Xavier Truant

Georges Cailletaud

Florent Fournier

Dit Chabert

Ivan Guillot

Serge Kruch
email: serge.kruch@onera.fr

Cyclic elastoplastic behaviour of 2198-T8 aluminium alloy welded panels

Keywords: Al 2 CuLi alloy, 2198-T8, Friction Stir Welding, Mechanical gradient behaviour, Digital Image Correlation, Cyclic plasticity Declarations

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Introduction

The FSW process was developed in 1991 by the Welding Institute [1]. FSW is a solid-state process without any filler material. The temperatures reached are below the melting temperature of the material. This makes it possible to weld hard weldable materials such as aluminium alloys of the 2XXX series [START_REF] Mishra | Friction Stir Welding and Processing[END_REF]. Today, it is used for airframes as an alternative to the riveting process. The third generation aluminium alloy 2198-T8 is considered in this study. The addition of lithium improves the mechanical properties compared to 2024 [START_REF] Alexopoulos | Fatigue behavior of the aeronautical Al-Li (2198) aluminum alloy under constant amplitude loading[END_REF] and reduces the weight of the structures. FSW welded samples have been tested under tensile mechanical loads in different studies [4, 5, 6, 7, 8 and 9]. Several approaches to determine the mechanical behaviour of a welded joint are proposed. One expensive and time-consuming approach is to mechanically test each zone using micro-samples cut across the joint [START_REF] Gallais | Joints soudés par friction malaxage d'alliages d'aluminium de la série 6xxx : caractérisation et modélisation[END_REF]. Another approach, adopted and developed in this study, proposes to take advantage of all the sophisticated features of the digital image correlation technique which allows the observation of the displacement field in different regions of the joint [4, 5, 7, 8 and 9].

Generally, experimental data are used to propose and identify constitutive equations to reproduce the mechanical behaviour observed on a 2198-T8 welded joint from simple tensile tests as in the work of Le Jolu [START_REF] Jolu | Influence des défauts de soudage sur le comportement plastique et la durée de vie en fatigue de soudures par friction-malaxage d'un alliage Al-Cu-Li[END_REF]. Here, we focus our analysis on the low cycle fatigue behaviour of welded joints which requires an additional degree of complexity in the modelling. It is recalled that a correct consideration of the cyclic behaviour of the material is a fundamental preliminary step in the modelling of fatigue crack initiation. However, this last point is not addressed in this paper. The observation of the microstructure of the base metal (BM) 2198-T8 allowed the quantification of T1 precipitates (Al 2 CuLi) with a transmission electron microscope [START_REF] Dorin | Mécanismes de durcissement structural par des précipités anisotropes dans un alliage Al-Cu-Li de troisième génération[END_REF]. The evolution of the T1 reinforcement precipitates is observed by Cavaliere [START_REF] Cavaliere | Al-Li plates joined by Friction Stir Welding: Mechanical and Microstructural Behavior[END_REF] in different areas of a 2198-T8 welded joint. An evolution of the number, shape and distribution of these precipitates is observed over the whole joint. In the work of Gao [13], this precipitation is also quantified for a welded joint in 2198-T8. A link between the yield strength of each zone of the joint and the resistance mechanisms, involved by precipitates or T1 dislocations, is proposed. The paper is divided in the following sections. First, experimental procedures are extensively described regarding the analysis of the microstructures along the welded joint as well as the mechanical tests and their instrumentations especially developed for these tests. Based on these experimental results, section 3 presents the mechanical model proposed to simulate as accurately as possible the cyclic behaviour of the welded joint

Experimental procedures and results

Welding Al-Cu-Li 2198-T8 samples

The 2198-T8 aluminium alloy under consideration is described in Table 1. The welding process is carried out at ONERA with a POWERSTIR 25T device from Crawford Swift Ltd, with a speed feed of 0.4. The tool used [START_REF] Krasnowski | Influence of the tool shape and weld configuration on microstructure and mechanical properties of the Al 6082 alloy FSW joints[END_REF] has a shoulder diameter of 13 mm and a pin diameter of 4.2 mm. The length of the pin is adjusted to be less than the thickness of the sheet which is about 3.18 mm. Welding is mainly carried out in the longitudinal direction of rolling, called L. However, since aluminium 2198-T8 is known to be highly anisotropic, welding is also carried out in the other two main rolling directions (called T for the transverse direction and D for the 45° direction). It is very important to study the effect of the welding direction in relation to the rolling direction of the sheet on the mechanical properties of the welded joints.

Elements

Regarding the width of the specimen affected by the welding process, it is possible to distinguish 4 zones each with a specific microstructure. From the left to right on the top of figure 1 (with a symmetry with respect to the centre of the welded zone), the base metal (BM) is the zone where the microstructure is not affected by the process. In the Thermal Affected Zone (TAZ), the microstructure is only influenced by the temperature. In the Thermo-Mechanical Affected Zone (TMAZ) an influence of temperature is observed as well as an important rotation of the microstructure. Finally, the Nugget Zone (NZ) is characterized by a completely recrystallized microstructure.

Microhardness analysis

Microhardness is measured through the welded joint with a Buehler device (Omnimet MHT 5101) and a 100g load. Measurements are taken halfway through the thickness of the sheet.

The evolution of the microhardness along the welded width, presented on figure 1, shows a maximum hardness in the base metal, away from the centre of the joint, of about 160 HV0.2. An important gradient is observed in the TAZ and TMAZ zones. Finally, the minimum hardness occurs in the nugget zone with a value of about 120 HV0.2. In order to describe more accurately this strong gradient the TAZ and TMAZ are divided in two parts. Based on these observations, the welded joint will be then characterized by 6 regions: BM, TAZ2, TAZ1, TMAZ2, TMAZ1 and NZ from the left to the right. To be more precise it is also convenient to distinguish the left hand side (advancing side) and the right hand side (retreating side), from the centre of the NZ, where the microstructure of the Thermo-Mechanically Affected zones may vary because of the rotation direction of the pin. However, from these first observations and the microhardness analysis, it is not possible to distinguish an influence of the rotation direction from both retreating and advancing sides

Microstructure quantification

Transmission electron microscopy (TEM) analyses were performed on 3 mm diameter and 750 µm disks were cut from different areas of the welded joint (figure 1). The slices were thinned down to about 100 µm by mechanical polishing and finally thinned by the classical twin-jet electropolishing apparatus with solution of nitric acid (1/3) in methanol (2/3) at -30 °C using a 12V tension. Microstructures are observed by transmission electron microscopy at 200kV with a JEOL 2000EX.

The three main precipitations most commonly observed on a very fine scale in the 2198 aluminium alloy in the T8 state are: '-Al3Li, '-Al2Cu and T1-Al2CuLi. The T1 precipitates that form very thin platelets of high aspect ratio on the {111} planes of the aluminium matrix are considered to play a key role in the strengthening of aged commercial Al-Cu-Li alloys [START_REF] Hardy | The phase sections at 500 and 350°C of Al rich Al-Cu-Li alloys[END_REF][START_REF] Steuwer | A combined approach to microstructure mapping of an Al-Li AA2199 friction stir weld[END_REF][START_REF] Gayle | Structure and properties during aging of an ultra-high strength aluminum-copper-lithium-silver-magnesium alloy[END_REF][START_REF] Donnadieu | Atomic structure of T1 precipitates in Al-Li-Cu alloys revisited with HAADF-STEM imaging and small-angle X-ray scattering[END_REF][START_REF] Deschamps | The influence of precipitation on plastic deformation of Al-Cu-Li alloys[END_REF][START_REF] Decreus | The influence of Cu/Li ratio on precipitation in Al-Cu-Li-x alloys[END_REF][START_REF] Zhang | Ageing response of a Al-Cu-Li 2198 alloy[END_REF][START_REF] Gao | Microstructural evolution and mechanical behavior of friction spot welded 2198-T8 Al-Li alloy during aging treatment[END_REF]. These T1 precipitates are roughly homogeneously distributed in the matrix and were easily imaged by conventional dark field TEM in different regions of the welded joint in order to be able to quantify them. The length and thickness of the T1 precipitates were measured from dark-field TEM micrographs of the particles viewed edge-on by selection of one diffuse streak corresponding to a (0002) T1 reflection parallel to the diffraction vector (111) of the aluminium matrix (Figure 1). The difficulty is to separate some precipitates that are too close, so the Fiji software [START_REF] Schindelin | Fiji: an open-source platform for biological-image analysis[END_REF] is used to extract the length as accurately as possible. The measurements were performed on ten dark field images from different grains of the thin foil, the values obtained correspond at least to the average of 1000 precipitates. For the base metal, i.e. far from the weld zone, the mean length of the precipitates T1 is 58±25 nm and the thickness 3.5±1.2 nm.

The values of the lengths and thicknesses of precipitate T1 measured in the thermal-affected zone 2 (TAZ2) are the same as in the BM (Figure 2.b), however the volume fraction decreases from 6.9% in the BM to 5.0% in TAZ2. In the TAZ1 zone, near the weld joint nugget, the photograph in Figure 2.C. shows that the number of T1 precipitates is much lower and that the microstructure is changing to smaller and thicker precipitates than in the BM.

It is then measured to be about 48.6 nm long and about 4.7 nm thick at a volume fraction of 2.7%. Precipitation is not quantified in TMAZ. Figure 3a shows a dark field of T1 precipitates as seen from above, which are almost hexagonal in shape. Figure 3b, which corresponds to a weak beam of the aluminium matrix in the same area, shows curved dislocations pinned on T1 precipitates. The precipitates are therefore bypassed by the dislocations and shear of these precipitates was not observed. The Nugget Zone (NZ) is characterized by total dissolution of the T1 precipitates in this zone, and there are only few helical dislocations (Figure 4) that originate from the precipitation on dislocations of vacancies formed at high temperature during welding. However, the thickness of the precipitates measured in TEM is only approximate as a result of the variation in the illumination of the precipitates in a dark field due to the presence of internal stresses in the thin foil. The results obtained in SAXS in the work of Decreus [START_REF] Decreus | Etude de la précipitation dans les alliages Al-Li-Cu de troisième génération : relations entre microstructures et propriétés mécaniques[END_REF] show a thickness of these precipitates of about 1.3 nm and a volume fraction of about 5.2% for the base metal. Consequently, the volume fraction, length and thickness of the precipitates are re-evaluated in the TAZ1 and TAZ2 zones in relation to the base metal data obtained in [START_REF] Decreus | Etude de la précipitation dans les alliages Al-Li-Cu de troisième génération : relations entre microstructures et propriétés mécaniques[END_REF]. The values are presented in Table 2. The density of dislocations is taken into account in this study thanks to the results of Gao [13]. Using X-ray diffraction and the Williamson-Hall method [START_REF] Williamson | X-ray line broadening from filed aluminium and wolfram[END_REF], a dislocation density of 2.1x10 14 m -2 and 10.1x10 14 m -2 can be measured in the nugget zone and in the base metal, respectively. Subsequently, these values will be used to determine a yield strength contribution in different areas of the welded joint.

