Alexander Mendler

Clément Ambroise Cadoret

Clément Freyssinet

Michael Döhler

Yann Lecieux

Laurent Mevel

Carlos Ventura

Minimum Localizable Damage for Stochastic Subspace-based Damage Diagnosis

Keywords: Ambient vibrations, statistical test, detectability, localization resolution, false localization alarms

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

Minimum Localizable Damage for Stochastic Subspace-based Damage Diagnosis

Alexander Mendler * Ambroise Cadoret * * Clément Freyssinet * * Michael Döhler * * * Yann Lecieux * * Laurent Mevel * * * Carlos Ventura *

INTRODUCTION

Modern societies depend on structural and mechanical systems, such as power plants, bridges, offshore platforms, defence systems, aircraft and spacecraft. Because many of these structures cannot be economically replaced or because they are exposed to natural hazards, monitoring techniques are developed to guarantee a safe operation, even beyond their original life span.

In the context of civil engineering structures, the process of implementing a damage diagnosis strategy is referred to as structural health monitoring (SHM). This process is divided into the operational evaluation of the structure, the observation through permanently installed sensors, the extraction of damage-sensitive features, and the subsequent damage diagnosis [START_REF] Farrar | Structural health monitoring: A machine learning perspective[END_REF], which is subdivided into damage detection, localization, quantification, and lifetime prognosis [START_REF] Rytter | Vibrational based inspection of civil engineering structures[END_REF]. Damage detection can be performed in an unsupervised learning mode, meaning no vibration data from the damaged structure is available. Damage localization and quantification, on the other hand, either require vibration from the damaged structure for training or a finite element (FE) model of the examined structure.

On one hand, FE models are generally over-parametrized, meaning changes in multiple structural parameters have a similar effect on the data-driven features. So, it is not possible to identify the structural parameter that has changed due to damage. This either makes the damage localization impossible or causes false localization alarms, meaning undamaged components are incorrectly classified as damaged. In the literature, multiple approaches are summarized to overcome this problem, for example, the collinearity index method [START_REF] Brun | Practical identifiability analysis of large environmental simulation models[END_REF], the column-pivoting method [START_REF] Velez-Reyes | Subset selection in identification, and application to speed and parameter estimation for induction machines[END_REF], the Gram-Schmidt orthogonalization [START_REF] Yao | Modeling Ethylene/Butene Copolymerization with Multi-site Catalysts: Parameter Estimability and Experimental Design[END_REF], a recursive approach based on principal component analysis as in [START_REF] Li | Selection of Model Parameters for Off-Line Parameter Estimation[END_REF] or other.

On the other hand, the use of FE models allows one to physically define damage and to focus the damage diagnosis on user-defined structural parameters. Moreover, it is possible to calculate the sensitivity of data-driven features toward model-based parameter changes and to extrapolate knowledge beyond the reference state of the structure. This paper explains how the localizable damage can be quantified in percent of a model-based parameter change (in, for example, material properties or crosssectional values) before damage actually occurs.

All studies in this paper are performed for the stochastic subspace-based damage diagnosis methods, which are capable of detecting [START_REF] Basseville | Subspace-based fault detection algorithms for vibration monitoring[END_REF] and localizing [START_REF] Basseville | Statistical model-based damage detection and localization: Subspace-based residuals and damage-to-noise sensitivity ratios[END_REF] structural changes based on statistical hypothesis tests. The problem of overparametrization is remedied by combining redundant parameters into parameter clusters [START_REF] Balmès | Statistical model-based damage localization: A combined subspace-based and substructuring approach[END_REF], where the number of clusters (the localization resolutions) depends on a userdefined input parameter [START_REF] Allahdadian | Towards robust statistical damage localization via model-based sensitivity clustering[END_REF]. More recently, a method was published to determine the minimum detectable damage based on vibration data from the undamaged structure and a FE model [START_REF] Mendler | A reliabilitybased approach to determine the minimum dectable damage for statistical damage detection[END_REF]. In the current paper, the method is extended to damage localization, where the mean test response for damaged and undamaged parameters is predicted. The approach has already been described in the automatic control community [START_REF] Mendler | Clustering of Redundant Parameters for Fault Isolation with Gaussian Residuals[END_REF] but has not been applied to larger civil engineering structures, such as bridges.

The paper is organized as follows: Section 2 recaps the stochastic subspace-based damage localization method. Section 3 introduces three optimization criteria to assess the localizability of damage. In Section 4, the approach is applied to a cable-stayed bridge followed by the conclusions in Section 5.

STATISTICAL DAMAGE LOCALIZATION

This section recaps the stochastic subspace-based damage diagnosis method. It is based on the asymptotic local approach to damage detection [START_REF] Benveniste | The asymptotic local approach to change detection and model validation[END_REF] in combination with the stochastic subspace-based residual vector as a damagesensitive criterion. The underlying vibration model is assumed to be linear and time-invariant, with white noise excitation and no periodic input signals.

Damage-sensitive Residual

The damage-sensitive feature is formed in the mathematical subspace of covariance functions. Covariance functions describe the similarity of output signals at different measurement channels. They can be estimated based on output data, which is indicated through the hat symbol:

Ri = 1 N N k=1 y κ+i y T κ ∈ R r×r ,
where i is the time lag, N is the number of samples in the measurement record and r is the number of channels. By arranging the output covariance estimates in block Hankel format

Ĥp+1,q =         R1 R2 . . . Rq R2 R3 . . . Rq+1 . . . . . . . . . . . . Rp+1 Rp+2 . . . Rp+q         (1)
a dynamic system with m degrees of freedom can be modelled. The parameters p and q are time lag parameters with min(pr, qr) ≥ n and often p + 1 = q. After applying a singular value decomposition

Ĥp+1,q = Û1 Û0 D1 0 0 D0 VT 1 VT 0 ,
the dynamic system matrices can be retrieved from the column space Û1 ∈ R (p+1)r×n using stochastic system identification [START_REF] Van Overschee | Subspace identification for linear systems: Theory, Implementation, Application[END_REF][START_REF] Peeters | Reference-based stochastic subspace identification for output-only analysis[END_REF], where n = 2m. The null space Û0 is orthogonal to the column space Û1 , so a damage-sensitive residual can be formed by pre-multiplying the left null space to the block Hankel matrix and by vectorizing the residual matrix, using the column-stacking operator vec(•), so [START_REF] Basseville | Subspace-based fault detection algorithms for vibration monitoring[END_REF] ε = vec( ÛT 0 Ĥp+1,q ).

(2)

Damage Parametrization

For a physical interpretation, damage has to be expressed in terms of model parameters and linked to the damagesensitive feature using sensitivity analysis. Using FE models, the health state can be expressed through model-based design parameters θ ∈ R H with (i.e., material properties, cross-sectional values, support conditions, or pre-stressing forces), and damage can be defined as a deviation from their reference values θ 0 , so ∆ = θθ 0 .

(3) Using a first-order Taylor series expansion, the residual's mean vector can be expressed in terms of the parameter vector as

E θ [ ε] ≈ E θ 0 [ ε] + J (θ -θ 0 ), (4)
where the first term approximates a zero vector E θ 0 [ ε] → 0 and the second term includes the Jacobian matrix J , which holds the feature's derivative with respect to structural parameters, that is,

J = ∂E θ [ ε] ∂θ θ=θ 0 . (5)
The Jacobian matrix links changes in the data-driven residual vector to model-based design parameters, which allows for a physical interpretation of changes in data. More information on the analytical computation of the Jacobian matrix for the residual from Eq. ( 2) can be found in the literature [START_REF] Balmès | Statistical model-based damage localization: A combined subspace-based and substructuring approach[END_REF][START_REF] Allahdadian | Towards robust statistical damage localization via model-based sensitivity clustering[END_REF].

Statistical Distributions of the Feature

Uncertainties due to stochastic loads, measurement noise, and limited sample size N have to be considered to guarantee a reliable damage diagnosis. It can be shown that the residual vector approximates a multi-dimensional normal distribution

ζ = √ N ε -→ N (0, Σ) (reference) N (J δ, Σ) (damaged) ( 6
)
where ζ is known as the Gaussian residual vector and δ = √ N (θ -θ 0 ) is the asymptotic change vector, cf. Eq. ( 4). Damage manifests itself in a changing parametrized mean vector. Uncertainties related to the estimation of the Gaussian residual, on the other hand, can be captured through the covariance matrix estimate [START_REF] Benveniste | The asymptotic local approach to change detection and model validation[END_REF]

Σ = 1 n b -1 n b k=1 ζ k ζ T k , (7)
which is calculated as a sample covariance with n b number of data blocks where ζ k is the residual evaluated on each block. The covariance captures the uncertainties in the estimation of the Gaussian residual.

Predicted Mean Test Response

Statistical hypothesis tests, such as the generalized likelihood ratio (GLR) can be applied for damage detection. The parametrized test statistic is defined as [START_REF] Basseville | Subspace-based fault detection algorithms for vibration monitoring[END_REF] t

= ζ T Σ -1 J J T Σ -1 J -1 J T Σ -1 ζ (8)
and follows a χ 2 -distribution with ν = rank(F ) degrees of freedom. The bracketed term in Eq. ( 8) is known as the Fisher information matrix F ∈ R H×H . It can be estimated based on reference data and should be seen as a measure for the detectability of damage with F = J T Σ -1 J .

