Constantinos Kitsos
email: ckitsos@laas.fr

Gildas Besancon
email: gildas.besancon@grenoble-inp.fr

Christophe Prieur
email: christophe.prieur@grenoble-inp.fr

Besanc Gildas

Christophe ¸on

Prieur

G Besanc

High-Gain Observer Design for a Class of Quasi-linear Integro-Differential Hyperbolic Systems -Application to an Epidemic Model

Keywords: high-gain observers, PIDEs, nonlocal terms, quasilinear hyperbolic systems, Lyapunov analysis, C 1 exponential stability, SIR epidemic models

I. INTRODUCTION

The classical high-gain observer design for finitedimensional nonlinear systems has been extensively studied in the literature and remains widely considered, see [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF] and references therein. In short, it relies on a single tuning coefficient, to be chosen large enough so as to ensure exponential -and possibly arbitrarily fast -convergence. High-gain observers apply to a large class of cases corresponding to uniformly observable systems [START_REF] Gauthier | Observability for any u(t) of a class of nonlinear systems[END_REF], [START_REF] Gauthier | A simple observer for nonlinear systems: applications to bioreactors[END_REF]. In the recent paper [START_REF] Kitsos | High-gain observer design for a class of hyperbolic systems of balance laws[END_REF], this approach was extended to a class of hyperbolic systems, for which first solutions to this "High-Gain Observer Design Problem" (H-GODP) have been proposed for a particular case of uniformly observable systems, written as an n×n quasi-linear hyperbolic system of balance laws and considering distributed measurements. In more recent works [START_REF] Kitsos | A high-gain observer for a class of 2x2 hyperbolic systems with C1 exponential convergence[END_REF], [START_REF] Kitsos | High-gain observers for a class of 2×2 quasilinear hyperbolic systems with 2 different velocities[END_REF] more general cases were considered, where 2 × 2 systems were written in an observable form with two distinct characteristic velocities, differentiating from [START_REF] Kitsos | High-gain observer design for a class of hyperbolic systems of balance laws[END_REF], where it was needed to consider identical ones. There exist other studies on observer design for hyperbolic systems in the literature, mainly considering the full state vector on the boundaries as measurement. Amongst others, one can refer to [START_REF] Di Meglio | Stabilization of a system of n + 1 coupled first-order hyperbolic linear PDEs with a single boundary input[END_REF] and [START_REF] Hasan | Boundary observer design for hyperbolic PDE-ODE cascade systems[END_REF] for the backstepping design, to [START_REF] Besanc ¸on | Sur la commande en dimension finie d'une classe de systèmes non linéaires de dimension infinie[END_REF] for a discretization approach, to [START_REF] Castillo | Boundary observers for linear and quasi-linear hyperbolic systems with application to flow control[END_REF] for direct infinite-dimension-based Lyapunov techniques (see also [START_REF] Besanc ¸on | Robust state estimation for a class of convection-diffusion-reaction systems[END_REF]), or to [START_REF] Nguyen | State and parameter estimation in 1-d hyperbolic PDEs based on an adjoint method[END_REF] for optimization methods. For semigroup-based methods, see [START_REF] Curtain | Finite-dimensional compensator design for parabolic distributed systems with point sensors and boundary input[END_REF], [START_REF] Christofides | Feedback control of hyperbolic pde systems[END_REF] and [START_REF] Schaum | A simple observer scheme for a class of 1-D semi-linear parabolic distributed parameter systems[END_REF].

The present paper aims at providing a solution to the H-GODP for a class of quasi-linear hyperbolic systems of balance laws, including nonlocal terms, i.e., systems of hyperbolic Partial Integro-Differential Equations (PIDEs), as an extension of [START_REF] Kitsos | High-gain observer design for a class of hyperbolic systems of balance laws[END_REF]. The class of systems that we consider, written in an appropriate canonical form, can be found in various cases, like chemical reactors [START_REF] Boskovic | Backstepping control of chemical tubular reactors[END_REF], where, by measuring the temperature of the reactor we would expect to estimate the concentrations of masses of chemicals, or some age-dependent epidemic models, where with the only measurement of the population that has recovered from a disease, the objective would be to estimate the remaining population groups arbitrarily fast. Stability and controllability of age-structured population models have been already studied (see [2, Chapter 1] and references therein, [START_REF] Schmidt | Yield trajectory tracking for hyperbolic age-structured population systems[END_REF] and in general PIDEs or PDEs with nonlocal terms have been considered for instance in [START_REF] Coron | Stabilization and controllability of first-order integro-differential hyperbolic equations[END_REF], [START_REF] Karafyllis | Stability of integral delay equations and stabilization of age-structured models[END_REF], [START_REF] Deutschmann | Backstepping-based boundary observer for a class of time-varying linear hyperbolic PIDEs[END_REF].

In the present approach, we consider systems having only one characteristic velocity, in the spirit of our former work of [START_REF] Kitsos | High-gain observer design for a class of hyperbolic systems of balance laws[END_REF]: this results from a certain commutative property in the proposed Lyapunov stability analysis. Note that a similar constraint can be found in some studies of stability problems as in [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF], [START_REF] Coron | Dissipative boundary conditions for onedimensional quasi-linear hyperbolic systems: Lyapunov stability for the C1-norm[END_REF] and in [START_REF] Prieur | Stability of switched linear hyperbolic systems by Lyapunov techniques[END_REF] (Proposition 2.1), and other works. This paper extends the first results of [START_REF] Kitsos | High-gain observer design for a class of hyperbolic systems of balance laws[END_REF] in different ways: first, a larger class of systems (PIDE instead of PDE) is considered, i.e., hyperbolic systems including nonlocal terms; then, the conditions on the nonlinearities are weakened (to locally Lipschitz ones); in addition, a more general result of stability (here in the C 1 , instead of the sup spatial norm) for the error equation is proven, with arguments inspired by [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF] and [START_REF] Coron | Dissipative boundary conditions for onedimensional quasi-linear hyperbolic systems: Lyapunov stability for the C1-norm[END_REF] for instance; and finally, we illustrate the applicability of this approach to an epidemic model.

The paper is organized as follows. The solution to the H-GODP, the sufficient conditions and detailed proofs are provided in Section II, where Theorem 1 constitutes our main result. In Section III an application to an age-dependent SIRtype epidemic model is proposed. Conclusions and perspectives are discussed in Section IV.

Notation: For a given w in R n , |w| denotes its usual Euclidean norm. For a given constant matrix A in R n×n , A denotes its transpose, |A| := sup {|Aw| , |w| = 1} is its induced norm and Sym(A) = A+A 2 stands for its symmetric part. By eig(A) we denote the minimum eigenvalue of a symmetric matrix A. By I n×n we define the identity matrix of dimension n. For given ξ : [0, +∞)×[0, L] → R n and time t ≥ 0 we use the notation ξ(t)(x) := ξ(t, x), for all x in [0, L]. For a continuous

(C 0 ) map [0, L] x → ξ(x) ∈ R n we adopt the notation ξ ∞ := max{|ξ(x)| , x ∈ [0, L]}. For a continuously differ- entiable (C 1 ) map [0, L] x → ξ(x) ∈ R n we adopt the notation ξ 1 := ξ ∞ + ξ x ∞ . For a mapping f (•, •), we use the difference operator given by ∆ ξ [f (•, F)] (ξ)(x) := f x, F[ ξ](x) -f (x, F[ξ](x)
), parametrized by ξ, where F denotes any chosen operator acting on ξ. By g we denote the usual derivative of a scalar function g. By D u f we denote the Jacobian of a differentiable mapping R n u → f (u) ∈ R n . For a differentiable in its second argument mapping f , by

D u f (•, F[u]
) , h we denote its Fréchet derivative w.r.t. u acting on h, where F is any Fréchet differentiable operator acting on functions u. Note that by the chain rule,

D u f (•, F[u]) = D w f (•, w)| w=F [u] D u F[u], where D u F[u]
denotes the Fréchet derivative of F. By B δ we denote the set B δ := u ∈ C 1 ([0, L]; R n ) : u 1 ≤ δ . By sgn(x) we denote the signum function sgn(x) = d dx |x|, when x = 0, with sgn(0) = 0.

II. MAIN OBSERVER RESULT

A. Problem Statement and Requirements

We are concerned with one-dimensional, first-order quasilinear hyperbolic system of balance laws, described by the following equations in a strip

Π := [0, +∞) × [0, L]; L > 0 ξ t (t, x) + λ(ξ 1 (t, x))ξ x (t, x) = Aξ(t, x) + f (x, F [ξ(t)] (x)) (1a)
where

ξ 1 • • • ξ n := ξ.
Consider also a distributed measurement, available in the output, of the form

y(t, x) = Cξ(t, x) (1b)
where y : [0, +∞) × [0, L] → R . We assume that the involved constant matrices satisfy the following particular structures:

A =       0 1 0 • • • 0 . . . . . . . . . . . . 1 0 • • • 0       , C = 1 0 • • • 0 ,
and the nonlinear balance terms are of the form

f (•, F [ξ]) =      f 1 (•, F 1 [ξ 1 ]) f 2 (•, F 2 [ξ 1 , ξ 2 ])
. . .

f n (•, F n [ξ 1 , . . . , ξ n ])      .
From the previous equations, we observe that the system satisfies some triangular structure, which presents an analogy to the finite-dimensional case (see [START_REF] Khalil | High-gain observers in nonlinear feedback control[END_REF]).

Remark 1: Notice that if f (•, F[ξ]) reduces to f (ξ) in (1a)
, implying that we have not any nonlocal terms, then we recover the more simple case of [START_REF] Kitsos | High-gain observer design for a class of hyperbolic systems of balance laws[END_REF].

We assume the following regularity for the dynamics:

• the characteristic velocity λ is continuously differentiable, i.e., λ in C 1 (R; R) and, without loss of generality, λ(ξ 1 ) > 0, for all ξ 1 in R (hyperbolicity condition).

• the nonlinear balance term f is continuously differen- tiable, i.e., f in C 1 ([0, L] × R n ; R n ). As a result, for every R > 0, there exists L R > 0, such that for every w, ŵ in R n , with |w|, | ŵ| ≤ R, max x∈[0,L] |f (x, w) -f (x, ŵ) | ≤ L R |w -ŵ|.
In addition, suppose that for every R > 0, there exists L R > 0, such that for every w,

ŵ in R n , with |w|, | ŵ| ≤ R, max x∈[0,L] |D w f (x, w) -D ŵf (x, ŵ) | ≤ L R |w -ŵ|. Furthermore, F : C 1 ([0, L]; R n ) → C 1 ([0, L]; R n ) is a
Fréchet differentiable mapping that can include terms g 1 (ξ), where g 1 in C 1 (R n ; R n ) and also nonlinear nonlocal terms (integral terms of Volterra type) of the form x 0 g 2 (ξ(s))ds, where g 2 in C 0 (R n ; R n ). More explicitly, suppose that for every R > 0, there exist

L 1,R , L 2,R , L 3,R , L 1,R , L 2,R , L 3,R > 0, such that for every ξ, ξ in C 0 ([0, L]; R n ), with ξ ∞ , ξ ∞ ≤ R, |∆ ξ [F](ξ)(x)| ≤ L 1,R |ξ(x) -ξ(x)| + L 2,R |ξ(L) - ξ(L)| + L 3,R L 0 |ξ(s) -ξ(s)|ds, and |∆ ξ D ξ [F](ξ)(x)| ≤ L 1,R |ξ(x) -ξ(x)| + L 2,R |ξ(L) -ξ(L)| + L 3,R L 0 |ξ(s) - ξ(s)|ds, for all x in [0, L].
We further consider initial and boundary conditions of the following general form

ξ(0, x) =ξ 0 (x), x ∈ [0, L] (2a) ξ(t, 0) =H (ξ(t, L)) , t ∈ [0, +∞) (2b)
where we assume the following regularity:

• initial condition ξ 0 is continuously differentiable and satisfies zero-order and one-order compatibility conditions (see [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF]App. B] for precise definition of compatibility conditions). The mapping H is continuously differentiable, i.e., H in C 1 (R n ; R n ). More explicitly, suppose that for every R > 0, there exist L 4,R , L 4,R > 0, such that for every ξ,

ξ in R n , with |ξ|, | ξ| ≤ R, |∆ ξ [H] (ξ)(L)| ≤ L 4,R |ξ(L) -ξ(L)|, and |∆ ξ [D ξ H] (ξ)(L)| ≤ L 4,R |ξ(L) -ξ(L)|. Remark 2:
The specific type of input/output relation (2b) on boundary conditions stands as a sufficient condition for the well-posedness of the observer. This will be shown in the sequel. Also, the main result of the present paper would remain unchanged if we considered the case where boundary conditions satisfy the relation ξ(t, 0) = h (t, ξ(t, L)), where h(•, •) is a C 1 map. This case would require only slight modifications.