Mechanical tests

The monotonic and cyclic tensile tests are performed at room temperature on a 100 kN LOS SYSTEM servohydraulic device from Losenhausen (Great Britain) Ltd. The cyclic tests are carried out on the same installation with a symmetrical load ratio (R = -1, stress controlled). An anti-buckling device has been designed as proposed by Khan et al. [START_REF] Khan | Experimental and numerical lifetime assessment of Al 2024 sheet[END_REF]. Effects of this device are not negligible and will be discussed below. For each sample, 4 successive load levels were tested (Figure 5). The base metal and welded joints were tested using either stress or strain control. For the welded specimens, digital image correlation (DIC) is used to measure the displacement fields, to calculate the total strains and thus to be able to obtain local stress-strain curves. This technique has already been used on inhomogeneous materials [START_REF] Reynolds | DIC for determination of weld and base metal constitutive behaviour[END_REF]. During the mechanical tests, images are taken with an Aramis 4M, CMOS, Titanar 5.6/100 camera (resolution 2352x1728 pixels). The grey level contrast is obtained by using a pattern of speckle marks on the surface of the sample, consisting of black and white spots. The ARAMIS [27] software is used to analyse the images, to measure the displacement fields and to calculate the strain fields. For DIC, subsets of 35x35 pixel images are chosen, which corresponds to an area of 0.6x0.6 mm2. Such a size of subsets is well suited to capture the gradient of mechanical properties across the welded joint with a fairly good accuracy.

Model identification

Experimental results

Figure 6 shows the resulting strain fields taken at two different times during the tensile test. Figure 7 plots the evolution of the longitudinal displacement and strain as a function of the position along the weld ligament, at different times. At the beginning of the tensile test, the displacement curve is linear across the weld joint. The experimental displacement data are evenly spaced and the deformation is homogeneous within the specimen. At the end of the tensile test, the shape of the displacement curve describes a sigmoidal curve. The displacement contours are no longer uniformly spaced. The strain in the region inside and near the weld varies from about 0.4 % outside the shoulder diameter to about 30 % in the centre of the nugget. The weld creates a strong inhomogeneity in the material properties and most of the plastic deformation is concentrated in the region of the weld. In summary, strains are localized in the weakest area of the sample which is, in our case, the centreline of the weld (Fig. 6, right). The stressstrain response evaluated by the local DIC at the centre of the weld shows a high level of plasticity while the base metal remains mainly elastic. DIC results make it possible to directly extract the constitutive stress-strain behaviour for each region of the welded joint. A first validation of these measurements consists in comparing an average of the local strains with the values recorded using a conventional extensometer (Figure 8a). Due to the fact that the local stress is not uniform and not measurable, it was decided to use the mean stress on the ligament (i.e. the force divided by the effective crosssection of the specimen) in order to be able to relate it to an experimental measurement. Despite the high strain levels that can be reached in the single tension tests, the strain plotted in all the figures of this paper is the engineering strain as it is explained in the next section of the paper.

The good correlation between the two measurements techniques (DIC and conventional) allows to validate the measurement of displacements and the calculation of the strain. Then the local strain values can be evaluated in other regions from the centre line of the joint (red curve in figure 8b) to the base metal (black curve in figure 8b). As seen previously, the centre of the welded joint exhibits a high level of plasticity while the base metal remains practically elastic. For the constitutive behaviour, the weld appears to have a lower yield strength and also a lower hardening modulus than the base metal. The experimental device being in place, cyclic tests are performed to characterize the local cyclic mechanical behaviour along the welded joint. The sample is loaded with several increasing cyclic loads, for instance 30 cycles at 300 MPa (global load), 20 cycles at 320 MPa, 10 cycles at 340 MPa and 5 cycles at 360 MPa. Figure 9 (left) shows the stress-strain behaviour of the nugget zone, taken directly from the DIC results. Again, the stress is assumed to be the value of the continuous engineering stress in each section of the specimen. For each stress level, only the first hysteresis loop is plotted for sake of clarity. As observed previously, the nugget area shows a high level of plasticity. Significant isotropic hardening is also observed in each zone of the welded joint. The evolution of the total plasticity range as a function of the number of cycles is presented in figure 9 (right). The nugget zone shows a greater isotropic hardening than the other regions. For the highest levels, it can be seen that the number of cycles performed is not sufficient to reach a stabilized hysteresis loop. These results will be improved with longer cyclic tests. These results validate the DIC technique to evaluate the stress-strain constitutive behaviour directly extracted from the different regions of the experimental sample. In particular, kinematic and isotropic hardenings can be clearly identified. However, the strain range over which the constitutive behaviour of a particular region of the weld is measured, is limited by the strength of the weakest area of the weld microstructure. Tensile test results show that, just prior to the specimen failure, the base metal remains almost elastic while the weld centreline has a high level of plasticity. Therefore, the evaluation of the cyclic constitutive behaviour of the base metal should be carried out on homogeneous samples without welds.

Mechanical behaviour modelling

The aluminium sheet is highly anisotropic due to the rolling manufacturing process [START_REF] Chen | Ductile Tearing of AA2198 Aluminum-Lithium Sheets for Aeronautic Application[END_REF]. Nevertheless, since we are considering, for the moment, samples with a welding direction in the rolling direction L, we decide to use, as a first approximation, an isotropic elastoplastic Chaboche's model using a Von Mises criterion [START_REF] Lemaitre | Mechanics of Solid Materials[END_REF][START_REF] Besson | Non-Linear Mechanics of Materials[END_REF][START_REF] Altenbach | Phenomenological yield and failure criteria[END_REF]. Equations are briefly presented below (bold symbols represent second order tensors).

The plastic criterion writes:

where represents the deviatoric part of the stress tensor. The plastic potential is characterized by a plastic threshold R 0 , 2 isotropic nonlinear hardenings of Voce type, Ri, and 3 terms of nonlinear kinematic hardening of Armstrong-Frederick type with backstresses Xj (there is no summation on repeated indices).

, , and are material parameters to be identified, is the cumulative plastic strain. Table 3 summarizes the results of the identification of these material parameters for the different areas of the welded joint as defined in section 2. Areas are chosen in order to reproduce as precisely as possible the gradient of the mechanical behaviour experimentally observed. The mechanical parameters are identified on a volume element, considering the local strains and the nominal stress. All the numerical developments have been implemented in the Finite Element suite Zset developed by Ecole des Mines and Onera. The results are presented and discussed in the next section with a finite element analysis on a 3D meshed sample.

Table 3 Identification of matrerial parameters

In order to achieve a good convergence of the numerical analysis, material parameters are expressly chosen not to vary excessively from one area to another. The plasticity threshold is 180 MPa in the nugget zone, the weakest area, and 340 MPa in the base metal, the strongest one. There is a fast (b=400) negative non-linear isotropic hardening 1 (NLIH 1) in the joint that enables a strong reduction of the elastic domain of the metal. The second NLIH 2, which is slower than the first one, allows a slow cyclic hardening in the centre of the joint (visible on figure 16) and disappears in the base metal. Different terms of non-linear kinematic hardenings (with backstresses X 1 , X 2 and X 3 ) are used to correctly reproduce the evolution of hysteresis loops during cycles which parameters are not influenced by the microstructure. As mentioned here before, the dissolution of the precipitates in the nugget zone induces a soft behaviour (see figure 1) characterized by both a lower yield stress and a stronger reduction of the yield surface as plasticity evolves.

Single tension tests performed on welded specimens show finite strains, particularly in the nugget zone. It would be necessary to perform the simulations in a finite strain formalism to be coherent with this observation. However, despite the fact that this formalism induces longer computational times and more difficult numerical convergences, the final goal of this study is to be able to predict the fatigue crack initiation in welded panels. The fatigue cyclic load levels are much smaller (see figure 16 for example) and it is not necessary to refine the constitutive equations to reproduce as well as possible experimental results for large strains. However the formalism introduced in the Zset finite element program is able to deal with finite strain formalisms with corotational formulations [START_REF] Green | A General Theory of Elastic-Plastic Continuum[END_REF][START_REF]On the Implementation of Inelastic Constitutive Equations with Special Reference to Large Deformation Problems[END_REF], but this option has not been considered in this project. More recent formalisms combining isotropic and kinematic hardenings in a finite strain framework are available in the literature [START_REF] Lion | Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological elements[END_REF][START_REF] Shutov | Finite strain viscoplasticity with nonlinear kinematic hardening: Phenomenological modeling and time integration[END_REF][START_REF] Vladimirov | Anisotropic finite elastoplasticity with nonlinear kinematic and isotropic hardening and application to sheet metal forming[END_REF][START_REF] Shutov | Models of nonlinear kinematic hardening based on different versions of the rate-independent Maxwell fluid[END_REF].

