Xiaoyi Mai

Romain Couillet

SEMI-SUPERVISED SPECTRAL CLUSTERING

Keywords: semi-supervised learning, spectral clustering, graphs, consistency

In this article, we propose a semi-supervised version of spectral clustering, a widespread graph-based unsupervised learning method. The semi-supervised spectral clustering has the advantage of producing consistent classification of data with sufficiently large number of labelled or unlabelled data, unlike classical graph-based semi-supervised methods which are only consistent on labelled data. Theoretical arguments are provided to support the proposition of this novel approach, as well as empirical evidence to confirm the theoretical claims and demonstrate its superiority over other graph-based semisupervised methods.

INTRODUCTION

Semi-supervised learning aims to maximize the learning ability by using both labelled and unlabelled data, typically a small amount of labelled data with abundant unlabelled data. In many real-life machine learning tasks, the labeling process is much more expensive than the collection of unabelled data, it is thus of great practical value to study the semisupervised approach. A successful semi-supervised learning method should be able to learn from both the labelled data and the underlying clusters in the unlabelled data. Combining these two types of information is conceptually more difficult than one-sided learning perspectives focused on labelled or unlabelled data. In this regard, some seemingly natural semisupervised approaches may have unexpected outcomes.

A recent random matrix analysis [START_REF] Mai | A random matrix analysis and improvement of semi-supervised learning for large dimensional data[END_REF] notably shows that in very high dimensions, a popular branch of graph-based semisupervised learning methods (here referred to as Laplacian regularization) fails to exploit the clustering structure of unlabelled data. Also, the results in [START_REF] Nadler | Semisupervised learning with the graph laplacian: The limit of infinite unlabelled data[END_REF] state that the Laplacian regularization approach converges to a trivial solution in the limit of infinite data. As such, the Laplacian regularization methods tend to disregard additional unlabelled data which we refer to as being inconsistent with respect to the unlabelled dataset. As a consequence, given sufficient unlabelled data, Laplacian regularization will be outperformed by spectral clustering, an unsupervised method which learns from the data graph by optimizing the same objective as Laplacian regularization but whose consistency is proven in [START_REF] Von Luxburg | Consistency of spectral clustering[END_REF].

The graph-based semi-supervised learning method presented in this article is based on spectral clustering, in order to ensure its consistency with respect to unlabelled data. An original strategy is proposed to incorporate the labelled data in the learning process. The proposed method is easy to implement with similar cost to spectral clustering. To demonstrate its efficiency as a semi-supervised learning algorithm, we provide a mathematical discussion along with experimental confirmations, all showing that the semi-supervised approach yields better partitions of data points than spectral clustering as a result of exploiting labelled data.

The reminder of the article is organized as follows. We start by introducing preliminary notions in Section 2. The proposed semi-supervised spectral clustering method is presented in Section 3, along with theoretical justification and corroborating simulations. Finally, we end the article with concluding remarks in Section 4.

PRELIMINARIES

We present the common setting of graph-based methods in Section 2.1. The principles and consistency properties of spectral clustering, on which the proposed method is based, are briefly explained and discussed in Section 2.2, followed by a section on the Laplacian regularization methods.

Problem Setup

Graph-based methods represent data points as vertices in a graph with their similarities reflected by the edges. In this paper, we consider undirected and weighted graphs G = (V, E) with set of vertices V = {x 1 , • • • , x n }, which is also the dataset, and set of edges E = {(x i , x j , w ij )|x i , x j ∈ V }, where (x i , x j , w ij ) denotes an edge of weight w ij between x i and x j , and w ii = 0. The degree d i of a node x i is defined as d i = ∑ j w ij . The adjacency matrix W of the graph is an n × n symmetric matrix with W ij = w ij , and the degree matrix D ∈ R n×n is a diagonal matrix with D ii = d i . The Laplacian matrix L is defined as L = D -W . The symmetric normalized form of the Laplacian matrix is

L s = D -1 2 LD -1 2 = I -W s where W s = D -1 2 W D -1 2
is the symmetric normalized adjacency matrix.

The objective of graph-based methods is to learn a data representation that is in accordance with the graph structure, meaning that data points connected with large weights have similar representations. This is usually achieved by minimizing a loss function:

1 2 n ∑ i,j=1 w ij (f i -f j ) 2 = f T Lf (1)
where f is the representation function. A normalized formulation of (1) writes as

1 2 n ∑ i,j=1 w ij ( f i √ d i - f j √ d j ) 2 = f T L s f. ( 2
)
It is easily seen that the minimization of (1) imposes data points connected with large weights to have close values on the representation function f , so does (2) for closely connected data points with similar degrees. In the following, we will refer to (1) and ( 2) as the graph smoothness penalty terms of f since their minimizations encourage f to have similar values on neighboring vertices in the graph.

Spectral Clustering

Evidently, the graph smoothness penalty term (1) is minimized when f has constant values on all data points, which is obviously a useless data representation for the partitioning.

Similarly for the normalized Laplacian regularizer (2) with the trivial solution f i = √ d i . Therefore, the spectral clustering method states the problem as minimizing (1) or [START_REF] Nadler | Semisupervised learning with the graph laplacian: The limit of infinite unlabelled data[END_REF] in the subspace orthogonal to the trivial solution for a fixed norm of f . This is equivalent to finding the eigenvector associated with the second smallest eigenvalue for L (unnormalized spectral clustering) or L s (normalized spectral clustering), as the trivial solutions are the eigenvectors of Laplacian matrices associated with the smallest eigenvalue 0. See [START_REF] Von | A tutorial on spectral clustering[END_REF] for more details.

The consistency properties of spectral clustering are investigated in [START_REF] Von Luxburg | Consistency of spectral clustering[END_REF], where the authors prove that under very mild assumptions, the clusters constructed by normalized spectral clustering converge to a limit clustering of the whole data space as the number of sampled data tends to infinity. It is also found that unnormalized spectral clustering is only consistent under strong additional assumptions, which are not always satisfied in real data.

Laplacian Regularization

The Laplacian regularization approach [START_REF] Zhu | Semisupervised learning using gaussian fields and harmonic functions[END_REF][START_REF] Zhou | Learning with local and global consistency[END_REF][START_REF] Delalleau | Efficient non-parametric function induction in semi-supervised learning[END_REF] is very similar to spectral clustering, in the sense that it searches the representation function f that minimizes the same graph smoothness penalty terms (1) and ( 2) as spectral clustering. The difference is that for Laplacian regularization, the representation function f has to also respect the labeling y [l] of the labelled data, which leads to imposing f

[l] = y [l]
1 [START_REF] Zhu | Semisupervised learning using gaussian fields and harmonic functions[END_REF], where f [l] is the subset of f corresponding to the labelled data. The Laplacian regularization algorithms consist thus in minimizing (1) or ( 2)

under the constraint f [l] = y [l] .
Although seemingly a perfectly natural way to learn the inherent clusters of the graph in a semi-supervised manner, the Laplacian regularization methods have some unexpected behaviors when dealing with large dimensional data, as demonstrated in [START_REF] Mai | A random matrix analysis and improvement of semi-supervised learning for large dimensional data[END_REF] by random matrix arguments. The main issue is that for datasets of sufficiently large dimension and size, an increase in the number of unlabelled data has negligible contribution to the performance of Laplacian regularization algorithms, implying that the Laplacian regularization approach is inconsistent with respect to unlabelled data. Another theoretical work [START_REF] Nadler | Semisupervised learning with the graph laplacian: The limit of infinite unlabelled data[END_REF] shows that the Laplacian regularization methods yield flat solutions of f [u] (f [u] being the part of f corresponding to the unlabelled data) in the limit of infinite unlabelled data. Since a flat f [u] does not conform to the inherent clusters in the graph, it is conjectured that the same inconsistency problem of high dimensional data also occurs in small dimensions. We refer the reader to [START_REF] Nadler | Semisupervised learning with the graph laplacian: The limit of infinite unlabelled data[END_REF] and [START_REF] Mai | A random matrix analysis and improvement of semi-supervised learning for large dimensional data[END_REF] for more details.

SEMI-SUPERVISED SPECTRAL CLUSTERING

Despite coming from the same idea of learning a data representation that is smooth on the graph by using the smoothness penalty terms (1) and ( 2), the Laplacian regularization approach does not share the same consistency properties as spectral clustering, as discussed in the above section. As an answer to this problem, we propose in this section a semisupervised adaptation of the spectral clustering method. The idea is to preserve the data representation learned from spectral clustering while making use of labelled data, so that the proposed method is consistent with respect to unlabelled data as in the spectral clustering case. The challenge is to ensure that the use of labelled data improves the performance of the proposed algorithm, i.e., it should outperform spectral clustering on the same datasets.

Here are some notations that will be used in the following. Data points

x i are separated into k classes C 1 , • • • , C k . For a ∈ {1, • • • , k}, let n a denote the number of data in class C a ,
with n [l]a labelled ones and n [u]a unlabelled ones. The total numbers of labelled and unlabelled data are

n [l] = ∑ a n [l]a , n [u] = ∑ a n [u]a .
The vector j a ∈ R n is defined as the indicator vector for data points in C a with [j a ] i = 1 if the data point

x i belongs to C a , otherwise [j a ] i = 0. Similarly, j [l] ∈ R n (resp., j [l] ∈ R n ) is the indicator vectors for labelled (resp., unlabelled) data points with [j [l] ] i = 1 if x i is labelled, oth- erwise [j a ] i = 0.
The same is understood with the indicator vectors j [l]a , j [u]a ∈ R n for labelled and unlabelled data in C a . The operator D(v) = D{v a } k a=1 is the diagonal matrix having v 1 , • • • , v k as its ordered diagonal elements.

Proposed Method

As explained in Section 2.2, spectral clustering consists in conducting first an eigendecomposition of the Laplacian matrix L or the normalized Laplacian matrix L s , then selecting a few eigenvectors associated with the smallest eigenvalues except the first one to construct feature vectors of data, before the final partitioning step. Since the normalized spectral clustering algorithm is consistent [START_REF] Von Luxburg | Consistency of spectral clustering[END_REF], so is the learned data representation (i.e., the eigenvectors). In order to maintain the consistency property of normalized spectral clustering, the natural move is to base the semi-supervised learning algorithm on the eigenvectors of L s .

In fact, there exist already such algorithms. Like spectral clustering, the manifold-based method [START_REF] Belkin | Using manifold stucture for partially labeled classification[END_REF] also uses the eigenvectors of Laplacian matrices V = {v 1 , • • • , v m } associated with the m smallest eigenvalues except the first one. Since eigenvectors with small eigenvalues are considered smooth on the graph, the manifold-base method constrains the representation function f to live in the subspace constructed by a certain number of eigenvectors with small eigenvalues, as a means to control the smoothness of f . This strategy can also be interpreted from a graph signal processing perspective [START_REF] Gadde | Active semi-supervised learning using sampling theory for graph signals[END_REF][START_REF] Anis | Asymptotic justification of bandlimited interpolation of graph signals for semi-supervised learning[END_REF], where it is justified as limiting the "frequency" of f . The function f = V a is determined by minimizing ∥f [l] -y [l] ∥, with a solved by the method of least squares.

The disadvantage of this method resides in the delicate choice of the number of selected eigenvectors. If the number is large, eigenvectors with relatively high eigenvalues are included, which induces a risk of hurting the smoothness of f ; on the other hand, a decrease in the number of selected eigenvector will lead to greater loss of information contained in the discarded eigenvectors.

We propose here an algorithm based also on the eigenvectors of Laplacian matrices, but without the aforementioned disadvantage of the manifold-based method. In other terms, the algorithm should focus on the smoothest eigenvectors, and in the meantime, allow to leverage the whole Laplacian matrix. Obviously, a perfect data representations is composed of the class indicator vectors j a . The consistency of normalized spectral clustering [START_REF] Von Luxburg | Consistency of spectral clustering[END_REF], as well as recent large dimensional arguments [START_REF] Couillet | Kernel spectral clustering of large dimensional data[END_REF], implies that there exist informative eigenvectors of L s which can be seen as a weighted sum of j a plus noise. As we already know the class of labelled data, we can denoise the eigenvectors by replacing their labelled subsets with weighted sums of j [l]a , the class indicator vectors for labelled data. In a subsequent step, the information on labelled points is propagated through the graph by multiplying the denoised eigenvectors with the normalized adjacency matrix W s , similarly to the label propagation procedure of [START_REF] Zhou | Learning with local and global consistency[END_REF][START_REF] Zhu | Learning from labeled and unlabeled data with label propagation[END_REF]. The proposed semi-supervised spectral clustering is summarized in Algorithm 1.