The problem that we address in this work is stated in the following definition.

Definition 1: (H-GODP) The High-Gain Observer Design Problem is solvable for a system given by (1a), (2) with output (1b) and solutions existing in Π, if there exists a well-posed observer system with solutions in Π, which estimates the state of (1a) with a convergence speed that can be arbitrarily tuned via a single parameter (high gain).

The assumption that follows is essential to assert the wellposedness of our system, along with an observer design requirement of forward completeness. Furthermore, it imposes global boundedeness of the classical solutions in the C 1 -norm, which is essential in the design of our nonlinear observer. The latter requirement is due to the quasi-linearity of the system (the dependence of λ on state ξ) and can be dropped for the case of semilinear systems, but then a stronger assumption on the nonlinear source terms would be imposed in its place. For more detailed presentation of the nature of this assumption, the reader can refer to [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF], [START_REF] Li | Boundary value problems for quasilinear hyperbolic systems[END_REF] and references therein, where sufficient conditions for the well posedness and existence of classical solutions for quasi-linear hyperbolic systems of balance laws are given.

Assumption 1: Consider a set M ⊂ C 1 ([0, L]; R) nonempty and bounded, consisting of functions satisfying zero-order and one-order compatibility conditions for problem (1a)- [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF]. Then for any initial condition ξ 0 in M, problem (1a)-( 2) admits a unique classical solution in C 1 ([0, +∞) × [0, L]; R n ). Moreover, there exists δ > 0, such that for all ξ 0 in M, ξ(t, •) 1 ≤ δ, for all t in [0, +∞).

Define now a

C 1 vector-valued function R n ζ → s δ (ζ) = s 1 δ (ζ 1 ), • • • , s n δ (ζ n )
, parametrized by δ, and satisfying the following properties.

For every δ > 0 and v, w in R n , such that |w| ≤ δ, there exists ω δ > 0, such that the following inequalities are satisfied

|s δ (v) -w| ≤ω δ |v -w|, |D v s δ (v) -D w s δ (w)| ≤ω δ |v -w|. (3a)
Moreover, there exists m δ > 0, such that for every v in R n ,

|s δ (v)|, |D v s δ (v)| ≤ m δ . (3b)
Note that a saturation-like function of the form

s i δ (ζ i ) = ζ i , |ζ i | ≤ δ sgn(ζ i ) (|ζ i | -δ) e -|ζi|+δ + δ , |ζ i | > δ (4)
satisfies [START_REF] Besanc ¸on | Sur la commande en dimension finie d'une classe de systèmes non linéaires de dimension infinie[END_REF] with

ω δ = m δ = √ n max e -1 + δ, 1 , ω δ = √ ne -3 .
We are now in a position to introduce our candidate observer dynamics and its boundary conditions for system (1)-( 2). Define, first, a diagonal matrix Θ by

Θ := diag θ, θ 2 , . . . , θ n , (5)
where θ > 1 is the candidate high-gain constant of the observer, which will be selected precisely later. Consider, also, a vector gain K in R n , such that A + KC is Hurwitz (we can always find such a K, due to the observability of the pair (A, C)). Note that for such a K, one can find a symmetric and positive definite n × n matrix P satisfying a quadratic Lyapunov equation of the following form

2Sym (P (A + KC)) = -I n×n . (6)
Let us remark that P satisfying (6) cannot be diagonal, since matrix A fails by its definition to be a diagonally stabilizable matrix. P will be used as the Lyapunov matrix in the Lyapunov functional used in the proof of the observer convergence. However, in stability analysis of general hyperbolic systems, see for instance [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF], the chosen Lyapunov functionals are diagonal, in order to commute with the matrix of the characteristic velocities. In the present case, we assume only one velocity and, thus, we do not need that P is diagonal.

With the previous assumptions, the observer that will solve the H-GODP can be given by the following equations on Π ξt

(t, x) + λ (y(t, x)) ξx (t, x) = A ξ(t, x) -ΘK y(t, x) -C ξ(t, x) + f x, F s δ ξ(t) (x) (7a) ξ(t, 0) = H s δ ξ(t, L) . (7b)
We can easily deduce some difference inequalities for the dynamics, to be invoked later, as a direct consequence of the regularity assumptions and properties (3) of s δ (•). Precisely, for any δ > 0, there exist constants

L 1,δ , L 2,δ , L 3,δ , L 4,δ , L 1,δ , L 2,δ , L 3,δ , L 4,δ > 0 depending on constants ω δ , ω δ , and m δ , such that for every ξ, ξ in C 0 ([0, L]; R n ),
with ξ ∞ ≤ δ, the following inequalities are satisfied for all x in [0, L]:

|∆ s δ ( ξ) [f (•, F)] (ξ)(x)| ≤ L 1,δ | ξ(x) -ξ(x)| + L 2,δ | ξ(L) -ξ(L)| + L 3,δ L 0 | ξ(s) -ξ(s)|ds, (8a)
|∆ s δ ( ξ) [H] (ξ)(L)| ≤ L 4,δ | ξ(L) -ξ(L)|, (8b)
|∆ s δ ( ξ) [D ξ f (•, F)] (ξ)(x)| ≤ L 1,δ | ξ(x) -ξ(x)| + L 2,δ | ξ(L) -ξ(L)| + L 3,δ L 0 | ξ(s) -ξ(s)|ds, (8c)
|∆ s δ ( ξ) [D ξ H] (ξ)(L)| ≤ L 4,δ | ξ(L) -ξ(L)|. (8d)
Remark 3: Assumption 1 guarantees the existence of a preassigned ball in C 1 in which system's solutions take values. This, in conjunction with the regularity assumptions, would allow us to avoid globally Lipschitz assumption for the nonlinear dynamics of our system, which in general constitutes a natural assumption in classical high-gain observer designs for finite dimensions. To avoid such a restricting assumption, we plug function s δ in observer dynamics [START_REF] Christofides | Feedback control of hyperbolic pde systems[END_REF]. Exploiting this function guarantees that observer system's dynamics are globally Lipschitz. We note also that, although it is not described explicitly by the observer's equations, we avoid injecting s δ in potential linear terms included in f and H. In that case, we set s δ ξ = ξ, since linear terms are globally Lipschitz.

The following lemma guarantees the existence of unique global classical solutions for the candidate observer. The proof of the lemma follows from classical arguments and the fact that nonlinearities in observer system are globally Lipschitz.

Lemma 1: (Existence/Uniqueness of Global Classical Solutions to the Observer System) Under the regularity assumptions for the dynamics and for any y in C 1 ([0, +∞) × [0, L]; R), the problem described by [START_REF] Christofides | Feedback control of hyperbolic pde systems[END_REF], with initial condition ξ0 (x) := ξ(0, x), for all x in [0, L], satisfying zero-order and one-order compatibility conditions, admits a unique classical solution in Π, i.e., there exists a unique solution ξ in C 1 ([0, +∞) × [0, L]; R n ) to [START_REF] Christofides | Feedback control of hyperbolic pde systems[END_REF]. Proof: See Appendix A for a detailed proof.

We are now in a position to present our main result on the solvability of the H-GODP.

Theorem 1: (Observer Convergence) Consider system (1a), (2), defined on Π with output (1b) and suppose that Assumption 1 holds for initial condition ξ 0 in M. Let also K in R n , chosen in such a way that A + KC is Hurwitz. Then, the H-GODP for system (1a) -(2) is solvable by system (7) for θ > 1 as a high gain and initial condition ξ0 in C 1 ([0, L]; R n ), with ξ0 (x) = ξ(0, x) satisfying zero-order and one-order compatibility conditions. This is a high-gain observer for ξ, in the sense that for θ large enough it admits a unique classical solution in Π on the one hand, providing an estimate for the state of system (1a) -(2) on the other hand. More precisely, for every κ > 0, there exists θ 0 ≥ 1, such that for every θ > θ 0 , the following inequality holds

ξ(t, •) -ξ(t, •) 1 ≤ e -κt ξ0 (•) -ξ 0 (•) 1 , ∀t ≥ 0 (9)
for some > 1, polynomial in θ. This theorem states that for system (1a) -( 2) with output (1b) we have a high-gain observer design providing an estimate of its full state, with a convergence rate adjustable via θ. The well posedness results from Lemma 1. The convergence result is established in the next subsection.

B. Observer Convergence Proof

This subsection is dedicated to the proof of Theorem 1. Consider the observer error e := ξ -ξ which satisfies the following equations e t (t, x) + λ(y(t, x))e x (t, x) = (A + ΘKC) e(t, x)

+∆ s δ ( ξ(t)) [f (•, F)] (ξ(t)) (x), (10)
e(t, 0) = ∆ s δ ( ξ(t)) [H] (ξ(t))(L). (11)
We now define the linearly transformed error by ε := Θ -1 e and we derive the following hyperbolic equations for ε on Π

ε t (t, x) + λ (y(t, x)) ε x (t, x) = θ (A + KC) ε(t, x) +Θ -1 ∆ s δ ( ξ(t)) [f (•, F)] (ξ(t))(x), (12)
ε(t, 0) = Θ -1 ∆ s δ ( ξ(t)) [H] (ξ(t))(L). (13)
At this point, let us introduce an operator K :

C 1 ([0, L]; R n ) → C 0 ([0, L]; R n×n ) defined by K[ξ] :=λ (Cξ) (λ(Cξ)) -1 I n×n C × -λ(Cξ) ∂ ∂x ξ + Aξ + f (•, F[ξ]) . (14)
Define also K ξ 1

: C 1 ([0, L]; R n ) → C 0 ([0, L]; R n ), parametrized by ξ ∈ C 0 ([0, L]; R n ) and given by K ξ 1 [ξ] := -K[ξ]Θ -1 ∆ s δ ( ξ) [f (•, F)] (ξ) + Θ -1 ∆ s δ ( ξ) [D ξ f (•, F)] (ξ), D ξ s δ ( ξ)ξ t + D u f (•, F[u])| u=s δ ( ξ) , ∆ ξ [D ξ s δ ](ξ)ξ t . ( 15
)
Next, by temporarily assuming that ε is of class C 2 , we derive the following hyperbolic equations for ε t , where we have also substituted the expression of ε x derived from [START_REF] Deutschmann | Backstepping-based boundary observer for a class of time-varying linear hyperbolic PIDEs[END_REF],

ε tt (t, x) + λ (y(t, x)) ε tx (t, x) = K[ξ(t)](x)ε t (t, x) + Θ -1 D u f (•, F [u])| u=s δ ( ξ(t)) , D ξ s δ ξ(t) Θε t (t) (x) + θ(A + KC)ε t (t, x) -θK[ξ(t)](x)(A + KC)ε(t, x) + K ξ(t) 1 [ξ(t)](x), (16)
ε t (t, 0) = Θ -1 ∆ s δ ( ξ(t)) [D ξ H] (ξ(t))(L)D ξ s δ ( ξ(t, L)) ×ξ t (t, L) + D u H (u)| u=s δ ( ξ(t)) (L)D ξ s δ ξ(t, L) Θε t (t, L) +D ξ H(ξ(t, L))∆ ξ(t) [D ξ s δ ](ξ(t))(L)ξ t (t, L) . ( 17
)
Remark 4: Due to cumbersome notation in the previous relations, we provide an example to make the calculations of the Fréchet derivative more clear. Let, for instance,

f (x, F[u](x)) = g(x)g 1 (u(x))+g 2 (u(L)) + x 0 g 3 (u(s)) ds. Then, D u f (•, F[u]) , u t (x) = g(x)D u g 1 (u) (x)u t (x) +D u g 2 (u) (L)u t (L) + x 0 D u g 3 (u) (s)u t (s)ds.
Now, notice that whenever ξ in B δ , due to continuity of λ(•), the fact that λ(ξ 1 ) > 0, for all ξ 1 in R and the fact that ξ ∞ ≤ δ, the quantities sup ξ∈B δ (λ(Cξ)), inf ξ∈B δ (λ(Cξ)) are positive and finite. In addition, whenever ξ in B δ and invoking the hyperbolic dynamics (1a), we can easily calculate constant δ 1 > 0, dependent on δ, such that

