Jean Auriol

SOME INSIGHTS ON THE PRACTICAL CONTROL OF HYPERBOLIC SYSTEMS

In this chapter, we present some recent developments in the practical control of hyperbolic systems. Although our primary focus lies on scalar balance laws, most of the results showcased herein can be extended to non-scalar systems or interconnected configurations. The control strategies proposed in this chapter are said to be practical as they are constructive, easily implementable, and offer degrees of freedom that can be tailored to satisfy specific performance requirements. We first design finite-time stabilizing controllers using the backstepping approach. Then, we show that finite-time stabilization can yield vanishing robustness margins. Therefore, we introduce tuning parameters in the design to allow for potential trade-offs between performance specifications (e.g., robustness and convergence rate). Finally, we consider possible approximations of the control law in view of practical implementation and give generic conditions to guarantee closedloop stability.

Introduction

Distributed parameter systems offer a natural way to represent industrial processes that involve the evolution of quantities in both time and space. Hyperbolic Partial Differential Equations (PDEs), in particular, are essential for mathematically describing transport phenomena with finite propagation speeds, such as the movement of matter, sound waves, and information. These equations serve as a fundamental framework for describing a wide range of complex systems, including wave propagation, traffic network systems, electric transmission lines, hydraulic channels, drilling devices, communication networks, smart structures, as well as multiscale and multiphysics systems [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF][START_REF] Aarsnes | Torsional vibrations with bit off bottom: Modeling, characterization and field data validation[END_REF][START_REF] Yu | Traffic Congestion Control by PDE Backstepping[END_REF][START_REF] Yu | Simultaneous downstream and upstream output-feedback stabilization of cascaded freeway traffic[END_REF]. Controlling and monitoring hyperbolic systems are difficult control engineering problems due to the distributed nature of the systems (time and space dependency) and the physical/economic infeasibility of placing sensors and actuators everywhere along the spatial domain. The stringent operating, environmental and economical requirements and the high mathematical complexity of these systems explain why traditional control methods exhibit a limited range of applicability and have not been successful at high technology readiness levels (TRLs) [START_REF] Auriol | Robust output feedback stabilization for two heterodirectional linear coupled hyperbolic PDEs[END_REF][START_REF] Curtain | An introduction to infinite-dimensional linear systems theory[END_REF]. Consequently, the theory of control for distributed parameter systems requires substantial advancements to achieve control and estimation objectives effectively. This explains the notable surge in scholarly publications dedicated to this subject over the past few decades.

In recent years, numerous theoretical approaches have emerged for designing boundary controllers and observers specifically tailored to one-dimensional linear balance law systems, a specific class of hyperbolic systems [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF]Chap. 5]. Among these approaches, one can find flatness-based controllers [START_REF] Meurer | Flatness-based feedback control of diffusion-convection-reaction systems via k-summable power series[END_REF][START_REF] Sagert | Backstepping and flatness approaches for stabilization of the stick-slip phenomenon for drilling[END_REF], optimization controllers [START_REF] Lions | Optimal control of systems governed by partial differential equations problèmes aux limites[END_REF] or Lyapunov-based controllers [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF][START_REF] Coron | Control and nonlinearity[END_REF][START_REF] Prieur | Lyapunov functions for switched linear hyperbolic systems[END_REF] that have for instance enabled the design of dissipative boundary conditions [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF][START_REF] Coron | Control and nonlinearity[END_REF]. In this context, the backstepping approach [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF] stands out as a constructive method with distinct characteristics. This technique employs an invertible change of variables, accomplished through an integral transformation, to map the original system onto a more amenable form known as the "target system." The backstepping approach allows for analysis, control, and observer design, resulting in explicit controllers expressed as functionals of the distributed states, akin to classical finite-dimensional counterparts. Initially introduced for a class of parabolic equations [START_REF] Smyshlyaev | Closed-form boundary state feedbacks for a class of 1-d partial integro-differential equations[END_REF], the backstepping approach has been extended to wave equations [START_REF] Krstic | Output-feedback stabilization of an unstable wave equation[END_REF] and linear or quasilinear hyperbolic systems [START_REF] Coron | Local exponential H 2 stabilization of a 2× 2 quasilinear hyperbolic system using backstepping[END_REF][START_REF] Auriol | Minimum time control of heterodirectional linear coupled hyperbolic PDEs[END_REF][START_REF] Coron | Finite-time boundary stabilization of general linear hyperbolic balance laws via Fredholm backstepping transformation[END_REF][START_REF] Deutscher | Periodic output regulation for general linear heterodirectional hyperbolic systems[END_REF]. A comprehensive historical account of the backstepping method for PDEs and its extensions can be found in [START_REF] Vazquez | Taking a step back: A brief history of PDE backstepping[END_REF]. For linear hyperbolic systems, one notable outcome of the backstepping controllers is their ability to partially address the finite-time stabilization and observability problems formulated in [START_REF] Russell | Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory[END_REF][START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions[END_REF] and further generalized by Tatsien Li in [START_REF] Li | Controllability and observability for quasilinear hyperbolic systems[END_REF]. For the latest advancements in optimal finite-time stabilization of homogeneous quasilinear hyperbolic systems, we refer the reader to [START_REF] Coron | Finite-time stabilization in optimal time of homogeneous quasilinear hyperbolic systems in one dimensional space[END_REF][START_REF] Coron | Null-controllability of linear hyperbolic systems in one dimensional space[END_REF].

The backstepping method encounters a major challenge in determining a suitable target system that strikes a balance between simplicity for control law design and the existence of a transformation mapping the original system to this target system. The choice of the target system significantly influences the closed-loop performance. However, the broader question of identifying reachable target systems remains an open problem. For balance laws systems, target systems are often selected to be finite-time stable [START_REF] Coron | Null-controllability of linear hyperbolic systems in one dimensional space[END_REF], thereby neglecting the robustness properties of the corresponding closed-loop systems [START_REF] Logemann | Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop[END_REF][START_REF] Michiels | Strong stability of neutral equations with an arbitrary delay dependency structure[END_REF]. These limitations in robustness can stem from various factors, including parameter uncertainties, disturbances acting on the system, measurement noise, neglected dynamics, or delays in the actuators. It has been observed that the introduction of even small time delays in the feedback loop can lead to instability for many feedback systems [START_REF] Datko | An example on the effect of time delays in boundary feedback stabilization of wave equations[END_REF][START_REF] Logemann | Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop[END_REF]. Controllers designed to ensure finite-time stability are typically non-strictly proper and thus possess zero robustness margins. In [START_REF] Auriol | Robust output feedback stabilization for two heterodirectional linear coupled hyperbolic PDEs[END_REF], the authors introduced tuning parameters in the design to allow for potential trade-offs between different specifications, such as delay-robustness and convergence rate. The key idea behind these modifications was to avoid the complete cancellation of reflections at the spatial domain boundaries of the PDE, guaranteeing robustness margins. A general procedure for robustification was proposed in [START_REF] Auriol | Robustification of stabilizing controllers for ODE-PDE-ODE systems: a filtering approach[END_REF]. This robustification was achieved by the design of appropriate filters, generalizing the approaches in [START_REF] Bou Saba | Strictly proper control design for the stabilization of 2 × 2 linear hyperbolic ODE-PDE-ODE systems[END_REF][START_REF] Auriol | Robust state-feedback stabilization of an underactuated network of interconnected n+ m hyperbolic PDE systems[END_REF]. To introduce additional natural tuning parameters (e.g., the dissipation rate) with clear energy interpretations, one could take advantage of Port-Hamiltonian Systems (PHS) theory [START_REF] Jacob | Linear port-Hamiltonian systems on infinite-dimensional spaces[END_REF]. PHS theory provides a comprehensive and modular energybased representation for multi-physical systems, taking into account their inherent physical properties such as passivity, dissipativity, and reversibility. This theory establishes a physical framework through which the closed-loop properties of the target systems can be parameterized. Initially developed for finite-dimensional systems [START_REF] Duindam | Modeling and control of complex physical systems: the port-Hamiltonian approach[END_REF], the PHS framework has been extended to encompass Partial Differential Equations [START_REF] Le Gorrec | Dirac structures and boundary control systems associated with skew-symmetric differential operators[END_REF][START_REF] Jacob | Linear port-Hamiltonian systems on infinite-dimensional spaces[END_REF]. The PHS framework proves particularly relevant in control design utilizing damping assignment or energy shaping methods. By leveraging the natural physical properties of the system, it becomes possible to define wellposed candidates for the target systems. The degrees of freedom induced by such a general class of target systems can then be tuned to fulfill a given set of performance criteria. This strategy has been successfully applied to simple test cases, as demonstrated in previous studies [START_REF] Ramirez | On backstepping boundary control for a class of linear port-hamiltonian systems[END_REF][START_REF] Redaud | In-domain damping assignment of a timoshenko-beam using state feedback boundary control[END_REF].

Finally, implementing these novel control strategies on real systems necessitates a thorough assessment of their computational costs and a comparative analysis against existing controllers, particularly PID controllers, in light of the advancements in computational systems, such as simple Arduino boards. In this context, it is essential to consider complexity reduction strategies to find the right compromise between performance and numerical complexity. This may imply approximating the controller by a finite-dimensional system (late lumping approach). The question of the convergence of late-lumping backstepping controllers has not been well-investigated, contrary to the approximation of the kernels themselves, e.g., in [START_REF] Jadachowski | An efficient implementation of backstepping observers for time-varying parabolic pdes[END_REF] using a trapezoidal rule or in [START_REF] Ascencio | Backstepping PDE design: A convex optimization approach[END_REF] using a sum-of-squares approach. In [START_REF] Woittennek | On approximation and implementation of transformation based feedback laws for distributed parameter systems[END_REF], a method for computing the bounded part of the control operator is proposed. It relies on a finite-dimensional approximation of the state and enables efficient computing of the feedback law. Recently, Lyapunov methods were applied to assess the convergence of approximated backstepping controllers [START_REF] Auriol | Late-lumping backstepping control of partial differential equations[END_REF][START_REF] Karafyllis | Input-to-state stability for PDEs[END_REF]. Additionally, learning-based approaches have emerged as an intriguing avenue, with approximation guarantees (such as DeepONet algorithms [START_REF] Deng | Convergence rate of deeponets for learning operators arising from advection-diffusion equations[END_REF][START_REF] Bhan | Operator learning for nonlinear adaptive control[END_REF]).