3: Compute the k + 1 eigenvectors v 0 , v 1 , • • • , v k corre- sponding to the k + 1 smallest eigenvalues of L s . 4: For a, b = 1, • • • , k, define m ab = v T a j [l]b /n [l]b . 5: For a = 1, • • • , k, compute va = ∑ k b=1 m ab j [l]b + D(j [u]
)v a and ṽa = W s va /∥W s va ∥.

6: Let Ṽ = [ṽ 1 , • • • , ṽk ] ∈ R n×k . 7: For i = 1, • • • , n, define ϕ i ∈ R k as the feature vector
for data point x i and affect it with the i-th row of Ṽ .

8: Cluster (ϕ i ) i=1,••• ,n with k-means algorithm (or other clustering techniques) into C 1 , • • • , C k .

Theoretical Arguments

The objective of this section is to provide more technical details to the intuitive justification of the semi-supervised spectral clustering method given in Section 3.1 and to explain the success of this semi-supervised approach in leveraging labelled data to improve the performance of spectral clustering.

In common statistical modeling, the data in the same class C a are considered to be drawn independently according to a certain probability measure P a . Since w ij measures the similarity between x i , x j , it is considered as a function of x i , x j , i.e., w ij = g(x i , x j ) where g is a symmetric function with g(x i , x j ) = g(x j , x i ). Therefore, for all pair of x i , x j belonging to the same combination of classes, g(x i , x j ) follow the same distribution. Otherwise speaking, the graph G = (V, E) is statistically invariant to the exchange of data points in the same class. The adjacency matrix W is consequently written as

W = E{W } + B = k ∑ a,b=1 q ab j a j T b + B
for some scalars q ab with q ab = q ba , and B is a noise matrix with E{B} = 0. And so is the normalized adjacency matrix W s :

W s = E{W s } + B ′ = k ∑ a,b=1 r ab j a j T b + B ′ . ( 3
)
for some scalars r ab with r ab = r ba , and B ′ is a noise matrix with E{B ′ } = 0.

The key steps of Algorithm 1 are the steps 4 and 5, in which the class information of labelled data are employed to modify the selected eigenvectors of L s , before they are used as feature vectors for the final clustering step. In order to understand the purpose of these key steps and to demonstrate the performance gain achieved by them, we consider a selected eigenvector v a of L s with λ a v a = L s v a , where λ a is the associated eigenvalue of v a . Recalling that L s = I -W s , we have

W s v a = (I -L s )v a = (1 -λ a )v a , ( 4
)
v a is thus an eigenvector of W s with the corresponding eigenvalue 1 -λ a . Following the discussion above, the data points in the same class are interchangeable on W s , it is easily deduced that v a can be expressed as

v a = E{v a } + β a = k ∑ b=1 c ab j b + β a . ( 5
)
for some scalars c ab , and β a is a noise vector with E{β a } = 0. Evidently, β a is useless to the recovery of the underlying data clusters, due to the fact E{β a } = 0. It is thus reasonable to assume that it produces no information when propagated though the graph, i.e., βa = W s β a is also non-informative noise.

Since labelled data within the same class as unlabelled ones share the same statistical properties, the quantities m ab defined in Algorithm 1 are estimators for c ab in (5); these estimators are asymptotically consistent as the number of labelled data grows. As such, the vector va computed in the step 5 can be written as

va = k ∑ b=1 c ab j b + k ∑ b=1 (m ab -c ab )j [l]b + D(j [u] )β a . (6)
To put it simply, va is an estimation of E{v a } plus the projection of β a on the unlabelled points. If c ab are satisfyingly estimated by m ab , we have

ṽa = W s va ≃ W s [ E{v a } + D(j [u] )β a ] . (7)
Finally, we obtain from (4), ( 5) and ( 7) that

(1 -λ a )v a ≃ ṽa + W s D(j [l] )β a . ( 8
)
As discussed before, the term L s D(j [l] )β a only introduces additional noise, v a is thus less informative than ṽa as features of data points, suggesting that the semi-supervised spectral clustering method has a superior learning ability than spectral clustering.

Experimentation

In this section, we provide simulations conducted on synthetic and real datasets. For simplicity, we focus here on binary tasks. The synthetic data are generated from a Gaussian mixture model, i.e., x i ∈ C a ⇔ x i ∼ N (µ a , C a ). The real datasets tested come from the MNIST database [START_REF] Lecun | The mnist database of handwritten digits[END_REF], a standard database of hand-written digits.

To guarantee the generality of the experimental results, we use simple settings. The similarities w ij are computed with a Gaussian kernel function: w ij = exp(-∥x i -x j ∥2 /p) where p is the dimension of data vectors x i . All possible numbers of selected eigenvectors are tested for the manifoldbased method in order to find the best performance 2 . Even though performance gains are observed when using multiple eigenvectors for the proposed spectral clustering algorithm, notably on MNIST datasets, we report only results with one eigenvector.

The first purpose of the experimentation is to verify empirically that the proposed semi-supervised spectral clustering algorithm learns indeed from both labelled and unlabelled data, by showing that it surpasses the unsupervised spectral clustering on the same datasets, as theoretically claimed in Section 1. To illustrate this point, the accuracy3 curves as a function of the ratio of labelled data n [l] /n are given in Figure 1 and Figure 2, where semi-supervised spectral clustering is shown to have a growing performance gain over spectral clustering as the ratio of labelled data increases.

= C 2 = I p , n = 600, n [l]1 = n [l]2 , n [u]1 = n [u]2 .
Averaged over 500 iterations.

[l]1 = n [l]2 , n [u]1 = n [u]2 .
Averaged over 500 iterations.

After confirming the proposed method as a valid semisupervised approach, we now move to compare it with the two most classic graph-based semi-supervised techniques: the Laplacian regularization method [START_REF] Zhu | Semisupervised learning using gaussian fields and harmonic functions[END_REF] and the manifoldbased method [START_REF] Belkin | Using manifold stucture for partially labeled classification[END_REF]. Figure 3 and Figure 4 show that unlike Laplacian regularization, the semi-supervised clustering algorithm and the manifold-based method both benefit from an increasing number of unlabelled data, with the manifoldbased method consistently outperformed by the proposed approach. It should be mentioned that the extremely poor performances displayed by the Laplacian approach are due to the very small n [l] of the tested datasets.

CONCLUDING REMARKS

Mixing labelled and unlabelled data makes the design of semi-supervised techniques an interesting but difficult question. Some common semi-supervised methods, such as Laplacian regularization, do not really learn from both types of data. This article proposes a semi-supervised spectral clustering method which is consistent with respect to both labelled and unlabelled data, with significantly better performance over other classical graph-based semi-supervised approaches.

The semi-supervised spectral clustering algorithm presented in this article contains some new critical steps, entirely different from existing graph-based semi-supervised strategies. A more systematic analysis, as a follow up of [START_REF] Mai | A random matrix analysis and improvement of semi-supervised learning for large dimensional data[END_REF][START_REF] Couillet | Kernel spectral clustering of large dimensional data[END_REF], on the asymptotic performances of the proposed method, is however needed to quantitatively evaluate further the gains as well as the limitations of the approach, which will be the subject of forthcoming investigations.

Algorithm 1 2 :

 12 Semi-supervised spectral clustering 1: Input: Normalized Laplacian matrix L s . Number k of classes. Class indicator vectors j [l]1 , • • • , j [l]k for labelled data. Output: Classification of the unlabelled dataset.

Fig. 1 .

 1 Fig. 1. Accuracy as a function of n [l] /n for 2-class Gaussian data vectors of dimension p = 200 with ∥µ 1 -µ 2 ∥ = 2 and C 1 = C 2 = I p , n = 600, n [l]1 = n [l]2 , n [u]1 = n [u]2 .Averaged over 500 iterations.

Fig. 2 .

 2 Fig. 2. Accuracy as a function of n[l] /n for 2-class MNIST data (6,9), n = 600, n [l]1 = n [l]2 , n [u]1 = n [u]2 .Averaged over 500 iterations.

Fig. 3 .Fig. 4 .

 34 Fig. 3. Accuracy as a function of n [u] for 2-class Gaussian data vectors of dimension p = 200 with ∥µ 1 -µ 2 ∥ = 2 andC 1 = C 2 = I p , n [l]1 = n [l]2 = 5, n [u]1 = n [u]2 .Averaged over 500 iterations.

This condition can be relaxed by adding a penalty term of the form ∥f[l] -y [l] ∥

to the optimization objective[START_REF] Chapelle | Semi-supervised learning[END_REF].

Thus the provided manifold performances can be seen as oracle ones

The accuracy refers to the proportion of correctly classified unlabelled data.


Christophe Roman

Delphine Bresch-Pietri

Christophe Prieur

Olivier Sename

Robustness to in-domain viscous damping of a collocated boundary adaptive feedback law for an anti-damped boundary wave PDE

In this paper, the robustness to model mismatch of a pre-existing collocated boundary adaptive feedback law is investigated. This control law was originally designed for an anti-damped pure wave Partial Differential Equation (PDE). Actuation and measurements are located at the same boundary. Adaptive terms account for uncertain parameters located at the anti-damped boundary, opposite to the collocated actuation and measurement. By extending and transforming the system state using, in particular, backstepping, this paper establishes that this controller is robust to sufficiently small in-domain damping. In particular both stability and attractivity (convergence) are established similarly as in the nominal case. Note moreover that, assuming that some parameters are known, the exponential stability to an attractor holds. Simulations are performed to illustrate the interest of this study to attenuate mechanical vibrations in an oil-drilling context.

I. INTRODUCTION

K EYSTONES of Partial Differential Equations include transport phenomena, the heat equation, and the wave propagation. This paper considers a special form of the latter.

The wave equation has been studied in several fields, mostly in physics regarding Maxwell equations and light propagation. Furthermore, premature failures of mechanical systems are often due to vibrations. These can be explained by the very deep structure of matter, i.e., the string-mass atomic interactions, and be modeled by a wave equation.

In this paper, we are interested in studying the robustness of a previously designed control law to a class of model mismatch. This question is of crucial interest from an application point of view. Indeed, in practice, such uncertainties are ubiquitous and robust control is thus of prime importance. However, obtaining a controller satisfying a priori robustness margins can turn out to be a very difficult problem. To handle such an objective, we propose to extend the use of a given adaptive collocated feedback law.

A. Wave PDE under consideration

We consider the following PDE system u tt (x,t) = u xx (x,t) -2λ u t (x,t)

(1) u x (1,t) = U(t)

(2) u tt (0,t) = a q u t (0,t) + a[u x (0,t)d]

in which λ is an in-domain damping coefficient, q > 0 is an unknown anti-damping boundary parameter, d > 0 is an unknown bias, a is a positive constant. The distributed variable of the system is u(x,t) ∈ R with x the spatial variable, and t the time variable. The controlled input of the system is the scalar U(t). For the case in which λ = 0, it has been proven (in [START_REF] Bresch-Pietri | Adaptive output feedback for oil drilling stick-slip instability modeled by wave PDE with anti-damped dynamic boundary[END_REF]) that all eigenvalues are on the complex right-half plan, i.e., all have non negative real parts. This system main characteristics are (i). the unknown unstable parametric dynamics in (3) opposite to the actuation (2) (ii). the in-domain viscous damping in [START_REF] Artstein | Linear systems with delayed controls: a reduction[END_REF]. The goal is to stabilize the velocity u t (x,t), in particular u t (0,t), using only the measurement of the collocated velocity u t (1,t) (and its history) despite in-domain damping and parameters uncertainties.

B. Related work

The distinctive feature of system (1)-( 3) is the dynamical boundary [START_REF] Bresch-Pietri | Adaptive output feedback for oil drilling stick-slip instability modeled by wave PDE with anti-damped dynamic boundary[END_REF]. Note that, there exist some works on the multidimensional wave equation with dynamic boundary conditions, for example [START_REF] Fourrier | Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions[END_REF] and [START_REF] Nicaise | Exponential stability of second-order evolution equations with structural damping and dynamic boundary delay feedback[END_REF] where they consider a 3D setting. However, the associated literature seldom focus on stabilization (feedback law design) but on well-posedness, stability and regularity as it is the case of both references. Moreover standard literature on the one-dimensional wave equation stabilization does not consider this type of dynamics boundary [START_REF] Bresch-Pietri | Adaptive output feedback for oil drilling stick-slip instability modeled by wave PDE with anti-damped dynamic boundary[END_REF]. The most usual boundary condition is of Diriclet type, as considered in [START_REF] Smyshlyaev | Boundary stabilization of a 1-D wave equation with in-domain antidamping[END_REF] where the exponential stabilization problem for a wave PDE with in-domain space-dependent term has been solved. As an intermediate step, some papers consider an anti-stable boundary. In [START_REF] Smyshlyaev | Boundary control of an anti-stable wave equation with anti-damping on the uncontrolled boundary[END_REF], a backstepping observer based feedback is presented. Then, in [START_REF] Krstic | Adaptive control of an anti-stable wave PDE[END_REF], considering that the antistable boundary parameters are unknown, an adaptive control has been proposed. However, none of the previously mentioned works can be straightforwardly extended to encompass the dynamical boundary condition [START_REF] Bresch-Pietri | Adaptive output feedback for oil drilling stick-slip instability modeled by wave PDE with anti-damped dynamic boundary[END_REF].