ξ t ∞ = -λ(ξ 1 )ξ x + Aξ + f (•, F[ξ]) ∞ ≤ δ 1 . (18)
By combining the above arguments, the fact that θ > 1, [START_REF] Inaba | Mathematical analysis of an age-structured SIR epidemic model with vertical transmission[END_REF], continuity of the involved mappings, and inequalities (8), we can easily calculate positive constants γ i,δ , i = 1, . . . , 6, such that whenever ξ in B δ , ξ in C 0 ([0, L]; R n ), the following inequalities are satisfied for all x in [0, L]

|Θ -1 D u f (•, F [u])| u=s δ ( ξ) , D ξ s δ ξ Θε t (x)| ≤ γ 1,δ |ε t (x)| + |ε t (L)| + L 0 |ε t (s)|ds , (19a)
|Θ -1 ∆ s δ ( ξ) [f (•, F)] (ξ) (x)| ≤ γ 2,δ (|ε(x)| + |ε(L)| + L 0 |ε(s)|ds , (19b)
|Θ -1 ∆ s δ ( ξ) [H] (ξ) (L)| ≤ γ 3,δ θ n-1 |ε(L)|, (19c)
|Θ -1 ∆ s δ ( ξ) [D ξ f (•, F)] (ξ) (x)| ≤ γ 4,δ (|ε(x)| + |ε(L)| + L 0 |ε(s)|ds , (19d)
|Θ -1 D u H [u]| u=s δ ( ξ) (L)D ξ s δ ξ(L) Θ| ≤ γ 5,δ θ n-1 , (19e)
|Θ -1 ∆ s δ ( ξ) [D ξ H] (ξ)(L)D ξ s δ ( ξ) +D ξ H(ξ(L))∆ ξ [D ξ s δ ](ξ)(L) | ≤ γ 6,δ θ n-1 |ε(L)|. (19f)
Define also constants

γ 7,δ := sup ξ∈B δ |λ (Cξ)|, γ 8,δ := inf ξ∈B δ λ(Cξ), γ 9,δ := sup ξ∈B δ λ(Cξ), (20a)
and derive the following bounds for all x in [0, L]

|K[ξ](x)| ≤γ 10,δ := δ 1 γ 7,δ γ 8,δ , (20b)
|K ξ 1 [ξ](x)| ≤γ 11,δ |ε(x)| + |ε(L)| + L 0 |ε(s)|ds ; (20c) γ 11,δ :=γ 10,δ γ 2,δ + δ 1 γ 4,δ .
Remark 5: Note here that all constants γ 1,δ , . . . , γ 11,δ are calculated by taking bounds of the involved mappings on B δ and depend on the global bound δ of the system's trajectories (and subsequently on δ 1 in (18)), constants in [START_REF] Coron | Dissipative boundary conditions for onedimensional quasi-linear hyperbolic systems: Lyapunov stability for the C1-norm[END_REF], the order of the system n, and are independent of the observer gain θ. More explicitly, γ 1,δ , γ 2,δ , γ 4,δ , and γ 11,δ being independent of θ is a direct consequence of the assumed triangularity of the involved mappings, similarly as in the classical high-gain observer designs [START_REF] Gauthier | A simple observer for nonlinear systems: applications to bioreactors[END_REF]. This property turns out to be sufficient for the solvability of the H-GODP. More precisely, in the sequel, while bounding the Lyapunov time-derivative from above, the independence of these parameters on θ shall not add positive terms with linear (or higher-order) dependency on θ. On the other hand, negative terms will appear depending linearly on θ as a direct consequence of the assumed observability of the pair (A, C). This will render the negativity of the Lyapunov time-derivative feasible, as this will be more clear in the forthcoming Lyapunov analysis.

To prove the exponential stability of the solution to the error system at the origin, we need to define a Lyapunov functional

W p : C 1 ([0, L]; R n ) → R by W p [ε] := L 0 π(x)exp (pµ θ,δ x) G p [ε](x)dx 1/p (21a)
where

G p : C 1 ([0, L]; R n ) → R is given by G p [ε] := ε P ε + ρ 0 ε t P ε t p (21b)
and ρ 0 in (0, 1) is a constant (to be chosen appropriately), p in N, P in R n×n is positive definite symmetric satisfying [START_REF] Castillo | Boundary observers for linear and quasi-linear hyperbolic systems with application to flow control[END_REF],

π : [0, L] → R is given by π(x) := (π -1) x L + 1; π := sup ξ∈B δ λ(Cξ) inf ξ∈B δ λ(Cξ) (22)
and constant µ θ,δ is given by

µ θ,δ := 1 L ln(µ δ θ 2n-2 ) (23a)
where

µ δ := |P | eig(P ) max γ 2 3,δ , γ 2 5,δ , γ 2 6,δ δ 2 1 , γ 5,δ γ 3,δ δ 1 . (23b)
Note here that, by its definition, π is bounded as follows

1 ≤ π(x) ≤ π, ∀x ∈ [0, L]. (24)
Let us also define functional V :

C 1 ([0, L]; R n ) → R by V[ε] := exp (µ θ,δ •) ε P ε ∞ + ρ 0 exp (µ θ,δ •) ε t P ε t ∞ . ( 25
)
The idea is to let p → ∞ in the end, to approximate V by W p . This convergence property is presented in details in Appendix B. By invoking Lemma 1 and Assumption 1, which establish global unique classical solutions for both observer system [START_REF] Christofides | Feedback control of hyperbolic pde systems[END_REF] and system (1a), ( 2) respectively, we are now in a position to define G p , W p , V : [0, +∞) → R by

G p (t) := G p [ε](t), W p (t) := W p [ε](t), V (t) := V[ε](t), ∀t ≥ 0. (26)
Calculating the time-derivative Ẇp along the classical solutions to ( 12) -( 13), ( 16) -( 17), we get

Ẇp = 1 p W 1-p p L 0 pπ(x) exp (pµ θ,δ x) G p-1 (x) × ε t (x) P ε(x) + ε (x)P ε t (x) + ρ 0 ε tt (x)P ε t (x) +ρ 0 ε t (x) P ε tt (x) dx. (27)
After substituting the dynamical equations ( 12) and ( 16) into the above equation and performing integration by parts, Ẇp can be written in the following form

Ẇp = W 1-p p 1 p T 1,p + 1 p T 2,p + T 3,p + T 4,p + T 5,p (28)
where

T 1,p := -π(L)λ(y(L)) exp (pµ θ,δ L) G p (L) +π(0)λ(y(0))G p (0), (29a)
T 2,p := L 0 ∂ x [π(x) exp (pµ θ,δ x) λ (y(x))] G p (x)dx, (29b) T 3,p := 2 L 0 π(x) exp (pµ θ,δ x) G p-1 (x) ε (x)P Θ -1 ×∆ s δ ( ξ) [f (•, F)] (ξ)(x) + ρ 0 ε t (x)P K ξ 1 [ξ](x) +ρ 0 ε t (x)Sym(P K[ξ](x))ε t (x) dx, (29c)
T 4,p := 2 L 0 π(x) exp (pµ θ,δ x) G p-1 (x)ρ 0 ε t (x) P Θ -1 × D u f (•, F [u])| u=s δ ( ξ) , D ξ s δ ξ Θε t (x)dx, (29d) T 5,p := θ L 0 π(x) exp (pµ θ,δ x) G p-1 (x) × 2ε Sym(P (A + KC))ε + 2ρ 0 ε t Sym(P (A + KC))ε t -ρ 0 ε t P K[ξ](A + KC)ε -ρ 0 ε (A + KC) K [ξ]P ε t dx. ( 29e
)
After substituting boundary equations ( 13) and ( 17) in T 1,p and by virtue of ( 8) and ( 22), using the previously calculated bounds and trivial inequalities, we can easily obtain the following inequality

T 1,p ≤ sup ξ∈Bδ (λ(Cξ))G p (L) -exp (pµ θ,δ L) + θ 2n-2 µ δ p (30)
and, subsequently, by (23a)

T 1,p ≤ 0. ( 31
)
By the fact that π(x) ≥ 1, for all x in [0, L], we can derive the following bound for T By exploiting ( 19)-( 20), T 3,p can be bounded as follows

T 3,p ≤ 2 L 0 π(x) exp (pµ θ,δ x) G p-1 (x)|P | × γ 2,δ |ε(x)| |ε(x)| + |ε(L)| + x 0 |ε(s)|ds +ρ 0 γ 11,δ |ε t (x)| |ε(x)| + |ε(L)| + x 0 |ε(s)|ds +ρ 0 γ 10,δ |ε t (x)| 2 dx ≤ L 0 π(x) exp (pµ θ,δ x) G p-1 (x) |P | eig(P ) × max {4γ 2,δ + ρ 0 γ 11,δ , 3γ 11,δ + γ 10,δ } G 1 (x)dx + L 0 π(x) exp (pµ θ,δ x) G p-1 (x) |P | eig(P ) × (γ 2,δ + ρ 0 γ 11,δ ) G 1 (L) + x 0 G 1 (s)ds dx ≤ ω 3,δ W p p + ω 4,δ exp (pµ θ,δ •) G 1 (•) ∞ W p-1 p-1 ≤ ω 3,δ W p p + ω 4,δ V W p-1 p-1 , (33)
Similarly, we can bound T 4,p as follows

T 4,p ≤ ω 5,δ W p p + ω 6,δ V W p-1 p-1 , (35)
where

ω 5,δ := 2 |P | eig(P ) γ 1,δ , ω 6,δ := 2 |P | eig(P ) γ 1,δ (1 + L).
The term T 5,p of Ẇp can be rewritten in the following way

T 5,p := -θ L 0 π(x) exp (pµ θ,δ x) G p-1 (x) × ε (x) ε t (x) Σ[ξ](x) ε(x) ε t (x) dx, ( 36
)
where, after utilizing (6), Σ :

B δ → C 0 [0, L]; R 2n×2n is given by Σ[ξ] := I n×n -ρ 0 (A + KC) K [ξ]P -ρ 0 P K[ξ](A + KC) ρ 0 I n×n . (37)
Now, we can easily verify (using Schur complement) that for all w ∈ R 2n \0, we have

inf ξ∈B δ w Σ[ξ]w |w| 2 > 0, if 0 < ρ 0 < min 1 |P | 2 |A + KC| 2 γ 2 10,δ , 1 . ( 38
)
It turns out that for every choice of matrices P and K satisfying ( 6), there always exists a ρ 0 , such that (38) is satisfied and this fact renders Σ positive. Consequently, for appropriate choice of ρ 0 , there exists σ δ > 0, such that

T 5,p ≤ -θ σ δ |P | W p p . (39)
Combining ( 31), ( 32), ( 33), (35), and ( 39) with ( 28), we obtain

Ẇp ≤ (-θω 7,δ + ω 8,δ ln(θ) + ω 9,δ )W p + (ω 4,δ + ω 6,δ ) W 1-p p W p-1 p-1 V, (40)
where

ω 7,δ := σ δ |P | , ω 8,δ := ω 2,δ (2n-2) L , ω 9,δ := ω 1,δ + ω 3,δ + ω 2,δ L | ln µ δ |.
Now, using Hölder's inequality, one can obtain

W p-1 p-1 ≤ W p-1 p π(•) 1/p ∞ .
Utilizing the above inequality, (40) gives

Ẇp ≤ (-θω 7,δ + ω 8,δ ln(θ) + ω 9,δ )W p + (ω 4,δ + ω 6,δ ) π1/p V. ( 41
)
We obtained the estimate (41) of Ẇp for ε of class C 2 , but, by invoking density arguments, the results remain valid with ε only of class C 1 (see [START_REF] Coron | Dissipative boundary conditions for onedimensional quasi-linear hyperbolic systems: Lyapunov stability for the C1-norm[END_REF] for further details). From (41), taking the limit as p → +∞ and using the fact that lim p→+∞ π1/p = 1, we get in the distribution sense in (0, +∞),

V (t) ≤ (-θω 7,δ + ω 8,δ ln(θ) + ω 10,δ ) V (t) (42)
where ω 10,δ := ω 4,δ + ω 6,δ + ω 9,δ . For more details on how to obtain the above estimate, see Properties 1 and 2 in Appendix B. Now, one can select the high gain θ, such that

θ > θ 0 ( 43
)
where θ 0 ≥ 1 is such that

-θω 7,δ + ω 8,δ ln(θ) + ω 10,δ ≤ -2κ δ , ∀θ > θ 0 ( 44
)
for some κ δ > 0. One can easily check that for any κ δ > 0, there always exists a θ 0 ≥ 1, dependent on the involved constants, such that the previous inequality is satisfied. Subsequently, (42) yields to the following differential inequality in the distribution sense in (0, +∞)

