REF] Infante | Boundary observability for the space semi-discretizations of the 1d wave equation[END_REF]. Recall that we know the regularity of v by Proposition 2.2. Direct calculations show that y is the solution of the system (1)-( 4). Based on the regularities of v and z, we deduce the following regularity for y: y ∈ C 0 [0, 1], H 1 (0, T ) ∩ C 0 [0, T ],W 1,1 (0, 1) . Since the function z is globally stable in finite time, according to [START_REF] Krstic | Boundary Control of PDEs[END_REF], the same applies to y.

We then established the following theorem:

Theorem 2.1. It is assumed that (6)-( 7) are satisfied and let T > 0. For (z 0 , z 1 ) ∈ H , there exists a control u ∈ H 2 ([0, T ]) such that the solution y is globally finite-time stable for the time t F :=

y ∈ C 0 [0, 1], H 1 (0, T ) ∩ C 0 [0, T ],W 1,1 (0, 1) .
Remark 2.1. As a consequence of this theorem, it is easy to see that the problem (1)-( 4) is null controllable for T ≥ t F . In fact, because of global stability, we deduce that

∀T ≥ t F , ∀s ∈ [0, 1], y(s, T ) = 0.

Numerical control

In this section we first construct an approximation for direct kernels using the method presented in [START_REF] Anfinsen | Adaptive control of linear2×2 hyperbolic systems[END_REF] and used in [START_REF] Wijnand | Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension[END_REF], then use the integral formula (45) to derive an approximation for inverse kernels using the trapezoidal formula. Once this is done, an approximation of the target state can be constructed by solving the transport equations using the upwind and downwind methods, and then deriving an approximation of the state system using a trapezoidal method as well. With this in hand, approximations of the spatial and temporal derivatives of the unknown z are obtained using explicit and simple formulae, as well as approximate control. Then, using a trapezoidal method for integrating the spatial derivative, we derive an approximation for the unknown z. Independently, we can construct an approximation to the elliptic problem using a finite difference method. Using these approximations we can return to our initial unknown y and find an approximation.

Direct kernel approximation

The method is based on [START_REF] Anfinsen | Adaptive control of linear2×2 hyperbolic systems[END_REF] which was used in [START_REF] Wijnand | Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension[END_REF] for the flexible cable case.

We are interested in the numerical approximation of Goursat's problem on the triangular domain

T = {(x, ξ) : 0 ≤ ξ ≤ x ≤ 1}.            λ(x)K uv x -λ(ξ)K uv ξ -λ ′ (ξ)K uv - λ ′ (ξ) 2 K uu = 0, λ(x)K uu x + λ(ξ)K uu ξ + λ ′ (ξ)K uu + λ ′ (ξ) 2 K uv = 0, K uv (x, x) = -λ ′ (x) 4λ(x) , K uu (x, 0) + K uv (x, 0) = 0.
For the other two parts of the direct kernel K vu and K v v , this is done in the same way with Proposition 2.4. We therefore focus on the above system.

We follow the method introduced in [START_REF] Anfinsen | Adaptive control of linear2×2 hyperbolic systems[END_REF], which consists in approximating the partial differential equations with an upwind finite difference scheme. We approximate the derivative part as follows

λ(x)K uv x -λ(ξ)K uv ξ ∼ λ(x) 2 + λ(ξ) 2 σ(x, ξ) K uv (x, ξ) -K uv (x -σ(x, ξ)ν 1 (x, ξ), ξ -σ(x, ξ)ν 2 (x, ξ)) λ(x)K uu x + λ(ξ)K uu ξ ∼ λ(x) 2 + λ(ξ) 2 σ ′ (x, ξ) K uu (x, ξ) -K uu (x -σ ′ (x, ξ)ν ′ 1 (x, ξ), ξ -σ ′ (x, ξ)ν ′ 2 (x, ξ))
where σ(x, ξ) > 0, σ ′ (x, ξ) > 0 are discretisation steps that we will specify in the following, ν 1 , ν 2 , ν ′ 1 and ν ′ 2 are the components of the unit vectors in the direction of the characteristics

ν(x, ξ) := 1 λ(x) 2 + λ(ξ) 2 λ(x) -λ(ξ) and ν ′ (x, ξ) := 1 λ(x) 2 + λ(ξ) 2 λ(x) λ(ξ) .
For N ≥ 1, we choose a step length ∆ := 1 N in both directions and let (x i , ξ j ) be approximation nodes where x i := i ∆, ξ j := j ∆ with 1 ≤ j ≤ i ≤ N . The approximation nodes on the triangle are shown in the following figure. Let σ i , j := σ(x i , ξ j ) a step that can vary depending on ∆ and σ ′ i , j := σ ′ (x i , ξ j ) an other step (we explain how to choose them below).

The discretisation of the previous approximations is written as

λ(x i )K uv x (x i , ξ j ) -λ(ξ j )K uv ξ (x i , ξ j ) ∼ λ(x i ) 2 + λ(ξ j ) 2 σ i , j K uv (x i , ξ j ) -K uv (x i -σ i , j ν 1 i , j , ξ j -σ i , j ν 2 i , j ) , λ(x i )K uu x (x i , ξ j ) + λ(ξ j )K uu ξ (x i , ξ j ) ∼ λ(x i ) 2 + λ(ξ j ) 2 σ ′ i , j K uu (x i , ξ j ) -K uu (x -σ ′ i , j ν ′ 1 i , j , ξ -σ ′ i , j ν ′ 1 i , j ) , with ν i , j = ν 1 i , j ν 1 i , j := 1 λ(x i ) 2 + λ(ξ j ) 2 λ(x i ) -λ(ξ j ) , ( 61
)
ν ′ i , j = ν ′ 1 i , j ν ′ 1 i , j := 1 λ(x i ) 2 + λ(ξ j ) 2 λ(x i ) λ(ξ j ) . ( 62
)
The steps σ i , j and σ ′ i , j are chosen according to the edge of the square shown in Figure 2 towards which the vectors -σ i , j ν i , j and -σ ′ i , j ν ′ i , j point to. This is explained in [START_REF] Anfinsen | Adaptive control of linear2×2 hyperbolic systems[END_REF] and details are given below. Here we present the method for K uv (for K uu , we reason in the same way), two cases can arise: either the vector -σ i , j ν i , j points to the left (case 1) or it points to the top (case 2) of the edge of the square shown in Figure 2. (Note that for K uu either the vector -σ ′ i , j ν ′ i , j points to the left or to the bottom of the edge of the square shown in Figure 2)

(i -1, j ) (i , j ) (i -1, j + 1) (i , j + 1) σν σν (a) Pour -σ i , j ν i , j (i -1, j -1) (i , j -1) (i -1, j ) (i , j ) σ ′ ν ′ σ ′ ν ′ (b) Pour -σ ′ i , j ν ′ i , j
Figure 2: Different scenarios for the vector direction -σ i , j ν i , j and -σ ′ i , j ν ′ i , j

