Claude Pasquier
email: claude.pasquier@unice.fr

KEYSTRACT: Single document keyphrase extraction using sentence clustering and Latent Dirichlet Allocation

This paper describes the design of KEYSTRACT, a system designed for keyphrase extraction from a single document. The principle of the algorithm is to cluster sentences of the documents in order to highlight parts of text that are semantically related. The clusters of sentences, that reflect the themes of the document, are then analyzed to find the main topics of the text. Finally, the most important words, or groups of words, from these topics are proposed as keyphrases. This method is evaluated on task number 5 (Automatic Keyphrase Extraction from Scientific Articles) of SemEval-2010: the 5th International Workshop on Semantic Evaluations.

Introduction

Keyphrases are words, or groups of words, that capture the key ideas of a document. They represent important information concerning a document and constitute an alternative, or a complement, to full-text indexing. Pertinent keyphrases are also useful to potential readers who can have a quick overview of the content of a document and can select easily which document to read. Usually, authors of academic papers are requested to give a set of keyphrases with the document. However, this manual annotation may be subjective and may not correspond to the keyphrases selected by independent readers. In addition, numerous documents that are available electronically do not have author assigned keyphrases.

Currently, the most powerful keyphrases extraction algorithms are based on supervised learning. These methods address the problem of associating keyphrases to documents as a classification task. However, the fact that this approach requires a corpus of similar documents, which is not always readily available, constitutes a major drawback. For example, if one encounters a new Web page, one might like to know quickly the main topics addressed. In this case, a domain-independent keyword extraction system that applies to a single document is needed.

The paper describes our keyphrase extraction algorithm from a single document. We show that our system performs well without the need for a corpus.

The paper is organized as follows. The next section describes the principles of our keyphrase extraction system. We present the main parts of the algorithm in section 3 and detail the methods in section 4. Finally, we evaluate the system's performance in section 5 and we conclude the paper.

Principles

When an author writes a document, he has to think first at the way he will present his ideas. Most of the time, he establishes a content summary that highlight the main topics of his text. Then he writes the content of the document by carefully selecting the most appropriate words to describe each topic. In this paper, we make the assumption that the words, or the set of words, that are representative of each topic constitute the keyphrases of the document. In the following of this paper, we call terms, the components of a documents that constitutes the vocabulary (see the detail of the identification of terms in subsection 4.3).

In statistical natural language processing, one common way of modeling the contributions of different topics to a document is to treat each topic as a probability distribution over words. Therefore, a document is considered as a probabilistic mixture of these topics [START_REF] Griffiths | Finding scientific topics[END_REF].

Generative models can be used to relate a set of observations (in our case, the terms used in a document) to a set of latent variables (the topics). A particular generative model, which is well suited for the modeling of text, is called Latent Dirichlet Allocation (LDA) [START_REF] Blei | Latent dirichlet allocation[END_REF]. LDA is mostly used to analyze text corpora. Given a set of documents, the algorithms describes each document as a mixture over topics, where each topic is characterized by a distribution over words.

The idea of KEYSTRACT is to perform first a clustering of the sentences of the document based on their semantic similarity. Intuitively, one can see each cluster as a part of the text dealing with semantically related content. Therefore, the initial document is divided into a set of clusters and LDA can then be applied on this new representation.

Algorithm

The algorithm is composed of 8 steps:

1. Identification and expansion of abbreviations.

2. Splitting the content of the document into m sentences.

3. Identification of the n unique terms in the document that are potential keyphrases.

4. Creation of a m × n sentence-term matrix X to identify the occurrences of the n terms within a collection of m sentences.

5. Dimension reduction to transform data in the high-dimensional matrix X to a space of fewer dimensions.

6. Data clustering performed in the reduced space. The result of the clustering is used to build a new representation of the source document, which is now considered as a set of clusters, with each cluster consisting of a bag of terms.

7. Execution of LDA on the new document representation.

8. Selection of best keyphrases by analyzing LDA's results.

Methods

KEYSTRACT is build on UIMA (Unstructured Information Management Architecture) (http://incubator.apache.org/uima/ ), a robust and flexible framework that facilitates interoperability between tools. The method process one document at a time by performing the steps described below.

abbreviations expansion

The program ExtractAbbrev [START_REF] Schwartz | A simple algorithm for identifying abbreviation definitions in biomedical text[END_REF] is used to identify abbreviations (short forms) and their corresponding definitions (long forms). Once abbreviations have been identified, each short form is replaced by its corresponding long form in the processed document.

sentences detection

Splitting the content of a document into sentences is an important step of the method. To perform this task, we used the OpenNLP's sentence detector module (http://opennlp.sourceforge.net/ ) trained on a corpus of general english texts.

terms identification

Word categories are identified by using the Ling-Pipe's general english part-of-speech (POS) tagger trained on the Brown Corpus (http://aliasi.com/lingpipe/ ). We leverage POS information to collect, for each sentence, nominal groups that are potential keyphrases. We use a very simple rule to identify nominal groups: a nominal group is an adjacent list of words tagged as determiner, adjective, noun, pronoun, gerund, present participle, past participle or adverb. These nominal groups represent the vocabulary of the document. They are called terms in this paper.

matrix creation

Let D = {d 1 , d 2 , . . . , d n } be the complete vocabulary set of the document identified in subsection 4.3 above. We build a m×n matrix X

= [x ik ]
where m is the number of sentences in the document, n is the number of terms and x ij is the weight of the jth term in the ith sentence.

The weight of a term in a sentence is the product of a local and global weight given by x ij = l ij × g j , where l ij is the local weight of term j within sentence i, and g j is the global weight of term j in the document. The local weighting function measures the importance of a term within a sentence and the global weighting function measures the importance of a term across the entire document. Three local weighting functions were investigated: term frequency, log of term frequency and binary.

Five global weighting functions were also investigated: Normal, GFIDF (Global frequency × Inverse document frequency, IDF (Inverse document frequency), Entropy and none (details of calculation can be found in [START_REF] Dumais | Improving the retrieval of information from external sources[END_REF] paper).

dimension reduction

The matrix X is a representation of a document in a high-dimensional space. Singular Value Decomposition (SVD) [START_REF] Forsythe | Computer Methods for Mathematical Computations[END_REF] and Non-Negative Matrix Factorization (NMF) [START_REF] Lee | Learning the parts of objects by non-negative matrix factorization[END_REF] are two matrix decomposition techniques that can be used to transform data in the high-dimensional space to a space of fewer dimensions.

With SVD, the original matrix X is decomposed as a factor of three other matrices U , Σ and V such as:

X = U ΣV T
where U is an m×m matrix, Σ is a m×n diagonal matrix with nonnegative real numbers on the diagonal, and V T denotes the transpose of V , an n × n matrix. It is often useful to approximate X using only r singular values (with r < min(m, n)), so that we have X = U r Σ k V T r + E, where E is an error or residual matrix, U r is an m × r matrix, Σ r is a k × r diagonal matrix, and V r is an n × r matrix.

NMF is a matrix factorization algorithm that decomposes a matrix with only positive elements into two positive elements matrices, with X = W H +E. Usually, only r components are fit, so E is an error or residual matrix, W is a non-negative m × r matrix and H is a non-negative r × n matrix. There are several ways in which W and H may be found. In our system, we use Lee and Seung's multiplicative update method [START_REF] Lee | Learning the parts of objects by non-negative matrix factorization[END_REF].

All matrix calculations are executed on the R software environment (http://cran.r-project.org/ ). The SVD decomposition of the matrix is computed by the base function svd() and the method nnmf () from package NMFN (http://cran.rproject.org/web/packages/NMFN/) is used for NMF decomposition.

sentences clustering

The clustering of sentences is performed in the reduced space by using the cosine similarity between sentence vectors. Several clustering techniques have been investigated: k-means clustering, Markov Cluster Process (MCL) [START_REF] Van Dongen | Graph clustering via a discrete uncoupling process[END_REF] and ClassDens [START_REF] Guénoche | Clustering by vertex density in a graph[END_REF].

However, while the latent semantic space derived by SVD does not provide a direct indication of the data partitions, with NMF, the cluster membership of each document can be easily identified, directly on the W matrix [START_REF] Xu | Document clustering based on non-negative matrix factorization[END_REF]. Each value w ij of matrix W , indicates, indeed, to which degree sentence i is associated with cluster j. If NMF was calculated with the rank r, then r clusters are represented on the matrix. We use a simple rule to determine the content of each cluster: sentence i belongs to cluster j if w ij > a max k∈{1...m} w ik . In our system, we fixed a = 0.1.

applying LDA on the new document representation

By using the result of the clustering, the source document is now represented by c clusters of terms.

The terms associated with a cluster c i is the sum of the terms belonging to all the sentences in the cluster. JGibbLDA (http://jgibblda.sourceforge.net/ ) is used to execute LDA on the new dataset. We tried to extract different numbers of topics t (with t ∈ {2, 5, 10, 20, 50, 100}) and we choose the Dirichlet hyperparameters such as α = 0.1 and β = 50/t. LDA inferences a topic model by estimating the cluster-topic distribution Θ and the topic-word distribution Φ [START_REF] Griffiths | Finding scientific topics[END_REF][START_REF] Blei | Latent dirichlet allocation[END_REF].

terms ranking and keyphrases selection

We assume that topics covering a significant portion of the document content are more important than those covering little content. To reflect this assumption, we calculate the importance of a term in the document (its score) with a function that takes into account the distribution of topics over clusters given by θ, the distribution of terms over topics given by Φ and the clusters' size.

score(i) = max j∈{1...n} (Φ ji c k=1 (Θ kj p(s(k)))
where score(i) represents the score of term i and s(k) is the size of the cluster k. We tested three different functions for p: the constant function p(i) = 1, the linear function p(i) = i and the exponential function

p(i) = i 2 .
When a score is attributed to each term of the vocabulary, our system simply selects the top terms with the highest score and propose them as keyphrases.

We evaluated our system on the trial and train data provided in order to estimate the best options and parameters to use. Numerous parameters have influence on the method: the weighting of the terms in the document matrix, the dimension reduction method used, the number of dimension retained, the clustering algorithm, the number of topics used to execute LDA and the way best keyphrases are selected.

The parameter that most affects the performance is the method used to perform the dimension reduction. In all cases, whatever the other parameters, NMF performs better. We found that using only 10 components for the factorization is sufficient. There was no significant performance increase by using more factors.

The second more important parameter is the clustering method used. When NMF is used, the best results were achieved by retrieving clusters from the W matrix. With SVD, ClassDens gets the best results. We tested the performance of kmeans clustering by specifying a number of clusters varying from 5 to 100. The best performances were achieved with a number of clusters ≥ 20. However, k-means scores a little bit below Class-Dens and MCL is found to be the worst method.

The choice of the global weighting function is also important. In our experiments, the use of IDF and no global weighting gave the worst results. This confirms the conclusion stated by [START_REF] Michael | A comparison of machine measures of text document similarity with human judgments[END_REF]. Entropy and normal weighting gave the best results but, in average, entropy performs a little better than normal weight. In the final version, the global weighting function used is entropy.

The last parameters that have a visible influence of the quality of extracted keyphrases is the selection of keyphrases from LDA's results. In our experiments, this is the exponential function that performs best. The remaining parameters do not have notable influence on the results. As already stated by [START_REF] Michael | A comparison of machine measures of text document similarity with human judgments[END_REF], the choice of local weighting function makes relatively little difference. Similarly, the number of topics used for LDA has little influence. In KEYSTRACT we used term frequency as local weighting and executed LDA with a number of expected topics of 10.

Conclusion

The performance of our system are in the average of other submitted systems. The F-scores for the top 15 candidates over reader-assigned keyword is 0.1775 while the F-score over both authorassigned and reader-assigned keywords is 0.1852. On the basis of these two measures, the system ranks 13th and 10th respectively out of 20 submitted systems.

