Jeanne Redaud

Jean Auriol

Yann Le Gorrec

In domain dissipation assignment of boundary controlled Port-Hamiltonian systems using backstepping

Keywords: Backstepping-based control design, hyperbolic PDE systems, distributed parameter systems, Port-Hamiltonian Systems

In this paper, we develop a systematic approach to stabilize a general class of hyperbolic systems while assigning them a specified closed-loop behavior with a clear energy interpretation. More precisely, we address in-domain dissipation assignment for boundary-controlled Port Hamiltonian systems. The controller is designed so that the closed-loop system behaves like a target system with a specified energy decay rate. The PHS framework is used to take advantage of the natural physical properties of the system to define well-posed, exponentially stable, and easily parametrizable target system candidates, thus resulting in modular controllers. Under some generic structural assumptions, we rewrite the considered Port Hamiltonian system in the Riemann coordinates. The control approach is then based on the backstepping methodology. We combine classical Volterra transformations with an innovative time-affine transform to map the original system to the desired target system. The proposed approach is applied to two test cases: a clamped string and a clamped Timoshenko beam. Both are illustrated in numerical simulations.

Introduction

In recent years, Port-Hamiltonian Systems (PHS) formulations have emerged as a powerful framework for the modeling and control of distributed parameter systems [START_REF] Van Der Schaft | Hamiltonian formulation of distributed-parameter systems with boundary energy flow[END_REF][START_REF] Le Gorrec | A semigroup approach to port Hamiltonian systems associated with linear skew symmetric operator[END_REF][START_REF] Le Gorrec | Dirac structures and boundary control systems associated with skew-symmetric differential operators[END_REF][START_REF] Villegas | A port-Hamiltonian approach to distributed param-625 eter systems[END_REF][START_REF] Duindam | Modeling and Control of Complex Physical Systems: The Port-Hamiltonian Approach[END_REF]. This formalism is particularly well suited to describe the dynamics of large-scale (multi)physics systems, such as those arising in fluid mechanics [START_REF] Serban | Modelling and control of fluid power systems using the port-Hamiltonian approach[END_REF], heat transfer [START_REF] Van Der Schaft | Port-635 Hamiltonian formulation of distributed parameter systems: Convective transport and phase transitions[END_REF], and structural mechanics [START_REF] Wu | Port-Hamiltonian modeling and control of beam vibration[END_REF]. The Port-Hamiltonian approach makes it possible to highlight and take advantage of the physical properties of the considered systems through a well-defined geometric structure. It has been successfully applied to a wide range of engineering applications, including the optimization of systems performances [START_REF] Dunn | Optimization-based design of a high-performance hybrid vehicle powertrain using the port-Hamiltonian framework[END_REF], systems stability analysis [START_REF] Augner | Stability and stabilization of infinitedimensional linear port-Hamiltonian systems[END_REF][START_REF] Bagherpour | Stability and energy-preserving boundary control of flexible me-650 chanical systems with distributed parameters using the port-Hamiltonian approach[END_REF], and the design of controllers for complex systems [START_REF] Ortega | Control by interconnection and standard passivity-based control of port-655 Hamiltonian systems[END_REF][START_REF] Macchelli | On the synthesis of boundary control laws for distributed port-Hamiltonian systems[END_REF]. Regarding linear hyperbolic PDEs, the Port-Hamiltonian framework can be used to prove the existence of solutions or to parametrize all the boundary conditions guaranteeing the well-posedness of the associated boundary control system [START_REF] Le Gorrec | Dirac structures and boundary control systems associated with skew-symmetric differential operators[END_REF][START_REF] Zwart | Well-posedness and regularity of hyperbolic boundary control systems on a onedimensional spatial domain[END_REF][START_REF] Macchelli | On the synthesis of boundary control laws for distributed port-Hamiltonian systems[END_REF]. It can also be used to design boundary controllers, exploiting the system's physical properties efficiently. For instance, the energy shaping method via control by interconnection was developed for boundarycontrolled port Hamiltonian systems [START_REF] Macchelli | On the synthesis of boundary control laws for distributed port-Hamiltonian systems[END_REF][START_REF] Macchelli | Exponential stabilization of port-Hamiltonian boundary control systems via energy shaping[END_REF]. It allows the modification of the closed-loop energy function and is, * Corresponding author: jeanne.redaud@centralesupelec.fr therefore, a design method with a clear physical interpretation.

In parallel, the backstepping methodology [START_REF] Krstic | Boundary control of PDEs. A course on backstepping designs[END_REF] has proven to be a powerful design method for boundary feedback control of interconnected PDE systems [START_REF] Anfinsen | Adaptive output feedback stabilization of n+m coupled linear hyperbolic PDEs with uncertain boundary conditions[END_REF][START_REF] Wang | Balancing and suppression of oscillations of tension and cage in dual-cable mining elevators[END_REF][START_REF] Redaud | Output-feedback control of an underactuated network of interconnected hyperbolic PDE-ODE systems[END_REF]. Based on 30 invertible state transformations (usually Volterra integral transforms), it consists of mapping the original system into a simpler form (called target system) amenable to analysis, control, and observer design [START_REF] Bou Saba | Stability analysis for a class of linear 2 × 2 hyperbolic PDEs using a backstepping transform[END_REF]. One of the main difficulties with the backstepping method lies in finding a suitable [START_REF] Hu | Boundary exponential stabilization of 1-D inhomogeneous quasi-linear hyperbolic systems[END_REF] target system. It should be simple enough to allow the design of the control law. Still, in the meantime, one must prove the existence of a transformation mapping the original system to this target system. The choice of the target system directly impacts the closed-loop performance. The 40 general question of reachable target systems is still an open problem. In the case of hyperbolic PDE systems, finitetime stable target systems are usually chosen [START_REF] Li | Strong (weak) exact controllability and strong (weak) exact observability for quasilinear hyperbolic systems[END_REF][START_REF] Coron | Finite-time boundary stabilization of general linear hyperbolic balance laws via Fredholm backstepping transformation[END_REF] as they often present a simple structure with amenable properties [START_REF] Auriol | An explicit mapping from linear first order hyperbolic PDEs to difference systems[END_REF]. However, this choice of target systems corresponds to a specific performance criterion (here, finite-time stability), 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]. 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 for two coupled hyperbolic equations, thus guaranteeing po-50 tential trade-offs between different specifications (namely delay-robustness and convergence rate). However, these parameters have a limited range of action since they only affect the system's boundary conditions. Recent contribu-tions focused on the backstepping exponential stabilization of a potentially unstable Timoshenko beam with an arbitrary decay rate [START_REF] Chen | Rapid stabilization of Timoshenko beam by PDE backstepping[END_REF], thereby introducing a class of parameterized controllers. These results are currently being extended to multi-layer beams [START_REF] Chen | Backstepping-based rapid stabilization of two-layer Timoshenko composite beams[END_REF]. Similar questions have been raised for parabolic systems with the design of prescribed-time controllers [START_REF] Wei | Prescribed-time stabilization of uncertain heat equation via boundary time-varying feedback and disturbance estimator[END_REF][START_REF] Steeves | Input delay compensation in prescribed-time of boundary-actuated reaction-diffusion PDEs[END_REF][START_REF] Espitia | Boundary time-varying feedbacks for fixed-time stabilization of constantparameter reaction-diffusion systems[END_REF], requiring the introduction of non-trivial target systems to achieve the desired specific closed-loop behavior. All these examples illustrate how adding tuning terms in the target systems allows for better closed-loop properties, such as arbitrarily large exponential decay rates or better robustness margins. More general target systems (and thus additional degrees of freedom) could be obtained by preserving dissipative in-domain couplings. This would require precise knowledge of the influence of the different parameters of the system in terms of stability properties. In this context, the Port-Hamiltonian approach could be advantageously used to define well-posed, exponentially stable target system candidates. A first attempt to take advantage of Port Hamiltonian formulations and their associated geometric structure for backstepping control design can be found in [START_REF] Ramirez | On backstepping boundary control for a class of linear port-Hamiltonian systems[END_REF][START_REF] Vu | Distributed and backstepping boundary controls to achieve IDA-PBC design[END_REF]. Recently, the backstepping method was successfully combined with PHS on simple test cases (wave equation and Timoshenko beam) to design state-feedback controllers that assign the distributed damping of the closedloop system, thus determining the decay rate of the solutions while reducing the associated control effort [START_REF] Redaud | Distributed damping assignment for a wave equation in the port-Hamiltonian framework[END_REF][START_REF] Redaud | In-domain damping assignment of a Timoshenko beam using state feedback boundary control[END_REF].

In this paper, we develop a systematic framework to stabilize a general class of hyperbolic PDE systems while assigning a specified closed-loop behavior with a clear energy interpretation. In this context, we take advantage of the PHS theory, which corresponds to a multi-physical and modular energy-based representation that considers the system's natural physical properties. These natural physical properties can then be advantageously used to define well-posed, exponentially stable, and easily parametrizable target system candidates. Therefore, the proposed approach will allow parameterizing controllers by employing physically inspired tuning parameters (seen as degrees of freedom). In the near future, these controllers could be tuned to fulfill a given set of performance specifications best. Such a methodology was initially developed for simple low-dimensional systems in [START_REF] Redaud | Distributed damping assignment for a wave equation in the port-Hamiltonian framework[END_REF][START_REF] Redaud | In-domain damping assignment of a Timoshenko beam using state feedback boundary control[END_REF]. Its extension to the general case requires introducing a new kind of transformation.

Our approach is the following. The original hyperbolic system is first expressed in the Port Hamiltonian framework to emphasize its inherent physical properties. In particular, we can easily express the evolution of the associated energy. Then, we can define the desired target system as a copy of the original PHS with modulated internal dissipation (in-domain damping), which implies a specific energy decay of the original system in closed-loop. In the meantime, we express these two systems in the Riemann coordinates (as balance laws) since this framework is required to apply the backstepping method. More precisely, in this set of coordinates, we can combine successive backstepping transformations [START_REF] Hu | Boundary exponential stabilization of 1-D inhomogeneous quasi-linear hyperbolic systems[END_REF] with an original time-affine transformation to map the original system to the target system, thus fulfilling our control objective. To illustrate 115 our methodology, we apply it to two test cases: distributed damping assignment for a wave equation and the stabilization of a Timoshenko beam. An interesting by-product of our analysis is that we can design controllers that can map a system of linear coupled balance laws [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF] to any 120 target system with the same structure but with arbitrary in-domain coupling terms. This allows for a more generic class of target system compared to existing ones in the literature [START_REF] Hu | Boundary exponential stabilization of 1-D inhomogeneous quasi-linear hyperbolic systems[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].

This article is organized as follows. We first present the beam. In both test cases, numerical simulations illustrate our results. Some concluding remarks and perspectives end this paper (Section 6).