• case 1 (the vector points upwards): Denote θ i , j the angle between the right-hand edge of the square and the vector -σ i , j ν i , j , see the Figure 3. From the data available and using the trigonometric formulae for a right-angled triangle, we choose

σ i , j := cos(θ i , j )∆ and d := σ i , j sin(θ i , j ). ( 63
)
• case 2 (the vector points to the left): In the same way, we note the angle θ i , j , see Figure 4. And in the same way as the previous case, we choose Step selection when the vector -σ i , j ν i , j points to the top of the square

σ i , j := sin(θ i , j )∆ and d := σ i , j cos(θ i , j ). ( 64
) (i -1, j ) (i , j ) (i -1, j + 1) (i , j + 1) σ i , j d ∆ θ i , j
(i -1, j ) (i , j ) (i -1, j + 1) (i , j + 1) σ i , j ∆ d θ i , j Figure 4:
Step selection when the vector -σ i , j ν i , j points to the left of the square We specify that σ ′ i , j is chosen in the same way. The value of K uv (x i -σ i , j ν i , j ,1 , y j -σ i , j ν i , j ,2 ) is obtained by linear interpolation at the points (x i -1 , ξ j ) and (x i -1 , ξ j -1 ) in the first case or at the points (x i -1 , ξ j -1 ) and (x i , ξ j -1 ) in the other case. This gives us the interpolations:

• case 1: K uv (x i -σ i , j ν i , j ,1 , ξ j -σ i , j ν i , j ,2 ) ∼ ∆-d ∆ K uv (x i , ξ j +1 ) + d ∆ K uv (x i -1 , ξ j +1 ) • case 2: K uv (x i -σ i , j ν i , j ,1 , ξ j -σ i , j ν i , j ,2 ) ∼ ∆-d ∆ K uv (x i -1 , ξ j ) + d ∆ K uv (x i -1 , ξ j +1 )
However, care must be taken in the case of the subdiagonal, since the point (x i -1 , ξ j +1 ) does not exist, so the weight obtained must be distributed over the other three summits (we note that for K uu , this problem does not appear). To do this, we use a barycentric combination of the weights at the four summits: we consider K uv (x i -1 , ξ j +1 ) as barycentre of the points K uv (x i , ξ j ),K uv (x i , ξ j +1 ) and K uv (x i -1 , ξ j ) assigned weights respectively 1, -1 and -1. This gives us:

K uv (x i -1 , ξ j +1 ) = -K uv (x i , ξ j ) + K uv (x i , ξ j +1 ) + K uv (x i -1 , ξ j ).
Similar approximations are made for K uu . Denote (ΠK uv ) j i and (ΠK uu ) j i the interpolations obtained. The discrete scheme we'll consider in the following is then written for 1 ≤ j ≤ i ≤ N -1:

                   λ(x i ) 2 +λ(ξ j ) 2 σ i , j (K uv ) j i -(ΠK uv ) j i -λ ′ (ξ j )(K uv ) j i - λ ′ (ξ j ) 2 (K uu ) j i = 0, λ(x i ) 2 +λ(y j ) 2 σ ′ i , j (K uu ) j i -(ΠK uu ) j i + λ(ξ j )(K uu ) j i + λ ′ (ξ) 2 (K uv ) j i = 0, (K uv ) i i = - λ ′ (x i ) 4λ(x i ) , (K uu ) 1 i + (K uv ) 1 i = 0. ( 65
)
The aim is to study the convergence of this scheme. Therefore, we consider its consistency and stability separately. To do this, especially for stability, we need to be more precise about the approximations.

If the vectors hit the left side of the square (we are in the case of the red lines in the Figure 2), we have for K uv (also for 1 ≤ j ≤ i ≤ N -1)

Σ i j -λ ′ (ξ j ) (K uv ) j i -Σ i j ∆ -d ∆ (K uv ) j i -1 -Σ i j d ∆ (K uv ) j +1 i -1 - λ ′ (ξ j ) 2 (K uu ) j i = 0 ( 66
)
Σ i i -1 1 + d ∆ -λ ′ (ξ i -1 ) (K uv ) i -1 i -Σ i i -1 (K uv ) i -1 i -1 -Σ i i -1 d ∆ (K uv ) i i - λ ′ (ξ i -1 ) 2 (K uu ) i -1 i = 0 (67) (K uv ) i i = f (x i ) ( 68
)
and for K uu (for 1 ≤ j ≤ i ≤ N -1)

Σ ′ i j + λ ′ (ξ j ) (K uu ) j i + Σ ′ i j d ∆ (K uu ) j -1 i -1 + Σ ′ i j ∆ -d ∆ (K uu ) j i -1 + λ ′ (ξ j ) 2 (K uv ) j i = 0 ( 69
)
Σ ′ i i + λ ′ (ξ i ) (K uu ) i i + Σ ′ i i d ∆ (K uu ) i -1 i -1 + λ ′ (ξ i ) 2 (K uv ) i i = 0 (70) (K uu ) 1 i = (K uv ) 1 i (71)
If the vectors hit the top side in (a) and the bottom side in (b) of the squares of Figure 2 (we are in the case of the blue lines), we have for K uv

Σ i j -λ ′ (ξ j ) (K uv ) j i -Σ i j ∆ -d ∆ (K uv ) j +1 i -Σ i j d ∆ (K uv ) j +1 i -1 - λ ′ (ξ j ) 2 (K uu ) j i = 0 ( 72
)
Σ i i -1 1 + d ∆ -λ ′ (ξ i -1 ) (K uv ) i -1 i -Σ i i -1 (K uv ) i i -Σ i i -1 d ∆ (K uv ) i -1 i -1 - λ ′ (ξ i -1 ) 2 (K uu ) i -1 i = 0 (73) (K uv ) i i = C B ( 74
)
and for K uu

Σ ′ i j + λ ′ (ξ j ) (K uu ) j i + Σ ′ i j d ∆ (K uu ) j -1 i -1 + Σ ′ i j ∆ -d ∆ (K uu ) j -1 i + λ ′ (ξ j ) 2 (K uv ) j i = 0 ( 75
)
Σ ′ i i + λ ′ (ξ i ) (K uu ) i i + Σ ′ i i d ∆ (K uu ) i -1 i -1 + λ ′ (ξ i ) 2 (K uv ) i i = 0 (76) (K uu ) 1 i = (K uv ) 1 i (77)

Consistency

We will study the consistency of K uv (for K uu , it can be treated in the same way). We assume that K uv ∈ C 2 (T ).