(9) Assuming that damage can be described through a single parameter change, where all other parameters remain unchanged, the mean test response (i.e. the non-centrality in the χ 2 -distribution) can be predicted as [START_REF] Mendler | A reliabilitybased approach to determine the minimum dectable damage for statistical damage detection[END_REF]

λ h = (∆θ h ) 2 F hh f s T. ( 10
)
The damage diagnosis is reliable if the mean test response is sufficiently large, with λ > λ min , because then, the test result is clear despite the uncertainties in the damagesensitive feature. The minimum mean test response depends on the width of the χ 2 -distribution (expressed through the number of degrees of freedom) and can, for example, be determined based on the allowable false-alarm rate and the prescribed probability of detection, which should be close to one-hundred percent [START_REF] Mendler | A reliabilitybased approach to determine the minimum dectable damage for statistical damage detection[END_REF].

Hierarchical Parameter Clustering

Damage localization is more challenging than damage detection due to the large number of structural parameters in FE models. Changes in multiple structural parameters have a similar effect on the damage-sensitive residual, and vice versa, it is not possible to identify the structural parameter that has changed. To remedy this, parameters with similar sensitivities are combined to parameter clusters which corresponds to a substructuring of the FE model. To preserve the statistical properties of the damage-sensitive feature throughout the clustering [START_REF] Balmès | Statistical model-based damage localization: A combined subspace-based and substructuring approach[END_REF], the Jacobian matrix is normalized as follows J = Σ -1/2 J . (11) The first step of the hierarchical clustering approach [START_REF] Allahdadian | Towards robust statistical damage localization via model-based sensitivity clustering[END_REF] is to evaluate the similarity of all sensitivity vectors using the dissimilarity measure

d ij = 1 - J T i J j || J i || • || J j || ,
where the second term is the cosine between the sensitivity vectors (the columns in the Jacobian matrix). Secondly, the two vectors with the shortest distance are selected. If some of the sensitivity vectors are already clustered, the complete-linkage criterion is applied [START_REF] Duda | Pattern classification[END_REF] to evaluate the critical distance D as the one with the maximum distance between individual sensitivity vectors

D(C a , C b ) = max{d ij : i ∈ C a , j ∈ C b }. ( 12
)
The two partitions with the shortest distance D are then combined through averaging, with the cluster centres

c k = 1 m k i∈C k J i , (13)
where m k is the number of parameters in cluster C k and k ∈ [1, . . . , K] is the number of clusters. The steps from Eq. ( 12)-( 13) are repeated until the distance drops below a user-defined threshold d thres . Ultimately, a clustered Jacobian matrix is obtained

J c = [c 1 • • • c K ] . (14)

Damage Localization Test

During damage detection, each structural parameter θ h is individually tested for changes. In the presented version, the minmax localization test evaluates the minimum likelihood of the tested parameter having changed against the maximum likelihood of changes in untested clusters, as this yields the localization result with the highest localization resolution. Mathematically, this concept is implemented through a geometrical projection as explained in the following. First, the Jacobian matrix is re-organized

J h ,h = [J h J c h] , (15)
where J h is the residual's sensitivity toward the tested parameter from Eq. ( 5) and J c h is the untested partition that includes the cluster centres of all parameters from Eq. ( 14) except the one that contains the tested parameter, so

J c h = c 1 . . . c k(h)-1 c k(h)+1 . . . c K .
Considering that the clustered Jacobian matrix is already normalized, see Eq. ( 11), the Fisher information for the reorganized Jacobian matrix can be expressed as

F h ,h = F h F c h h F c hh F c hh = J T h Σ -1 J h J T h J c h J cT h J h J cT h J c h . (16)
Subsequently, the Gaussian residual ζ is projected onto the tested and untested partitions,

ζ h = J T h Σ -1 ζ, ζ c h = J cT h Σ -1/2 ζ, (17)
and a geometrical projection is applied to obtain the robust residual and its Fisher information [START_REF] Döhler | Fault detection, isolation and quantification from Gaussian residuals with application to structural damage diagnosis[END_REF]

ζ * h = ζ h -F c h hF c-1 hh ζ c h, (18)
F * h = F h -F c h hF c-1 hh F c hh , (19)
respectively. This operation preserves the residual's information content regarding the tested parameter and makes it blind to changes in untested partitions. Ultimately the minmax test statistic is evaluated through

t * h = ζ * T h F * -1 h ζ * h , (20)
which is χ 2 -distributed with ν = 1 degree of freedom and a mean test response (i.e. a non-centrality) of

λ h = δ T h F * h δ h . (21)
The mean test response can be predicted using Eq. ( 10) after exchanging the Fisher information F hh with the robust Fisher information F * h from Eq. [START_REF] Mendler | Minimum diagnosable damage and optimal sensor placement for structural health monitoring[END_REF]. Equivalently, the mean test response of a tested but undamaged parameter can be predicted.

Problem Statement

Stochastic subspace-based damage diagnosis is based on statistical hypothesis tests, such as the general likelihood ratio and the minmax localization test. The tests are datadriven but a FE model is required to physically define damage, to evaluate the sensitivity towards structural parameter changes, and to localize damage. The model itself is assumed to be perfect and is not updated during the damage diagnosis. FE models are over-parametrized, meaning changes in multiple structural parameters have a similar effect on the damage-sensitive feature. Structural parameters with similar sensitivity vectors have to be combined to parameter clusters, which corresponds to a substructuring of the FE model. The damage localization resolution is a user-defined input parameter and is defined through the cut-off value d thres . It appears intuitive to strive for a high damage localization resolution but the next chapter will show that an increased localization resolution can lead to negative effects, that is, a lower damage detectability and false localization alarms.

OPTIMAL DAMAGE LOCALIZATION

This section summarizes an approach from the literature [START_REF] Mendler | Clustering of Redundant Parameters for Fault Isolation with Gaussian Residuals[END_REF] and presents three optimization criteria to find the minimum localizable damage, namely, the damage localization resolution, the minimum detectable damage, and false localization alarms. Moreover, a multi-objective optimization strategy is introduced to find the optimal compromise between the three criteria and to determine the optimum settings for damage localization.

Damage Localization Resolution

The first criterion is the damage localization resolution. The resolution depends on the number of clusters K because, even though a single parameter change is assumed, undamaged parameters also show a significant test response if they are within the same cluster. A low number of clusters corresponds to a low localization resolution, and a high number of clusters to a high resolution. The higher the resolution, the more accurate the damage localization. Therefore, an objective function can be formed as follows

f 1 (K) = K -K b K g -K b , ( 22
)
where K is the number of clusters, and g and b indices for a good and bad number of clusters. The normalization through a lower bound K b = 2 and an upper bound K g = rank(F ) yields in a objective function between zero and one with values close to zero reflecting an ideal localization resolution.

Minimum Detectable Damage

The second criterion is the minimum detectable damage, that is, the minimum change in local structural design parameters that can reliably be detected based on changes in global damage-sensitive features. It can be calculated based on reference data by rearranging the predictive formula from Eq. ( 10) for the relative parameter change and substituting the minmax Fisher information from Eq. ( 19)

∆ h (K) = 1 θ 0 h λ min N • F * h . ( 23
)
The diagnosis is considered "reliable" if the probability of false alarms is low and the probability of detection is high which is equivalent to requiring a minimum mean test response λ = λ min . Equivalently, a damage is "detectable" if it is significant under consideration of typical uncertainties and empirical safety thresholds. The lower the detectable damage, the more sensitive the damage diagnosis. Likewise, the decisive parameter is the one with the highest detectable damage ∆ max (K) = max{∆ 1 (K), ..., ∆ H (K)}, as this corresponds to the parameter with the lowest damage detectability. An objective function can be formed as follows

f 2 (K) = ∆ max (K) -∆ g ∆ b -∆ g , (24)
with a lower bound of ∆ g = 0% and a upper bound of ∆ b = 100%.

False Localization Alarms

The third criterion is the false alarm susceptibility. A false localization alarm is defined as a significant test response for an unchanged parameter that is not within the same cluster as the parameter that has changed due to damage. False alarms occur for inappropriate cluster settings, meaning if the cluster centres are not representative of individual sensitivity vectors or if the sensitivity estimates have not fully converged toward the true value.

The previous section explained how the mean test response could be predicted based on the Fisher information Eq. [START_REF] Allahdadian | Towards robust statistical damage localization via model-based sensitivity clustering[END_REF]. A reliable damage localization requires that no false localization alarms exceeds the test response for the parameters that have actually changed. Consequently, it is sensible to evaluate the relative test response, using the non-centrality ratio NCR = λ h,h /λ h ,h .

(25) The NCR describes the ratio of the predicted mean test response for a tested but unchanged parameter λ h,h to the one of a tested and changed parameter λ h ,h , cf. Eq. ( 10) and [START_REF] Mendler | Minimum diagnosable damage and optimal sensor placement for structural health monitoring[END_REF]. More details can be found in the literature [START_REF] Mendler | Clustering of Redundant Parameters for Fault Isolation with Gaussian Residuals[END_REF].

Whether or not false localization alarms are significant is determined by introducing a threshold value of, for example, 90%. That means that the NCR must not exceed 90% for any undamaged parameter. To form an optimization criterion the false alarms are evaluated for all possible damage scenarios (i.e. changes in all structural parameters θ h with h = 1, . . . , H), and the number of scenarios N sc is counted that exhibits false localization alarms. Then, the objective function can then be defined as

f 3 (K) = N sc (K) -N g N b -N g , (26)
where N g and N b are lower and upper bounds that allow the user to weigh the importance of false alarms.