Notations

For any n ∈ N * , for any compact set K, We denote C 1 (K; R n ) the space of real differentiable functions defined on K with values in R n and a continuous derivative, C 1 (K; R n ) + its subset of strictly positive functions, and C 1 pw (K; R n ) the set of piecewise differentiable functions with continuous derivative. We denote D + n , the set of diagonal matrices in R n×n with positive coefficients. The notation I n stands for the n × n identity matrix (if the dimensions are not ambiguous, the subindex will be omitted). Following classical notations from the Port-Hamiltonian framework, we define the Hilbert space

χ n = H 1 ([0, 1]; R n ) equipped with the inner-product < u, v > χn = 1 2 1 0 u(z) H(z)v(z)dz, with H ∈ D + n . Note that the χ n -norm is equivalent 145 to the classical L 2 -norm. Let τ > 0 be a positive fixed time delay. We denote D τ [t] = H 1 ([-τ, 0], R) the Ba- nach space of H 1 real-valued functions mapping the in- terval [-τ, 0] into R. For any function φ : [-τ, ∞) → R, we define its associated partial trajectory φ [t] ∈ D τ by 150 φ [t] (θ) = φ(t + θ), -τ ≤ θ ≤ 0. The associated norm is 2 given by φ [t] τ = 0 -τ φ(t + θ) 2 dθ 1 2 .
The unit square [0, 1] 2 is denoted S. Its lower (resp. upper) triangular part is denoted

T -= {(x, y) ∈ [0, 1] 2 | 0 ≤ y ≤ x} (resp. T + ).
For any λ, µ, a > 0, we denote

T + λ = {(x, y) ∈ [0, 1] × [0, 1 λ ] | 0 ≤ y ≤ x λ }, T - µ = {(x, y) ∈ [0, 1] × [0, 1 µ ] | 0 ≤ y ≤ 1 µ (1 -x)}, and P + a,λ = {(x, y) ∈ [0, 1] × [0, a + 1 λ ] | x λ ≤ y ≤ a + x λ )}, P - a,µ = {(x, y) ∈ [0, 1] × [0, a + 1 µ ] | 1-x µ ≤ y ≤ a + 1-x µ )
} two parallelogram domains. We use the standard notation for the Kronecker symbol δ ij . = 1 if i = j, 0 else. We extend this definition with δ i j . = 1 if i j, 0, where denote any order relation.

Problem under consideration and control objective

As explained in the introduction, the main objective of this paper is to stabilize a general class of hyperbolic systems while assigning a specified closed-loop behavior with a clear energy interpretation. More precisely, we aim to introduce degrees of freedom in the design to obtain a class of easily parametrizable closed-loop systems. The controller will be designed such that it is possible to map the closed-loop system to this desired target system using invertible transformations, thus guaranteeing equivalent properties. We take advantage of the PHS framework to develop this class of attainable (exponentially stable) systems, which can help identify naturally dissipative terms. Consequently, this implies that the desired target system we want to reach has to be expressed in this framework. In this paper, we also chose to express the original hyperbolic system in the PHS framework, as this can give us a clear energy interpretation of the open-loop behavior. However, we emphasize that our methodology could be applied to original systems that do not necessarily adhere to the PHS paradigm as long as they can be expressed in the so-called Riemann coordinates. Indeed, to develop our methodology, we need to rewrite the PHS systems in the Riemann coordinates and actually show that it is possible to design a control law such that we can map any controlled hyperbolic PDE system in the Riemann coordinates to any arbitrary hyperbolic system with an analogous structure but modified in-domain coupling terms. In the rest of this section, we first introduce the system under consideration in the PHS framework before giving its expression in the Riemann coordinates. We then properly state our control objective and introduce the class of target systems we want to reach. In this paper, we consider the boundary control of Port-Hamiltonian systems [START_REF] Jacob | Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces[END_REF] defined on a one-dimensional do-main z ∈ [0, 1] by

∂x ∂t = P 1 ∂ ∂z (H(z)x(t, z)) + (P 0 -Π 0 )H(z)x(t, z), (1)
where x(t, z) ∈ χ 2n (n ∈ N\{0}) is the vector of energy variables defined on [0, +∞) × [0, 1], H is a symmetric and Lipschitz continuous coercive matrix-valued function defined on [0, 1], P 1 is a full rank matrix such that P 1 = P 1 ∈ R 2n×2n , the matrix P 0 verifies P 0 = -P 0 ∈ R 2n×2n and Π 0 ∈ R 2n×2n verifies Π 0 = Π 0 ∈ R 2n×2n . The boundary inputs/ouputs are defined by

u ∂ (t) = W B H(1)x(t, 1) H(0)x(t, 0) , (2)
y ∂ (t) = W C H(1)x(t, 1) H(0)x(t, 0), (3)
where

W B , W C ∈ R 2n×4n with W B = 1 √ 2 (Ξ -+ Ξ + P 1 ) 1 √ 2 (Ξ --Ξ + P 1 ) , (4)
W C = 1 √ 2 (Ξ + + Ξ -P 1 ) 1 √ 2 (Ξ + -Ξ -P 1 ) , (5)
and Ξ + and Ξ -in R 2n×2n satisfy

Ξ -Ξ + + Ξ + Ξ -= 0, and Ξ -Ξ -+ Ξ + Ξ + = I 2n . (6)
For all t ≥ 0, we define the total energy of the system E(x) as

E(x(t)) = 1 2 1 0 x (t, z)H(z)x(t, z) dz = x(t, •) 2 χ .
We have [START_REF] Jacob | Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces[END_REF] ∂E

(x(t)) ∂t = y ∂ (t)u ∂ (t) - 1 0 x (t, z)Π 0 x(t, z)dz.
It has been shown in [START_REF] Le Gorrec | A semigroup approach to port Hamiltonian systems associated with linear skew symmetric operator[END_REF] that if Π 0 is semi-definite positive, the system (1-3) defines a boundary control system [START_REF] Curtain | An introduction to infinite-dimensional linear systems theory[END_REF]. Under some conditions on Π 0 (corresponding to some damping in velocity coordinates), it could be shown that the open-loop system is asymptotically or exponentially 205 stable [START_REF] Jacob | Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces[END_REF]. However, we emphasize that in this paper, we only assume Π 0 = Π T 0 . Therefore, the system we consider may be undamped or anti-damped and unstable. In this study, we make the following assumption: Assumption 1. One boundary is fully actuated, i.e.

u ∂ (t) = u 0 (t) 0
where u 0 is the control input, and the matrix W B is block 210 diagonal or block anti-diagonal, the different blocks being of dimension n × 2n.

Assumption 1 means that one boundary is fully actuated and the other is set to zero, implying the original system can be written into the hyperbolic PDE form of Section 3 (Riemann coordinates). Note that this assumption is verified for numerous physical applications as clamped/actuated strings (described in Section 2.1.3 and Section 4) or clampedactuated Timoshenko beams (described in Section 5). Without any loss of generality, we consider that the actuated boundary is in z = 1.

2.1.2. Hyperbolic system in the Riemann coordinates Here, we reformulate the system (1)-( 3) in the Riemann coordinates. Indeed, one of the key ingredients of the methodology we present in this paper is the backstepping approach that has mostly been developed for systems of balance laws with diagonal velocity matrices [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF][START_REF] Coron | Finite-time boundary stabilization of general linear hyperbolic balance laws via Fredholm backstepping transformation[END_REF][START_REF] Auriol | Minimum time control of heterodirectional linear coupled hyperbolic PDEs[END_REF]. Therefore, it is of specific interest to have an alternate representation of the system (1)-( 3) in this framework. The matrix P 1 being full rank and H(z) coercive, P 1 H(z) is diagonalizable, i.e., there exist a matrix-valued function

Q 1 (z) ∈ R 2n×2n , a diagonal matrix-valued function Λ(z) defined on [0, 1], such that ∀ z ∈ [0, 1], P 1 H(z) = Q 1 (z)Λ(z)Q -1 1 (z).
For all t > 0, we first define a new set of variables

ζ(t, •) = Q -1 1 (•)x(t, •) ∈ χ 2n .
It satisfies a set of transport PDEs with in-domain coupling terms

∂ζ ∂t (t, z) + Λ(z) ∂ζ ∂z (t, z) = Σ(z)ζ(t, z), (7)
u ∂ (t) = W B H(1)Q 1 (1)ζ(t, 1) H(0)Q 1 (0)ζ(t, 0) , (8)
with

Σ(z) = [Q -1 1 (z)(P 1 ∂H ∂z +(P 0 -Π 0 ))H(z)- ∂Q -1 1 (z) ∂z
]Q 1 (z). When applying the backstepping approach, the matrix Σ will be required to have zero diagonal terms. To fulfill this constraint, we use an exponential change of variables [START_REF] Coron | Local exponential H 2 stabilization of a 2×2 quasilinear hyperbolic system using backstepping[END_REF]. For all z ∈ [0, 1], define A(z) ∈ D + 2n with diagonal terms given by

A ii (z) = e Ii(z) , with I i (z) = - z 0 Σ ii (s) Λ ii (s) ds. (9)
The matrix-valued function A is invertible, and we can define a new state variable by

ξ(t, z) = A(z)ζ(t, z). It satisfies ∂ξ ∂t (t, z) + Λ(z) ∂ξ ∂z (t, z) = σ(z)ξ(t, z), (10)
u ∂ (t) = W B H(1)Q 1 (1)A -1 (1)ξ(t, 1) H(0)Q 1 (0)ξ(t, 0) , (11)
with

σ ij (z) = [A(z)Σ(z)A -1 (z)
] ij for i = j, and 0 else.

Example of a clamped-string (wave equation)

We now illustrate the previous concepts with an example to familiarize the reader with the previous concepts. Consider a vibrating string clamped at the first end (z = 0) and actuated at the other end (z = 1). We denote the vertical position of the string at point z and time t > 0 as w(t, z). It satisfies

ρ(z) ∂ 2 w ∂t 2 (t, z) = ∂ ∂z E(z) ∂w ∂x (t, z) -κ(z) ∂w ∂t (t, z), (12)
with ρ(z), E(z) ∈ C 1 ([0, 1]; R) + being the mass density and Young's modulus, which are here space-dependent. The term κ(z) ∈ C 0 ([0, 1]) corresponds to a damping term. Its sign impacts the global behavior of the system. For instance, a negative κ corresponds to a destabilizing antidamping action. Initially, the position of the string is given by w(z, 0) = w 0 (z) ∈ C 1 ([0, 1]; R), and its speed by ∂w ∂t | t=0 (z) = w 1 (z), with w 0 (0) = 0 = w 1 (0). We first rewrite the model as a Port Hamiltonian system. The energy state variables x = [x 1 , x 2 ] ∈ χ 2 are defined by

x 1 (t, z) = ∂w ∂z (t, z), x 2 (t, z) = ρ(z) ∂w ∂t (t, z), (13)
where x 1 (t, z) (resp. x 2 (t, z)) corresponds to the strain (resp. to the momentum). The state (x 1 , x 2 ) satisfies

∂ ∂t x 1 x 2 = 0 ∂ ∂z 1 ρ(z) • ∂ ∂z (E(z)•) -c(z) x 1 x 2 , (14)
with c(z) = κ(z) ρ(z) . The first end of the string is clamped while the other end is actuated such that

x 2 (0, t) = 0, E(1)x 1 (1, t) = u 0 (t), (15)
where u 0 (t) is the control input. The Hamiltonian density is given for all z ∈ [0, 1] by

H(z) = diag(E(z), 1 ρ(z) ) ∈ D + 2 .
We have P 0 = 0, P 1 = 0 1 1 0 and Π 0 = 0 0 0 κ . The boundary conditions rewrite as in equation ( 2) with

W B = 0 1 0 0 0 0 1 0 . It can be checked that W B verifies equa- tion (4) with Ξ -= 1 2 0 1 1 0 , and Ξ + = 1 2 1 0 0 -1 .
This system verifies Assumption 1.