In this chapter, we provide a comprehensive overview of recent advancements in the practical control of hyperbolic systems. While our focus is primarily on scalar balance laws, most presented results can be extended to more complex systems, including non-scalar and interconnected systems. The control strategies proposed in this study are said to be practical, as they are constructive, easily implementable, and offer degrees of freedom that can be tailored to satisfy specific performance requirements. Our methodology builds upon the backstepping approach and incorporates frequential analysis techniques. This chapter serves as a tutorial and is an introduction to the methods developed by the author for the control and stabilization of balance law systems.

This chapter is organized as follows. Section 2 presents the system under consideration and introduces the control objectives. We present in Section 3 a general introduction to the backstepping approach and show how this methodology can be used to design a stabilizing controller for the class of system under consideration. We analyze the robustness properties of the proposed control law in Section 4. We first show that finite-time stabilization can lead to vanishing robustness margins. We then propose some adjustments to obtain a robust control law. In Section 5, we show that it is possible to use the backstepping approach to map the original system to a large class of target systems. This allows introducing additional degrees of freedom in the design that can then be used to improve the closed-loop behavior. Section 6 focuses on model reduction aspects and how the proposed controllers can be approximated to be implemented on real systems while guaranteeing closed-loop stability. Some concluding remarks are finally given in Section 7.

Notations:

We denote L 2 ([0, 1], R) the space of real-valued square-integrable functions defined on [0, 1] with the standard L 2 norm, i.e., for any f ∈ L 2 ([0, 1], R), we have

||f || L 2 = 1 0 f 2 (x)dx 1 2
.

By convention, for any

u ∈ L 2 ([0, 1], R) and v ∈ L 2 ([0, 1], R), we denote ||(u, v)|| L 2 = ||u|| 2 L 2 + ||v|| 2 L 2 .
We denote H 1 ([0, 1], R) the one-dimensional Sobolev space, i.e., the subset of functions f in L 2 ([0, 1], R) such that f and its weak derivative of order one have a finite L 2 norm. We denote C 0 ([0, 1], R) the space of real-valued continuous functions defined on [0, 1]. We denote s, the Laplace variable. The partial derivative of a multivariable function f with respect to the variable x will be denoted ∂f ∂x .

2. System under consideration 2.1. Balance law equations. In this chapter, we consider the following general class of linear hyperbolic systems, which appears in Saint-Venant equations, heat exchanger equations, and other linear hyperbolic balance laws [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF]:

∂u ∂t (t, x) + λ ∂u ∂x (t, x) = σ ++ (x)u(t, x) + σ +-(x)v(t, x), (1)
∂v ∂t (t, x) -µ ∂v ∂x (t, x) = σ -+ (x)u(t, x) + σ --(x)v(t, x), (2)
with the linear boundary conditions

u(t, 0) = qv(t, 0), v(t, 1) = ρu(t, 1) + V (t), (3)
where the velocities λ and µ are assumed to be strictly positive. The results we present in this chapter could be extended to spatially-varying velocities at the cost of technical computations. The boundary couplings q and ρ are constant and are respectively called distal reflection (reflection at the unactuated boundary) and proximal reflection (reflection at the actuated boundary). The in-domain couplings σ •• belong to C 0 ([0, 1], R). The states u and v have values in R. They are evolving in

{(t, x)| t > 0, x ∈ [0, 1]}. The control input V (t) belongs to R. The initial condition is denoted (u 0 , v 0 ) ∈ (H 1 ([0, 1], R)) 2 .
The following theorem assesses the well-posedness of the open-loop system Theorem 1. Consider that V (t) = 0 for all t ≥ 0. For every initial condition

(u 0 , v 0 ) ∈ (H 1 ([0, 1], R)) 2 that verifies the compatibility conditions u 0 (0) = qv 0 (0), v 0 (1) = ρu 0 (1), (4)
there exists one and one only

(u, v) ∈ C 1 ([0, ∞), (L 2 ([0, 1], R)) 2 ) ∩ C 0 ([0, ∞), (H 1 ([0, 1], R)) 2 ),
which is a solution to the open-loop Cauchy problem (1)- [START_REF] Aarsnes | Torsional vibrations with bit off bottom: Modeling, characterization and field data validation[END_REF]. Moreover, there exists κ 0 > 0 such that for every (u 0 , v 0 ) ∈ (H 1 ([0, 1], R)) 2 satisfying the compatibility conditions (4), the unique solution verifies

||(u(t, •), v(t, •))|| L 2 ≤ κ 0 e κ0t ||(u 0 , v 0 )|| L 2 , ∀t ∈ [0, ∞).
(

) 5
Proof. The proof can be adjusted from [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF]Theorem A.1,Theorem A.6]. It relies on Lumer-Philipps theorem [START_REF] Lumer | Dissipative operators in a banach space[END_REF][START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]. □

This theorem (and most of the results presented in the chapter) could be adjusted to deal with weak solutions to the Cauchy problem (1)-( 2), as shown in [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF]. However, this would be at the cost of involved technical computations. It is important to emphasize that Theorem 1 only shows the well-posedness in open-loop. The closed-loop well-posedness could be shown using the admissibility of the control operator [START_REF] Coron | Control and nonlinearity[END_REF] (the proposed control law will be time-continuous) or adjusting the proof of Theorem 1. Moreover, the different control strategies proposed throughout the manuscript rely mainly on the backstepping methodology [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF]. Using invertible and bounded transformations, the closed-loop system can be mapped to a target system for which usual well-posedness results (as Theorem 1) can be straightforwardly applied. Consequently, the original and target systems share equivalent stability properties. This is why the well-posedness of the closed-loop system will not be discussed. The compatibility condition (4) may be changed in closed-loop to encompass the effect of the feedback law that has been designed. Thus, it may appear artificial and rather stringent, as it requires very specific values of the initial conditions. However, as shown in [START_REF] Coron | Local exponential H 2 stabilization of a 2× 2 quasilinear hyperbolic system using backstepping[END_REF], it is possible to modify our control law in a way that, without losing the stabilizing properties, does not require any specific values in the initial values beyond the natural conditions. 2.2. Simplification of the system. Our first objective is to remove the growth terms σ ++ (x) and σ --(x) in equations ( 1)-( 2) to simplify the equations. Let us consider the invertible exponential transformation defined for all t > 0 and all x ∈ [0, 1] by

ū(t, x) = u(t, x)e -x 0 σ ++ (ν) λ dν , v(t, x) = v(t, x)e + x 0 σ --(ν) µ dν , (6)
which is well-defined since the functions σ --(•) and σ ++ (•) are continuous. This yields

∂ ū ∂t (t, x) + λ ∂ ū ∂x (t, x) = ( ∂u ∂t (t, x) + λ ∂u ∂x u(t, x) -σ ++ (x)u(t, x))e -x 0 σ ++ (ν) λ dν = σ +-(x)v(t, x)e -x 0 σ ++ (ν) λ dν = σ +-(x)e -x 0 σ ++ (ν) λ dν e -x 0 σ --(ν) µ dν v(t, x) = σ + (x)v(t, x), (7)
where we have denoted σ

+ (x) = e -x 0 σ ++ (ν) λ dν e -x 0 σ --(ν) µ dν σ +-(x). Similarly, de- noting σ -(x) = e + x 0 σ ++ (ν) λ dν e + x 0 σ --(ν) µ dν σ +-(x), we obtain ∂v ∂t (t, x) -µ ∂v ∂x (t, x) = σ-(x)ū(t, x). (8)
The boundary conditions (3) rewrites

ū(t, 0) = qv(t, 0), v(t, 1) = ρu(t, 1) + V (t), (9)
with ρ = ρe

+ 1 0 σ ++ (ν) λ dν+ 1 0 σ --(ν) µ dν and V (t) = e + 1 0 σ --(ν) µ dν V (t)
. Consequently, the system (1)-( 2) can be rewritten as

∂u ∂t (t, x) + λ ∂u ∂r (t, x) = σ + (x)v(t, x), (10)
∂v ∂t (t, x) -µ ∂v ∂x (t, x) = σ -(x)u(t, x), (11)
with the boundary conditions u(t, 0) = qv(t, 0), v(t, 1) = ρu(t, 1) + V (t).

(

) 12
Note that we have removed the bars in this new system for clarity. The initial conditions still satisfy the appropriate compatibility conditions. In this chapter, we consider that q ̸ = 0. Although the results we present can be extended in the case q = 0, choosing q ̸ = 0 simplifies the exposition of the different results. The system ( 10)-( 12) is schematically pictured in Figure 1.

Figure 1. Schematic representation of the system (10)-( 12)

u(t, x) v(t, x) σ - σ + q ρ V (t) 0 1 x 2.3. Control objectives.
Let us first recall the following definition Definition 1 ([22]). The hyperbolic system (10)-( 12) is exponentially stable in the sense of the L 2 -norm if there exist ν > 0, and C > 0 such that, for every initial condition (u 0 , v 0 ) ∈ (H 1 ([0, 1], R)) 2 satisfying the compatibility conditions (4), the L 2 -solution of the problem (10)- [START_REF] Auriol | Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time[END_REF] satisfies

||(u(t, •), v(t, •))|| L 2 ≤ Ce -νt ||(u 0 (•), v 0 (•))|| L 2 . ( 13
)
Note that although the solution of the system (10)- [START_REF] Auriol | Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time[END_REF] lies in the H 1 -space, we consider the stability in the sense of the L 2 -norm that somehow relates to the energy of the system. One of the objectives of this chapter is to design a feedback control law

V = K[(u, v)], where K : (H 1 ([0, 1], R)) 2 → R is a linear operator such that
• The state (u, v) of the resulting feedback system (10)-( 12) is exponentially stable (stabilization problem). • The resulting feedback system (10)-( 12) is robustly stable with respect to small delays in the loop (delay robustness), i.e., there exists δ ⋆ > 0 such that for any δ ∈ [0, δ ⋆ ], the control law V (t -δ) still stabilizes (10)- [START_REF] Auriol | Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time[END_REF].