Pareto Optimization

Finding the minimum localizable damage is a multiobjective optimization problem. It depends on the localization resolution, the minimum detectable damage, and false localization alarms. After scaling with user-defined lower and upper bounds, all three criteria are equally important. For example, a high damage detectability is useless if the localization resolution is low with K = 1, because it is merely possible to detect the presence of damage but not its location. At the same time, a high localization resolution is meaningless if the false localization alarms are excessive, because the actual damage location remains hidden. The localization resolution and the damage detectability are conflicting optimization criteria, meaning it is not possible to improve one criterion without degrading the other.

Dealing with conflicting optimization criteria of equal importance is known as Pareto optimization. The optimal solution is the one that simultaneously optimizes the localization resolution f 1 from Eq.( 22), the damage detectability from Eq. ( 24) and the false alarm susceptibility from Eq. ( 26). One way to find the optimal solution is to define a compromise function as the Euclidean distance in the three-dimensional optimization space min

K f = f 1 (K) 2 + f 2 (K) 2 + f 3 (K) 2 (27) s.t. f 1 (K) < 1, f 2 (K) < 1, f 3 (K) < 1.
The optimal compromise can then be found by selecting the number of clusters (the localization resolution) with a global minimum in the compromise function f from Eq. ( 27).

NUMERICAL APPLICATION

In this section, the developed methods are applied to the numerical model of a cable-stayed bridge [START_REF] Mendler | Minimum diagnosable damage and optimal sensor placement for structural health monitoring[END_REF]. The goal is to evaluate the optimal substructure arrangement to localize the failure of stay cables and to validate the prediction of the mean test response for damaged and undamaged cables, that is, false localization alarms.

Model Bridge Description

The specimen is a scaled-down laboratory model of the St. Nazaire cable-stayed bridge in France (scale 1:200), described in [START_REF] Cadoret | Fault detection using modal analysis[END_REF]. The real bridge and the laboratory model are shown in Fig. 1 and Fig. 2. The model is split into the main span and two side spans with a length of 2.02 m and 79 cm, respectively. The bridge deck is a steel box girder, continuous over all three spans, with a width of 7.5 cm. The cross-section is composed of foam encased by a trapezoidal steel sheet. The piers at the cable end spans are made of aluminum. The two Ashaped towers, also made of aluminum, rise 33 cm over the steel deck, where the two aluminum posts of each tower are connected at the top through a steel sheet that includes fixtures for stay cables. Each tower sits on top of a monolithic tower pier with two vertical pillars, a transverse

Finite Element Model

The stochastic subspace-based damage diagnosis method is data-driven, but a FE model is required to parametrize damage and to calculate the sensitivities of the damagesensitive residual toward structural design parameters. Since the case study in this paper is based on the numerical vibration data, the model is also used for vibration generation.

The FE model from Fig. 3 was created and calibrated by the Department for Civil and Mechanical Engineering in Nantes [START_REF] Cadoret | Fault detection using modal analysis[END_REF]. It is designed in MATLAB ® using customcode for 3-D beam and truss elements. The continuous deck is discretized into 51 nodes and 50 beam elements with identical cross-sections and material properties over all three spans (Group 1 in Table 1). The piers are split into pier pillars (Group 2) and pier caps (Group 3). The pylons are split into six groups: the piers (Group 4), the V-shaped pier caps (Group 5 and 6), the A-shaped towers (Group 7 and 8), and the vertical cable anchor elements made of metal (Group 9). The stay cables are the only elements that are modelled as trusses (Group 10). In total, the model exhibits 1,038 degrees of freedom. All six degrees of freedom at the two tower foundations and the four pier foundations are fixed, reducing the number of unrestrained degrees of freedom to 1,002.

The model calibration is conducted based on experimental data from the laboratory bridge model from Fig. 2. High uncertainties are expected in the modelling of the bridge deck, because the properties of the epoxy glue and the foam infill are not specified. First, the natural frequencies are extracted using stochastic subspace-based system identification, and subsequently, the cross sectional values of the deck are updated (i.e., the moment of inertia, and the mass density) until a satisfactory match of numerical and experimental frequencies is obtained [START_REF] Cadoret | Fault detection using modal analysis[END_REF].

The considered natural frequencies and mode shapes of the FE model are summarized in Fig. 4. The mode shape of Mode 2 is similar to Mode 3, but it exhibits no bending components in the towers and a more pronounced longitudinal movement of the deck. No. For vibration generation, the mechanical system matrices are extracted from the model, the modal damping matrix is set up based on an assumed value of 1% critical damping for each mode, and transformed into a state space model in modal coordinates. This allows for the individual modes of vibration to be selected that contribute to the vibration record. Subsequently, a white noise input signal is applied to the vertical degrees of freedom of the deck beam, a transient analysis is run, and the output at the sensor locations from Fig. 3 and4 is stored after superimposing it with uniformly distributed measurement noise with a magnitude corresponding to 5% of the output variance. All other signal processing parameters are summarized in Table 2. (28) The stiffness contribution of three-dimensional truss elements is a linear function of the cross-section A, the modulus of elasticity E, and the cable length L, meaning a change linearization through the Jacobian matrix does not introduce bias.

ρ E A Iy Iz J [ kg m 3 ] [ M N m 2 ] [cm 2 ] [cm 4 ] [cm 4 ] [cm 4 ]

Signal Processing

K loc el = EA/L 1 -1 -1 1 (29)
b) Sensitivity Computation. The Jacobian matrix links changes in the data-driven residual to structural design parameters [START_REF] Balmès | Statistical model-based damage localization: A combined subspace-based and substructuring approach[END_REF][START_REF] Allahdadian | Towards robust statistical damage localization via model-based sensitivity clustering[END_REF]. For its computation, the first six modes of vibration are considered, see Fig. 4. The pole derivatives are computed analytically, and so are the mode shape derivatives. The mode shape derivatives are computed using the modal approach [START_REF] Sutter | Comparison of several methods for calculating vibration mode shape derivatives[END_REF], which theoretically requires a loop over all modes of vibration of the FE model. However, the derivatives converge rapidly, and for simplification, only the first 30 modes are considered in this study.

Optimal Substructures for Damage Localization

Before the optimal substructure arrangement can be found, the main input parameters for the damage diagnosis have to be defined, including the time lag parameters p, q, the system order n, and the number of blocks n b or the block length N b = N 0 /n b for the covariance estimation, where N 0 = T f s = 612, 000 is the number of samples from the reference state. For conciseness, all input parameters for the damage diagnosis are summarized in Table 3. Finding the optimal substructure arrangement is a multiobjective optimization problem. Figure 5 visualizes all three objective functions from Eq. ( 22), (24), and (26) for a varying number of parameter clusters, together with the compromise function from Eq. ( 27). The individual objective functions can be interpreted as follows:

Damage Diagnosis

• f 1 is the localization resolution (the dashed line). Damage localization with a single parameter cluster is meaningless, so the worst solution with f 1 = 1 is achieved for two parameter clusters. With an increasing number of clusters, the localization resolution linearly improves with an optimal value of f 1 = 0 for K g = rank(F ) = 25 clusters. • f 2 is the damage detectability (the solid black line).

The detectability is high for a low number of clusters, but for more than six clusters, it gradually worsens with a distinct jump for more than 20 parameter clusters. This jump indicates that the minimum detectable damage for the parameters with the lowest damage detectability exceeds 100%. • f 3 is the false alarm susceptibility (the dotted line).

False alarms are zero for one or 36 clusters and unequal to zero for most other cases. The solution close to the allegedly optimal point with 20 clusters exhibits a high number of false alarms, which underlines the importance of the false alarm susceptibility as an optimization criterion.

Figure 5 also contains the compromise function (the thick solid black line). It exhibits minima for 16 or 18 parameter clusters, meaning these cluster settings simultaneously maximize the localization resolution and the damage detectability, and minimize false localization alarms. The solution with 18 clusters is visualized in Fig. 6; for this cluster setting, the cable pairs in each cable fan are defined as separate substructures, so the damaged cable pair can be clearly identified. However, due to the symmetry of the bridge, it is not possible to identify the cable fan in which damage has occurred.

Result Validation

All previous studies in this paper are based on vibration data from the undamaged structure and a FE model.

To validate the predictions for the minimum detectable damage from Eq. ( 23) and the mean test response of undamaged parameters (i.e. the NCR) from Eq. ( 25), structural damage is now simulated in the FE model by decreasing the cross-sectional area in individual cables pairs by 50%. Since 36 cable pairs are monitored, there are 36 possible damage scenarios, one for each monitoring parameter from Eq. (28).

First, damage scenario P3 is discussed, with a 50% reduction in the cross-section area of cable pair P3. The damage localization results for this scenario is are shown in Figure 7. It appears that the predicted mean test response (plus sign) is close to the measured value (white bar) for the damaged cable pair P3. Cable pair P34 is in the same cluster as cable pair P3, so the test response was to be expected. All other cable pairs are not within the same cluster, but the mean test responses (grey bars) can be predicted with sufficient accuracy. Cable pairs P1, P2, and P4, as well as P33, P35 and P36, exceed the threshold of 90% which is why this scenario is classified as a false alarm scenario.