We now rewrite the PHS system ( 14)-( 15) in the Riemann

coordinates. Let us define Λ(z) = λ(z) 0 0 -λ(z)
, where

λ(z) = E(z)/ρ(z) ∈ R + .
To simplify the notations, we introduce the following functions r(z) = E(z)ρ(z) and

δ(z) = ρ ρ + E E (z), δ 3 1 (z) = 3 ρ ρ -E E (z), δ 1 3 (z) = ρ ρ -3 E E (z)
. Since the matrix P 1 H(z) admits two opposite eigenvalues ±λ(z), we have

P 1 H(z) = Q 1 (z)Λ(z)Q 1 (z) -1 ,
where

Q 1 (z) = 1 √ 2 1 - 1 √ ρ(z)E(z) ρ(z)E(z) 1 . The new state variables ζ = [ζ + , ζ -] = Q -1 1 [x 1 ,
x 2 ] satisfy equation [START_REF] Van Der Schaft | Port-635 Hamiltonian formulation of distributed parameter systems: Convective transport and phase transitions[END_REF] with in-domain spatially varying continuous coupling terms defined by

           Σ 11 (z) = 1 2 -c(z) + λ 2 δ 1 3 (z) , Σ 12 (z) = 1 2ρ(z) c(z) λ(z) -1 2 δ(z) , Σ 21 (z) = E(z) 2 c(z) λ(z) + 1 2 δ(z) , Σ 22 (z) = 1 2 -c(z) -λ 2 δ 3 1 (z) ,
The diagonal coupling terms Σ 11 and Σ 22 can be suppressed by introducing the following exponential change of coordinates

[ξ + (t, z), ξ -(t, z)] = A(z)[ζ + (t, z), ζ -(t, z)] ,
with

A(z) = diag(f (z)e Ic(z) , g(z)e -Ic(z) ), where ∀z ∈ [0, 1], f (z) = E(z)λ(z) E(0)λ(0) , g(z) = λ(z)ρ(0) ρ(z)λ(0)
and

I c (z) = z 0 c(s)
2λ(s) ds. Henceforth, the new variables ξ = [ξ + , ξ -] satisfy the hyperbolic PDEs ( 10)- [START_REF] Bagherpour | Stability and energy-preserving boundary control of flexible me-650 chanical systems with distributed parameters using the port-Hamiltonian approach[END_REF], where the scalar in-domain coupling terms are defined by

     σ 11 (z) = σ 22 (z) = 0, σ 12 (z) = 1 2r(0) e 2Ic(z) (c(z) -λ(z) 2 δ(z)), σ 21 (z) = r(0) 2 e -2Ic(z) (c(z) + λ(z) 2 δ(z)),
Finally, we have the following boundary conditions ξ + (t, 0) = 1 r(0) ξ -(t, 0) and ξ -(t, 1) = -r(1)e -2Ic(1) ξ + (t, 1) + u(t), where the control input is given by

u(t) = g(1)e -Ic(1) √ 2 λ(1)
u 0 (t).

Control objective 2.2.1. Target system in the PHS framework

This paper aims to stabilize the system (1) while guaranteeing a specific closed-loop behavior with a clear energy interpretation. We want to impose a specific decay rate on the energy function E. The expected outcome is to obtain a class of easily parametrizable closed-loop systems with tuning parameters (degrees of freedom). Then, the associated controllers could be tuned in future works to best fulfill a given set of performance specifications. The desired target systems correspond to the original system (1) but with modified in-domain damping terms. More precisely, we want to obtain the following closed-loop behavior

∂ x ∂t = P 1 ∂ ∂z (Hx(t, z)) + P0 -Π0 (Hx(t, z)) , (16)
where x ∈ χ 2n and P0 = -P 0 ∈ R 2n×2n , Π0 ∈ R 2n×2n satisfying Π0 + Π 0 ≥ 0. The boundary conditions are given by

W B H(1)x(t, 1) H(0)x(t, 0) = 0 2n , with W B 0 I 2n I 2n 0 W B ≥ 0. ( 17
)
where W B ∈ R 2n×4n has a structure analogous to the one of W B (i.e., it is block diagonal or bloc anti-diagonal due to Assumption 1). Note that for the sake of simplicity, Π0 has been chosen constant but it could have been chosen as a function of z, i.e. Π0 (z) is a real matrix-valued function satisfying Π0 (z) + Π 0 (z) ≥ 0, ∀z ∈ [0, 1] and chosen such that the target system is exponentially stable. The energy function associated to this target system satisfies

dE(x(t)) dt = - 1 0 x (t, z) Π0 x(t, z)dz ≤ 0. (18)
Therefore, in closed-loop, the energy decay is determined 230 by the matrix Π0 . Since Π0 is chosen as positive definite, the system is exponentially stable. Yet, in most examples, not all the states need to be damped. Following the approach proposed in [START_REF] Mattioni | A Lyapunov approach for 760 the exponential stability of a damped Timoshenko beam[END_REF], the associated exponential decay rate of the energy of the target system can be computed 235 using a Lyapunov analysis. This could be a way of choosing the desired target matrix Π0

Target system in the Riemann coordinates

Similarly to what we have done above, we can rewrite the target system ( 16)-( 17) in the Riemann coordinates, which will be more amenable to apply backstepping transformations. More precisely, we define ζ(t, z) = Q -1 1 (z)x(t, z), where the matrix Q 1 is defined in Section 2.1.2. This new state satisfies the set of PDEs

∂ ζ ∂t (t, z) + Λ(z) ∂ ζ ∂z (t, z) = Σ(z) ζ(t, z), (19)
0 2n = W B H(1)Q 1 (1) ζ(t, 1) H(0)Q 1 (0) ζ(t, 0) , ( 20
) with Σ(z) = [Q -1 1 (z)(P 1 ∂H ∂z +( P0 -Π0 )H(z))+ ∂Q -1 1 (z) ∂z ]Q 1 (z). For all z ∈ [0, 1], define Ā(z) ∈ D + 2n with Āii (z) = e Īi(z) , with Īi (z) = - z 0 Σii (s) Λ ii (s) ds. (21)
The matrix-valued function is invertible, and we can define the new state variable by ξ(t,

z) = Ā(z) ζ(t, z) ∂ ξ ∂t (t, z) + Λ(z) ∂ ξ ∂z (t, z) = Σ(z) ξ(t, z), (22)
0 2n = W B H(1)Q 1 (1) Ā-1 (1) ξ(t, 1) H(0)Q 1 (0) ξ(t, 0) , (23)
with σij (z) = [ Ā(z) Σ(z) Ā-1 (z)] ij for i = j, and 0 else. We can now proceed further with the control design. For the sake of simplicity, we now assume that the Hamiltonian H is constant (the physical parameters are spaceindependent). The following approach can be adapted to the space-dependent case to the price of more technical computations. [START_REF] Ortega | Control by interconnection and standard passivity-based control of port-655 Hamiltonian systems[END_REF]. For this system, our objective is to impose a specific decay rate to the energy of the system E, using a distributed damping assignment. More precisely, we want to make the dynamics of x equivalent to the dynamics of x = [x 1 , x2 ] satisfying ( 16)-( 17) with P0 = 0 and

Π0 (z) = 0 0 0 -K(z)
, with a new strictly positive spacevarying damping term K(z) > 0. Consequently, the energy of the closed-loop system will exponentially decrease with a decay rate given by K.

We can then rewrite the system x in the Riemann coordinates. Using the same notations as the ones given in Section 2.1.

3, we define [ ζ+ , ζ-] = Q -1 [x 1 , x2 ] . Next,
we use an exponential variable change to suppress the diagonal in-domain couplings. Define for all z ∈ [0, 1],

Ā(z) = diag(f (z)e I K (z) , g(z)e -I K (z) ), I K (z) = z 0 K(s) 2λ(s) ds.
The new state [ ξ+ , ξ-] = Ā[ ζ+ , ζ-] satisfies heterodirectional transport equations of form ( 22)-( 23) with indomain couplings defined by

     σ11 (z) = σ22 (z) = 0, σ12 (z) = 1 2r(0) e 2I K (z) (K(z) -λ(z) 2 δ(z)), σ21 (z) = r(0) 2 e -2I K (z) (K(z) + λ(z) 2 δ(z)).
The objective is now to design a control law such that the closed-loop system can be mapped to this desired target system. 250

Mapping of an hyperbolic + m PDE system to an arbitrary target system

As explained in Section 2.2, the objective of this paper is to map the original Port-Hamiltonian system (1)-( 2) to the target Port-Hamiltonian system ( 16)-( 17), thus ensuring closed-loop stability with a specific decay rate for the energy of the system. The parameter Π0 in equation( 16) can therefore be seen as a degree of freedom reflecting the closed-loop system's dissipativity. This objective somehow corresponds to in-domain damping assignment [START_REF] Redaud | Distributed damping assignment for a wave equation in the port-Hamiltonian framework[END_REF][START_REF] Redaud | In-domain damping assignment of a Timoshenko beam using state feedback boundary control[END_REF]. In this section, we show how we can use the backstepping methodology to map the original system (1)-( 2) to the target system ( 16)- [START_REF] Anfinsen | Adaptive output feedback stabilization of n+m coupled linear hyperbolic PDEs with uncertain boundary conditions[END_REF]. As explained before, we consider the equivalent problem of mapping the system (10)- [START_REF] Bagherpour | Stability and energy-preserving boundary control of flexible me-650 chanical systems with distributed parameters using the port-Hamiltonian approach[END_REF] (that corresponds to (1)-( 2) in the Riemann coordinates) 265 to the target system ( 22)- [START_REF] Auriol | An explicit mapping from linear first order hyperbolic PDEs to difference systems[END_REF] (that corresponds to ( 16)- [START_REF] Anfinsen | Adaptive output feedback stabilization of n+m coupled linear hyperbolic PDEs with uncertain boundary conditions[END_REF] in the Riemann coordinates). Although the original hyperbolic system we consider in Section 2 is expressed in the Port Hamiltonian framework, the result we have in this paper does not actually require this original system 270 to adhere to the PHS paradigm. More precisely, we show that for any arbitrary system of linear balance laws (with boundary actuation), it is possible to design a controller such that we can map the closed-loop system to any target system with a similar structure but whose source terms can 275 be arbitrarily chosen. This result is in itself a significant breakthrough as it gives the possibility of parametrizable target systems for the backstepping approach. As emphasized in Section 2, the PHS framework can indicate appropriate ones among these parametrizable target systems.

280

More precisely, we consider the following ( + m) × ( + m) hyperbolic PDE system [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF] ∂ξ

+ ∂t + Λ + ∂ξ + ∂z = σ ++ (z)ξ + + σ +-(z)ξ -, (24)
∂ξ - ∂t -Λ -∂ξ - ∂z = σ -+ (z)ξ + + σ --(z)ξ -, (25)
where ξ + is a vector of dimension > 0 and ξ -a vector of dimension m > 0. The velocity matrices are defined by Λ = diag(Λ + , -Λ -) with Λ + = diag(λ 1 , ..., λ ), Λ -= diag(µ 1 , ...µ m ) with λ 1 > ... > λ > 0, µ 1 > ... > µ m > 0.