A control law that satisfies these two constraints is said to delay-robustly stabilize (in the sense of [START_REF] Logemann | Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop[END_REF]) system (10)- [START_REF] Auriol | Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time[END_REF]. For system (10)- [START_REF] Auriol | Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time[END_REF], it has been shown in [START_REF] Li | Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems[END_REF] that there exists a state-feedback control law that ensures the finite time stabilization in the sense of the L 2 -norm. This finite time corresponds to τ = 1 λ + 1 µ . It has been proved [START_REF] Russell | On boundary-value controllability of linear symmetric hyperbolic systems[END_REF][START_REF] Russell | Nonharmonic fourier series in the control theory of distributed parameter systems[END_REF] that this minimum time is "critical" in the sense that it is, in general, impossible to satisfy the given initial and terminal conditions if less time is allowed. Moreover, for this specific case of two equations, it has been proved that the control law that ensures finite-time boundary stabilization in the minimum time is unique [START_REF] Russell | Control theory of hyperbolic equations related to certain questions in harmonic analysis and spectral theory[END_REF]. The proof of this result is based on the explicit evolution of the Riemann invariants along the characteristics (see [START_REF] Li | Controllability and observability for quasilinear hyperbolic systems[END_REF] for details) and does not allow for an explicit design of the control law. In the next section (Section 3), we will use the backstepping method to design such a finite-time controller. We will then show in Section 4 that such a controller may not ensure delay-robustness and present some adjustments to delay-robustly stabilize system (10)-( 12).

3. Stabilization of system ( 10)-( 12): an introduction to the backstepping approach

The backstepping approach is a control method initially developed for nonlinear finite-dimensional systems (sometimes referred to as "backstepping by integrator") [START_REF] Krstic | Nonlinear and adaptive control design[END_REF]. While such a method cannot be directly extended to infinite dimensional systems, it has been possible to propose an alternative version for systems of partial differential equations. It was first introduced to stabilize systems of parabolic equations [START_REF] Smyshlyaev | Closed-form boundary state feedbacks for a class of 1-d partial integro-differential equations[END_REF] before being extended to wave equations [START_REF] Krstic | Output-feedback stabilization of an unstable wave equation[END_REF] and hyperbolic linear (and quasilinear) systems [START_REF] Auriol | Minimum time control of heterodirectional linear coupled hyperbolic PDEs[END_REF][START_REF] Coron | Finite-time boundary stabilization of general linear hyperbolic balance laws via Fredholm backstepping transformation[END_REF][START_REF] Deutscher | Periodic output regulation for general linear heterodirectional hyperbolic systems[END_REF][START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic PDEs[END_REF]. In this section, we present the main ideas inherent to this method to design a stabilizing controller for the scalar system of balance laws ( 10)-( 12). The reader is referred to [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF] for a deeper introduction to the backstepping method.

3.1. General ideas. The main idea behind the backstepping methodology is to perform a change of variables using a Volterra integral operator to transform the original system into a target system, amenable to analysis, control, and observer design. Such a transformation must be boundedly invertible to guarantee equivalent stability properties between the two systems (original and target). In particular, the stability of the target system must imply the stability of the original system. This results in explicit controllers, similar to classical finite-dimensional counterparts, expressed as functionals of the distributed states. In what follows, we will denote (α(t, x), β(t, x)) the state of the target system. These new variables are defined from the variables (u(t, x), v(t, x)) using the following Volterra transformations

α(t, x) = u(t, x) + x 0 K uu (x, y)u(t, y) + K uv (x, y)v(t, y)dy, (14)
β(t, x) = v(t, x) + x 0 K vu (x, y)u(t, y) + K vv (x, y)v(t, y)dy, (15)
where the functions K •• are called the kernels of the transformation. They are continuous functions defined on the triangular domain

T = {(x, y) ∈ [0, 1] 2 , y ≤ x}.
As we will see later, these kernels must satisfy a set of partial differential equations (the so-called kernel equations) that depend on the chosen target system. If we show the existence of a solution for these kernel equations, this will imply the existence of a transformation mapping the original system to the desired target system.

3.2. Target system. One of the main difficulties with the backstepping method is finding a suitable target system. It should be simple enough to allow the design of the control law. Still, in the meantime, we must prove the existence of a transformation mapping the original system to this target system (which means showing the existence of a solution to the associated kernel equations). The choice of the target system directly impacts the closed-loop performance. The general question of reachable target systems is still an open problem. For hyperbolic equations, they were usually chosen as finite-time stable [START_REF] Li | Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems[END_REF]. In the case of the original system ( 10)-( 12), the target system can be defined as [START_REF] Coron | Local exponential H 2 stabilization of a 2× 2 quasilinear hyperbolic system using backstepping[END_REF]:

∂α ∂t (t, x) + λ ∂α ∂x (t, x) = 0, (16)
∂β ∂t (t, x) -µ ∂β ∂x (t, x) = 0, (17)
with the boundary conditions

α(t, 0) =qβ(t, 0), (18)
β(t, 1) =ρα(t, 1) + 1 0 (K vu (1, y) -ρK uu (1, y))u(t, y)dy + 1 0 (K vv (1, y) -ρK uv (1, y))v(t, y)dy + V (t). (19)
Note that the second boundary condition depends on the original set of variables. However, this target system is satisfying since the in-domain coupling terms (σ + and σ -) have been moved to the actuated boundary. They now take the form of integral couplings. The PDEs now read as transport equations. Therefore, we can choose the control input V (t) as

V (t) = -ρα(t, 1) - 1 0 (K vu (1, y) -ρK uu (1, y))u(t, y)dy - 1 0 (K vv (1, y) -ρK uv (1, y))v(t, y)ds = -ρv(t, 1) - 1 0 K vu (1, y)u(t, y) + K vv (1, y)v(t, y)ds, (20)
so that the system ( 16)-( 19) rewrites as

∂α ∂t (t, x) + λ ∂α ∂x (t, x) = 0, ∂β ∂t (t, x) -µ ∂β ∂x (t, x) = 0, (21)
with the boundary conditions

α(t, 0) = qβ(t, 0), β(t, 1) = 0. (22)
This resulting system is exponentially stable [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF]. Applying the method of characteristics, the states α and β converge to 0 in finite time equal to τ = 1 λ + 1 µ . Indeed, for all x ∈ [0, 1] and all t > τ , we have

β(t, x) = β(t -1-x µ , 1) = 0 and α(t, x) = α(t -x λ , 0) = qβ(t -x λ , 0) = 0.
Of course, for the above expressions to make sense, it is first necessary to prove the existence of a transformation of the form ( 14)-( 15) that maps the system ( 10)-( 12) into the system ( 16)- [START_REF] Auriol | Closed-loop tool face control with the bit off-bottom[END_REF]. It is important to note that only the terms in the domain and the boundary condition at x = 1 have been modified. Transport velocities and the proximal reflection q remain unchanged. The target system ( 16)-( 19) is pictured in Figure 2.

Kernel equations.

We now show that it is possible to map the system (10)-(12) into the system ( 16)-( 19) using the transformation ( 14)- [START_REF] Auriol | Robust output feedback stabilization for two heterodirectional linear coupled hyperbolic PDEs[END_REF]. Let us first notice that the boundary condition of system ( 18)-( 19) are always satisfied, independently of the choice of the kernels K •• . Let us focus first on the β-transformation (equation [START_REF] Auriol | Robust output feedback stabilization for two heterodirectional linear coupled hyperbolic PDEs[END_REF]). The computations we present below can be easily adjusted for the αtransformation (equation ( 14)). Differentiating equation [START_REF] Auriol | Robust output feedback stabilization for two heterodirectional linear coupled hyperbolic PDEs[END_REF] with respect to space Figure 2. Schematic representation of the target system ( 16)-( 19)

u(t, x) v(t, x) q 0 0 1 x
and using Leibniz's formula, we obtain

∂β ∂x (t, x) = ∂v ∂x (t, x) + K vu (x, x)u(t, x) + K vv (x, x)v(t, x) + x 0 ∂K vu ∂x (x, y)u(t, y) + ∂K vv ∂x (x, y)v(t, y)dy.
Differentiating equation ( 15) with respect to t and using that (u, v) is a solution of ( 10)-( 12), we obtain

∂β ∂t (t, x) = ∂v ∂t (t, x) + x 0 K vu (x, y) ∂u ∂t (t, y) + K vv (x, y) ∂v ∂t (t, y)dy = µ ∂v ∂x (t, x) + σ -u(t, x) + x 0 -λK vu (x, y) ∂u ∂y (t, y) + K vu (x, y)
•σ + (y)v(t, y) + µK vv (x, y) ∂v ∂y (t, y) + K vv (x, y)σ -(y)u(t, y)dy .

Integrating by parts, and using that u(t, 0) = qv(t, 0) (equation ( 12)), we get

∂β ∂t (t, x) = µ ∂v ∂x (t, x) + σ -(x)u(t, x) -λK vu (x, x)u(t, x) + λqK vu (x, 0) v(t, 0) + µK vv (x, x)v(t, x) -K vv (x, 0)v(t, 0) + x 0 λ ∂K vu ∂y (x, y)u(t, y) +K vu (x, y)σ + (y)v(t, y) -µ ∂K vv ∂y (x, y)v(t, y) + K vv (x, y)σ -u(t, y)dy .
Consequently, we obtain

∂β ∂t (t, x) -µ ∂β ∂x (t, x) = σ -(x) -(λ + µ)K vu (x, x) u(t, x) + (λqK vu (x, 0) -µK vv (x, 0)) v(t, 0) + x 0 λ ∂K vu ∂x (x, y) -µ ∂K vu ∂y (x, y) + K vv (x, y)σ -(y) • u(t, y) -µ ∂K vv ∂x (x, y) + µ ∂K vv ∂y (x, y) -K vu (x, y)σ + (y) v(t, y)ds.
To obtain equation ( 17), we choose K vu and K vv such that

µ ∂K vu ∂x (x, y) -λ ∂K vu ∂y (x, y) = σ -(y)K vv (x, y), (23)
µ ∂K vv ∂x (x, y) + µ ∂K vv ∂x (x, y) = σ + (y)K vu (x, y), (24)
along with the boundary conditions

K vu (x, x) = σ -(x) λ + µ , K vv (x, 0) = qλ µ K vu (x, 0). (25)
Similarly, we can differentiate equation ( 14) with respect to space and time to obtain the following set of equations that the kernels K vu and K vv should satisfy

λ ∂K uu ∂x (x, y) + λ ∂K uu ∂y (x, y) = -σ -(y)K uv (x, y), (26)
λ ∂K uv ∂x (x, y) -µ ∂K uv ∂y (x, y) = -σ + (y)K uu (x, y), (27)
with the boundary conditions

K uv (x, x) = - σ + (x) λ + µ , K uu (x, 0) = µ qλ K uv (x, 0). (28)
We have the following theorem [START_REF] Coron | Local exponential H 2 stabilization of a 2× 2 quasilinear hyperbolic system using backstepping[END_REF] Theorem 2. [31, Theorem A.1] The system (23)-( 28) admits a unique continuous solution on T .