Secondly, the validation study is repeated for damage scenario P7, P13, and P19, see Figure 7. The predictions appear to be accurate for both changed parameters and unchanged parameters (false localization alarms).

Ultimately, comparing the localization results for the damage scenarios from Figure 7 highlights the importance of all three optimization criteria. Where the localization resolution and the damage detectability are conflicting criteria (meaning it is not possible to improve one without degrading the other), the false alarm susceptibility is not conflicting. Nonetheless, it should be considered as an optimization criterion, because it critically affects the localizability of damage. For example, failure of cable pair P3 leads to significant test responses for three adjacent cable pairs (P1, P2, and P4) and three more on the opposite side of the bridge, with NCRs over 95%. Therefore, it cannot be said with certainty whether P2, P3, P34, or P35 is damaged. Scenario P7 is even more ambiguous, as it exhibits test responses for 24 unchanged parameters. With the current settings, both scenarios would be classified as false alarm scenarios. To apply a more severe penalty to scenarios such as P7, the threshold value for the NCR could be lowered from 90% to 50%. On the other hand, damages in cable pair P13 and in P19 can be more clearly localized. To conclude, for a fixed sensors layout, the localizability of some damages is higher than for others, where localizability is a compromise between high damage localization resolution, high detectability, and few false localization alarms.

CONCLUSION

This paper describes an approach to determine the minimum localizable damage, which is defined as the minimum damage that can be localized reliably. It presents mathematical formulas to assess the detectability of damage and the false alarm susceptibility. Moreover, it is demonstrated that an increasing damage localization resolution leads to lower damage detectability, and a changing susceptibility for false localization alarms. This paper employs a subspace-based residual vector as a damage-sensitive feature, but the method is readily applicable to any damagesensitive feature with Gaussian properties.

The analysis is based on vibration data from the undamaged structure and a finite element model. Therefore, the developed method allows the user to evaluate the performance of the damage localization algorithm before damage actually occurs. This is particularly important for unique structures and unprecedented damaged events, where no data is available for different damage scenarios.

Being able to evaluate the performance of the damage localization is essential to optimize the sensor layout and to convince decision-makers of the value of implementing a SHM system.

Figure 1 .

 1 Figure 1. St. Nazaire Bridge, Nantes.

Figure 2 .

 2 Figure 2. Laboratory model. beam, and a solid piece that expands to the foundations. The 72 steel cables, connect the upper part of the tower to the deck and form two cable planes in fan-arrangement.

Figure 3 .

 3 Figure 3. Finite element model in Matlab [19].

f 1 =

 1 35.7 Hz, 1 1 = 1.0 [%] f 2 = 53.8 Hz, 1 2 = 1.0 [%] f 3 = 61.2 Hz, 1 3 = 1.0 [%] f 4 = 91.5 Hz, 1 4 = 1.0 [%] f 5 = 119.9 Hz, 1 5 = 1.0 [%] f 6 = 150.9 Hz, 1 6 = 1.0 [%]

Figure 4 .

 4 Figure 4. Modes of vibration for damage diagnosis

2 Figure 5 .

 25 Figure 5. Objective functions.

Figure 6 .

 6 Figure 6. Substructure arrangement.

Table 1 .

 1 FE model properties.

	1	37,000 210,000	0.2	2.1	0.18	2.3
	2	2,700	69,000	2.0	0.17	0.67	0.83
	3	7,800	210,000	2.4	0.8	0.29	1.1
	4	2,700	69,000	18	120	6.0	130
	5	2,700	69,000	9.0	15	3.0	18
	6	2,700	69,000	9.0	15	3.0	18
	7	2,700	69,000	0.24	0.0032 0.0072	0.01
	8	2,700	69,000	0.24	0.0032 0.0072	0.01
	9	7,800	210,000	0.24	0.0032 0.0072	0.01
	10	7,800	210,000	77

Table 2 .

 2 Input parameters a) Damage Parametrization. Damage is defined as a change in the cross-section A of the cables. The same monitoring parameters are assigned to the cables on both sides of the deck, reducing the number of parameters from 72 to 36 with the monitoring vector θ = [A 1 . . . A 36 ] .

	Measurement quantity	acceleration
	Sampling frequency	340 Hz
	Reference data length	30 min
	Training/testing data	30 min

Table 3 .

 3 Input parameters


G Vergara-Hermosilla

G Leugering

Y Wang

BOUNDARY CONTROLLABILITY OF A SYSTEM MODELLING A PARTIALLY IMMERSED OBSTACLE

Keywords: 2010 Mathematics Subject Classification. 35L50, 35L65, 93B05, 93C20 Nonlinear shallow water equations, exact controllability, nodal profile controllability, boundary control, semi-global piecewise C 1 solution

In this paper, we address the problem of boundary controllability for the one-dimensional nonlinear shallow water system, describing the free surface flow of water as well as the flow under a fixed gate structure. The system of differential equations considered can be interpreted as a simplified model of a particular type of wave energy device converter called oscillating water column. The physical requirements naturally lead to the problem of exact controllability in a prescribed region. In particular, we use the concept of nodal profile controllability in which at a given point (the node) timedependent profiles for the states are required to be reachable by boundary controls. By rewriting the system into a hyperbolic system with nonlocal boundary conditions, we at first establish the semi-global classical solutions of the system, then get the local controllability and nodal profile using a constructive method. In addition, based on this constructive process, we provide an algorithmic concept to calculate the required boundary control function for generating a solution for solving these control problem.

1. Introduction 1.1. Context. Free surface interactions with fixed or floating structures have been intensively studied by the mathematical community in the last years with respect to modelling, well-posedness, numerical simulations, etc. Recently, Lannes proposed in [START_REF] Lannes | On the dynamics of floating structures[END_REF] a new formulation of water-waves problem in order to take into account the presence of a floating body. More precisely, in his work Lannes implemented a method for the full water wave equations and for reduced asymptotic models, such as the Boussinesq and the nonlinear shallow-water equations, where the pressure exerted by the fluid on the partially immersed structure appears as a Lagrange multiplier associated with the constraint that under the floating structure, the surface of the fluid coincides with the bottom of the structure. In the case of the nonlinear shallow water equations, the resulting fluid-structure model with vertical lateral walls has been studied in [START_REF] Lannes | On the dynamics of floating structures[END_REF] and in the more general case of non vertical walls in [START_REF] Iguchi | Hyperbolic free boundary problems and applications to wave-structure interactions[END_REF]. An extension to a system modelling a floating structure on a viscous shallow water regime has been recently studied by Maity et al. in [START_REF] Maity | Analysis of a simplified model of rigid structure floating in a viscous fluid[END_REF], and by Matignon, Tucsnak and Vergara-Hermosilla in [START_REF] Vergara-Hermosilla | Well-posedness and input-output stability for a system modelling rigid structures floating in a viscous fluid[END_REF] and [START_REF]Asymptotic behaviour of a system modelling rigid structures floating in a viscous fluid[END_REF].

Bresch, Lannes and Métivier in [START_REF] Bresch | Waves interacting with a partially immersed obstacle in the boussinesq regime[END_REF] treat the derivation and mathematical analysis of a fluid-structure interaction problem with a configuration where the motion of the fluid is governed by the Boussinesq system, which is a dispersive perturbation of the hyperbolic nonlinear shallow water equations, and in the presence of a fixed partially immersed obstacle. They showed that the fluid-structure interaction problem can be reduced to a transmission problem.

In a similar way and motivated for mathematical modeling and simulations of a specific type of wave energy converting device, the so-called oscillating water column (OWC) device, Bocchi, He and Vergara-Hermosilla discuss in [START_REF] Bocchi | Modelling and simulation of a wave energy converter[END_REF] about the fluid-structure interaction of the partially submerged fixed wall structure of the OWC device, considering the nonlinear shallow water equations to describe the fluid movement, and obtain explicit transmission conditions for the system and respective reduced transmission problems. Recently, Bocchi, He and Vergara-Hermosilla propose in [START_REF]Well-posedness of a nonlinear shallow water model for an oscillating water column with timedependent air pressure[END_REF] a new and general approach on the mathematical modelling of the OWC, where include the presence of the timedependent air pressure in the device and prove a local well-posedness result in a Sobolev setting. Hence, by considering their results, in this work we deep on a particular kind of boundary controllability on the transmission problems studied in [START_REF] Bocchi | Modelling and simulation of a wave energy converter[END_REF] on an equivalent physical configuration. More precisely, we deal with the exact boundary controllability of nodal profile on a system modelling a structure partially immersed in a fluid governed by the nonlinear shallow water equations [START_REF] Lannes | On the dynamics of floating structures[END_REF], considering a discontinuity in the height of the fluid bottom and the transmission conditions developed in [START_REF] Bocchi | Modelling and simulation of a wave energy converter[END_REF]. This physical situation is presented in a graphical sketch in Figure 1.

This kind of boundary controllability was motivated by practical applications on gas networks and introduced recently in the literature by Gugat, Herty and Schleper, in [START_REF] Gugat | Flow control in gas networks: exact controllability to a given demand[END_REF]. Their new approach was almost immediately generalized by Li to general 1-D first order quasilinear hyperbolic systems with general nonlinear boundary conditions in [START_REF] Li | Exact boundary controllability of nodal profile for quasilinear hyperbolic systems[END_REF].