The in-domain coupling σ •• are continuously differentiable functions. For all 1 ≤ i ≤ and all 1 ≤ k ≤ m, we assume σ ++ ii (•) = 0, and σ -- kk (•) = 0. We have the boundary conditions [START_REF] Chen | Rapid stabilization of Timoshenko beam by PDE backstepping[END_REF] with coupling matrices Q 0 ∈ R m× , R 1 ∈ R ×m , and where u(t) is the control input. An attentive reader would have noticed that the system (24)-( 26) is more generic than the system (10)- [START_REF] Bagherpour | Stability and energy-preserving boundary control of flexible me-650 chanical systems with distributed parameters using the port-Hamiltonian approach[END_REF] introduced in Section 2.1.2, since here and m are not necessarily equal. Indeed, the system (10)-285 [START_REF] Bagherpour | Stability and energy-preserving boundary control of flexible me-650 chanical systems with distributed parameters using the port-Hamiltonian approach[END_REF] was obtained from the Port Hamiltonian system (1) and consequently the number of rightward propagating states was equal to the number of leftward propagating states, i.e. we had = m = n that is + m = 2n. Again, this emphasizes the generality of the results we give in 290 this Section. The initial condition of the system is denoted (ξ + 0 , ξ - 0 ) ∈ χ +m and satisfies the compatibility conditions ξ + 0 (0

ξ + (t, 0) = Q 0 ξ -(t, 0), ξ -(t, 1) = R 1 ξ + (t, 1) + u(t),
) = Q 0 ξ - 0 (0), ξ - 0 (1) = R 1 ξ + 0 (1)
. Therefore, the open-loop system is well-posed [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF]. We chose to state the well-posedness of the open-loop system in the 295 state space χ +m (therefore requiring appropriate compatibility conditions) to avoid dealing with the weak formulation. However, we believe all the results we present in the rest of the paper can be extended to L 2 functions (and in that case, no compatibility condition would be required). Since the control operator is admissible, the closed-loop system remains well-posed for a continuous feedback law. Moreover, any stabilizing control law can be dynamically modified to satisfy the compatibility conditions in closed-loop while guaranteeing exponential stability [START_REF] Coron | Local exponential H 2 stabilization of a 2×2 quasilinear hyperbolic system using backstepping[END_REF]. The system ( 24)-( 26) admits a unique solution, whose state is denoted ξ = [ξ + , ξ -] ∈ χ m+ and defined for (t, z) ∈ [0, +∞) × [0, 1]. System ( 24)-( 26) is schematically represented in Figure 1. Our control objective is to find a state-feedback control law u(t) such that the system (24)-( 26) behaves in closedloop as the following target system

∂ ξ+ ∂t + Λ + ∂ ξ+ ∂z = σ++ (z) ξ+ + σ+-(z) ξ-, (27)
∂ ξ- ∂t -Λ -∂ ξ- ∂z = σ-+ (z) ξ+ + σ--(z) ξ-, (28)
with boundary conditions

ξ+ (t, 0) = Q 0 ξ-(t, 0), ξ-(t, 1) = R1 ξ+ (t, 1), ( 29
)
where σ•• are arbitrary continuously differentiable functions (the matrices σ++ , σ--have no term on their diagonal) and R1 is a constant matrix. In other words, we want to design a feedback controller such that the system ( 24)-( 26) can be mapped to the target system ( 22)-( 23) by means of appropriate transformations that will guarantee that the two systems share the same stability properties. Again, we emphasize that system ( 22)-( 23) is a particular case of system ( 27)- [START_REF] Steeves | Input delay compensation in prescribed-time of boundary-actuated reaction-diffusion PDEs[END_REF]. In that sense, the PHS approach provides adequate σ•• and R1 to guarantee the exponential stability of the target system while imposing the energy decay rate in [START_REF] Wang | Balancing and suppression of oscillations of tension and cage in dual-cable mining elevators[END_REF]. Note that the boundary couplings at the unactuated boundary z = 0 and the velocity matrices are not modified. 325

Control strategy

We describe below the three steps we follow to fulfill our control objective:

1. In Section 3.3, we introduce a classical backstepping Volterra transform [START_REF] Hu | Boundary exponential stabilization of 1-D inhomogeneous quasi-linear hyperbolic systems[END_REF] (K) to map the system ξ to Inspired by [START_REF] Hu | Boundary exponential stabilization of 1-D inhomogeneous quasi-linear hyperbolic systems[END_REF][START_REF] Coron | Finite-time boundary stabilization of general linear hyperbolic balance laws via Fredholm backstepping transformation[END_REF], we use a classical Volterra transform to map the ξ-system (24)-( 26) to a simpler system. More precisely, we for all t > 0 and z ∈ [0, 1], we define the state γ(t, z) by

γ(t, z) = ξ(t, z) - z 0 K(z, y)ξ(t, y)dy, (30)
where the kernels

K = K ++ K +- K -+ K --
belong to the space

C 1 pw (T -; R ( +m)×( +m)
). They satisfy the following partial differential equations

Λ + ∂K ++ ∂z + ∂K ++ ∂y Λ + = -K ++ σ ++ (y) -K +-σ -+ (y), Λ + ∂K +- ∂z - ∂K +- ∂y Λ -= -K ++ σ +-(y) -K +-σ --(y), Λ -∂K -+ ∂z - ∂K -+ ∂y Λ + = K -+ σ ++ (y) + K --σ -+ (y), Λ -∂K -- ∂z + ∂K -- ∂y Λ -= K -+ σ +-(y) + K --σ --(y). ( 31
)
with the boundary conditions,

Λ + K ++ (z, z) -K ++ (z, z)Λ + = σ ++ (z), Λ + K +-(z, z) + K +-(z, z)Λ -= σ +-(z), Λ -K -+ (z, z) + K -+ (z, z)Λ + = -σ -+ (z), Λ -K --(z, z) -K --(z, z)Λ -= -σ --(z). (32)
We add the following boundary condition for i ≤ j,

K -- ij (z, 0) = K -+ (z, 0)Λ + Q 0 (Λ -) -1 ij . (33)
If the boundary coupling matrix Q 0 is invertible, we can choose

K ++ ij (z, 0) = K +-(z, 0)Λ -(Λ + Q 0 ) -1 ij , i ≤ j, (34)
otherwise, we set K ++ ij (z, 0) = 0, i ≤ j. Finally, we impose arbitrary conditions in z = 1 for K ±± ij when i > j. For instance, we can choose

K ++ ij (1, y) = σ ++ ij (1) λ i -λ j , K -- ij (1, y) = σ -- ij (1) µ j -µ i . ( 35
)
The well-posedness of the kernel equation ( 31)-( 35) is assessed in the following lemma:

Lemma 1. The system defined by (31)-( 35) admits a unique solution in C 1 pw (T -; R ( +m)×( +m) ). Therefore, the Volterra integral transform defined by (30) is boundedly invertible. More precisely, for all t > 0 and all z ∈ [0, 1], we have

ξ(t, z) = γ(t, z) - z 0 L(z, y)γ(t, y))dy, (36)
where the kernel

L ∈ C 1 pw (T -; R ( +m)×( +m) ) satisfies L(z, y) = -K(z, y) + z y K(z, s)L(s, y)ds. (37)
Proof. The proof can be adjusted from [START_REF] Hu | Boundary exponential stabilization of 1-D inhomogeneous quasi-linear hyperbolic systems[END_REF]Theorem A.1]. The regularity of the kernels is a consequence of the regularity of the coupling terms.

360 Differentiating [START_REF] Espitia | Boundary time-varying feedbacks for fixed-time stabilization of constantparameter reaction-diffusion systems[END_REF] with respect to time and space, and injecting therein the dynamics ( 24)-( 26), we can show that the state γ = [γ + , γ -] satisfies

∂γ + ∂t (t, z) + Λ + ∂γ + ∂z (t, z) = Γ + (z)γ -(t, 0), ( 38
)
∂γ - ∂t (t, z) -Λ -∂γ - ∂z (t, z) = Γ -(z)γ -(t, 0), (39)
with the boundary conditions

γ + (t, 0) = Q 0 γ -(t, 0), (40)
γ -(t, 1) = R 1 γ + (t, 1) + u(t) + I 1 (t). ( 41
)
The integral term I 1 (t) is given by

I 1 (t) = 1 0 (L -+ (1, y) -R 1 L ++ (1, y))γ + (t, y) (42)
+ (L --(1, y) -R 1 L +-(1, y))γ -(t, y)dy,
while the coupling matrix-valued functions are defined by

Γ + ij (z) = 0, for j ≤ i if Q0 invertible K +-(z, 0)Λ --K ++ (z, 0)Λ + Q0 ij , else, (43)
Γ - ij (z) = 0, if j ≤ i, K --(z, 0)Λ --K -+ (z, 0)Λ + Q0 ij , else. (44)
Note that the matrix Γ -is strictly lower triangular. The advantage of the system (38)-( 41) relies on the triangular structure of the matrix Γ -, as it creates a cascade structure in the system.

Remark 1. In most contributions dealing with the back-365 stepping stabilization of + m systems, the boundary conditions for the kernels K ++ ij (equation [START_REF] Redaud | In-domain damping assignment of a Timoshenko beam using state feedback boundary control[END_REF]) are usually set to zero or to arbitrary values [START_REF] Hu | Boundary exponential stabilization of 1-D inhomogeneous quasi-linear hyperbolic systems[END_REF]. However, having Q 0 invertible and choosing the kernel boundary condition as in equation [START_REF] Redaud | In-domain damping assignment of a Timoshenko beam using state feedback boundary control[END_REF] effectively allows for simplifying the target 370 system as it induces a triangular structure for the matrix Γ + . This type of kernel boundary condition is analogous to what has been done in [START_REF] Coron | Local exponential H 2 stabilization of a 2×2 quasilinear hyperbolic system using backstepping[END_REF] in the case of two equations.

Second Volterra Integral transform: simplify the ξsystem 375

Similarly to what we have done for the ξ-system, we use a Volterra integral transform of the second kind to map the target system ( 27)-( 29) to a simpler system (46)-(49). More precisely, for all t > 0 and all z ∈ [0, 1], we define the state γ as

γ(t, z) = ξ(t, z) - z 0 K(z, y) ξ(t, y)dy. ( 45
)
where the kernels K = K++ K+-K-+ K--belong to the space C 1 pw (T -; R ( +m)×( +m ) satisfy analogous equations to the ones given by ( 31)-( 33) (except that the terms σ •• are replaced by functions σ•• in equations ( 31)-( 32),( 35)). The transformation ( 45) is boundedly invertible, and the kernels of the associated inverse transform are denoted L. Differentiating [START_REF] Di Meglio | Stabilization of coupled linear heterodirectional hyperbolic PDE-ODE systems[END_REF] with respect to time and space, and since [ ξ+ , ξ-] ∈ C 1 ([0, +∞); H 1 ([0, 1], R ×m )) is the unique solution of ( 27)-( 29), we can show that the new state γ = [γ + , γ-] satisfies

∂γ + ∂t (t, z) + Λ + ∂γ + ∂z (t, z) = Γ+ (z)γ -(t, 0), ( 46
)
∂γ - ∂t (t, z) -Λ -∂γ - ∂z (t, z) = Γ-(z)γ -(t, 0), (47)
with boundary conditions

γ+ (t, 0) = Q 0 γ-(t, 0), (48)
γ-(t, 1) = R1 γ+ (t, 1) + I 2 (t).