However, one has to note that KEYSTRACT uses only the information available from a single document. Therefore, the algorithm described here is an interesting alternative to supervised learning methods when no corpus of similar documents is available.


domain: math spi

text:
Jeanne Redaud

Jean Auriol

Silviu-Iulian Niculescu

Stabilizing Integral Delay Dynamics and Hyperbolic Systems using a Fredholm Transformation

In this paper, we design a stabilizing statefeedback control law for a system represented by a general class of integral delay equations. Under an appropriate spectral controllability assumption, an implementable control law is proposed. The approach is constructive and makes use of the well-known backstepping methodology. Due to the integral terms present in the original system, the proposed problem requires a Fredholm transform, which is not always invertible. The invertibility of this transformation is proved using an operator formulation. In particular, we show that this invertibility property is a consequence of spectral controllability. The existence of the kernels defining the Fredholm transform is proved similarly by showing that they satisfy an invertible integral equation. Some test case simulations complete the paper.

I. INTRODUCTION

In this paper, we design a state-feedback control law that stabilizes a general type of Integral Delay Equations (IDE). They can represent a wide variety of systems. Indeed, numerous engineering or biological models involve transport, communication, or measurement delays [START_REF] Niculescu | Delay effects on stability: a robust control approach[END_REF]. In particular, such delays naturally appear when considering networks of interconnected subsystems. More specifically, IDEs naturally arise when modeling propagation of electric pulses in the heart dynamics [START_REF] Courtemanche | A delay equation representation of pulse circulation on a ring in excitable media[END_REF], for instance, population dynamics or biomedical systems, as epidemics, using transmission delays in their representation [START_REF] Cooke | A periodicity threshold theorem for epidemics and population growth[END_REF].

When analyzing such systems, one of the difficulties to consider is that they combine pointwise and distributed delays. In most cases, such delays may induce instabilities of the closed-loop system or bad performances of outputfeedback control laws. Various problems arise in implementing control laws that include distributed delays [START_REF] Niculescu | On delay-dependent stability under model transformations of some neutral linear systems[END_REF]. Initially, various ideas from complex analysis have been used to find necessary and sufficient stabilizability and controllability conditions [START_REF] Hale | Strong stabilization of neutral functional differential equations[END_REF], [START_REF] Bellman | Differential-difference equations[END_REF]. However, these conditions guarantee the existence of a stabilizing state feedback control, but are in most of the cases not operational.

It has been shown in recent contributions that a large class of hyperbolic systems can be rewritten as difference equations with distributed delays, for instance in [START_REF] Auriol | An explicit mapping from linear first order hyperbolic PDEs to difference systems[END_REF], using a backstepping approach [START_REF] Krstic | Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF]. Delay terms are implied by the transport phenomena inherent to hyperbolic systems and to in-domain couplings (convection terms, for instance). Interestingly, when considering under-actuated systems of hyperbolic equations [START_REF] Auriol | Backstepping stabilization of an underactuated 1+2 linear hyperbolic system with a proper control[END_REF], such distributed delays also appear in the actuation, thus emphasizing the existing links between hyperbolic systems and IDEs. In this context, it is interesting to explore the cases further when specific methods developed for hyperbolic systems can be applied to stabilize IDEs.

jeanne.redaud@centralesupelec.fr

In this paper, we propose a constructive approach to design an implementable control law that stabilizes the considered class of IDEs. More precisely, the contribution of the paper is three-fold. First, unlike classical approaches, we use a Partial Differential Equation (PDE) control-oriented framework to stabilize the considered class of IDEs. Then, the proposed approach requires the introduction of an appropriate Fredholm integral transformation. To the best of the authors' knowledge, the existence of this transformation represents a novelty in the literature and corresponds to a generalization of the constructive backstepping technique. Finally, it is worth mentioning that our methodology can be extended to underactuated hyperbolic systems.

The proposed approach can be resumed as follows. We first define a comparison system1 [START_REF] Niculescu | Delay effects on stability: a robust control approach[END_REF] described by a set of hyperbolic PDEs including control inputs. We will show that its closed-loop stability properties imply those of the original IDE system. Next, the use of the backstepping methodology allows mapping this hyperbolic system to a stable target system and the design of a stabilizing control law. Unlike traditional approaches [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic PDEs[END_REF], the specific structure of the original system requires the use of a Fredholm integral transform. Contrary to Volterra transformations, the existence and invertibility of such a transformation are not guaranteed. Several results in the literature deal with the invertibility of Fredholm transforms when kernels have a specific structure [START_REF] Coron | Finite-time boundary stabilization of general linear hyperbolic balance laws via Fredholm backstepping transformation[END_REF], [START_REF] Bribiesca-Argomedo | Backstepping-forwarding control and observation for hyperbolic pdes with fredholm integrals[END_REF]. As these conditions are not fulfilled here, we use an operator framework, as suggested by [START_REF] Coron | Stabilization and controllability of first-order integro-differential hyperbolic equations[END_REF]. More precisely, we show that the well-posedness and invertiblity of our transform represent a consequence of a natural spectral controllability assumption.

Notations: Inspired by [START_REF] Hale | Introduction to functional differential equations[END_REF], for any fixed τ > 0, we denote

D τ = H 1 ([-τ, 0], R) the Banach space of H 1 real-valued functions mapping the interval [-τ, 0] into R. For a function φ : [-τ, ∞) → R, its partial trajectory φ [t] ∈ D τ is defined by φ [t] (θ) = φ(t + θ), -τ ≤ θ ≤ 0. The associated L 2 norm is given by ||φ [t] || L 2 = 0 -τ φ(t + θ) 2 dθ 1 2 . We denote S = [0, 1] 2 , T -= {(x, y) ∈ [0, 1] 2 , x ≥ y} and T + = {(x, y) ∈ [0, 1] 2 , x ≤ y}.

II. PROBLEM UNDER CONSIDERATION

A. Time-delay formulation

Consider two known positive delays τ 0 > 0 and τ 1 > 0 and let us define the bounded linear operators L i :

D τi → R, i ∈ {0, 1}, by L i : φ [t] → a i φ [t] (-τ i ) + τi 0 f i (ν)φ [t] (-ν)dν, (1)
1 see, e.g., [START_REF] Niculescu | Delay effects on stability: a robust control approach[END_REF] and the references therein for some discussions on delay systems with a 0 ∈ (-1, 1), a 1 ∈ R\{0} and f 0 , f 1 two piecewise continuous functions. Consider the following time-delay equation defined for all t ≥ 0 by

φ [t] = L 0 φ [t] + L 1 V [t] , (2)
with initial data given by φ 0 = φ 0 ∈ D τ0 . The function V ∈ D τ1 corresponds to the actuation, with V (t) = 0, ∀ t ≤ 0.

A function φ : [-τ 0 , ∞) → R is called a solution of the initial value problem (2) if φ 0 = φ 0 , and if (2) is satisfied for t ≥ 0. Since the coefficient |a 0 | is strictly inferior to one, the open-loop system only has a finite number of unstable roots [START_REF] Hale | Introduction to functional differential equations[END_REF]. More precisely, the principal part of the system has to be exponentially stable. If such a condition is not fulfilled, then it is impossible to delayrobustly stabilize the system (2) [START_REF] Logemann | Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop[END_REF]. Due to the distributed delay term ( τ0 0 f 0 φ(t -ν)dν), the open-loop system may be unstable. The objective of this paper is to design a control law V (t) that exponentially stabilizes the system in the sense of the D τ0 -norm. More precisely, we want to design a control law such that ∃ ν > 0, C 0 > 0, ∀φ 0 ∈ D τ0 , all solutions of the closed-loop (2) satisfy

||φ [t] || L 2 ([-τ1,0]) ≤ C 0 e -νt ||φ 0 || L 2 ([-τ1,0]) .
Note that the actuation in (2) appears through both pointwise and distributed delay terms. It has been seldom studied in the literature [START_REF] Bekiaris-Liberis | Stability of predictor-based feedback for nonlinear systems with distributed input delay[END_REF], [START_REF] Ponomarev | Reduction-based robustness analysis of linear predictor feedback for distributed input delays[END_REF], and is a major difference compared to existing results. However, we assume here that there is (at least) a pointwise delay on the actuation since a 1 = 0. Obviously, the difficulties to stabilize equation ( 2) are related to the simultaneous presence of a distributed-delay term on the actuation and on the state. Let us formally take the Laplace transform of equation (2) (zero initial condition). We have F 0 (s)φ(s) = F 1 (s)V (s), where the holomorphic function F 0 and F 1 are defined by

F 0 (s) = 1 -a 0 e -τ0s - τ0 0 f 0 (ν)e -νs dν, (3)
F 1 (s) = a 1 e -τ1s + τ1 0 f 1 (ν)e -νs dν. (4)
To guarantee the possibility to stabilize system (2), we make the following (spectral) controllability assumption Assumption 1: Spectral controllability [START_REF] Mounier | Algebraic interpretations of the spectral controllability of a linear delay system[END_REF], [START_REF] Pandolfi | Stabilization of neutral functional differential equations[END_REF] For all s ∈ C, rank[F 0 (s), F 1 (s)] = 1. In other words, this assumption says that F 0 and F 1 cannot simultaneously vanish.

B. Control strategy

The control strategy we propose is as follows:

• first, we rewrite the integral delay equation ( 2) as a hyperbolic system, and prove that this hyperbolic system shares the same stability properties; • second, using an invertible Fredholm integral transform, we map this hyperbolic system to a stable target system. The invertibility of the backstepping transformation is shown using an operator framework; • next, to show the existence of the backstepping transformation, we rewrite the kernel equations as integral equations. Again, the existence of a solution to these Fredholm integral equations is shown using an operator framework; • finally, this leads to a stabilizing state-feedback control law for the original system.

III. PDE FORMULATION

In this section, we introduce a simple hyperbolic system that will be useful to conclude to the stability of the original system (2). Indeed, its stability properties imply appropriate stability properties for the "original" time-delay system (2). In other words, this system may be seen as a comparison system for (2). Its well-posedness is shown using an operator formulation. Finally, we conclude this section by giving some general results on Fredholm transformations.

A. A new hyperbolic system

In this section, we introduce a new hyperbolic system. It corresponds to two conservation laws, with integral terms acting at one boundary.

1) System equations:

Define λ = 1 τ0 , µ = 1 τ1 , ρ = a 0 and the piecewise continuous functions N u , N v by N u (x) = 1 λ f 0 ( 1 λ x) and N v (x) = 1 µa1 f 1 ( 1 µ (1 -x)). Consider the following hyperbolic system of state (u(t, x), v(t, x)) T with (t, x) ∈ R + * × [0, 1] ∂ t u(t, x) + λ∂ x u(t, x) = 0, ( 5
) ∂ t v(t, x) -µ∂ x v(t, x) = 0, (6)
with the boundary conditions

u(t, 0) = v(t, 0) + ρu(t, 1) + 1 0 N u (y)u(t, y) + N v (y)v(t, y)dy, (7)
v(t, 1) = V (t) = a 1 V (t). (8)
The initial conditions u 0 and v 0 are defined for all x ∈ [0, 1] by u 0 (x) = u(0, x) = φ 0 (-x λ ) and v 0 (x) = v(0, x) = 0. They belong to H 1 ([0, 1], R). System ( 5)-( 8) corresponds to a balance law system [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF] actuated through one boundary.