Proof. The proof is quite classical (see [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic PDEs[END_REF], [START_REF] John | Continuous dependence on data for solutions of partial differential equations with a prescribed bound[END_REF] and [START_REF] Whitham | Linear and nonlinear waves[END_REF]) and consists in writing the integral equations associated to equations ( 23)-( 28) using the method of characteristics. These integral equations are then solved using the method of successive approximations. □ Theorem 2 proves the existence of a unique solution to system ( 23)-( 28) and, consequently, the existence of the transformation ( 14)-( 15) that maps the original system (10)- [START_REF] Auriol | Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time[END_REF] to the system ( 16)- [START_REF] Auriol | Closed-loop tool face control with the bit off-bottom[END_REF]. In the case of constant coefficients σ + and σ -, it is possible to obtain an explicit solution to the kernel equations ( 23)-( 28) using Bessel and Marcum Q-functions [START_REF] Vazquez | Marcum Q-functions and explicit kernels for stabilization of 2× 2 linear hyperbolic systems with constant coefficients[END_REF]. In the general case, only a numerical approximated solution is usually available.

3.4.

Invertibility of the transformation. The transformation ( 14)-( 15) is a Volterra transformation. Therefore, it is invertible [START_REF] Yoshida | Lectures on differential and integral equations[END_REF]. More precisely, there exist continuous functions L αα , L βα , L αβ , L ββ such that for all t ≥ 0 and all x ∈ [0, 1]

u(t, x) = α(t, x) + x 0 L αα (x, y)α(t, y) + L αβ (x, y)β(t, y)dy, (29) v(t, x) = β(t, x) + x 0 L βα (x, y)α(t, y) + L ββ (x, y)β(t, y)dy. ( 30
)
One can easily show that the functions

L •• defined on T = {(x, y) ∈ [0, 1] 2 , y ≤ x} verify the set of PDEs λ ∂L αα ∂x (x, y) + λ ∂L αα ∂y (x, y) = σ + (x)L βα (x, y), (31)
λ ∂L αβ ∂x (x, y) -µ ∂L αβ ∂y (x, y) = σ + (x)L ββ (x, y), (32)
µ ∂L βα ∂x (x, y) -λ ∂L βα ∂y (x, y) = -σ -(x)L αα (x, y), (33)
µ ∂L ββ ∂x (x, y) + µ ∂L ββ ∂y (x, y) = -σ -(x)L αβ (x, y), (34)
with the boundary conditions

L αα (x, 0) = µ qλ L αβ (x, 0), L ββ (x, 0) = qλ µ L βα (x, 0) (35)
L αβ (x, x) = σ + (x) λ + µ , L βα (x, x) = - σ -(x) λ + µ . ( 36
)
Remark 1. Using this inverse transformation, the boundary condition [START_REF] Auriol | Closed-loop tool face control with the bit off-bottom[END_REF] can be rewritten as

β(t, 1) = ρα(t, 1) - 1 0 N α (ξ)α(t, ξ) + N β (ξ)β(t, ξ)dξ + V (t), (37)
where

N α (ξ) = L βα (1, ξ) -ρL αα (1, ξ), N β (ξ) = L ββ (1, ξ) -ρL αβ (1, ξ). (38)
The functions K •• and L •• are continuous and defined on a compact set. Consequently, they are uniformly bounded by a constant M > 0. We have the following lemma Lemma 1. There exist two constants κ 0 > 0 and κ 1 > 0 such that for all t > 0

κ 0 ||(α(t, •), β(t, •))|| L 2 ≤ ||(u(t, •), v(t, •))|| L 2 ≤ κ 1 ||(α(t, •), β(t, •))|| L 2 (39)
Proof. The proof is straightforward and is omitted here. □ 3.5. Control law and stabilization.

Theorem 3. The closed-loop system (10)- [START_REF] Auriol | Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time[END_REF] with the control input [START_REF] Banks | Approximation in LQR problems for infinite dimensional systems with unbounded input operators[END_REF] is exponentially stable in the sense of the L 2 -norm. Moreover, for all t > 1 λ + 1 µ , for all x ∈ [0, 1], u(t, x) = 0 and v(t, x) = 0.

Proof. The complete proof can be found in [START_REF] Coron | Local exponential H 2 stabilization of a 2× 2 quasilinear hyperbolic system using backstepping[END_REF]. Since the target system ( 16)- [START_REF] Auriol | Closed-loop tool face control with the bit off-bottom[END_REF] with the control law ( 20) is L 2 -exponentially stable, there exists κ > 0 and ν > 0

such that ||(α(t, •), β(t, •))|| L 2 ≤ κe -νt ||(α(0, •), β(0, •))|| L 2 .
Consequently, using equation [START_REF] Di Meglio | Stabilization of coupled linear heterodirectional hyperbolic PDE-ODE systems[END_REF], we obtain

||(u(t, •), v(t, •))|| L 2 ≤ κ 1 κ 0 κe -νt ||(u(0, •), v(0, •))|| L 2 ,
which implies the exponential stability of the system (10)- [START_REF] Auriol | Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time[END_REF]. The finite-time convergence of the state (α, β) to zero is a direct consequence of the method of characteristics applied on the system ( 16)- [START_REF] Auriol | Closed-loop tool face control with the bit off-bottom[END_REF]. This directly implies the finite-time convergence of the state (u, v) to zero. □

The control law [START_REF] Banks | Approximation in LQR problems for infinite dimensional systems with unbounded input operators[END_REF] ensures the stabilization of the original system (10)- [START_REF] Auriol | Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time[END_REF]. However, the control law ( 20) is a full-state feedback as it requires the value of the state across the domain. Full-state distributed measurements are rarely available in practice, as only a finite number of sensors can be used. Backstepping-based observers can be designed to reconstruct the state and design an appropriate outputfeedback controller [START_REF] Auriol | Robust design of backstepping controllers for systems of linear hyperbolic PDEs[END_REF]. This is not presented here.

Remark 2.

If the coefficient q is zero, the above method is invalid since the boundary condition (28) cannot be properly defined.However, it can be adjusted, slightly modifying the target system by

∂α ∂t (t, x) + λ ∂α ∂x (t, x) = g(x)β(t, 0), ∂β ∂t (t, x) -µ ∂β ∂x (t, x). ( 40
)
and choosing the control law V (t) such that

α(t, 0) = qβ(t, 0), β(t, 1) = 0. ( 41
)
Again, the system (40)-( 41) is a cascade system which is L 2 exponentially stable and converges in finite time to zero. The kernel equations resulting from the backstepping transformation are identical to (23)-( 28) except that the boundary condition for K uu has been changed by K uu (x, 0) = h(x) where h is a continuous function that can be chosen as desired. Note that this has no impact on the resulting feedback law since it does not change the expression of K vu and K vv used in (20).

3.6. Simulation results and robustness. We illustrate below this result with simulations. The numerical parameters of the system are chosen as follows

λ = 1, µ = 2, σ +-= 1.7, σ -+ = 0.8, q = -0.8, ρ = 1. (42)
These parameters are chosen to make the open-loop system unstable (as seen in Figure 3). The space domain [0, 1] is discretized with a mesh of 100 points. The solution to the kernel equations ( 23)-( 28) is obtained using a fixed-point algorithm [START_REF] Auriol | Minimum time control of heterodirectional linear coupled hyperbolic PDEs[END_REF].

The simulations use a classical finite volume method based on a Godunov scheme. The integral term in the control law [START_REF] Banks | Approximation in LQR problems for infinite dimensional systems with unbounded input operators[END_REF] is computed using a trapezoidal method with a precision corresponding to the mesh size used to compute the kernels. We have pictured in Figure 3 the time-evolution of the L 2 -norm of the system (10)- [START_REF] Auriol | Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time[END_REF] in open-loop and closed-loop using the control law [START_REF] Banks | Approximation in LQR problems for infinite dimensional systems with unbounded input operators[END_REF]. As expected, although the open-loop system is unstable, the closed-loop system is finite-time stable. The convergence time corresponds to the expected one:

1 λ + 1 µ = 3 2 s.
Let us now introduce a small delay in the actuation path δ = 0.03s. As seen in Figure 4, the L 2 -norm of the state (u, v) no longer converges to zero. This means that the control law does not delay-robustly stabilize the system (10)- [START_REF] Auriol | Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time[END_REF]. Focusing on finite-time stability, we have neglected the robustness aspects. In the next section, we will analyze the robustness properties of the control law [START_REF] Banks | Approximation in LQR problems for infinite dimensional systems with unbounded input operators[END_REF] and modify it to guarantee delay-robust stabilization.