(49)

The integral term I 2 (t) given by

I 2 (t) = 1 0 ( L-+ (1, y) -R1 L++ (1, y))γ + (t, y) + ( L--(1, y) -R1 L+-(1, y))γ -(t, y)dy. The matrix Γ+ ∈ C 1 pw ([0, 1], R ×m ) and the matrix Γ-∈ C 1 pw ([0, 1], R m×m ) are defined by Γ+ ij (z) = 0, for j ≤ i, if Q0 invertible K+-(z, 0)Λ --K++ (z, 0)Λ + Q0 ij , else, (50)
Γ- ij (z) = 0, if j ≤ i, K--(z, 0)Λ --K-+ (z, 0)Λ + Q0 ij , else. ( 51
)
Note that the matrix Γis also strictly lower triangular.

Time-space affine change of variable

In this section, we now aim at mapping the system ( 38)-(41) to the system (46)-(49). Define for all z ∈ [0, 1], t ≥ t * .

= mτ > 0, the time-affine change of variables by

γ- i (t, z) = γ - i (t, z) + 1-z µ i 0   i-1 j=1 F - ij (z, y)γ - j (t -y, 0) + i-1 j=2 H - ij (z, y)γ - j (t -y, 0)   dy, 1 ≤ i ≤ m, (52)
γ+ i (t, z) = γ + i (t, z) + z λ i 0   m j=1 F + ij (z, y)γ - j (t -y, 0) + m j=2 H + ij (z, y)γ - j (t -y, 0)dy + 1 µm + z λ i z λ i m-1 j=1 M + ij (z, y)γ - j (t -y, 0) + m-1 j=2 N + ij (z, y)γ - j (t -y, 0)   dy. 1 ≤ i ≤ , (53)
Notice that this transformation requires past values of the boundary state γ(., 0). Therefore, it is only defined for t > t * so we can guarantee that the terms (t -y) that appear in the different integrals are always positive. Also note that the first component of the leftward convecting state is not modified: ∀z ∈ [0, 1], γ-

1 (t, z) = γ - 1 (t, z). For all 1 ≤ i ≤ , 1 ≤ j ≤ m (resp. 2 ≤ j ≤ m), F + ij , (resp. H + ij ) is a real-valued function in C 1 pw (T + λi ; R), and for all 1 ≤ i ≤ , 1 ≤ j ≤ m -1 (resp. 2 ≤ j ≤ m -1), M + ij (resp. N + ij ) is a real-valued function defined in C 1 pw (P + 1 µm ,λi ; R). For all 1 ≤ i ≤ m, 1 ≤ j ≤ i -1 (resp. 2 ≤ j ≤ i -1), the function F - ij (resp. H - ij ) is defined in C 1 pw (T - µi ; R).
The different kernels are defined as follows

for 1 ≤ i ≤ , ∀(z, y) ∈ T + λi , (54)
F + ij (z, y) = δ j1 Γ+ i1 (z -λ i y) -Γ + ij (z -λ i y), (1 ≤ j ≤ m), H + ij (z, y) = Γ+ ij (z -λ i y), (2 ≤ j ≤ m), for 2 ≤ i ≤ m, ∀(z, y) ∈ T - µi , (55)
F - ij (z, y) = δ j1 Γ- i1 (z + µ i y) -Γ - ij (z + µ i y), (1 ≤ j ≤ i -1), H - ij (z, y) = Γ- ij (z + µ i y), (2 ≤ j ≤ i -1), for 1 ≤ i ≤ , ∀(z, y) ∈ P + 1 µm ,λi , 1 ≤ j ≤ m -1 (56) M + ij (z, y) = m k=j+1 1 [ z λ i , 1 µ k + z λ i ] (y)(Q 0 ) ik F -(0, y - z λ i ), N + ij (z, y) = m k=j+1 1 [ z λ i , 1 µ k + z λ i ] (y)(Q 0 ) ik H -(0, y - z λ i ).
The right-hand side of the proposed time-affine transformation (52)-( 53) depends on γ-. However, due to the strict triangular structure of the transformation, the states 380 γi and γ+ i are properly defined. With straightforward but tedious computations, we could get rid of the γ(., 0) dependency in (52)-(53). More precisely, we have the following lemma Lemma 2. For all i ≤ m, there exist piecewise continuous functions

F - ij (1 ≤ j < i), M - ij (1 ≤ j ≤ i -2) respec- tively defined on T - µi and P - bij ,µi (with b ij = i-1 k=j+1 1 µ k ); and F + ij (1 ≤ j ≤ m), M + ij , (1 ≤ j ≤ m -1)
, for i ≤ , respectively defined on T + λi and P + τ,λi , such that the transformation (52)-( 53) can be rewritten as

γ- i (t, z) = γ - i (t, z) + 1-z µ i 0 i-1 j=1 F - ij (z, y)γ - j (t -y, 0)dy + i-2 j=1 bij + 1-z µ i 1-z µ i M - ij (z, y)γ - j (t -y, 0)dy, ( 57
)
γ+ i (t, z) = γ + i (t, z) + z λ i 0 m j=1 F + ij (z, y)γ - j (t -y, 0)dy + m-1 j=1 Aj + z λ i z λ i M + ij (z, y)γ - j (t -y, 0)dy, ( 58
)
where

A j = 1 µm + m k=j+1 1 µ k .
Proof. The proof is straightforward and is omitted.

Differentiating ( 52)-( 53) with respect to time and space and integrating by parts, we can easily verify that, for t ≥ t * , the transformation ( 52)-( 53) maps the solution of ( 38)-( 41) to the solution of ( 46)-( 49).

Let us now introduce the space χ .

= H 1 ([0, 1], R 2n ) × D τ , where D τ . = D τ1 × D τ2 × ... × D τm .
The delays τ i are defined as τ i = (m -i + 1)τ . We emphasize that although the transformation ( 52)-( 53) can be defined on χ, we actually consider here a specific case, where the function in D τ [t] corresponds to delayed values of (a subpart of) the boundary state at time t. This property will be crucial to show that ( 38)-( 41) and ( 46)-( 49) have equivalent stability properties. The following theorem states the invertibility of the transformation ( 52)-( 53) Theorem 1. The transformation (52)-( 53) is boundedly invertible on χ.

Proof. The invertibility of the transformation ( 52)-( 53) can be straightforwardly shown by induction componentwise. Consider the proposition P i , defined for 1 ≤ i ≤ m by P i : " For all t ∈ [mτ, +∞), γi (t, .) can be expressed as a causal function of γ - j (t, .) ∈ H 1 ([0, 1], R l+m ), γ - j (., 0) ∈ D τj (1 ≤ j ≤ i) and γj (., 0) ∈ D τ j (1 ≤ j < i)." Since γ - 1 (t, z) = γ-1 (t, z), proposition P 1 is true. Next, assume P j is satisfied for j < i, and let us prove that P i is true. By induction, the terms in (52) are well defined. In particular, we have

γ- i (t, 0) = γ - i (t, 0) known on [t -τ i , t] + 1 µ i 0 i-1 j=1 F - ij (0, y)γ - j (t -y, 0) + i-1 j=2 H - ij (0, y)γ - j (t -y, 0) known on [t -τ j + 1 µi , t] ⊂ [t -τ i , t] dy,
such that P i is satisfied. The inverse transform is immediately given by 53) is a well-defined affine transform.

γ - i (t, z) = γ- i (t, z) + 1-z µ i 0 i-1 j=1 F - ij (z, y)γ - j (t -y, 0) + i-1 j=2 H- ij (z, y)γ - j (t -y, 0)dy, ( 59
) with H- ij (z, y) = -F - ij (z, y) and F - ij (z, y) = -δ j1 F - i1 (z, y)- δ j>1 H - ij (z, y). Next, the transform (
Since γ-can be expressed as a function of γ -, its invertibility is straightforward.

Stabilizing control law

We can finally design the adequate control input u(t) such that ( 24)-( 26) and ( 27)-( 29) share equivalent stability properties. For all t > 2t , consider the control law

u(t) = R1 γ+ (t, 1) -R 1 γ + (t, 1) -I 1 (t) + I 2 (t). ( 60
)
By convention, we set u(t) = 0 if t ≤ 2t . We choose the bound 2t to avoid useless case distinctions in the different 410 proofs (the bound t is necessary to define the time-affine transformation ( 52)-(53) properly). It is possible to express the control law (60) as a function of the original state ξ using the different transforms. Let us now prove that the closed-loop system ξ and the target system ξ have 415 equivalent asymptotic stability properties. We first have the following lemma Lemma 3. There exists two constants κ 0 and κ 1 , such that for any t > t * , and 0 < r < min( 1 µ1 ,

1 2m Γ -2 ∞ ), κ0 γ - [t+r] (•, 0) r ≤ γ -(t, .) L 2 ≤ κ1 γ - [t+ 1 µm ] (•, 0) 1 µm . (61)
The exponential stability of γ -(t, •) in the sense of the L 2norm is equivalent to the exponential stability of γ - [t] (., 0) in the sense of the D 1 µm -norm.

420

Proof. The proof of this lemma is inspired by [START_REF] Auriol | An explicit mapping from linear first order hyperbolic PDEs to difference systems[END_REF]. The right-hand side of inequality (61) can be obtained by rewriting γ -(t, z) as future values of γ -(•, 0). In what follows, C will be an overloaded constant. Using the methods of characteristics, we have, for t > t * ,

γ -(t, .) 2 L 2 = 1 0 m i=1 γ - i (t, ν) 2 dν by definition = m i=1 1 0 (γ - i (t + ν µ i , 0) + ν µ i 0 i-1 k=1 Γ - ik (ν -µ i s)γ - k (t + s, 0)ds) 2 dν ≤ 2( m i=1 1 0 γ -2 i (t + ν µ i , 0)dν + m i=1 1 0 ν µ i 0 i-1 k=1 Γ -2 ik (ν -µ i s)γ -2 k (t + s, 0)dsdν) ≤ 2(µ 1 1 µm 0 m i=1 γ -2 i (t + s, 0)ds + m 1 µ i 0 m k=1 Γ -2 ∞ γ -2 k (t + s, 0)ds) ≤ C γ - [t+ 1 µm ] (., 0) 2 1 µm
using a time translation.