2) Operator formulation and well-posedness: In order to motivate the definition of a weak solution for the Cauchy problem ( 5)-( 6) with the initial condition (u 0 , v 0 ), we multiply ( 5)-( 6) by smooth test functions (ϕ, ψ) and integrate by parts. This leads to the following:

Definition 1: Let us consider a time T > 0, V ∈ L 2 ([0, T ]). We say that (u, v) is a (weak) solution to (5)- (8) if (u, v) ∈ (C 0 ([0, T ]; L 2 (0, 1))) 2 and T 0 1 0 (-ϕ t (t, x) -λϕ x (t, x) -λN u (x)ϕ(t, 0))u(t, x) + (-ψ t (t, x) + µψ x (t, x) -λN v (x)ϕ(t, 0))v(t, x)dxdt + 1 0 ϕ(T, x)u(T, x) -ϕ(0, x)u(0, x) + ψ(T, x)v(T, x) -ψ(0, x)v(0, x)dx - T 0 µψ(t, 1)V (t)dt = 0, (9) for every (ϕ, ψ) ∈ C 1 ([0, T ] × [0, 1]) 2 such that ϕ(t, 1) = ρϕ(t, 0), ψ(t, 0) = λ µ ϕ(t, 0). (10)
We can now rewrite the system ( 5)-( 8) in the abstract form

d dt u v = A u v + BV, (11)
where we can identify the operators A and B through their adjoints by taking formally the canonical scalar product of [START_REF] Di Meglio | Stabilization of coupled linear heterodirectional hyperbolic pde-ode systems[END_REF] with smooth test functions (ϕ, ψ) and comparing with [START_REF] Coron | Finite-time boundary stabilization of general linear hyperbolic balance laws via Fredholm backstepping transformation[END_REF]. The operator A is thus defined by

A :D(A) ⊂ (L 2 (0, 1)) 2 → (L 2 (0, 1)) 2 u v -→ -λu x (x) µv x (x) , (12)
with

D(A) = {(u, v) ∈ (H 1 (0, 1)) 2 |u(0) = v(0) + ρu(1) + 1 0 N u (y)u(y) + N v (y)v(y), v(1) = 0}.
The operator A is well posed and densely defined. Adjusting the approach of [3, Appendix A], it is possible to show that A generates a C 0 -semigroup. Its adjoint A * is

A * :D(A * ) ⊂ (L 2 (0, 1)) 2 → (L 2 (0, 1)) 2 u v -→ λu x (x) + λN u (x)u(0) -µv x (x) + λN v (x)u(0) , (13)
with

D(A * ) = {(u, v) ∈ (H 1 (0, 1)) 2 |u(1) = ρu(0), v(0) = λ µ u(0)}. The operator B ∈ L(R, D(A * ) ) is defined by < BV, u v >= µv(1)V . Note that B is well defined since
BV is continuous on H 1 (0, 1) (due to the trace theorem and the fact that the graph norm associated to

D(A ) is equivalent to the H 1 -norm). Its adjoint B * ∈ L(D(A * ), R)
is then defined by

B * u v = µv(1). (14)
Following the approach of [START_REF] Coron | Stabilization and controllability of first-order integro-differential hyperbolic equations[END_REF], we can show that B is admissible and consequently that for every V ∈ L 2 ([0, T ]), there exists a unique solution to ( 5)-( 8) (u, v) ∈ C 0 ([0, T ]; L 2 (0, 1)). This is omitted here due to space restriction. We will later show that the feedback law V is actually continuous since it will correspond to a bounded feedback operator.

B. Stability properties

We now show that the stability of the system ( 5)-( 8) implies the stability of system (2). We have the following Lemma 1: Consider ( 5)-( 8) with any feedback law V (t). If the solution of ( 5)-( 8) exponentially converges to zero (with L 2 -norm), then the solution of (2) exponentially converges to zero (with L 2 ([-τ 0 , 0]) norm).

Proof: Let us define z(t) = u(t, 0). Applying the method of characteristics, we show that z is the solution of (2), and that the L 2 -exponential stability of (u, v) implies the L 2 ([-τ 0 , 0]) exponential stability of z. Due to Lemma 1, it is sufficient to design a feedback law that stabilizes (5)- [START_REF] Coron | Stabilization and controllability of first-order integro-differential hyperbolic equations[END_REF]. Note that such a feedback law will be expressed as a function of u, v but can be causally expressed as a function of z and past values of itself using u(t, x) = z(t-τ 0 x) and v(t, x) = V (t-τ 1 (1-x)). As a consequence, we have the following lemma Lemma 2: Consider the operators A * verifying (13) and B * verifying [START_REF] Hale | Introduction to functional differential equations[END_REF]. Under Assumption 1 we have,

∀s ∈ C, ker(s -A * ) ∩ ker(B * ) = {0}. ( 15
)
The proof is omitted due to space restrictions. Note that condition [START_REF] Hu | Control of homodirectional and general heterodirectional linear coupled hyperbolic PDEs[END_REF] is a controllability condition that can be found in [START_REF] Coron | Stabilization and controllability of first-order integro-differential hyperbolic equations[END_REF]. It is also introduced by [START_REF] Fattorini | Some remarks on complete controllability[END_REF] in a much larger setting.

C. Invertibility of some Fredholm integral operators

The stabilization of the PDE system ( 5)-( 8) will be done using a backstepping transformation of Fredholm type, since it offers more degrees of freedom than the traditionally used Volterra transforms. More precisely, we will consider an integral operator T : (L 2 (0, 1)) 2 → (L 2 (0, 1)) 2 defined by

T α(x) β(x) = α(x) β(x) - 1 0 K(x, y) α(y) β(y) dy, ( 16
)
where K is a bounded function defined on the unit square S = {(x, y) ∈ [0, 1] 2 }. Unlike Volterra integral transformation, Fredholm transformations are not always invertible [START_REF] Yoshida | Lectures on differential and integral equations[END_REF]. The following lemma (adjusted from [8, Lemma 2.2, Proposition 2.6]) guarantees their invertibility under several conditions. Lemma 3: Consider a Fredholm integral operator T : (L 2 (0, 1)) 2 → (L 2 (0, 1)) 2 as defined by [START_REF] Krstic | Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF]. Assume that 1) ker(T ) ⊂ D(A * ), 2) ker(T ) ⊂ ker(B * ), 3) ∀z ∈ ker(T ), T A * z = 0 4) ker(s -A * ) ∩ ker(B * ) = {0}, for every s ∈ C. Then, the operator T is invertible.

Proof: The proof follows the steps of [8, Lemma 2.2, Proposition 2.6], and can be directly derived from [START_REF] Redaud | Observer design for a class of delay systems using a fredholm transform[END_REF].

IV. DESIGN OF A STABILIZING CONTROL LAW

In this section, we use a Fredholm integral operator of form [START_REF] Krstic | Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF] to map the initial hyperbolic system to an exponentially stable target system. First, we define the candidate target system. Then, we give the equations satisfied by its kernels. Finally, proving the invertibility of the Fredholm integral operator, we can define a full-state feedback control law stabilizing the initial hyperbolic system.

A. Presentation of the target system

We define a target state (α, β), that satisfies

∂ t α + λ∂ x α = 0, ∂ t β -µ∂ x β = 0, (17)
D(A 0 ) = {(α, β) ∈ H 1 (0, 1) 2 | α(0) = β(0) + ρα(1), β(1) = 0}. Its adjoint A * 0 is defined by A * 0 :D(A * ) ⊂ (L 2 (0, 1)) 2 → (L 2 (0, 1)) 2 α β -→ λα x (x) -µβ x (x) . (19)

B. Kernel equations

To map the original system ( 5)-( 8) to the target system (17)-( 18), we use a Fredholm integral operator P of the form [START_REF] Krstic | Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF], such that u(x) v(x) = P α(x) β(x) . The corresponding kernels are denoted K ij = (K) ij , where (i, j) ∈ {1, 2} 2 . They are piecewise-continuous fucntions defined on S. Deriving the transformation with respect to time and space and integrating by parts, we obtain the following kernel equations,

∂ ξ K 11 (x, ξ) + ∂ x K 11 (x, ξ) = 0, (20)
µ∂ ξ K 12 (x, ξ) -λ∂ x K 12 (x, ξ) = 0, (21)
λ∂ ξ K 21 (x, ξ) -µ∂ x K 21 (x, ξ) = 0, (22) ∂ ξ K 22 (x, ξ) + ∂ x K 22 (x, ξ) = 0, (23)
with the boundary conditions λK 11 (x, 0) = µK 12 (x, 0), K 11 (x, 1) = ρK 11 (x, 0), (24) µK 22 (x, 0) = λK 21 (x, 0), K 21 (x, 1) = ρK 21 (x, 0), (25)

K 12 (x, 1) = 0, K 22 (x, 1) = 0. (26)
The two boundary conditions [START_REF] Yoshida | Lectures on differential and integral equations[END_REF] are necessary to guarantee the invertibility of P . Evaluating ( 16) in x = 0, we also have

N u (ξ) - 1 0 N u (ν)K 11 (ν, ξ) + N v (ν)K 21 (ν, ξ)dν = -K 11 (0, ξ) + K 21 (0, ξ) + ρK 11 (1, ξ), (27)
N v (ξ) - 1 0 N u (ν)K 12 (ν, ξ) + N v (ν)K 22 (ν, ξ)dν = -K 12 (0, ξ) + K 22 (0, ξ) + ρK 12 (1, ξ). ( 28
)
Theorem 1: The set of equations ( 20)-( 26) admits a unique bounded solution on the unit square S. Since the kernels are defined by a complex set of equations (b.c ( 27)-( 28)) we cannot apply classical methods [START_REF] Di Meglio | Stabilization of coupled linear heterodirectional hyperbolic pde-ode systems[END_REF] to prove the well-posedness. The proof of Theorem 1 requires technical computations, and is deferred to Section V.

C. Invertibility of the Fredholm transform

We now show the invertibility of the Fredholm integral backstepping transform P . We have the following theorem.

Theorem 2: Consider the Fredholm integral operator P of the form [START_REF] Krstic | Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF], whose kernels are defined by ( 20)- [START_REF] Yoshida | Lectures on differential and integral equations[END_REF]. Then P is invertible.

Proof: The adjoint operator P * associated to P , is also of the form [START_REF] Krstic | Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF]. We have

P * ( α(x) β(x) ) = α(x) β(x) - 1 0 K T (y, x) α(y) β(y) dy.
Since P maps the original system ( 5)-( 8) to the target system ( 17)-( 18), we have for all z ∈ ker(B * ), P * A * z = A * 0 P * z [START_REF] Coron | Stabilization and controllability of first-order integro-differential hyperbolic equations[END_REF]. Direct computations show that P * satisfies the conditions of Lemma 3. It is therefore invertible, and so is P . Since the kernel equations are well-posed by Theorem 1, the inverse operator P -1 associated to P is uniquely defined by

P -1 u(x) v(x) = u(x) v(x) + 1 0 L(x, y) u(y) v(y) dy, ( 29
)
where L verifies L(x, y) = K(x, y) + 1 0 K(x, ν)L(ν, y)dν.

D. Stabilizing control law

Using the inverse transform (29), evaluated in x = 1, we can design a full-state feedback controller V (t):

V (t) = - 1 0 L 21 (1, ν)u(ν, t) + L 22 (1, ν)v(ν, t)dν. (30)
Theorem 3: The state-feedback control law V (t) defined by (30) exponentially stabilizes the hyperbolic system ( 5)-( 8) in the sense of the L 2 -norm. Thus, the control law V (t) = 1 a1 V (t) exponentially stabilizes the integral delay dynamics (2) in the sense of the D τ0 -norm.

Proof: With the control law V (t) defined by (30), the hyperbolic system ( 5)-( 8) is equivalent to the target system ( 17)-( 18). The target system is exponentially stable in the sense of the L 2 -norm, and so is the initial system ( 5)- [START_REF] Coron | Stabilization and controllability of first-order integro-differential hyperbolic equations[END_REF]. Lemma 1 allows us to conclude the proof.