In order to make the expressions clearer for the reader, we set

s i , j := λ(x i ) 2 + λ(ξ j ) 2 σ i , j (78)
and we define τ j i , the consistency error at node (x i , ξ j ), by:

τ j i := s i , j K uv (x i , ξ j ) -ΠK uv (x i , ξ j ) -λ(x i )K uv x (x i , ξ j ) + λ(ξ j )K uv ξ (x i , ξ j ).
We can see that

|τ j i | ≤ s i , j K uv (x i , ξ j ) -K uv (x i -σ i , j ν 1 i , j , ξ j -σ i , j ν 2 i , j ) -λ(x i )K uv x (x i , ξ j ) + λ(ξ j )K uv ξ (x i , ξ j ) + s i , j K uv (x i -σ i , j ν 1 i , j , ξ j -σ i , j ν 2 i , j ) -ΠK uv (x i , ξ j ) . ( 79
)
The second term in the right member is O(∆) because s i , j ≤ C ∆ (see ( 78)) and by classical linear interpolation, the remainder of this term is O(∆ 2 ) (see for example [START_REF] Quarteroni | Méthodes Numériques: Algorithmes, analyse et applications[END_REF]). For the first term on the right-hand side, using the Taylor-Young formula (two variables) of order two, we have

K uv (x i -σ i , j ν 1 i , j , ξ j -σ i , j ν 1 i , j ) = K uv (x i , ξ j ) -σ i , j ν 1 i , j K uv x (x i , ξ j ) -σ i , j ν 2 i , j K uv ξ (x i , ξ j ) + 1 2 (σ i , j ν 1 i , j ) 2 (K uv ) 2 xx + 2σ 2 i , j ν 1 i , j ν 2 i , j (K uv ) 2 xξ + (σ i , j ν 2 i , j ) 2 (K uv ) 2 y y + o ∥σ i , j ν i , j ∥ 2 .
Or, σ i , j ν i , j = O(∆) by definition of the terms (see ( 61), ( 64) and ( 63)).

Thus, the first term on the right-hand side of ( 79) is O(∆) and the same applies to |τ j i |. We conclude that the scheme is consistent to order one.

Stability and convergence

Let's show the stability of the scheme (65). For simplicity, let's denote by f the function f

(x) = -λ ′ (x) 4λ(x)
representing the boundary condition for K uv .

To study the stability of the scheme, we denote by X k the vector composed by the (K

uv ) i -k i i =k+1,...,N
and X ′k the one composed of the (K uu ) i -k i i =k+1,...,N . Due to (68) (or equivalently (74)), we have

∥X 0 ∥ ∞ ≤ ∥ f ∥ ∞ .
For X 1 , due to (67) or (73) (depending on which side of the square is hit), we have

diag Σ i i -1 1 + d ∆ -λ ′ i -1 X 1 =    a 2 b 2 . . . . . . a N b N    X 0 + diag λ ′ i -1 2 X ′1
where the a i , b i are equals to

Σ i i -1 or Σ i i -1 d ∆ .
Hence, by inverting the left hand side matrix which is allowed because the diagonal elements are non-zero, we obtain

X 1 =    a ′ 2 b ′ 2 . . . . . . a ′ N b ′ N    X 0 + diag   1
Σ i i -1 1 + d ∆ -λ ′ i -1 λ ′ i -1 2   X ′1
where

a ′ i = 1 Σ i i -1 1+ d ∆ -λ ′ i -1 a i and b ′ i = 1 Σ i i -1 1+ d ∆ -λ ′ i -1 b i
We are looking for the infinity norm in the sense that we take the supremum on the indexes i , knowing that d ∆ ≤ 1, so we have to estimate the absolute value of the terms

1 Σ i i -1 1+ d ∆ -λ ′ i -1
.

Or, for small value of ∆, there exists a constant independent of ∆ such that

|Σ i i -1 1 + d ∆ -λ ′ i -1 | ≥ C |Σ i i -1 1 + d ∆ | ≥ C |Σ i i -1 |.
We conclude by the same arguments that the infinity norm of the first right matrix is O(1) and that of the second right matrix is O(∆). We can then conclude that there are two constants

C 1 ,C 2 > 0 independent of ∆ such that ∥X 1 ∥ ∞ ≤ C 1 ∥X 0 ∥ ∞ +C 2 ∆∥X ′1 ∥ ∞ .
For the other terms,

diag Σ i i -k -λ ′ i -k X k =    a k+1 b k+1 . . . . . . a N b N    X k-1 +     0 Σ k+11 d ∆ . . . . . . 0 Σ N N -k d ∆     X k-2 + diag λ ′ i -k 2 X ′k
where the a i , b i are equals to 0 or

Σ i i -k ∆-d ∆ .
As before, we can invert the left-hand side matrix to obtain the estimate for some constants

C 1 ,C 2 ,C 3 > 0 independent of ∆ ∥X k ∥ ∞ ≤ C 1 ∥X k-1 ∥ ∞ +C 2 ∥X k-2 ∥ ∞ +C 3 ∆∥X ′k ∥ ∞ .
Thus, we can show by induction that there exist two constants

C 1 > 0,C 2 ≥ 0 independent of ∆ such that ∥X k ∥ ∞ ≤ C 1 ∥ f ∥ ∞ +C 2 ∆∥K uu ∆ ∥ ∞ . ( 80
)
For K uu , if we denote Y k the vector composed of the (K uu ) k i i =k,...,N and Y ′k the one composed by the (K uv ) k i i =k,...,N . Due to (71) (or equivalently (77)), we have

∥Y 1 ∥ ∞ ≤ ∥Y ′1 ∥ ∞ .
For the other terms, we can write

      Σ ′ kk + λ ′ k a k+1 Σ ′ k+1k + λ ′ k . . . . . . a N Σ ′ N k + λ ′ k       Y k =       Σ ′ kk 0 Σ ′ k+1k d ∆ b k+1 . . . . . . Σ ′ N k d ∆ b N       Y k-1 + diag λ ′ k 2 Y ′k
where the a i , b i are equals to 0 or Σ i ,k ∆-d ∆ . Let M k be the left-hand side matrix. This matrix is invertible because the determinant is non-zero due to the fact that the diagonal coefficients are non-zero. Let's estimate the infinity norm of M -1 k . We will need the following lemma: Lemma 3.1. Let A be a bidiagonal matrix written as

A =       b 1 a 2 b 2 . . . . . . a n b n      
If b i ̸ = 0 for all i then A is invertible and the coefficients of A -1 are given by

(A -1 ) i j = (-1) i + j j Π k=i +1 a k j Π k=i b k , j < i (A -1 ) i i = 1 b i (A -1 ) i j = 0, j > i
Applying this result to M k , we can increase the terms

|(M -1 k ) i j | by 1 |Σ ′ i ,k
| so that they are O(∆). We can apply the Lemma 3.1 to obtain the estimate (for some positive constants C 1 ,C 2 independent of ∆)

∥Y k ∥ ∞ ≤ C 1 ∥Y 0 ∥ ∞ +C 2 ∆∥Y ′k ∥ ∞
which can be reshaped to deduce that there exists a constant C > 0 independent of ∆ such that