V (t) ≤ -2κ δ V (t) (45)
and by the comparison lemma, we get

V (t) ≤ e -2κ δ t V (0), ∀t ≥ 0. ( 46
)
Now, by the dynamics [START_REF] Deutschmann | Backstepping-based boundary observer for a class of time-varying linear hyperbolic PIDEs[END_REF], in conjunction with [START_REF] Karafyllis | Stability of integral delay equations and stabilization of age-structured models[END_REF], [START_REF] Kermack | A contribution to the mathematical theory of epidemics[END_REF], we can obtain the following inequalities

ε t ∞ ≥ γ 8,δ ε x ∞ -s δ,θ ε ∞ ε t ∞ ≤ γ 9,δ ε x ∞ + s δ,θ ε ∞
where s δ,θ := θ|A + KC| + γ 2,δ (2 + L). Invoking these inequalites, also (23a), estimate (46) and the following inequality,

ρ 0 2 e µ θ,δ -|µ θ,δ | 2 L eig(P ) ( ε ∞ + ε t ∞ ) 2 ≤ V ≤ e µ θ,δ +|µ θ,δ | 2 L |P | ( ε ∞ + ε t ∞ ) 2 we obtain ε 1 ≤ δ,θ e -κ δ t ε 0 1 , ∀t ≥ 0, (47)
where ε 0 (x) := ε(0, x) and

δ,θ := |P | ρ 0 eig(P ) (µ δ ) 1 2L θ n-1 L × max s δ,θ + 1, 1 γ 8,δ max {1 + 2s δ,θ , 2γ 9,δ } .
By (47), we derive the following estimate, which holds for every t ≥ 0

ξ(t, •) -ξ(t, •) 1 ≤ ¯ δ,θ e -κ δ t ξ0 -ξ 0 1 , (48)
where ¯ δ,θ := θ n-1 δ,θ . Concluding, we solved the H-GODP for (1a)-( 1b)-( 2) by designing an exponential in the C 1 -norm high-gain observer of adjustable convergence rate κ δ , dependent on the selection of θ. The proof of Theorem 1 is complete.

Remark 6: Notice that the H-GODP is not solvable in case of boundary measurement, instead of internal measurement as in (1b). First, arbitrary convergence condition would not be fulfilled, since a boundary-based observer for hyperbolic systems would experience a limitation with respect to convergence speed. The rate of convergence is limited by a minimal observation time which depends on the size of the domain L and the characteristic velocities in that case (see [START_REF] Li | Exact boundary observability for quasilinear hyperbolic systems[END_REF] for minimum time of observability due to transport phenomena, and [START_REF] Deutschmann | Backstepping-based boundary observer for a class of time-varying linear hyperbolic PIDEs[END_REF] for comments on the convergence of boundary observers). Second, boundary measurement of the whole state vector would be required, instead for just the first state, in order to end up in some dissipativity in the boundary conditions of the error system (as in [START_REF] Castillo | Boundary observers for linear and quasi-linear hyperbolic systems with application to flow control[END_REF] for instance). In [START_REF] Coron | Local exponential stabilization of a 2x2 quasilinear hyperbolic system using backstepping[END_REF], a backstepping design is proposed for some kind of dual control problem via boundary control of only one state, but in our observer problem, because of the generality of our boundary conditions, we cannot achieve their dissipativity by one observation only, which would lead to stability of the observation error system (see [START_REF] Coron | Dissipative boundary conditions for onedimensional quasi-linear hyperbolic systems: Lyapunov stability for the C1-norm[END_REF] about linking dissipativity of boundary conditions with stability). In general, the problem of controllability and observability for cascade systems with reduced numbers of controls and observations, less than the number of states, is a hard problem, see for instance [START_REF] Alabau-Boussouira | Internal controllability of first order quasilinear hyperbolic systems with a reduced number of controls[END_REF].

Remark 7: The observer design proposed in this work relies on a continuous measurement in the whole space, see (1b). This means that a direct application a priori needs the system to have such measurements. In addition, due to the absence of localized coupling between the nonlinear PDEs, as the coupling matrix A is constant, it is hard to see how a possibly localized measurement in a part of the domain could be used in order to guarantee the existence of a damping term as (A + KC) in [START_REF] Deutschmann | Backstepping-based boundary observer for a class of time-varying linear hyperbolic PIDEs[END_REF], that would render the Lyapunov derivative negative. However, some improvement for practical implementation could be to consider an output of the form (1b), but with C being substituted by C(x) := c(x)C, where c(x) = N i=1 e -ai(x-xi) 2 , with a i sufficiently large. This would correspond to a continuous measurement again, but that would have higher contribution around points x i , i = 1, . . . , N inside the domain and less contribution far from them. Then, the damping term would be A+K(x)C(x), with K(x) chosen such that A + K(x)C(x) is Hurwitz. Such improvements will be a subject of future works.

III. APPLICATION

As an illustrative example, let us consider the case of an epidemic model. In the field of epidemiology, mathematical models are currently used to explain epidemic phenomena and to assess vaccination strategies through control mechanisms. For infectious diseases, a fundamental model was formulated by Kermack and McKendrick in 1927 [20]. This model classifies population into three groups: (i) the individuals who are uninfected and susceptible (S) of catching the disease, (ii) the individuals who are infected (I) by the concerned pathogen, (iii) the recovered (R) individuals who have acquired a permanent immunity to the disease. In the case when the age of patients is taken into account, S(t, x), I(t, x), R(t, x) represent the age distribution of the population of each group at time t. As a result, the integral from x 1 to x 2 of S, I and R is the number of individuals of each group with ages between x 1 and x 2 .

The dynamics of the disease propagation in the population are then described by the following set of hyperbolic PIDEs on Π := [0, +∞) × [0, L] (see [START_REF] Bastin | Stability and boundary stabilization of 1-d hyperbolic systems[END_REF]Chapter 1] and also [17, Chapter III] for detailed presentation of such systems) ∂ t S(t, x) + ∂ x S(t, x) + µ(x)S(t, x) + G[S(t), I(t)](x) = 0, (49a)

∂ t I(t, x) + ∂ x I(t, x) + (γ(x) + µ(x)) I(t, x) -G[S(t), I(t)](x) = 0, (49b) ∂ t R(t, x) + ∂ x R(t, x)+µ(x)R(t, x) -γ(x)I(t, x) = 0, (49c)
where G[S(t), I(t)](x) := β(x)S(t, x) L 0 I(t, s)ds represents the disease transmission rate by contact between susceptible and infected individuals which is assumed to be proportional to the sizes of both groups. β(x) > 0 is the age-dependent transmission coefficient between all infected individuals and susceptibles having age x. The maximal life duration in the considered population is denoted by L and, thus, S(t, L) = I(t, L) = R(t, L) = 0. Parameter µ(x) > 0 denotes the natural age-dependent per capita death rate in the population and γ(x) > 0 is the age-dependent rate at which infected individuals recover from the disease. Parameters µ, β, and γ are assumed twice continuously differentiable.

We consider the following boundary conditions

S(t, 0) = B(t), I(t, 0) = 0, R(t, 0) = 0. (50)
Here, B(t) stands for the inflow of newborn individuals in the susceptible part of the population at time t. Let us assume that the number of people in the group R of recovered patients between ages 0 and x, for every age x ∈ [0, L] and time t ≥ 0, is the system's output

y(t, x) = x 0 R(t, s)ds (51)
This is consistent with the remarks in the recent paper [START_REF] Park | A Note on Observation Processes in Epidemic Models[END_REF] about the COVID-19 pandemic. An alternative measurement, however, can be the number of infected patients

x 0 I(t, s)ds. We perform the following coordinates invertible transformation, in order to write (49) in the appropriate form for observer design

ξ 1 (t, x) = x 0 R(t, s)ds, (52a)
Using these coordinates, (49) is written as follows:

∂ t ξ 1 (t, x) + ∂ x ξ 1 (t, x) = ξ 2 (t, x) + x 0 µ (s)ξ 1 (t, s)ds -µ(x)ξ 1 (t, x), ( 53a
)
∂ t ξ 2 (t, x) + ∂ x ξ 2 (t, x) = ξ 3 (t, x) + x 0 (κ 1 (s) -κ 2 (s)) ξ 2 (t, s)ds + (κ 2 (x) -κ 1 (x)) ξ 2 (t, x), (53b)
∂ t ξ 3 (t, x) + ∂ x ξ 3 (t, x) = ξ 3 (t, x)g 1 [ξ 2 (t), ξ 3 (t)](x) - x 0 (k 3 (s) -µ (s) -g 2 [ξ 2 (t)](L)β (s)) ξ 3 (t, s)ds + g 2 [ξ 2 (t)](L)B(t)β(0)γ(0), (53c) y(t, x) = Cξ(t, x), (53d) ξ(t, 0) = 0, (53e)
where

κ 1 (x) := µ(x) + γ(x), κ 2 (x) := γ (x) γ(x) , κ 3 (x) := (β(x)γ(x)) β(x)γ(x) , g 1 [ξ 2 (t), ξ 3 (t)](x) := κ 3 (x) -µ(x) -β(x)g 2 [ξ 2 (t)](L) + 1 g 2 [ξ 2 (t)](L) L 0 κ 1 (s) γ(s) ξ 2 (t, s)ds - κ 1 (L) γ(L) ξ 2 (t, L) + L 0 γ (s) γ 2 (s) ξ 3 (t, s)ds + ξ 3 (t, L) γ(L) , g 2 [ξ 2 (t)](L) := ξ 2 (t, L) γ(L) + L 0 γ (s) γ 2 (s) ξ 2 (t, s)ds, C = 1 0 0 .
Notice that hyperbolic dynamics (53) of the transformed system satisfy a simpler version of structure (1a) (semilinear, with λ = 1), with H = 0 in (2b). Note also, that g 2 [ξ 2 (t)](L), being equal to L 0 I(t, b)db, is positive, because of the positiveness of system (53) and, therefore, mapping g 1 is well defined.

Choose now some initial conditions, such that Assumption 1 of previous section is satisfied for system (53) with δ = 20, we meet sufficient conditions for solvability of the H-GODP for this particular choice of initial conditions (existence 49) is established in [START_REF] Inaba | Mathematical analysis of an age-structured SIR epidemic model with vertical transmission[END_REF]).

The high-gain observer dynamics, as in [START_REF] Christofides | Feedback control of hyperbolic pde systems[END_REF], are given by

∂ t ξ1 (t, x) + ∂ x ξ1 (t, x) = ξ2 (t, x) + x 0 µ (s) ξ1 (t, s)ds -µ(x) ξ1 (t, x) -θk 1 (y(t, x) -ξ1 (t, x)), (54a)
∂ t ξ2 (t, x) + ∂ x ξ2 (t, x) = ξ3 (t, x) + x 0 (κ 1 (s) -κ 2 (s)) ξ2 (t, s)ds + (κ 2 (x) -κ 1 (x)) ξ2 (t, x) -θ 2 k 2 (y(t, x) -ξ1 (t, x)), (54b)
∂ t ξ3 (t, x) + ∂ x ξ3 (t, x) = s 3 δ ( ξ3 (t, x))g 1 s 2 δ ξ2 (t) , s 3 δ ξ3 (t) (x) - x 0 k 3 (s) -µ (s) -g 2 s 2 δ ξ2 (t) (L)β (s) s 3 δ ξ3 (t, s) ds +g 2 ξ2 (t) (L)B(t)β(0)γ(0) -θ 3 k 3 (y(t, x) -ξ1 (t, x)), (54c)
ξ(t, 0) = 0, (54d)
where s i δ ξi , i = 2, 3 are given by (4). We assume for the sake of simplicity that the inflow of the newborns B(t) is not time-varying, but zero. This time invariance simplification is made in order to meet system's (1a) time invariance. The case where B is time-varying is not included in this framework, but it is not hard to adjust the present analysis to this case.

In Figure 1 we represent the considered functions µ, γ, and β (academic choices) and we choose L = 100.

We now apply Theorem 1, with θ = 20 and K = k 1 k 2 k 3 = -2 -1 -1 , which establishes the solvability of the H-GODP for system (53) with observer (54). After inversing the coordinates transformation, the convergence of the observation errors between states S, I, R and Ŝ, Î, R (the observer states in the original coordinates) is guaranteed by Theorem 1.

In Figure 2 we represent the output ξ 1 . In Figures 345we see the observation errors for each of the original states S, I and R, after choosing arbitrary observer's initial conditions satisfying compatibility conditions.

Remark 8: In Theorem 1 we proved a type of stability in the C 1 spatial norm. It turns out that this is essential for the particular case of systems that we study in the current section and it is due to the nonlocal transformation we performed in (52). More precisely, the exponential stability of the spatial derivative of the error between (53) and (54) induces the exponential stability in the sup-norm for the estimation error of the initial system (49).