Connection between microstructure and mechanical parameters

The yield strength increment for each zone of the welded joint, including the elastic limit of the metal and the different nonlinear kinematic and isotropic hardenings, can be expressed as the sum of the precipitation strengthening (

), the dislocation strengthening ( ), the grain refinement strengthening ( ), the intrinsic yield strength of the metal ( ) and the solid solution elements ( ):

Firstly, the yield strength due to plates precipitates (T 1 ) can be evaluated with the equations proposed by Starink et al. [START_REF] Starink | Microstructure and strengthening of Al-Li-Cu-Mg alloys and MMCs: II. Modelling of yield strength[END_REF] and Seidman et al. [START_REF] Seidman | Precipitation strengthening at ambient and elevated temperatures oh heat-treatable Al(Sc) alloys[END_REF] (1)

(2)

The difference between equations 1 and 2 depends on the size of the precipitates. The Orowan mechanism states that precipitates are cut by dislocations if the mean radius of the precipitates is less than a critical radius, then equation 1 is used. If the mean radius of the precipitates is greater than this critical value, then the precipitates are bypassed by dislocations and equation 2 is used. The equations are composed by: M, the Taylor factor [START_REF] Seidman | Precipitation strengthening at ambient and elevated temperatures oh heat-treatable Al(Sc) alloys[END_REF], , the anti-phase boundary energy for precipitates [START_REF] Hansen | Boundary strengthening over five length scales[END_REF], b, the Burgers vector [START_REF] Woo | Grain structure and dislocation density measurements in a friction-stir welded aluminium alloy using X-ray peak profile analysis[END_REF], , the volume fraction of precipitates in each zone of the joint, , the mean radius of the precipitates and , the line tension of the dislocation [START_REF] Huang | Strengthening mechanisms associated with T1 particles in 2 Al-Li-Cu alloys[END_REF]. This critical radius drives the transition between the shearing and bypassing mechanisms. It is obtained when equations 1 and 2 are equal. Then, the yield strength (

) is obtained with:

Next, the contribution of the yield strength due to dislocation densities across the welded joint is calculated using the Taylor equation [START_REF] Hansen | Boundary strengthening over five length scales[END_REF]:

where  is the geometric factor of polycrystalline aluminium [START_REF] Huang | Strengthening mechanisms associated with T1 particles in 2 Al-Li-Cu alloys[END_REF], G is the shear modulus of aluminium [START_REF] Woo | Grain structure and dislocation density measurements in a friction-stir welded aluminium alloy using X-ray peak profile analysis[END_REF] and  is the dislocation density measured with X-ray spectrometry profiles. The values of dislocation densities used in this work are given by the work of Gao [13].

Thirdly, there is the contribution of the yield strength induced by the diminishing grain size in the material. This contribution is given by the Hall-Petch relation [START_REF] Armstrong | Past to Present Nanoscale Connections[END_REF]:

where K is a material parameter to be adjusted and D is the average grain diameter in each area of the joint. The smaller the diameter, the greater the strength. Therefore, yield strength is estimated only in the area of the nugget zone where the microstructure is fully recrystallized. Fourthly, the elements of the solid solution contribute to determine the yield strength in the joint. This contribution is determined by the Fleischer and Labusch equation [START_REF] Fleisgher | Solution hardening[END_REF][START_REF] Labusch | A statistical theory of solid solution hardening[END_REF].

where is the critical shear stress of the polycrystalline aluminium [START_REF] Fleisgher | Solution hardening[END_REF], k and z are material parameters, T is the temperature, E is the modulus of elasticity, is the jump frequency of a dislocation on a solid solution atom and is the initial dislocation velocity [START_REF] Labusch | A statistical theory of solid solution hardening[END_REF]. This contribution being too difficult to calculate, it is therefore estimated by an inverse technique knowing all the other contributions and the increment of the experimental yield strength.

Finally, the lattice stress is a constant parameter as proposed by Peierls-Nabarro [START_REF] Peierls | The size of a dislocation[END_REF][START_REF] Nabarro | Dislocations in a simple cubic lattice[END_REF] Parameters for T1 precipitates are not measured in the TMAZ1 and TMAZ2 zones, so a linearization of the precipitate contribution is performed between the TAZ and the Nugget zone. There is a logical decay from the base metal, where precipitation plays an important role in the hardening of the material, to the nugget zone, where the precipitation is fully dissolved. The contribution of the yield strength due to grain refinement is calculated only for the equiaxed grains in the nugget zone, because the grains are very long in the other zones and the Hall-Petch relationship depends on their diameter. Finally, the contribution of the yield strength of the solid solution elements increases slightly up to the centre of the joint. The value of this contribution is overestimated because many other types of precipitates are not taken into account. Nevertheless, the determination of these contributions to the material strength makes it possible to relate the mechanical parameters of the model to the evolution of the microstructure through the welded joint. These spatial evolutions allow to propose a constitutive model capable of reproducing the gradient of the mechanical properties and to take into account the effect of the microstructure in the different zones of the joint. This model is tested in the following section.

Finite element model validation

In a first step to validate the model, numerical simulations are carried out on a 3D sample with a perpendicular joint with respect to the loading direction. A first symmetry is carried out with the P1 plane to reduce the simulation time (figure 10). Moreover, the mechanical behaviour of the joint has only been identified for the forward side (AS), so the calculations consider a second symmetry with the P2 plane. However, the experimental results for the AS and the reverse side (RS) are compared with the simulations for each area of the welded joint.

The mesh used to calculate this sample is characterized by 23 511 nodes and 5 376 quadratic hexahedral and quadratic prismatic full integrated elements. A special procedure has been developed to introduce the evolution of the material parameters as a function of the position of the integration points in the structure in order to take into account the influence of the welded zone and to avoid strong discontinuities from one zone to another. This specific numerical procedure induced very stable calculations without convergence issues.

The anti-buckling device used during the experimental tests prevents the observation and analysis of the entire surface of the sample. Thus, only a useful area is chosen to compare experimental results (Figure 3). First of all, the tensile test is simulated with imposed force loading. Results of the strain fields are presented for three different times corresponding to three different mechanical behaviours of the welded joint (figure 11). At 50 s during the tensile test, the mechanical behaviour of the structure remains elastic. At 75 s, the welded joint is barely deformed, whereas the nugget zone is already plastically deformed by almost 2%. And finally, at 98 s, at the maximum load of the tensile test, the nugget zone reaches a plastic strain of 11.5%. 12. For a nominal stress of about 300 MPa, the sample surface presents a homogeneous triaxiality of about 0.3. This corresponds to a state of uniaxial stress in the different areas of the joint and confirms the choice of using the uniaxial nominal stress to identify the parameters of the constitutive equations. However, at the end of the test and in the TAZ, the triaxiality increases and the mechanical behaviour reaches a multi-axial state. In order to improve the mechanical model, the parameters must be identified again by considering the stresses in the three directions. Figure 13 shows a comparison of strain fields obtained experimentally with the DIC method and by the finite element calculation for the single tension test. The strain gradient is similar in both cases and the contours at 1% and 1.5% of deformation allow to distinguish a similar evolution of the deformation gradient between the nugget zone and the base metal. After this first validation, the strain fields can also be compared for a cyclic test. An example is shown in Figure 14 when the welded structure is loaded at the maximum compressive load. The nominal stress is approximately -300 MPa. The strain fields plotted for an overall strain of -0.5% and 0.1% allow defining two distinct zones. The first one gathers the maximum plastic strain in the nugget zone. The second one is located in the Thermalaffected zone with a maximum of the compressive strain about -0.7%. Results of the simulation show that this model is capable of reproducing the gradients of the mechanical behaviour of the joint even if the maximum strain levels don't match exactly the experimental results. D). The comparisons show no big differences between them up to a strain of about 7%, corresponding to a nominal stress of about 350 MPa, for the nugget zone (Figure 15 left). The isotropic criterion is adequate for this zone but does not allow to reproduce the maximum strain difference observed between the three directions L, T and D. These curves explain the difference in strain levels observed between the experimental and numerical strain fields (Figure 13). Within the TAZ1 zone, a significant difference appears for a nominal stress of about 300 MPa (Figure 15 on the right). Moving from the nugget zone, where the microstructure is fully recrystallized, to the base metal, which is strongly anisotropic, the assumption of an isotropic model is partially correct. For the analysis of the welded joint loaded in fatigue, a maximum plastic deformation of about 2% will be applied in the nugget zone. Thus, in this case, the identified isotropic model will be appropriate. For higher deformations, the limits of this model, as perceived through the tensile test, show a material too stiff for the TAZ1 zone.. Hereinafter, the cyclic test is simulated with a symmetrical load and 4 different load levels. The cyclic loading applied on each level allows the investigation of the isotropic hardening of the material. For the sake of clarity only the first and last cycle of each load level are plotted in these figures.

As it has been detailed before, the constitutive equations consist of three terms of non-linear kinematic hardening which allow a good representation of the hysteresis loops of the cycles. The first stress level in the nugget area (Figure 16) shows the possible effects of the anti-buckling device on the structure with a relatively poor representation of the material behaviour in compression. This device prevents the behaviour of the nugget area from reaching the minimum strain imposed experimentally in compression. Nevertheless, the difference in the plastic strain obtained during the unloading path is approximately the same experimentally and numerically. Another effect observed in the experimental result, but not reproduced numerically, is the change in the Young's modulus during the cyclic loading, which is more noticeable in the TAZ1 zone at the fourth stress level (Figure 17). The 2 non-linear isotropic hardenings used in the model reproduce correctly the closure of the loops for the first level. This allows to obtain the correct strains for the three following levels. Numerically, a saturation of the hardening is observed after about 30 cycles in the first level. Experimentally, a new hardening, of small amplitude, is observed for the other levels (see Figure 6), however, the model is already saturated and cannot reproduce this effect. Another difficulty arising during the identification and validation process of the constitutive equations relies in the flat hardening appearing close to the saturation. This particular situation may induce a large discrepancy on the strain levels for a small stress perturbation as it can be observed in figure 16 for the first cycle in compression or the 60th cycle in tension. Despite these difficulties, the results obtained with the proposed model, varying the material parameters along the different welded zones, are quite accurate.

Conclusion

This study is concerned with the cyclic mechanical characterization and modelling of 2198-T8 sheets welded by the FSW process. Microstructural analysis of the welded joints, combined with hardness measurements taken across the joint, showed significant changes in the microstructure resulting from this process. Transverse specimens 3.18 mm thick were machined from the welds and mechanically tested with the weld direction perpendicular to the loading direction. Tensile and cyclic tests were carried out at room temperature. Analysis of the results obtained by DIC confirmed that this technique is capable of providing the displacement field along the entire length of the welded joint and thus allows the calculation of the strain field. Based on these experimental results, an initially isotropic constitutive model of the welded joint was proposed and identified. The 3D finite element simulations of more complex loads have shown that this formalism is not sufficient to reproduce the mechanical behaviour of all the zones of the welded joint under multi-axial conditions. Next steps, already in progress [START_REF] Truant | Etude et modélisation du comportement mécanique de panneaux de structure soudés par frictionmalaxage (FSW)[END_REF], are as follows:

-propose more complex constitutive equations in order to reproduce both the isotropic behaviour of the nugget zone and the anisotropic behaviour of the base metal and the other zones of the structure.