∥Y k ∥ ∞ ≤ C ∥K uv ∆ ∥ ∞ . (81)
So combining (80) et (81), one can have

∥K uv ∆ ∥ ∞ ≤ C 1 ∥ f ∥ ∞ +C 2 ∆∥K uu ∆ ∥ ∞ ≤ C ′ 1 ∥ f ∥ ∞ +C ′ 2 ∆∥K uv ∆ ∥ ∞
We can now deduce that there exist a constant C > 0 for ∆ small enough such that

∥K uv ∆ ∥ ∞ ≤ C ∥ f ∥ ∞ .
And in the same way, one can have

∥K uu ∆ ∥ ∞ ≤ C ∥ f ∥ ∞ .
So that the stability of the scheme is obtain. Because of the stability, the scheme is then be convergent to order one thanks to Lax's Theorem [START_REF] Lax | Survey of the stability of linear finite difference equations[END_REF]. More precisely, we have the following theorem.

Theorem 3.1. The numerical scheme (65) converges to order one and, in particular, we have the following estimate: there exists a constant C > 0 independent of the discretisation step such that

K uv ∆ K uu ∆ - K uv K uu ∞ ≤ C ∆.
Remark 3.1. One can show a similar result for the pair K vu and K v v . So that we have a scheme convergent to order one for the complete direct kernels approximation.

We have tested the convergence numerically. We have considered λ(x) = e -4x for this simulation and the aim was to obtain K uv (x, ξ) = 1 + x -ξ and K uu = -1x + 2ξ which satisfy the boundary conditions. We have plotted the logarithm of the error as a function of the logarithm of the number of points considered, which gives us a straight line with slope -1, showing convergence to order one. The curve obtained is shown in the Figure 5.

A numerical approach to inverse kernels

We follow the method of [START_REF] Anfinsen | Adaptive control of linear2×2 hyperbolic systems[END_REF] used in [START_REF] Wijnand | Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension[END_REF]. To do this, we use the formula (45) which we recall here:

L(x, ξ) = K (x, ξ) + x ξ K (x, σ)L(σ, ξ)d σ.
Thus, we note that L(x, x) = K (x, x). For the approximation, we use the trapezoidal method to approximate the integral that appears in the previous formula. As with direct kernels, we use the same step in x and in ξ, noted ∆ (the same as for the direct kernel scheme) and let x i := i ∆, ξ j := j ∆ for i and j satisfying 0 ≤ ξ j ≤ x i ≤ 1 considering the same nodes as in Figure 1. Let m i , j := ij be the relevant number of subdivisions of the interval [ξ j , x i ]. Thus, the discretisation is written:

L j i = K j i + x i -ξ j 2m i , j i -1 k= j K k i L j k + K k+1 i L j k+1 (83)
Let's take a look at the convergence of this scheme. We assume that L ∈ C 2 (T ).

|L(x

i , ξ j ) -L j i | = K (x i , ξ j ) + x i ξ j K (x i , σ)L(σ, ξ j )d σ -K j i + x i -ξ j 2m i , j i -1 k= j K k i L j k + K k+1 i L j k+1 ≤ |K (x i , ξ j ) -K j i | + x i ξ j K (x i , σ)L(σ, ξ j )d σ - x i -ξ j 2m i , j i -1 k= j K k i L j k + K k+1 i L j k+1
In addition, we note that for the second term in the right member, using the triangular inequality, is increased by the following terms

x i ξ j K (x i , σ)L(σ, ξ j )d σ - x i -ξ j 2m i , j i -1 k= j K (x i , ξ k )L(x k , ξ j ) + K (x i , ξ k+1 )L(x k+1 , ξ j ) ( 84
) x i -ξ j 2m i , j i -1 k= j |L(x k , ξ j ) -L j k | • |K k i | + |L(x k+1 , ξ j ) -L j k+1 | • |K k+1 i | ( 85
) x i -ξ j 2m i , j i -1 k= j |K (x i , ξ k ) -K k i | • |L(x k , ξ j )| + |K (x i , ξ k+1 ) -K k+1 i | • |L(x k+1 , ξ j )| (86)
Or, from the fact that L ∈ C 2 (T ), the expression (84) is O(∆ 2 ) and we have |L(x k , ξ j )| ≤ C . Moreover, since the scheme for direct kernels is of order one (see Theorem 3.1) and since

x i -ξ j 2m i , j ≤ 1 2m i , j
, the expression (86) is then O(∆).

Thus, there exists a constant C > 0 such that we have the inequality

|L(x i , ξ j ) -L j i | ≤ C ∆ + |x i -ξ j | 2m i , j i -1 k= j |L(x k , ξ j ) -L j k | • |K k i | + |L(x k+1 , ξ j ) -L j k+1 | • |K k+1 i | (87)
which can be rewritten as follows:

1

- |x i -ξ j | 2m i , j |K i i | |L(x i , ξ j ) -L j i | ≤ C ∆ + |x i -ξ j | 2m i , j |L(x j , ξ j ) -L j j | • |K j i | + 2 i -1 k= j +1 |L(x k , ξ j ) -L j k | • |K k i | .
From the formula (45), we deduce that |L(x j , ξ j ) -

L j j | = |K (x j , ξ j ) -K j j |.
So, thanks to Theorem 3.1, we observe that there exists a constant C > 0 such that |L(x j , ξ j ) -

L j j | • |K j i | ≤ C ∆.
We therefore deduce that there exists a constant C > 0 such that

1 - |x i -ξ j | 2m i , j |K i i | |L(x i , ξ j ) -L j i | ≤ C ∆ + |x i -ξ j | m i , j i -1 k=1 |L(x k , ξ j ) -L j k | • |K k i |.
From the definition of m i , j and the approximation nodes, we infer that

|x i -ξ j | m i , j
= ∆ and so we can get the following inequality:

1 - |x i -ξ j | 2m i , j |K i i | |L(x i , ξ j ) -L j i | ≤ C ∆ + ∥K ∥ ∞ ∆ i -1 k=1 |L(x k , ξ j ) -L j k |. (88)
We're almost ready to conclude with the discrete Grönwall lemma. In order to use it, we need to know that the term

|x i -ξ j | 2m i , j
|K i i | to be increase by 1. Or,

|x i -ξ j | 2m i , j |K i i | < 1 ⇔ |K i i | < 2 ∆ ⇔ |K i i | < 2N
This last inequality is verified for N sufficiently large, according to the stability of the discrete scheme of direct kernels (65). We therefore deduce that there exists a constant 0 < C ′ < 1 such that

|L(x i , ξ j ) -L j i | ≤ C C ′ ∆ + ∥K ∥ ∞ C ′ ∆ i -1 k=1 |L(x k , ξ j ) -L j k |. (89)
Applying the discrete Grönwall lemma (see for example [START_REF] Quarteroni | Numerical Approximation of Partial Differential Equations[END_REF]), we then have the estimate:

|L(x i , ξ j ) -L j i | ≤ C C ′ ∆ exp ∥K ∥ ∞ C ′ ∆(i -1) . (90)
Or, the quantity ∆(i -1) is O(1) for all i ≥ 1 and we can conclude that there is a constant C > 0 independent of the step such that

|L(x i , ξ j ) -L j i | ≤ C ∆.
From this we deduce that this scheme converges to order one, and we can state the following theorem.