Similarly, we can obtain the left-hand side of inequality (61)

r 0 m i=1 γ - i (t + s, 0) 2 ds = r 0 m i=1 (γ - i (t, µ i s) - s 0 i-1 k=1 Γ - ik (µ i (s -ν))γ - k (t + ν, 0)dν) 2 ds ≤ 2 m i=1 µir 0 1 µ i γ -2 i (t, s)ds + m i=1 r 0 s 0 Γ -2 ∞ i-1<m k=1 γ -2 k (t + ν, 0)dνds ≤ 2 m i=1 1 0 1 µ i γ -2 i (t, s)ds + m i=1 r 0 r 0 Γ -2 ∞ i-1<m k=1 γ -2 k (t + ν, 0)dνds ≤ 2 µ m γ -(t, .) 2 L 2 + 2mr Γ -2 ∞ γ - [t+r] (., 0) 2 r =⇒ 0 < µ m 2 (1 -2mr Γ -2 ∞ ) γ - [t+r] (., 0) 2 r ≤ γ -(t, .) 2 L 2 since r < 1 2m Γ -2 ∞
Let us now show that the exponential stability of γ -(t, •) in the sense of the L 2 -norm is equivalent to the exponential stability of γ -

[t] (., 0) in the sense of the D 1 µm -norm. Let us consider first that γ -(., 0) is exponentially stable in the sense of the D 1 µm -norm. By definition, for any η > 0, there exists C 0 > 0, ν > 0 such that for all t > max{t * , η},

γ - [t] (., 0) 1 µm ≤ C 0 e -νt γ - [η] (., 0) 1 µm . Then, for all t > t * , γ -(t, .) L 2 ≤ κ 1 γ - [t+ 1 µm ] (•, 0) 1 µm by (61) ≤ C 0 e -ν µm κ 1 e -νt γ - [η] (., 0) 1 µm .
We can decompose the term γ - [η] (., 0) 1 µm using a finite number of terms defined on intervals of length r (where r is defined in the statement Lemma 3). Define n r = max k∈N (kr ≤ 1 µm ). We have

γ - [η] (., 0) 2 1 µm = m i=1 0 -1 µm γ -2 i (η + s, 0)ds ≤ m i=1 nr k=0 -kr -(k+1)r γ -2 i (η + s, 0)ds = m i=1 nr k=0 0 -r γ -2 i (η + s -kr, 0)ds = nr k=0 γ - [η-kr] (., 0) 2 r ≤ nr k=0 1 κ 2 0 γ -(η -(k + 1)r, .) 2 L 2 (62)
Choosing η such that n r + 1 ≤ η < 2t and since the system (38)-( 41) is well-posed, there exists κ > 0, such

that γ -(η -(k + 1)r, .) L 2 ≤ κ γ - 0 L 2 . We finally obtain γ -(t, .) L 2 ≤ C 0 κ 1 κ 2 0 κ(n r + 1)e -ν µm e -νt γ - 0 2 L 2 .
This implies the exponential stability of γ -in the sense of the L 2 -norm. We can use the same arguments to prove that the exponential stability of γ -in the sense of the L 2 -norm implies the exponential stability of γ -(., 0) in the sense of the D 1 µm -norm.

425

We can finally show the equivalence between the original system ξ and the target system ξ Theorem 2. Let choose the functions σ±± and σ∓± such that for any initial conditions ( ξ+

0 (z), ξ- 0 (z)) ∈ H 1 ([0, 1], R ( +m)
) satisfying the appropriate compatibility conditions, 430 the solutions of ( 27)-( 29) are exponentially stable in the sense of the L 2 -norm. Then, for any initial conditions

(ξ + 0 (z), ξ - 0 (z)) ∈ H 1 ([0, 1], R (l+m)
) satisfying the appropriate compatibility conditions, the solutions of ( 24)-( 26) with the control input defined by (60) are also exponentially 435 stable in the sense of the L 2 -norm.

Proof. We know that the L 2 -exponential stability of the system (38)-( 41) is equivalent to the L 2 -exponential stability of the system ( 24)-( 26) [START_REF] Hu | Boundary exponential stabilization of 1-D inhomogeneous quasi-linear hyperbolic systems[END_REF]. Similarly, we know that the L 2 -exponential stability of the system ( 46)-( 49) is equivalent to the L 2 -exponential stability of the system ( 27)-( 29). Therefore we only need to show that the L 2 -exponential stability of the system (38)-( 41) is equivalent to the L 2 -exponential stability of the system (46)-(49). Let us assume first that the system (38)-( 41) is L 2exponentially stable. Consequently, there exists C 0 , ν > 0, such that for all t ≥ 2t ,

(γ + (t, .), γ -(t, .)) L 2 ≤ C 0 e -νt (γ + (η, .), γ -(η, .)) L 2 ,
where η is such that n r + 1 ≤ η < 2t . Using transformation (57), we have for all t > 2t

1 0 m i=1 γ-2 i (t, s)ds = 1 0 m i=1   γ - i (t, s) + 1-s µ i 0 i-1 j=1 F - ij (s, y)γ - j (t -y, 0)dy + i-2 j=1 bij + 1-s µ i 1-s µ i M - ij (s, y)γ - j (t -y, 0)dy   2 ds, ≤ C( 1 0 m i=1 γ -2 i (t, s)ds + m F -2 ∞ γ - [t] (., 0) 2 1 µm + m M -2 ∞ γ - [t] (., 0) 2 m µm ), ( 63
)
where C is a constant. Using inequality (62), there exists

k 1 > 0 such that γ - [t] (., 0) 1 µm ≤ k 1 e -νt (γ + (η, .), γ -(η, .)) L 2 .
Similarly, we can show that there exists k 2 > 0 such that γ - [t] (., 0) m µm ≤ k 2 e -νt (γ + (η, .), γ -(η, .)) L 2 . Injecting in equation ( 63), there exists a constant C 1 > 0 such that for all t > 2t , we have γ-(t, .) L 2 ≤ C 1 e -νt (γ + (η, .), γ -(η, .)) L 2 .

Similarly, using transformation (53), we can show the existence of a constant C 2 such that for all t > 0 γ+ (t, .)

L 2 ≤ C 2 e -νt (γ + (η, .), γ -(η, .)) L 2 .
We now need to bound (γ + (η, .), γ -(η, .)) L 2 by a term that depends on (γ + (0, .), γ-(0, .)) L 2 . Due to transformation (59), we have

1 0 m i=1 γ -2 i (η, s)ds ≤ C 1 0 m i=1 γ-2 i (η, s)ds + m F -2 ∞ γ- [η] (., 0) 2 1 µm + m M -2 ∞ γ- [η] (., 0) 2 m µm . ( 64
)
Therefore, adjusting inequality (62) and using the wellposedness of the system ( 46)-( 49), we obtain

(γ -(η, .)) L 2 ≤ C 3 (γ + (0, .), γ-(0, .)) L 2 ,
where C 3 is a positive constant. Similar computations give

(γ + (η, .)) L 2 ≤ C 4 (γ + (0, .), γ-(0, .)) L 2 ,
where C 4 is a positive constant. Consequently, there exists C 5 > 0 such that for all t > 2t (γ + (t, .), γ-(t, .)) L 2 ≤ C 5 e -νt (γ + (0, .), γ-(0, .)) L 2 .

This last inequality still holds when t ≤ 2t due to the well-posedness of the system ( 46)-(49). Consequently, the system ( 46)-( 49) is exponentially stable. The converse can be easily proved using the inverse transformation (59).

Remark 2. The control law requires the knowledge of distributed values of the state ξ. It is, therefore, necessary to design a state observer that can be coupled with the proposed state feedback controller to obtain an output-feedback control law. Well-posed backstepping observers have already been designed in the literature for hyperbolic systems (see [START_REF] Auriol | Minimum time control of heterodirectional linear coupled hyperbolic PDEs[END_REF][START_REF] Auriol | Robust output feedback stabilization for two heterodirectional linear coupled hyperbolic PDEs[END_REF] for instance). The resulting closed-loop system would be well-posed, as the control input would remain continuous. The closed-loop exponential stability would be guaranteed due to the input-to-state stability of the original linear hyperbolic system [START_REF] Lamare | Robust output regulation of 2 × 2 hyperbolic systems: Control law and input-to-state stability[END_REF]. Obviously, the observer proposed in [START_REF] Auriol | Minimum time control of heterodirectional linear coupled hyperbolic PDEs[END_REF] is not the only possible observer, and we could add degrees of freedom while designing the observer, adjusting the approach presented in the current paper.

Remark 3. The proposed time-affine transformation (52)-( 53) is only defined for t > t * = mτ . This time t corresponds to propagation of the "information" at the boundary. Since the control input is set to zero for t ≤ 2t , this may deteriorate the transient properties of the system. Following [START_REF] Auriol | Minimum time control of heterodirectional linear coupled hyperbolic PDEs[END_REF], the delay t may be reduced to a smaller value 460 such as max{ 1 µi } = 1 µm , to the price of a more intricated transformation.

Remark 4. So far, we have not discussed the robustness properties of the proposed control law (60). It has been

465

shown in [START_REF] Auriol | Delayrobust control design for two heterodirectional linear coupled hyperbolic PDEs[END_REF] that canceling the boundary coupling terms in equation (60) could lead to vanishing robustness margins. Thus, depending on the matrices R 1 and R1 , the closedloop system may not be robust to parameter uncertainties or input delays. Although the complete robustness analysis 470 of the closed-loop system is out of the scope of the paper, we can mention several methods to guarantee the existence of robustness margins:

• Choose the desired target system such that the conditions of [START_REF] Auriol | An explicit mapping from linear first order hyperbolic PDEs to difference systems[END_REF]Theorem 5] are satisfied. This means 475 that we can only cancel a small part of the boundary reflection terms (in the Riemann coordinates). However, this condition needs to be expressed in the PHS framework.

• Low-pass filter the control law with an appropriate fil-480 ter, as shown in [START_REF] Auriol | Robustification of stabilizing controllers for ODE-PDE-ODE systems: a filtering approach[END_REF]. With this solution, the closedloop performance may not be exactly the same as the ones of the desired target system (at least for the high frequencies), but the control transfer function will be strictly proper, thus guaranteeing the existence of ro-485 bustness margins [START_REF] Auriol | Robustification of stabilizing controllers for ODE-PDE-ODE systems: a filtering approach[END_REF][START_REF] Curtain | An introduction to infinite-dimensional linear systems theory[END_REF].

Application to the distributed damping assignment for a wave equation

In this section, we apply our strategy to perform distributed damping assignment for a clamped string (wave 490 equation). The results we give below are inspired by [START_REF] Redaud | Distributed damping assignment for a wave equation in the port-Hamiltonian framework[END_REF]. We consider the vibrating string introduced in Section 2.1.3 and Section 2.2.3 and its expression in the Riemann coordinates.