-simulate the cyclic behaviour of welded structures (real components or complex biaxial specimens) with the new constitutive equations -analyse the impact of the friction stir welding process on the crack initiation and propagation

Fig 1

 1 Fig 1 Micro hardness through the welded joint and the thin foils taken in different region of the joint

Fig 2

 2 Fig 2 T1 precipitation, (a.) Base Metal (BM), (b.) Thermal Affected Zone 2 (TAZ2) and (c.) Thermal Affected Zone 1 (TAZ1)

Fig 3

 3 Fig 3 T1 precipitation in TAZ. (a.) dark field of T1 precipitates as seen from above, and (b.) [111] weak beam of the aluminium matrix.

Figure 4 :

 4 Figure 4: weak beam of the aluminium matrix, total dissolution of the T1 precipitation into the Nugget Zone (NZ)

Fig 5

 5 Fig 5 Anti-buckling device and the base metal specimen (left), overall mechanical behaviour of the base metal for symmetric loadings and four different strain levels (right)

Fig 6 Fig 7

 67 Fig 6 Results of strain fields at two different times (50 s and 98 s) during the tensile test

Fig 8 a

 8 Fig 8 a) Comparison between the DIC assessed strain values and those obtained by the conventional technique, b) Assessment of the local strain values at 3 different regions of the welded joint

Fig 9

 9 Fig 9 Local behaviour of the nugget zone obtained with DIC for symmetrical loads (left) and observation of cyclic hardening for three different zones in the welded joint and for four stress levels (right)

Fig 10

 10 Fig 10 Meshed specimen with boundaries conditions and observed zone

Fig 12

 12 Fig 12 Triaxiality stress calculated on the useful zone of the specimen

Fig 13

 13 Fig 13 Comparison of strain fields for a single tension test. Experimental (left) and numerical (right) results at 75s

Fig 14

 14 Fig 14 Comparison of strain fields for a cyclic test. Experimental (left) and numerical (right) results at 75s (compression load) Another way to verify the accuracy of the model with respect to experimental results is to analyze the local strain -nominal stress curves in different areas of the welded joint. The nugget zone, where strains are maximum (figure 15 on the left), and the TAZ1 zone, located far from the centre of the joint and therefore with less plastic strains (figure 15 on the right), are presented and compared in the different rolling directions of the material (T, L andD). The comparisons show no big differences between them up to a strain of about 7%, corresponding to a nominal stress of about 350 MPa, for the nugget zone (Figure15 left). The isotropic criterion is adequate for this zone but does not allow to reproduce the maximum strain difference observed between the three directions L, T and D. These curves explain the difference in strain levels observed between the experimental and numerical strain fields (Figure13). Within the TAZ1 zone, a significant difference appears for a nominal stress of about 300 MPa (Figure15on the right). Moving from the nugget zone, where the microstructure is fully recrystallized, to the base metal, which is strongly anisotropic, the assumption of an isotropic model is partially correct. For the analysis of the welded joint loaded in fatigue, a maximum plastic deformation of about 2% will be applied in the nugget zone. Thus, in this case, the identified isotropic model will be appropriate. For higher deformations, the limits of this model, as perceived through the tensile test, show a material too stiff for the TAZ1 zone..

Fig 15

 15 Fig 15 Nugget Point (Left) and TAZ1 Point (Right)

Fig 16

 16 Fig 16 Hysteresis loops in the nugget ZoneFinally, the hysteresis loops are less open in the TAZ1 zone (Figure17) than in the Nugget one where the strains are higher. As for the tensile test, the maximum strain concentrations are located in the Nugget Zone. This fact forces the TAZ1 zone to remain in a quasi-elastic state with very slightly open loops, making more difficult the identification of the mechanical parameters of the model. Improvements can be achieved to reproduce the full mechanical behaviour of the different zones of the joint. The first one would be to take into account the anisotropic behaviour of the thermal affected area and the base metal.

Fig 17

 17 Fig 17 Hysteresis loops in the Thermal Affected Zone

Table 1

 1

		Al	Cu	Li	Ag	Mg	Zr	Fe	Si	Zn	Ti
	wt%	Bal.	3.2	0.97	0.32	0.31	0.11	0.04	0.03	0.02	0.02

2198-T8 aluminium alloy composition in wt%

Table 2

 2 Microstructure

parameters in different zones of the joint (A and R refers to Advancing and Retreating sides respectively)

Table 4

 4 [13]. All the contributions are estimated and transcribed in table 4: Microstructure parameters in different zone of the joint

	(MPa)	Base Metal	TAZ 2	TAZ 1	TMAZ 2	TMAZ 1	Nugget Zone
			exp	488	488	448	428	398	388
			p		182	154	113	80	40	0
			l			30	30	30	30	30	30
		H 	P	0	0	0	0	0	102
			dis	145	145	145	145	145	66
			ss		131	159	160	173	183	190

Acknowledgements

The authors would like to acknowledge Constellium for providing aluminium sheet metal.

This research was supported integrally by Onera (Office National d'Etudes et de Recherches Aérospatiales)

Conflicts of interest/

The datasets generated during the current study are not publicly available due to legal restrictions.

Code availability (software application or custom code):

Numerical analysis was performed with the Finite Element suite Zset developed by


Swann Marx
email: swann.marx@ls2n.fr

Lucas Brivadis
email: lucas.brivadis@univ-lyon1.fr

Daniele Astolfi
email: daniele.astolfi@univ-lyon1.fr

Forwarding techniques for the global stabilization of dissipative infinite-dimensional systems coupled with an ODE

Keywords: Forwarding design, Abstract systems, Semigroup theory, Nonlinear systems

This paper deals with the stabilization of a coupled system composed by an infinite-dimensional system and an ODE. Moreover, the control, which appears in the dynamics of the ODE, is subject to a general class of nonlinearities. Such a situation may arise, for instance, when the actuator admits a dynamics. The openloop ODE is exponentially stable and the open-loop infinite-dimensional system is dissipative, i.e., the energy is nonincreasing, but its equilibrium point is not necessarily attractive. The feedback design is based on an extension of a finite-dimensional method, namely the forwarding method. We propose some sufficient conditions that imply the well-posedness and the global asymptotic stability of the closed-loop system. As illustration, we apply these results to a transport equation coupled with an ODE.

Introduction

This paper deals with the stabilization problem of systems in cascade, in which the first subsystem is an ordinary differential equation (ODE), and the second one is an infinitedimensional system. Another term to denote the structure under consideration is the feedforward-form, that has been studied deeply in the context of finite-dimensional systems, see, e.g., [START_REF] Mazenc | Adding integrations, saturated controls, and stabilization for feedforward systems[END_REF].

There exist many systems in such a form. One interesting example is the PIcontroller case [START_REF] Terrand-Jeanne | Adding integral action for open-loop exponentially stable semigroups and application to boundary control of pde systems[END_REF], where an additional finite-dimensional dynamic is added to achieve the regulation problem for an infinite-dimensional system. In our case, it is not added to obtain better performances: the finite-dimensional dynamics is imposed. It may represent, for instance, the dynamics of the actuator of the infinite-dimensional system to be controlled. Moreover, we suppose that our control is nonlinear. This situation may arise when considering saturating controllers [START_REF] Marx | Stability analysis of dissipative systems subject to nonlinear damping via Lyapunov techniques[END_REF][START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]. Finally, another example of application of the considered class of systems in feedforward form is the case in which an infinite-dimensional internal-model based regulator is used to solve a robust output regulation problem for a finite-dimensional system [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimum-phase systems[END_REF][START_REF] Califano | A stability analysis based on dissipativity of linear and nonlinear repetitive control[END_REF][START_REF] Paunonen | Stability and robust regulation of passive linear systems[END_REF][START_REF] Paunonen | Internal model theory for distributed parameter systems[END_REF][START_REF] Weiss | Repetitive control of MIMO systems using h∞ design[END_REF]. In this case, our proposed methodology can be employed to solve the stabilization problem.

Designing an explicit stabilizing feedback law in the infinite-dimensional context is not an easy task. To the best of our knowledge, very few is known nowadays. The backstepping method [START_REF] Krstic | Boundary Control of PDEs: A Course on Backstepping Designs[END_REF] has been proved to be applicable for many PDEs, even coupled PDE/ODE systems [START_REF] Auriol | Delay-robust stabilization of a hyperbolic PDE-ODE system[END_REF][START_REF] Tang | Stabilization for a coupled PDE-ODE control system[END_REF]. Note, however, that the method applies only on linear PDEs. It does not allow to stabilize globally nonlinear PDEs, but only locally around the desired equilibrium [START_REF] Cerpa | Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition[END_REF][START_REF] Kang | Boundary constrained control of delayed nonlinear Schrodinger equation[END_REF]. Moreover, the backstepping method gives raise to a kernel function representing the gain of the feedback law and which solves itself a PDE that might be difficult in some cases to compute numerically. It is also worthy mentionning the construction of Lyapunov functionals with Legendre polynomials for coupled PDE/ODE systems, which leads to a hierarchy of Linear Matrix Inequalities (LMIs), and that has been applied on many systems such as the transport/ODE system [START_REF] Safi | Tractable sufficient stability conditions for a system coupling linear transport and differential equations[END_REF], the wave/ODE system [START_REF] Barreau | Lyapunov stability analysis of a string equation coupled with an ordinary differential system[END_REF] or the heat/ODE system [START_REF] Baudouin | Stability analysis of a system coupled to a heat equation[END_REF]. Note however that the Legendre method has been only applied on linear systems in 1D.

In the semigroup theory (see, e.g., [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF] for an introduction to the linear theory and [START_REF] Miyadera | Nonlinear semigroups[END_REF] for the nonlinear one), there exists also some design method for the stabilization of linear systems, but it is difficult most of the time to obtain explicit gains for the feedback law. Let us mention the method provided in [START_REF] Urquiza | Rapid exponential feedback stabilization with unbounded control operators[END_REF], which allows to achieve rapid stabilization of the closed-loop system and which relies on the Grammian operator. This method has been applied on the linear Korteweg-de Vries (KdV) equation [START_REF] Cerpa | Rapid exponential stabilization for a linear Korteweg-de Vries equation[END_REF]. For coupled systems, it is also worthy mentionning some researchs related to the ouput regulation [START_REF] Paunonen | Stability and robust regulation of passive linear systems[END_REF][START_REF] Paunonen | Internal model theory for distributed parameter systems[END_REF], where some coupled systems arise when adding the dynamics, or even [START_REF] Feng | Actuator dynamics compensation in stabilization of abstract linear systems[END_REF], which deals with similar systems that the one we are facing with, but where the actuator dynamics is linear.