Theorem 3.2. The numerical scheme (83) converges to order one and, in particular, we have the following estimate: there exists a constant C > 0 independent of the discretisation step such that

∥L ∆ -L∥ ∞ ≤ C ∆.
For the simulations, we take the data from Paragraph 3.1 and obtain the following representations.

A discrete scheme for the target state

We are interested in the numerical approximation of the partial differential equations of the target state:

     α t = -λ(x)α x α(0, t ) = -β(0, t ) α(x, 0) = α 0 (x) and      β t = λ(x)β x β(1, t ) = 0 β(x, 0) = β 0 (x)
Let N x , N t ≥ 1, we denote ∆ x and ∆ t respectively the space and the time steps. Next, we set (x j , t n ) the approximation nodes where x j := ∆ x j , t n := ∆ t n . First, we approximate β using a first-order scheme of the downwind type, then α using a first-order upwind scheme (see for example [START_REF] Sainsaulieu | Calcul scientifique: cours et exercices corrigés pour le 2e cycle et les écoles d'ingénieurs[END_REF][START_REF] Ciarlet | Introduction à l'analyse numérique matricielle et à l'optimisation[END_REF]). The discrete schemes are then written :

       α n j = α n-1 j -λ(x j ) ∆ t ∆ x α n-1 j -α n-1 j -1 α n 0 = -β n 0 α 0 j = α 0 (x j ) et        β n j = β n-1 j + λ(x j ) ∆ t ∆ x β n-1 j +1 -β n-1 j β n N x = 0 β 0 j = β 0 (x j ) (91)
It's well known (see for example [START_REF] Sainsaulieu | Calcul scientifique: cours et exercices corrigés pour le 2e cycle et les écoles d'ingénieurs[END_REF][START_REF] Ciarlet | Introduction à l'analyse numérique matricielle et à l'optimisation[END_REF]) that the schemes are consistent to order one in time and space and stable under the CFL condition (for Courant Friedrichs Lewy)

∥λ∥ ∞ ∆ t ∆ x ≤ 1. (92)
For convergence, we use Lax's theorem [START_REF] Lax | Survey of the stability of linear finite difference equations[END_REF] to deduce that these schemes both converge (in infinite norms) to order one.

Theorem 3.3. The schemes (91) converge to order one in space and time under the CFL condition (92).

For the simulations, since the construction of the target state does not depend on the coefficients of the equation ( 1), the same result is obtained in the two cases presented in Paragraph 3.1. We choose the initial data β 0 (x) = 1x and α 0 (x) = x. This gives us the representations in Figure 8.

The discrete elliptic problem

We are interested in the numerical approximation of the elliptic problem:

-(av x ) x + c v = 0, v(0) = v(1) = 1.
-(au x ) x + cu = c, u(0) = u(1) = 0. ( 93
)
Here we follow the approach of [START_REF] Quarteroni | Méthodes Numériques: Algorithmes, analyse et applications[END_REF] to write a discretising scheme of (93). Let h = 1 N , N ≥ 1, the discretisation step, x j = j h the approximation points and x j + 1 2 = x j +x j +1 2

= j h + h 2 , the midpoints. Finally, we set a j := a(x j ), u j := u(x j ), c j := c(x j ) et a j ± 1 2 := a(x j ± 1 2 ). The discrete scheme is therefore

-1 h 2 a j -1 2 u j -1 -a j + 1 2 + a j -1 2 u j + a j + 1 2 u j +1 + c j u j = c j , u 0 = u N = 0. (94)
This scheme can be written in matrix form

A h u h = C 0 where A h := 1 h 2 tridiag N -1 (a, d , a) + diag N -1 (c) and C 0 := diag N -1 (c) (to avoid confusion with constants) with          a = -a 3 2 , a 5 2 , ..., a N -3 2 T ∈ R N -2 d = a 1 2 + a 3 2 , a 3 2 + a 5 2 , ..., a N -3 2 + a N -1 2 T ∈ R N -1 c = (c 1 , c 2 , ..., c N -1 ) T ∈ R N -1
From the fact that a, c > 0 we deduce that the matrix A h is symmetric positive definite (and with a strictly dominant diagonal) and therefore invertible. The discrete solution therefore exists and is unique.

Convergence

We suppose that u ∈ C 3 (0, 1) and au ′ ∈ C 3 (0, 1).

Note R j , the consistency error at point x j , the quantity

R j := 1 h 2 a j -1 2 u(x j -1 ) -a j + 1 2 + a j -1 2 u(x j ) + a j + 1 2 u(x j +1 ) -au ′ ′ (x j )
So that the following theorem is satisfied [START_REF] Quarteroni | Méthodes Numériques: Algorithmes, analyse et applications[END_REF].

Theorem 3.4. The discrete scheme (94) converges to order one (in space).

In the cases presented in Paragraph 3.1, we have v ≡ 1 and so here we haven't made any representations.

Numerical control of the wave equation

Return to the state variable

The state variable is given by the formula

w(x, t ) = γ(x, t ) + x 0 L(x, ξ)γ(ξ, t )d ξ.
As for Paragraph 3.2 on the approximation of inverse kernels, we approximate the integral by the trapezoidal method and adopt the notations and discretisation method of Paragraph 3.3 on the target state.

Adapting the calculations of Paragraph 3.2 to inverse kernels, we find

|w(x j , t n ) -w n j | = γ(x j , t n ) + x j 0 L(x j , σ)γ(σ, t n )d σ -γ n j + x j 2 j j -1 k=1 L k j γ n k + L k+1 j γ n k+1 ≤ |γ(x j , t n ) -γ n j | + x j 0 L(x j , σ)γ(σ, t n )d σ - x j 2 j j -1 k=1 L k j γ n k + L k+1 j γ n k+1 .
We assume that L ∈ C 2 (T ), γ ∈ C 2 ((0, 1) × (0, T )). Because of the stability of the schemes for the target state (see Paragraph 3.3), their convergence to order one in time and space according to Theorem 3.3 and using arguments similar to those of Paragraph 3.2, we deduce that we have convergence to order one in time and space.

Theorem 3.5. The discrete scheme obtained by approximation using the trapezoid method converges to order one in time and space.