V. WELL-POSEDNESS OF THE KERNEL EQUATIONS

In this section, we prove the existence of a unique solution to the kernels equations ( 20)- [START_REF] Yoshida | Lectures on differential and integral equations[END_REF]. We rewrite these equations as two integral equations. Using Lemma 3, we show that the associated integral operator is invertible. This guarantees the existence and uniqueness of a solution. In what follows, we only consider the case λ > µ. The case λ ≤ µ can be done following the same ideas. Due to space restrictions, we only give a sketch of the different proofs.

A. Kernels reduction

The first step of our proof consists in showing that the existence and uniqueness of K 12 (0, ξ) and K 21 (0, ξ) on [0, 1] implies the one of all the kernels on S. We have Lemma 4: For all (x, ξ) ∈ S, (i, j) ∈ {1, 2} 2 , K ij (x, ξ) can be expressed as functions of K 12 (0, •) and K 21 (0, •).

Proof: Applying the method of characteristics on ( 20)-( 23), we can express K ij (x, ξ) as functions on the corresponding boundary values. Using the boundary conditions ( 24)-( 26), and iterating the procedure, we can express all the kernels as functions of K 12 (x, 0) and K 21 (x, 0). Direct computations give

K 11 (x, ξ) = µ λ (1 [0,ξ] (x)ρK 12 (x -ξ + 1, 0) + 1 [ξ,1] (x)K 12 (x -ξ, 0)), (31)
K 22 (x, ξ) = λ µ 1 [ξ,1] (x)K 21 (x -ξ, 0). ( 32
)
To deal with the remaining kernels, we need to have a closer look to their definition domain and to the corresponding characteristic lines. Define p as the unique integer verifying p µ λ ≤ 1 < (p + 1) µ λ . We can divide S into different subdomains D k , k ∈ 0, p + 1 , as illustrated on Figure 1. More precisely, we have: Using the fact that K 21 (x, 1) = ρK 21 (x, 0) and integrating along the characteristic lines, we obtain by iteration

• D 0 = {0 ≤ ξ ≤ 1, 0 ≤ x ≤ µ λ (1 -ξ)}, • ∀k ∈ 1, p -1 , D k = {0 ≤ ξ ≤ 1, µ λ (k -ξ) ≤ x < µ λ (k + 1 -ξ)}, • D p = {0 ≤ ξ ≤ 1, µ λ (p -ξ) ≤ x < min(1, 1 -µ λ ξ)}, • D p+1 = {0 ≤ ξ ≤ 1, min(1, 1 -µ λ ξ) ≤ x ≤ 1}.
∀(x, ξ) ∈ D k,k≤p+1 , K 21 (x, ξ) = ρ k K 21 (0, ξ -k + λ µ x). Similarly, we have K 12 (x, ξ) = 1 [0, λ µ (1-ξ)] (x)1 [0,1] (x)K 12 (0, ξ + µ λ x
). This concludes the proof.

B. Integral formulation

We can now rewrite ( 27)-(28) as two integral equations. Henceforth, the existence and uniqueness of K 12 (0, ξ) and K 21 (0, ξ) will guarantee the well-posedness of the kernel equations according to Lemma 4. Notice first that K 11 (0, ξ) -ρK 11 (1, ξ) = 0. Using the expressions of the different kernels obtain in Section V-A, we have

K 21 (0, ξ) =N u (ξ) - 1 0 I 11 (ξ, ν)K 12 (0, ν) + I 12 (ξ, ν)K 21 (0, ν)dν, (33)
K 12 (0, ξ) = -N v (ξ) + 1 [0,1-µ λ ] (ξ)ρK 12 (0, ξ + µ λ ) (34)
+ 1 0 I 21 (ξ, ν)K 12 (0, ν) + I 22 (ξ, ν)K 21 (0, ν)dν,
where

I 11 (ξ, ν) = 1 [0, µ λ (1-ξ)] (ν)N u (ξ + λ µ ν) + 1 [ µ λ (1-ξ), µ λ ] (ν)ρN u (ξ -1 + λ µ ν), (35)
I 12 (ξ, ν) = µ λ [1 [ξ,1] (ν)N v ( µ λ (ν -ξ)) + p k=1 1 [0, λ µ -k+ξ] (ν)ρ k N v ( µ λ (ν -ξ + k)) (36)
+ 1 [p+1+ µ λ ,1] (ξ)1 [0, λ µ -(p+1)+ξ] (ν)ρ p+1 Nv( µ λ (ν -ξ + p + 1))], I 21 (ξ, ν) = 1 [ξ,ξ+ µ λ ] (ν)1 [0,1] (ν) λ µ N u ( λ µ (ν -ξ)), (37)
I 22 (ξ, ν) = p k=0 1 [0, λ µ (1-ξ)-k] (ν)ρ k N v (ξ + µ λ (ν + k)).
(38) The expressions of I ij are obtained using Fubini's theorem, and the computations are quite technical. Due to space restrictions, we could not give the intermediate steps.

C. Operator formulation 1) Change of variable:

We aim to rewrite (33)-(34) using an integral operator of form [START_REF] Krstic | Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF]. To do so, we need to get rid of the term 1 [0,1-µ λ ] (ξ)ρK 12 (0, ξ + µ λ ) in (34). Let y be a bounded function, and define ȳ, s.t ∀ξ ∈ [0, 1] we have

ỹ(ξ) = y(ξ) -1 [0,1-µ λ ] (ξ)ρy(ξ + µ λ ). (39)
This yields the following lemma Lemma 5: The operator . defined by (39) is invertible. More precisely, we have

y(ξ) = p k=0 ρ k 1 [0,1-k µ λ ] (ξ)ỹ(ξ + k µ λ ) (40)
Using this change of variables, we can rewrite (34)-(33) as

K 21 (0, ξ) = N u (ξ) - 1 0 Ĩ11 (ξ, ν) K12 (0, ν) + I 12 (ξ, ν)K 21 (0, ν)dν, (41)
K12 (0, ξ) = -N v (ξ) + 1 0 Ĩ21 (ξ, ν) K12 (0, ν) + I 22 (ξ, ν)K 21 (0, ν)dν. (42)
with Ĩj1 , j ∈ {1, 2} defined by

Ĩj1 (ξ, ν) = p k=0 ρ k 1 [ µ λ k,1] (ν)I j1 (ν - µ λ k). (43)
To show that the integral equations ( 41)-( 42) have a unique solution, we use an operator formalism.

2) Definition of the integral operator Q: Introduce the Fredholm integral operator Q acting on L 2 (0, 1) 2 by

Q : f g -→ f g -R f g , ( 44
)
where R is the integral operator defined by

R f (ξ) g(ξ) = 1 0 -I 12 (ξ, ν)f (ν) + Ĩ11 (ξ, ν)g(ν)dν 1 0 -I 22 (ξ, ν)f (ν) + Ĩ21 (ξ, ν)g(ν)dν , (45)
s.t (41)-(42) rewrites Q K 21 (0, •) -K12 (0, •) = N u N v . Thus, if
we prove that Q is invertible, we will get that (41)-( 42) admits a unique solution.

3) Invertibility of operator Q: Lemma 6: The operator Q defined by equations ( 44)-( 45) is invertible.

Proof: The proof is done by showing that assumptions 1) to 4) of Lemma 3 are verified for the operator Q. Indeed, due to the regularity of the kernels, Q can be seen as an operator acting on the space L 2 (0, 1) 2 . Assumption 4) derives from the spectral controllability of the system, and was proven in Lemma 2. Assumptions 1), 2) are proved by evaluating the four coupling terms in ξ = 0 and ξ = 1. To prove that ker(Q) is stable by A * (Assumption 3), we compute the derivative of functions (f, g) ∈ ker(Q) on one side, and we integrate by parts in the integral terms on the other side. The complete proof is technical as we need to be careful to correctly define the integration domains due to the presence of characteristic functions in the integral terms. It is omitted here due to space constraints. Since N u and N v are piecewise-continuous, so are K 21 (0, •) and K12 (0, •).

D. Conclusion: well-posedness of the kernel equations

We have all elements to give the proof of Theorem 1. Proof: The invertibility of operator Q given by Lemma 6 implies the existence and uniqueness of K12 (0, ξ), K 21 (0, ξ). Their regularity is a consequence of the regularity of the functions N u and N v . Using Lemma 5 and (40), we have access to the boundary values K 12 (0, ξ) for all ξ ∈ [0, 1]. According to Lemma 4, the four kernels K ij are then uniquely defined on S. This concludes the proof.

VI. SIMULATION RESULTS

In this section, we give some simulation results to illustrate our approach. Consider the hyperbolic system (1), with coefficients τ 0 = 0.5, τ 1 = 1, a 0 = 0.7 < 1, a 1 = 1, and constant integral coupling terms f 0 = 0.8, f 1 = -0.35. The control strategy is implemented using Matlab. We simulated our system on a time scale of 5s, with 51 space-discretization points in [0, 1]. The initial condition is a constant functions φ 0 = 1. We choose to work on the equivalent PDE system, since analyzing the L 2 -norm of this system will show the D τ0 -stability of z but also the convergence to zero of the control law (which is crucial for implementation and which would not be guaranteed if the actuator dynamics were directly compensated in (1)). First, the kernels K ij and L ij are computed offline using the fixed-point algorithm (successive approximation technique). The integral terms are approximated using the trapezoidal method. Then, the evolution of the hyperbolic system ( 5)-( 8) is computed using a Godounov Scheme [START_REF] Leveque | Finite volume methods for hyperbolic problems[END_REF]. The control effort is computed at each time step using (30). In Figure 2, we pictured the Fig. 2. Evolution of the L 2 -norm of system ( 5)- [START_REF] Coron | Stabilization and controllability of first-order integro-differential hyperbolic equations[END_REF]. evolution of the L 2 -norm of system ( 5)-( 8), in open-loop and closed-loop. As illustrated by the blue curve, the coefficient values were chosen such that the system is unstable in open loop. As expected, in presence of the control law (30), the system becomes exponentially stable (red curve).

VII. CONCLUDING REMARKS

In this paper, we proposed a new approach to stabilize a general class of integral delay equations, under a spectral controllability condition. In the proposed methodology, we have first rewritten the delay equation as a scalar hyperbolic system that shares the same stability properties (comparison system). We then used the backstepping approach to map this PDE system to a simple (exponentially stable) target system. However, the configuration considered in this paper required a Fredholm transform, which is not always invertible (contrary to Volterra transforms ordinarily used). We proved the invertibility of the Fredholm transform using an operator framework inspired by [START_REF] Coron | Stabilization and controllability of first-order integro-differential hyperbolic equations[END_REF]. The well-posedness of the kernels defining the Fredholm transform was proved using similar tools. It appeared that invertibility was a consequence of the assumed spectral controllability condition. Finally, some simulations illustrated the relevance of this approach to stabilize the integral delay dynamics. The proposed approach could be extended to the stabilization of underactuated PDE systems [START_REF] Auriol | Backstepping stabilization of an underactuated 1+2 linear hyperbolic system with a proper control[END_REF] and underactuated networks of interconnected PDE systems with actuation only available at the in-between boundary. However, the proposed strategy could not be used for industrial applications, since control law (30) requires the knowledge of all the state. We believe that an observer could be design for such systems using a similar approaches. More generally, this paper paves the way for future contributions on networks with actuation inside the graph structure. We believe that this approach could be combined with [START_REF] Redaud | Output-feedback control of an underactuated network of interconnected hyperbolic pde-ode systems[END_REF] to tackle a wider diversity of physical systems with an arbitrary number of PDEs or ODEs. In future contributions, we will try to extend our results to non-scalar systems. This will lead us to a general methodology for stabilizing integral delay dynamics and underactuated hyperbolic systems.

  with b.c α(t, 0) = β(t, 0) + ρα(t, 1), β(t, 1) = 0. (18) It is obviously exponentially stable in the sense of the L 2norm. It can be rewritten using an abstract formulation as d dt α β = A 0 α β where A 0 satisfies (12), and is defined on

Fig. 1 .