It is important then to emphasize on the fact that, in this paper, we aim at proposing a design methods to stabilize globally coupled systems admitting a class of input nonlinearities. A typical example of such a nonlinearity is the saturation, but we consider a larger class of nonlinearity, namely cone-bounded nonlinearities, as in [START_REF] Marx | Cone-bounded feedback laws for m-dissipative operators on Hilbert spaces[END_REF]. Note that very few is known about methods for the stabilization of nonlinear PDEs, but let us mention [START_REF] Marx | Global stabilization of a Korteweg-de Vries equation with a saturating distributed control[END_REF], which deals with the stabilization of a nonlinear KdV subject to a saturation, [START_REF] Chitour | L p -asymptotic stability analysis of a 1d wave equation with a nonlinear damping[END_REF][START_REF] Prieur | Wave equation with cone-bounded control laws[END_REF], which addresses the stabilization problem of some wave equations with saturated controller; or [START_REF] Marx | Stability analysis of dissipative systems subject to nonlinear damping via Lyapunov techniques[END_REF][START_REF] Slemrod | Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control[END_REF], which builds Lyapunov functionals for abstract systems. However, these papers do not deal with coupled systems. Let us however mention [START_REF] Daafouz | Nonlinear control of a coupled pde/ode system modeling a switched power converter with a transmission line[END_REF], where a coupled PDE/ODE system is considered with a saturated input.

As mentionned earlier, our method is nonlinear. It is based on the so-called forwarding method, which has been first introduced in the finite-dimensional context [START_REF] Benachour | Forwarding design with prescribed local behavior[END_REF][START_REF] Kaliora | On the stabilization of feedforward systems with bounded control[END_REF][START_REF] Mazenc | Adding integrations, saturated controls, and stabilization for feedforward systems[END_REF], and it applies on cascade systems, with particular stability properties: the first subsystem needs to be globally stabilizable at the origin, while the second one is dissipative. The forwarding method has been applied in other contexts than the stabilization one, such as in the design of integral-based controllers [START_REF] Astolfi | Integral action in output feedback for multi-input multi-output nonlinear systems[END_REF]. To the best of our knowledge, this method has been extended in an infinite-dimensional setting only in [START_REF] Terrand-Jeanne | Adding integral action for open-loop exponentially stable semigroups and application to boundary control of pde systems[END_REF], but, as mentioned earlier, the context is different.

The forwarding method that we use gives raise to a gain which is a solution to an infinite-dimensional version of the well-known Sylvester equation. We show in this paper that this gain is not difficult to obtain in some simple cases, since it can be obtained as the solution of some well-know boundary value problems. Our method needs some approximate observability condition to apply the LaSalle's Invariance principle. This condition might seem difficult to check in general, but we also provide a sufficient condition for having such a property provided that the infinite-dimensional system under consideration is conservative (but not attractive at the origin). This condition is known in the output-regulation context as a nonresonance condition see, e.g., [START_REF] Isidori | Robust autonomous guidance: an internal model approach[END_REF]Chapter 1.4]). Let us however mention that we are not able to impose a decay rate for the trajectory of the closed-loop system, in contrast with the backstepping method.

The paper is organized as follows. In Section 2, we introduce the system under consideration together with the functional setting which will be used all along the paper. In Section 3, we explain the forwarding approach and introduce the infinitedimensional version of the Sylvester equation. In Section 4, we state and prove the well-posedness of the closed-loop system by using some nonlinear semigroup theory results. In Section 5, we state and prove the global asymptotic stability result. Section 6 provides and proves a sufficient condition for the approximate observability in infinite for our system in cascade. Finally, Section 7 collects some further research lines to be followed.

Notation: Set R + = [0, ∞). We denote by | • | the Euclidean norm and by • the induced matrix norm. Given two Hilbert spaces H 1 and H 2 , the space L(H 1 , H 2 ) denotes the space of functions bounded from H 1 to H 2 , and L(H 1 ) = L(H 1 , H 1 ). Given N ∈ N, for a function w : (t, x) ∈ R + × [0, 1] → w(t, x) ∈ R N , the notation w t (resp. w x ) denotes the partial derivative of w with respect to the variable t (resp. with respect to the variable x). We keep this notation both for the weak and the strong definition of partial derivatives.

Problem Statement

Let H be a Hilbert space equipped with a scalar product •, • H and the corresponding norm • H . In this paper we are interested in the stabilization (at the origin) problem for systems that can be described as a cascade of two systems reading as follows

           d dt z = Az + Bσ(u), y = Cz, d dt w = Sw + Γ y, z(0) = z 0 , w(0) = w 0 , (1)
where In the following, we use the state space X := R n × H equipped with the norm defined by (z, w) X = |z| + w H . We also consider the following set of assumptions, on which our results rely.

A ∈ R n×n , B ∈ R m×n , C ∈ R n×p , u ∈ R m is

Assumption 1

The following statements hold.

(i) The operator S : D(S) ⊆ H → H generates a strongly continuous semigroup of contractions, that is denoted by (T(t)) t≥0 . (ii) The matrix A is Hurwitz. (iii) The spectra of S and A are disjoint and nonempty.

In system (1), the state component z lives in R n , while w is a (possibly infinitedimensional) state living in the Hilbert space H. System (1) can be viewed as an infinite-dimensional control system in which the z-dynamics represent the actuator's dynamics. This class of systems may arise also in output regulation and repetitive control problems (see, e.g., [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimum-phase systems[END_REF][START_REF] Paunonen | Internal model theory for distributed parameter systems[END_REF][START_REF] Weiss | Repetitive control of MIMO systems using h∞ design[END_REF]): in these cases, the w-dynamics represents the state of a dynamical feedback which is used to achieve a desired control goal for the controlled finite-dimensional z-subsystem.

Definition 1 (Cone-bounded nonlinearity)

A continuous function σ : R m → R m is said to be a cone-bounded nonlinearity if it satisfies the following properties.

1. For all s ∈ R m , σ(s) = 0 if and only if s = 0.

It is monotone1, which means that, for every

s 1 , s 2 ∈ R m (σ(s 1 ) -σ(s 2 )) (s 1 -s 2 ) ≥ 0.
(2)

3. It is linearly bounded, that is: there exists a positive constant L such that, for all

s ∈ R m |σ(s)| ≤ L|s|. (3)
The following examples are borrowed from [START_REF] Marx | Cone-bounded feedback laws for m-dissipative operators on Hilbert spaces[END_REF].

Example 1 (Examples of cone-bounded nonlinearities)

1. Any linear mapping σ(s) = µs, where µ is a positive constant, is a cone-bounded nonlinearity. 2. The so-called saturation function for s ∈ R:

sat ū(s) :=      -ū if s ≤ -ū, s if |s| ≤ ū, ū if s ≥ ū, ∀s ∈ R ( 4
)
where ū is a positive constant, is a cone-bounded nonlinearity. By using (4), we can also define a saturation function when s ∈ R m as follows:

sat U (s) :=    sat ū1 (s 1 ) . . . sat ūm (s m )    , ∀ s := (s 1 , . . . , s m ) ∈ R m (5)
where U := (ū 1 , . . . , ūm ) is a vector of positive constants ūi > 0, for all i = 1, . . . , m. The saturation function ( 5) is still a cone-bounded nonlinearity.

The function

σ(s) : s ∈ R → sat ū(ϕ(s)), (6)
where ū > 1 and ϕ is defined as follows

ϕ(s) :=      -|s| -1 -1 if s < -1, s if |s| ≤ 1, √ s -1 + 1 if s > 1 (7)
is also a cone-bounded nonlinearity, but it is not globally nor locally Lipschitz, because of the square root function in its definition.

Remark 1 (About A Hurwitz) It is well-known that in presence of input nonlinearities, global stabilization of the origin of a finite-dimensional system of the form

d dt z = Az + Bσ(u)
is not possible without any restrictive condition on the spectrum of A and the stabilizability property of the pair (A, B). For instance, if σ is a saturation function, a necessary condition is that the eigenvalues of A must not have positive real parts [36, Chapter 1.6.2.1, Theorem 1.2]. In our context, due to the presence of the infinite-dimensional properties of system (1), we suppose a slightly more stringent condition in Assumption 1-(ii), that is, A Hurwitz. An extra condition on B will be required later on, in Theorem 2. Note, however, that, in absence of input nonlinearity, that is, σ(u) = u, Assumption 1-(ii) can be replaced by supposing the pair (A, B) to be stabilizable.

We provide now an example of a transport equation coupled with an ODE which corresponds to our functional setting. Such an example will be also used in the following sections as an illustration of the proposed design.

Example 2 (Transport equations coupled with an ODE) Consider

           ż(t) = Az(t) + Bσ(u(t)), t ∈ R + w t (t, x) + Λw x (t, x) = 0, (t, x) ∈ R + × [0, 1] w + (t, 0) = D 0 w -(t, 0), t ∈ R + , w -(t, 1) = D 1 w + (t, 1) + Cz(t), t ∈ R + z(0) = z 0 , w(0, x) = w 0 (x), x ∈ [0, 1], (8)
where The state space is chosen to be

Λ = diag(λ 1 , . . . , λ N ) is such that λ i > 0 (resp. λ i < 0),
X := R n × H, H = L 2 (0, 1; R N ).
The operator S is given by Sw := -Λw with domain D(S)

:= {w ∈ H 1 (0, 1; R N ) | w + (0) = D 0 w -(0), w -(1) = D 1 w + (1)
}, and the operator Γ is the delta function at x = 1 in L(R p , H -1 ), i.e., v, Γ y D(S * ),H-1 = v(1)y for all y ∈ R p and v ∈ D(S * ) where •, • D(S * ),H-1 is the dual product. Now, under some assumptions on the matrices D 0 and D 1 to be given later on, let us prove that Asssumption 1 (i) is satisfied. First, by [32, Theorem 3.1.], the operator S generates a strongly continuous semigroup (for all D 0 and D 1 ). Then, S is dissipative if and only if S generates a strongly continuous semigroup of contractions. To do so, let us introduce the following real inner product on H: for all w 1 , w 2 ∈ H,

w 1 , w 2 H := 1 0 w 1 (x) Λw 2 (x)dx, Λ := diag 1 |λ 1 | , . . . , 1 |λ N | .
Showing that S is dissipative reduces to prove that Sw, w H ≤ 0. We have

Sw, w H = - 1 0 (w + ) (x) w + (x)dx + 1 0 (w -) (x) w -(x)dx,
One deduces by integration by parts that

Sw, w H = 1 2 w -(0) D 0 D 0 w -(0) -w -(0) w -(0) + 1 2 w + (1) D 1 D 1 w + (1) -w + (1) w + (1) .
Hence, if matrices D 0 and D 1 satisfy D 0 D 0 -I M 0 and D 1 D 1 -I N -M 0, one can deduce that S is dissipative. Conversely, if these matrix inequalities are not satisfied, one can pick w -(0) and w + (1) such that Sw, w H ≥ 0, hence S does not generate a strongly continuous semigroup of contractions.