The approximate control

From formulas

W (t ) = 1 0 L βα (1, ξ)α(ξ, t ) + L ββ (1, ξ)β(ξ, t )d ξ, u(t ) = - λ(1) 2 t F t w 1 (1, τ) + W (τ)d τ,
using the trapezoid method for any t ≥ 0, one can have the discretisation: From the trapezoid method, the first term (97) is O(∆ 2 t ). For the second term (98), by convergence of the scheme for the state variable system (see Theorem 3.5), the term |w 1 (1, t k ) -(w 1 ) k N | is O(∆ t ) and it remains to study the term in W . To do this, we write

W n = 1 2m m-1 k=1 (L βα ) k N α n k + (L ββ ) k N β n k + (L βα ) k+1 N α n k+1 + (L ββ ) k+1 N β n k+1 , (95)
u n = - λ(1)
W (t k ) -W k = 1 0 L βα (1, ξ)α(ξ, t k ) + L ββ (1, ξ)β(ξ, t k )d ξ - 1 2m m-1 k=1 (L βα ) j N α k j + (L ββ ) j N β k j + (L βα ) j +1 N α k j +1 + (L ββ ) j +1 N β k j +1
expression that can be increased by the sum of the term For the term (99), from [START_REF] Quarteroni | Méthodes Numériques: Algorithmes, analyse et applications[END_REF], using a triangulation of T h and composite formulas, one can show that the trapezoidal method is of order two in space and time. For the sum term in the equation (100), we'll look at the first term that appears, the others are treated in the same way: L βα (1, ξ j )α(x j , t k ) -(L βα ) j N α k j ≤ L βα (1, ξ j )α(x j , t k ) -L βα (1, ξ j )α k j + L βα (1, ξ j )α k j -(L βα ) j N α k j According to the regularity of L βα and the scheme convergence for the target system, the first term is O(∆ t , ∆ x ). The same holds for the second term, since the scheme for the inverse kernel converges to order one. (see Theorem 3.2) and thanks to the stability of the scheme for the target system, the term |α k j | is O(1). This gives us the following theorem.

Theorem 3.6. Under the CFL condition (92), the scheme (96) for approximate control converges to order one (in time).

We consider the various cases presented in Paragraph 3.1, namely the case where a(s) = 1 + s and the case where a(s) = 1+2 cos 2 (5πs). The representations obtained are shown in Figure 9 in which the finite-time stability is well illustrated.

Approximation of the unknown z and its derivatives

From the formulas z x (x, t ) = 1 2 λ(x) (w 2 (x, t )w 1 (x, t )) , where (w 1 ) n j et (w 1 ) n j are the approximations of the state variable introduced in the previous subsection 3.5.1. Thus, according to the triangular inequality and the convergence to order one in time and space of the scheme for the state variable (see Theorem 3.5), we deduce that these schemes also converge to order one in time and space.

From the formula z(x, t ) =

x 0 z x (s, t )d s + u(t ), using the trapezoid formula to approximate the integral, we obtain the discretisation (always keeping the same discretisation steps)

z n j = x j 2 j j -1 k=1 (z x ) n k + (z x ) n k+1 + u n . ( 101
)
For convergence, once again, we can observe that we have

x j 0 z x (s, t n )d s - x j 2 j j -1 k=1 (z x ) n k + (z x ) n k+1 ≤ x j 0 z x (s, t n )d s - x j 2 j j -1 k=1 z x (x j , t n ) + z x (x k+1 , t n ) + x j 2 j j -1 k=1 |z x (x k , t n ) -(z x ) n k | + |z x (x k+1 , t n ) -(z x ) n k+1 |
We assume that z x ∈ C 2 ((0, 1) × (0, T )). We thus deduce that this term is O(∆ t , ∆ x ) since the scheme for z x converges to order one in space and time, and the trapezoidal method converges to order 2, as established for the control. Since the scheme for the control u converges to order one in time according to Theorem 3.6, combining this with the above gives the same convergence for this scheme. We therefore have the following convergence theorem.

Theorem 3.7. The scheme for the unknown z described in (101) is convergent to order one in time and space under the CFL condition (92).

1 0 2 λ

 2 (x) dx of the problem (1)-(4) with the regularity:

Figure 1 :

 1 Figure 1: Discretisation of the triangle in the case N = 4

Figure 3 :

 3 Figure 3: Step selection when the vector -σ i , j ν i , j points to the top of the square

Figure 5 :Figure 6 :

 56 Figure 5: Order of convergence for direct kernels

Figure 7 :

 7 Figure 7: Discrete problem solution representation for inverse kernels

Figure 8 :

 8 Figure 8: Discrete solution representation for target state

Figure 9 : 1 2

 91 Figure 9: Discrete control representation

  ) -(w 1 ) k N | + |w 1 (1, t k+1 ) -(w 1 ) k+1 N | + |W (t k ) -W k | + |W (t k+1 ) -W k+1 | (98)

								2	t F -t n 2m ′	k=1 m ′ -1	(w 1 ) k N + W k + (w 1 ) k+1 N + W k+1 .	(96)
	For convergence, once again, we observe that we can increase the term
				2 λ(1)	t	t F	w 1 (1, τ) + W (τ)d τ -	t F -t n 2m ′	k=1 m ′ -1	(w 1 ) k N + W k + (w 1 ) k+1 N + W k+1
	using the triangular inequality by the sum of the terms
	t	t F	w 1 (1, τ) + W (τ)d τ -	t F -t n 2m ′	k=1 m ′ -1	w 1 (1, t k ) + W (t k ) + w 1 (1, t k+1 ) + W (t k+1 )	(97)
	t F -t n 2m ′	k=1 m ′ -1	|w 1 (1, t k

1 0

 1 L βα (1, ξ)α(ξ, t k ) + L ββ (1, ξ)β(ξ, t k )d ξ βα (1, ξ j )α(x j , t k ) + L ββ (1, ξ j )β(x j , t k ) + L βα (1, ξ j +1 )α(x j +1 , t k ) + L ββ (1, ξ j +1 )β(x j +1 , t k )

	-	1 2m	m-1 k=1	L (99)
	and of the term
	|	2m 1	|	m-1

k=1 |L βα (1, ξ j )α(x j , t k ) -(L βα ) j N α k j | + |L ββ (1, ξ j )β(x j , t k ) -(L ββ ) j N β k j | + |L βα (1, ξ j +1 )α(x j +1 , t k ) -(L βα ) j +1 N α k j +1 | + |L ββ (1, ξ j +1 )β(x j +1 , t k ) -(L ββ ) j +1 N β k j +1 | (100)

Numerical simulations

We now illustrate how to obtain kernels in two special cases. In both cases the coefficients b, c, ρ of (1) will satisfy b ≡ c ≡ 0 and ρ ≡ 1. The coefficient a of the main part of [START_REF] Alabau-Boussouira | Finite-time stabilization of a network of strings[END_REF], in the first case, will have the form a(s) = 1 + s to simplify the calculation and to be close to the study of [START_REF] Wijnand | Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension[END_REF], and in the second case, it will have the form a(s) = 1 + 2 cos 2 (5πs) found, for example, in [START_REF] Biccari | Propagation of one-and two-dimensional discrete waves under finite difference approximation[END_REF]. Note that in both cases the solution v of the elliptic problem ( 14)-( 15) is constant and v ≡ 1.