Some of the works focusing on such a problem are [START_REF] Andréa-Novel | Exponential stabilization of an overhead crane with flexible cable via a backstepping approach[END_REF], [START_REF] Conrad | Strong stability of a model of an overhead crane[END_REF], [START_REF] Böhm | Modeling and boundary control of a hanging cable immersed in water[END_REF], [START_REF] Meurer | Tracking control design for a wave equation with dynamic boundary conditions modeling a piezoelectric stack actuator[END_REF] and [START_REF] Sagert | Backstepping and flatness approaches for stabilization of the stick-slip phenomenon for drilling[END_REF]. In [START_REF] Andréa-Novel | Exponential stabilization of an overhead crane with flexible cable via a backstepping approach[END_REF], the dynamical boundary is reconsidered as a PDE-ODE coupling. However, the control input is located at the dynamical boundary. Here, it is considered at the opposite boundary. In [START_REF] Conrad | Strong stability of a model of an overhead crane[END_REF], both boundaries are dynamical but the wave in-domain damping is not considered. In [START_REF] Böhm | Modeling and boundary control of a hanging cable immersed in water[END_REF], a boundary control problem, inspired by a hanging cable immersed in water is tackled. However, they do not consider viscous distributed term neither in the wave equation nor in the dynamic boundary condition, which is opposite to the Dirichlet attraction. Moreover, the controller is a full-state feedback. In [START_REF] Meurer | Tracking control design for a wave equation with dynamic boundary conditions modeling a piezoelectric stack actuator[END_REF], the wave equation is considered to model a piezoelectric stack actuator. The paper establishes the flatness of a wave PDE without distributed term and dynamic boundary conditions. Moreover, using this property an output control law is suggested. Finally, [START_REF] Sagert | Backstepping and flatness approaches for stabilization of the stick-slip phenomenon for drilling[END_REF] investigates a problem similar to ours and proposes a full-state backstepping controller. But it requires the parameters a, d, q, and λ to be known and full-state measurement.

C. Paper contributions

We are here interested in the analysis of a control law requiring only the parameter a to be known, and using only measurements at the controlled boundary (x = 1). This paper is the logical continuation of [START_REF] Roman | Robustness of an adaptive output feedback for an anti-damped boundary wave PDE in presence of in-domain viscous damping[END_REF] which establishes the robustness to in-domain viscous damping of the design in [START_REF] Bresch-Pietri | Adaptive output feedback for oil drilling stick-slip instability modeled by wave PDE with anti-damped dynamic boundary[END_REF]. [START_REF] Bresch-Pietri | Adaptive output feedback for oil drilling stick-slip instability modeled by wave PDE with anti-damped dynamic boundary[END_REF] presented a boundary velocities output adaptive feedback, i.e., using the measurement of u t (1,t) and u t (0,t).

This paper brings further developments to [START_REF]Adaptive output-feedback for wave PDE with anti-damping -Application to surface-based control of oil drilling stick-slip instability[END_REF] where a collocated boundary adaptive control was designed, i.e., using the measurement of u t (1,t) only. Our purpose is to prove that such a control law is robust with respect to small enough indomain damping.

In other words, our objective is to prove that the system (1)-( 3) is stable with the feedback law proposed in [START_REF]Adaptive output-feedback for wave PDE with anti-damping -Application to surface-based control of oil drilling stick-slip instability[END_REF].

With this aim of view, we propose to build an extended system. The main difficulty in carrying out this construction is that the in-domain damping presence leads to states coupling. Moreover, this coupling modifies numerous parts of the analysis presented in [START_REF]Adaptive output-feedback for wave PDE with anti-damping -Application to surface-based control of oil drilling stick-slip instability[END_REF], leading to unmatched adaptive error terms.

All these difficulties are tackled with a careful choice of the extended variables. The distributed states coupling is handled by the introduction of an estimated variable, associated to the backstepping methodology. By adding an extra state, the adaptive error term error is handled, at the expense of regularity.

The paper is organized as follows. In Section II, the problem under consideration is stated and the control design is detailed. In Section III, our robustness result is presented, i.e., the main result of this paper. Section IV is devoted to its proof. Finally, the interest of this result is illustrated through simulations of the angular velocity regulation in a drilling vibrations model.

D. Notations

In this paper, | • | is the Euclidean norm and u(•) is the spatial L 2 -norm of a functional [0, 1] x → u(x, •), which is denoted as u(•) = 1 0 u(x, •) 2 dx (4)
Sometimes, when the context is clear, the abusive notation u , will be used to denote u(•) . Moreover, when the context is clear, the notation u(t) will be used to denote u(.,t).

For (a, b) ∈ R 2 such that a < b, let us define the standard projector operator on the interval [a, b] as a function of two scalar arguments f (denoting the parameter being updated) and g (denoting the nominal update law) in the following manner

PROJ [a,b] ( f , g) = g    0 if f = a and g < 0 0 if f = b and g > 0 1 otherwise (5)

II. PROBLEM STATEMENT AND CONTROL DESIGN

In this section, we present the problem under consideration, as well as the adaptive control law we chose to study.

Let us recall that the main features of the considered wave PDE are the in-domain damping, the unknown anti-damped boundary dynamics opposite to the control. The objective is to control the system velocity u t , using the controlled boundary measurement u t (1,t) only. The following assumption will be needed throughout the paper.

Assumption 1: There exist known constants q, q, d and d such that q < q, d < d and q ∈ [q, q], d ∈ [d, d].

A. Presentation of the control and adaptive laws

We consider the following control law, which was developed in [START_REF]Adaptive output-feedback for wave PDE with anti-damping -Application to surface-based control of oil drilling stick-slip instability[END_REF] for the system (1)-( 3) assuming λ = 0,

U(t) = -u t (1,t) + d(t) -(c 0 + q(t) -1) e 2a( q(t)-1) µ(t) + a t t-2 e a( q(t)-1)(t-τ) η(τ) -d(t) dτ (6)
in which c 0 > 0 is a tuning constant, q is an estimate of the unknown parameter q, d is an estimate of the unknown parameters d, and µ and η are defined as

µ(t) = 1 2 u t (1,t) + u t (1,t -2) -u x (1,t) + u x (1,t -2) (7) η(t) = U(t) + u t (1,t) (8)
The parameter adaptation laws are ˙ q(t) = aγ q 1 + N(t) PROJ [q,q] q(t), µ(t) µ(t) + b 1 (c 0 + q(t) -1)

× t t-2 e (a( q(t)-1)+ 1 2 )(τ-t+2) σ (τ,t)dτ (9) ˙ d(t) = aγ d 1 + N(t) PROJ [d,d] d(t), -µ(t) -b 1 (c 0 + q(t) -1) × t t-2
e (a( q(t)-1)+ 1 2 )(τ-t+2) σ (τ,t)dτ [START_REF] Ioannou | Robust Adaptive Control[END_REF] in which

N(t) =µ(t) 2 + b 1 t t-2 e τ-t 2 +1 σ (τ,t) 2 dτ + b 2 t t-1 e τ-t+1 (2µ(τ) -η(τ -2) + d(t)) 2 dτ (11) σ (τ,t) =η(τ) -d(t) + (c 0 + q(t) -1) e a( q(t)-1)(τ-t+2) µ(t) + a τ t-2 e a( q(t)-1)(τ-χ) (η(χ) -d(t))dχ (12)
The tuning parameters of the control law are c 0 , b 1 , b 2 , γ q and γ d . In the case where the adaptive parameters are known (i.e., q = q and d = d) and without in-domain damping (λ = 0), c 0 represents the closed-loop decay rate of the velocity at u t (0,t).

As it has been said, the adaptive control law needs only the knowledge of u t (1,t), its history, and the value of the parameters a. Indeed, using the controlled boundary of the system (2) one gets that µ defined in [START_REF] Curtain | An introduction to infinite-dimensional linear systems theory[END_REF] can be expressed as

µ(t) = 1 2 U(t) + u t (1,t) -U(t -2) + u t (1,t -2) (13)
The remaining parameters, i.e., c 0 , b 1 , b 2 , γ q and γ d are tuning parameters and thus are set by the user. As it is usual in prediction-based design, the control law needs the history of itself and the history of the output (here u t (1,t)) on a two units of time window. In the application in view, this is not a problem to use u t (1,t) and its past values but, actually, the key issue is to avoid using u t (0,t). The aforementioned control law is thus well defined. Indeed, using (2), ( 6), ( 7) and ( 8), one gets

U(t) 1 - c 0 + q(t) -1 2 e 2a( q(t)-1) = -u t (1,t) 1 + c 0 + q(t) -1 2 × e 2a( q(t)-1) + d(t) -(c 0 + q(t) -1) e 2a( q(t)-1) 2 (u t (1,t -2) +U(t -2)) + a t t-2 e a( q(t)-1)(t-τ) [U(τ) + u t (1, τ) -d(t)]dτ (14)
Now when c 0 + q(t)-1 2 e 2a( q(t)-1) = 1, the control law satisfies a Volterra integral of the second kind and therefore is welldefined. Moreover, in the opposite case where c 0 + q(t) -1 = 2e 2a( q(t)-1) > 0, the time derivative of ( 14) gives also a Volterra integral of the second kind and therefore is well-defined. Nevertheless, the considered solution needs to be regular enough for u tt (1, •) to have meaning. This is the case as we detail in Section III.

B. Discussion on the control law design

First, let us enter into the details of the control law [START_REF] Conrad | Strong stability of a model of an overhead crane[END_REF]. Define the Riemann variables

ζ (x,t) = u t (x,t) + u x (x,t) -d(t) (15) ω(x,t) = u t (x,t) -u x (x,t) + d(t) (16)
along with

W (t) = U(t) + u t (1,t) -d(t) (17) v(t) = u t (0,t) (18)
Then, one can express the system (1)-(3) as

v(t) = a(q -1)v(t) + a[ζ (0,t) -d(t)] (19)
ζ t (x,t) = ζ x (x,t) - ˙ d(t) -λ (ζ + ω) (20)
ζ (1,t) = W (t) (21)
ω t (x,t) = -ω x (x,t) + ˙ d(t) -λ (ζ + ω) (22) ω(0,t) = 2v(t) -ζ (0,t) (23)
in which d(t) = dd(t). These equations represent two coupled transport phenomena with distributed terms, coupled with the ODE [START_REF] Ritto | Non-linear dynamics of a drillstring with uncertain model of the bit-rock interaction[END_REF]. Note that q is an unknown constant which we do not use neither in the control law nor the parameters update law. ( 19)-( 23) is only a reformulation of the model at stake which indeed does depend on q, as q is a parameter of the model.

In the case where λ = 0, note that the two transport phenomena [START_REF] Roman | Backstepping observer based-control for an anti-damped boundary wave PDE in presence of in-domain viscous damping[END_REF] and [START_REF] Sagert | Backstepping and flatness approaches for stabilization of the stick-slip phenomenon for drilling[END_REF] are not coupled anymore. Then, for any x, the variable ζ (x,t) can be expressed as a delayed value of the boundary (21) (applying Lemma 6 in Appendix A-1 to [START_REF] Roman | Backstepping observer based-control for an anti-damped boundary wave PDE in presence of in-domain viscous damping[END_REF]). This enables us to consider [START_REF] Ritto | Non-linear dynamics of a drillstring with uncertain model of the bit-rock interaction[END_REF] as an input-delay system, as studied in [START_REF] Artstein | Linear systems with delayed controls: a reduction[END_REF], [START_REF] Kwon | Feedback stabilization of linear systems with delayed control[END_REF], and [START_REF] Manitius | Finite spectrum assignment problem for systems with delays[END_REF].