Our objective in this paper is to propose a design procedure to stabilize [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimum-phase systems[END_REF]. We follow a forwarding strategy, that has been applied in the finite-dimensional context in many situations [START_REF] Mazenc | Adding integrations, saturated controls, and stabilization for feedforward systems[END_REF], and that has been extended recently to some infinite-dimensional systems in a different context [START_REF] Terrand-Jeanne | Adding integral action for open-loop exponentially stable semigroups and application to boundary control of pde systems[END_REF].

The Forwarding Approach

Inspired by the forwarding strategy and in particular by the change of coordinate approach proposed in [25, Section IV], we look for an operator M :

R n → H -1 satisfying SM -M A = -Γ C, (9)
where S is understood as the extension of the operator S in H -1 . Since the spectra of A and S are disjoint (see Assumption 1 (iii)), we conclude that there exists a unique M : R n → H -1 solution of ( 9), see [

∈ R n SM z -M Az -µM z + M zµ = -Γ Cz.
By a simple computation, one can prove that, for all z ∈ R n ,

M z = (µI H -S) -1 M (µI n -A)z + (µI H -S) -1 Γ Cz. (10)
Since (µI H -S) -1 ∈ L(H -1 , H), one can deduce from the above expression that M z ∈ H for all z ∈ R n .

Example 3 (Sylvester equation for (8)) In Example 2, the Sylvester equation ( 9) can be rewritten as follows:

   -ΛM (x) -M (x)A = 0, x ∈ [0, 1] M + (0) = D 0 M -(0) M -(1) = D 1 M + (1) + C
The solution M : [0, 1] → R N ×m to this two-boundary value problem can be computed explicitly using some vectorization operators. Consider the case of the one-dimensional transport equation driven by a scalar ODE:

       ż = -az + σ(u), t ∈ R + w t (t, x) + λw x (t, x) = 0, (t, x) ∈ R + × [0, 1] w(t, 0) = w(t, 1) + cz, t ∈ R + z(0) = z 0 , w(0, x) = w 0 (x), x ∈ [0, 1], (11)
with a, c and λ three positive constants. System [START_REF] Cheverry | Handbook of Spectral Theory[END_REF] can be seen as a subcase of (8) with N = 2 and n = m = p = 1.

Indeed, taking λ = 2λ 1 = -2λ 2 , D 0 = D 1 = 1, A = -a, B = 1, C = c and w 0 (x) = w + 0 (2x) if x < 1 2 , w - 0 (2(1 -x)) if x > 1 2 ,
any (z, (w + , w -)) satisfies ( 8) if and only if (z, w) is a solution of [START_REF] Cheverry | Handbook of Spectral Theory[END_REF], where

w(t, x) = w + (t, 2x) if x < 1 2 , w -(t, 2(1 -x)) if x > 1 2 .
Then M is the unique solution of

λM (x) = aM (x), x ∈ [0, 1] M (0) = M (1) + c, (12)
i.e., for all x ∈ [0, 1],

M (x) = c 1 -exp a λ exp a λ x ,
Now go back to the general case [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimum-phase systems[END_REF].The fact that A is Hurwitz implies that there exists a symmetric positive definite matrix P satisfying

P A + A P = -I R n . (13)
We denote by p min > 0 (resp. p max > 0) its smallest (resp. largest) eigenvalue. Following the forwarding approach, let us first introduce the following candidate Lyapunov functional

V (z, w) := z P z + w -M z 2 H , ( 14
)
where M is the operator previously defined in [START_REF] Cerpa | Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition[END_REF]. Then V induces a scalar product on X defined by

z 1 w 1 , z 2 w 2 V := z 1 P z 2 + w 1 -M z 1 , w 2 -M z 2 H . (15)
The norm induced by V , denoted in the following by • V , is equivalent to • X . Indeed, for all (z, w) ∈ X, we have:

V (z, w) = z P z + w 2 H + M z 2 H -2 w, M z ≤ p max |z| 2 + 2( w 2 H + M z 2 H ) ≤ max 2, p max + 2 M 2 L(R n ,H) (z, w) 2 X and V (z, w) = z P z + w 2 H + M z 2 H -2 w, M z ≥ p min |z| 2 + (1 -ε) w 2 H + 1 - 1 ε M z 2 H ≥ p min + 1 - 1 ε M 2 L(R n ,H) |z| 2 + (1 -ε) w 2 H ≥ min p min 2 , p min p min + 2 M 2 L(R n ,H) (z, w) 2 X by choosing ε = 2 M 2 L(R n ,H) p min + 2 M 2 L(R n ,H) .
Let (z, w) be a sufficiently regular solution to (1), i.e., (z, w) ∈ D(A), where

D(A) := {(z, w) ∈ X | Sw + Γ Cz ∈ H}. (16)
Since (T(t)) t≥0 is a strongly continuous semigroup of contractions, S is a dissipative operator. Then,

d dt V (z, w) = 2z P (Az + Bσ(u)) + Sw -M Az + Γ Cz, w -M z H + w -M z, Sw -M Az + Γ Cz H -M Bσ(u), w -M z H -w -M z, M Bσ(u) H , = -z z + 2z P Bσ(u)
(by ( 9) and ( 13))

+ Sw -SM z, w -M z H + w -M z, Sw -SM z H -M Bσ(u), w -M z H -w -M z, M Bσ(u) H , ≤ -z z + 2z P Bσ(u) (since S is dissipative) -M Bσ(u), w -M z H -w -M z, M Bσ(u) H , ≤ -z z + 2 z P -(M * (w -M z)) Bσ(u). (17)
Hence, a natural candidate control u is:

u = -B [P z -M * (w -M z)] (18)
yielding d dt V (z, w) ≤ -z z -2u σ(u) ≤ 0.
The previous inequality being not strict, i.e., we don't have any negative term in (z, w) X , we cannot conclude any attractivity result without a more precise analysis.

This will be done in Section 5. The closed-loop system (1)- [START_REF] Kaliora | On the stabilization of feedforward systems with bounded control[END_REF] given by

           d dt z = Az + Bσ -B (P z -M * (w -M z)) , d dt w = Sw + Γ Cz, z(0) = z 0 , w(0) = w 0 , (19)
can be rewritten as

   d dt ζ = A(ζ), ζ(0) = ζ 0 , (20)
where ζ = z w and

A(ζ) = Az + Bσ -B (P z -M * (w -M z)) Sw + Γ Cz .
The domain of A is D(A) (given in ( 16)) equipped with the graph norm. In the next section, we first study the existence and uniqueness of solutions of [START_REF] Kang | Boundary constrained control of delayed nonlinear Schrodinger equation[END_REF].

Example 4 (Control design for (11)) For system [START_REF] Cheverry | Handbook of Spectral Theory[END_REF], the control law ( 18) can be explicitly design by using the function M defined in [START_REF] Chitour | L p -asymptotic stability analysis of a 1d wave equation with a nonlinear damping[END_REF]. In particular, in this case, M * is given by M * : L 2 (0, 1) w → 1 0 M (x)w(x)dx ∈ R. Hence, the feedback (18) reads, for all t ≥ 0

u(t) = - 1 2a z + 1 0 M (x)[w(t, x) -M (x)z]dx.
See also [START_REF] Marx | Forwarding design for stabilization of a coupled transport equation/ode with a cone-bounded input nonlinearity[END_REF].

Remark 2 (Nonlinear design and small control) Note that the feedback law ( 18) can be also modified in

u = ψ -B P z -M * (w -M z) ,
with ψ being any desired function so that the composition σ •ψ is still a cone-bounded function satisfying Definition 1. Indeed, the resulting closed-loop system would still read as [START_REF] Kang | Boundary constrained control of delayed nonlinear Schrodinger equation[END_REF] and all the forthcoming results apply. For instance, if u takes value in R, it suffices to select ψ as any cone-bounded function satisfying Definition 1. Recall indeed that the composition of two monotonic functions f, g : R n → R, is still a monotonic function. The other two properties in Definition 1 are trivially verified. Such a choice can be of particular interest for the design of saturated feedback laws with small magnitude (that is, small energy), see, e.g., [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]. Indeed, by selecting ψ as a saturation function of the form (5), it turns out that the saturation level Ū can be chosen arbitrarily small to reduce the efforts on the actuators (although, as a drawback, this may also sensitively reduce the converge rate). Such a feature, highly desirable from a practical point of view, is a typical feature of forwarding approach for finite-dimensional systems (see, e.g., [START_REF] Kaliora | On the stabilization of feedforward systems with bounded control[END_REF]) which is still preserved in our context, and that is very hard to retain when considering pole-placement techniques (e.g., [START_REF] Feng | Actuator dynamics compensation in stabilization of abstract linear systems[END_REF][START_REF] Paunonen | Internal model theory for distributed parameter systems[END_REF]) or backstepping design (see, e.g., [START_REF] Auriol | Delay-robust stabilization of a hyperbolic PDE-ODE system[END_REF][START_REF] Krstic | Boundary Control of PDEs: A Course on Backstepping Designs[END_REF][START_REF] Tang | Stabilization for a coupled PDE-ODE control system[END_REF]) in the infinite-dimensional framework.

Well-posedness Result

We are now in position to state our first result, which deals with the well-posedness of system [START_REF] Kang | Boundary constrained control of delayed nonlinear Schrodinger equation[END_REF].

Theorem 1 (Well-posedness and Lyapunov stability of ( 19)) Suppose Assumption 1 is satisfied. Then, the following statements hold.