ACKNOWLEDGEMENT

The first author would like to thank Fabian Wirth (University of Passau) and Roxanne Jackson for fruitful discussions on this topic.

IV. CONCLUSION

In this paper we designed a high-gain observer for a class of observable systems of hyperbolic PIDEs with distributed measurement. This result constitutes an extension of the highgain observer design for finite-dimensional systems to a class of PIDEs and, also, an extension of our previous works towards this direction. We proved the exponential decay of the observer error in the C 1 -norm step by step by first choosing an appropriate Lyapunov functional and we illustrated our methodology with its application to an epidemic model. The extension of this methodology to more general cases of hyperbolic systems with distinct characteristic velocities will be subject to our future work. Now, choosing ξ1 , ξ2 in C 0 Π τ0 ; R n with identical initial conditions, from the mean value theorem, we obtain the following: This, in conjunction with (55), directly leads to the following:

sup (t,x)∈Π τ 0 |f x, F s δ ξ1 (t) (x) -f x, F s δ ξ2 (t) (x) | ≤ sup s∈[0,1] D u f (•, F [s δ (
sup (t,x)∈Π τ 0 |G ξ1 -G ξ2 | ≤ τ 0 q 0 sup (t,x)∈Π τ 0 | ξ1 -ξ2 |;
q 0 := max (t,x)∈Π τ 0 |Φ(t, t 0 (t, x))|L f (1 + L H ).

Now, choosing τ 0 = (2q 0 ) -1 and applying the Banach fixedpoint theorem, we prove the existence and uniqueness of a C 0 Π τ0 ; R n solution ξ. In order to prove the existence and uniqueness of a continuous solution in Π T , for arbitrary T , we follow the previous procedure in T /τ 0 steps, iterating the local existence-uniqueness result in domains (Π jτ0 ∩Π T )\ Π (j-1)τ0 , where j ≤ T /τ 0 . Next, we follow the same procedure to prove the existenceuniqueness of C , when t 0 (t, x) = 0.

(59)

We prove the existence-uniqueness of continuous solutions ξx in Π τ1 , for some τ 1 satisfying condition (57) with τ 0 replaced by τ 1 . Combining observer system's dynamics (7a) and (59),

Taking into account that ξ is a known continuous function, we apply the operator defined by the right hand side of (59) to continuous functions ∂ x ξ1 and ∂ x ξ2 , starting from the same initial conditions. We can, therefore, obtain the estimate sup (t,x)∈Π ((λ • y)(t, x)) -1 .

Choosing τ 1 , such that q 1 τ 1 < 1 and in conjuction with the fact that ξ is C 1 function in both arguments (as a result of (7a)), we obtain the existence-uniqueness of classical solutions in Π τ1 . Iterating the previous local existence-uniqueness result in domains (Π jτ1 ∩ Π T ) \ Π (j-1)τ1 , where j ≤ T /τ 1 , we obtain classical solutions in Π T and since T is arbitrary, we obtain unique classical solutions in Π. The proof is complete.

APPENDIX B PROPERTIES OF THE LYAPUNOV FUNCTIONAL

We prove here two properties, which we have invoked in the stability proof of Theorem 1.

Property 1: For every continuous functions (or more generally, L ∞ functions, where • ∞ denotes the ess.sup-norm) ε, ε t : [0, L] → R n ; L > 0, matrix P , π : [0, L] → R as in [START_REF] Kitsos | High-gain observer design for a class of hyperbolic systems of balance laws[END_REF], ρ 0 > 0, and µ in R, we have

G p := ε P ε + ρ 0 ε t P ε t p .

(62b)

where ω 3

 3 ,δ := |P | eig(P ) max {4γ 2,δ + ρ 0 γ 11,δ , 3γ 11,δ + γ 10,δ } , ω 4,δ := |P | eig(P ) (γ 2,δ + ρ 0 γ 11,δ )(1 + L).

ξ 2 (

 2

Fig. 1 .

 1 Fig. 1. Transmission, recovery, and death rates as functions of age

Fig. 2 .Fig. 4 .

 24 Fig. 2. Time and age evolution of simulated system's output

Fig. 5 .

 5 Fig. 5. Time and age evolution of the third estimation error

  u)]| u= ξ1 +s( ξ2 -ξ1 ) op ×| D u s δ (u)| u= ξ1 +s( ξ2 -ξ1 ) | sup (t,x)∈Π τ 0 | ξ1 (t, x) -ξ2 (t, x)| ≤ L f sup (t,x)∈Π τ 0 | ξ1 (t, x) -ξ2 (t, x)|.

×,

  D u f (z, F [u])| u=s δ ( ξ(s))(z) , D ξ s δ ξ(s) (z) × ξz (s, z) -∂ z (λ • y)(s, z) ξz (s, z) -ΘKy z (s, z))]| z=ω(s;t,x) ds, R 1 [ ξ](t, x) when t 0 (t, x) = 0 and R 1 [ ξ](t, x) = (λ • y) -1 (τ, 0) × f 0, F s δ ξ(τ ) (0) -D u H(u)| u=s δ ( ξ) (τ, L) ×D ξ s δ ξ(τ ) (L) ξτ (τ, L) τ =t0(t,x)

  we get the following expression for the right boundary:ξt (t, L) = -(λ • y)(t, L) ξx (t, L) + A ξ(t, L) -ΘK y(t, L) -C ξ(t, L) + f L, F s δ ξ(t) (L) = -(λ • y)(t, L) Φ (t, s) D u f (z, F [u])| u=s δ ( ξ(s))(z) , D ξ s δ ξ(s) (z) ξz (s, z) -∂ z (λ • y)(s, z) ξz (s, z)-ΘKy z (s, z))]| z=ω(s;t,L) ds + (A + ΘKC) ξ(t, L)-ΘKy(t, L) + f L, F s δ ξ(t) (L) .

  lim p→+∞ W p = e µ• ε P ε + ρ 0 ε t P ε t ∞ (61)where W p , as in (21a), is given byW p := L 0 π(x)e pµx G p (x)dx 1/p ;

  2,p T 2,p ≤ (ω 1,δ + p|µ θ,δ |ω 2,δ )W p

						p	(32)
	where	ω 1,δ :=	|P |δγ 7,δ eig(P )	,	ω 2,δ :=	|P |γ 9,δ eig(P )	.

  1 solutions in Π T , by considering the problem for ξx .

	By (55), we obtain:
	ξx (t, x) =	d dx	Φ (t, t t
			t0(t,x)

0 (t, x)) R[ ξ](t, x) +Φ (t, t 0 (t, x)) R 1 [ ξ](t, x) + Φ(t, s)

  τ 1 |∂ x ξ1 -∂ x ξ2 | ≤ τ 1 q 1 sup

	|∂ x	ξ1 -∂ x	ξ2 |;
	(t,x)∈Π τ 1
	q 1 := max

(t,x)∈Π τ 1 |Φ(t, t 0 (t, x))| (L f + max (t,x)∈Π τ 1 ∂ x |(λ • y)(t, x)| 1 + L H max (t,x)∈Π τ 1 ((λ • y)(t, x)) × max (t,x)∈Π τ 1

APPENDIX A PROOF OF LEMMA 1

To prove the existence and uniquenes of global classical solutions to the observer system, which is a semilinar system of hyperbolic PIDEs, we follow a fixed-point methodology inspired by [START_REF] Kmit | Classical solvability of nonlinear initial-boundary problems for first-order hyperbolic systems[END_REF] (Theorem 2.1), where a similar result is proven for a system of semilinear hyperbolic PDEs.

We first represent [START_REF] Christofides | Feedback control of hyperbolic pde systems[END_REF] in its integral form, as follows:

where ω(s; t, x) denotes the characteristic (for characteristic velocity λ • y) passing through (t, x) ∈ Π, t 0 (t, x) is the smallest value of s at which x 0 = ω(s; t, x) reaches ∂Π and by Φ(t, t 0 ) we denote the fundamental matrix of A + ΘKC. More precisely, ω(s; t, x) is the unique solution of

It is sufficient to prove this lemma in Π T := (0, T ) × (0, L) for some fixed T > 0.

In conjunction with our regularity assumptions, it is easy to see that for every δ > 0, there exist L f , L H ≥ 0, such that

First, we prove that there exists a unique continuous solution in Π τ0 , for some τ 0 > 0, such that:

From the integral representation (55) we obtain the following equation for t in [0, τ 0 ]:

To prove this, define, first for ¯ > 0 arbitratily small, the set

Then, for all p in [1, +∞),

where µ(B ¯ ) denotes the Lebesgue measure of B ¯ .

The above gives

Furthermore,

This implies

since Lπ is finite. By virtue of ( 65) and (67), we obtain (61). Property 2:

in the weak * topology σ L ∞ (0, +∞), L 1 (0, +∞) ,

where by σ(X, X * ) we denote the weak * topology on X, with X * being the topological dual of X. In 2010 he joined the Gipsa-lab, Grenoble, France where he is currently a senior researcher of the CNRS. His current research interests include nonlinear control theory, hybrid systems, and control of partial differential equations.


Lorenzo Spadini

Aline Navel

Jean M F Martins

Erwann Vince

Isabelle Lamy

Soil aggregates: a scale to investigate the densities of metal and proton reactive sites of organic matter and clay phases in soil

scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In dilute conditions, non-redox trace metals (TM) adsorbed on soil are distributed over accessible surface sites of key reactive organic and mineral phases in relation to site densities and sorption strengths. Despite the importance of such processes for environmental quality purposes, determining the distribution of metals in whole soils remains a difficult task. Research on pure organic and mineral phases has allowed the development of conceptual models of metal retention on relevant soil phases such as organic matter [START_REF] Hiemstra | A surface structural approach to ion adsorption: the charge distribution (CD) model[END_REF][START_REF] Tipping | Humic ion-binding model VI: an improved description of the interactions of protons and metal ions with humic substances[END_REF], clay [START_REF] Tournassat | The titration of clay minerals II. Structure-based model and implications for clay reactivity[END_REF]) and X-ray amorphous iron oxides [START_REF] Dzombak | Surface Complexation Modeling: Hydrous Ferric Oxide[END_REF][START_REF] Spadini | Structure and stability of Cd 2+ surface complexes on ferric oxides: 1. Results from EXAFS spectroscopy[END_REF]. However, the transfer of this knowledge from pure phases to natural soils is difficult because complex and coated phases coexist and control water fractionation (mobile and immobile), which modify the accessibility of water and reactivity of the phases (e.g. Jocteur [START_REF] Monrozier | Distribution of microorganisms and fate of xenobiotic molecules in unsaturated soil environments[END_REF]. Experimental sequential or selective extraction processes that aim to associate extracted metals with specific mineral or organic carrier phases give limited information [START_REF] Zimmerman | Heavy metal and trace metal analysis in soil by sequential extraction: a review of procedures[END_REF]. Theoretical distributions of metals originating from speciation modelling are more promising. For instance, [START_REF] Weng | Contribution of individual sorbents to the control of heavy metal activity in sandy soil[END_REF] modelled the distribution of sorbed copper on the organic, clay and oxide contents of L. Spadini a , A.Navel a , J.M.F.Martins a , E.Vince a & I. Lamy b a IGE UMR 5001 / UGA / CNRS / G-INP/ IRD, CS 40700, 38 058, Grenoble Cedex 9, France, b INRA / AgroParisTech, UMR 1402 ECOSYS, pôle Ecotoxicologie, RD10, F-78026, Versailles Cedex, France soil types that varied in their pH and mineral and organic contents. Important variables need to be identified for modelling, but their experimental verification is lacking. Site density of reactive phases is one of these essential variables, but it is difficult to determine experimentally. For example, illite is classically used in modelling to represent the entire soil clay content, and its site density is fixed arbitrarily with no convincing experimental basis. For reactive iron oxides, a surface area of 600 m 2 g -1 is proposed as a default for model input data; this value originated from titrated fresh ferrihydrite precipitates [START_REF] Dzombak | Surface Complexation Modeling: Hydrous Ferric Oxide[END_REF]. The surface area of this solid, however, evolves rapidly over time in the laboratory; therefore, the representativeness of such synthetic compounds in situ is questionable (Schwertmann & Cornell, 2000). Finally, humic or fulvic compounds are generally used to represent reactivity of the soil's organic phase, but some organic matter is in the form of bioclasts or particulate organic matter with lower site density and different reactivities [START_REF] Sebastia | Buffer capacity and Cu affinity of soil particulate organic matter (POM) size fractions[END_REF].