As is well known, the usual exact boundary controllability that asks the solution to the system under certain boundary controls to satisfy a given final state at a suitably large time t = T , however, the exact boundary controllability of nodal profile, requires that the value of solution satisfies the given profiles on one or more nodes for t ≥ T by using boundary controls. This approach can be established by means of a constructive method with modular structure, by using the following three ingredients: existence and uniqueness of semi-global classical solution to the mixed initial-boundary value problem, exchanging the role of the the space variable x and time variable t, and the uniqueness of classical solution to the one-sided mixed initial-boundary value problem. This approach was synthesized in a systematic way in the book of Li et al. [START_REF] Li | Exact boundary controllability of nodal profile for quasilinear hyperbolic systems[END_REF], where their also deal with the local exact boundary controllability of nodal profile on a tree-like network with general topology. For more details, see [START_REF] Gu | Exact boundary controllability of nodal profile for quasilinear hyperbolic systems in a tree-like network[END_REF], [START_REF] Wang | Exact boundary controllability of nodal profile for 1-d quasilinear wave equations[END_REF] and [START_REF] Wang | Exact boundary controllability of nodal profile for quasilinear wave equations in a planar tree-like network of strings[END_REF]. 1.2. General settings. Mathematically speaking, in this work we consider an incompressible, irrotational, inviscid and homogeneous fluid in a shallow water regime. The motion of the fluid is governed by the 1D nonlinear shallow water equations [START_REF] Lannes | On the dynamics of floating structures[END_REF], which

ζ(t, x) h 0 h 1 h w 0 l 0 + r l 1 s 0 Incoming waves Air z x l 0 -r -l h 2 Figure 1. Configuration. are given by (1)    ∂ t ζ + ∂ x q = 0, ∂ t q + ∂ x q 2 h + gh∂ x ζ = 0,
where ζ(t, x) is free surface elevation, h(t, x) is the fluid height given by h b + ζ with h b denoting the height of the bottom of the fluid, q(t, x) is the horizontal discharge defined by

(2) q(t, x) := ζ(t,x) -h b u(t, x, z)dz,
with u(t, x, z) denoting the horizontal component of the fluid velocity vector field. In the following, we consider the interval (-l, l 1 ) as spacial domain, and we assume that the height of the bottom h b is given by

h b = h 0 , x ∈ (-l, 0), h 1 , x ∈ [0, l 1 ),
where h 1 and h 0 are real constants such that h 1 < h 0 . Furthermore, we assume that (3) q h < gh.

For our analysis, we divide the domain of the problem (-l, l 1 ) into two parts: the interior domain I = (l 0 -r, l 0 + r), and its complement E = (-l, l 1 )\I, called exterior domain, and which is the union of three intervals

E 0 ∪ E 1 ∪ E 2 with (4) E 0 = (-l, 0), E 1 = (0, l 0 -r), E 2 = (l 0 + r, l 1 ). Let (5) ζ(t, x) = ζ i (t, x), x ∈ E i , i = 0, 1, 2, ζ w ∈ R, x ∈ I.
where ζ w is the parameterization of the bottom of the solid. Therefore, the fluid heights are defined by ( 6)

h(t, x) = h i (t, x) := hi + ζ i (t, x), x ∈ E i , i = 0, 1, 2, h w ∈ R, x ∈ I,
where hi (i = 0, 1, 2) are the fluid height at rest with h1 = h2 = h0 -s, and s is the height of the step.

h w = h1 + ζ w . (7) q(t, x) = q i (t, x), x ∈ E i , i = 0, 1, 2, q w (t), x ∈ I.
where ∂ t q w (t) = -hw ρ ∂ x P w with the fluid density ρ and an unknown surface pressure P w . Thus, the coupled system of nonlinear shallow water equations for x ∈ E i can be written as

(8)    ∂ t ζ i + ∂ x q i = 0, ∂ t q i + ∂ x q 2 i h i + gh i ∂ x ζ i = 0, i = 0, 1, 2,
with the boundary conditions given as

(9) x = -l : ζ 0 = f (t), (10)
x = l 1 : q 2 = 0, where f (t) denotes a prescribed boundary function or a boundary control to be determined.

In addition, we consider the transmission conditions developed in [START_REF] Bocchi | Modelling and simulation of a wave energy converter[END_REF], wich read as follows: [START_REF] Li | Exact boundary controllability of nodal profile for quasilinear hyperbolic systems[END_REF] x = 0 :

ζ 0 (t, 0) = ζ 1 (t, 0), q 0 (t, 0) = q 1 (t, 0), (12)
x = l 0 ± r :

       q 2 (t, l 0 + r) = q 1 (t, l 0 -r) = q w (t), q 2 2 2h 2 2 + gζ 2 x=l 0 +r - q 2 1 2h 2 1 + gζ 1 x=l 0 -r = -α d dt q w (t),
where α = 2r hw . The initial data of the system is given at rest, [START_REF] Li | Local exact boundary controllability for nonlinear vibrating string equations[END_REF] ζ(0, x) = 0, x ∈ E, ζ w , x ∈ I. and q(0, x) = 0.

1.3. Outline of the paper. In Section 2 we present the main results about exact controllability of final data given on the whole or a part of fluid domain at a given finite time T > 0 (see in §2.1), and exact controllability of nodal profile for a given demand at one end x = l 1 (See in §2.2). In §3.1 we prove some results relative to the well-posedness result of semi-global C 1 solutions, and in §3.2 we prove the main theorems of the paper by considering a constructive method with modular structure. Finally, in §2.3, we proposed an algorithm that synthesise the ideas of the proof of Theorem 2.4, which could be utilised to implement numerically our results on exact controllability of nodal profile.

Main results

Now we come to our results about control problems for the model given in [START_REF] Li | Controllability and observability for quasilinear hyperbolic systems[END_REF]. In order to ensure the well-posedness of quasilinear system ( 8) with ( 9)-( 12), we consider the exact controllability problem in the neighbourhood of the equilibrium state. In the following, Theorem 2.1 and Theorem 2.4 show the local exact controllability and of nodal profile near the rest state given as [START_REF] Li | Local exact boundary controllability for nonlinear vibrating string equations[END_REF], respectively. The corresponding results near other equilibrium state can be obtained in a similar way, we explain with some words in Remark 2.2.

2.1.

Exact controllability with one boundary control. The aim of exact controllability for system ( 8) is looking for a boundary control f (t) acting at x = -l though the boundary condition [START_REF] Li | Boundary value problems for quasilinear hyperbolic systems[END_REF], such that the prescribed final data (ζ iT (x), q iT (x)) is attained in the entire water regime at a given time T :

(14) U i (T, x) := (ζ i , q i )(T, x) = (ζ iT (x), q iT (x)), x ∈ E i , i = 0, 1, 2,
where the U i is generated by the control f as a solution of system ( 8)-( 12) with the initial condition:

(15) (ζ i , q i )(0, x) = (ζ i0 (x), q i0 (x)), x ∈ E i , i = 0, 1, 2.
In order to ensure the existence and uniqueness of piece-wise C 1 semi-global solution for this problem, here we give some assumptions on the initial data and final data in advance:

For i = 0, 1, 2, (S1) ζ i0 (x), q i0 (x) are C 1 functions with small norm (ζ i0 , q i0 ) C 1 [E i ]×C 1 [E i ] ; (S2) ζ i0 (x)
, q i0 (x) satisfy the C 1 compatibility at the points (t, x) = (0, -l), (0, l 1 ) and the piecewise C 1 compatibility at the adjoint points (t, x) = (0, 0), (0, l 0 -r), (0, l 0 + r); Accordingly, (S1')

ζ iT (x), q iT (x) are C 1 functions with small norm (ζ iT , q iT ) C 1 [E i ]×C 1 [E i ] ; (S2') ζ iT (x)
, q iT (x) satisfy the C 1 compatibility at the points (t, x) = (T, -l), (T, l 1 ) and the piecewise C 1 compatibility at the adjoint points (t, x) = (T, 0), (T, l 0 -r), (T, l 0 + r).

The following result states the existence of such a control, which can be constructed explicitly in Section 3. .

For any given initial data (ζ i0 , q i0 ) and final data (ζ iT , q iT ) satisfying the assumptions (S1)-( S2) and (S1')-(S2'), respectively, there exists a boundary control f (t) with small norm f (t) C 1 [0,T ] , such that mixed initial-boundary value problem for equation [START_REF] Li | Controllability and observability for quasilinear hyperbolic systems[END_REF] with the initial condition (15), the boundary conditions ( 9)-( 10) and the transmission conditions (11)-( 12) admits a unique piecewise

C 1 solution (ζ i , q i ) = (ζ i , q i )(t, x) (i = 0, 1, 2) with small piecewise C 1 × C 1 norm on the domain R i (T ) = {(t, x) | 0 ≤ t ≤ T, x ∈ E i },
which exactly satisfies the desired final condition (14).

In the above results, we consider only the steady-state at rest, but we can easily establish the corresponding local exact controllability in the neighborhood of a stationary subsonic continuously differentiable state.