1. For every initial conditions (z 0 , w 0 ) ∈ X, there exists a unique weak solution (z, w) ∈ C 0 (R + ; X) to [START_REF] Kang | Boundary constrained control of delayed nonlinear Schrodinger equation[END_REF]. Moreover, for all t ≥ 0,

(z(t), w(t)) V ≤ (z 0 , w 0 ) V . ( 21
)
2. For every initial conditions (z 0 , w 0 ) ∈ D(A), there exists a unique strong solution [START_REF] Kang | Boundary constrained control of delayed nonlinear Schrodinger equation[END_REF]. Moreover, for all t ≥ 0,

(z, w) ∈ C 1 (R + ; X) ∩ C 0 (R + ; D(A)) to
(z(t), w(t)) 2 V + A(z(t), w(t)) 2 V ≤ (z 0 , w 0 ) 2 V + A(z 0 , w 0 ) 2 V . ( 22
)
If one proves that the operator A defined in ( 20) is a m-dissipative operator on (X, • X ), according to the definition below, one can apply the result provided by [START_REF] Miyadera | Nonlinear semigroups[END_REF]Corollary 3.7,Theorem 4.20], and conclude that the statements of Theorem 1 hold.

Definition 2 (m-dissipative operators) An operator

A : D(A) ⊂ X → X is said to be m-dissipative if and only if • The operator A is dissipative, i.e. A(ζ 1 ) -A(ζ 2 ), ζ 1 -ζ 2 V ≤ 0, ∀ζ 1 , ζ 2 ∈ D(A). (23)
• The operator A is maximal, i.e. there exists λ 0 > 0 such that (equivalently, for all

λ 0 > 0) Ran(λ 0 I X -A) = X, ( 24
)
where I X is the identity operator over the Hilbert space X, and Ran is the range operator.

Proof (Theorem 1) The proof of Theorem 1 relies on the so-called semigroup theory (see, e.g., [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] or [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF] for an introduction to this theory in the linear case, and [START_REF] Miyadera | Nonlinear semigroups[END_REF] in the nonlinear case). The proof is divided in two steps: we first prove that A is dissipative and second that A is maximal. First step: A is dissipative. To prove this result, we use the scalar product introduced in [START_REF] Daafouz | Nonlinear control of a coupled pde/ode system modeling a switched power converter with a transmission line[END_REF], that is equivalent to the standard scalar product used for X. Let

ζ 1 = z 1 w 1 and ζ 2 := z 2 w 2 be in D(A). We introduce z = z 1 -z 2 and w = w 1 -w 2 , u i = -B (P z i -M * (w i -M z i )). Since (z, w) ∈ D(A) and because S w + Γ C z ∈ H, one can compute A(ζ 1 ) - A(ζ 2 ), ζ 1 -ζ 2 V . We obtain: A(ζ 1 ) -A(ζ 2 ), ζ 1 -ζ 2 V (25) = z A P z + (σ(u 1 ) -σ(u 2 )) B P z + S w -(M A -Γ C)z -M B(σ(u 1 ) -σ(u 2 )), w -M z H = - 1 2 |z| 2 + (σ(u 1 ) -σ(u 2
)) B P z (by ( 9))

+ S( w -M z) -M B(σ(u 1 ) -σ(u 2 )), w -M z H ≤ - 1 2 |z| 2 + (σ(u 1 ) -σ(u 2 )) B P z (S is dissipative) -(σ(u 1 ) -σ(u 2 )) B M * ( w -M z) - 1 2 |z| 2 -(σ(u 1 ) -σ(u 2 )) (u 1 -u 2 ) ≤ 0. ( 26
)
where in the last step we used the monotony of σ given in [START_REF] Astolfi | Integral action in output feedback for multi-input multi-output nonlinear systems[END_REF]. Thus A is a dissipative operator, concluding the first step of the proof.

Second step:

A is a maximal operator. Proving that A is maximal reduces to show that, given a positive constant λ, for all ζ ∈ X, there exists ζ ∈ D(A) such that

(λI X -A) ζ = ζ. (27)
The constant λ will be selected later on. Let (z, w) ∈ X. We seek (z, w)

∈ D(A) such that λz -Az -Bσ B (-P z + M * ( w -M z) = z λ w -S w -Γ C z = w. ( 28
)
Since S is a dissipative operator, the real parts of its eigenvalues are non-positive, and thus any positive constant λ is in the resolvent of S. Then, the operator (λI H -S) is invertible. We have w = (λI

H -S) -1 [w + Γ C z]. (29)
We can therefore rewrite the first line of (28) as an equation depending only on z and the given data w and z.

λz

-Az -Bσ B (-P z + M * ((λI H -S) -1 [w + Γ C z] -M z)) = z. ( 30
)
Since A is Hurwitz, one may rewrite the latter equation as follows

z = (λI n -A) -1 Bσ B (-P z + M * ((I H -S) -1 [w + Γ C z] -M z)) + z . (31)
We prove that there exists a solution to this equation by using a fixed-point strategy. Let us define the operator

F : R n → R n z → F (z), (32)
where F is defined by

F (z) := (λI n -A) -1 z+ Bσ B (-P z + M * ((λI H -S) -1 [w + Γ C z] -M z)) = (λI n -A) -1 z+ Bσ B -P z + M * [(λI H -S) -1 (w -M (λI n -A)z)] ,
by [START_REF] Cerpa | Rapid exponential stabilization for a linear Korteweg-de Vries equation[END_REF].

Our aim is to prove that this operator admits at least one fixed-point, which will prove that there exists a solution to [START_REF] Prieur | Wave equation with cone-bounded control laws[END_REF]. To this end, we apply the Schauder fixed-point theorem [START_REF] Coron | Control and Nonlinearity[END_REF]Theorem B.19].

Consider the ball B R ⊂ X centered at 0 of radius R > 0. If for every z ∈ B R , we succeed to prove that F (z) ∈ B R , then the Schauder fixed-point theorem applies (since B R is convex and compact), and the proof is achieved. We have

|F (z)| ≤ (λI n -A) -1 L B 2 P + M * (λI H -S) -1 M (λI n -A) |z| + (λI n -A) -1 L B 2 |M * (λI H -S) -1 w| + (λI n -A) -1 |z|. (33)
Now choose λ sufficiently large to apply the Schauder fixed-point theorem.

If λ > A , then (λI n -A) -1 ≤ 1 λ-A (see, e.g., [38, Lemma 2.2.6]). Hence (λI n - A) -1 → 0 as λ → +∞. Moreover, (λI H -S) -1 M (λI n -A) remains bounded. Indeed, (λI n -A) ≤ λ + A and (λI H -S) -1 ≤ m λ-ω by [38, Corollary 2.3.3]
for some positive constants m, ω and all λ > ω. We can therefore select λ sufficiently large such that

(λI n -A) -1 L B 2 P + M * (λI H -S) -1 M (λI n -A) ≤ 1 2
Then, if one selects R satisfying

(λI n -A) -1 L B 2 |M * (λI H -S) -1 w| + (λI n -A) -1 |z| ≤ R 2 we obtain |F (z)| ≤ R.
Applying Schauder fixed-point theorem [START_REF] Coron | Control and Nonlinearity[END_REF]Theorem B.19.], the operator F admits a fixed-point, which implies that the operator A is maximal. Since A is dissipative and maximal, we obtain that A is m-dissipative, which concludes the proof of Theorem 1.

Global Asymptotic Stability Result

To prove the global asymptotic stability of the origin of ( 19), we will invoke an infinite-dimensional version of the LaSalle's Invariance Principle [START_REF] Slemrod | Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control[END_REF]Theorem 3.1.].

But such a result is not sufficient to deduce the global asymptotic stability of the closedloop system. As in the finite-dimensional context, we need an observability condition for [START_REF] Kang | Boundary constrained control of delayed nonlinear Schrodinger equation[END_REF]. To this end, we recall below the definition of infinite-time approximate observability given in [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]Definition 6.5.1.]. Definition 3 (Infinite-time approximate observability) Let H and Y be two real Hilbert spaces, S : D(S) ⊂ H → H be the generator of a strongly continuous semigroup (T(t)) t≥0 and C ∈ L(X, Y ). Let ψ ∈ L(D(S), L 2 ([0, ∞); Y )) be defined by Then, the origin of ( 19) is globally asymptotically stable in X, that is, for every initial condition (z 0 , w 0 ) ∈ X, the origin is Lyapunov stable and lim t→+∞ (z(t), w(t)) X = 0.

(ψζ 0 )(t) = CT(t)ζ 0 , ∀ζ 0 ∈ D(S) (34

Remark 3 (Compact resolvent)

According to [START_REF] Cheverry | Handbook of Spectral Theory[END_REF]Proposition 4.24], it is worthy mentionning that there is an equivalence between the compactness of the resolvent and the precompactness of the positive orbit, a property required to apply LaSalle's Invariance Principle [START_REF] Slemrod | Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control[END_REF]Theorem 3].

Remark 4

According to [14, Theorem 4.1.5], since S is unbounded and B M * takes values in a finite-dimensional space, the pair (S, B M * ) cannot be exactly observable. For this reason, we rely on an approximate observability hypothesis.

Remark 5 Checking whether (v) holds might be difficult in practice, except in some simple cases, as the one provided in Example 5. Unfortunately, this condition introduces the operator M , defined in [START_REF] Cerpa | Rapid stabilization for a Korteweg-de Vries equation from the left Dirichlet boundary condition[END_REF], and which does not depend directly on the data of the initial problem given in [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimum-phase systems[END_REF]. Nevertheless, in the case where S is skew-adjoint with compact resolvent, we provide a necessary condition which does not depend on M , see below in Proposition 1.