Design of robust control laws for hyperbolic systems

As seen in the previous section (Figure 4), the control law (20) may have vanishing robustness margins. This is not surprising since it has been observed that for many feedback systems, the introduction of arbitrarily small time delays in the loop may cause instability for any feedback law (see [START_REF] Datko | An example on the effect of time delays in boundary feedback stabilization of wave equations[END_REF][START_REF] Logemann | Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop[END_REF]). In this section, we analyze the robustness properties of the control law [START_REF] Banks | Approximation in LQR problems for infinite dimensional systems with unbounded input operators[END_REF] and propose some adjustments to guarantee the existence of robustness margins. The transformation ( 14)-( 15) can still be applied, as it does not directly induce the design of the control law. Indeed, the system ( 16)-( 19) is more amenable for analysis since it consists of two coupled transport equations. Using Remark 1, we can map the system (10)-( 12) to the target 10)-( 12) in closed-loop using the control law [START_REF] Banks | Approximation in LQR problems for infinite dimensional systems with unbounded input operators[END_REF] with no delay (red) and with a 0.03s delay in the actuation path (dashed blue)

system ∂α ∂t (t, x) + λ ∂α ∂x (t, x) = 0, ∂β ∂t (t, x) -µ ∂β ∂x (t, x) = 0, (43)
β(t, 1) = ρα(t, 1) - 1 0 N α (ξ)α(t, ξ) + N β (ξ)β(t, ξ)dξ + V (t), (44)
Applying the method of characteristics on equations ( 43)-( 44), we obtain for all x ∈ [0, 1] and for all t > 0

α(t, x) = qβ(t - x λ - 1 µ , 1), β(t, x) = β(t - 1 -x µ , 1).
Consequently, combining these equations and equation ( 44), we get:

β(t, 1) = qρβ(t -τ, 1) - τ 0 Ñ (ν)β(t -ν, 1)dν + V (t), (45)
where we recall that τ = 1 λ + 1 µ and where Ñ is defined by

Ñ (ν) = qλN α (λν - λ µ )1 [ 1 µ ,τ ] (ν) + µN β (1 -µν)1 [0, 1 µ ] (ν). (46)
This invertible coordinate change enables us to rewrite β as the solution of a difference equation with distributed delays, more precisely, an Integral Delay Equation (IDE). For a single scalar equation, such connections between hyperbolic PDEs and IDEs were already established in [START_REF] Karafyllis | On the relation of delay equations to first-order hyperbolic partial differential equations[END_REF]. Furthermore, in [START_REF] Karafyllis | On the relation of delay equations to first-order hyperbolic partial differential equations[END_REF], the authors have stated that the stability analysis is easier when converting the PDEs to this time-delay form. These connections were then generalized for two balance law equations in [START_REF] Auriol | Delay-robust control design for heterodirectional linear coupled hyperbolic PDEs[END_REF][START_REF] Auriol | Robust output feedback stabilization for two heterodirectional linear coupled hyperbolic PDEs[END_REF] and in [START_REF] Auriol | An explicit mapping from linear first order hyperbolic PDEs to difference systems[END_REF] for non-scalar systems. Next, we show how this representation can be amenable to robustness analysis.

4.1.

Open-loop analysis. It has been observed (see [START_REF] Datko | An example on the effect of time delays in boundary feedback stabilization of wave equations[END_REF][START_REF] Logemann | Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop[END_REF]) that if the open-loop transfer function of a feedback system has an infinite number of poles in the complex right-half plane, then the introduction of arbitrarily small time delays in the loop may cause instability for any feedback control law. Therefore, it is first essential to analyze the open-loop behavior of the system ( 43)- [START_REF] Hale | Strong stabilization of neutral functional differential equations[END_REF], or equivalently of the system [START_REF] Hale | Introduction to functional differential equations[END_REF]. Taking the Laplace transform of the open-loop system (45), we obtain

1 -ρqe -τ s + τ 0 Ñ (y)e -ys dy = 0. (47)
The next theorem shows how the properties of the characteristic equation ( 47) relate to the stability properties of equation ( 45)

Theorem 4. The system (45) is exponentially stable if and only if there exists η > 0 such that all solution of the characteristic equation (47) satisfy Re(s) < -η.

Proof. The proof is obtained with minor adjustments from [START_REF] Carvalho | On quadratic liapunov functionals for linear difference equations[END_REF]. □

We then have the following lemma Proof. Let us denote

F (s) = 1 -ρqe -sτ , H(s) = τ 0 Ñ (ξ)e -ξs dξ.
Using Riemann-Lebesgue's lemma, we obtain

|H(s)| → 0 as |s| → ∞, Re(s) > 0.
The function F has an infinite number of zeros whose real parts are equal to ln(|ρq|) 2τ . We can then apply [6, Lemma 3] to conclude that F + H has infinite zeros with positive real parts. This concludes the proof. □

We can actually get a slightly stronger result with the following lemma (whose proof is omitted) Lemma 3. If |ρq| ≥ 1, then, for any η > 0, the characteristic equation [START_REF] Jackson | The theory of approximation[END_REF] has an infinite number of zeros whose real parts are greater than -η.

We can finally state the following theorem Theorem 5. If |ρq| ≥ 1, system (10)-( 12) cannot be exponentially delay-robustly stabilized.

Proof. Due to Lemma 2, when |ρq| > 1, the open-loop transfer function of system ( 10)-( 12) has an infinite number of poles with a positive real part. Thus, system ( 10)-( 12) cannot be delay-robustly stabilized (see [START_REF] Logemann | Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop[END_REF]Theorem 1.2]). When |ρq| = 1, due to Lemma 3, there is an asymptotic chain of zeros along the imaginary axis. Thus, the system cannot be exponentially robustly stabilized [START_REF] Hale | Introduction to functional differential equations[END_REF]. □

This result shows that if the product |ρq| is larger than one, one cannot find a controller whose delay margin is non-zero. One should be aware that this condition does not hold for most practical applications. Models of the form ( 10)-( 12) are simplistic and do not capture phenomena that would be susceptible to making the delay margins non-null. For instance, they may neglect the diffusivity stemming from Kelvin-Voigt damping. Therefore, although the delay-robustness margins for such systems would be poor, it remains theoretically possible to stabilize them robustly.

The end of finite-time stabilization?

As seen in Figure 4, even when |ρq| < 1, the introduction of a small delay in the control input [START_REF] Banks | Approximation in LQR problems for infinite dimensional systems with unbounded input operators[END_REF] can induce zero robustness margins. Before designing a robust controller, we must understand the cause of non-robustness. The control law V (t) defined by equation ( 20) can be expressed for all t ≥ τ as

V (t) = -qρβ(t -τ, 1) + τ 0 Ñ (ν)β(t -ν, 1)dν, (48)
such that equation (45) rewrites β(t, 1) = 0. In the presence of a small delay δ > 0 in the actuation path, equation [START_REF] Hale | Introduction to functional differential equations[END_REF] rewrites

β(t, 1) =qρ(β(t -τ, 1) -β(t -τ -δ, 1) + τ 0 Ñ (ν)(β(t -ν, 1) -β(t -ν -δ, 1))dν, (49)
We then have the following theorem Theorem 6. If |ρq| > 1 2 , then the system (10)-( 12) with the delayed backstepping control law V (t -δ) defined by equation ( 48) is unstable for any δ > 0.

Proof. The characteristic equation associated with equation (49) rewrites

F (s) = H(s), (50)
where

F (s) = 1 -ρqe -τ s + ρqe -(τ +δ)s , H(s) = τ 0 Ñ (ν)e -νs (1 -e -δs )dν.
Since 2|ρq| > 1, the function F (s) has an infinite number of zeros in the complex right half-plane [START_REF] Hale | Strong stabilization of neutral functional differential equations[END_REF]. The function H(s) is strictly proper due Riemann-Lebesgues' lemma, i.e. |H(s)| converges to zero for |s| large enough (with a positive real part). Due to Rouché's theorem, we have that F -H has at least one root whose real part is strictly positive. This concludes the proof. □

Note the proof can be adjusted when |ρq| = 1 2 to show that the delayed control law cannot exponentially stabilize the system (10)-( 12) (the stabilization can at best be asymptotical). Theorem 6 implies that the backstepping controller (20) cannot be used in practical applications (for which there is always a small delay in the loop). Therefore, one must give up finite-time convergence to obtain a tractable implementation for the controller stabilizing system (10)- [START_REF] Auriol | Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time[END_REF]. These observations are consistent with reports by industrial practitioners on the limitations of the impedance matching method. The impedance matching method (see [START_REF] Aarsnes | Limits of controller performance in the heave disturbance attenuation problem[END_REF][START_REF] Aarsnes | Modeling and avoidance of heaveinduced resonances in offshore drilling[END_REF][START_REF] Egeland | Modeling and simulation for automatic control[END_REF]) consists in matching the load impedance to the characteristic line impedance. For instance, in [START_REF] Kyllingstad | A new stick-slip prevention system[END_REF], the authors designed a controller preventing stick-slip oscillations, a class of undesired torsional oscillations characterized by a series of stick (a cessation of bit rotation) and slip (a sudden release of rotational energy) that often occur in drilling devices. They observed that canceling the proximal reflection coefficient can change the dynamics of the string in a way that makes the system unstable. All in all, controllers of the form (20) significantly trade off delay-robustness for performance, making them likely to be unusable.

Robustification of the control law.

In this section, we present some methods to overcome the robustness limitation of the controller (20) while maintaining an analogous control structure. The control law [START_REF] Banks | Approximation in LQR problems for infinite dimensional systems with unbounded input operators[END_REF] is composed of two parts:

(1) the integral part whose objective is to remove the effect of in-domain couplings (2) the term -ρu(t, 1) whose objective is to cancel the proximal reflection and to ensure finite-time convergence.

As seen above, the robustness issue is due to the term -ρu(t, 1) in the control law. Indeed, the resulting feedback control operator may not be strictly proper [START_REF] Curtain | An introduction to infinite-dimensional linear systems theory[END_REF][START_REF] Curtain | Transfer functions of distributed parameter systems: A tutorial[END_REF], thus inducing vanishing robustness margins. We present below two solutions to modify the control law and guarantee robustness margins, thereby giving up finite time stabilization.

Introduction of degrees of freedom.

A first solution is to cancel only a part of the reflection terms in the PDE, using a convolutional procedure as performed, e.g., in [START_REF] Auriol | Delay-robust stabilization of an n + m hyperbolic PDE-ODE system[END_REF][START_REF] Woittennek | Flatness based feedback design for hyperbolic distributed parameter systems with spatially varying coefficients[END_REF][START_REF] Meurer | Flatness-based feedback control of diffusion-convection-reaction systems via k-summable power series[END_REF]. More precisely, consider the control law

V 2 (t) = -ρu(t, 1) + (ρ -ρ) 1 0 K uu (1, y)u(t, y) + K uv (1, y)v(t, y)dξ - 1 0 K vu (1, y)u(t, y) + K vv (1, y)v(t, y)dy, (51)
where the coefficient ρ is chosen such that

|ρ| < 1 -|ρq| |q| . ( 52
)
The target system ( 43)-( 44) now rewrites

∂α ∂t (t, x) + λ ∂α ∂x (t, x) = 0, ∂β ∂t (t, x) -µ ∂β ∂x (t, x) = 0, (53)
α(t, 0) = qβ(t, 0), β(t, 1) = (ρ -ρ)α(t, 1), (54)
Due to equation ( 52), we have |q(ρ -ρ)|. Therefore, the target system ( 43)-( 44) is exponentially stable. The coefficient ρ can be interpreted as a tuning parameter, enabling a trade-off between performance (convergence rate) and robustness with respect to delays. We have the following theorem.