Furthermore, when λ = 0, using Lemma 6 in Appendix A-1, one can get from ( 7)

µ(t) = u t (0,t -1) (24)
Note that, considering q = q and d = d, using ( 6) and [START_REF] Nicaise | Exponential stability of second-order evolution equations with structural damping and dynamic boundary delay feedback[END_REF], one can prove that

W (t) = -(c 0 + q -1)u t (0,t + 1) (25)
Indeed, ( 6) is a prediction starting from µ(t) = u t (0,t -1) (according to [START_REF] Saldivar | Stick-Slip Oscillations in Oilwell Drillstrings: Distributed Parameter and Neutral Type Retarded Model Approaches[END_REF]) over a two units of time horizon: one for the measurement delay [START_REF] Saldivar | Stick-Slip Oscillations in Oilwell Drillstrings: Distributed Parameter and Neutral Type Retarded Model Approaches[END_REF] and one for the input delay resulting from [START_REF] Roman | Backstepping observer based-control for an anti-damped boundary wave PDE in presence of in-domain viscous damping[END_REF]. Then, the closed-loop system state v satisfies v(t) = -ac 0 v(t), for t 1 [START_REF] Smyshlyaev | Boundary control of an anti-stable wave equation with anti-damping on the uncontrolled boundary[END_REF] Therefore, in the nominal case λ = 0, q = q and d = d, exponential stabilization is achieved. In the general case, applying the certainty equivalence principle, (6) follows.

The objective of this control law is to stabilize the system (1)-(3) using the controlled boundary velocity measurement u t (1,t) only. This is a challenging objective, since lots of control techniques, such as backstepping, require full-state measurement, which, for a PDE system, is seldom the case in practice. Thus, many approaches use an observer (see [START_REF] Roman | Backstepping observer based-control for an anti-damped boundary wave PDE in presence of in-domain viscous damping[END_REF] when both boundary velocities are measured). To our knowledge, such an observer has not been designed yet for the framework under consideration. For this reason, we rely on the reconstruction of a delayed value of the boundary velocity [START_REF] Curtain | An introduction to infinite-dimensional linear systems theory[END_REF] (which satisfies [START_REF] Saldivar | Stick-Slip Oscillations in Oilwell Drillstrings: Distributed Parameter and Neutral Type Retarded Model Approaches[END_REF] only when λ = 0). Nevertheless, any physical relevant model has dissipative terms. Indeed, in the application considered in Section V, λ = 0 means that the drillstring is in a vacuum field. This emphasizes the need of the present study in order to use the considered control in real applications. Moreover, the prediction-based design is related to the Smith predictor, generally speaking the Smith predictor is not known to perform effectively with respect to model mismatch. Furthermore, λ = 0 prevents the use of a prediction-based design. However, note that λ = 0 could help for other control design, e.g. passivity. Intuitively, damping should help to stabilize the system. However as the control technique used in [START_REF]Adaptive output-feedback for wave PDE with anti-damping -Application to surface-based control of oil drilling stick-slip instability[END_REF] is based on a delay representation of the system which only holds without damping, the closedloop analysis with damping is complexified here as it involves coupled PDEs.

The adaptive laws (9)-( 10) result from a Lyapunov-based design, i.e., from indirect adaptive control. Moreover ˙ d and ˙ q are bounded by definition, due to the normalization term N in ( 9)- [START_REF] Ioannou | Robust Adaptive Control[END_REF]. Besides, the projector operator allows to limit the estimated variable within its boundaries (for more details on adaptive control paradigm see [START_REF] Ioannou | Robust Adaptive Control[END_REF]).

Finally, note that the control law is robust with respect to a constant input disturbance, due to the structure of [START_REF] Ioannou | Robust Adaptive Control[END_REF], which can be seen as an integral term.

C. Problem under consideration

We wish to study the robustness of the control law [START_REF] Conrad | Strong stability of a model of an overhead crane[END_REF] designed in [START_REF]Adaptive output-feedback for wave PDE with anti-damping -Application to surface-based control of oil drilling stick-slip instability[END_REF] with respect to λ , i.e., in-domain damping model mismatch. We are looking for a condition on λ such that the system (1)-( 3), using the previous control [START_REF] Conrad | Strong stability of a model of an overhead crane[END_REF] and adaptive laws ( 9)-( 10) (designed for λ = 0), is still Lyapunov stable and convergent in a sense detailed in the following section (see Theorem 1).

III. MAIN RESULT

In the following, the system state is denoted as

X (t) =[u(t), u t (t), u(0,t), u t (0,t), q -q(t), d -d(t)] T ∈ H 2 (0, 1) × H 1 (0, 1) × R 4 (27)
Indeed, due to the dynamical boundary (3), u(0,t) and u t (0,t) need to be considered, in addition to (u, u t ), to guarantee the well-posedness of the system (see [START_REF] Andréa-Novel | Exponential stabilization of an overhead crane with flexible cable via a backstepping approach[END_REF]). Furthermore, both adaptive estimation terms are included in the state due to the choice of the (adaptive) dynamical control law. Nevertheless, as the goal of the control law is the control of velocity, we introduce two vectors

X s (t) = u x (•,t), u t (•,t), u t (0,t), q -q(t), d -d(t) (28) X a (t) = u x (•,t), u t (•,t), u t (0,t), d -d(t) (29)
The first vector is the partial state for which stabilization is achieved. The second one satisfies a convergence property, as emphasized below.

Theorem 1: Consider the closed-loop system consisting of the plant (1)-(3) satisfying Assumption 1, the control law (6) and the parameter estimation laws (9)- [START_REF] Ioannou | Robust Adaptive Control[END_REF]. Define the functionals Γ, Ξ and ϒ as

Γ(X s (t)) =u t (0,t) 2 + u t (t) 2 + u x (t) -d 2 + u xt (t) 2 + u xx (t) 2 + (q -q(t)) 2 + (d -d(t)) 2 (30) Ξ(X s (t)) = max s∈[0,3] Γ(X s (t -s)) (31) ϒ(X a (t)) =u t (0,t) 2 + u t (t) 2 + u x (t) -d 2 + (d -d(t)) 2 (32)
Then, for all c 0 > 0, there exist b

2 (c 0 ) > 0, b 1 (c 0 , b 2 ) > 0, γ(c 0 , b 1 , b 2 ) > 0, such that, for • b 2 ∈ (0, b 2 ), • b 1 ∈ (b 1 , ∞), • γ d , γ q ∈ (0, γ), there exists λ (c 0 , b 1 , b 2 , γ, X s (0)) > 0 such that, when λ ∈ (0, λ ), it follows, Ξ(X s (t)) R(e ρΞ(X s (0)) -1) (33)
and

lim t→∞ ϒ(X a (t)) = 0 (34)
for suitable R > 0 and ρ > 0.

The parameters c 0 , b 1 , b 2 , γ q and γ d are tuning parameters for the control law and adaptive laws. The scalar b 2 , b 1 , and γ are the parameter bounds. λ is the upper bound of the indomain viscous coefficient λ for which the system (1)-( 3) is stable with respect to the subspace defined as the kernel of Ξ(•). This is what (33) means.

As usual in adaptive control [START_REF] Ioannou | Robust Adaptive Control[END_REF] (e.g. see [START_REF]Output-feedback adaptive control of a wave PDE with boundary anti-damping[END_REF] for a application of it to the wave equation), this result includes two distinct properties, (i) stability in term of the functional Ξ and (ii) convergence (attractivity) in term of the functional ϒ. This is due to the fact that adaptive estimation terms may be stable but not necessarily asymptotically stable, as the term qq.

Note that ∀c ∈ R, u(x,t) = dx + c fulfills ϒ(X (t)) = 0. Recall that the objective of the considered adaptive control law is to stabilize u t (.,t). As the application considered in Section V is the control of torsional vibration, we do not need to control the angular position. This is a feature inherited from the control law in [START_REF]Adaptive output-feedback for wave PDE with anti-damping -Application to surface-based control of oil drilling stick-slip instability[END_REF] which is usual for the considered control problem, e.g. [START_REF] Saldivar | The control of drilling vibrations: A coupled PDE-ODE modeling approach[END_REF], [START_REF] Sagert | Backstepping and flatness approaches for stabilization of the stick-slip phenomenon for drilling[END_REF], and [START_REF] Roman | Backstepping observer based-control for an anti-damped boundary wave PDE in presence of in-domain viscous damping[END_REF]. The presented method just extends the result in [START_REF]Adaptive output-feedback for wave PDE with anti-damping -Application to surface-based control of oil drilling stick-slip instability[END_REF] for in-domain viscous damping mismatch, it does not change the goal of the considered adaptive control law. Note that, if one is interested on the position stabilization, one could study the robustness mismatch of the adaptive control design in [START_REF]Output-feedback adaptive control of a wave PDE with boundary anti-damping[END_REF]. It is worth noticing that U(t) = d for u(x,t) = dx+c, ∀c ∈ R. This is consistent with the fact that d can be seen as a feedfoward bias, and the adaptive control law ˙ d as an integral control. It is worth noticing that, if the adaptation parameter q is perfectly known ( q = q), then the exponential stability of the closed-loop system (1)-( 3) with the control law ( 6) and the adaptive law [START_REF] Ioannou | Robust Adaptive Control[END_REF] in terms of the functional Ξ follows ( see Lemma 2). The associated attractor is the kernel of Ξ(•).

Comment of the closed-loop system well-posedness

Even if the well-posedness of the closed-loop system is not tackled in this paper, the idea of its proof can be stated in three steps, as follows (i). For each solution of the original system (of state X defined in ( 27)) there is a solution of the Extended Target system (of state X e defined in Section IV-A6). Therefore if one proves that the extended system is well-posed, then so is the original system. Note that the Extended Target system is a coupled hyperbolic PDEs plus ODEs system. (ii). In [START_REF] Peralta | Local well-posedness of a class of hyperbolic pde-ode systems on a bounded interval[END_REF], the authors study the local well-posedness of hyperbolic PDE-ODE systems. The Extended Target system (of state X e defined in Section IV-A6) is close to the one considered in [START_REF] Peralta | Local well-posedness of a class of hyperbolic pde-ode systems on a bounded interval[END_REF]. Indeed, the first order hyperbolic PDE source terms, which are depending on ODE variables, can be canceled using a change of variable (like the one in the proof in Appendix A-1 (121)). (iii). Using the particular property of the Extended Target system (of state X e ), i.e., diagonal, Lipschitz and marginally stable, we can use Theorem 5.2 in [START_REF] Peralta | Local well-posedness of a class of hyperbolic pde-ode systems on a bounded interval[END_REF], which states the local well-posedness. Then using the finite time blowup criterion from Theorem 5.3 in [START_REF] Peralta | Local well-posedness of a class of hyperbolic pde-ode systems on a bounded interval[END_REF], and the stability property from Lemma 2 in Section IV-B, one is able to conclude the global well-posedness of the Extended Target system (of state X e defined in Section IV-A6).

IV. PROOF OF THEOREM 1

The method proposed in this section to prove Theorem 1 is to define an extended system (of state X e ), whose stability implies the stability of the original system in the sense of (33) in Theorem 1.

The proof is organized as follows. First, in Section Section IV-A, an extension of the system referred to as the Extended Target system, is presented. Second, we define a corresponding Lyapunov functional V (X e ), the stability of which is proved in Section IV-B. Then, in Section IV-C, two lemmas detail the equivalence properties between V (X e )) and Ξ(X s ) introduced in (31). Finally, the convergence with respect to the functional ϒ, defined in (32), is established in Section IV-D, and the proof of Theorem 1 is concluded in Section IV-E.

A. Extended Target system

The different steps to build the Extended Target system are listed below.

• First the system (1)-( 3) is reformulated using the Riemann invariants into coupled transport phenomena with an ODE ( 19)-( 23) • The two transport phenomena are divided into two dynamics: the Estimated system and the Auxiliary system (Section IV-A1). • Delayed state variables are introduced to handle the fact that the control design involves an output delay from ( 7) and ( 24) (Section IV-A2). • A backstepping transformation is performed on a subset of these delay variables and the corresponding Target system is computed in Section IV-A3. • A new state variable is added to handle an adaptive error term in the Lyapunov analysis (Section IV-A4). • Finally, the system is extended with the time-and spacederivatives of the Estimated and the Auxiliary systems (Section IV-A5). The last subsection (Section IV-A6) is a summary of the Extended Target system.

The following subsection are organized this way: first we define the variables we use to extend the system, second we compute their associated dynamics, and lastly we comment about their use.