Research on metal reactivity in bulk soil has not, as yet, taken account of soil aggregates although they are key components in soil structure. Soil aggregates of a given size class are specific for several properties relevant to soil function, such as (i) mineral and organic composition [START_REF] Tisdall | Organic matter and water-stable aggregates in soils[END_REF][START_REF] Ducaroir | Evidence of trace metal association with soil organic matter using particle size fractionation after physical dispersion treatment[END_REF][START_REF] Balabane | Aggregation and carbon storage in silty soil using physical fractionation techniques[END_REF], (ii) metal distribution [START_REF] Besnard | Influence of organic amendments on copper distribution among particle-size and density fractions in Champagne vineyard soils[END_REF][START_REF] Quenea | Interactions between metals and soil organic matter in various particle size fractions of soil contaminated with waste water[END_REF][START_REF] Navel | Effect of long term organic amendments and vegetation of vineyard soils on the microscale distribution and biogeochemistry of copper[END_REF]) and (iii) overall microbial functioning [START_REF] Chenu | Short-term changes in the spatial distribution of microorganisms in soil aggregates as affected by glucose addition[END_REF][START_REF] Constancias | Microscale evidence for a high decrease of soil bacterial density and diversity by cropping[END_REF]. An aggregate size class thus represents a soil domain of closely related chemical, physical and biological characteristics.

In this research, we assumed that determining reactivities of trace metals at the scale of aggregates could improve our understanding of metal distributions and more generally of biogeochemical processes in soils. We combined wet chemical methods to determine reactivity with gentle physical fractionation methods that would preserve the integrity of the soil's microstructure [START_REF] Jocteur Monrozier | Components and microbial biomass content of size fractions in soils of contrasting aggregation[END_REF][START_REF] Six | A history of research on the link between (micro)aggregates, soil biota, and soil organic matter dynamics[END_REF][START_REF] Navel | Effect of long term organic amendments and vegetation of vineyard soils on the microscale distribution and biogeochemistry of copper[END_REF]. The overall approach adopted in this study was as follows:

1. The objective was to determine the distribution of non-redox sensitive and surface complex-forming TM such as Cu(II), Zn(II), Pb(II) and Cd(II). Copper was chosen as a test element because of its tendency to precipitate little in soil and strong association with surface-forming complexes with 'soft' acids, such as those existing on organic matter [START_REF] Kirpichtchikova | Speciation and solubility of heavy metals in contaminated soil using X-ray microfluorescence, EXAFS spectroscopy, chemical extraction, and thermodynamic modeling[END_REF][START_REF] Jacobson | Electron microprobe and synchrotron X-ray fluorescence mapping of the heterogeneous distribution of copper in high-copper vineyard soils[END_REF][START_REF] Causse | Xanthan exopolysaccharide: Cu 2+ complexes affected from the pH-dependent conformational state; implications for environmentally relevant biopolymers[END_REF]. 2. Organic matter (OM) and clay are the two main soil components that determine reactivity, which is widely accepted for non-tropical soils (i.e. in the absence of large amounts of reactive iron (hydr)oxide phases). Thus, the total carbon concentration (because of the filamentous nature of organic OM) and the 'partial' extraction of Al (because of the particulate nature of Al minerals) can be taken as representative measures of site densities of organic matter and Al-bearing secondary mineral phases (clays and Al oxides) in soil. In this study we used a silty loam vineyard soil (Mâcon, France) commonly found in France, which has been the object of a long-term experiment of controlled organic inputs [START_REF] Lejon | Relationships between soil organic status and microbial community density and genetic structure in two agricultural soils submitted to various types of organic management[END_REF]. 3. The total concentration of weak acids of soil solids, easily accessed from titrimetric measures, is a measure of the total concentration of metal complexation sites. In that, we followed classical reactivity determinations of pure organic and mineral phases [START_REF] Dzombak | Surface Complexation Modeling: Hydrous Ferric Oxide[END_REF][START_REF] Hiemstra | A surface structural approach to ion adsorption: the charge distribution (CD) model[END_REF][START_REF] Spadini | Hydrous ferric oxide: evaluation of Cd-HFO surface complexation models combining Cd K EXAFS data, potentiometric titration results, and surface site structures identified from mineralogical knowledge[END_REF] or organic phases [START_REF] Tipping | Humic ion-binding model VI: an improved description of the interactions of protons and metal ions with humic substances[END_REF][START_REF] Milne | Generic NICA-Donnan model parameters for metal-ion binding by humic substances[END_REF][START_REF] Borrok | A universal surface complexation framework for modeling proton binding onto bacterial surfaces in geologic settings[END_REF]. 4. The determination of total weak acid concentrations, total carbon and extracted Al in five soil size fractions (> 250, 250-63, 63-20, 20-2 and < 2 μm), assuming constant site densities of organic carbon and extracted Al in all fractions, allowed their determination through linear combination analysis. The contrast in distribution of Al compared with OM phases in both coarse and fine fractions of the soil ensured a robust fit. 5. Highly reactive soil organic matter consists mainly of degraded organic matter such as humic acids and microbial compounds (e.g. extracellular polymeric substances, [START_REF] Causse | Xanthan exopolysaccharide: acid-base reactivity related to structure and conformation. A model for understanding the reactivity of degraded and colloidal soil organic matter[END_REF].

Microbes that have high site densities can also be considered as highly reactive organic matter [START_REF] Borrok | A universal surface complexation framework for modeling proton binding onto bacterial surfaces in geologic settings[END_REF][START_REF] Guiné | Approche expérimentale et théorique de la réactivité de surface et de la mobilité des colloïdes bactériens en milieu poreux: impact sur le transfert accéléré de métaux lourds et intérêt pour la bio-dépollution des sols[END_REF]. In contrast, particulate organic matter (POM) is comprised of clasts, which have lower site densities, and is essentially composed of non-reactive cellulose and poorly reactive lignin. In this study, in order to determine site densities of the POM size classes, we compared the same soil with or without long-term organic matter amendment. We considered that the site density of organic matter present in soil not amended with organic matter was comparable to that of highly reactive organic matter (e.g. humic acids), such as that determined in pure phase studies.

Materials and methods

Soil characteristics and sampling

A vineyard field site at Mâcon Clessé (Burgundy, France, 46 ∘ 24 ′ 59 ′′ N, 4 ∘ 48 ′ 54 ′′ E) was the object of a long-term field amendment experiment. From 1991 to 2001, specific plots were amended with conifer compost (called amended soil, 100 m 3 ha -1 every 3-4 years), whereas others remained unamended (control soil). Amendment with conifer compost modifies soil physical and chemical properties compared with the control soil [START_REF] Navel | Effect of long term organic amendments and vegetation of vineyard soils on the microscale distribution and biogeochemistry of copper[END_REF]. Copper fungicide (Bordeaux mixture, i.e. a hydrated mixture of CuSO 4 and Ca(OH) 2 ) was applied twice a year, which increased soil copper content in the top 5 cm of the soil from 15 (standard deviation, 5) to 70 (27) mg Cu kg -1 dry soil. The first value was determined from two soil cores sampled between 25-cm and 60-cm depth, whereas the second value was obtained from 112 surface soil samples [START_REF] Navel | Distribution, spéciation, impact et transfert du cuivre dans un sol sous vigne: rôle de la structuration spatiale et du statut organique[END_REF]. The data inside the parenthesis represent the standard deviation of the copper concentration obtained from the copper analysis of the replicated field samples. This soil has been described as a carbonate-free deep Eutric Cambisol (FAO, 1998), with a fine clay loam texture containing 12% sand, 57% silt and 31% clay [START_REF] Navel | Distribution, spéciation, impact et transfert du cuivre dans un sol sous vigne: rôle de la structuration spatiale et du statut organique[END_REF][START_REF] Bartoli | Impact of organic inputs on wettability characteristics and structural stability in silty vineyard topsoil[END_REF]. Exposure to long-term aqua regia led to extracted contents of 31 and 51 g kg -1 Fe and 28 and 48 g kg -1 Al in the control and OM-amended soils, respectively. Soil pH was 6.8 for the control and 6.6 for the OM-amended soils [START_REF] Sebastia | Changes in soil organic matter chemical properties after organic amendments[END_REF][START_REF] Navel | Effect of long term organic amendments and vegetation of vineyard soils on the microscale distribution and biogeochemistry of copper[END_REF].

No Mn nodules were present, which could otherwise play a role, together with organic matter and clay, in the complexation of Cu. The soil was sampled in February 2008 [START_REF] Navel | Effect of long term organic amendments and vegetation of vineyard soils on the microscale distribution and biogeochemistry of copper[END_REF]. Four replicates of each of the two soil treatments (control and OM amended) were sampled randomly from each of the four plots of each treatment and combined to obtain a representative composite sample. Each soil sample was thoroughly homogenized in a concrete mixer, air-dried and sieved to 4 mm.

Soil fractionation

The fractionation protocol gently separates the soil aggregates in order to preserve the natural aggregation of the soil (Jocteur [START_REF] Jocteur Monrozier | Components and microbial biomass content of size fractions in soils of contrasting aggregation[END_REF]. Thirty grams dw (dry weight) of the soils were gently dispersed in 200 ml of distilled water in 500-ml bottles by shaking at 250 revolutions per minute (rpm) for 1 hour. Particles > 20 μm that had settled were gently resuspended in 200 ml of water. This procedure was repeated three times. The remaining particles > 20 μm were then separated by wet sieving into the following three size fractions: 4 mm to 250 μm (identified as > 250 μm), 250-63 μm and < 63 μm. The 63-20 μm fraction was obtained by natural sedimentation for about 10 minutes [START_REF] Navel | Effect of long term organic amendments and vegetation of vineyard soils on the microscale distribution and biogeochemistry of copper[END_REF]. Subsequently, the < 20 μm particles in the supernatant solutions were allowed to settle by natural sedimentation for 5 to 6 hours to separate the 20-2 μm and the < 2 μm fractions. Finally, the <2 μm solid fraction was collected by centrifugation at 11 000 g. The quality of the fractionation procedure was evaluated by laser granulometry, performed with an automated Mastersizer 2000 (Malvern Instruments Ltd, Malvern, UK) as described by [START_REF] Navel | Effect of long term organic amendments and vegetation of vineyard soils on the microscale distribution and biogeochemistry of copper[END_REF]. The five fractions recovered from all samples were air-dried to maintain the best possible cohesion of the aggregates. In these experiments, the dissolved concentrations of all elements of interest (C, H, Al and Cu) were relatively small compared with the soil content, so that loss during fractionation was disregarded. The pH values of the soil fractions made on three replicates of each fraction of the composite soil from coarse to fine followed the order 6.5 (0.2), 6.7 (0.1), 6.7 (0.2), 7.0 (0.1) and 7.0 (0.1) for the control and 6.8 (0.1), 6.8 (0.1), 7.0 (0.2), 7.2 (0.2) and 7.1 (0.2) for the amended soil. These pH values were smaller than those measured for soil fractionated without air-drying [START_REF] Navel | Distribution, spéciation, impact et transfert du cuivre dans un sol sous vigne: rôle de la structuration spatiale et du statut organique[END_REF].

Chemical and extraction analysis

Aluminium extraction. For both clays and Al oxides, the surface site density depends on the crystal size: the larger are the crystals, the lower is the site density. A mixture of differently sized minerals may be present. For this reason, the total Al in soil might not be ascribed to the Al-mineral site density. Rather, this density was related to the amount of acid-leached Al by aqua regia (1/3 HNO 3 /HCl, v/v ratio), because this acid only partly dissolves clays and only marginally dissolves tectosilicates [START_REF] Krause | Comparative study of different digestion procedures using supplementary analytical methods for multielement-screening of more than 50 elements in sediments of the river Elbe[END_REF][START_REF] Chen | Comparison of three aqua regia digestion methods for twenty Florida soils[END_REF][START_REF] Ryan | Stepwise effects of the BCR sequential chemical extraction procedure on dissolution and metal release from common ferromagnesian clay minerals: a combined solution chemistry and X-ray powder diffraction study[END_REF]. The Al-containing primary phases will not dissolve readily, and other acid-sensitive Al sources can be reasonably excluded in this soil. The X-ray analysis indicated the absence of well-crystallized Fe and Al oxide phases. In the aluminium leachates, E Al was determined in duplicate for bulk and fractionated soil samples. Briefly, 5 ml of reagent and 0.5 g ground dry soil were incubated at 70 ∘ C for 14 hours, filtered and diluted with 50 ml distilled water for ICP-AES (3000 DV, Perkin Elmer, Paris, France) measurements. Standard solutions were prepared in the same aqua regia matrix.