 1 Fig. 1. Different domains in the expression of K 21

Jeanne Redaud, Jean Auriol, and Silviu-Iulian Niculescu are with Université Paris-Saclay, CNRS, Inria, CentraleSupélec, Laboratoire des Signaux et Systèmes, 91190, Gif-sur-Yvette, France. Corresponding author:


text:
Jeanne Redaud
email: jeanne.redaud@centralesupelec.fr

Jean Auriol

Silviu-Iulian Niculescu

Observer Design for a Class of Delay Systems Using a Fredholm Transform

Keywords: infinite dimensional systems, observer design, backstepping methodology

In this paper, we design an observer for a system represented by a general class of Integral Delay Equations (IDE). This class of equations encompasses various systems, from chemical or biological processes to the propagation of electric pulses on excitable media. The available measurement corresponds to a discrete and distributed-delayed value of the state. Under an appropriate spectral observability assumption, an implementable observer is proposed. Inspired by the fact that first-order hyperbolic Partial Derivative Equations (PDE) and timedelay systems are closely related, we use a PDE formulation and the well-known backstepping methodology for the design. However, due to integral terms in the dynamics, the use of a Fredholm integral transform is required. We prove its existence and invertibility by using an operator framework. Both proofs derive from the spectral observability assumption. Some test case simulations end the paper.

INTRODUCTION

In this paper, we design an observer for a dynamical system represented by a general class of Integral Delay Equations (IDEs). Such systems are encountered in a wide range of engineering applications, where transport phenomena occur. Delays also naturally appear when considering interconnected and/or networked systems. They are often sources of instabilities in the closed-loop system and can induce poor performance for the resulting feedback control laws [START_REF] Gu | Survey on recent results in the stability and control of time-delay systems[END_REF]. Time-delay systems have then been the object of many research works [START_REF] Niculescu | Delay effects on stability: a robust control approach[END_REF]. With the objective of designing efficient output-feedback (and thus implementable) control laws for such systems, interest for observers has grown significantly in the last decades. For an overview on the subject, the reader is referred to [START_REF] Sename | New trends in design of observers for time-delay systems[END_REF]. Even though time-delay systems are inherently infinite-dimensional systems, results on finite-dimensional systems can still be a relevant source of inspiration. Conversions between different representations are of great use to take advantage of analysis tools or algorithms built for one formulation [START_REF] Peet | Representation of networks and systems with delay: DDEs, DDFs, ODE-PDEs and PIEs[END_REF]. More specifically, a duality between observation and output feedback for general classes of linear systems with distributed and pointwise delays has been studied [START_REF] Sun | Duality between observation and output feedback for linear systems with multiple time delays[END_REF]. However, it led to restrictive stability assumptions. Though, the control law design framework can still inspire observer designs. In particular, [START_REF] Redaud | Stabilizing integral delay dynamics and hyperbolic systems using a fredholm transformation[END_REF] used an approach based on a comparison with a hyperbolic PDE system. Indeed, such systems can be seen as neutral differential equations [START_REF] Hale | Introduction to functional differential equations[END_REF]; [START_REF] Auriol | An explicit mapping from linear first order hyperbolic PDEs to difference systems[END_REF] that present a structure similar to one of the classes of IDEs under consideration. This representation allows the use of the well-known backstepping methodology that has been proved to be an efficient tool to design observers [START_REF] Krstic | Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays[END_REF]. More precisely, the contribution of the paper is two-fold. First, unlike classical observer design approaches for timedelay systems [START_REF] Klamka | Observer for linear feedback control of systems with distributed delays in controls and outputs[END_REF], we use a PDE framework. Then, we introduce an appropriate Fredholm integral transform, similar to the one used in [START_REF] Redaud | Stabilizing integral delay dynamics and hyperbolic systems using a fredholm transformation[END_REF].

To the best of the authors' knowledge, the operator framework used to prove the existence of the observer gains is a novelty in the literature. It gives a constructive way to design implementable observers. The proposed approach can be resumed as follows. First, we use a new representation of the delay system based on the state interpretation. This infinite-dimensional linear system class will allow us to use an operator framework. We then define a hyperbolic system and show that its boundaries satisfy the same equations as our initial delay system.

Next, we define an observer state and the resulting error system associated with this hyperbolic system. We use the backstepping methodology to map it to a stable target system. However, the integral terms in the delay system lead to coupling terms in the hyperbolic comparison system, and a Fredholm integral transform must be used. Unlike Volterra transforms traditionally used, the invertibility of such transforms is not guaranteed [START_REF] Yoshida | Lectures on differential and integral equations[END_REF]. Utilizing an operator theory framework [START_REF] Coron | Stabilization and controllability of first-order integro-differential hyperbolic equations[END_REF], we show that the invertibility of the Fredholm transform and the well-posedness of the kernel equations naturally derive from a spectral observability assumption. The existence of observer gains making the estimated state converging towards the actual state is then guaranteed. The paper's layout is as follows: in Section 2, we present the general class of time-delay equations considered herein and describe our strategy relying on a spectral observability assumption. Then we show in Section 3 that this equation can be reformulated as a hyperbolic system, for which we design an observer. We use the backstepping methodology with a Fredholm transform to map the error system to a stable target system. Next, we introduce the operator framework and show the well-posedness of the kernel equations. We use the same approach to prove that the Fredholm transform is invertible. Then, we define the observer gains to guarantee the convergence of the estimated state towards the actual state. We illustrate this result by simulations in Section 5. Finally, we end this paper with some concluding remarks in Section 6. Notations : In this paper, we denote D τ = H 1 ([-τ, 0], R) the Banach space of H 1 real-valued functions taking values in the compact [-τ, 0], for any fixed time-delay τ > 0. For any function φ : [-τ, ∞) → R, we define an associated partial trajectory φ

[t] ∈ D τ by φ [t] (θ) = φ(t + θ), -τ ≤ θ ≤ 0.
We use s as the Laplace variable. For all a, b, ν ∈ [0, 1], the characteristic function is denoted by

1 [a,b] (ν)(equals 1 if ν ∈ [a, b], 0 else).

PROBLEM UNDER CONSIDERATION

2.1 Presentation of the system Time-delay formulation Consider the neutral type equation defined for all t ≥ 0 by

z(t) + az(t -τ 1 ) + τ1 0 N 1 (ν)z(t -ν)dν = 0, (1)
where τ 1 is a strictly positive delay, and N 1 a piecewise continuous. The parameter a is a constant that belongs to ] -1, 1[. The initial data is given by z 0 ∈ D τ1 . A function z : [-τ 1 , ∞) → R is called a solution of the initial value problem (1) if z(0) = z 0 , and if (1) is satisfied for t ≥ 0. System (1) corresponds to a difference equation with a pointwise-delay term and a distributed-delay term. The class of system (1) can model a wide variety of systems in biology, chemistry, epidemiology, physics, or engineering sciences [START_REF] Niculescu | Delay effects on stability: a robust control approach[END_REF]. The condition |a| < 1 guarantees that the open-loop system has a finite number of unstable roots [START_REF] Hale | Introduction to functional differential equations[END_REF]. If |a| > 1, any arbitrary small delay would destabilize the potential estimation [START_REF] Logemann | Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop[END_REF]. Thus, it would not be possible to implement the resulting observer.

Our objective is to design an observer for the time-delay system (1) using the measurement

y(t) = z(t -τ 2 ) + τ2 0 N 2 (ν)z(t -ν)dν, (2)
with τ 2 > 0 and N 2 a piecewise continuous function. More precisely, using the delayed output y(t), we want to design an observer ẑ such that ||z(t)ẑ(t)|| L 2 ([-max{τ1,τ2},0] ---→ t→∞ 0. The measurement (2) corresponds the knowledge of a diffuse value of the state (distributed delay). An application example can be found with complex systems, whose control is ensured by multiple sensors. Measurements of each sensor are in most cases available after a specific acquisition, processing, and transportation time that varies from one sensor to another. Moreover, all sensors do not acquire data at the same frequency. One challenge is then to synchronize the different outputs [START_REF] Bak | Location estimation using delayed measurements[END_REF]. Multiple outputs processing could result in a measurement of form (2), where a sum approximates the integral term for highfrequency acquisition.

Spectral observability assumption

To guarantee the possibility to design an observer for (1)-( 2), we need an observability assumption [START_REF] Mounier | Algebraic interpretations of the spectral controllability of a linear delay system[END_REF]. This condition is obtained considering the spectral properties of the system. Taking the Laplace transform of (1)-( 2), we have

z(s)(1 + ae -τ1s + τ1 0 N 1 (ν)e -νs dν) = F(s)z(s) = 0 y(s) = (e -τ2s + τ2 0 N 2 (ν)e -νs dν)z(s) = O(s)z(s). (3)
Therefore, we obtain the following observability condition.

Hypothesis 1. Spectral observability For all s ∈ C, rank[F(s), O(s)] = 1.
It guarantees that F and O are not simultaneously equal to 0. A parallel can be drawn with the spectral controllability condition given in [START_REF] Redaud | Stabilizing integral delay dynamics and hyperbolic systems using a fredholm transformation[END_REF], since observability and controllability are dual problems.

Observer design strategy

The strategy to design the observer states as follows:

• first, we rewrite the integral delay equation ( 1) as a hyperbolic PDE system. The distributed measurement (2) corresponds to a boundary measurement for this new hyperbolic system; • second, we propose an observer for this PDE system as a copy of the original dynamics with some tunable output injection gains; • then, we use a Fredholm backstepping integral transform that maps the associated error system to a stable target system. This transformation gives a suitable choice for the observer gains; • next, we show that the existence of this backstepping transform is equivalent to the existence of a solution for some integral equations. We use an operator framework to establish the existence of a solution for this set of equations; • finally, the invertibility of the backstepping transform is shown using the same operator framework.

PDE FORMULATION

This section introduces a simple hyperbolic system that will be used as a comparison system [START_REF] Niculescu | Delay effects on stability: a robust control approach[END_REF], to simplify the design of the observer. For this PDE system, the measurement (2) rewrites as a pointwise boundary measurement. We show that the estimation of the PDE state implies the estimation of the initial state z.

Comparison system

Consider the following hyperbolic system, defined by

∂ t u(t, x) + λ∂ x u(t, x) = h 1 (x)u(t, 1), (4) ∂ t v(t, x) -µ∂ x v(t, x) = h 2 (x)u(t, 1), (5)
where (u(t, x), v(t, x)) T is the state of the system, the different arguments evolving in {(t, x) s.t. t > 0, x ∈ [0, 1]} with the boundary conditions u(t, 0) = au(t, 1), v(t, 1) = u(t, 1), ( 6)

and where λ = 1 τ1 , µ = 1 τ2 , and ∀ν ∈

[0, 1], h 1 (ν) = N 1 (τ 1 (1 -ν)), h 2 (ν) = N 2 (τ 2 ν).
The initial conditions u 0 and v 0 belong to H 1 ([0, 1], R). They are defined for all x ∈ [0, 1] by u 0 (x) = u(0, x) = z 0 (-1-x λ ) and v 0 (x) = v(0, x) = 0. We consider that we have access to the measurement y 1 (t) = v(t, 0). We will show below that y 1 (t) corresponds to y(t). We now rewrite the system (4)-( 6) in the abstract form

d dt u v = A u v
, where the operator A is defined by

A :D(A) ⊂ (L 2 (0, 1)) 2 → (L 2 (0, 1)) 2 u v -→ -λu x (x) + h 1 (x)u(1) µv x (x) + h 2 (x)u(1) , (7)
with

D(A) = {(u, v) ∈ (H 1 (0, 1)) 2 |u(0) = au(1), v(1) = u(1)}.
The operator A is well posed and densely defined [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF] (the solution remains in H 1 and the trace operator is properly defined). We now show that the system (4)-( 6) is a comparison system and that the measurement y 1 corresponds to y. Indeed, defining z(t) = u(t, 1) and applying the method of characteristics to ( 4)-( 6), we obtain

u(t, 1) = z(t) = az(t - 1 λ ) + 1 λ 0 h 1 (1 -λs)z(t -s)ds, y 1 (t) = z(t - 1 µ ) + 1 µ 0 h 2 (µs)z(t -s)ds.
Thus, u(., 1) is solution of (1), while y 1 satisfies (2).