Control design 495

We can apply the methodology presented in Section 3 to design a stabilizing control law that guarantees the desired damping assignment. Note that in this low-dimensional case ( = m = 1), there is no need for the time-affine transformation (52)-(53), and the approach given in Section 3 is a bit superfluous, though the idea behind remains the same. In particular, we have t = 0. Indeed, we can directly map the ξ-system (24)-(26) to the system ( 22)-( 23) using the backstepping transformation

ξ+ ξ-= ξ + ξ -- z 0 K ++ K +- K -+ K --(z, y) ξ + ξ -(y)dy,
where the kernels K ±∓ are uniquely defined on T -as the solution of the following set of equations

λ(z) ∂K ++ ∂z + ∂λ(y)K ++ ∂y = σ+-(z)K -+ -σ -+ (y)K +-, λ(z) ∂K +- ∂z - ∂λ(y)K +- ∂y = σ+-(z)K ---σ +-(y)K ++ , λ(z) ∂K -+ ∂z - ∂λ(y)K -+ ∂y = σ -+ (y)K ---σ-+ (z)K ++ , λ(z) ∂K -- ∂z + ∂λ(y)K -- ∂y = σ +-(y)K -+ -σ-+ (z)K +-,
with boundary conditions

K +-(z, z) = σ +--σ+- 2λ(z) , K ++ (z, 0) = q -1 0 K +-(z, 0), K -+ (z, z) = σ-+ -σ -+ 2λ(z) , K --(z, 0) = q 0 K -+ (z, 0).
The well-posedness of the kernel equations is proved in [START_REF] Di Meglio | Stabilization of coupled linear heterodirectional hyperbolic PDE-ODE systems[END_REF]. The set of above equations admits a unique piecewise continuous solution on T -. The control input is then given by

u 0 (t) = λ(1) √ 2g(1)e -Ic(1) (a 1 -r 1 )ξ + (1, t) (65)
+ 1 0 (K -+ (1, y) -a 1 K ++ (1, y))ξ + (y) + (K --(1, y) -a 1 K +-(1, y))ξ -(y)dy .
Remark 5. In this specific low-dimensional case, we could have directly mapped the PHS system (14) to the desired target system x using the transformation

x1 x2 = C(z) x 1 x 2 - z 0 N ++ N +- N -+ N --(z, y) x 1 x 2 (y)dy.
with, for all z ∈ [0, 1], I(z) = I c (z) -I K (z) and

C(z) = cosh (I(z)) -1 γ(x) sinh (I(z)) -γ(z) sinh (I(z))
cosh (I(z)) .

Simulation results

In this section, we illustrate the performances of the proposed control approach with Matlab simulations. We consider a soft PVC string of length 1m, with constant physical parameters ρ = 1.35 × 10 3 kg.m -3 , E = 0.9 GPa. Its initial position is w 0 (z) = 0.1 sin( 2x π ), and no speed. We simulated system ( 24)-( 26) on a time horizon of 20s using a Godunov Scheme [START_REF] Leveque | Finite volume methods for hyperbolic problems[END_REF] (with a Courant-Friedrichs-Lewy condition CF L = 0.99). The space domain [0, 1] is discretized with a mesh of 200 points. Beforehand, the ker-505 nels K ±∓ are computed offline using a fixed-point algorithm. The control input is computed at each time step using (65). We consider a case where the string is naturally slightly damped (c = 0.1). We want to artificially assign

Application to the distributed damping assignment for a Timoshenko Beam

In this section, we apply our strategy to perform distributed damping assignment for a Timoshenko beam. This model is usually used to represent compliant mechanical structures such as cantilevers or flexible endoscopes [START_REF] Wu | Optimal actuator location for electro-active polymer actuated endoscope[END_REF].

It takes into account shear deformation and rotational inertia of the structure. This second test case is more generic than the one presented in Section 4, since the system under consideration is of larger dimension. The results from this section are inspired by [START_REF] Redaud | In-domain damping assignment of a Timoshenko beam using state feedback boundary control[END_REF].

Reformulation in the Port-Hamiltonian framework

We consider the clamped actuated Timoshenko beam model proposed in [START_REF] Timoshenko | Vibration problems in engineering[END_REF]. We denote w(t, z) (resp. φ(t, z)) the transverse normalized displacement (resp. rotation angle) defined on [0, ∞) × [0, 1]. From the balance equations on the momenta, they satisfy

ρ ∂ 2 w ∂t 2 (t, z) = ∂ ∂z K s ∂w ∂z (t, z) -φ(t, z) , (66)
I ρ ∂ 2 φ ∂t 2 (t, z) = ∂ ∂z EI ∂φ ∂z + K s ∂w ∂z -φ . (67)
For the sake of simplicity, we assume that all physical parameters (mass per length unit ρ, rotary moment of inertia of a cross-section I ρ , Young's modulus of elasticity E, moment of inertia I and shear modulus K s ) are constant. The initial position of the beam is given by w

(z, 0) = w 0 (z) ∈ C 1 ([0, 1]), φ(z, 0) = φ 0 (z) ∈ C 1 ([0, 1]).
Following the Port-Hamiltonian framework, we define the energy state variables

x = [x 1 , x 2 ] by x 1 (t, z) = ∂w ∂x (t, z) -φ(t, z) ∂φ ∂x (t, z) , x 2 (t, z) = ρ ∂w ∂t (t, z) I ρ ∂φ ∂t (t, z)
, where x 1 (t, z) represents shear and angular displacements, while x 2 (t, z) represents momentum and angular momentum. The state x belongs to χ 2+2 = χ 4 . The Hamiltonian density matrix H is given by

H = diag K s , EI, 1 ρ , 1 I ρ ∈ D + 4 . ( 68
)
No movement is allowed at the clamped end, while the opposite end is fully actuated, such that

x 2 (t, 0) = 0 R 2 , K s 0 0 EI x 1 (t, 1) = u 0 (t) ∈ R 2 , ( 69
)
where u 0 (t) is the control input. The system (66)-( 69) fits in the general framework presented in Section 2 and verifies Assumption 1. As in Section 4, our objective is to impose a specific decay rate to the energy of the system E, using a distributed damping assignment. The target system is defined by equations ( 16)-( 17) where P0 = P 0 and Π0 is given by Π0 = diag(0, 0, c 3 ρ, c 4 I ρ ).

More precisely, we want the closed-loop system to behave as if there were arbitrary distributed damping terms c 3 , c 4 > 0. In that case, due to equation ( 18), the energy of the system decays as

dE dt = - 1 0 [ c 3 ρ , c 4 I ρ ]x 2 2 (t, z) dz. (70)

Riemann invariants

To apply the results of Section 3, we need to rewrite the system (66) in the Riemann coordinates. In what follows, we define the transport velocities λ = Ks ρ > 0, µ = EI Iρ > 0 and the matrices Λ + = Λ -.

= Λ = diag(λ, µ).

We assume (without any loss of generality) that λ > µ. We also define R = diag( λ Ks , 1 µIρ ) ∈ D + 2 , and α . = c3 2λ , β . = c4 2µ . The matrix P 1 H ∈ R 4×4 is invertible with four distinct real eigenvalues {±λ, ±µ}. It is diagonalizable, such that we have P 1 H = Qdiag(-Λ, Λ)Q -1 with

Q = 1 √ 2 -R I 2 I 2 R -1 ∈ R 4×4 . ( 71
)
Original system Let us define the new state ξ = Q -1 x, decomposed into ξ = [ξ + , ξ -] ∈ χ 4 . It verifies system ( 24)-( 26 The control input is given by u(t) = √ 2u 0 (t).

beam with physical parameters given in Table 1. It is initially at rest at a position w 0 (z) = 0.1 sin( π 2 z), φ 0 (z) = 0. Using the above control strategy, we want the closed-loop beam to behave as ( 16)-( 17) with c 3 = 0.5, c 4 = 0.8SI.

We simulate system ( 24)-( 26) on the time interval [0, 15]s 555 using a Godunov Scheme [START_REF] Leveque | Finite volume methods for hyperbolic problems[END_REF] (CF L = 1 and dx = 0.02m The evolution of the control effort is represented in Figure 7. Note that here it only equals zero for t < t . After the control input is applied, the position of the closed-loop system converges quickly to an equilibrium. As illustrated in Figure 8 (blue), its energy decreases at the same rate as 565 the target system (dotted red). Since the control law (72) requires the computation of state γ, γ at each time step, it is, therefore, more computationally expensive than traditional PI controllers. However, we believe that our approach could yield potential interesting trade-offs between 570 performance and control effort. This will be the purpose of future research works.

Conclusion

In this paper, we developed a systematic framework to stabilize a general class of hyperbolic systems while as-575 signing a specified closed-loop behavior with a clear energy interpretation. We took advantage of the PHS framework to emphasize the system's natural physical properties and Hamiltonian system in the Riemann coordinates. Then, we combined classical Volterra transformations with an innovative time-affine transform to map the original system to the desired target system. We finally illustrated the proposed approach on two application cases: clamped string with space-varying coefficients and a clamped Timoshenko beam. Interestingly, a by-product of this paper is to give a general class of reachable target systems for balance laws equations. Introducing degrees of freedom in the control design to obtain modular controllers is crucial to 595 guarantee potential trade-offs between different specifications (namely delay-robustness and convergence rate). For instance, preserving naturally dissipative terms while imposing a specific energy decay should allow a reduced control effort compared to traditional approaches. If the qual-600 itative effect of the tuning parameters can be understood in the PHS framework, we believe it is crucial to assess and quantify the performance of the resulting controllers with respect to a given set of specifications. This set of performance criteria should be defined in terms of prac-605 tically 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. We also aim to generalize our approach to more 610 complex interconnected systems.

2. 1 .

 1 Original system(s) under consideration 2.1.1. Hyperbolic system in the PHS framework

245 2 . 2 . 3 .

 223 Example of a clamped-string (wave equation) Consider the wave equation example introduced in equation

Figure 1 :

 1 Figure 1: Schematic representation of an hyperbolic PDE system

330a

  simpler target system γ for which most of the indomain coupling terms have been moved at the actuated boundary; 2. Similarly, in Section 3.4, we use a second classical backstepping Volterra transform K of the same form, 335to map the target system ξ to the system γ, for which most of the in-domain coupling terms have been moved at the boundary z = 1.3. Finally, in Section 3.5, we use a specific invertible time-affine transform F to map the system γ to the 340 system γ. This transformation is defined after a specific critical transport time t * .Composing the different transforms, it becomes straightforward to design the corresponding feedback law in Section 3.6 and map the system ξ to the system ξ. Although 345 it should be possible to write a single transformation encompassing the three transformations we propose, the resulting kernel equations become highly involved, and we did not manage to prove the existence of such a transformation in the general case. Introducing two intermediate 350 systems (38)-(41) and (46)-(49) helped us reaching the ultimate control objective stated in the previous section. A schematic representation of the control strategy is given in Figure 2.

Figure 2 :

 2 Figure 2: Overall strategy

Figure 3 :

 3 Figure 3: 3D-displacement w(t, z) in open-loop (top), and closedloop (bottom) using (65).

Figure 4 :

 4 Figure 4: Evolution of the energy E(t).

  and closed-loop systems in Figure4. With the proposed control input (blue), the energy of the closed-loop system decays faster than the natural open-loop decay (red). It 520 follows the reference energy decay of an open-loop system with in-domain damping K (dotted red). We represented in Figure5the evolution of the full-state feedback (65). As expected, the control effort goes to zero.

Figure 5 :

 5 Figure 5: Evolution of the control effort u(t).

  ) with boundary coupling terms Q 0 = -R -1 and R 1 = R, while the in-domain coupling matrices are given by

As illustrated in Figure 6 ,

 6 the position of the open-loop system oscillates around an equilibrium. Its energy is approximately constant due to the absence of dissipative terms. It is represented in Figure 8 (red).