1) Estimated and Auxiliary systems

Consider the following distributed variables

ζ (x,t) = ζ (x,t) + λ 1 x (ζ (χ,t + x -χ) + ω(χ,t + x -χ))dχ (35) ω(x,t) = ω(x,t) + λ x 0 (ζ (χ,t -x + χ) + ω(χ,t -x + χ))dχ -λ 1 0 (ζ (χ,t -x -χ) + ω(χ,t -x -χ))dχ (36) ζ (x,t) = ζ (x,t) -ζ (x,t) (37) ω(x,t) = ω(x,t) -ω(x,t) (38) Proposition 1: The state v (18) of the ODE (19) is solution of v(t) = a(q -1)v(t) + a[ ζ (0,t) + ζ (0,t) -d(t)]
(39)

The Estimated system satisfies

ζ t (x,t) = ζ x (x,t) - ˙ d(t) (40) ζ (1,t) = W (t) (41) ω t (x,t) = -ω x (x,t) + ˙ d(t) (42) ω(0,t) = 2g(t) -ζ (0,t) (43)
in which g(t) = v(t), and the Auxiliary system is

ζ t (x,t) = ζ x (x,t) -λ ( ζ + ω + ζ + ω)(x,t) (44)
ζ (1,t) = 0 (45)

ω t (x,t) = -ω x (x,t) -λ ( ζ + ω + ζ + ω)(x,t) (46) ω(0,t) = -ζ (0,t) (47)
Proof : From ( 19) and the definitions (35)-(38) one obtains (39).

From the definitions (35)-( 36), considering the Riemann invariants ( 20)-( 23) one gets (40)-( 43). ( 44)-( 47) are obtained in a similar manner.

We choose to denote g = v in (43) because v can be considered as a state and it can be express by other state variables. Therefore we kept v to denote the state, and g denote the values of v with respect to other state variables.

These definitions follow the same ideas as in [START_REF] Roman | Robustness of an adaptive output feedback for an anti-damped boundary wave PDE in presence of in-domain viscous damping[END_REF]. The system ( 20)-( 23) is decomposed in two parts. The first part (40)-( 43) is referred to as the Estimated system and has the same decoupled dynamics as the Nominal system (when λ = 0 in ( 20)-( 23)). It can be seen as the part where the control acts on, while the second part (44)-(47) encapsulates all the remaining dynamics. This second system is referred to as the Auxiliary system.

2) Delayed states

As the control law (6) uses a two units of time window, let us introduce some variable function of delayed state variables

δ (x,t) = ζ (x,t -1) + d(t -1) -d(t) (48) β (x,t) = ω(x,t -1) -d(t -1) + d(t) + ω(1,t -x) (49) δ (x,t) = ζ (0,t + x -1) (50) α(x,t) = δ (2x,t), x ∈ [0, 1/2] ζ (2x -1,t), x ∈ [1/2, 1] (51)
Proposition 2: The variables µ, β , α, and δ satisfy

μ(t) = a(q -1)µ(t) + a[ α(0,t) + δ (0,t) -d(t)] - a(q -1) 2 ω(1,t) + ω t (1,t) 2 (52) β t (x,t) = -β x (x,t) + ˙ d(t) (53)
β (0,t) = 2µ(t) -α(0,t) (54) 2 α t (x,t) = α x (x,t) -2 ˙ d(t) (55) α(1,t) = W (t) (56)
δ t (x,t) = δ x (x,t) (57) δ (1,t) = ζ (0,t) (58)
Proof : From the definition (7) of µ, using the definition ( 15)-( 16), along with the definitions (37)-( 38), applying Lemma 6 in Appendix A-1 for the transport equation ( 40) and (42) and finally using the definition [START_REF] Peralta | Local well-posedness of a class of hyperbolic pde-ode systems on a bounded interval[END_REF], one gets that

µ(t) = v(t -1) + 1 2 ω(1,t) (59)
According to the ODE satisfied by v (39), and using δ , and α, i.e., (50), and (51), one gets the following ODE at time t -1

v(t -1) = a(q -1)v(t -1) + a[ α(0,t) + δ (0,t) -d(t)] (60)
Moreover, the time derivative of (59) gives (52). Now, from the definition of β (49), using the transport equation ( 42) and the associated boundary condition (43) along with the expression (59), one gets (53) and (54).

Using the definition (48), the transport equation ( 40), and the associated boundary condition (41), one gets (55) and (56).

Taking space and time derivatives of definition (50), one obtains the transport equation (57) associated to the boundary condition (58).

In the following, we give some comments on these additional states.

First, µ (52) is considered instead of v because the state the control law (6) depends on µ.

Besides, the variable δ accounts for the input delay ζ (0,t -1) in (60). Similarly the variable β represents the delay of the variable ω.

Furthermore, the state variable α represents the history of ζ over a two units of time window. The idea behind gathering ζ and δ into (51) is to obtain a unique distributed variable to perform a backstepping transformation (see the next section). However, we still need the variable ζ for two reasons. First, the boundary condition (43) depends on ζ (0,t). Second, in the last part of the reformulation, we need to consider the derivatives of the Estimated and Auxiliary systems (40)-(47) which are depending on ζ so it eases the analysis.

Finally, one can observe that the last term of (52) is not expressed in the current set of variable, i.e., α, β , δ , ζ , ω, ζ , ω, µ, q, and d. This is why the system representation is extended considering state derivatives in Section IV-A5.

3) Backstepping transformation

Before presenting the Target system, the control law is reformulated as follows.

Claim 1: W defined in (17) can be expressed as W (t) = -(c 0 + q(t) -1) e 2a( q(t)-1) µ(t)

+ 2a 1 0 e 2a( q(t)-1)(1-χ) α(χ,t)dχ (61)
Proof : From ( 6)-( 8) and [START_REF] Nicaise | Exponential stability of second-order evolution equations with structural damping and dynamic boundary delay feedback[END_REF], with the change of variable χ = t + 2x -2, one gets

W (t) = -(c 0 + q(t) -1) e 2a( q(t)-1) µ(t) + 2a 1 0 e 2a( q(t)-1)(1-x) × W (t + 2x -2) + d(t + 2x -2) -d(t) dx (62)
Then, applying Lemma 6 in Appendix A-1 with y = 1 on (55) and using (56), one obtains (61). Consider the following backstepping transformation of α z(x,t) = α(x,t) + (c 0 + q(t) -1) e 2a( q(t)-1)x µ(t) + 2a

x 0 e 2a( q(t)-1)(x-χ) α(χ,t)dχ (63)

Lemma 1: The backstepping transformation (63) together with the control law (61) transform the plant (52)-(56) into the following Target system

μ(t) = -ac 0 µ(t) + a[ z(0,t) + δ (0,t) + µ(t) q(t) -d(t)] - a 2 (q -1) ω(1,t) + ω t (1,t) 2 (64) 2 z t (x,t) = z x (x,t) + ˙ q(t)g q (x,t) + ˙ d(t)g d (x,t) + q(t)µ(t) -d(t) + δ (0,t) + [- (q -1) 2 ω(1,t) + ω t (1,t) 2a ] h(x,t) (65)
z(1,t) = 0 (66)

β t (x,t) = -β x (x,t) + ˙ d(t) (67)
β (0,t) = (1 + c 0 + q(t))µ(t) -z(0,t) (68)
in which

g d (x,t) = -2 -4a(c 0 + q(t) -1)
x 0 e 2a( q(t)-1)(x-χ) dχ (69) h(x,t) = 2a(c 0 + q(t) -1)e 2a( q(t)-1)x (70) g q (x,t) = 2e 2a( q(t)-1)x µ(t) + 4a

x 0 e 2a( q(t)-1)(x-s) α(s,t)ds + (c 0 + q(t) -1) 4axe 2a( q(t)-1)x µ(t)

+ 8a 2 x 0 (x -χ)e 2a( q(t)-1)(x-χ) α(χ,t)dχ (71)
and in which α can be expressed via the inverse backstepping transformation α(x,t) = z(x,t) -(c 0 + q(t) -1) e -2ac 0 x µ(t)

+ 2a x 0 e -2ac 0 (x-χ) z(χ,t)dχ (72)
Proof : The proof is established from the time and space derivatives of (63), using the expressions (55)-( 56) and (52). As the computations are fairly standard, they are omitted.

4) Adaptive error term

Consider the following variable

d µ (t) = -ac 0 µ(t) -a d(t) (73)
Proposition 3: The dynamics of d µ is

˙ d µ (t) = -ac 0 d µ (t) -a 2 c 0 z(0,t) + δ (0,t) + µ(t) q(t) - 1 2 (q -1) ω(1,t) + 1 2a ω t (1,t) + a ˙ d(t) (74)
and g in (43) can be expressed as g(t) =e a(q-1) µ(t)e a(q-1) ω(1,t) 2

+ a 1 0 e a(q-1)(1-χ) [ α(χ/2,t) + δ (χ,t)]dχ + c 0 µ(t) + d µ (t) a e a(q-1) -1 q -1 (75)
in which α is expressed using the inverse backstepping transformation (72).

Proof : Computing the time derivative of (73) and using (64), one gets (74).

Using a prediction of the ODE (60), Lemma 6 in Appendix A-1 for the transport phenomena (55) and (57), and thanks to the definition (59), one obtains

g(t) = v(t) = e a(q-1) µ(t) - e a(q-1) 2 ω(1,t) + a 1 0 e a(q-1)(1-χ) × [ α(χ/2,t) + δ (χ,t)]dχ -d(t) e a(q-1) -1 q -1 (76)
Finally, using (73), one gets (75).

A first objective of this proposition is to reformulate the function g in terms of the variables of interest, that is µ, z, β , ζ , ω, ζ , ω, δ , q, and d. A second objectif is to handle the last term of (76) involving a d(t). Usually, the update law ˙ d could be designed to cope with this term. However, in the nominal case λ = 0 in [START_REF]Adaptive output-feedback for wave PDE with anti-damping -Application to surface-based control of oil drilling stick-slip instability[END_REF], the Lyapunov analysis was carried out in a cascaded manner. First the stability of z, µ, and β is established. Secondly using Grönwall's inequality the stability of ω is obtained (for more details see [START_REF]Adaptive output-feedback for wave PDE with anti-damping -Application to surface-based control of oil drilling stick-slip instability[END_REF]).

Here, we cannot use the same developments, due the interconnections between the variables generated by the in-domain damping. This compels us to perform a Lyapunov analysis of the entire system, and to deal with the d term appearing in (76). The additional variable d µ is thus introduced to overcome this difficulty.

5) Introduction of the first-order derivatives of the Estimated and Auxilary systems

Note that (64) and (74) are both depending on ω t (1,t). To get that type of term in the Lyapunov analysis we have to take under consideration the dynamic of ω t . As presented in the sequel, the dynamic of ω t is coupled with the dynamic of ζ t , ω x and ζ x . Therefore we extend the system by considering derivatives of the state variables of the Estimated and Auxiliary systems (40)-(47), as follows.

Proposition 4: The dynamics of ζ x and ω x are

ζ xt (x,t) = ζ xx (x,t) (77)
ζ x (1,t) = Ẇ (t) + ˙ d(t) (78)
ω xt (x,t) = -ω xx (x,t) (79)
ω x (0,t) = -2 ġ(t) + ζ x (0,t) (80)
in which the expressions of the time derivative of g and W are given in Appendix B. The dynamics of ζ t and ω t are

ζ tt (x,t) = ζ xt (x,t) -λ ( ζ t + ω t + ζ x -ω x )(x,t) (81) ζ t (1,t) = 0 (82) ω tt (x,t) = -ω xt (x,t) -λ ( ζ t + ω t + ζ x -ω x )(x,t) (83)
ω t (0,t) = -ζ t (0,t) (84)
Proof : Taking a space-derivative of the Estimated system transport (40) and (42), one gets respectively (77) and (79). From (40) evaluated for x = 1 and the boundary condition (41), one gets (78). With a similar argument on (42) and ( 43), (80) follows.

Adding the Estimated system transport phenomena (40) and (42), one obtains

ζ t + ω t = ζ x -ω x (85)
Thus, the time derivative of the Auxiliary system (44)-( 47) gives ( 81)-( 84).

The state variables of the Extended Target system are detailed explicitly in the next section.

6) Summary of the Extended Target system

To summarize, the Extended Target system whose state is

X e (t) = [µ(t), z(t), β (t), δ (t), ζ (t), ω(t), ζ (t), ω(t), d µ (t), ζ x (t), ω x (t), ζ t (t), ω t (t), q(t), d(t)] T ∈ {R, L 2 (0, 1) 3 , H 1 (0, 1) 4 , R, L 2 (0, 1) 4 , R 2 } (86)
consists of (64)-( 68), (57)-( 58), ( 40)-( 47), (77)-(84), and (74), with the update laws ( 9)- [START_REF] Ioannou | Robust Adaptive Control[END_REF]. Figure 1 illustrates the different steps detailed previously to build the Extended Target system.

B. Lyapunov analysis

This section focus on the stability analysis of the Extended Target system. First, we formulate the following claim Claim 2: The estimation laws (9)-( 10) can be rewritten as ˙ q(t) = aγ q 1 + N(t) PROJ [q,q] q(t), µ(t) µ(t) + 2b 1 (87)

× (c 0 + q(t) -1)
1 0 e 2(a( q(t)-1)+ 1 2 )x z(x,t)dx

˙ d(t) = - aγ d 1 + N(t) PROJ [d,d] d(t), µ(t) + 2b 1 (88)
× (c 0 + q(t) -1)

1 0 e 2(a( q(t)-1)+ 1 2 )x z(x,t)dx

N(t) = µ(t) 2 + 2b 1 1 0 e x z(x,t) 2 dx + b 2 1 0 e 1-x β (x,t) 2 dx (89)
in which β is defined in (49) and z in (63).

Proof : The proof follows arguments similar to the ones of Claim 1 in Section IV-A3. Then, one is able to establish the following key lemma.

Lemma 2: Let us consider the Lyapunov functional For all c 0 > 0, there exist b

V (X e (t)) = V 1 (t) +V 2 (t) +V 3 (t) +V 4 (t) +V 5 (t) (90) with V 1 (t) = log (1 + N(t)) + q(t) 2 γ q + d(t) 2 γ d (91) V 2 (t) = b 3 1 0 e x ζ (x,t) 2 dx + b 4 1 0 e 1-x ω(x,t) 2 dx (92)
2 (c 0 ) > 0, b 1 (c 0 , b 2 ) > 0, γ(c 0 , b 1 , b 2 ) > 0, such that, for all • b 2 ∈ (0, b 2 ), • b 1 ∈ (b 1 , ∞), • γ d , γ q ∈ (0, γ), there exist b i > 0, i ∈ {3, ..., 12} and λ (c 0 , b 1 , b 2 , γ, X (0)) > 0,
such that, for all λ ∈ (0, λ ), it follows

V (t) - ς 1 + N(t) z(0,t) 2 + δ (0,t) 2 + µ(t) 2 + d µ (t) 2 + ζ (0,t) 2 + ζ (0,t) 2 + z 2 + β 2 + ζ 2 + ζ 2 + ω 2 + ω 2 + δ 2 + ζ x 2 + ζ x 2 + ω t 2 + ω t 2 ( 96
)
for a suitable ς > 0, and V (X e (t)) V (X e (0)), t 0 (97)

Proof : The proof is given in Appendix C

C. Relation between the functionals Γ(X s ) and V (X e )

In order to establish the stability condition (33) of Theorem 1, we formulate the following two lemmas.

Lemma 3: Consider Γ defined in (30) and V defined in (90). There exists R > 0, such that Γ(X s (t)) R(e V (X e (t)) -1)

(98)

Proof : The details of the proof are given in Appendix D.

To establish Theorem 1, we also need to bound V by a function of Γ. This is the purpose of the next lemma. Lemma 4: Consider Γ defined in (30) and V defined in (90). There exists ρ > 0 such that

V (X e (t)) ρ max s∈[0,3] Γ(X s (t -s)) (99)
Proof : The proof is given in Appendix D

D. Convergence analysis

We conclude on the convergence with respect to the functional ϒ (32).

Lemma 5: v(t), ω(t) , ζ (t) , ζ (t) , and ω(t) tend to zero as t tends to infinity. The proof, based on Barbalat's lemma, is presented in Appendix E.

E. Conclusion on the proof of Theorem 1

Gathering (98) from Lemma 3 and (99) from Lemma 4, one gets the existence of R > 0 and ρ > 0 such that Γ(X s (t)) R(e ρ max s∈[0,3] Γ(X s (-s)) -1), (100)

Then, (100) along with (97) from Lemma 2 give the stability result (33) in Theorem 1. Finally, using Lemma 5, ( 15)-( 16) and ( 37)-( 38), it follows that u t (0,t) = v(t), u t (t) and u xd(t) and also dd(t) tend to zero as t tends to infinity. In other words, (34) holds. This concludes the proof of Theorem 1.

V. APPLICATION TO DRILLING TORSIONAL VIBRATIONS

For illustration purposes, the control law presented in Section II is applied to a nonlinear drilling model, presented in the next section.

A. Drilling context and nonlinear model

One of the possible applications of the control law (6) associated with the wave equation model ( 1)-( 3) is to attenuate the torsional vibrations occurring in drilling facilities (e.g. [START_REF] Bresch-Pietri | Adaptive output feedback for oil drilling stick-slip instability modeled by wave PDE with anti-damped dynamic boundary[END_REF], [START_REF]Adaptive output-feedback for wave PDE with anti-damping -Application to surface-based control of oil drilling stick-slip instability[END_REF], and [START_REF] Sagert | Backstepping and flatness approaches for stabilization of the stick-slip phenomenon for drilling[END_REF]). Such vibrations can lead to the so called stickslip phenomenon [START_REF] Jansen | Active damping of self-excited torsional vibrations in oil well drillstrings[END_REF]. Indeed, the friction at the bottom of the hole, between the rock and the drillbit, forces sometimes the bit to stop, while the surface is still rotating. After some time, the bit will start moving again at velocity higher than the top velocity (see Figure 5 before 15 s). This torsional dynamics can be modeled by a wave equation with a nonlinear boundary condition u tt (0,t) = aF u t (0,t) + au x (0,t) (see [START_REF] Saldivar | The control of drilling vibrations: A coupled PDE-ODE modeling approach[END_REF], [START_REF] Saldivar | Stick-Slip Oscillations in Oilwell Drillstrings: Distributed Parameter and Neutral Type Retarded Model Approaches[END_REF]), accounting for the friction between the drillbit and These value are taken from [START_REF] Sagert | Backstepping and flatness approaches for stabilization of the stick-slip phenomenon for drilling[END_REF] and [START_REF] Jansen | Nonlinear dynamics of oilwell drillstrings[END_REF]. The friction phenomenon is described by the model in [START_REF] Zhang | Sliding mode control of rotatory drilling system with stick slip oscillations[END_REF].

TABLE I PARAMETERS VALUES OF THE NONLINEAR MODEL USED IN SIMULATION

the rock. Even if there exist phenomenological expressions of this friction F (see [START_REF] Zhang | Sliding mode control of rotatory drilling system with stick slip oscillations[END_REF], [START_REF] Saldivar | Stick-Slip Oscillations in Oilwell Drillstrings: Distributed Parameter and Neutral Type Retarded Model Approaches[END_REF] and [START_REF] Ritto | Non-linear dynamics of a drillstring with uncertain model of the bit-rock interaction[END_REF]), they depend on some parameters, such as the weight on the bit, drilling mud properties, and the nature of the rock. So they may change during operation. This is the reason why using an adaptive controller is of high interest for this application.

Following [START_REF] Saldivar | Stick-Slip Oscillations in Oilwell Drillstrings: Distributed Parameter and Neutral Type Retarded Model Approaches[END_REF], the nonlinear dynamical model of the drillstring rotatory angle is denoted θ (ξ , t) at length ξ and time t. By convention, the top boundary is at ξ = 0 and the bottom boundary at ξ = L, and the torsional dynamics can be modeled by

GJθ ξ ξ (ξ , t) -Iθ tt (ξ , t) -Λθ t (ξ , t) = 0 (101)
along with the boundary conditions GJθ ξ (0, t) = c α θ t (0, t) -Ω(t) (102)

I b θ tt (L, t) = -GJθ ξ (L, t) -T BIT (θ t (L, t)) (103)
in which Ω(t) is the angular velocity of the rotatory table rotor the actual actuator at time t, T BIT is the nonlinear rock-onthe-bit friction term and other constants are listed in Table I. Note that there exist alternative models of this phenomenon, such as proposed in [START_REF] Saldivar | The control of drilling vibrations: A coupled PDE-ODE modeling approach[END_REF] which takes into account the axial vibrations.

Following [START_REF] Sagert | Backstepping and flatness approaches for stabilization of the stick-slip phenomenon for drilling[END_REF], we consider the changes of variables

ξ = L(1 -x) (104)
t = L I GJ t ∆ = c t t (105) u(x,t) = θ (ξ , t) (106)
U(t) = Lc α GJ Ω(c t t) - 1 c t u t (1,t) (107)
F(•) = - L GJ T BIT • c t (108)
with the following constants

a = L I I B , 2λ = ΛL √ GJI (109)
This allows to rewrite (101)-(103) as

u tt (x,t) = u xx (x,t) -2λ u t (x,t) (110) u x (1,t) = U(t) (111) u tt (0,t) = aF(u t (0,t)) + au x (0,t) (112)
Now, we consider a first order Taylor approximation of F(.) around an equilibrium u ref t , i.e., F(u t (0,t)) = q(u t (0,t) -

u ref t ) -d (113)
in which d = -F(u ref t ) and q = Ḟ(u ref t ). Then, assuming u ref t = 0 for the sake of conciseness, the torsional dynamics model can be reformulated under the form (1)-(3). For simulation, the friction term T BIT (.) is taken as (see [START_REF] Zhang | Sliding mode control of rotatory drilling system with stick slip oscillations[END_REF])

T BIT (χ) = γ χ + 2T tob π α 1 χe -α 2 |χ| + arctan(α 3 χ) (114)
where the parameter values are gathered in Table I.

The drill pipe is a hollow cylinder of outer diameters D d p and inner diameter d d p . The drill is in contact with a viscous field, of viscous dynamical coefficient µ m . Direct computation gives the associated damping coefficient

Λ = µ m π 2 (D 2 d p + d 2 d p ) (115)
In Figure 2, we provide a brief illustration of the simulation scheme.

System (101)-(103)

θ t (L, t) θ t (0, t)
Adaptive feedback ( 6)-( 11) Simulation are performed in the sequel for λ = 0 (nominal case), λ = 0.45 and λ = 0.9. It is worth noting that all simulations presented below have been performed using the nonlinear model ( 101)-(103).

c t u t (1,t) u ref t (107) Ω(t)
U(t + ) 1/c t µ(t) µ un (t)

B. Velocity regulation of the nonlinear model

Here, the control law (6) and adaptation laws ( 9)-( 11) are not used for stabilization but for regulation, i.e., we wish to stabilize u t (x,t)u ref t instead of u t (x,t). Note that in all simulation cases, the control law is turned on at t = 15 s. c 0 is chosen such as ac 0 = 1. Besides, we chose b 2 = 10 -4 , b 1 = 1, γ d = 0.5 and γ q = 0.01.

The reference u ref t is taken such that the unnormalized desired velocity is θ ref t = 5 rad/s. The top and bottom velocities θ t (0,t) and θ t (L,t) along with the unnormalized equivalent of µ defined as

µ un (t) = µ(t) c t (116)
are all displayed in Figures 3,5, and 8 for λ = 0, 0.45, 0.9 respectively. The estimations of d and q are displayed in Figure 7 for the case λ = 0.45. Fig. 3. Simulation of the top, bottom and delayed bottom estimated velocities for λ = 0, the nominal case for which the control has been developed (see [START_REF]Adaptive output-feedback for wave PDE with anti-damping -Application to surface-based control of oil drilling stick-slip instability[END_REF]). One can notice that the oscillations existing in the openloop phase, i.e., before 15 s in Figures 3 andFigure 5, are mitigated by the application of the control law and regulation is obtained.

Time t [s]

In Figure 3, as excepted when λ = 0 from ( 24), (106), and (116), one gets µ un (t) = θ t (L, tc t ). Moreover, the top and bottom velocities θ t (0, t) and θ t (L, t) reach 95% of the reference, at respectively 23.54 and 23.37 s, which are reasonable performance compared to the oscillation period. The time responses of the associated control laws Ω(t) and U(t) are displayed in Figure 4. Fig. 6. Time response of the control laws Ω(t) and U(t) in the case λ = 0.45.

Time t [s]

In Figure 5 (λ = 0.45), the top and bottom velocities θ t (0, t) and θ t (L, t) reach 95% of the reference at respectively 22.00 and 21.74 s. One observes that these settling times are close to the nominal case. Here, as excepted from (86), (106), and (116), µ un (t) is only an approximation of θ t (L, tc t ). The time responses of the associated control laws Ω(t) and U(t) are displayed in Figure 6.

The fact that the system for λ = 0.45 in Figure 5 has a settling time lower that in the nominal case (λ = 0 in Figure 3) can probably be explained by the fact that the respective velocity values are not the same when the controller is turned on. Moreover, notice that the magnitude of the oscillations before t = 15 s is lower in Figure 5 (case λ = 0.45) than in Figure 3 (case λ = 0), which is clearly explained by the fact that the in-domain damping adds dissipation. In Figure 7, as expected, d(t) converges to d. Even if velocity regulation is obtained, one can observe that the estimation q(t) does not converge to the value of q. Note that this latter observation does not contradict the conclusion of Theorem 1. This is deeply linked with our control goal, i.e., regulation (stabilization). Indeed, the control laws associated with this goal are not frequentially "rich" enough (as pictured in Figure 4 and Figure 6) for q(t) to converge towards q. In Figure 8, a high damping value is considered (λ = 0.9).

Consequently, the open-loop system does not exhibit an oscillatory behavior. Therefore, the benefits of the feedback controller used after 15 s are quite reduced. However, this simulation highlights the robustness capabilities of this control law since the closed-loop convergence is still well-achieved.

VI. CONCLUSION

We propose here a theoretical study of the mismatch robustness of an adaptive prediction-based control design. We wish to emphasize that the different ideas may be transposed to similar contexts. Finally, the interest of this result has been illustrated through simulations.

Future interesting works could include the study of robustness to other types of model mismatch, such as Kelvin-Voigt damping (a distributed term in u xxt ) which represents the dissipation within the matter.

Also, a future path to explore could be to develop control algorithms taking explicitly into account in-domain damping. A first step has been done in this sense in [START_REF] Roman | Backstepping observer based-control for an anti-damped boundary wave PDE in presence of in-domain viscous damping[END_REF], but could be generalized by including adaptive control.

APPENDIX A INTERMEDIATE RESULTS

1) Relationship between transport equation and delay

Lemma 6: Consider

f t (x,t) + c f x (x,t) = F (x,t) (117) f (0,t) = f (t) if c > 0, and f (1,t) = f (t) if c < 0 (118) f (., 0) = f 0 (119) in which F ∈ L 2 ((0, ∞) × (0, 1)), f ∈ L 2 (0, ∞), f 0 ∈ L 2 (0, 1)
, and c ∈ R\{0}. There exists a unique weak solution f ∈ L 2 ((0, ∞) × (0, 1)) for the abstract Cauchy problem resulting from (117)-(119). This solution satisfies, for all x ∈ [0, 1], t ∈ [0, ∞), and s such that 0 c(st) + x 1

f (c(s -t) + x, s) = f (x,t) + s t F (c(τ -t) + x, τ)dτ (120)
Note that we can deduce for any x the values of the distributed state f (x,t) with the knowledge of its boundary and F . Proof : Without lack of generality, consider c < 0 and the following variable

m(x,t) = f (x,t) - x 0 1 c F (s,t + s -x c )ds (121)
One gets that, if

F ∈ L 2 ((0, ∞) × (0, 1)), m ∈ L 2 ((0, ∞) × (0, 1)) ⇔ f ∈ L 2 ((0, ∞) × (0, 1)) (122)
Moreover, m satisfies m t (x,t) + cm x (x,t) = 0 (123)

m(1,t) = f (t) - 1 0 1 c F (s,t + s -1 c )ds (124) m(x, 0) = f 0 (x) - x 0 1 c F (s, s -x c )ds (125)
which is a standard transport equation. Following [START_REF] Curtain | An introduction to infinite-dimensional linear systems theory[END_REF] Example 2.2.4 or Exercise 3.14, this system is well-posed and its solution satisfies m(x,t) = m(1,t -x c ) which, in turns, implies (120).

2) Intermediate result for Lyapunov stability Proposition 5: Consider x i ∈ L 2 (R), (i m, i, m ∈ N), and V a positive definite functional of x 2 i i=1...m . Assume there exists a increasing function K such that

∀t 0, V (t) - m ∑ i=1 (a i -K(V (t))x i (t) 2 (126)
in which ∀i ∈ 0, m , a i > 0.

If ∀i, a i -K(V (0)) >0 then ∀i, ∀t 0, a i -K(V (t)) > 0 Proof : For the sake of simplicity, take p = 1 (similar arguments hold for the general case). By contradiction, assume the existence of t 0 > 0 such that a 1 -K(V (t 0 )) 0. By continuity, there exists at least one t ∈ [0,t 0 ] such that a 1 -K(V (t)) = 0. We denote t 1 > 0 the smallest. One has

∀t ∈ [0,t 1 [, a 1 -K(V (t)) > 0 (127)
and thus V (t 1 ) < V (0) according to (126). Consequently, a 1 = K(V (t 1 )) < K(V (0)) as K is an increasing function, which is a contradiction.

APPENDIX B EXPRESSIONS OF ġ AND Ẇ Proposition 6: The time derivative of g (75), can be expressed as

ġ(t) = e a(q-1) a[ z(0,t) + δ (0,t) + µ(t) q(t)] + d µ (t) - a 2 (a -1) ω(1,t) + e a(q-1)(1-χ) α(χ/2) 2 + δ (χ,t) 1 χ=0 + a 1 0 e a(q-1)(1-χ) a(q -1) α(χ/2,t) 2 + δ (χ,t) dχ (128)
The time derivative of W defined in (17) can be expressed as Ẇ (t) = -˙ q(t) e 2a( q(t)-1) µ(t) + 2a 1 0 e 2a( q(t)-1)(1-χ) α(χ,t)dχ -(c 0 + q(t) -1) 2a ˙ q(t)e 2a( q-1) µ(t) + e 2a( q(t)-1) a[ z(0,t)

+ δ (0,t) + µ(t) q(t)] + d µ (t) - a 2 (q -1) ω(1,t) + ω t (1,t) 2 + 2a 1 0 e 2a( q(t)-1(1-χ) 2a ˙ q(t)(1 -χ) α(χ,t) - ˙ d(t) (129)
+ a( q -1) α(χ,t) dχ + a e 2a( q(t)-1(1-χ) α(χ,t)

1 χ=0
in which α can be expressed using the inverse backstepping transformation (72).

Proof : Taking the time-derivative of (76), using the transports (55) and (57), applying integration by part on α x and δ x , and expressing d with (73), one gets (128).

From the time derivative of (61), in a similar way, one gets (129).

APPENDIX C LYAPUNOV FUNCTIONAL ANALYSIS

Before stating the proof of Lemma 2, one establishes the following proposition.

Proposition 7: There exist positive M 0 , M ω , M ωt , M i (b 1 ), i ∈ {1, 2, ..., 8}, C i , i ∈ {1, 2, ..., 8} such that

V1 (t) 1 1 + N(t) -ac 0 µ(t) 2 + 4a c 0 z(0,t) 2 + 4a c 0 δ (0,t) 2 + a c 0 (q -1) 2 ω(1,t) 2 + 1 ac 0 ω t (1,t) 2 + b 1 [-z(0,t) 2 - z 2 4 + γ q M 1 (b 1 )(µ(t) 2 + z 2 ) + γ d M 2 (b 1 )(µ(t) 2 + z 2 ) + M 0 δ (0,t) 2 + M ω ω(1,t) 2 + M ωt ω t (1,t) 2 ] + b 2 [-β (1,t) 2 + 2e(1 + c 0 + q(t)) 2 µ(t) 2 + 2e z(0,t) 2 -β 2 + γ d M 3 (b 1 )(µ(t) 2 + z 2 + β 2 )] (130)
V2 (t) b 3 -ζ (0,t) 2 -ζ 2 +C 1 [µ(t) 2 + z 2 ] + γ d M 4 (b 1 )[µ(t) 2 + z 2 + ζ 2 ] + b 4 -ω(1,t) 2 -ω 2 + 2e ζ (0,t) 2 +C 2 µ(t) 2 + d µ (t) 2 + z 2 + δ 2 + ω(1,t) 2 ] + γ d M 5 (b 1 )[µ(t) 2 + z 2 + ω 2 ] (131) V3 (t) b 5 -ζ (0,t) 2 -(1 + 2λ ) ζ 2 + λC 3 ζ 2 + ζ 2 + ω 2 + ω 2 + b 6 -ω(1,t) 2 + e ζ (0,t) 2 -(1 + 2λ ) ω 2 + λC 4 ω 2 + ζ 2 + ω 2 + ζ 2 + b 7 e ζ (0,t) 2 -δ (0,t) 2 -δ 2 (132) V5 (t) b 9 -ζ x (0,t) 2 -ζ x 2 +C 6 [ ω t (1,t) 2 + µ(t) 2 + d µ (t) 2 + z(0,t) 2 + z 2 + δ (0,t) 2 + ω(1,t) 2 ] + γ d M 7 (b 1 )[µ(t) 2 + z 2 ] + γ q M 8 (b 1 )[µ(t) 2 + z 2 ] + b 10 -ω x (1,t) 2 -ω x 2 + 3e ζ x (0,t) 2 +C 7 [µ(t) 2 + d µ (t) 2 + z(0,t) 2 + z 2 + δ 2 + δ (0,t) 2 + ω(1,t) 2 + ζ (0,t) 2 ] + b 11 -ζ t (0,t) 2 -(1 + 2λ ) ζ t 2 + λC 8 ζ t 2 + ζ x 2 + ω x 2 + ω t 2 + b 12 -ω t (1,t) 2 -(1 + 2λ ) ω t 2 + e ζ t (0,t) 2 + λC 9 ω t 2 + ζ x 2 + ω x 2 + ζ t 2 (133) V4 (t) b 8 -ac 0 d µ (t) 2 + γ d M 6 (b 1 )[µ(t) 2 + z 2 ] (134) +C 5 [µ 2 + z(0,t) 2 + δ (0,t) 2 + ω(1,t) 2 + ω t (1,t) 2 ]
Proof : For simplicity, we only detail the proof of (130), as the remaining inequalities can be obtained with similar arguments. From the definition of V 1 (t) one gets,

V1 (t) = Ṅ(t) 1 + N(t) -2 q(t) ˙ q(t) γ q -2 d(t) ˙ d(t) γ d (135)
From the expression of N (89), it follows that

Ṅ(t) = 2 μ(t)µ(t) + 4b 1 1 0 e x z z t dx + 2b 2 1 0 e 1-x β β t dx (136)
Then, using (64), and Young inequality, one obtains

2 μ(t)µ(t) -ac 0 µ(t) 2 + 4a c 0 z(0,t) 2 + 4a c 0 δ (0,t) 2 + a c 0 (q -1) 2 × ω(1,t) 2 + 1 ac 0 ω t (1,t) 2 + 2aµ(t)(µ(t) q(t) -d(t)) (137)
Finally using ( 65)-( 68), and Young and Cauchy-Schwarz inequalities, with integrations by parts, one obtains

4 1 0 e x z z t dx -z(0,t) 2 - z 2 4 + γ q M 1 (b 1 )(µ(t) 2 + z 2 ) + γ d M 2 (b 1 )(µ(t) 2 + z 2 ) + M 0 δ (0,t) 2 + M ω ω(1,t) 2 + M ωt ω t (1,t) 2 ] + 2a(c 0 + q(t) -1) × 1 0 e 2(a( q(t)-1)+ 1 2 )x z(x,t)[µ(t) q(t) -d(t)]dx (138) 2 1 0 e 1-x β β t dx -β (1,t) 2 + 2e(1 + c 0 + q(t)) 2 µ(t) 2 (139) + 2e z(0,t) 2 -β 2 + γ d M 3 (b 1 )(µ(t) 2 + z 2 + β 2 )
Gathering ( 135)-(139), and using Claim 2 concludes the proof of (130).