Observer and error systems

Let us design an observer for the hyperbolic system (4)-( 6). The observer state (û(t, x), v(t, x)) satisfies the same dynamics as the original hyperbolic system (4)-( 6), in which we add output injection gains. We have

∂ t û(t, x) + λ∂ x û(t, x) = h 1 (x)û(t, 1) -g 1 (x)(v(t, 0) -y(t)), ∂ t v(t, x) -µ∂ x v(t, x) = h 2 (x)û(t, 1) -g 2 (x)(v(t, 0) -y(t)),
with the boundary conditions û(t, 0) = aû(t, 1) v(t, 1) = û(t, 1), (8) where the observer gains g 1 and g 2 are two real-value functions defined on [0, 1] that will be determined later. They belong to H 1 ([0, 1]) The initial conditions û(0, •) and v(0, •) are arbitrarily H 1 functions. We now define the error between the observer and the state as ũ = u -û, ṽ = v -v. The error state satisfies the set of equations ∂ t ũ(t, x) + λ∂ x ũ(t, x) = h 1 (x)ũ(t, 1) -g 1 (x)ṽ(t, 0), (9) ∂ t ṽ(t, x) -µ∂ x ṽ(t, x) = h 2 (x)ũ(t, 1) -g 2 (x)ṽ(t, 0), (10) with the boundary conditions ũ(t, 0) = aũ(t, 1) ṽ(t, 1) = ũ(t, 1).

(11) The objective is to design the observer gains g 1 , g 2 , so that the error system is exponentially stable in the sense of the L 2 -norm. More precisely, we want to show that ∃ν > 0, C > 0, ||(ũ, ṽ)|| L 2 (0,1) ≤ Ce -νt ||(ũ 0 , ṽ0 )|| L 2 (0,1) . Indeed, denoting ẑ(t) = û(t, 1), we will show that the convergence of (ũ, ṽ) to zero implies the convergence of ẑ to z. We can rewrite the error system ( 9)-( 11) using the operator abstract form

d dt ũ ṽ = A ũ ṽ + GC ũ ṽ , (12)
where G = (g 1 , g 2 ) T corresponds to the product with the observer gains and where the operator C is the trace operator defined on D(A) by

C :D(A) ⊂ (L 2 (0, 1)) 2 → R (ũ ṽ) T -→ ṽ(0). ( 13
)

Observability properties

The following lemma is a consequence of Hypothesis 1 Lemma 2. Under Hypothesis 1, consider the operators A and C respectively defined by ( 7) and ( 13). Then, for any s ∈ C, we have ker(s

-A) ∩ ker(C) = {0}. ( 14
)
The proof is omitted due to space restrictions.

Insights on Fredholm integral operators

The stabilization of the error system ( 9)-( 11) will be done using a backstepping transform of the Fredholm type. Indeed, only a transformation of the Fredholm type gives enough degrees of freedom to map the error system to a stable target system. More precisely, we consider an integral operator T :

(L 2 ([0, 1])) 2 → (L 2 ([0, 1])) 2 defined by T u(x) v(x) = u(x) v(x) - 1 0 N (x, ν) u(ν) v(ν) dν, ( 15
)
where N is a bounded function defined on the unit square S = {(x, y) ∈ [0, 1] 2 }. Unlike Volterra integral transformations, Fredholm transformations are not always invertible [START_REF] Yoshida | Lectures on differential and integral equations[END_REF]. The following lemma (adjusted from [START_REF] Coron | Stabilization and controllability of first-order integro-differential hyperbolic equations[END_REF], Lemma 2.2, Proposition 2.6) guarantees (under several conditions) the invertibility of such an integral operator. Lemma 3. Consider a Fredholm integral operator T : Then, the operator T is invertible.

(L 2 ([0, 1])) 2 → (L 2 ([0, 1]))
Proof. The proof follows the steps of [START_REF] Coron | Stabilization and controllability of first-order integro-differential hyperbolic equations[END_REF], Lemma 2.2, Proposition 2.6). Since the integral part of T is a compact operator, the Fredholm alternative [START_REF] Brezis | Functional analysis, Sobolev spaces and partial differential equations[END_REF] implies that dim ker(T ) < ∞. Suppose that ker(T ) = {0}. Due to assumption (a), Az is welldefined, and assumption (c) implies that ker(T ) is stable by A, that is to say, for all z ∈ ker(T ), Az ∈ ker(T ). Since ker(T ) is finite-dimensional and not reduced to {0}, the restriction A | ker(T ) of A to ker(T ) has at least one eigenvalue ν ∈ C. Let ζ be the corresponding eigenfunction. Thus, ζ ∈ ker(ν -A) and ζ ∈ ker(C) (due to assumption (b)). This is in contradiction with assumption (d). Thus, ker(T ) = {0} and T is injective. Using the Fredholm alternative, we obtain that T is invertible.

OBSERVER DESIGN

In this section, we design the observer gains g 1 , g 2 to stabilize the error system ( 9)-( 11). First, we use a Fredholm integral transform to map ( 9)-( 11) to a stable target system. We show that the existence of such a transform is implied by the existence of solutions for some integral equations. Finally, using the operator framework introduced in Section 3, we prove that the proposed Fredholm integral operator is invertible.

Presentation of the target system

We now follow the backstepping methodology to map ( 9)-( 11) to a stable target system with equivalent stability properties. Consider the candidate target system ∂ t α(t, x) + λ∂ x α(t, x) = 0, ( 16)

∂ t β(t, x) -µ∂ x β(t, x) = 0, (17)
with the boundary conditions α(t, 0) = aα(t, 1) β(t, 1) = α(t, 1).

(18) This target system ( 16)-( 18) is exponentially stable in the sense of the L 2 -norm, since |a| < 1 [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF]. Let us define

α(t, x) = ũ(t, x) - 1 0 L 11 (x, ν)ũ(t, ν) + L 12 (x, ν)ṽ(t, ν)dν, β(t, x) = ṽ(t, x) - 1 0 L 21 (x, ν)ũ(t, ν) + L 22 (x, ν)ṽ(t, ν)dν,
with L ij , i, j ∈ {1, 2} four bounded functions defined on S. We want to prove that the existence of some kernels such that this transform maps the original system ( 9)-( 11) to the target system ( 16)-( 18). Then, if this transform is invertible, the exponential stability of ( 16)-( 18) will imply the one of ( 9)-(11).

Kernel equations

Let us define the kernels L ij by

∂ x L 11 (x, ξ) + ∂ ξ L 11 (x, ξ) = 0, (19) λ∂ x L 12 (x, ξ) -µ∂ ξ L 12 (x, ξ) = 0, (20) µ∂ x L 21 (x, ξ) -λ∂ ξ L 21 (x, ξ) = 0, (21) ∂ x L 22 (x, ξ) + ∂ ξ L 22 (x, ξ) = 0, ( 22
) where we have h 1 (x) + λ(L 11 (x, 1) -aL 11 (x, 0)) -µL 12 (x, 1) = 1 0 L 11 (x, ν)h 1 (ν) + L 12 (x, ν)h 2 (ν)dν, (23) h 2 (x) + λ(L 21 (x, 1) -aL 21 (x, 0)) -µL 22 (x, 1) = 1 0 L 21 (x, ν)h 1 (ν) + L 22 (x, ν)h 2 (ν)dν, (24)
and the boundary conditions

L 11 (0, ξ) = aL 21 (1, ξ), L 12 (0, ξ) = aL 12 (1, ξ), (25) L 11 (1, ξ) = L 21 (1, ξ), L 22 (1, ξ) = L 12 (1, ξ), (26) L 21 (0, ξ) = 0, L 22 (0, ξ) = 0.
(27) These two additional boundary conditions ( 27) are necessary to guarantee the invertibility of the Fredholm integral transform. Finally, let us define the observer gains as the solutions of the integral equations 19)-( 27) admit a solution, then, following the backstepping approach, we prove that ( 9)-( 11) can be mapped to ( 16)-( 18). Then, if the transformation is invertible, the two systems will have equivalent stability properties. Finally, note that the existence of the functions g 1 and g 2 defined by (28) will be a consequence of the invertibility of the integral operator.

g 1 (x) g 2 (x) - 1 0 L 11 (x, ν) L 12 (x, ν) L 21 (x, ν) L 22 (x, ν) g 1 (ν) g 2 (ν) dν = µ L 12 (x, 0) L 22 (x, 0) (28) If (

Well-posedness of the kernel equations

In this section, we prove the existence and uniqueness of a solution to the kernels equations ( 19)-( 27). We have the following theorem Theorem 4. The set of equations ( 19)-( 27) admits a unique bounded on the unit square S.

The proof of this theorem will be given in the rest of this section and requires several intermediate results. First, we show that the knowledge of the two functions L 12 (., 1) and L 21 (., 1) is sufficient to properly define the four kernels on S. Next, we rewrite the integral equations ( 23)-( 24) using an integral operator acting on these two boundary functions. Then, we show that this integral operator is invertible using Lemma 3. Consequently, it will imply that ( 19)-( 27) admit a unique bounded solution. In what follows, we will only consider the case λ ≤ µ. The case λ > µ can be done following the same ideas. Due to space restrictions, we will not give computational details.

Kernels reduction First, we show that the existence and uniqueness of all the kernels on S is implied by the one of L 12 (x, 1) and L 21 (x, 1) on [0, 1]. Lemma 5. For all (x, ξ) ∈ S, (i, j) ∈ {1, 2} 2 , L i,j (x, ξ) can be expressed as functions of L 12 (•, 1) and L 21 (•, 1).

Proof. We apply the method of characteristics to the transport equations ( 19)-( 22) to express the kernels as functions of their boundary values. We obtain

L 11 (x, ξ) =    L 21 (1 - µ λ (x -ξ), 1) if ξ ≤ x aL 21 (1 - µ λ (1 + ξ -x), 1) else (29) L 12 (x, ξ) =          L 12 (x - λ µ (1 -ξ), 1) if max{0, 1 -µ λ x} ≤ ξ ≤ 1 aL 12 (1 + x - λ µ (1 -ξ), 1) else (30) L 21 (x, ξ) = 0 if 0 ≤ ξ ≤ min{1 -λ µ x, 1} L 21 (x - µ λ (1 -ξ), 1) else (31) L 22 (x, ξ) =    L 12 (1 - λ µ (x -ξ), 1) if 0 ≤ ξ ≤ x 0 else (32)
In particular, we get L 21 (1, ξ) = 0 ∀ξ ≤ 1 -λ µ , as illustrated on Figure 1. Thus, we can compute L ij (x, ξ), (i, j) ∈ {1, 2} 2 on S from the values of L 12 (x, 1) and L 21 (x, 1) on [0, 1].