Example 5 (Checking (v) for [START_REF] Cheverry | Handbook of Spectral Theory[END_REF]) We aim at proving that property (v) holds for the system [START_REF] Cheverry | Handbook of Spectral Theory[END_REF] introduced in Example 3. Let ζ 0 = (z 0 , w 0 ) ∈ Ker ψ where ψ is given by ( 34) and C = B M * . In this case, M * is given by M * : L 2 (0, 1)

w → 1 0 M (x)w(x)dx ∈ R. Recalling that B = 1, the equation B M * T(t)w 0 (x) = 0 for all t ≥ 0 reduces to 1 0 M (x)w(t, x)dx = 0, ( 35
)
where w is the solution to

   w t + λw x = 0, (t, x) ∈ R + × [0, 1], w(t, 0) = w(t, 1), t ∈ R + w(0, x) = w 0 (x), x ∈ [0, 1]. (36)
Using [START_REF] Chitour | L p -asymptotic stability analysis of a 1d wave equation with a nonlinear damping[END_REF], one obtains therefore

1 0 M (x)w(t, x)dx = λ a 1 0 M (x)w(t, x)dx. (37)
Performing an integration by parts leads to λ a

1 0 M (x)w(t, x)dx = λ a [M (1)w(t, 1) -M (0)w(t, 0)] - 1 0 M (x)w x (t, x)dx .
By noting that (35) implies 1 0 M (x)w t (t, x)dx = 0, we obtain

1 0 M (x)w x (t, x)dx = 0.
Using the boundary conditions of ( 36) and ( 12), one can deduce that w(t, 1) = 0. Then, following the proof of [START_REF] Marx | Forwarding design for stabilization of a coupled transport equation/ode with a cone-bounded input nonlinearity[END_REF]Theorem 2], one can prove that w(t, x) = 0, which shows that (v) holds for [START_REF] Cheverry | Handbook of Spectral Theory[END_REF]. Moreover, from standard Sobolev injections results, the canonical embedding from D(S) into H is compact, i.e., S has compact resolvent. Then, Theorem 2 applies, and (0, 0) in R n × H is globally asymptotically stable in the R n × H-topology.

Proof (Theorem 2)

We prove the statement of the theorem for initial conditions (z 0 , w 0 ) in D(A). The result follows for all initial conditions in X by a standard density argument (see e.g. [21, Lemma 1])). Consider the Lyapunov function defined in [START_REF] Curtain | An Introduction to Infinite-Dimensional Systems Theory[END_REF]. Denoting u = -B (P z -M * (w -M z)), and performing the same computations than in [START_REF] Isidori | Robust autonomous guidance: an internal model approach[END_REF], with the control defined in [START_REF] Kaliora | On the stabilization of feedforward systems with bounded control[END_REF], one therefore obtains

dV dt (z, w) ≤ -z z -2u σ(u) (38)
for every (z 0 , w 0 ) ∈ D(A). Then, for all t ≥ 0,

V (z(t), w(t)) -V (z 0 , w 0 ) ≤ - t 0 z(s) z(s)ds -2 t 0 u(s) σ(u(s))ds. ( 39
)
This implies that

∞ 0 z(s) z(s)ds < +∞, ∞ 0 u(s) σ(u(s))ds < +∞. (40)
Since the trajectories of ( 19) are bounded in D(A) (20), and σ is linearly bounded, then z and

s → z(s) P -M * (w(s) -M z(s)) Bσ B [P z(s) -M * (w(s) -M z(s))]
are also linearly bounded. Hence, applying Barbalat's Lemma and using Definition 1, we have

lim t→+∞ z(t) = 0, lim t→+∞ B M * w(s) = 0. (41)
Let ω(z 0 , w 0 ) be the ω-limit set of the initial condition (z 0 , w 0 ) ∈ D(A), that is, is the set of all (z , w ) ∈ D(A) such that there exists an increasing sequence of time (t n ) n≥0 such that z(t n ) → z and w(t n ) → w in H as n goes to infinity. Since A is a m-dissipative operator, the trajectory (z, w) is bounded in D(A). The injection of D(A) in X being compact, if ω(z 0 , w 0 ) = {(0, 0)}, then (z, w) converges to the origin in the X-topology. Since (z, w) is bounded in time and since the injection of D(A) in X is compact, the positive orbit {(z(t), w(t)) | t ≥ 0} is precompact in X. Therefore, according to the LaSalle's Invariance Principle for infinite-dimensional systems [34, Theorem 3.1.], ω(z 0 , w 0 ) is a non-empty compact set of X that is invariant to the flow of [START_REF] Kang | Boundary constrained control of delayed nonlinear Schrodinger equation[END_REF]. Hence, for any initial condition (z 0 , w 0 ) ∈ ω(z 0 , w 0 ), the corresponding solution (z , w

) ∈ C 1 ((0, ∞); X) × C 0 ((0, ∞); D(A)) of (19) satisfies          d dt w = Sw w (0) = w 0 B M * w = 0, z = 0 (42)
Since the pair (S, B M * ) is approximately observable in infinite-time, this implies that w 0 = 0. Therefore, ω(z 0 , w 0 ) = {(0, 0)}, which means that (z, w) converges to (0, 0) in the X-topology. Moreover, according to Theorem 1, (0, 0) is Lyapunov stable in the X-topology. Thus, (0, 0) is globally asymptotically stable, which concludes the proof of Theorem 2.

Observability Result for Skew-Adjoint Operators

As mentioned earlier in Remark 5, we state now some sufficient conditions to verify the observability requirement of item (v) in Theorem 2. We focus on the case in which S is a skew-adjoint operator.

Proposition 1 Suppose that, in addition to Assumption 1, the following properties hold:

(vi) The operator S is skew-adjoint and has compact resolvent; (vii) The pair (S * , Γ * ) is approximately observable in infinite-time;

(viii) For all eigenvalue µ of S,

rank A -µI n B C 0 = n + p. ( 43
)
Then, the pair (S, B M * ) is approximately observable in infinite-time.

Remark 6 (About item (iii).) Note that with item (ii), i.e. A Hurwitz, and (vi), i.e. S skew-adjoint, item (iii), i.e. the spectra of A is S are disjoint and nonempty, is trivially satisfied.

Proof (Proposition 1) Pick ϕ ∈ D(S) an eigenvector of S and denote -µ its corresponding eigenvalue. In order to apply [38, Proposition 6.9.1], let us assume that B M * ϕ = 0 and look for a contradiction. Suppose that ϕ = 0. Since S is skew-adjoint, ϕ is also an eigenvector of S * , with corresponding eigenvalue µ. Then, according to ( 9 Example 6 (The property (v) for (8)) Consider again system (8) introduced in Example 2. By some standard Sobolev injection theorems, it is easy to prove that the canonical embedding from D(S) to H is compact, i.e., S has compact resolvent. Therefore, item (iv) is satisfied. Moreover, assume that M = N 2 and D 0 and D 1 are orthogonal, which ensures that the corresponding operator S is skew-adjoint, as noticed in [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions[END_REF]. In [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions[END_REF]Theorem 3.2], the exact controllability2 of the pair (S, Γ ) is proven, which, in turns, implies approximate observability in infinite-time of the pair (S * , Γ * ). Therefore, items (vi) and (vii) of Proposition 1 are satisfied. Hence, if A, B and C are given so that the conditions (viii) is also satisfied, Proposition 1 does apply and hence also the condition (v) is satisfied, namely Theorem 2 does apply.

Conclusions

In this paper, we have provided a methodology to globally stabilize a cascade system composed by a dissipative infinite-dimensional system and an ODE. This methodology relies on the so-called forwarding approach, which needs to introduce an infinite dimensional the Sylvester equation. Under an appropriate observability assumption, we prove that our design leads to a stabilizing feedback law. We also have provided a necessary condition implying this observability property. The proposed methodology can apply to other topics than the stabilization such as output regulation [START_REF] Paunonen | Stability and robust regulation of passive linear systems[END_REF][START_REF] Paunonen | Internal model theory for distributed parameter systems[END_REF] or the repetitive control [START_REF] Astolfi | Repetitive control design based on forwarding for nonlinear minimum-phase systems[END_REF][START_REF] Weiss | Repetitive control of MIMO systems using h∞ design[END_REF].

As further research lines, we may consider more complicated systems, such as nonlinear infinite-dimensional dissipative systems. Let us mention for example the Korteweg-de Vries equation [START_REF] Cerpa | Control of a Korteweg-de Vries equation: a tutorial[END_REF]. In this case, we would not have to face with a Sylvester equation, which is a static equation, but with another dynamical nonlinear PDEs, for which the proof of the existence and the uniqueness of solution is not an easy task. Finally, another possible extension, can be to generalize the proposed forwarding approaches to different cascades of systems, for instance, in which each subsystem is infinite-dimensional.

  ),ϕ * SM -ϕ * M A = -ϕ * Γ C, i.e., ϕ * M (µI n -A) + ϕ * Γ C = 0. µI n -A B C 0 has full row rank, Γ * ϕ = 0.According to [38, Proposition 6.9.1], this contradicts the infinite-time observability of the pair (S * , Γ * ). This concludes the proof of Proposition 1.

  Defining H -1 as the completion of H with respect to the norm w -1 := (βI H -S) -1 w H , where β is in the resolvent of S, we suppose that Γ ∈ L(R p , H -1 ) (i.e. Γ is a bounded operator from R p to H -1 ).Recall thatH -1 is the dual of D(S * ) = {w ∈ H | sup v∈D(S)\{0} | Sv, w H |/ v H }with respect to the pivot space H.

	the control input and
	σ : R m → R m is a cone-bounded nonlinearity, that we will define in a rigorous way
	below. In system (1), S : D(S) ⊆ H → H is a (possibly unbounded) operator, with
	D(S) densely defined in H.

  for every i ∈ {1, . . . , k} (resp. λ i > 0 for every i ∈ {k + 1, . . . N }), w + (t, x) (resp. w -(t, x)) corresponds to the k first components of w (resp. to the N -k last components of w), C, D 0 and D 1 are matrices of appropriate dimension. The matrix A is Hurwitz and the function σ is supposed to be a cone-bounded nonlinearity. The boundary control problem (8) can be rephrased as (1) according to[START_REF] Tucsnak | Observation and control for operator semigroups[END_REF] Chapter 10]. Following the steps of[START_REF] Tucsnak | Observation and control for operator semigroups[END_REF] Example 10.1.9], operators S and Γ can be determined as follows.

  ) for all t ≥ 0. Then, the pair (S, C) is approximately observable in infinite-time if and only if Ker ψ = {0}.

	Theorem 2 (Global asymptotic stability) Suppose that Assumption 1 holds. More-
	over, assume the following:
	(iv) S has compact resolvent;
	(v) (S, B M

*

) is approximately observable in infinite-time.

This terminology is used in functional analysis to denote monotone operators. If m = 1, then σ is nondecreasing.

See [38, Definition 11.1.1.] for a definition.

This research was partially supported by the French Grant ANR ODISSE (ANR-19-CE48-0004-01) and was also conducted in the framework of the regional programme "Atlanstic 2020, Research, Education and Innovation in Pays de la Loire", supported by the French Region Pays de la Loire and the European Regional Development Fund.

S. Marx LS2N,