Proof of Lemma 2: Gathering inequalities (130)-(133) of Proposition 7, we obtain

V (X e (t)) ∑ V ∈S δ V f V (.) V 2 (140) in which S = {µ, z , β , δ , ζ , ω , ζ , ω , d µ , ζ x , ω x , ζ t , ω t , z|0, β |1, δ |0 ζ |0, ω|1, ζ |0, ω|1, ζ x |0, ω x |1, ζ t |0, ω t |1} (141)
where ω|1 denote the boundary of ω for x = 1, i.e., ω(1,t)

δ V = 1 1+N(t) , if V ∈ {z|0, µ, z , β , β |1} 1 ( 142
)
and with Note that, we used that 1 + N(t) e V (t) (from the definition of V (90)).

f z|0 (b i ,V (t)) = 4a c 0 -b 1 + 2eb 2 + [b 8 C 5 + b 9 C 6 + b 10 C 7 ]e V (t) (143) f δ |0 (b i ) = 4a c 0 + b 1 M 0 -b 7 + b 8 C 5 + b 9 C 6 + b 10 C 7 (144) f µ (b i ,V (t)) = -ac 0 + b 1 γ q M 1 + b 1 γ d M 2 + 2eb 2 (1 + c 0 + q) 2 + b 2 γ d M 3 + e V (t) [b 3 C 1 + b 3 γ d M 4 + b 4 C 2 + b 4 γ d M 5 + b 8 γ d M 6 + b 8 C 5 + b 9 C 6 + b 9 γ q M 7 + b 9 γ d M 8 + b 10 C 7 ] (145) f d µ (b i ) = b 4 C 2 -ac 0 b 8 + b 9 C 6 + b 10 C 7 (146)
A sufficient condition for the stability of the Extended Target system of state X e (86) is therefore the existence of parameters b i , γ q , γ d , and λ such that

f V (.) < 0, V ∈ S (164)
This is the condition we investigate in the following. With this aim in view, we follow the following procedure to select the different parameters

γ q and γ d are chosen, according to (145), ( 150), ( 151), (152), and (154), as

γ q + γ d < min ac 0 -2eb 2 (1 + c 0 + q) 2 b 1 M 1 + b 1 M 2 + b 2 M 3 , ( 172
) b 1 4(b 1 M 1 + b 1 M 2 + b 2 M 3 ) , 1 M 3 , 1 M 4 , 1 M 5
Note that, if b i < 1, i ∈ {3, 4, 8, 9, 10}, this implies the existence of M(X e (0)) > 0 independent of b i , i ∈ {3, 4, 8, 9, 10}, such that V (X e (0)) M(X e (0)) which is simply denoted M in the following. Consequently, the parameter b 3 is fixed, according to (145) and (150), as1 Using Proposition 5 in Appendix A-2, with this choice of parameters, one obtains the existence of ϑ > 0 such that f V (.) < -ϑ for all V ∈ S . From (164), this implies that (96) holds which concludes the proof of Lemma 2.

Fig. 1 .

 1 Fig. 1. Schematic view of the successive transformations and extensions to obtain the Extended Target system, variables of which are encircled.

V 3 ( 1 0e 1 -b 11 1 0 1 0e 1 -

 311111 x (x,t) 2 dx + b 10 x ω x (x,t) 2 dx + e x ζ t (x,t) 2 dx + b 12 x ω t (x,t) 2 dx (95)in which N is expressed as (89), and b 3 , ..., b 12 > 0.

Fig. 2 .

 2 Fig. 2. Simulation scheme.

Fig. 4 .

 4 Fig. 4. Time response of the control laws Ω(t) and U(t) in the case λ = 0.

Fig. 5 .

 5 Fig. 5. Simulation of the top, and bottom and delayed bottom estimated velocities for λ = 0.45.

Fig. 7 .

 7 Fig. 7. Evolution of the parameter estimates d and q for λ = 0.45.

Fig. 8 .

 8 Fig.8. Simulation of the top, bottom and delayed bottom estimated velocities, for λ = 0.9.

6 + b 8 C 5 + b 9 C 6 +

 68596 f ζ |0 (b i ) =b 3 + 2eb 4(147)f ζ |0 (b i ) =b 5 + eb 6 + eb 7 + b 10 C 7 (148) f ω|1 (b i ) = a c 0 (q -1) 2 + b 1 M ω + b 4 C 2b b 10 C 7 (149) f z (b i ,V (t)) = -b 1 4 + b 1 γ q M 1 + b 1 γ d M 2 + b 2 γ d M 3 + e V (t) [b 3 C 1 + b 3 γ d M 4 + b 4 C 2 + b 4 γ d M 5 + b 8 γ d M 6 + b 9 C 6 + b 9 γ q M 7 + b 9 γ d M 8 + b 10 C 7 ] (150) f β (b i ) = -b 2 + b 2 γ d M 3 (151) f ζ (b i , λ ) = -b 3 + b 3 γ d M 4 + b 5 λC 3 + b 6 λC 4 (152) f ζ (b i , λ ) = -b 5 (1 + 2λ ) + b 5 λC 3 + b 6 λC 4 (153) f ω (b i , λ ) = -b 4 + b 4 γ d M 5 + b 5 λC 3 + b 6 λC 4 (154) f ω (b i , λ ) = -b 6 (1 + 2λ ) + b 5 λC 3 + b 6 λC 4 (155) f δ (b i ) = b 4 C 2b 7 + b 10 C 7 (156) f ζ x |0 (b i = -b 9 + 3eb 10 (157)f ζ t |0 (b i ) = -b 11 + eb 12 (158)f ω t |1 (b i ) = 1 ac 0 + b 1 M ωt + b 8 C 5 + b 9 C 6b 12(159)f ζ x (b i , λ ) = -b 9 + b 11 λC 8 + b 12 λC 9 (160) f ζ t (b i , λ ) = -b 11 (1 + 2λ ) + b 11 λC 8 + b 12 λC 9 (161) f ω x (b i , λ ) = -b 10 + b 11 λC 8 + b 12 λC 9 (162) f ω t (b i , λ ) = -b 12 (1 + 2λ ) + b 11 λC 8 + b 12 λC 9 (163)

b 2 →b

 2 b 1 → (b 6 , b 7 , b 12 ) → (b 11 , b 5 ) → (γ q , γ d ) → (b 3 , b 8 ) → (b 9 , b 4 ) → b 10 First, from (143) and (145) we choose b 2 and b 1 such that b 2 < ac 0 2e(1 + c 0 + q) 2 (165) 144), (149), (159) we choose respectively b 7 , b 6 , and b 12 such that 11 is taken, according to (157), such that b 11 > eb 12 (170) and from (148) we choose b 5 as b 5 > e(b 6 + b 7 )

b 3 <- 5 From 7 , b 5 - 7 , b 7 -b 4 C 2 C 7 ,

 3575777 min 1, f µ (b 3,4,8,9,10 = 0, M) e M [C 1 + γ d M 4 ] , f z (b 3,4,8,9,10 = 0, M) e M [C 1 + γ d M 4 ] (173) Then, b 8 is taken according to (143)-(145), (149), (150), and (159), as b 8 < min 1, f z|0 (b 4,8,9,10 = 0, M) e M C 5 , f δ |0 (b 4,8,9,10 = 0) e M C 5 , f µ (b 4,8,9,10 = 0, M) e M [γ d M 6 +C 5 ] , f ω|1 (b 4,8,9,10 = 0) f z (b 4,8,9,10 = 0, M) e M γ d M 6 , f ω t |1 (b 4,8,9,10 = 0) C (143)-(146), (149), (150) and (159), b 9 is fixed asb 9 < min 1, f z|0 (b 4,9,10 = 0, M) C 6 e M , f δ |0 (b 4,9,10 ) C 6 e M , f µ (b 4,9,10 = 0, M) e M [C 6 + γ q M 7 + γ d M 8 ] , f d µ (b 4,9,10 = 0) C 6 , f ω|1 (b 4,9,10 = 0) C 6 , f z (b 4,9,10 = 0, M) e M [C 6 + γ q M 7 + γ d M 8 ] , f ω t |1 (b 4,9,10 = 0) C 6(175)From (145)-(146), (147), (149), (150), and (156), b 4 is chosen with respect tob 4 < min 1, f µ (b 4,10 = 0, M) C 2 + γ d M 5 , f d µ (b 4,10 = 0) (b 4,10 = 0, M) C 2 + γ d M 5 (176)The parameters b 10 is fixed according to (143)-(146), (148), (149), (150), (156), (157), and (159) asb 10 < min 1, f z|0 (b 10 = 0, M) C 7 e M , f δ |0 (b 10 = 0) C 7 e M , f µ (b 10 = 0, M) C 7 e M , f d µ (b 10 = 0) C eb 6eb 7 C 7 , f ω|1 (b 10 = 0) Cf z (b 10 = 0,upper bound for λ is found according to (152)-(155), and (160)-(163) asλ < min b 3 (1γ q M 4 ) b 5 C 3 + b 6 C 4 , b 5 b 5 (C 3 -2) + b 6 C 4 , (178) b 4b 4 γ q M 5 b 5 C 3 + b 6 C 4 , b 6 b 5 C 3 + b 6 C 4 -2b 6 b 9 b 11 C 8 + b 12 C 9 , b 11 b 11 (C 8 -2) + b 12 C 9 ,b 10 b 11 C 8 + b 12 C 9 , b 12 b 11 C 8 + b 12 (C 9 -2) by assuming that C 3 > 2, C 4 > 2, C 8 > 2 and C 9 > 2, which can be taken as such.

Univ. Grenoble Alpes, CNRS, Grenoble-INP, GIPSA-lab, F-38000, Grenoble, France (name.surmane@gipsa-lab.fr)

MINES ParisTech, PSL Research University CAS -Centre Automatique et Systèmes, 60 bd Saint Michel, 75006 Paris, FRANCE (delphine.breschpietri@mines-paristech.fr)

By f µ (b 3,4,8,9,10 = 0, M) we mean f µ (b 1 , b

, b

= 0, b

= 0, b

, b

, b

, b

= 0, b

= 0, b

= 0, b

, b

,V = M)).

APPENDIX D PROOFS OF LEMMAS 3 AND 4

Proof of Lemma 3: We start by observing that u t (0, τ) = e a(q-1)(τ-t-1) (µ(t) -

+ 2a τ+1-t 0 e a(q-1)(τ+1-t-χ) [ α(χ/2,t) + δ (χ,t)d(t)]dχ which can be obtained by arguments similar to those used to prove (76). Then, using the fact that

ζ (0,t) = - Therefore, from (179) and (183), and using Young inequality on (72), there exists C 1 > 0 such that

which gives the existence of r 1 > 0 satisfying

Furthermore, from the definition of the Riemann variables, ( 15)-( 16) and ( 37)-(38), one gets

and, applying Young's, Cauchy-Schwarz's inequalities, one can get

Therefore, from (190)-(193)

for a given r 2 > 0. Finally

Consequently, gathering (185), (194)-(195), we obtain (98). This concludes the proof of Lemma 3.

Proof of Lemma 4: From the definition of V (90), one obtains Then, from the backstepping transformation (63) and from the definition of α (51), there exists

Besides, according to the definition of β (49), one obtains the existence of C 3 > 0

Furthermore, from the definition of d µ (73) one writes

Finally, by the definition of ζ , ω, ζ , and ω, (35)-(38) and using the expression (186), one gets

and concludes, applying Cauchy-Schwarz's inequality, that 205) 207)

Similarly, there exist positive constants C 5 , C 6 , C 7 , C 8 , and C 9 such that

Thus, gathering inequalities (196)-(200), and (205)-(213), it holds for a suitable ρ > 0

This concludes the proof of Lemma 4.

APPENDIX E PROOF OF LEMMA 5

From (97), one can easily get that N(t), q(t), d(t), and V 2 (t), V 3 (t) and V 4 (t) are uniformly bounded for t 0. Therefore,

and ω t (t) are also uniformly bounded for t 0.

From there, applying Young's inequality to ( 9) and [START_REF] Ioannou | Robust Adaptive Control[END_REF], one can obtain that ˙ q(t) and ˙ d(t) are uniformly bounded for t 0. Similarly, applying Cauchy-Schwarz's inequality to [START_REF] Conrad | Strong stability of a model of an overhead crane[END_REF], one can obtain that ζ (1,t) and thus α(1,t) are uniformly bounded for t 0. Moreover, using Lemma 6 in Appendix A-1, ζ (x,t) is also uniformly bounded for t 1 + x and, in particular, ζ (0,t) is uniformly bounded for t 1. Similarly, using Lemma 6, α(x,t) is also uniformly bounded for t 2(1x) and, in particular, α(0,t) is uniformly bounded for t 2.

Further, using the fact that

and from (44), it holds, for x ∈ [0, 1],

Using ( 9)-( 10), Cauchy-Schwarz's inequality and the previous considerations, it is straightforward that the right-hand terms in the previous equations are all uniformly bounded for t 2. Finally, integrating (96) from 0 to ∞, it follows that µ(t), d µ (t), ω(t) , ζ (t) , ζ (t) , ω(t) , ζ t , and ω t are square integrable. Therefore, with (183), so is ω(1,t), and finally v(t). Moreover, as d µ (t) and µ(t) are square integrable, so is d(t).

We conclude this proof with Barbalat's lemma.