Remark 2.2. The exact controllability result given in Theorem 2.1 still holds for initial states in a C 1 -neighborhood of a stationary subsonic continuously differentiable state

(q is , ζ is ), i = 0, 1, 2. For x ∈ E i , i = 0, 1, 2, the stationary solutions (q is (x), ζ is (x)) are given by (17) q is (x) = const, (18) d dx q 2 is hi + ζ is + g( hi + ζ is ) d dx (ζ is ) = 0.
Remark 2.3. The result given in Theorem 2.1 illustrates that after a finite time, the control given at one end x = -l can effect the state functions ζ and q in the entire space horizon x ∈ E. In fact, this result can be generalized to a special case where, the final condition ( 14) is replaced by a demand given on a part of the horizon:

(19) U 2 (T, x) := (ζ 2 , q 2 )(T, x) = (ζ 2T (x), q 2T (x)), x ∈ E 2 ,
where ζ 2T (x), q 2T (x) are regarded as given final state functions in the 'indoor water regime' E 2 . Now, the infimum controllability time (22) should be modified by a smaller lower bound:

(20) T > l g h0 + 2l 1 -l 0 -3r g h1
.

A limit case, with a given demand only at the end node x = l 1 , suggests a controllability problem of nodal profile, which will be shown in details in the following subsection.

2.2.

Exact controllability of nodal profile: a given demand at the end x = l 1 .

Stimulated by some practical applications, Gugat et al. [START_REF] Gugat | Flow control in gas networks: exact controllability to a given demand[END_REF] and Li [START_REF] Li | Exact boundary controllability of nodal profile for quasilinear hyperbolic systems[END_REF] proposed in 2010 another kind of exact boundary controllability, called the nodal profile control. This kind of controllability does not ask to exactly attain any given final state at a suitable time t = T by means of boundary controls, instead it asks the state to exactly fit any given profile on a node after a suitable time t = T by means of boundary controls.

For equation ( 8), we consider a given demand in fluid height and horizontal discharge at the end x = l 1 by :

(21) U 2 (t, l 1 ) := (ζ 2 , q 2 )(t, l 1 ) = ( h2 + ζ B (t), q B (t)), t ∈ [T, T ].
It's worth to mention that the given nodal profile function

U B := ( h2 + ζ B (t), q B (t))
should be compatible with the boundary condition [START_REF] Li | Exact boundary controllability of nodal profile for quasilinear hyperbolic systems[END_REF] in the time interval [0, T ] at the node x = l 1 , which implies that the demand q B must be set as 0. While, the other one, ζ B (t), can be chosen as any given C 1 function of time after a finite time T .

In the following, we give a positive answer to the exact controllability of nodal profile:

(ζ i , q i ) = (ζ i , q i )(t, x) (i = 0, 1, 2) with small C 1 × C 1 norm on the domain R i (T ) = {(t, x) | 0 ≤ t ≤ T,
x ∈ E i }, which exactly satisfies the given nodal profile condition at the end x = l 1 :

(23) ζ 2 (t, l 1 ) = ζ B (t), ∀t ∈ [T, T ].
Remark 2.5. In the above results, we consider only the steady state at rest, but we can easily replace it with any other steady state that is compatible with the boundary and transmission conditions (10)-( 12) and establish local exact controllability of nodal profile near a given subsonic continuously differentiable stationary state (q is (x), ζ is (x)), x ∈ E i , i = 0, 1, 2: There exists a C 1 -neighborhood of the stationary state such that for all initial data in this neighborhood that satisfies the C 1 -compatibility conditions and for any given smooth subsonic desired fluid height h2 + ζ B (t, l 1 ) and discharge profile q B (t, l 1 ) at the end x = l 1 that is in a sufficiently small C 1 -neighborhood of the boundary data corresponding to the stationary state we can construct a continuously differentiable control f = f (t) such that the demand is fulfilled exactly for all t ∈ [T, T ]. Moreover, this control generates a continuously differentiable system state in the entire domain.

Proofs

3.1. Existence and uniqueness of semi-global C 1 solution. We reduce the system in the exterior domain E to a compact form by introducing the couple U i = (ζ i , q i ) T , i = 0, 1, 2:

(24)

∂ t U i + A i (U i )∂ x U i = 0, x ∈ E i with (25) A i (U i ) = 0 1 gh i - q 2 i h 2 i 2q i h i
, which has two distinct eigenvalues:

(26) λ - i = q i h i -gh i < 0 < λ + i = q i h i + gh i
and the corresponding left eigenvectors can be taken as

(27) l - i = (-gh i - q i h i , 1) T , l + i = ( gh i - q i h i , 1) T ,
Introduce the Riemann invariants to the nonlinear shallow water equations (28)

       L i = 2 gh i (t, x) -g hi - q i h i , R i = 2 gh i (t, x) -g hi + q i h i . i = 0, 1, 2,
Then, (28) can be equivalently rewritten in (29)

     gh i = R i + L i 4 + g hi , q i h i = R i -L i 2 . i = 0, 1, 2, Thus, (30)
         ζ i = ( R i +L i 4 + g hi ) 2 g -hi , q i = R i -L i 2 ( R i +L i 4 + g hi ) 2 g . i = 0, 1, 2,
The 1D nonlinear shallow water equations in E can be rewritten as the following diagonal form:

(31) ∂ t L i + λ - i ∂ x L i = 0, ∂ t R i + λ + i ∂ x R i = 0, i = 0, 1, 2.
Obviously, for each i = 0, 1, 2, L i (t, x) is the Riemann invariant corresponding to the negative eigenvalue λ - i , while R i (t, x) is the Riemann invariant corresponding to the positive eigenvalue λ + i . λ - i (resp. λ + i ) is the entering (resp. departing) characteristic on the right-side boundary, while λ - i (resp. λ + i ) is the departing (resp. entering) characteristic on the left-side boundary. In order to guarantee the well-posedness of the mixed initial-boundary value problem on this coupled network, the boundary conditions on each boundary must satisfy (see [START_REF] Li | Boundary value problems for quasilinear hyperbolic systems[END_REF]):

(1) the number of the boundary conditions must be equal to that of the entering characteristics;

(2) the boundary conditions can be written in the form that the Riemann invariants corresponding to the entering characteristics can be explicitly expressed by all other Riemann invariants (corresponding to the departing characteristics).

The boundary conditions ( 9) and ( 10) can be rewritten in Riemann invariants as

(32) x = -l : R 0 = 2 g(f + h0 ) -g h0 - R 0 -L 0 2 .

and (33)

x = l 1 : R 2 -L 2 = 0, which imply that

(34) x = -l : L 0 = 3R 0 -4 g(f + h0 ) -g h0 and (35) x = l 1 : R 2 = L 2 ,
where R 0 and L 2 are the Riemann invariants corresponding to entering characteristic on x = -l and x = l 1 , respectively, and

(36) f C 1 [0,T ] small ⇐⇒ g(f + h0 ) -g h0 C 1 [0,T ]
small.

At x = 0, the Riemann invariants corresponding to entering characteristic are R 0 and L 1 . Let (37)

F 1 = ζ 0 -ζ 1 , F 2 = q 0 -q 1 .
By a direct calculation and noting ( 29)-(30), we have

(38) ∂F 1 ∂R 0 = ∂ζ 0 ∂R 0 = 1 g • 2( L 0 + R 0 4 + g h0 ) • 1 4 = √ gh 0 2g , Similarly, (39)
∂F 1 ∂L 1 = - ∂ζ 1 ∂L 1 = - √ gh 1 2g , ∂F 2 ∂R 0 = ∂q 0 ∂R 0 = h 0 2 + q 0 2h 0 g gh 0 = √ h 0 2 √ g ( gh 0 + q 0 h 0 ) > 0, ∂F 2 ∂L 1 = - ∂q 1 ∂L 1 = h 1 2 + q 1 2h 1 g gh 1 = √ h 1 2 √ g ( gh 1 + q 1 h 1 ) > 0, Thus, (40) ∂(F 1 , F 2 ) ∂(R 0 , L 1 ) (R 0 ,L 1 )=(0,0) = √ h 0 h 1 4g ( gh 1 + q 1 h 1 ) + √ h 0 h 1 4g ( gh 0 + q 0 h 0 ) (ζ 0 ,ζ 1 ,q 0 ,q 1 )=(0,0,0,0) = h0 h1 4g ( g h1 + g h0 ) = 0,
then by implicit function theorem, in a neighborhood of (R 0 , L 1 ) = (0, 0), the transmission conditions ( 11) can be equivalently rewritten as

(41) R 0 = F1 (R 1 , L 0 ), L 1 = F2 (R 1 , L 0 )
with F1 (0, 0) ≡ F2 (0, 0) ≡ 0.

Similarly, at x = l 0 ±r, the Riemann invariants corresponding to entering characteristic are R 1 and L 2 . Using the initial data and integral from 0 to t, the second transmission condition in [START_REF] Li | Semi-global c1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems[END_REF] can be replaced by a nonlocal condition as follows (42)

-αq w (t) = -αq 1 (0, l 0 -r) + t 0 q 2 2 2h 2 2 + gζ 2 x=l 0 +r - q 2 1 2h 2 1 + gζ 1 x=l 0 -r dτ = t 0 q 2 2 2h 2 2 + gζ 2 x=l 0 +r - q 2 1 2h 2 1 + gζ 1 x=l 0 -r dτ
Then, by the first transmission condition in [START_REF] Li | Semi-global c1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems[END_REF], we rewrite ( 12) into (43)

     q 1 (t, l 0 -r) = q 2 (t, l 0 + r), -αq 1 (t, l 0 -r) = t 0 q 2 2 2h 2 2 + gζ 2 x=l 0 +r - q 2 1 2h 2 1 + gζ 1 x=l 0 -r dτ.
Now, we introduce two boundary functions G 1 and G 2 :

(44)

     G 1 := q 1 (t, l 0 -r) -q 2 (t, l 0 + r), G 2 := αq 1 (t, l 0 -r) + t 0 q 2 2 2h 2 2 + gζ 2 x=l 0 +r - q 2 1 2h 2 1 + gζ 1 x=l 0 -r dτ.
In fact, by substituting (30) into (44), G 1 and G 2 can be also regarded as functions with respect to Riemann invariants R 1 , L 1 , R 2 , L 2 . For simplicity, we denote R 1 (t, l 0r), L 1 (t, l 0 -r) and R 2 (t, l 0 + r), L 2 (t, l 0 + r) by R 1 , L 1 and R 2 , L 2 , respectively, in the following expression:

(45)

G 1 = G 1 (R 1 , L 1 , R 2 , L 2 ) = R 1 -L 1 2 R 1 +L 1 4 + g h1 2 g - R 2 -L 2 2 R 2 +L 2 4 + g h2 2 g , (46)
G 2 = G 2 R 1 , L 1 , t 0 G(R 1 , L 1 , R 2 , L 2 ) dτ =α R 1 -L 1 2 ( R 1 +L 1 4 + g h1 ) 2 g + t 0 G(R 1 , L 1 , R 2 , L 2 ) dτ,
in which G is the function determined by

q 2 2 2h 2 2 + gζ 2 x=l 0 +r - q 2 1 2h 2 1 + gζ 1 x=l 0 -r
, and

t 0 G(R 1 , L 1 , R 2 , L 2 ) dτ is regarded as a new variable of G 2 . Thus, if (47)
∂(G 1 , G 2 ) ∂(R 1 , L 2 ) (R 1 ,L 2 )=(0,0) = 0,
then by implicit function theorem, (43) can be solved as

       R 1 (t) = Ĝ1 (L 1 , R 2 , t 0 G(L 1 , R 1 , L 2 , R 2 )dτ ), L 2 (t) = Ĝ2 (L 1 , R 2 , t 0 G(L 1 , R 1 , L 2 , R 2 )dτ ). (48)
Indeed, by calculation, we obtain (49)

∂G 1 ∂R 1 = ∂q 1 ∂R 1 = √ h 1 2 √ g ( gh 1 + q 1 h 1 ) > 0, ∂G 1 ∂L 2 = - ∂q 2 ∂L 2 = √ h 2 2 √ g ( gh 2 + q 2 h 2 ) > 0, ∂G 2 ∂R 1 = α ∂q 1 ∂R 1 = α √ h 1 2 √ g ( gh 1 + q 1 h 1 ) > 0, ∂G 2 ∂R 2 = 0. Therefore, (50)
∂(G 1 , G 2 ) ∂(R 1 , L 2 ) = -α √ h 1 h 2 4g ( gh 1 + q 1 h 1 )( gh 2 + q 2 h 2 ) < 0,
which implies (47). Using a method similar to [START_REF] Li | Boundary value problems for quasilinear hyperbolic systems[END_REF], we get the existence and uniqueness of piecewise local C 1 solution to the mixed initial-boundary value problem ( 8) and ( 9)- [START_REF] Li | Semi-global c1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems[END_REF]. And then, by a method similar to [START_REF] Li | Controllability and observability for quasilinear hyperbolic systems[END_REF], [START_REF] Li | Semi-global c1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems[END_REF], and the generalized result for a kind of non-local boundary condition as Lemma 4.1 in [START_REF] Li | Local exact boundary controllability for nonlinear vibrating string equations[END_REF], we can obtain the corresponding results on existence and uniqueness of piecewise semi-global C 1 solution.

Theorem 3.1. Let T > 0 be given. For any given initial data

(ζ i0 , q i0 ) with small norm ζ i0 (•), q i0 (•) C 1 [E i ]×C 1 [E i ] (i = 0, 1, 2) and boundary function f (t) with small norm f (•) C 1 [0,T ]
, satisfying the conditions (S1)-(S2), the forward mixed initial-boundary value problem of the shallow water system (8) on the connected water regime with the initial condition (15), the boundary conditions (9)-( 10) and the interface conditions (11)-( 12) admits a unique semi-global piecewise

C 1 solution U i = (ζ i (t, x), q i (t, x))(i = 0, 1, 2) with small norm i=0,1,2 (ζ i (•, •), q i (•, •)) C 1 [R i (T )]×C 1 [R i (T )] on the domain R(T ) = i=0,1,2 R i (T ) = i=0,1,2 {(t, x) | 0 ≤ t ≤ T, x ∈ E i }.
Similarly, for the backward mixed initial-boundary value problem, we have Theorem 3.2. Let T > 0 be given. For any given initial data (ζ iT , q iT ) and boundary function f (t) with small norm f (•) C 1 [0,T ] , satisfying the conditions (S1')-(S2'), the backward mixed initial-boundary value problem of shallow water system (8) with the final condition (14), the boundary conditions ( 9)-( 10) and the interface conditions (11)-( 12) admits a unique semi-global piecewise

C 1 solution U i = (ζ i (t, x), q i (t, x))(i = 0, 1, 2) with small norm i=0,1,2 (ζ i (•, •), q i (•, •)) C 1 [R i (T )]×C 1 [R i (T )] on the domain R(T ) = i=0,1,2 R i (T ) = i=0,1,2 {(t, x)|0 ≤ t ≤ T, x ∈ E i }.
3.2. Proof of exact controllability. Constructive method. In order to prove Theorem 2.1, by means of the constructive method with modular structure, it suffices to prove the following.

Lemma 3.3. Let T > 0 be defined by (22). For any given initial data (ζ i0 , q i0 ) and final data (ζ iT , q iT ) with small norm

(ζ i0 , q i0 ) C 1 [E i ]×C 1 [E i ] and (ζ iT , q iT ) C 1 [E i ]×C 1 [E i ] ,
the nonlinear system (8) with (9)-( 10) and ( 11)-( 12) admits a piecewise C 1 solution (ζ i , q i ) = (ζ i , q i )(t, x)(i = 0, 1, 2) with small C 1 × C 1 norm on the domain R i (T ), which satisfies simultaneously the initial condition [START_REF] Vergara-Hermosilla | Well-posedness and input-output stability for a system modelling rigid structures floating in a viscous fluid[END_REF] and the final condition [START_REF] Maity | Analysis of a simplified model of rigid structure floating in a viscous fluid[END_REF].

Proof. By ( 22) and h1 = h2 , there exists an 0 > 0 so small that (51)

1 2 T > sup |ζ 0 |+|q 0 |≤ 0 l g h0 + sup |ζ 1 |+|q 1 |≤ 0 l 0 -r g h1 + sup |ζ 2 |+|q 2 |≤ 0 l 1 -l 0 -r g h2 Let (52) T 1 = sup |ζ 0 |+|q 0 |≤ 0 l g h0 , T 2 = sup |ζ 1 |+|q 1 |≤ 0 l 0 -r g h1 , T 3 = sup |ζ 2 |+|q 2 |≤ 0 l 1 -l 0 -r g h2
.

Step 1:

On the domain R f = {(t, x)|0 ≤ t ≤ T 1 + T 2 + T 3 , x ∈ E} and R b = {(t, x)|0 ≤ t ≤ T -(T 1 + T 2 + T 3 ),
x ∈ E}, by solving the corresponding forward and backward mixed problem [START_REF] Li | Controllability and observability for quasilinear hyperbolic systems[END_REF] with the initial data (15) (or the final data ( 14)), boundary condition [START_REF] Li | Exact boundary controllability of nodal profile for quasilinear hyperbolic systems[END_REF] at x = l 1 , transmission conditions (11)- [START_REF] Li | Semi-global c1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems[END_REF] and artificial boundary condition (9) at x = 0 (in which f can be taken as any C 1 function with C 1 norm). By Theorem 3.1 and Theorem 3.2, there exist unique piecewise C 1 solutions U if = (ζ if , q if )(t, x) and U ib = (ζ ib , q ib )(t, x) (i = 0, 1, 2), respectively. Furthermore, we get the trace ζ 2f (t, l 1 ), q 2f (t, l 1 ) and ζ 2b (t, l 1 ), q 2b (t, l 1 ) at x = l 1 .

Step 4: Solve the following leftward problem from x = l 0 -r to x = 0 to get the solution

U 1 = U 1 (t, x) in the domain {0 ≤ t ≤ T, x ∈ E 1 }: (59)          ∂ t U 1 + A 1 (U 1 )∂ x U 1 = 0, x = l 0 -r : U 1 = ( ζ1 (t), q1 (t)), 0 ≤ t ≤ T, t = 0 : q 1 (0, x) = 0, x ∈ E 1 , t = T : ζ 1 (T, x) = ζ 1T (t), x ∈ E 1 ,
where ( ζ1 (t), q1 (t)) is obtained from Step 3 and ζ 2T (t) is given by the final data [START_REF] Maity | Analysis of a simplified model of rigid structure floating in a viscous fluid[END_REF].