Operator formulation

We now rewrite the integral equations ( 23)-( 24) as functions of L 12 (x, 1) and L 21 (x, 1). First, we notice that ( 19) and the boundary condition ( 25) imply that L 11 (x, 1) -aL 11 (x, 0) = 0. Thus, (23) rewrites

µL 12 (x, 1)+ 1 0 L 11 (x, ν)h 1 (ν)+L 12 (x, ν)h 2 (ν)dν = h 1 (x).
Using the different kernel equations, we finally obtain

µL 12 (x, 1) -λL 21 (x, 1) + 1 0 I(x, ν) L 12 (ν, 1) L 21 (ν, 1) = h 1 (x) h 2 (x) (33
) where the four components (I(x, ν)) ij are defined by

I 11 (x, ν) = 1 [0, λ µ ] (x)(1 [x+1-λ µ ,1] a µ λ h 2 (1 + µ λ (ν -1 -x)) + 1 [0,x] (ν) µ λ h 2 (1 + µ λ (ν -x))) + 1 [ λ µ ,1] (x)1 [x-λ µ ,x] (ν) µ λ h 2 (1 + µ λ (ν -x)) (34) I 12 (x, ν) = 1 [0, λ µ ] (x)(1 [0,1-µ λ x] a λ µ h 1 (1 -x - λ µ (1 -ν)) + 1 [1-µ λ x,1] (ν) λ µ h 1 (x - λ µ (1 -ν))) + 1 [ λ µ ,1] (x) λ µ h 1 (x - λ µ (1 -ν)) (35)
I 21 (x, ν) = 1 [1-λ µ x,1] (ν) µ λ h 2 (x - µ λ (1 -ν)) (36)
I 22 (x, ν) = 1 [0,x] (ν) λ µ h 1 (1 - λ µ (x -ν)) (37)
To rewrite (33) as an integral operator Q of the form (15), we define the new variables f (x) g(x) = µL 12 (x, 1) -λL 21 (x, 1) .

With kernels

N (x, ν) =    - 1 µ I 11 (x, ν) 1 λ I 12 (x, ν) - 1 µ I 21 (x, ν) 1 λ I 22 (x, ν)    (38) the Fredholm operator Q satisfies Q f g = h 1 h 2 . Thus,
by proving that Q is invertible, we will show that the integral equations ( 23)-( 24) admit a unique solution.

Invertibility of the operator Q Lemma 6. The Fredholm integral operator Q of form ( 15), with kernels defined by ( 38) is invertible.

Proof. We only need to prove that assumptions (a) - Following the same operator framework, we now show Theorem 7. The Fredholm integral transform P with kernels defined by ( 19)-( 27) is invertible. Consequently, the observer gains g 1 and g 2 are properly defined in (28).

Proof. Since the operator P is of form (15) with

N (x, ν) = L 11 (x, ν) L 12 (x, ν) L 21 (x, ν) L 22 (x, ν) , (39)
we can use Lemma 3 to show its invertibility. As before, assumption (d) derives from the spectral observability hypothesis 1. The boundary conditions ( 25)-( 27) imply that assumptions (a) -(b) hold. Assumption (c) is a direct consequence of ( 19)-( 24) and assumptions (a) -(b). Using equations (28), the gain observers g 1 , g 2 satisfy

P g 1 (x) g 2 (x) = µL 12 (x, 0) µL 22 (x, 0) . ( 40
)
Since P is invertible, the observer gains exist and are uniquely defined on [0, 1]. Moreover, it is straightforward to show that they belong to H 1 .

Convergence of the observer

We can now end this section with the design of an observer for the initial system (1)-( 2).

Theorem 8. Any solution (ũ, ṽ) of ( 9)-( 11) converges towards 0 in the sense of the L 2 -norm.

Proof. First, system ( 16)-( 18) is exponentially stable in the sense of the L 2 -norm. Since the backstepping transform is invertible by Theorem 7, system (9)-( 11) shares equivalent stability properties. Indeed, the inverse transform is of form (15) with bounded kernels.

Thus, the error system ( 9)-( 11) is exponentially stable. The observer state defined with gains g 1 , g 2 converges towards the initial state (u, v). More precisely, we can show that ũ(t, 1) converges towards 0. Thus, z(t) exponentially converges towards zero. We can thus compute an observer ẑ(t) for ( 1)-(2) as ẑ(t) = û(t, 1) that converges towards z.

SIMULATION RESULTS

In this section, we give some simulation results to illustrate the performance of our observer design for the time-delay system (1). Following the PDE approach presented in this paper, we simulate system (1) on Matlab using the hyperbolic comparison system (4)-( 6). The evolution of the initial and observer states are computed using a Godounov scheme [START_REF] Leveque | Finite volume methods for hyperbolic problems[END_REF]. We chose the parameter values τ 1 = 1 > τ 2 = 0.5, a = 0.8, and N 1 = 0.4, N 2 = -0.5 constant functions. They were selected such that the system is slightly unstable, as illustrated on Figure 2. One can verify that Hypothesis 1 is satisfied for these parameter values. We compute the evolution of the time-delay system over 10s, with a time-discretization of 0.01s. The initial conditions are z(0) = 1, ẑ(0) = 0.5. The kernels L ij and the observer gain functions g 1 , g 2 are previously computed using a fixed-point algorithm (successive approximation technique). Their definition domains are discretized with a space mesh equal to 0.02. The integral terms are numerically approximated by the trapezoidal method. Plotting the evolution of the norm of the error state z, we observe that it towards zero, the observer (red curve) converges towards the real value (blue curve) as expected.

CONCLUDING REMARKS

This paper proposed a new approach to design an observer for a general class of integral delay equations under a spectral observability condition. Following the methodology proposed in [START_REF] Redaud | Stabilizing integral delay dynamics and hyperbolic systems using a fredholm transformation[END_REF], we used a scalar hyperbolic system (sharing the same observability properties) as a comparison system. We designed an observer for this PDE system using the backstepping approach to map the resulting error system to a stable target system. Since the existence of distributed delay terms required the backstepping transform to be of Fredholm type, invertibility needed to be proven. To do so, we used an operator framework inspired by [START_REF] Coron | Stabilization and controllability of first-order integro-differential hyperbolic equations[END_REF]. Interestingly, the existence of solutions to the kernels equations depended on the invertibility of a similar operator. The wellposedness of the kernels defining the Fredholm transform was thus proved using the same results. The invertibility of both integral operators was a consequence of the assumed spectral observability condition. Finally, some simulations illustrated the relevance of this approach to design an observer for a time-delay system with delayed output.

In future contributions, we will try to extend our results to non-scalar systems. It will lead us to a general methodology for designing observers for integral delay dynamics and underactuated hyperbolic systems.

  2 as defined by (15). Assume that (a) ker(T ) ⊂ D(A), (b) ker(T ) ⊂ ker(C), (c) ∀z ∈ ker(T ), T Az = 0, (d) ∀s ∈ C, ker(s -A) ∩ ker(C) = {0}.

Fig. 1 .

 1 Fig. 1. Representation of kernels L ij on S

  (d) of Lemma 3 are satisfied for the operator Q. By evaluating the coupling terms (34)-(37), we immediately get assumptions (a)-(b). Assumption (d) is satisfied under the spectral observability Hypothesis 1. In order to prove that assumption (c) holds, we use assumptions (a) -(b), integration by parts and changes of variables.It is straightforward to show that (f, g) belongs to H 1 ([0, 1]) (due to the regularity of the functions h Coron et al. (2016)). We can now conclude the proof of Theorem 4. The boundary functions L 12 (x, 1) and L 21 (x, 1) exist and are unique on [0, 1] by Lemma 6. They define the four kernels L ij , (i, j) ∈ {1, 2} 2 on their definition domain S by Lemma 5.4.4 Invertibility of the Fredholm transform and existence of the observer gains g 1 and g 2

Fig. 2 .

 2 Fig.2. Evolution of the time-delay system (1) z(t), ẑ(t).


Insights into carbon-nanotube-assisted electrooxidation of polycyclic aromatic hydrocarbons for mediated bioelectrocatalysis

A series of polycyclic aromatics, naphthalene, phenanthrene, perylene, pyrene, 1-pyrenebutyric acid Nhydroxysuccinimide ester (pyrene NHS) and coronene, were immobilized via π stacking on carbon nanotube (CNT) electrodes and electro-oxidized in aqueous solutions. The obtained quinones were characterized and evaluated for the mediated electron transfer with FAD dependent glucose dehydrogenase during catalytic glucose oxidation.

Polycyclic aromatics and their derivatives are widely used to functionalize carbon nanotubes (CNTs) and graphene 1,

2

. Pyrene [START_REF] Chen | [END_REF] and perylene 4 are within the most famous examples related to easy chemical modification possibilities to attach any desired function. Their ability to form rapidly stable interaction with the pi-system of these carbon materials under ambient condition has the advantage that even sensitive compounds like biomolecules can be immobilized 5 . Other polycyclic aromatic hydrocarbons and derivatives were principally used for solubilization of CNTs 6 and graphene 7 or, in case of coronene, as stacked fluorescent fillers inside CNTs 8 . Unmodified polycyclic aromatic hydrocarbons do not show any electrochemical activity, but they can be electro-oxidized in organic media at high potentials forming oligomers and polymers 9, 10 . Barathi et al. studied comprehensively the electro oxidation of anthracene in buffer solutions and demonstrated the formation of the redox active 9,10-anthraquinone 11 . More recently, we reported the formation of electroactive quinones with pyrene on CNTs when electro-oxidized in aqueous media 12 . The obtained pyrenediones showed efficient electron transfers with FAD dependent glucose dehydrogenases (FAD-GDH) and represents a great candidate as mediator in enzymatic biofuel cell anodes. FAD -GDH is a glucose-oxidizing enzyme which has been proposed as a promising alternative to glucose oxidase in biosensors and biofuel cells 13, 14 . FAD-GDH has several advantages compared to the commonly used glucose oxidase, such as a tightly-bound FAD cofactor and no activity towards reduction of O 2 into H 2 O 2 . The electron transfer between the FAD cofactor and the electrode is quite challenging since this cofactor is deeply embedded inside the protein shell and the maximum distance for direct electron transfer is largely overpassed. Therefore, the only way to obtain efficient electron transfer is the use of redox mediators. Most redox mediators for FAD-GDH are based on osmium complexes [15][16][17] and quinones [18][19][20] . In this work, we extended the CNT-assisted electrooxidation to a range of polycyclic aromatic hydrocarbons from naphthalene (two aromatic rings) to coronene (seven aromatic rings) including pyrene derivatives. We demonstrate that all compounds can be electrooxidized at 1.2 V vs SCE but with different synthetic yields and redox potentials, suitable or not for mediated electron transfer with FAD-GDH. The MWCNT electrodes were modified with the polyaromatic hydrocarbons via drop casting as described in the ESI. These electrodes were then electro-oxidized in aqueous electrolyte solution at pH 2 by cycling between -0.2 and 1.2 V vs SCE at a scan rate of 100 mV s -1 . Figure S1 shows the recorded cyclic voltammograms of all studied compounds. The irreversible oxidation observed at 1.1 V corresponds to the oxidation of each aromatic hydrocarbon 12 . This irreversible oxidation decreases after each scan and in each case, a formation and evolution of a redox signal can be observed. The corresponding E 0 app values (vs SCE and NHE) for each compound are found in Table S1. The intensity of the redox signals can be ascribed to the yield of electro synthesized species. Figure S2 shows the determined yields in combination with the obtained surface coverage as a function of the amount of the deposited hydrocarbon. The highest yields between 8% and 16% were obtained with the pyrenes and perylene while naphthalene, phenanthene, and coronene were below 4%. The optimal surface coverages were obtained for all compounds after deposition of 20 µL of a 10 mmol L -1 solution (CH 2 Cl 2 ). It has to be noted that no clear morphology changes of the CNT electrode during the electrochemical modifications appeared (Figure S3). The obtained electrodes were then characterized in neutral buffer solution. The CVs in Figure 1 show the redox systems of the adsorbed oxygenated species for each studied aromatic hydrocarbon. While the electro oxidized naphthalene and phenanthrene show respectively identic redox potentials at E 0 app = -0.17 V and -0.19 V as the commercially available 1,4-naphthoquinone and 9,10-phenanthrenequinone, pyrene, pyrene-NHS, perylene, and coronene show redox potentials at -0.07 V, -0.10 V, -0.14 V, and -0.08 V vs SCE, respectively. The electro oxidation of pyrene is perfectly repeatable as reported in our previous work 12 . Its derivative pyrene-NHS has a slightly lower redox potential which can be ascribed to the electron pushing effect of the butyl-NHS function. For naphthalene, phenanthrene, and perylene, a second less pronounced redox system is formed which could partly be identified (see section below). Pyrene and its NHS derivative show clearly higher peak separations (ΔE p ) than the other studied compounds. This might be related to the higher electro-oxidation yields resulting in lower diffusion rates. Since the oxidoreduction of quinones is a 2 electrons / 2 protons process (Scheme S1), we observed a linear dependency of the redox potentials on the pH values which correspond nicely to the thermodynamic slope 21 . The obtained data about electro-oxidation yields, peak separations and pH dependencies are summarized in Table S1. It should be noted that the presence of a CNTs deposit plays a crucial role since only negligible oxidation yields could be observed with pyrene as reference compound on unmodified glassy carbon electrodes. With the aim to study the properties of these electrogenerated compounds as mediators for the electron transfer with FAD-GDH, stability tests were done by stimulating aging via repeated voltammetric cycling. After 100 cycles between -0.4 V and 0.4 V vs SCE for electro oxidized pyrenes, perylene, and coronene, and between -0.4 V and 0.1 V for electro oxidized phenanthrene, the two studied perylene and pyrene diones showed best stabilities with signal losses below 5%. Naphthoquinone and oxidized coronene also presented good stabilities with signal losses between 10% and 15%, and almost 30% of the phenanthrenequinone signal disappeared after cycling. These differences in stabilities might be related to an improved water solubility of the formed hydroquinones or to side reactions during the two electron transfers provoking the loss of electroactivity. To get an idea about the structure of the electroactive species via 1 H NMR spectroscopy, 10 MWCNT electrodes were prepared for each polyaromatic hydrocarbon to get sufficient amount of the electro-oxidized compound by immersing and rinsing the resulting electrodes with at total 0.75 mL of CDCl 3 . Figure 2A shows the spectrum of the electro-oxidized naphthalene. Beside the starting product (protons Ha and Hb) and its impurities (#), identified on the spectrum of purchased naphthalene (not shown), the signals can be assigned to 1,4-naphthoquinone. Other oxides or quinones like 1,2-naphthoquinone could not be resolved. The spectrum of electro oxidized phenanthrene (Figure 2B) shows in majority the starting material. The additional peaks indicate the presence of 9,10-phenanthrenequinone. The other, unassignable peaks (#) correspond to impurities in the starting material. Electro-oxidizing pyrene (Figure 2C) led to the two isomers 1,6-pyrenedione and 1,8-pyrenedione. It has to be noted that the starting material pyrene cannot be identified in this spectrum, which is most likely due to the excellent π-stacking interaction with the CNT sidewall resisting the rinsing steps. The spectrum of the products recovered from the electro-oxidized pyrene-NHS electrodes (Figure 2D) shows in majority the signature of the starting material. The remaining less pronounced peaks can be attributed to 1,8-pyrenedione-NHS and to 1,6-pyrenedione-NHS. Beside the signature of the starting material, the main product after the electro oxidation of perylene (Figure 2E) is 3,10-perylenedione. Less resolved peaks at 7.53 ppm and 6.80 ppm ( §) may be assigned to 3,4,9,10-perylenetetrone but noise and partial covering of more pronounced peaks do not allow a clear assignment. The identified products after electro oxidation of coronene (Figure 2F) are principally the starting product, 1,2coronenedione, and 1,10-coronenedione. However, it remains a certain ambiguity with the assignment of the last compound 1,10-coronenedione. For instance, proton Hh should give a singlet due to the absence of vicinal protons but a triplet is observed. Nonetheless, the interpretation of all spectra was assisted by gradient-selected COSY (gCOSY) spectra and simulation of all possible diones, tetrones, and hexones using the online tool nmrdb.org 22 which led us to conclude for the designated compounds as most likely formed quinones. More NMR data for each compound can be found in the ESI. All modified MWCNT electrodes were then studied towards mediated electron transfer with FAD-GDH immobilized via drop casting. Cyclic voltammograms were recorded between -0.4 V and 0.4 V vs SCE to determine the efficiency of electrocatalytic oxidation of glucose (Figure 3). 3A). The OCP, or onset potential, was determined by extrapolation of the linear part of the anodic wave to the potential axis at 0 A. This is consistent with previous studies using 1,4-naphthoquinone 12 , taking into account the low electrooxidation yield and the resulting surface coverage.

Phenanthrenedione electrodes show a slightly higher current density with 0.9 mA cm -2 at 0.2 V with an onset potential at -0.24 V (Figure 3B). These performances are clearly overpassed with pyrenedione electrodes which gave a current density of 7.5 mA cm -2 and an OCP at -0.14 V (Figure 3C). Interestingly, with similar electro oxidation yield and surface coverage, pyrene-NHS dione provided less than the half of electrocatalytic current density with 3 mA cm -2 but a slightly lower OCP of -0.18 V (Figure 3D). Despite the corresponding redox potential for mediated electron transfers with FAD-GDH, the CVs of both perylenedione and coronenedione did not show any or negligible catalytic currents and are therefore not presented here. We ascribe this incapability to transfer electrons to the size of the compounds and possible steric hindrance to interact with the catalytic site of the enzyme. For the other electrodes, we further stimulated again aging of the electrodes by cycling 100 times to evaluate the stability of electro oxidized compounds for mediated electron transfer. All core compounds lost more than 70% of their initial current densities with up to 90% for naphthoquinone. The exception is pyrenedione-NHS which lost less than 13%. This is not surprising since the majority of such losses are related to the desorption of the enzyme 12 . Pyrene-NHS has become a standard linker for the covalent grafting of proteins onto graphene and carbon nanotubes and has been used for two decades [START_REF] Chen | [END_REF] . Obviously, electro oxidation of the pyrene units does not affect the activated ester function which forms amides with the enzymes' amine functions and results in clearly enhanced stabilities for mediated electron transfers and thus catalytic current densities.

These determined performances, however, do not reflect the reality of the capability of these formed quinones for the electron transfer with FAD-GDH since the electro oxidation yield and the resulting surface coverages are different. To harmonize the obtained data and to get a better view about the performances, we plotted the respective redox potentials (E ½ ) as a function of the normalized catalytic current densities considering the surface coverage (Figure 4A). With this plot, a clear tendency of electron transfer rates in relation with the redox potential can be observed, where the derivatives of perylene and coronene are excluded due to the size and steric hindrance effects toward FAD-GDH. It appears that the electrochemical properties of pyrenedione allow enhanced electron transfer kinetics at slightly higher but still competitive redox potential for enzymatic biofuel cells. The slightly lower potential of pyrene-NHS matches nicely the trend of the curve but some steric effect of the NHS group vs the access to the prosthetic group of the enzyme can also be imagined. The tendency of higher electron transfer kinetics related to higher overpotentials is perfectly in line with other reports 23 . This mediator was further investigated for its analytical performances in a biosensor configuration (Figure S4). Chronoamperometric measurements at 0.2 V vs. SCE led to a step wise increase of the catalytic current upon glucose injections until saturation at 200 mmol L -1 . Figure S4B shows the anodic current response of the biosensors as a function of glucose concentration. The sensitivity of this setup, determined by the slope of the linear part of the calibration curve, is 0.44 mAM -1 cm -2

. A limit-of-detection of 0.11 mmol was determined. An apparent Michaelis Menten constant of K app M = 23 mmol L -1 is in good agreement with the report of Tsujimura et al. 24 The sensor performances are quite satisfying with a good sensitivity and a linear range covering the glucose concentration range of interest for diabetes patients. To profit from the beneficial properties of pyrenedione in terms of electron transfer efficiency and of pyrene-NHS dione in terms of stability, we used mixtures of these two compounds since the optimal electro oxidation conditions are identic. Figure 4B shows the CV of the first and 100 th scan of the optimized pyrene/pyrene-NHS mixture (90% / 10%). The recorded catalytic current density is 5.9 mA cm -2 at 0.2 V with a relative standard deviation of 15%. This is below the performance of pyrenedione but twice as higher as pyrene-NHS dione only. There is a clearer improvement in terms of stability; after 100 cycles, the loss of electrocatalytic current density is just 13.7% which is by far more stable than pyrenedione only (> 75 %). In conclusion, a series of polyaromatic hydrocarbons, immobilized on MWCNT electrodes were electro oxidized in aqueous solution and the obtained quinones were characterized using 1 H NMR spectroscopy and cyclic voltammetry. The formed pyrenedione showed to be a more appropriate redox mediator for the electron transfer with the enzyme FAD GDH, giving excellent catalytic current densities at a reasonable half-cell voltage enabling a good cell voltage when combined with an appropriate (bio) cathode. Furthermore, the possibility to electro oxidize its derivative pyrene-NHS, keeping intact the activated ester function, allows to graft covalently the enzyme leading to clearly enhanced stabilities for bioelectrocatalytic oxidation of glucose. In combination with pyrenedione, glucose-oxidizing bioelectrodes were obtained with high current densities and high operational stability.

Figure 1 : 1 ,

 11 Figure 1: Cyclic voltammograms (100 scans) for the characterization and evaluation of the stability of electro oxidized naphthalene (A), phenanthrene (B), pyrene (C), pyrene-NHS (D), perylene, (E), and coronene (F). The CVs were recorded in phosphate buffer solution (0.1 mol L -1 , pH 7) at a scan rate of 100 mV s -1

Figure 2 : 1 H

 21 Figure 2: 1 H NMR spectra of the recovered oxidized species after electro oxidation of (A) naphthalene, (B) phenanthrene, (C) pyrene, (D) pyrene-NHS, (E) perylene, and (F) coronene. (* = 13 C satellite peaks of CDCl 3 ; # = residues in the starting product)

Figure 3 :

 3 Figure 3: Cyclic voltammograms using FAD GDH modified electrodes with the respective electrooxidized compound. (black) CV before addition of 200 mmolL -1 glucose, (red) after glucose addition, (green) 100th cycle for (A) naphthoquinone, (B) phenanthrenequinone, (C) pyrenedione, and (D) pyrene-NHS dione. The CVs were recorded in phosphate buffer solution (0.1 mol L -1 , pH 7) at a scan rate of 2 mV s -1

Figure 4 : 1 ,

 41 Figure 4: (A) Plot of the surface coverage normalized catalytic current densities as a function of the redox potential of all studied electro oxidized polyaromatic hydrocarbons. (B) Cyclic voltammograms using FAD GDH modified electrodes with a 90% -10% mixture of electro oxidized pyrene and pyrene-NHS. (black) CV before addition of 200 mmolL -1 glucose, (red) first cycle after glucose addition, (green) 100 th

Editing; Alan Le Goff: Writing, Data Curation; Michael Holzinger: Conceptualization, Supervision, Data Curation, Writing. There are no conflicts to declare The authors wish to acknowledge the support from the platform Chimie NanoBio ICMG FR 2607 (PCN-ICMG) and from the LabEx ARCANE (ANR-11-LABX-0003-01 and CBH-EUR-GS, ANR-17-EURE-0003).The authors particularly thank the ANR -JST program, project Mocca-Cell (ANR-15-JTIC-0002-01), and the Institut Carnot PolyNat (CARN 0007-01) for financial support. E.M.G. thanks CNPq for the grants.