In the first case, where a(s) = 1 + s, it can be shown by direct calculation that we obtain the expression for the variable x as a function of s:

and conversely s depending on x: s(x) = e ln(x)-1 .

With this in hand, one can derive an expression for the function λ defined in [START_REF] Majda | Disappearing solutions for the dissipative wave equation[END_REF]:

.

For the other case in which the coefficient a is given by a(s) = 1+2 cos 2 (5πs), the variable x is given as a function of the variable s by 3

from which we can deduce the reciprocal:

We can then deduce the expression:

, with the variable s defined by (82). We choose N = 200 and we use the discrete scheme (65). The solution for this scheme is shown in Figure 6.

The result in the first case is close to that of [START_REF] Wijnand | Finite-time stabilization of an overhead crane with a flexible cable submitted to an affine tension[END_REF] and the second case allows us to show that we can also consider oscillatory function as in [START_REF] Biccari | Propagation of one-and two-dimensional discrete waves under finite difference approximation[END_REF].

Approximation of the initial unknown and conclusion on convergence

Recall that the initial unknown is defined by y(x, t ) = v(x)z(x, t ). Using the previous notations, the discrete scheme is written as y n j = v j z n j , where v j is approximated in Paragraph 3.4 and z n j in Paragraph 3.5.3. Hence,

Or, v h = A -1 h c so ∥v h ∥ ∞ ≤ C and from the regularity of z, we have ∥z∥ ∞ ≤ C . Thus, convergences for the elliptic scheme seen in Theorem 3.4 and the variable z in Theorem 3.7, we infer that the scheme for the variable y converges to order one. We remind you that in the cases presented in Paragraph 3.1, for all j , n, v j = 1 and so y n j = z n j for all j , n. The discrete solution obtained in both cases is shown below.

Acknowledgments:

The author would like to thank Lionel Rosier for this interesting topic, Henrik Anfinsen and Marc Wijnand for their help with the numerical calculation of the kernels and the approximate control and starting unknown, respectively, and for providing some MATLAB codes.


M.-L Aix

D J Bicout

PROTOCOLE D'ÉVALUATION ET D'UTILISATION D'UNE STATION LOW-COST DE MESURE DES PARTICULES FINES

Keywords: capteurs low-cost, PM2.5, calibration / KEYWORDS: low-cost sensors, PM2.5, calibration

Protocole for evaluation and use of a low-cost particulate matter monitoring station RÉSUMÉ Les capteurs « low-cost » pourraient révolutionner la mesure des particules fines (PM) mais il paraît nécessaire d'évaluer leurs performances. Ce travail propose un protocole de montage, d'évaluation et d'utilisation d'une station comportant un capteur PMS7003.

Les performances de mesure des PM2.5 (PM de taille <2,5µm) sont comparées aux directives de l'Environmental Protection Agency américaine (EPA). Il apparaît que certaines métriques (pente et NRMSE) sont non conformes en l'absence de calibration. Une calibration linéaire les ramène dans les limites de l'EPA et une calibration mécanistique améliore encore la précision. Les RMSEs sont alors de 0.90 µg/m 3 (échelle horaire) et 0.48 µg/m 3 (journée).

INTRODUCTION

Ces dernières années, nous assistons à une croissance de l'intérêt pour les microcapteurs optiques "lowcost". Parmi ceux-ci, le PMS7003 est l'un des plus utilisés pour la mesure des particules fines (PM). Il est plébiscité à la fois par les citoyens, les organismes officiels de mesure de qualité de l'air et la communauté scientifique. Il paraît donc important de mieux connaître ses performances et d'évaluer l'incertitude associée à ses mesures. S'appuyant sur une méthodologie développée par l'EPA (Duvall et al., 2021) et déjà utilisée récemment dans une étude (Zimmerman, 2022), notre travail vise à donner un aperçu des performances et des limites d'un concept de station de qualité de l'air utilisant le PMS7003. Notre objectif est de fournir un mode d'emploi couvrant le montage, l'évaluation des performances et le déploiement de ce type de dispositif. Nous nous focalisons ici sur la mesure des particules fines de taille inférieure à 2,5µm (PM2.5). L'approche décrite ci-dessus peut s'adapter et s'appliquer à d'autres dispositifs similaires.

MONTAGE

Le dispositif de mesure des PM que nous avons mis en oeuvre comporte deux capteurs low-cost : le PMS 7003 (Plantower), capteur optique de PM, ainsi que le DHT22 (MaxDetect), capteur d'humidité relative et de température. Ils sont connectés à un microcontrôleur Wi-Fi ESP8266 qui collecte les informations et les transforme selon un algorithme. Des fils de couleur sont utilisés pour connecter les composants entre eux et un boîtier en polycarbonate IP66 les protège des intempéries. Fin 2020, le coût total d'une telle station avoisinait les 65€. Pour la réalisation des objectifs de ce travail, nous avons construit 9 stations en partenariat avec Atmo Auvergne-Rhône-Alpes (Atmo AuRA).

PERFORMANCES

Pour évaluer les performances des stations, nous avons réalisé une expérience de colocation sur la station des Frênes (Grenoble) en collaboration avec Atmo AuRA. Il s'était agi alors de comparer les valeurs des capteurs à celles remontées par le Fidas ® 200 (Palas GmbH) d'Atmo. Ce processus de colocation s'est étalé sur 8 mois, du 28 janvier 2021 au 29 septembre 2021.

Critères de l'EPA (Duvall et al., 2021)

L'EPA est l'agence qui implémente les lois fédérales de protection de l'environnement aux États-Unis. C'est le cas par exemple du Clean Air Act, qui règlemente l'ensemble des émissions atmosphériques. Dès 1998, le Congrès a demandé à l'EPA d'élargir ses recherches sur les effets sanitaires des PM. Depuis, l'EPA soutient activement la recherche sur les PM. Particulièrement active dans le domaine des capteurs low-cost, elle a défini 5 critères permettant de comparer les valeurs des capteurs à celles d'une référence : le coefficient de détermination (R 2 ), la pente, l'ordonnée à l'origine, la racine de l'erreur quadratique moyenne (RMSE) et la racine de l'erreur quadratique moyenne normalisée (NRMSE). Le R 2 rend compte de la relation linéaire entre le capteur et la référence, tandis que le RMSE reflète l'écart entre les mesures des stations low-cost et celles de la référence ; il indique donc la précision du capteur. Enfin, l'EPA utilise d'autres indicateurs, obtenus en comparant les valeurs des différents capteurs low-cost entre elles, indépendamment de la référence. C'est par exemple le cas du coefficient de variation (CV) et de l'écart-type (SD). Les standards de l'EPA pour les capteurs (repris en gris sur la Figure 2) sont les suivants : R 2 ≥ 0.7, RMSE ≤ 7 µg/m 3 ou NRMSE ≤ 30%, SD ≤ 5 µg/m 3 ou CV ≤ 30%, pente = 1 ± 0.35 et -5 ≤ intercept ≤ 5 µg/m 3 (Duvall et al., 2021).

Avant calibration

Nous avons retiré du jeu de données les périodes de dust sahariens et celles où l'humidité relative (HR) dépassait 98%. Lorsque l'on étudie les données des stations à la fréquence horaire, les métriques telles que la pente et le NRMSE sortent des standards de l'EPA (Figure 2, partie "Avant calibration") ; les autres métriques restant conformes. Lorsque l'on passe à une fréquence journalière, on constate que le NRMSE s'améliore, mais reste non conforme. La pente évolue peu. Il est donc nécessaire de procéder à une calibration des stations. Par ailleurs, le passage à une fréquence journalière permet une amélioration de l'ensemble des autres métriques, sauf l'ordonnée à l'origine (intercept) qui est légèrement dégradée.

Calibration

Dans un souci de simplicité et de transparence, nous avons utilisé deux modèles empiriques : (1) un modèle linéaire, (2) un modèle mécanistique prenant en compte l'HR et la température mesurées par le capteur. Ces modèles ont été ajustés sur 75% du jeu de données et testés sur les 25% restants (Figure 1).

Après calibration

La régression linéaire permet de corriger efficacement les concentrations remontées par les capteurs. L'ensemble des métriques devient conforme aux exigences de l'EPA (Figure 2, partie "Après calibration linéaire"). Au niveau horaire, on observe une légère baisse de 3.4% du R 2 médian. L'utilisation de fréquences journalières offre des métriques plus performantes sans dégradation du R 2 et avec une amélioration de 40% du RMSE. La calibration mécanistique au niveau horaire permet quant à elle, par rapport à la régression linéaire, une amélioration de 7,4% du RMSE et de 3,5% du R 2 . On observe en revanche une légère dégradation des indicateurs suivants : CV et écart-type. Au niveau journalier, on observe une hausse de 0,4% du R 2 médian et une amélioration de 20,7% du RMSE par rapport à la régression linéaire.

DÉPLOIEMENT

Une fois calibrées, les stations low-cost peuvent être déployées pour l'expérimentation. De manière générale, les conditions environnementales dans lesquelles sont installées les stations doivent être aussi proches que possible de celles de la colocation :

• Le lieu de l'installation doit être situé près d'une alimentation et à portée d'un réseau Wi-Fi.

• Les dispositifs doivent être éloignés de zones fumeurs ou de chantiers.

• La hauteur d'accroche doit être similaire à celle du site d'accueil.

• Aucun obstacle, autre que le support, ne doit se trouver dans un périmètre de 1m.

• Il est recommandé de ne pas orienter les capteurs plein sud et de les abriter (exposition au soleil).

Enfin, nous avons constaté une baisse des performances de mesure de certains capteurs d'humidité relative (DHT22) après 7 mois, une partie des capteurs donnant continuellement des valeurs égales à 100%. Nous recommandons donc l'utilisation de capteurs d'humidité et de température plus robustes (Bosch BME280, Sensirion SHT85…).

CONCLUSION

Il ressort de cette expérience qu'une simple calibration linéaire suffit à rendre les métriques de nos stations low-cost conformes aux recommandations de l'EPA en termes de performances. Une calibration mécanistique améliore significativement la précision des capteurs mais entraîne une très légère dégradation du CV et de l'écart-type, restants néanmoins dans les plages de l'EPA. À l'échelle horaire, la calibration mécanistique permet d'atteindre un RMSE de 0.90 µg/m 3 et un R 2 médian de 0.96, ce qui est très satisfaisant. À l'échelle journalière, il est même possible d'obtenir un RMSE de 0.48 µg/m 3 et un R 2 de 0.99 par correction mécanistique. De manière générale, nous constatons qu'il est plus facile de satisfaire les critères de performance EPA lorsque l'on transpose les concentrations à une échelle journalière, plutôt qu'à une échelle horaire.

RÉFÉRENCES

REMERCIEMENTS

Marie-Laure Aix est soutenue par une bourse du Ministère de l'Éducation Nationale de l'Enseignement Supérieur, de la Recherche et de l'Innovation. Elle réalise sa thèse au sein de l'École Doctorale Ingénierie pour la Santé, la Cognition et l'Environnement (ED ISCE) de l'Université Grenoble Alpes. Nous tenons à remercier chaleureusement Atmo Auvergne-Rhône-Alpes pour l'assistance au montage et à la colocation des stations dans le cadre de notre partenariat.

Figure 1 .

 1 Figure 1. Découpage de la période de colocation Les formules des modèles (Tableau 1 ci-dessous), dont les coefficients sont issus du meilleur ajustement, ont été obtenues avec le logiciel « RStudio Server 2021.09.1 ». Tableau 1. Formules de calibration (PM2.5 adj = concentration en PM2.5 ajustée (corrigée) du capteur, RHint = humidité relative interne mesurée par le capteur DHT22, Tint = température interne mesurée par le DHT22).

Figure 2 .

 2 Figure 2. Indicateurs de performance des stations low-cost (zones grisées : standards EPA, lignes en gras : valeurs cibles, "Heure" = fréquences horaires, "Jour" = fréquences journalières). Le rouge est utilisé lorsque les métriques sortent des limites recommandées par l'EPA, le vert correspondant à une conformité.

  Aix, M.-L., Schmitz, S., Bicout, D. J. (2023). Open-source calibration of low-cost sensors for high-quality monitoring of fine particulate matter [Document soumis pour publication] Duvall, R., A. Clements, G. Hagler, A. Kamal, Vasu Kilaru, L. Goodman, S. Frederick, K. Johnson Barkjohn, I. VonWald, D. Greene, T. Dye. (2021). Performance Testing Protocols, Metrics, and Target Values for Fine Particulate Matter Air Sensors: Use in Ambient, Outdoor, Fixed Site, Non-Regulatory Supplemental and Informational Monitoring Applications. ://cfpub.epa.gov/si/si_public_record_Report.cfm?dirEntryId=350785&Lab=CEMM Zimmerman, N. (2022). Tutorial: Guidelines for implementing low-cost sensor networks for aerosol monitoring. Journal of Aerosol Science, 159, 105872. https://doi.org/10.1016/j.jaerosci.2021.105872

	U.S.	EPA	Office	of	Research	and	Development,	Washington,	DC,	EPA/600/R-20/280.

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