The problem (59) is well-posed and admits a solution U 1 = (ζ 1 , q 1 )(t, x) on the domain R 1 := {(t, x)|0 ≤ t ≤ T, x ∈ E 1 }. Then, we get the values of trace ζ 1 (t, 0) and q 1 (t, 0).

Step 5: By the transmission conditions (11) at x = 0, it is easy to get (60) ( ζ0 , q0 ) := (ζ 0 (t, 0), q 0 (t, 0)) = (ζ 1 (t, 0), q 1 (t, 0)).

Similarly, we solve the leftward problem from x = 0 to x = -l to get the solution

U 0 = U 0 (t, x) in the domain {0 ≤ t ≤ T, x ∈ E 0 }: (61)          ∂ t U 0 + A 1 (U 0 )∂ x U 0 = 0, x = 0 : U 0 = ( ζ0 (t), q0 (t)), 0 ≤ t ≤ T, t = 0 : q 0 (0, x) = 0, x ∈ E 0 , t = T : ζ 0 (T, x) = ζ 1T (t),
x ∈ E 0 .

Then we take the boundary control f (t) as the trace ζ 0 (t, -l). We can verify that the solutions U i = U i (t, x), i = 0, 1, 2 constructed in the Step 2, 4, 5 satisfy all requirements of the Lemma.

Proof of Theorem 2.4.

Proof. The prescribed boundary data ζ B (t) and a fixed control function f (t) in (8) would generate an overdetermined initial-boundary value problem. To prove Theorem (2.4), we will find a piecewise C 1 solution (ζ i (t, x), q i (t, x)) for system (8) on the domain R(T ), which satisfies simultaneously the initial condition [START_REF] Vergara-Hermosilla | Well-posedness and input-output stability for a system modelling rigid structures floating in a viscous fluid[END_REF], the interface conditions ( 11)-( 12), the null flux boundary condition [START_REF] Li | Exact boundary controllability of nodal profile for quasilinear hyperbolic systems[END_REF], and the given nodal profile (23). Substituting this solution into the boundary condition [START_REF] Li | Boundary value problems for quasilinear hyperbolic systems[END_REF], we obtain the desired boundary control f (t).

We construct a solution to the control problem using the following steps explained in more detail below, where T 1 , T 2 and T 3 are defined as in (52).

Step 1: We construct the solution

U if = (ζ if , q if )(t, x) on the domain R f = {(t, x) | 0 ≤ t ≤ T 1 +T 2 +T 3 ,
x ∈ E} by solving system [START_REF] Li | Controllability and observability for quasilinear hyperbolic systems[END_REF] with the initial data [START_REF] Vergara-Hermosilla | Well-posedness and input-output stability for a system modelling rigid structures floating in a viscous fluid[END_REF], null flux boundary condition (10) at x = l 1 , transmission conditions (11)-( 12) and the boundary condition ζ 0 (t, -l) = F (t), where F is an artificial given function satisfying the C 1 -compatibility condition at (t, x) = (0, -l).

Step 2: From the forward solution U if , we denote the trace at x = l 1 as (ζ 2 (t, l 1 ), q 2 (t, l 1 )) = (ζ Bf (t), 0), thus ζ Bf is a C 1 function in the time interval [0, T 1 +T 2 +T 3 ] and q 2 (t, l 1 ) ≡ 0 is determined by the boundary condition. Thus, we can find a

C 1 function ζ2 (t) ∈ C 1 [0, T ] such that (62) ζ2 (t) = ζ Bf (t, l 1 ), 0 ≤ t ≤ T 1 + T 2 + T 3 , ζ B (t), T ≤ t ≤ T ,
where ζ B is the given demand.

Step 3: We first solve the leftward problem (53) in the domain R 2 to get the solution U 2 = (ζ 2 , q 2 )(t, x) and then do the same construction procedure shown as the Step 3, Step 4 and Step 5 in the proof of Lemma 3.3 to get solution U i = (ζ i , q i )(t, x), i = 1, 0 in the domain R 1 and R 0 , respectively.

Step 4: Using the trace ζ 0 (t, -l) from the solution U 0 = (ζ 0 , q 0 )(t, x), we finally compute the control function f (t) = ζ 0 (t, -l).

Step 5: Using the uniqueness theorem, we verify that: at time t = 0, U i (0, x) ≡ U if (0, x), x ∈ E i . Thus, the solution U i (i = 0, 1, 2) is the piecewise C 1 solution to solve the overdetermined initial-boundary value problem 3.4. Numerical algorithm. In this Subsection we present an algorithm (see in Algorithm 1) that could be utilised to implement numerically our results on the exact controllability of nodal profile of the system modelling the partially immersed structure considered. More precisely, we synthesize the ideas of the proof of Theorem 2.4 in a consecutive scheme, which gives rise to the following algorithm in order to obtain numerical approximations of the control f (t) associated with our problem. The implementation, however, is beyond the scope of this article and is deferred to a forthcoming publication.

Algorithm 1: Nodal profile controllability

Input: Initial data ζ(0, x), q(0, x), and a boundary condition ζ(t, -l) satisfying the C 1 -compatibility for the C 1 function at (0, -l). Output: Boundary control f (t).

Steps: 1: By considering the system [START_REF] Li | Controllability and observability for quasilinear hyperbolic systems[END_REF], to obtain the forward solution U if (t, x) on the domain R f (T ) and compute the trace at x = l 1 : (ζ 2 (t, l 1 ), q 2 (t, l 1 )) = (ζ Bf (t), 0). 3: By considering the system (53), to compute the leftward solution U 2 (t, x) on the domain R 2 and obtain the value of the traces ζ 2 (t, l 0 + r) and q 2 (t, l 0 + r) at x = l 0 + r. 4: By considering the values of the traces obtained in Step 3 and the transmission conditions ( 12) at x = l 0 ± r, obtain the trace q 1 (t, l 0 -r) and solve the nonlinear equation (57) in order to get the trace ζ 1 (t, l 0 -r). 5: By considering the system (59) and the values of the traces obtained in Step 4 to compute the leftward solution U 1 (t, x) on the domain R 1 . 6: By considering the solution obtained in Step 5 and the transmission conditions [START_REF] Li | Exact boundary controllability of nodal profile for quasilinear hyperbolic systems[END_REF] obtain the traces ζ 0 (t, 0) and q 0 (t, 0). 7: By considering the traces obtained in Step 6 and the system (61) obtain the leftward solution U 0 (t, x) on the domain R 0 and obtain the trace ζ 0 (t, -l). 8: To compute the control f (t) = ζ 0 by using the trace computed in the Step 7.

Theorem 2 . 4 .

 24 (Exact Controllability of Nodal Profile). Let let T be an arbitrarily given number satisfying T > T . Then for any given initial data (ζ i0 , q i0 ) satisfying assumptions (S1)-(S2), and for any given demand of the surface elevation ζ B (t) with small norm ζ B C 1 [T,T ] , there exists a boundary control f (t) ∈ C 1 [0, T ], such that the mixed initial-boundary value problem for equation (8) with initial condition (15), boundary conditions (9)-(10) and transmission conditions (11)-(12) admits a unique piecewise C 1 solution

2 :

 2 Considering the given demand ζ B to compute the values of the function ζ2(t) ∈ C 1 [0, T ] such that ζ2 (t) = ζ Bf (t, l 1 ), if 0 ≤ t ≤ T 1 + T 2 + T 3 or ζ2 (t) = ζ B (t), if T ≤ t ≤ T .

Acknowledgments

The first author has been partially supported by the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No.765579-ConFlex. The second and third authors are supported by DFG L595/31-1.

Step 2: Solve the following leftward problem from x = l 1 to x = l 0 + r to get the solution U 2 = U 2 (t, x) in the domain {0 ≤ t ≤ T, x ∈ E 2 }:

and ζ 2T (t) is given by the final data [START_REF] Maity | Analysis of a simplified model of rigid structure floating in a viscous fluid[END_REF].

The problem (53) is well-posed (by exchanging the role of t and x in hyperbolic system) and admits a solution

Then, we get the value of trace ζ 2 (t, l 0 + r) and q 2 (t, l 0 + r) at x = l 0 + r.

Step 3: With the known function ζ 2 (t, l 0 + r) and q 2 (t, l 0 + r), we try to determine the ζ 1 (t, l 0 -r) and q 1 (t, l 0 -r) by the transmission conditions (12) at x = l 0 ± r.

At first, by the continuity condition, we have (55) q 1 (t, l 0 -r) = q 2 (t, l 0 + r) and (56)

Substituting the value of q 1 (t, l 0 -r), q 2 (t, l 0 + r), d dt q w (t) and ζ 2 (t, l 0 + r) into the second formula of [START_REF] Li | Semi-global c1 solution to the mixed initial-boundary value problem for quasilinear hyperbolic systems[END_REF], it becomes an equation for ζ 1 (t, l 0 -r) that ( 57)

provided with the hyperbolicity near the rest state. Thus, we can solve (57) near the rest state and uniquely determine ζ 1 (t, l 0 -r) by Implicit Theorem.

We denote the values of ζ 1 (t, l 0 -r) and q 1 (t, l 0 -r) obtained in above as ζ1 (t) and q1 (t). Hence, we can verify that (1) ζ1 (t) and q1 (t) are both C 1 functions;

(2) U 1 = ( ζ1 (t), q1 (t)) is compatible with the value of U 1f (t, l 0 -r) and U 1b (t, l 0 -r) on the corresponding intervals.