Theorem 7. Consider the control law V 2 defined by [START_REF] Karafyllis | On the relation of delay equations to first-order hyperbolic partial differential equations[END_REF] with ρ satisfying [START_REF] Karafyllis | Input-to-state stability for PDEs[END_REF]. This control law delay-robustly stabilizes the system (10)- [START_REF] Auriol | Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time[END_REF].

Proof. Consider a positive delay δ. Slightly adjusting the method used to derive (45), we get the following equation satisfied by the output β(t, 1).

β(t, 1) = qρβ(t -τ, 1) -q ρβ(t -τ -δ, 1) - τ 0 Ñ (ν)(β(t -ν, 1) -β(t -δ -ν, 1))dν, ( 55
)
where Ñ is defined by [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic PDEs[END_REF]. Taking the Laplace, we obtain the following characteristic equation

F (s) = 1 -qρe -τ s + ρqe -(τ +δ)s -I(s, δ) = 0,
where I(s, δ) is defined by

I(s, δ) = τ 0 Ñ (ν)(e -νs -e -(ν+δ)s )dν
Let us now consider a complex number s such that Re(s) ≥ 0. We then have

|F (s)| ≥ |1 -qρe -τ s + ρqe -(τ +δ)s | -|I(s, δ)| ≥ 1 -|qρe -τ s | -|ρqe -(τ +δ)s | -|I(s, δ)| ≥ 1 -|qρ| -|ρq| -|I(s, δ)|.
Since ρ satisfies (52), there exists ϵ 0 > 0 such that

1 -|qρ| -|ρq| > ϵ 0 .
Let us now focus on the term I(s, δ). Due to Riemann-Lebesgues' lemma, we have

∀|s| > M 0 , | τ 0 N (ν)e -νs dν| < ϵ 0 2
We can now choose δ 0 small enough such that for any δ ≤ δ 0 , for all complex s such that |s| ≤ M 0 , |I(s, δ)| < ϵ 0 . With this choice of δ 0 , one can easily check that, ∀δ ≤ δ 0 , ∀s ∈ C such that Re(s) ≥ 0, |I(s, δ)| < ϵ 0 Consequently, for δ ≤ δ 0 , we have

|F (s)| > 0.
It means that for 0 < δ ≤ δ 0 , the function F (s) does not have any root whose real part is positive. Moreover, there is no asymptotic chain of zeros in zero [START_REF] Hale | Strong stabilization of neutral functional differential equations[END_REF]. Consequently, equation ( 55) is exponentially stable. This concludes the proof. □

Although such a procedure is now somehow standard, it presents the drawback of not distinguishing the effects of high and low frequencies in terms of stability and robustness. In addition, the implementation of such an approach can be challenging when considering networks of balance laws with many subsystems [START_REF] Auriol | Robust state-feedback stabilization of an underactuated network of interconnected n+ m hyperbolic PDE systems[END_REF]. 4.3.2. Filtering of the control law. An alternative robustifying approach has been proposed in [START_REF] Bou Saba | Strictly proper control design for the stabilization of 2 × 2 linear hyperbolic ODE-PDE-ODE systems[END_REF][START_REF] Auriol | Robust state-feedback stabilization of an underactuated network of interconnected n+ m hyperbolic PDE systems[END_REF][START_REF] Auriol | Robustification of stabilizing controllers for ODE-PDE-ODE systems: a filtering approach[END_REF]. It combines the proposed non-proper control law [START_REF] Banks | Approximation in LQR problems for infinite dimensional systems with unbounded input operators[END_REF] with a well-tuned low-pass filter. The resulting control law then becomes strictly proper, which guarantees the existence of robustness margins. This filtering technique can simplify the design of stabilizing controllers for the proposed class of systems as it dissociates the stabilization problem from the robustness problem. More precisely, when applied conjointly with the existing controller, the low-pass filter will ensure the delay-robust stability of the closed-loop dynamics. The filters can be designed by leveraging the fact that robustness issues appear at high frequencies. We give below sufficient conditions that ensure robust stabilization provided the original control laws ensure exponential stabilization. The analysis will be carried out on the Laplace domain. Let us consider a positive constant M . Let us consider a low-pass filter w(s) (i.e. w(s) → 1 as |s| → 0 and |w(s)| → 0 as |s| → +∞) with relative degree 1, such that for all s ∈ C + , we have |w(s

)| < 1, |(1 -w(s))| < 1 and if |s| ≤ M |1 -w(s)| < 1 |ρqe -τ s + τ 0 Ñ (ν)e -νs ds| + 1 . ( 56
)
Note that it is always possible to find such a filter. One can for instance consider the low-pass filter defined for all s ∈ C + by w ν0 (s) = 56) is satisfied if |s| ≤ M . Consider the control law V f defined in the Laplace domain by V f (s) = w(s)V (s), where the control law V (t) is defined by equation [START_REF] Banks | Approximation in LQR problems for infinite dimensional systems with unbounded input operators[END_REF]. Then, there exists M > 0 such that the control law delay-robustly stabilizes the system (10)- [START_REF] Auriol | Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time[END_REF].

Proof. Using equation [START_REF] Hale | Introduction to functional differential equations[END_REF], the characteristic equation associated with the closedloop system rewrites

1 = (1 -w(s))(ρqe -τ s + τ 0 Ñ (ν)e -νs dν). ( 57
)
Due to equation [START_REF] Kyllingstad | A new stick-slip prevention system[END_REF], there exists η > 0 such that (57) does not admit any solution whose real part is greater than -η. Thus, the closed-loop system is exponentially stable. Finally, the control law V f (s) is strictly proper. Adjusting the proof from [33, Theorem 9.2.6], we can show that it is robust to small delays in the loop. □

Note that having a strictly proper control law is sufficient to guarantee the existence of robustness margins. However, it is not a necessary condition. As seen above, canceling a part of the reflection terms while guaranteeing robustness margins is possible. The proposed filtering technique considerably simplifies the design of delay-robust controllers since the filters can be plugged into the stabilizing control law. It extends the approach derived in [START_REF] Morgul | Robust stabilization of the wave equation against small delays[END_REF][START_REF] Morgul | On the stabilization and stability robustness against small delays of some damped wave equations[END_REF] for wave equations. 4.4. Simulation results. We illustrate below this result with simulations. The numerical parameters of the system are identical to the ones of Section 3.6. Since |ρq| = 0.8, we must choose |ρ| < 0.25. We have pictured in Figure 5, the time evolution of the L 2 -norm of the system (10)-( 12) in closed-loop using the control law [START_REF] Karafyllis | On the relation of delay equations to first-order hyperbolic partial differential equations[END_REF] in the presence of a 0.05s delay for different values ρ (ρ = 0, ρ = 0.1 and ρ = 0.2). In all cases, the control law guarantees delay-robust stabilization. However, this improvement in terms of delay margin comes at the cost of a diminution of the convergence rate as we have lost finite-time stabilization. The larger ρ is, the smaller the corresponding delay margin will be, but in the meantime, the better will the convergence rate be. This illustrates the trade-off between performance and robustness. In Figure 6, we have pictured the evolution of the L 2 -norm of the closed-loop system in the presence of a 0.05s delay for the control law (51) (ρ = 0.2) and using the filtered control law introduced in Theorem 8. We considered two filters with different bandwiths: 125 rad.s -1 and 50 rad.s -1 . The associated control efforts have been pictured in Figure 7. As expected, for the filtered controllers, the states of the system converge to zero despite the presence of the input delay. The convergence rate is better than the one obtained using the control law [START_REF] Karafyllis | On the relation of delay equations to first-order hyperbolic partial differential equations[END_REF]. This is because only high-frequency dynamics are affected by the filter. Moreover, the associated control effort is smaller. However, we have not analyzed the associated qualitative effect on the robustness margins or the impact of the different tuning parameters with respect to other performance specifications. This should be the purpose of future contributions. Extensions. The proposed approaches can be easily extended to show the robustness with respect to additive, multiplicative or left-coprime-factor uncertainties [START_REF] Curtain | An introduction to infinite-dimensional linear systems theory[END_REF]. Input-to-State Stability (ISS) has been shown in [START_REF] Auriol | Robust output feedback stabilization for two heterodirectional linear coupled hyperbolic PDEs[END_REF] for a general class of disturbances, thus extending the results from [START_REF] Deutscher | Backstepping design of robust state feedback regulators for linear 2 × 2 hyperbolic systems[END_REF][START_REF] Deutscher | Finite-time output regulation for linear 2 × 2 hyperbolic systems using backstepping[END_REF] where the disturbance signal is generated by an exosystem of finite dimension or the results from [START_REF] Lamare | Adding an integrator to backstepping: Output disturbances rejection for linear hyperbolic systems[END_REF][START_REF] Lamare | Control of 2 × 2 linear hyperbolic systems: Backsteppingbased trajectory generation and PI-based tracking[END_REF] where only smooth disturbances were considered. Moreover, in [START_REF] Auriol | Robust output feedback stabilization for two heterodirectional linear coupled hyperbolic PDEs[END_REF], the authors introduced different tuning parameters in the design, thus guaranteeing potential trade-offs between different specifications (namely delay-robustness convergence rate, noise sensitivity, and disturbance rejection). The gained robustness comes at the price of degraded performance. It is important to emphasize that the resulting control laws introduce different degrees of freedom, but the choice of these tuning parameters and the underlying trade-offs still have to be qualitatively and quantitatively analyzed. Regarding the filtering approach, it can be extended to a large class of systems. In [START_REF] Auriol | Robustification of stabilizing controllers for ODE-PDE-ODE systems: a filtering approach[END_REF], the authors give general conditions under which it is possible to low-pass filter the (potentially non-strictly proper) control laws for a broad class of PDE systems coupled with ODEs. Finally, in [START_REF] Auriol | Mean-square exponential stabilization of coupled hyperbolic systems with random parameters[END_REF], the authors showed the mean-square exponential robustness of the control law [START_REF] Karafyllis | On the relation of delay equations to first-order hyperbolic partial differential equations[END_REF] in the presence of stochastic uncertainties. The proof relies on a specific Lyapunov analysis.

5. Towards parametrizable target systems.

One of the primary challenges associated with the backstepping method revolves around identifying a suitable target system. It should be simple enough to allow the design of the control law. Still, in the meantime, we must prove the existence of a transformation mapping the original system to this target system. The target system's choice significantly influences the closed-loop system's overall performance. The question of identifying general reachable target systems remains an unresolved issue. In the case of hyperbolic equations, these target systems have traditionally been selected as finite-time stable [START_REF] Coron | Null-controllability of linear hyperbolic systems in one dimensional space[END_REF], thereby shadowing the robustness properties of the corresponding closed-loop systems [START_REF] Logemann | Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop[END_REF][START_REF] Michiels | Strong stability of neutral equations with an arbitrary delay dependency structure[END_REF], as seen in the previous section. In [START_REF] Auriol | Robust output feedback stabilization for two heterodirectional linear coupled hyperbolic PDEs[END_REF], we introduced tuning parameters in the design, thus guaranteeing potential trade-offs between different specifications (namely delayrobustness and convergence rate). Nevertheless, the influence of these tuning parameters is limited, as they only impact the system's boundary conditions. General target systems (and thus additional degrees of freedom) could be obtained by preserving dissipative in-domain couplings while imposing a specific energy decay. This should imply a reduced control effort compared to traditional approaches. 5.1. Desired target system. In this section, we show how we can define an appropriate backstepping transformation to map the system (10)- [START_REF] Auriol | Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time[END_REF] to any arbitrary target system with an analogous structure. More precisely, we will denote (ū, v) as the state of the target system. It verifies

∂ ū ∂t (t, x) + λ ∂ ū ∂x (t, x) = σ+ (x)v(t, x), (58)
∂v ∂t (t, x) -µ ∂v ∂x (t, x) = σ-(x)ū(t, x), (59)
with the boundary conditions ū(t, 0) = qv(t, 0), v(t, 1) = ρū(t, 1),

where the parameters q, ρ, and the continuous functions σ-and σ+ can be arbi- trarily chosen. We emphasize that the parameters λ, µ, and q remain unchanged.

This control objective somehow corresponds to in-domain damping assignment [START_REF] Redaud | In-domain damping assignment of a timoshenko-beam using state feedback boundary control[END_REF][START_REF] Redaud | Distributed damping assignment for a wave equation in the port-hamiltonian framework[END_REF]. We do not detail in this chapter how to adequately choose the parameters of the target system. However, in this context, the Port-Hamiltonian approach (PHS) [START_REF] Le Gorrec | Dirac structures and boundary control systems associated with skew-symmetric differential operators[END_REF] could be a path to follow. The PHS framework was initially developed for finite-dimensional systems [START_REF] Duindam | Modeling and control of complex physical systems: the port-Hamiltonian approach[END_REF] and was then extended to PDEs [START_REF] Le Gorrec | Dirac structures and boundary control systems associated with skew-symmetric differential operators[END_REF][START_REF] Jacob | Linear port-Hamiltonian systems on infinite-dimensional spaces[END_REF]. It is particularly relevant for control design using damping assignment or energy shaping methods. The natural physical properties of the system can be advantageously used to define well-posed, exponentially stable target system candidates. Such a strategy was successfully applied on simple test cases [START_REF] Ramirez | On backstepping boundary control for a class of linear port-hamiltonian systems[END_REF][START_REF] Redaud | In-domain damping assignment of a timoshenko-beam using state feedback boundary control[END_REF]. Therefore, the PHS framework could help us to introduce degrees of freedom in the design and obtain a class of easily parametrizable (exponentially stable) closed-loop target systems. Other tuning parameters could also be added by following the robustness approach proposed in [START_REF] Auriol | Robust output feedback stabilization for two heterodirectional linear coupled hyperbolic PDEs[END_REF]. Then, we could use the backstepping method to map the original system ( 10)-( 12) to this specific target system.

5.2. Backstepping transformation. We now map the system ( 10)-( 12) to the target system ( 58)- [START_REF] Le Gorrec | Dirac structures and boundary control systems associated with skew-symmetric differential operators[END_REF]. Consider the following backstepping transformation

ū(t, x) = u(t, x) + x 0 Kuu (x, y)u(t, y) + Kuv (x, y)v(t, y)dy, (61)
v(t, x) = v(t, x) + x 0 Kvu (x, y)u(t, y) + Kvv (x, y)v(t, y)dy, (62)
where the kernels Kuu , Kuv , Kvu and Kvv are continuous functions defined on the triangular domain T = {(x, y) ∈ [0, 1] 2 , y ≤ x}. Differentiating the transformation ( 61)-( 62) with respect to time and space, we can show that the kernels verify the following set of equations

Kuv (x, x) = σ + (x) -σ+ (x) λ + µ , Kuu (x, 0) = q -1 Kuv (x, 0), Kvu (x, x) = σ-(x) -σ -(x) λ + µ , Kvv (x, 0) = q Kvu (x, 0).
The well-posedness of this set of kernel equations is proved in [START_REF] Di Meglio | Stabilization of coupled linear heterodirectional hyperbolic PDE-ODE systems[END_REF]. Therefore, a unique continuous solution exists on T . The control input is then given by

V (t) = (ρ -ρ)u(t, 1)-
5.3. Simulation results. We illustrate below the results of this section with simulations. The numerical parameters of the system are identical to the ones of Section 3.6. We have pictured in Figure 5 the time evolution of the L 2 -norm of the system (10)-( 12) in closed-loop using the control law (51) (with ρ = 0) and using the control law [START_REF] Lions | Optimal control of systems governed by partial differential equations problèmes aux limites[END_REF]. For this latter case, we have chosen ρ = ρ, σ+ = 0.7, and σ-= 0, so the closed-loop system is exponentially stable. We have pictured in Figure 5 the associated control efforts. As can be seen, the two controllers guarantee the L 2stability in closed-loop. Moreover, the exponential decay rates are almost identical (even if the maximum value reached during the transient is larger when using the control law ( 63)). However, the control effort is smaller for the control law [START_REF] Lions | Optimal control of systems governed by partial differential equations problèmes aux limites[END_REF]. Therefore, preserving some naturally dissipative in-domain coupling terms in the target system can guarantee a similar convergence rate while reducing the associated control effort. The next step would be to develop analytical techniques to quantify the performance of our output-feedback laws regarding a given set of performance specifications, such as disturbance rejection, robustness margins, noise sensitivity, or convergence rate. A thorough literature review and classification of existing criteria for finite-dimensional systems is needed to separate candidates for distributed parameter systems. Furthermore, these concepts must be tailored to the specific requirements of infinite-dimensional systems: the concept of phase margin is, for instance, an insufficient metric for infinite-dimensional systems since significant dynamics can be spread over a wide frequency range and with more esoteric behavior of systems that are not strictly proper [START_REF] Curtain | Transfer functions of distributed parameter systems: A tutorial[END_REF][START_REF] Logemann | Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop[END_REF]. The success of classical control design algorithms (such as P, PI, and PID controllers) in industrial engineering applications can be attributed to their simplicity and low computation burden. For instance, Soft-Speed and Soft-Torque, the industry-standard products that handle torsional vibrations in drilling devices, rely on PI regulators [START_REF] Dwars | Recent advances in soft torque rotary systems[END_REF]. On the other hand, implementing infinite-dimensional outputfeedback controllers (as the backstepping control law given in ( 20)) requires expertise and computational power. As explained in this chapter, the motivation for considering such complex control strategies stems from the desire for improved performance by explicitly considering delays and high-frequency content in the design. While advancements in electronic technology have enabled the use of computationally demanding techniques, evaluating the computational efforts inherent to these complex control strategies is crucial. Indeed, performing real-time state estimation for hyperbolic systems using rapidly converging observers, such as those based on backstepping, is computationally expensive and, in many cases, prohibitive. For instance, the method presented in [START_REF] Redaud | Output-feedback control of an underactuated network of interconnected hyperbolic PDE-ODE systems[END_REF] to stabilize interconnections of scalar hyperbolic systems needs to compute state predictions for each subsystem composing the interconnection, which is time-consuming. This numerical burden may explode when considering complex PDE networks. Consequently, evaluating the computational costs of the control methods we design appears crucial.

To leverage the numerical effort induced by these controllers, it may be necessary to approximate them (e.g., by finite-dimensional systems). The approximation scheme should nonetheless guarantee satisfying closed-loop properties (and, in particular, closed-loop stability). As the last step in the design is to approximate the controller by a finite-dimensional or lumped parameter system, such kind of direct approach is sometimes referred to as late-lumping. This denomination emphasizes the distinction with early-lumping strategies where the control input is obtained using a finite-dimensional approximation of the PDE [START_REF] Banks | Approximation in LQR problems for infinite dimensional systems with unbounded input operators[END_REF][START_REF] Banks | The linear regulator problem for parabolic systems[END_REF][START_REF] Morris | Control of systems governed by partial differential equations[END_REF]. Different late-lumping approximation methods have been suggested in [START_REF] Ecklebe | Approximation and implementation of transformation based feedback laws for distributed parameter systems[END_REF] or [START_REF] Auriol | Late-lumping backstepping control of partial differential equations[END_REF]. Recently, machine-learning approximations (based on the DeepONet algorithm) have been successfully tested in [START_REF] Shi | Machine learning accelerated PDE backstepping observers[END_REF] on simple examples. Some stability guarantees have been given in [START_REF] Bhan | Operator learning for nonlinear adaptive control[END_REF] using Lyapunov functions. In this section, we briefly present some tools to obtain sufficient conditions guaranteeing the convergence of the backstepping-based late-lumping controller [START_REF] Karafyllis | On the relation of delay equations to first-order hyperbolic partial differential equations[END_REF] when considering an approximation of the control law V (t).

6.1. Approximation of the control input. Consider the system (10)-( 12) with the sequence of control input V n (t) that corresponds to an approximation of the nominal control input V (t) defined by equation [START_REF] Karafyllis | On the relation of delay equations to first-order hyperbolic partial differential equations[END_REF]. We assume that |ρq| < 1, as explained in Section 4. To simplify the computations, we consider that ρ = 0. Applying the backstepping transformations ( 14)-( 15), we obtain the target system

∂α ∂t (t, x) + λ ∂α ∂x (t, x) = 0, ∂β ∂t (t, x) -µ ∂β ∂x (t, x) = 0, (64)
with the boundary conditions

α(t, 0) =qβ(t, 0), β(t, 1) = ρα(t, 1) + V n (t) -V (t). (65)
6.2. Lyapunov analysis. To analyze the stability of the target system ( 64)-( 65), we consider the following Lyapunov function

W (t) = 1 0 e -νx λ α 2 (t, x) + q 2 e νx µ β 2 (t, x)dx, (66)
where ν > 0. The function W is equivalent to the square L 2 -norm of the state (α, β) and consequently to the square of L 2 -norm of the state (u, v). Taking the derivative with respect to time and integrating by parts, we obtain Ẇ (t) = -νW (t) -e -ν α 2 (t, 1) + q 2 e ν β 2 (t, 1).

Using the boundary condition ( 65) and Young's inequality, we have

Ẇ (t) ≤ -νW (t) + e -ν (e 2ν q 2 ρ 2 2ϵ -1)α 2 (t, 1) + q 2 e ν ϵ 2 (V n (t) -V (t)) 2 ,
where ϵ is a positive constant. Let us now choose ν and ϵ such that e 2ν q 2 ρ 2 2ϵ < 1, which is always possible since |ρq| < 1. Let us denote C = q 2 e ν ϵ 2 . We obtain

Ẇ (t) ≤ -νW (t) + C(V n (t) -V (t)) 2 . ( 67
)
We then have the following approximation result Theorem 9. Assume that the approximating control law V n (t) uniformly converges to V (t) when n → ∞ such that there exists a sequence C n such that

|V n (t) -V (t)| ≤ C n ||(u, v)|| 2 L 2 , (68)
C n → 0, when n → 0.

Then, there exists N > 0 such that for all n ≥ N , the system (10)-( 12) with the control input V n (t) is exponentially stable.

Proof. For n large enough, equation (67) rewrites Ẇ (t) ≤ -ηW (t), where η > 0. Thus, the function W exponentially converges to zero, and so does the state (u, v). □

The properties (68)-( 69) are verified for several approximation schemes such as Galerkin approximations and some machine-learning algorithms (e.g., DeepONet). However, generalizing the proposed approach to more general systems requires the design of appropriate Lyapunov functions. In this context, the methodology presented in [START_REF] Auriol | On input-to-state stability of linear difference equations and its characterization with a Lyapunov functional[END_REF] seems promising. This result is a first step toward practical applications of late-lumping controllers for hyperbolic systems. The question of the latelumping backstepping controller-observer or the extension to systems of larger dimensions has not been considered so far. 6.3. Simulation results. We illustrate below the results of this section with simulations. The numerical parameters of the system are identical to the ones of Section 3.6. As this section is given for illustrating purposes, we only focus here on the control aspects, neglecting the design of any potential observer. However, to reflect that we do not have fully-distributed measurements, we assume that only an approximation of the state is available to design the law. More precisely, we consider a Galerkin approximation of the system (10)- [START_REF] Auriol | Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time[END_REF] (see [START_REF] Lasiecka | Ritz-galerkin approximation of the time optimal boundary control problem for parabolic systems with dirichlet boundary conditions[END_REF][START_REF] Morris | Design of finite-dimensional controllers for infinite-dimensional systems by approximation[END_REF][START_REF] Orner | A design procedure for a class of distributed parameter control systems[END_REF] for details on the Galerkin approximation) and only assume that a finite number of modes n are available to design the control law. Consider the family ϕ k defined for all k = 1, 2 . . ., by

ϕ k (x) = ϕ 1 k (x) ϕ 2 k (x) = 1 kπ cos(kπx) cos(kπx) (70)
Define ϕ 0 and ϕ 0,1 as ϕ 0 (x) = 1 0 T , ϕ 0,1 (x) = 0 1

T ( 71
)
For n ∈ N, we now define χ n = span{span i=-n,••• ,n {ϕ i }, ϕ 0,1 } and denote P n the orthogonal projection onto χ n . The space χ n is equipped with the H 1 -norm. We consider that only the approximated state (u n , v n ) = P n (u, v) is available for control design. The corresponding backstepping controller (defined by equation ( 51)) is denoted V n . Due to Jackson's theorem [START_REF] Jackson | The theory of approximation[END_REF],[74, Exercise 1.5.14], we can easily show that this approximation scheme satisfies the requirements of Theorem 9. We compare in Figure 10 the time evolution of the L 2 -norm of the system (10)-( 12) in closed-loop using the control law (51) using the approximated states (u n , v n ) for different values of n. The real system is still simulated using the previous finite volume method. When the number of modes used in the approximation scheme is too small, the system is not stabilized anymore. However, when this number of modes increases, we have closed-loop stability as stated in Theorem 9 (although the convergence rate can be impacted by the number of modes).

Concluding remarks

This chapter presents a comprehensive overview of recent developments for the practical control of scalar balance law systems using the backstepping approach. The main idea of the backstepping method is to map the original system to a target system with amenable properties. Although these target systems were originally chosen to be finite-time stable, we showed that finite-time stability could induce vanishing robustness margins. Therefore, we proposed a robustness analysis using a time-delay representation of the hyperbolic system under consideration. Such a representation allowed us to introduce degrees of freedom in the design to guarantee potential trade-offs between convergence rate and robustness margins. Interestingly, the proposed approach can be extended to more complex systems, including non-scalar systems [START_REF] Auriol | An explicit mapping from linear first order hyperbolic PDEs to difference systems[END_REF], interconnected systems [START_REF] Redaud | Stabilizing output-feedback control law for hyperbolic systems using a fredholm transformation[END_REF][START_REF] Redaud | Output-feedback control of an underactuated network of interconnected hyperbolic PDE-ODE systems[END_REF] or systems actuated at both boundaries [START_REF] Auriol | Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time[END_REF][START_REF] Auriol | Two-sided boundary stabilization of heterodirectional linear coupled hyperbolic pdes[END_REF]. We showed in this chapter that it was possible to map the original system to any arbitrary target system using appropriate backstepping transformations. Consequently, general target systems (and thus additional degrees of freedom) could be obtained by preserving dissipative in-domain couplings. This would require precise knowledge of their influence in terms of stability. The Port-Hamiltonian approach could be a path to follow. If the qualitative in closed-loop using the control law (51) and the approximation scheme P n for different values of n.

effect of these tuning parameters can be understood in the PHS framework, it is crucial to assess and quantify the performance of the resulting controllers for a given set of specifications. This set of performance criteria should be defined in terms of practically relevant properties for industrial applications, e.g., sensitivity, robustness margins, smoothness of the state, or convergence rate. Such a complete performance analysis has to be developed. It will be the purpose of our future work. Finally, we focused on the questions related to implementing the proposed controllers in real test cases. Unlike simple PID control laws, the control strategies we develop explicitly consider the network structure and the distributed dynamics of each subsystem that compose it. Therefore, they may require some computational effort to be implemented. The incentive to do so is a performance objective: explicitly taking into account the delays and high-frequency content in the model should lead to overall increased performance. As a counterpart, the numerical burden may explode with the complexity of the system. Moreover, the implementation of the proposed control laws on micro-controllers requires numerical approximations whose effects in terms of stability and convergence should be verified. We gave generic conditions under which model reduction strategies (late-lumping approaches) can still guarantee closed-loop stability. The presented results are a first step toward practical applications of backstepping controllers. The question of the late-lumping backstepping controller-observer or the extension to systems of larger dimensions has yet to be considered. Although simulations against highfidelity simulators have been performed for some applications cases, such as the control of torsional vibrations in drilling devices [START_REF] Auriol | Sensing and computational frameworks for improving drillstring dynamics estimation[END_REF][START_REF] Auriol | Closed-loop tool face control with the bit off-bottom[END_REF], it is necessary to develop a systematic model-reduction methodology before deploying, demonstrating and validating the proposed techniques developed our experimental setups and real test cases.

Figure 3 .

 3 Figure 3. Time evolution of the L 2 -norm of the system (10)-[START_REF] Auriol | Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time[END_REF] in open-loop (dashed blue) and in closed-loop using the control law (20) (red)

Figure 4 .

 4 Figure 4. Time evolution of the L 2 -norm of the system (10)-(12) in

Lemma 2 .

 2 If |ρq| > 1, then the characteristic equation (47) has infinite zeros with a non-negative real part.

1 ( 8 .

 18 1+ν0s) . We immediately have |w ν0 | < 1 and |1 -w ν0 | < 1. Since the set S = {s ∈ C + , |s| ≤ M } is compact, we can define M = inf s∈S 1 |ρqe -τ s + τ 0 Ñ (ν)e -νs ds|+1 > 0. Choosing ν 0 = M M (1+ M ) , we directly obtain |1 -w ν0 (s)| < M . Indeed |1 + ν 0 s| ≥ 1 -|ν 0 s|, which implies | ν0s 1+ν0s | < M . We have the following theorem Theorem Consider a low-pass filter w(s) such that |1 -w(s)| < 1, |w(s)| < 1 and such that equation (

Figure 5 .Figure 6 .Figure 7 .

 567 Figure 5. Time evolution of the L 2 -norm of the system (10)-(12) in closed-loop using the control law (51) in presence of a 0.05s delay in the actuation path for different values of ρ

  y) + λ ∂ Kuv ∂y (x, y) = σ+ (z) Kvu (x, y) -σ -(y) Kuv (x, y), λ ∂ Kuv ∂x (x, y) -µ ∂ Kuv ∂y (x, y) = σ+ (z) Kvv (x, y) -σ + (y) Kuu (x, y), µ ∂ Kvu ∂x (x, y) -λ ∂ Kvu ∂y (x, y) = σ -(y) Kvv (x, y) -σ-(x) Kuu (x, y), µ ∂ Kvv ∂x (x, y) + µ ∂ Kvv ∂y (x, y) = σ + (y) Kvu (x, y) -σ-(x) Kuv (x, y),with boundary conditions

1 0( 1 0(

 11 Kvu (1, y) -ρ Kuu (1, y))u(t, y)dy -Kvv (1, y) -ρ Kuv (1, y))v(t, y)dy.

Figure 8 .

 8 Figure 8. Time evolution of the L 2 -norm of the system (10)-(12) in closed-loop using the control law (51) (with ρ = 0) and using the control law[START_REF] Lions | Optimal control of systems governed by partial differential equations problèmes aux limites[END_REF]

Figure 9 .

 9 Figure 9. Time evolution of the control effort for the control law (51) (with ρ = 0) and the control law[START_REF] Lions | Optimal control of systems governed by partial differential equations problèmes aux limites[END_REF]

Figure 10 .

 10 Figure 10. Time evolution of the L 2 -norm of the system (10)-[START_REF] Auriol | Two-sided boundary stabilization of two linear hyperbolic PDEs in minimum time[END_REF]