Organic matter analysis. For highly reactive (HR) organic matter, we assumed that total organic carbon content (T OC ) was proportional to complexation site density for each fraction, suggesting a unique reactivity of HR OM among fractions. This accords with recent findings demonstrating the overall high and similar site densities of various soil reactive organic compounds: humic and fulvic acids [START_REF] Milne | Generic NICA-Donnan model parameters for proton binding by humic substances[END_REF], bacterial exopolysaccharides and biofilms [START_REF] Causse | Xanthan exopolysaccharide: acid-base reactivity related to structure and conformation. A model for understanding the reactivity of degraded and colloidal soil organic matter[END_REF], bacterial cells [START_REF] Borrok | A universal surface complexation framework for modeling proton binding onto bacterial surfaces in geologic settings[END_REF] and dissolved organic matter [START_REF] Hruška | Is a universal model of organic acidity possible: comparison of the acid/base properties of dissolved organic carbon in the boreal and temperate zones[END_REF]. The site densities of these compounds have a small range between 1 and 5.7 mmol g -1 dw , [START_REF] Causse | Xanthan exopolysaccharide: acid-base reactivity related to structure and conformation. A model for understanding the reactivity of degraded and colloidal soil organic matter[END_REF], allowing us to define a single descriptor for all these HR organic constituents. The analysis of the POM fraction of the soil studied is described in [START_REF] Sebastia | Changes in soil organic matter chemical properties after organic amendments[END_REF]. For both HR and POM OM, the T OC was determined by analysing dried and crushed duplicate 20 mg samples of bulk soil and size fractions, respectively, inserted into a FlashEA1112/FLASH 2000 Elemental Analyser equipped with a thermal conductivity detector (Thermo Scientific, Paris, France).

Determination of pH.

The pH values of bulk and fractionated soil samples were measured at a 1/10 solid/liquid mass ratio with a calibrated combined pH electrode (Metrohm, Paris, France).

Copper extraction. Total copper concentrations (T Cu ) were deter- mined in duplicate in aqua regia extracts of bulk and fractionated soil samples, following the same procedure as used for aluminium determination. Standard solutions were prepared in the same aqua regia matrix. The quantification limit for Cu determined in triplicate was 20 μg l -1 .

Acid-base titrimetry. This study aimed to determine the total concentrations of weakly acidic acid-base reactive sites of the soil and its fractions. The fraction-specific variable T Hs is defined as the number of protons released into the solution from surfaces at pH 4 and 10:

T Hs = H S (pH (10)) -H S (pH (4)) , ( 1
)
where H S represents the concentration of surface protons released into the solution, referenced to an arbitrary charging state of that surface [START_REF] Herbelin | FITEQL 4.0: A Computer Program for Determination of Chemical Equilibrium Constants from Experimental Data[END_REF], and T Hs is determined by acid-base titrations at a fixed 0.1 m ionic strength. Experimental details, the rationale for working at fixed ionic strength, the pH electrode calibration, the recognition of mineral dissolution and precipitation issues in titrations and issues related to cation exchange reactions are addressed in the Supporting Information.

Modelling

Modelling was based on four basic assumptions. The first assumption was that weak acids are metal complexation sites. It is assumed that at pH 4 and 10 (the weak acid range) each surface site exchanges one proton against one copper ion. Based on that simplified assumption, the experimentally determinable T Hs (mol Hs g -1 dw ) equals the total concentration of Cu(II) complexing surface sites. The second basic assumption is that in all fractions, there are three sources for T Hs : (i) soil OM (T Hs-C ), (ii) Al-bearing secondary mineral phases (clay and Al oxides, T Hs-Al ) and (iii) residual non-clay, non-Al, non-OM phases (T Hs-Res ).

The third assumption was that both carbon-and Al-bearing mineral phases have a unique mass-based site density in all fractions, which is, D Hs-C (mol Hs g -1 C ) for carbon and D Hs-Al for Al-bearing secondary mineral phases.

For organic carbon, a linear relation between total organic carbon T OC [g C g -1 dw ] and site density is assumed:

T Hs-C = T OC × D Hs-C , ( 2
)
based on (i) the almost exclusive presence of aged HR OM in the non-amended soil and (ii) the uniform water accessibility of this polymeric phase in all soil fractions. In contrast to carbon in clays and oxides, site density does not relate to total Al, but to the specific surface area. This may be approached by stripping a limited number of reactive Al surface layers; that is, the measured concentration of partially extracted Al E Al . A unique site density, D Hs-Al , of the extracted Al is necessarily assumed in all fractions, impeded by a unique dominant Al phase in all fractions or by a constant mixture of Al minerals in all fractions. Conceptually, both statements may be considered valid in considering large aggregates as a corpus comprised of smaller aggregates.

Finally, T Hs-calc was fitted to the experimentally determined T Hs in all soil fractions by adjusting the fraction-independent densities D Hs-C and D Hs-Al . Differences between T OC compared with E Al in the fractions ensured an unambiguous result from the following calculation:

T Hs-calc = T Hs-C + T Hs-Al + T Hs-Res = T OC × D Hs-C + E Al × D Hs-Al + T Hs-Res .
(3)

Here, T Hs-Res is considered constant among the fractions. As shown below, its concentration is too small to affect the two fitted site densities.

In OM-amended soil, the increase in POM content affects the overall OM reactivity. The T Hs-C (mmol g -1 f ) is consequently split into two terms, one for high-reactivity carbon (HC), T Hs-HC , and one for low-reactivity carbon (LC, i.e. POM carbon), T Hs-LC , as follows:

T Hs-C = T Hs-HC + T Hs-LC = T OC-HC × D Hs-HC + T OC-LC × D Hs-LC-f (4)
The total carbon concentration is expressed as the sum of the high and low organic reactivity contributions:

T OC = T OC-HC + T OC-LC , ( 5
)
and each contribution is associated with its density, D Hs-HC and D Hs-LC-f . Concerning the HR contribution: there was almost no POM in the control soil (i.e. no low-reactivity carbon). Consequently, the site density of organic carbon determined in this soil (D Hs-C ) corresponds to that of highly reactive carbon, and might equal the corresponding density in the amended soil: D Hs-HC = D Hs-C . The density D Hs-HC was consequently fixed to the value D Hs-C obtained in the control soil.

For the LR contribution: in the amended soil, POM is expected to be present as two coarse fractions of > 250 μm and 250-63 μm. The site densities of the two fractions vary with the POM size because the specific surface of 'closed' (i.e. non-porous) solids increases with decreasing volume. Thus, in contrast to D Hs-HC the POM site density D Hs-LC-f is fraction specific. A site density was then calculated for each of the two coarse POM fractions.

The T Hs-LC was obtained by potentiometric titration of POM isolated from soil through flotation in water, whereas T OC-LC (g C g -1 POM ) was obtained by total carbon analysis of POM. 'Clean' POM (i.e. POM without mineral impurities) should have a T OC-LC value close to 0.5 g C g -1 POM , because a factor of two separates carbon and organic matter masses in soil [START_REF] Pribyl | A critical review of the conventional SOC to SOM conversion factor[END_REF]. This relation enables correction for the presence of mineral impurities in floating POM. Consequently, this approach can provide the final comparison of site densities for size-specific POMs with those of HR carbon and extracted Al. Dissolved matter (copper and OM) was not considered because its contribution is negligible compared with that of the solid moieties. Scanning electron microscope figures are provided in the Supporting Information.

Results

Characteristics of bulk soils and soil size fractions

Table 1 presents the physical and chemical characteristics of the size fractions of amended (conifer compost) and control (not amended) soils. The two dominant size fractions in terms of mass are the 63-20 and 20-2 μm (Table 1), which is consistent with the clay loam texture of the soil.

For the control soil, T OC concentrations (Table 1) were larger (35 and 24 mg g -1 f ; f fraction) in the two coarse fractions, whereas T OC , T Cu , T Hs , T Hs-calc and E Al : total organic carbon, total copper, total measured and calculated proton exchange capacity and acid-extractable aluminium concentrations, respectively (kg -1 f , g -1 f : per kg or g fraction mass). Parentheses represent standard deviations from triplicate experiments.

the distribution of E Al , representing Al from partially extracted Al phases (clay and Al oxides), showed the reverse pattern and was largest in the two fine fractions (43.5 and 18.9 mg g -1 f ), which was expected for this clay-rich soil. Contrasting distributions of Al and carbon were observed in the two coarsest and two finest size fractions (> 250 and 250-63 μm and 20-2 and < 2 μm, respectively), which was consistent with the expected reactivity and distribution of clay and OM in such a clay loam soil. In contrast to T OC and E Al distributions, total copper concentrations, T Cu , were similar in all soil size fractions (around 100 μg g -1 , Table 1), except for the 63-20-μm fraction, where T Cu was significantly less, which was similar to T OC and E Al .

Despite variable clay and organic matter contents in the five soil size fractions, T Cu appeared quite homogeneous in all fractions, except the poorly reactive 63-20-μm fraction. This result shows that copper reacted with both clay and OM. As for T Cu , the T Hs values of the two coarsest and two finest size fractions were similar (about 0.25 mmol g -1 f ), whereas that of the 63-20-μm fraction was small (0.08 mmol g -1 f ). The similarity between T Cu and T Hs supports our hypothesis that copper binds preferentially to acid-base reactive sites. In the 63-20-μm fraction, T Cu and T Hs concentrations were small, despite being the more abundant fraction in the soil. This is consistent with the fact that the silt fraction usually comprises poorly reactive constituents, including primary minerals characterized by small specific surface areas.

For the amended soil, the patterns of distribution of T Hs , T OC , T Cu and E Al were similar to those of the control soil (Table 1), suggesting similar Cu-binding modes in both soils. In absolute terms, T Hs , T OC and T Cu values increased in the amended soil compared with the control. The long-term organic amendment of the clay loam soil thus increased its global soil reactivity. At the scale of the size fractions, this was particularly valid for the two coarse fractions (> 250 and 250-63 μm) in which exogenous organic matter accumulates preferentially [START_REF] Jocteur Monrozier | Components and microbial biomass content of size fractions in soils of contrasting aggregation[END_REF][START_REF] Oorts | Carbon and nitrogen stocks in relation to organic matter fractions, aggregation and pore size distribution in no-tillage and conventional tillage in northern France[END_REF].

The ratios of T Cu /T OC and T Hs /T OC are shown in Figure 1. From the coarse to the fine fractions, the ratios increased, except for the 63-20-μm fraction of both the control and the amended soils. This tendency to increase with decreasing fraction size suggests that both Cu and protons do not react exclusively with OM, but rather with both OM and clay. In comparison, the T Cu /T Hs ratio was relatively constant between all fractions in both soils: within a standard deviation of 23% (Table 1). This relative constancy confirms that copper is bound to acid-base reactive and metal-complexing surface sites, whatever the nature of the reactive phase. This also confirms that non-protonable sites, such as cation exchange sites, are poorly involved in Cu retention, [START_REF] Covelo | Simultaneous sorption and desorption of Cd, Cr, Cu, Ni, Pb, and Zn in acid soils: II. Soil ranking and influence of soil characteristics[END_REF].

In the control soil, both T Cu /T OC and T Hs /T OC ratios of the 63-20-μm fraction were relatively large compared with the other fractions. This suggests that, in addition to clay and OM, other proton and copper reactive phases exist. Overall the 63-20-μm fraction had the smallest site density for protons and copper (T Hs , T Cu ), with particularly small carbon and clay contents (small T OC , E Al ). Thus, this fraction explicitly and sensitively revealed the presence of other Cu reactive sites (i.e. non-clay and non-OM phases with probably minor site densities). These sites are referred to here as 'reactive residues' (T Hs-Res ).

Modelling phase reactivity in the control soil

Modelling involved varying the global D Hs-C , D Hs-Al and T Hs-Res model parameters in order to adjust T Hs-calc to T Hs as closely as possible in all fractions (Equation (3), Table 1). Figure 2(a) shows that T Hs-calc follows the trend of T Hs among the fractions, suggesting the validity of our conceptual approach. Figure 2(a) also presents the calculated phase-specific T Hs-C , T Hs-Al and T Hs-Res reactivity contributions. As expected, there was a relative increase in T Hs-C in the two coarse fractions compared with the two fine fractions, whereas the opposite was observed for clay. Fitting the model to the data without considering T Hs-Res led to an acceptable fit in all fractions, except for the 63-20-μm fraction because of its low global reactivity. The T Hs-Res value was determined in the 63-20-μm fraction and considered to exist equally in all other fractions. This clearly resulted in a close agreement between T Hs and T Hs-calc in this fraction. We considered this T Hs-Res contribution to be background reactivity present in all fractions. It represented about half of the reactivity of the 63-20-μm fraction and constituted a minor contribution to Cu reactivity in the other soil size fractions.