  560

Figure 6 :

 6 Figure 6: 3D-displacement w(t, z) in open-loop (top) and closed-loop (bottom) using (72).

Figure 7 :Figure 8 :

 78 Figure 7: Evolution of the control effort u ∂ (t)

Table 1 :

 1 ). Numerical values for simulation

	Param.	Description	Value
	E	elastic modulus	1kg m -1 s -2
	I	second moment of area	0.5m 4
	I ρ	rotary moment of inertia	0.9kg m
	K s	shear modulus	1.2kg m s -2
	ρ	linear density	0.9 kg m -1

Target System

We perform the same change of variables to rewrite the target system ( 16)- [START_REF] Anfinsen | Adaptive output feedback stabilization of n+m coupled linear hyperbolic PDEs with uncertain boundary conditions[END_REF] in the Riemann coordinates. We define first ζ = Q -1 X, where ζ = [ ζ+ , ζ-] ∈ χ 4 . This new state verifies system ( 27)- [START_REF] Steeves | Input delay compensation in prescribed-time of boundary-actuated reaction-diffusion PDEs[END_REF], where the coupling terms are defined by σ++ = 1 2

For all z ∈ [0, 1], we define Ā(z) = diag(e αz , e βz , e -αz , e -βz ).

To remove the diagonal terms of σ++ and σ--, we now consider the state ξ = ξ+ , ξ-= Āζ . It verifies system ( 27)-( 29), with the boundary coupling R1 = diag(e -2α , e -2β )R.

The space-dependant in-domain coupling terms are defined by

Ks µIρ e (α-β)z 0 .

Control design

We can now apply the methodology presented in Section 3 to design a stabilizing control law that guarantees the desired damping assignment.

Following the approach presented in Section 3.5, we define the transformation F : χ 4 → χ 4 such that γ = F(γ), by ( 52)-(53). The resulting control input reads

with u(t) defined by (60). It could be rewritten with dis-545 tributed values of the original states x using the different transforms. More details can be found in [START_REF] Redaud | In-domain damping assignment of a Timoshenko beam using state feedback boundary control[END_REF].

Simulation results

We now illustrate the performance of our control strategy with simulation results. We consider a Timoshenko


Inrae Researcher

Co

Pierre-André Garambois

François Colleoni

Benjamin Renard

Hydrology-Hydraulics Numerical Modeling Data Assimilation Stochastic Hydrology Uncertainty Quantification Hydrology Numerical Modeling

• An open source, Python library interfacing the differentiable Fortran Spatially distributed Modelling and ASsimilation for Hydrology

• Providing a variety of user-friendly routines for hydrological modeling, including data preprocessing, high-dimensional optimization tools, sensitivity and signatures analysis, all interfaced with an efficient Fortran solver

• Offering a range of advanced optimization techniques, including Variational Data Assimilation (VDA), Bayesian estimation, and Artificial Neural Network (ANN) approaches, using an adjoint model generated with an automatic differentiation engine

• A gridded mesh and modular design

• Different components such as: snow module, surface interception, production, transfer, percolation functions on each cell

• Different routing models with a cell-to-cell numerical scheme 6 SMASH GR-like conceptual hydrological model structure [START_REF] Perrin | Improvement of a parsimonious model for streamflow simulation[END_REF]

Context

Estimating parameters of spatially distributed hydrological models, requiring spatial constraints given relatively sparse discharge data, in view to model ungauged catchments, and uncertainty correction:

→ New approach (HDA-PR) for learning spatially distributed parameters of a differentiable hydrological model from physical descriptors using multivariate regression and artificial neural network, with adjoint-based gradient and high dimensional optimization algorithms → Application of the method in a multi-gauge calibration setup on flash flood prone area with complex hydrological responses → Uncertainty quantification and correction, learnable hydrological operators, both in the structural modeling and the content of the data

Learnable regionalization

Previous works on regionalization within the calibration process:

Learnable regionalization

• Rainfall-runoff model:

• Full forward model with regionalization mapping:

with a regionalization function for mapping physical descriptors onto unknown conceptual hydrological parameters:

where 𝞺 be the optimizable parameters of the regionalization mapping • Two regionalization mappings are considered: multivariate polynomial regression and multilayer perceptron

Learnable regionalization

• Inverse problem:

• Gradient computation:

The first term can be computed via the automatic differentiation applied to the Fortran code and the second is obtained by analytical calculus applicable given the explicit architecture of the neural network

Results

One year of observed and simulated discharges (in m³/s) at one gauged (Y4615020) and one pseudo-ungauged catchment (Y5215020)

11 The spatio-temporal error of simulated discharges is assumed to follow a Gaussian distribution with learnable parameters such as mean and standard deviation. Then, these two parameters are estimated using a bidirectional Long Short-Term Memory (LSTM) network to improve the model's ability to capture temporal dependencies and spatial patterns in uncertainty modeling

Two

Ongoing works and perspectives

Ongoing works:

• Testing of HDA-PR and uncertainty correction with LSTM at national scales and on other continent • Study of effective descriptor selection along with multi-gauge cost functions explicitly accounting for data uncertainties Perspectives:

• Introduction of learnable production and transfer operators in the forward hydrological model (e.g., neural ordinary differential equations (neural ODEs) in Hoge et al., 2022) • Regionalization of differentiable integrated hydrological-hydraulic networks models [START_REF] Pujol | Multi-dimensional hydrological-hydraulic model with variational data assimilation for river networks and floodplains[END_REF]

  study areas on the map on France: Ardèche (left) andMedEst (right) Radial plots of the NSE (optimal value = 1) in gauged catchments(left) and pseudo-ungauged catchments (right) Calibration period: 2006-2016 Validation period: 2016-2018ResultsRelative error (optimal value = 0) of 4 flood event signatures: base flow (Ebf), flood flow (Eff), runoff coefficient (Erc) and peak flow (Epf), evaluated using 49 flood events at gauged catchments for temporal validation(Temp Val) and 19 flood events at pseudo-ungauged catchments for spatio-temporal validation (Spatio-Temp Val) Sub-figure a: Maps of input descriptors. Sub-figure b: Calibrated hydrological parameters maps for three non-uniform regionalization methods. Sub-figure c: Linear correlation between descriptor and parameter for the three regionalization methods 12 Flood performances (Huynh et al., 2023) Stochastic process for uncertainty correction Recent works on streamflow correction with LSTM: Cho & Kim (2022); Hashemi et al. (2023) A proposed stochastic process to correct simulation responses addresses: • Observation error • Structure modeling error where:

THANK YOU

Contact: ngo-nghi-truyen.huynh@inrae.fr or pierre-andre.Garambois@inrae.fr


Emilie Rouzies

Claire Lauvernet

Arthur Vidard

Multi-sources data assimilation in a spatialized model of water and pesticide transfers

Development of the PESHMELBA model

We have a dream that one day PESHMELBA will be used as a decision-making tool to set up management scenarios and to identify an optimal landscape configuration for pesticide transfer mitigation.

We have a dream that one day PESHMELBA will be used as a decision-making tool to set up management scenarios and to identify an optimal landscape configuration for pesticide transfer mitigation. This is our objective...but before, it is necessary to quantify and reduce the uncertainty associated to PESHMELBA output variables.

We have a dream that one day PESHMELBA will be used as a decision-making tool to set up management scenarios and to identify an optimal landscape configuration for pesticide transfer mitigation. This is our objective...but before, it is necessary to quantify and reduce the uncertainty associated to PESHMELBA output variables. 1. Use PESHMELBA to generate a "True" reference simulation

Add perturbation to generate virtual observations

Emilie Rouzies JMSC 2022 8 / 14

Obj 1: assimilate satellite soil surface moisture images to correct PESHMELBA moisture profile and input parameters.

  Development of the PESHMELBA model (Rouzies et al. 2019) to simulate pesticide transfers and fate on small agricultural catchments Simulations of heterogenous landscapes composed of plots, vegetative filter zones, hedges, ditches and rivers Development of the PESHMELBA model (Rouzies et al. 2019) to simulate pesticide transfers and fate on small agricultural catchments Simulations of heterogenous landscapes composed of plots, vegetative filter zones, hedges, ditches and rivers Modular structure to explore landscape management scenarios Spatialized, process-oriented model : water transfers on surface and subsurface + pesticide advection, adsorption and degradation

PhD objectives 1 .""

 1 Quantify: performing an uncertainty analysis and a sensitivity analysis of the model 2. Reduce: performing data assimilation to integrate different sources of data: soil moisture images, ERT measurements and in-situ data of pesticide concentration First attempt of DA in the PESHMELBA model: let's keep it simple...but realistic! (types of landscape elements, number of parameters, climate conditions...) DA in the PESHMELBA model: let's keep it simple...but realistic! (types of landscape elements, number of parameters, climate conditions...) Surface moisture images got from the synergic use of Sentinel-1 and Sentinel-2 data (El Hajj et al. 2017) One observation of mean moisture in the top 5 cm per landscape element per time step Freq. of observation: 144h, obs. error : assumed Gaussian, std ∼ 0.02 cm 3 cm -3 ( ! may highly differ on vineyard!) In-situ moisture profiles Moisture profiles from EMI measurements or probe. Assumption : 2m-moisture profile on some landscape elements, obs. error : assumed Gaussian, std ∼ 0.02 cm 3 cm -3 "Approximation of the true state of a physical system at a given time by combining time-distributed observations with a dynamic model in an optimal way" Approximation of the true state of a physical system at a given time by combining time-distributed observations with a dynamic model in an optimal way" (Asch et al. 2016) Objectives Improve moisture dynamics modelling both in surface and subsurface Improve estimation of pesticide export at the outlet Estimate input parameters (θ sat ) that would be set for the exploration of landscape Approximation of the true state of a physical system at a given time by combining time-distributed observations with a dynamic model in an optimal way" (Asch et al. 2016) Objectives Improve moisture dynamics modelling both in surface and subsurface Improve estimation of pesticide export at the outlet Estimate input parameters (θ sat ) that would be set for the exploration of landscape management scenarios ⇒ Joint-estimation abilities are investigated

First

  step: twin experiments to set and validate the DA framework ( ! Reanal- ysis context). assimilate satellite soil surface moisture images to correct PESHMELBA moisture profile and input parameters.

First

  step: twin experiments to set and validate the DA framework ( ! Reanal- ysis context).

First

  step: twin experiments to set and validate the DA framework ( ! Reanal- ysis context).

  assimilate satellite soil surface moisture images to correct PESHMELBA moisture profile and input parameters.

First

  step: twin experiments to set and validate the DA framework ( ! Reanal- ysis context). assimilate satellite soil surface moisture images to correct PESHMELBA moisture profile and input parameters.

First

  step: twin experiments to set and validate the DA framework ( ! Reanal- ysis context).

Ensemble Smoother with Multiple Data Assimilation (Emerick and Reynolds 2013)

  Ensemble method that inherits from Kalman Filter Iterative smoother well suited to parameter estimation problems in non linear contexts

	Obj 1: assimilate satellite soil surface moisture images to correct PESHMELBA
	moisture profile and input parameters.
	First step: twin experiments to set and
	validate the DA framework ( ! Reanal-
	ysis context).
	Emilie Rouzies Emilie Rouzies Emilie Rouzies	JMSC 2022 JMSC 2022 JMSC 2022	6 / 14 7 / 14 8 / 14

Obj 1: assimilate satellite soil surface moisture images to correct PESHMELBA moisture profile and input parameters.

Emilie RouziesJMSC 2022

Emilie RouziesJMSC 2022 9 / 14

Data assimilation of satellite moisture images

Good correction of surface moisture and surface parameters Corrections do not propagate to subsurface (lack of correlations between surface and subsurface compartments)