Figure 2(a) shows that T Hs-calc > T Hs in the > 250-μm fraction. This is the expected tendency for POM present in the coarse fractions. The low site density of this type of OM compared with that of HR organic matter led to T Hs-calc being overestimated. The analysis of the amended soil, which contained far more POM, consequently required different types of OM to be considered.

Modelling of the phase reactivity of the amended soil

Organic amendment induced the doubling of T OC from 15.3 g C kg -1 dw (control soil) to 30.3 g C kg -1 dw (amended soil, Table 1) at the sampling date (February 2008). The T OC-amended /T OC-control ratio was 3.1, 2.1, 1.7, 1.2 and 1.4 in the five fractions (from the coarsest to the finest, respectively), confirming that exogenous OM accumulates preferentially in coarse soil fractions before being progressively incorporated into the microaggregates [START_REF] Navel | Effect of long term organic amendments and vegetation of vineyard soils on the microscale distribution and biogeochemistry of copper[END_REF]). The T Hs-amended /T Hs-control ratio decreased correspondingly in these five fractions (1.6, 1.3, 1.3, 1.2 and 1.0), although the decrease was smaller than for the T OC-amended /T OC-control ratio. In the > 250-μm fraction, a three-fold increase of T OC (34.7 to 107.6 mg g -1 , non-amended to amended soil) induced only 60% of the increase in reactivity of H S . This is consistent with the incorporation into the soil of poorly reactive OM (i.e. POM) during the period of amendment. The application of Equation (3) to the amended soil, using as a first approximation the densities, D Hs-C and D Hs-Al , fitted to the control soil, resulted in large T Hs-calc-amended values (Figure 2(b)) for the amended soil compared with its experimental value T Hs . This presupposes the exclusive presence of highly reactive OM in all fractions, which clearly did not apply to the amended soil. The excess reactivity calculated was larger in the > 250-μm fraction than in the 250-63-μm fraction, indicating greater reactivity of the POM present in the latter. This is consistent with the progressive degradation of POM, leading to the presence of smaller OM particles. This fits the expected size-reactivity relation of POM and confirms former findings of POM reactivities measured in this soil [START_REF] Sebastia | Buffer capacity and Cu affinity of soil particulate organic matter (POM) size fractions[END_REF].

Discussion

Site densities of soil Al mineral phases

In the control soil, the calculated D Hs-Al , representing clay and Al oxides, was 0.005 or 0.135 mol mol -1

Al . Thus, one of the 7.4 extracted Al atoms can be considered acid-base reactive. This means that the acid dissolved more than one surface layer. Schematically, one may consider that an average of seven consecutive surface layers was stripped from reactive Al minerals. However, E Al represented about 1.7% only of the soil mass (Table 1), which is a small fraction of the total Al contained in the clay of this soil (31% clay, [START_REF] Bartoli | Impact of organic inputs on wettability characteristics and structural stability in silty vineyard topsoil[END_REF]. Thus, only a minor fraction of clay and or Al oxides, or both, was dissolved, as expected. Under these conditions, the extracted Al can be considered to represent a measure of the density of clay edges and Al oxide sites. The aqua regia reactant might be too strong, however, a diluted form could be used in further studies.

Site densities of organic matter phases in soil

In the control soil, the site density, D Hs-C , of the measured organic carbon was 6.94 mmol g C -1 or 83.3 mmol mol C -1 . Values of 7.74 and 5.70 mmol g OM -1 calculated from pure-phase studies are commonly accepted for fulvic and humic acids, respectively [START_REF] Milne | Generic NICA-Donnan model parameters for proton binding by humic substances[END_REF], which can be converted to 15.48 and 11.4 mmol g C -1

using a conversion factor of 0.5 g C g OM -1 [START_REF] Pribyl | A critical review of the conventional SOC to SOM conversion factor[END_REF][START_REF] Causse | Xanthan exopolysaccharide: acid-base reactivity related to structure and conformation. A model for understanding the reactivity of degraded and colloidal soil organic matter[END_REF]. The D Hs-C is thus about half of the site densities observed for fulvic and humic acids, which are among the most reactive OM types in soil. These results show that the HR OM assessed at the scale of aggregates compares with purified HR OM in terms of site density and that aqueous solutes can easily access these sites in the soil studied.

In the same amended soil, [START_REF] Sebastia | Buffer capacity and Cu affinity of soil particulate organic matter (POM) size fractions[END_REF] found T OC-LC values of 0.38 (0.01) g C g -1 POM and 0.28 (0.03) g C g -1 POM for the 2000-200-μm and 200-50-μm POM size-fractions, respectively. These fractions can be compared with the > 250-μm and 250-63-μm soil size fractions of the present study. The decrease in T OC-LC with particle size relates to increasing mineral impurities in POM. 'Clean' POM should have a carbon/particle mass ratio of 0.53 g C g -1 OM , regardless of the size because this ratio characterizes both wood and humic substances, the principal constituents of POM [START_REF] Volkoff | Caractéristiques biochimiques des acides humiques des sols tropicaux du Brésil. Distinction fondamentale entre les sols équatoriaux et les sols des régions à climat tropical contrasté[END_REF][START_REF] Causse | Xanthan exopolysaccharide: acid-base reactivity related to structure and conformation. A model for understanding the reactivity of degraded and colloidal soil organic matter[END_REF]. Consequently, about half of the POM mass in the soil of this study relates to mineral impurities. The T Hs-LC , the acid-base exchange capacities of the two POM fractions, were retrieved from the same study of the same soil [START_REF] Sebastia | Buffer capacity and Cu affinity of soil particulate organic matter (POM) size fractions[END_REF]: 0.219 and 0.627 mmol g -1 POM , respectively. The POM site densities, D Hs-LC-f = T Hs-LC / T OC-LC , were 0.57 and 2.25 mmol g -1 C for the coarse and fine POM fractions, respectively. These values are 12 and 3.1 times less, respectively, than D Hs-HC , the site density of HR carbon (6.94 mmol g -1 C ). It follows that one organic carbon out of 12 (for HC), out of 40 (for low-reactive carbon LC in the 250-63 μm fraction) and out of 132 (for LC in the > 250 μm fraction) has an acid-base reactive surface group.

More generally, the results of our study accord with previous findings [START_REF] Besnard | Influence of organic amendments on copper distribution among particle-size and density fractions in Champagne vineyard soils[END_REF][START_REF] Six | A history of research on the link between (micro)aggregates, soil biota, and soil organic matter dynamics[END_REF][START_REF] Oorts | Carbon and nitrogen stocks in relation to organic matter fractions, aggregation and pore size distribution in no-tillage and conventional tillage in northern France[END_REF][START_REF] Sebastia | Changes in soil organic matter chemical properties after organic amendments[END_REF], which showed that the size distribution of POM evolves with time, but not the reactivity of a given size fraction. Conceptually, the POM fragments of particles become smaller and thus more reactive as they are degraded by soil organisms, but they remain less reactive than HR OM. Previous μ-SXRF studies have shown that soil POM can concentrate copper locally [START_REF] Jacobson | Electron microprobe and synchrotron X-ray fluorescence mapping of the heterogeneous distribution of copper in high-copper vineyard soils[END_REF], suggesting POM is an entity of 'greatest reactivity' in soil, which was not the case in the present study. Such spectroscopic methods reveal only the volumetric density of sorbed metals (in this case copper). When interpreting such measurements, one must keep in mind that locally concentrated copper sorbed on POM particles levels off from the high-reactivity OM, which is 'diluted' in the relatively weak reactive clay phase. Metals associated with highly reactive OM are then likely to be less visible by spectroscopic methods. Extractive and spectroscopic methods thus appear to be compatible and complementary because they each provide a specific focus on different definitions of site density.

The method presented in this study also enables the distribution of OM and site densities of LR and HR OM to be assessed. The T OC-HC was obtained by scaling T Hs-calc to T Hs , keeping the other reactivity contributions unchanged. The high-reactivity carbon (i.e. T Hs-HC ) represented 87.2 and 87.4% of the organic carbon-based proton reactivity T Hs-C in the > 250-μm and 250-63-μm fractions, respectively (Figure 2b, thin columns). These values represent 36 and 69% of the total carbon mass in these fractions. The remaining mass of organic carbon relates to low-reactivity OM (POM). In the > 250-μm soil fraction, the high-reactivity OM is dominant in reactivity compared with POM, but not in mass. Because of uncertainties in the measurements, we considered this statement to be qualitatively justifiable only.

Extraction of other mineral phases

In this study a robust result was obtained when restricting the analysis to OM and clay in order to take into account their different distributions among the coarse and fine soil fractions, and also because we limited the number of phases determined to two in the control soil and three in the amended soil. This approach could be extended to other soil phases. For example, Fe could be partially extracted as with Al and the data interpreted similarly by relating linearly the proportion of partially extracted Fe to water-accessible surface sites of Fe minerals. Carbonated phases are sensitive to strong acids and would require a specific treatment. With acid-base titrations the presence of acid-sensitive phases such as carbonate can be assessed easily by a steep increase in the slope of H S plotted against pH at the known pH of dissolution (pH ∼ 5.5). In such cases the protocol would have to be adapted as described in the Supporting Information section.

Application of the calculation approach to a series of soil types

The calculation approach we described may be applied to a series of bulk soils as shown by [START_REF] Weng | Contribution of individual sorbents to the control of heavy metal activity in sandy soil[END_REF], rather than to fractions of a single soil. That means comparing the C and Al contents between the different soils instead of between the fractions of a single soil. However, one would have to take into account the two main assumptions: an implicit assumption that unique Al and C site densities prevail in the different soils and an explicit assumption concerning uniform leaching behaviour of Al mineral phases among the different soils. Such strong assumptions apply more to fractions of a unique soil than to different soils.

Conclusion and perspectives

In this study, the acid-extractable Al, the OM content and the acid-base exchange capacity in the fractions of a silty clay soil were combined linearly. This approach succeeded in determining site densities of clayey and organic phases in the soil studied. By comparing soil plots variably amended with organic matter, the method enabled the site densities of highly reactive organic matter compared with low-reactive organic matter of two POM size classes to be quantified.

Our results were consistent with those reported in the literature in terms of site densities of highly reactive OM. This suggests that an approach based on the analysis of soil size fractions enables the determination of metal-reactive site densities of the main reactive phases in soil in which OM and clay are assumed as the dominant reactive phases, which is the case in most soils. Therefore, we propose to validate future multisite modelling studies with such approaches, which enable site densities to be determined instead of being arbitrarily fixed. The calculation approach is also potentially applicable to a series of bulk soils instead of a series of soil size fractions provided that some basic working hypotheses are validated first. The method may also be expanded to include additional phases such as iron oxides when working with iron-rich soil. In our study the number of phases determined was limited to the dominant reactive phases of the silty loam soil (HR and LR OM and clay). This enabled site densities of the dominantly reactive phases of this soil to be determined unambiguously.

Figure 1

 1 Figure1Total copper (T Cu ), organic carbon (T OC ), acid-base exchange capacities (T Hs ) and extractable aluminium (E Al ) ratios of the fractions of (a) the control soil and (b) the soil amended with conifer compost. Bars represent standard errors of the mean of triplicate experiments.

Figure 2

 2 Figure 2 Experimental (⧫) and fitted (total bar height) acid-base capacities of the fractions: (a) control and (b) amended soil. The total heights of large bars are the sum of highly reactive carbon (white, T Hs-C ), Al minerals such as clay (grey, T Hs-clay ) and residual phase (black, T Hs-res ) contributions. Thin bars are scaled to the experimental acid-base capacities by introducing low-reactivity carbon (LC, densely dotted, with a size-class-specific site density) below the highly reactive carbon (HC, sparsely dotted).

Table 1

 1 Characteristics of control and amended soils

	Mass	T OC	T Cu	T Hs	T Hs-calc	E Al

Acknowledgements

We acknowledge the French National Centre for Scientific Research (CNRS INSU) for its financial support through the EC2CO-Cytrix funding (Project MOBIPO-Cu). We thank Philippe Crozier from the Chambre d'Agriculture de Saône-et-Loire and Daniel Lacroze, the owner of the study site, for providing field site access and permitting soil sampling. The IGE is part of Labex OSUG@2020 (ANR10 LABX56).

Supporting Information

The following supporting information is available in the online version of this article: