Hugues Prétrel

William Plumecocq

EXPERIMENTAL AND NUMERICAL STUDY OF FIRE EVENT INVOLVING TWO SIMULTANEOUS FIRE SOURCES IN CONFINED AND VENTILATED COMPARTMENTS

The study deals with fire events involving two simultaneous fire sources in a confined and ventilated set of enclosures. Within the framework of fire safety assessment in nuclear facilities, it aims at investigating two configurations, one with two fire rooms separated by an adjacent empty room and a second one with two adjacent fire rooms. The objective is to analyse how a secondary fire may influence a primary fire, the flows at doorways and the overall smoke propagation. Some experimental results obtained for two large scale fire tests performed with gas burners are discussed. A specific analysis of the level of temperature in the fire rooms and in the adjacent room is presented. Numerical simulations with a zone approach are performed and satisfactory agreements are found between calculations and experimental data. The study shows that the distance between fires in case of multiple fire event is an important parameter. Close fires lead to an increase in aggression around the fire place whereas distance fire contributes to maximize the smoke temperature and therefore the aggression in the neighbouring area around fire places.

INTRODUCTION

Fire event involving several fire sources simultaneously is a concern for fire safety assessment activities as well as for fire protection design. Multiple fires in the same environment make more complex the fire scenario and the associated physical phenomena. These scenarios are specific and the resulting physics should be better understood as well as reliably predicted. Such scenarios are encountered in open industrial plans with many potential fire sources [START_REF] Vasanth | Multiple pool fires: Occurrence, simulation, modeling and management[END_REF] . They are also of concern in case of post-earthquake events for which several sources may ignite simultaneously. Fire event in tunnels or in parking places are also frequent scenarios involving several sources. A particular feature of multiple fire source events is the extent of the damage they can cause. Therefore, they require particular attention in terms of fire safety assessment and fire protection design. Although this type of fire event remains frequent, the amount of researches remains rather weak in comparison to stand alone fire scenarios. A first issue in the understanding of multiple fire events is the fire merging process. This process is closely related to the distance between fires as well as to the environmental conditions, like wind that can favour or not the merging process. First studies appear in the sixties and investigate experimentally the burning of several sources leading to mass fire [START_REF] Putnam | A model study of the interaction of multiple turbulent diffusion flames[END_REF][START_REF] Huffman | Interaction effects of multiple pool fires[END_REF] or arranged in array of sources close to each other [START_REF] Thomas | Buoyant diffusion flames: some measurements of air entrainment, heat transfer and flame merging[END_REF][START_REF] Weng | Study on flame height of merged flame from multiple fire sources[END_REF][START_REF] Kamikawa | Experimental study of merged flames from multifire sources in propane and wood crib burners[END_REF][START_REF] Weng | Experimental study on merged flame characteristics from multifire sources with wood cribs[END_REF] or with only two fires [START_REF] Vasanth | CFD simulation of pool fires situated at differing elevation[END_REF][START_REF] Gao | Study on thermal stratification of an enclosure containing two interacting turbulent buoyant plumes of equal strength[END_REF] . The results of these experiments show that the fire resulting from the merging of several single sources can lead to much more severe fires that the arithmetic sum of each single fire. Criteria for having merging flame and the features of the resulting fire (flame height and temperature, air entrainment) are investigated with different fire sources (pool fire, wood cribs, gas fire). Multiple fire scenarios may also lead to non-merging flame in case of sufficiently large distances between fires. For such scenarios, main scientific issues concern the interaction between fires and the associated smoke propagation. Configurations of interest are mainly fires in confined spaces as in tunnels, parking places or enclosures (ventilated or not). The specific topics of interest concern the effective radiative heat fluxes emitted by several fires and the smoke propagation induced in the vicinity of the fires as well as far away (vertical stratification, room filling, flows through openings) [START_REF] Wan | Effects of pool size and spacing on burning rate and flame height of two square heptane pool fires[END_REF][START_REF] Wan | Experimental study on horizontal gas temperature distribution of two propane diffusion flames impinging on an unconfined ceiling[END_REF][START_REF] Zhao | Study of the fire characteristics for multi-source fires in the confined corridor[END_REF][START_REF] Ji | Experimental study on flame merging behaviors from two pool fires along the longitudinal centerline of model tunnel with natural ventilation[END_REF][START_REF] Heidarinejad | A comprehensive study of two fire sources in a road tunnel considering different arrangement of obstacles[END_REF] .

The present research focuses on the second issue and in particular on scenarios with several forced ventilated enclosures where two fires may occur simultaneously. The objective is to study smoke propagation in case of multiple fire scenario in a set of ventilated rooms. Applications concern fire safety assessment in nuclear power plant. A basic configuration has been defined with two fires in an arrangement of three rooms connected together with doorways, the whole set being connected to a ventilation network (Figure 1). Two fire scenarios have been investigated. A first one with two fire rooms separated by an adjacent empty room and a second one with two adjacent fire rooms. The objective is to investigate how a secondary fire (fire N°2 in Figure 1) may influence a primary fire, especially the flow exchanges at doorways and the overall smoke propagation. The analysis focuses on the specific smoke propagation and the room temperature distribution.

The study comprises three stages. First, an experimental study, based on a propane gas fire, has been performed to characterise temperature rise in both configurations. Then, test results are compared to numerical simulation performed with the SYLVIA zone fire model. A third part proposes extrapolation analysis from SYLVIA predictions for other practical configurations in order to enlarge the understanding of multi-fire scenarios. 2). Walls consist in 35 cm or 40 cm thick re-enforced concrete and the ceiling is covered with silicate of calcium panels to thermally insulate the wall. The ventilation network consists in one admission line in each room and one exhaust line located in the room without fire (L2 or L1 according to the configuration as illustrated in Figure 1). The inlet and exhaust sections of 0.18 m 2 are situated in the upper part of the room at about 0.80 m from the ceiling. The ventilation lines are connected to an industrial ventilation network equipped with blowing and exhaust fans. The fire sources are propane gas burners located at the center of the rooms.

Propane flow rate was measured with a mass flow-meter. The corresponding fire heat release rate (HRR) is determined considering the combustion efficiency equal to one and the effective combustion enthalpy equal to 46.4 MJ/kg. Air flow rates in ventilation lines were measured with average Pitot probe devices, connected to pressure transducers. Gas temperature and pressure as well as gas species concentrations were also measured in the ventilation lines. Rooms were equipped with a large set of sensors for measuring gas and wall temperatures, radiative and total heat fluxes, velocity at doorways, gas species concentrations. The present work only focused on gas temperature. Temperature stratification in rooms was measured from 1.5 mm K-type thermocouples (uncertainty of +/-4°C) positioned on five vertical masts located at the center of the room and distributed in the four corners. Video cameras were also positioned in each room. The study is based on two sets of experiments corresponding to configurations A and B illustrated in Figure 1. Fires are set in rooms L3 and L1 for configuration A and in rooms L3 and L2 for configuration B. The difference between the two configurations is therefore the distance between fires. For configuration A, fire rooms are separated with an adjacent room without fire, whereas the fire rooms are contiguous for configuration B. Regarding the ventilation set-up, each room is equipped with an admission line in both configurations. The exhaust line is located in the room with no fire (L2 for configuration A and L1 for configuration B).

For both configurations, the varying parameter is the secondary fire HRR. It mimics a situation with a main fire in one room and a secondary fire occurring simultaneously in another room. For all experiments, the fire heat release rate in room L3 is set to a constant level, about 278 kW, the varying parameter being the second fire HRR (in room L1 for configuration A or in room L2 for configuration B) that varies between 71 kW and 350 kW (see Table 1). The first test corresponds to a reference test with only one fire in room L3, at constant fire HRR of 279 kW.

Table 1: test matrix with the fire heat release rate during the steady phase (B1 tests correspond to configuration A and B2 tests correspond to configuration B, the test names are those given during the experimental program).

Prior to ignition, the ventilation regime is set to the same operating conditions for all experiments with an air flow rate of 1200 m 3 /h in each room and a resulting flow rate of 3600 m 3 /h at the exhaust (in room L2 or L1). The flow rate corresponds to a renewal rate of Tr = q v /V = 10.9 h -1 (with the volume of one room V = 110 m 3 ). The test procedure is as follows: first the ventilation network is set to the targeted ventilation flow rate. Then gas burners are turned on simultaneously in the two rooms for a period between 1200 s to 1400 s such as steady state is reached.

Results

As an example, Figure 3 presents time variations of fire HRRs in rooms L3 and L2 for configuration B tests. Fire HRRs are constant with time during all experiment. For configuration B, fire HRR is constant in room L3 and varies from one test to another for room L2. Figure 4 presents an example of gas temperature rise in the three rooms for test B2P3 (HRR L3 = 271 kW and HRR L2 = 141 kW). Gas temperature rises up to a quasi-steady state after time 1000 s. The temperature is greater in the room having the largest fire HRR (here room L3 with 278 kW). The adjacent room with no fire gives the lowest temperature. The temperature in the second fire room L2 with 141 kW gives temperatures between the two extrema. For each room, average vertical temperature profile is determined considering the four masts located in the room corners. These average temperature profiles in each room at t=1000 s for the two configurations are presented in Figure 5. For all tests, the temperature profile shows a vertical thermal stratification with higher temperature in the upper part of rooms. However, these stratifications differ between rooms. Temperature profiles in the L3 fire room with constant fire HRR of about 278 kW remain similar for all tests with two temperature gradients, the first one with the strongest gradient (about 100 °C / m) in the first meter from the floor, and the other one above with a gradient of about 40 °C / m. Moreover, the amplitude of temperature increases slightly because of the global increase of the total fire heat release rate HRR tot = HRR L3 + HRR L1 or L2 . In both configurations, the second fire contributes to increase the temperature in the primary fire room. Even if fires are in separate rooms, they do not behave independently. Temperature profile in the adjacent room with no fire shows different shapes between configuration A and B. The vertical temperature gradient is constant in configuration A (with about 12 °C/m) whereas the gradient increases with the increase in fire HRR for configuration B (from 5 to 15 °C/m). These results show that the relative position of the two fire sources modifies the thermal consequences in the adjacent room. In the secondary fire room, the vertical temperature profile shows also a constant gradient which increases with the fire HRR. It is worth noting that for all rooms, temperature profile does not show two homogenous layer stratification but layers with constant gradient. These are typical behaviours observed in confined spaces. In order to compare the scenarios with each other, analysis focuses on the maximum temperature reached in each room (at the highest positon in the room). This maximum temperature is plotted as a function of the total fire HRR for the two configurations in Figure 6. For both configurations, the room having the highest temperature increase is the fire room where the HRR varies between tests (L1 for configuration A and L2 for configuration B). The temperature in the adjacent room with no fire also increases because of the increase of the total fire HRR tot . In this case, the convective heat is mainly propagated through the doorway. As previously noticed, the temperature in the primary fire room L3 does not remain constant despite a constant fire HRR, but also increases. The secondary fire room also influences the temperature in the primary fire room by limiting the release of smoke through the doorways. The comparison between the two configurations, A and B, is illustrated in Figure 7. Configuration B leads to a higher temperature in the primary fire room L3. A second fire in a room contiguous to the main fire room contributes to limit smoke propagation at the doorway and therefore to confine heat in the fire room L3. This configuration is the more unfavourable for temperature in the fire room. Conversely, configuration A leads to a higher maximum temperature in the adjacent room with no fire in enhancing smoke propagation from both fires in the adjacent room. From the point of view of temperature in the adjacent room without fire, the configuration A is the more severe scenario. However, if the concern is the maximum temperature whatever the room, configuration B is the worst scenario with the highest maximum temperature in the fire room L3. The relative position of two fire sources modifies the smoke propagation. In configuration A, the smoke produced by the two fires is released by two doorways whereas it is released by only one doorway in configuration B.

Figure 7: Effect of the configuration on the maximum temperature in, from left to right, the emptied adjacent room (L1 or L2), the secondary fire room (L2 or L1) and the primary fire room (L3) versus the total fire heat release rate.

COMPARISON WITH ZONE CODE MODELING

In order to assess the capability of the SYLVIA zone code to take into account multiple fire sources in confined and mechanically ventilated enclosures, test data are compared to numerical simulations. The zone code approach has been chosen for its capability to simulate several complex and large scale scenarios with limited computational times.

Code description and inputs

The SYLVIA software [START_REF] Plumecocq | Numerical method for determining water droplets size distributions of spray nozzles using a two-zone model[END_REF] was developed at IRSN. It is designed to predict the behaviors of mechanical/natural ventilation, fire growth, hot gas and smoke propagation, and airborne contamination transfer in confined and mechanically ventilated enclosures. It is a simulation tool for calculating the consequences of multi-compartment fires in industrial and nuclear facilities, which are equipped with a full ventilation network (including ducts, room leaks, fire dampers, fans, horizontal/vertical openings, filters…). The fire design of the SYLVIA software belongs to the wellknown two-zone fire modeling [START_REF] Karlsson | Enclosure Fire Dynamics[END_REF][START_REF] Quintiere | Fundamental of Fire Phenomena[END_REF] . In such software, each compartment is described with two Lagrangian control volumes separated by a thermal interface (Figure 8). Mass and energy balances are performed in each zone: the lower zone simulating the fresh gas and the upper zone simulating the combustion products and the gas entrained by the plume. In a two-zone approach, a plume feeds the upper zone of the fire room, whose volume increases, which has the effect of lowering the interface and leading to the under-oxygenation of the fire source if the gas flow in the exhaust duct or at the level of openings of the fire room is not sufficient to remove all gases supplied by the plume. The features of ambient quantities (pressure, temperature, mass fraction of species) in each control volume are assumed to be uniform at a given time. These control volumes can be connected by various ventilation components representing either natural or mechanical ventilation. The convective/conductive/radiative heat transfers inside the compartments and in the ventilation network (walls, gas phase) are fully modelled and play a major role for predicting temperature and pressure, especially in rooms and ventilation ducts near the fire. The fire itself is described by correlations (plume, flame height...) available in the literature [START_REF] Karlsson | Enclosure Fire Dynamics[END_REF][START_REF] Quintiere | Fundamental of Fire Phenomena[END_REF] . Furthermore, the SYLVIA software has additional specific models for simulating a water spray system, the condensation/adsorption of gaseous species (as water vapour) on walls, and the transport and deposition of aerosol particles (soot, dust) on solid surfaces as ventilation ducts, room walls or High Efficiency Particulate Air filters. The design of ventilation network is based on the node-and-branch method [START_REF] Nishio | Development of the FACE computer code to evaluate the safety of an air ventilation system during a postulated solvent fire in a nuclear fuel reprocessing plant[END_REF] . A node is a point (no volume) or a simple zone (with a fixed volume) in the ventilation network, where the different physical quantities (pressure, temperature, particle and gaseous species mass fractions) are considered uniform. A point can be considered as a simple junction between two or more ducts or linked with an exterior environment (boundary condition node). A zone, which leads to inertial effects under transient conditions due to the fixed volume, may involve a single room (as the fire compartment) or a node between two branches. A branch represents a passive element of the network (for instance, duct, filter, fire damper, leak, etc.) or an active element (for instance, fan, motor-driven valve, etc.). A mass flow rate is assumed to be uniform in 0D branches and is a function of the pressure difference at its boundaries (nodes). A given branch is always connected to two nodes, but there is no limit on the number of branches connected to a given node.

SYLVIA modelling of the ventilation network of the DIVA facility is reported in Figure 9. Various nodes and branches of the modelled ventilation network correspond to aeraulic elements for which experimental measurements (pressure, temperature, volume flowrate) were performed. Thus, this modelling allows reproducing at best the experimental aeraulic resistivity of the ventilation network of the fire tests in the simulations.

Determination of zone code variables from experimental data

Validation is made on upper and lower gas layer temperatures. Experimental gas layer temperatures are obtained from regression methods applied to mean vertical temperature profile in each room for a given time. Several regression methods are available for processing vertical temperature profiles. In the present work, the method used is the one proposed by Pretrel & al [START_REF] Prétrel | New developments in the use of data regression methods for characterization of vertical fire smoke stratification[END_REF] which considers the thermal stratification as two layers, a lower one with constant temperature (the lower temperature) and an upper one with a constant vertical temperature gradient (linear increase of temperature with elevation). This theoretical description is the one that best fits experimental temperature profiles, ie, that minimizes the difference between theoretical and experimental profiles. For each test, experimental upper and lower temperatures as well as the interface height are determined for each time step, in the three rooms and for all tests. The upper layer temperature is the average temperature of the upper layer.

Validation

An example of comparison is presented in Figure 10 for test B1P4 (configuration A with a fire HRR of 277 kW in room L3 and a fire HRR of 210 kW in room L2). Temporal variations of upper (T sup ) and lower (T inf ) temperatures and the interface height (hi) in each room obtained from tests and from SYLVIA predictions are compared. The agreement is satisfactory for the upper layer temperature, and this is true for all the tests. Regarding the lower layer temperature and interface height, larger discrepancies between test data and simulations are reported. This is mainly due to the fact that these scenarios lead to small lower layer and consequently to a large discrepancy in predicting its characteristics, thickness and temperature. Then for each test, the upper layer temperature at t=1000 s is compared between test data and SYLVIA predictions, and the whole set of data are presented in Figure 11. Predictions and tests reproduce the same behaviour previously analysed in Figure 6 for the maximum experimental temperature. The average discrepancies between simulations and test data are 13% and 6% for configurations A and B respectively, which are considered as satisfactory.

In addition, the comparison of the calculated upper layer temperatures between the two configurations A and B is also presented in Figure 12. Variations of the predicted upper layer temperature show the same trends as for the experimental maximum temperature. However, the differences observed during the tests for the maximum temperatures between the two configurations are not as pronounced with calculated layer temperatures. With SYLVIA predictions, both configurations give similar results regarding the upper layer temperature. This result indicates that in terms of mean temperature (average temperature of the upper layer), the two configurations are similar. Differences between the two configurations concern more specifically local variations of temperature especially near the ceiling that cannot be highlighted by the two layer simulations.. Nevertheless, an interesting result emerges from the comparison between test data and SYLVIA predictions: a correlation seems to exist between the experimental maximum temperature and the predicted upper layer temperature. An issue when performing zone code calculation is the relation between the upper layer temperature predicted and the real maximum temperature that can be expected in the enclosures. As illustrated in Figure 13, the maximum gain of temperature measured during tests (T max -T ref ) is compared to the gain of temperature predicted by SYLVIA code (T sup -T ref ). The maximum temperature (near the ceiling) is about 1.47 greater than the upper layer temperature. This ratio certainly depends on the type of thermal stratification reported for these fire scenarios and other stratifications could have given different values. Nevertheless, this analysis leads to a practical criterion that can be considered for assessing maximum gas temperature from zone code calculations in the experimental configurations used for this study.

SIMULATIONS OF COMPLEMENTARY CONFIGURATIONS

Based on the previous validation, other multiple fire scenarios are considered only from zone code predictions. The aim of these simulations is to analyse some other configurations in order to see whether similar conclusions can be drawn concerning multiple fire scenarios.

Scenario B inverted

A first configuration is the scenario B in switching rooms L2 and L3 in order to investigate the effect of fire relative positions. This scenario named "B inverted" consists in locating the fire with larger HRR in room L2 and the varying fire HRR in room L3. These predictions aims at investigating if the position of the primary fire has an influence on the results given with scenario B. Results are presented in Figure 14 with a comparison of the upper layer temperature increase in each room. As expected from the previous analysis, there is nearly no difference in the two fire rooms. Whatever the relative position of the two fires regarding the open doorway, the mean smoke temperature (ie. the upper layer temperature) is identical. Differences are only noticed on the adjacent room with no fire. The scenario that leads to the highest temperature in the adjacent room is the one where the strongest fire is contiguous to the adjacent room. Smoke propagation through the doorway is more effective and then a greater amount of convective heat is transported towards the adjacent room. Regarding smoke temperature in the adjacent room, multiple scenarios sorted from the more to the less severe scenario are scenarios A, B inverted and B. Doorways are an efficient means for propagating smoke and the doorway near to the greatest fire source is more favourable for smoke propagation. A second set of simulations is performed with configurations A and B but using heptane instead of propane gas (used in the experiment). This change in the fuel type aims at considering more realistic configurations rather than propane gas fire suitable when performing parametric analysis with large scale fire tests. Regarding the numerical modelling, the change of fuel implies other parameters for predicting the effect of oxygen depletion on the burning rate. Simulations presented in the validation section are performed with the same fire HRR in each room. Results are compared in Figure 15. Very few differences are noticed between the two set of simulations. The change of fuel does not modify the conclusions presented in the previous section.

Figure 15: Effect of the effect of fuel on the upper layer temperature in, from left to right, the emptied adjacent room (L1 or L2), the secondary fire room (L2 or L1) and the primary fire room (L3) versus the total fire heat release rate

CONCLUSION

The objective of this study was to investigate the behaviour of multiple fire sources in a confined and ventilated set of enclosures. Scenarios with two fires have been considered with various distances between fires. Experimental results obtained from fire tests with propane burner in DIVA facility have been analysed. A comparison with simulation results obtained with the SYLVIA zone code was then proposed in order to validate the predicting tools for such scenarios. In a final part, simulations of other scenarios have been discussed to see whether similar conclusions could be drawn concerning multiple fire sources.

The study highlights that the relative position of fires in case of multiple fire scenarios is of importance for the consequences of a fire event. Scenarios with fire sources located far apart may contribute to reach higher temperatures in adjacent rooms. Fire interactions are weak and heat efficiently propagates in adjacent rooms. Scenarios with close fire sources may contribute to reach higher temperatures in the fire rooms. Fire interactions are stronger and areas where fire places are located may lead to higher temperatures than in cases where the fire sources are far apart.

A second important conclusion concerning multiple fire scenarios relates to local effects. In the experimental configurations used for this study, the differences between the multiple fire scenarios have mainly local effects (maximum temperature near the ceiling for instance). But in terms of global behaviour, the fire event behaves in a fairly similar way as the same fire scenario with one equivalent fire source instead of several sources. In term of layer temperature, the zone code approach is appropriate and IRSN SYLVIA code gives satisfactory results. However, if the local effects are of interest, CFD simulations are necessary.

This study brings also information regarding the relationship between the upper layer temperature given by the zone code prediction and the effective maximum temperature expected during a fire event. In the experimental configurations used for this study, the simulations show that the maximum measured temperature is about 1.5 times higher than the upper layer temperature predicted with a zone code approach.

Figure 1 :

 1 Figure 1: Schematic of the two configurations.

Figure 2 :

 2 Figure 2: Pictures of the three rooms L1, L2 and L3 in DIVA facility

Figure 3 :

 3 Figure 3: Time variation of the fire HRRs in the two rooms, L3 and L2 for configuration B.

Figure 4 :

 4 Figure 4: Time variation of gas temperature in one mast in room L1, L2 and L3 for test B2P3.

Figure 5 :

 5 Figure 5: Average temperature profiles in each room at time t=1000 s for configurations A (first row) and B (second row).

Figure 6 :

 6 Figure 6: Maximum temperature in the three rooms versus the total fire heat release rate for configurations A (left plot) and B (right plot)

Figure 8 :

 8 Figure 8: Two-zone approach of fire compartments.

Figure 9 :

 9 Figure 9: SYLVIA modelling of the ventilation network of the fire tests.

Figure 10 :

 10 Figure 10: Comparison between experiment (plain line) and simulation (dotted line) for the upper lower and interface height (from left to right) for test B1P4 (configuration A) (if h i =0, then T inf is the temperature of the first temperature probe).

Figure 11 :Figure 12 :

 1112 Figure 11: Upper layer temperature from simulations (SYL) and tests (EXP) at time t=1000 s in the three room and for configuration A (left) and B (right)

Figure 13 :

 13 Figure 13: Correlation between the upper temperature predicted with SYLVIA code and maximum experimental temperature measured in the room

Figure 14 :

 14 Figure 14: Effect of the configuration (B and B inverted) on the upper layer temperature in, from left to right, the adjacent room with no fire (L1), the secondary fire room (L2 or L3) and the primary fire room (L3 or L2) versus the total fire heat release rate


Nicolas Vanspranghe
email: nicolas.vanspranghe@tuni.fi.

Lucas Brivadis
email: lucas.brivadis@centralesupelec.fr.

Output regulation of infinite-dimensional nonlinear systems: a forwarding approach for contraction semigroups *

Keywords: O utput regulation, infinite-dimensional control systems, nonlinear contraction semigroups, forwarding design, semilinear systems. AMS subject classifications. 9 3C20, 93C10, 93B52

This paper deals with the problem of robust output regulation of systems governed by nonlinear contraction semigroups. After adding an integral action to the system, we design a feedback law based on the so-called forwarding approach. For small constant perturbations, we give sufficient conditions for the existence of a locally exponentially stable equilibrium at which the output coincides with the reference. Under additional assumptions, global asymptotic stability is achieved. All these conditions are investigated in the case of semilinear systems, and examples of application are given.

Introduction

Robust output regulation in one of the oldest problems in control theory. It consists in designing a feedback law which ensures that the output of a system tracks a given reference, even in the presence of external disturbances. If a feedback law can be designed to stabilize the system at some target point then a natural strategy to regulate the output robustly with respect to constant disturbances consists in adding an integral action to the system. Then, one must find a new feedback law that stabilizes the augmented system (i.e., the state and the output integrator) at some equilibrium point even in the presence of constant disturbances. The integrator guarantees that the output is at the reference when the full state is at the equilibrium.

While the theory of output regulation for linear finite-dimensional systems is well-understood since the seminal work of E. Davidson [START_REF] Davison | The robust control of a servomechanism problem for linear time-invariant multivariable systems[END_REF][START_REF] Davison | A generalization of the output control of linear multivariable systems with unmeasurable arbitrary disturbances[END_REF] (see also [START_REF] Francis | The internal model principle for linear multivariable regulators[END_REF]), many problems remain open in the case of nonlinear and/or infinite-dimensional systems. In [START_REF] Pohjolainen | Robust multivariable PI-controller for infinite dimensional systems[END_REF][START_REF] Pohjolainen | Robust controller for systems with exponentially stable strongly continuous semigroups[END_REF], S. Pohjolainen proposed an extension of the finite-dimensional linear robust regulation theory to the infinite-dimensional context. In particular, a controller based on an output integrator was investigated. More recently, this work has been continued in [BLGS00, [START_REF] Rebarber | Internal model based tracking and disturbance rejection for stable well-posed systems[END_REF][START_REF] Paunonen | Internal model theory for distributed parameter systems[END_REF][START_REF] Paunonen | Controller design for robust output regulation of regular linear systems[END_REF][START_REF] Paunonen | Stability and robust regulation of passive linear systems[END_REF] where an internal model principle was developed to reject a wider class of disturbances (and not only constant ones). Other works focusing on output regulation of particular linear partial differential equations include [START_REF] Guo | Adaptive error feedback regulation problem for 1D wave equation[END_REF][START_REF] Lhachemi | PI regulation of a reaction-diffusion equation with delayed boundary control[END_REF][START_REF] Guo | Output regulation for a heat equation with unknown exosystem[END_REF], to mention only a few. In comparison, very few attempts to tackle infinite-dimensional nonlinear output regulation problems exist in the literature, especially in an abstract framework. For instance, [START_REF] Logemann | Time-varying and adaptive integral control of infinite-dimensional regular linear systems with input nonlinearities[END_REF] dealt with single-input singleoutput linear distributed parameter systems subject to some input nonlinearity, while [START_REF] Natarajan | Approximate local output regulation for nonlinear distributed parameter systems[END_REF] focused on local approximate regulation of Lipschitz perturbations of linear systems.

In the finite-dimensional context, a strategy to take nonlinearities into account is the so-called forwarding approach, devised by F. Mazenc and L. Praly in [START_REF] Mazenc | Adding integrations, saturated controls, and stabilization for feedforward systems[END_REF] and developed with different purposes in [START_REF] Kaliora | On the stabilization of feedforward systems with bounded control[END_REF][START_REF] Sofiane Benachour | Forwarding design with prescribed local behavior[END_REF][START_REF] Astolfi | Integral action in output feedback for multiinput multi-output nonlinear systems[END_REF][START_REF] Poulain | Robust asymptotic stabilization of nonlinear systems by state feedback[END_REF]. We propose to approach the issue of constant output regulation via the forwarding technique. Few extensions of this methodology in the infinitedimensional setting exist. Control systems investigated in [START_REF] Terrand-Jeanne | Adding integral action for open-loop exponentially stable semigroups and application to boundary control of PDE systems[END_REF][START_REF] Marx | Forwarding-Lyapunov design for the stabilization of coupled ODEs and exponentially stable PDEs[END_REF] are linear. The article [START_REF] Marx | Forwarding techniques for the global stabilization of dissipative infinite-dimensional systems coupled with an ODE[END_REF] dealt with cone-bounded nonlinearities (such as saturation) applied to the input. More recently, [START_REF] Balogoun | ISS Lyapunov strictification via observer design and integral action control for a Korteweg-de Vries equation[END_REF] dealt with the regulation of the linearized Korteweg-de Vries equation, and local regulation of the nonlinear model with the same feedback law was proved. In the recent work [START_REF] Giaccagli | Sufficient conditions for global integral action via incremental forwarding for inputaffine nonlinear systems[END_REF], the authors followed a contraction approach to achieve the output regulation of nonlinear finite-dimensional systems by means of an incremental forwarding approach. Inspired by this work, we focus on systems of contraction, namely, such that the distance between any pair of trajectories is exponentially converging towards zero. While we do not restrict ourselves to Lipschitz perturbations of linear systems (which we will refer to as semilinear systems), we pay special attention to this case.

Organization of the paper In the first section of the paper, we state more precisely the class of systems under consideration and the output regulation problem. Then, we give some insights on the forwarding strategy we use to design a stabilizing feedback law. Main results are stated and discussed in Section 3, and proved in Section 4. Some examples of application are provided in Section 5.

Notation If H is a Hilbert space endowed with the norm • H , B H (x, r) denotes the open ball of (H, • H ) centered at x ∈ H of radius r > 0. Let Z be a Hilbert space. We denote by C(H, Z) the space of continuous maps from H to Z and by C 1 (H, Z) the space of continuously Fréchet differentiable maps. We also denote by L(H, Z) the Banach space of bounded linear operators from H to Z. Given L in L(H, Z), the adjoint operator L * is uniquely defined in L(Z, H) by (Lx, y) Z = (x, L * y) H , where (•, •) denotes the scalar product. We shall say that a map is locally Lipschitz continuous if it is Lipschitz continuous on every bounded set. Given a H-valued map A defined on some subset D(A) of H, we say that w is a strong solution to the abstract evolution equation dw/dt + A(w) = 0 on [0, T ] if w is absolutely continuous with respect to the topology of H, takes values in D(A), and solves the differential equation almost everywhere (a.e.) on (0, T ). A weak solution w is a limit of such strong solutions in C([0, T ], H).

Problem statement

Systems under consideration

Consider a nonlinear infinite-dimensional controlled system with measured output of the form:

dw dt + A(w) = Bu(t), (2.1a) y = Cw, (2.1b)
where w is the state of the system lying in some real Hilbert space H, y is the measured output lying in some real Hilbert space Z, u is the control input lying in some real Hilbert space U , A : D(A) → H is a singled-valued maximal monotone operator defined on the dense subset D(A) ⊂ H such that A(0) = 0, B : U → H is a bounded linear map, and C : D(C) → Z is a (potentially unbounded) linear map defined on the (dense) subspace D(C) ⊃ D(A). According to [Sho13, Chapter 4, Proposition 3.1], -A is the generator of a nonlinear strongly continuous contraction semigroup over H, denoted by {T t }. In order to address the problem of output regulation, we assume that A satisfies the following assumption.

Hypothesis 2.1 (Monotonicity). The nonlinear operator A is strongly monotone, i.e., there exists α > 0 such that

(A(w 1 ) -A(w 2 ), w 1 -w 2 ) H α w 1 -w 2 2 H , ∀w 1 , w 2 ∈ D(A).
(2.2) Hypothesis 2.1 implies that the semigroup {T t } satisfies

T t w 1 -T t w 2 H exp(-αt) w 1 -w 2 H , ∀w 1 , w 2 ∈ H, ∀t 0. (2.3)
We address the problem of regulating the output y of (2.1) at some constant reference y ref ∈ Z in presence of constant disturbances d ∈ H acting on the dynamics of the system. We proceed by adding an integrator to the output of the system, so that the resulting full system may be written as follows:

dw dt + A(w) = Bu(t) + d, (2.4a) dz dt = Cw -y ref , (2.4b)
where z lying in Z is the output integrator. The reason for that strategy is the following. Assume that a feedback law ϕ : H × Z → U may be designed, so that (2.4) in closed-loop with

u(t) = ϕ(w(t), z(t)) is asymptotically stable at (0, 0) ∈ H × Z when [d, y ref ] = (0, 0).
Then, formally, one can expect that for small perturbations of this system, i.e., for small values of [d, y ref ], the closed-loop is still asymptotically stable at some new equilibrium point [w ⋆ , z ⋆ ] near (0, 0) ∈ H × Z. Roughly speaking, this would imply that the dynamics of z, namely dz/dt = Cwy ref , still tend towards zero, that is, the output y = Cw is regulated at y ref .

In this paper, we propose a design strategy of such a feedback law ϕ based on the so-called forwarding approach, for which we make precise this formal reasoning.

Forwarding design

Following the forwarding design [START_REF] Mazenc | Adding integrations, saturated controls, and stabilization for feedforward systems[END_REF], we consider a candidate Lyapunov function of the form:

V (w, z) = 1 2 w 2 H + ρ 2 z -M(w) 2 Z (2.5)
where w 2 H /2 corresponds to the Lyapunov function associated to the contraction semigroup {T t }, M : H → Z is some Fréchet differentiable nonlinear map that vanishes at zero to be tuned, and ρ is some positive constant to be fixed (large enough) later. Let us compute the derivative of V along a strong solution [w, z] of the open-loop system (2.4) in the case where [d,

y ref ] = [0, 0]: d dt V (w, z) = -(w, A(w)) H + (w, Bu(t)) H + ρ(z -M(w), dM(w)A(w) + Cw -dM(w)Bu(t))) Z .
(2.6) Under Hypothesis 2.1, we have (w, A(w)) H α w 2 H . The key idea of the forwarding design is to choose a nonlinear map M satisfying the following functional equation, which guarantees that the graph of M is positively invariant under the uncontrolled dynamics (2.4):

M(0) = 0,
(2.7a) dM(w)A(w) + Cw = 0, ∀w ∈ D(A).

(2.7b)

In order to follow the forwarding approach, let us consider the next hypothesis.

Hypothesis 2.2. The functional equation (2.7) admits a continuously Fréchet differentiable solution M ∈ C 1 (H, Z).

For such a map M, we obtain:

d dt V (w, z) -α w 2 H + (B * w, u(t)) U -ρ(B * dM(w) * [z -M(w)], u(t)) U . (2.8)
This suggests to set the feedback law as

u(t) = B * dM(w) * [z -M(w)].
(2.9)

Then, applying Young's inequality, we get that for all ε > 0,

d dt V (w, z) -α w 2 H + ε 2 B 2 L(U,H) w 2 H + 1 2ε u(t) 2 U -ρ u(t) 2 U .
(2.10)

In particular, setting ρ = B 2 L(U,H) /α and ε = 1/ρ yields

d dt V (w, z) - α 2 w 2 H - ρ 2 u(t) 2 U 0, (2.11)
so that V is indeed a Lyapunov function. These formal computations give an insight on the feedback law (2.9) to choose and on the Lyapunov function (2.5) to consider in order to stabilize (2.4). However, many important questions remain open, in particular concerning the well-posedness of the closed-loop, the stability properties in presence of [d, y ref ], and the feasibility of the functional equation (2.7).

Remark 2.3. Checking Hypothesis 2.2 on a general nonlinear system of the form (2.1) is difficult in general. Moreover, since M is used in feedback law (2.9), an expression of M in terms of parameters of the system should be given in order to implement the controller. For these two reasons, we investigate later in the paper the case of semilinear systems, for which we give sufficient conditions on A for Hypothesis 2.2 to be satisfied, as well as an expression of M in terms of A and C.

Main results

We now state our results on the closed-loop system (2.4)-(2.9).

Well-posedness of the closed-loop dynamics

Our first result concerns the existence and uniqueness of solutions to the closed-loop equations (2.4)-(2.9).

Theorem 3.1 (Well-posedness). Assume Hypotheses 2.1 and 2.2 hold and dM is locally Lipschitz continuous. Then, (i) For all [d, y ref ] ∈ H × Z and all [w 0 , z 0 ] ∈ H × Z, the closed-loop system (2.4)-(2.9) admits a unique maximal weak solution [w, z] satisfying the initial condition [w(0), z(0)] = [w 0 , z 0 ] and defined on [0, T max ) for some T max ∈ (0, +∞];

(ii) Moreover, if w 0 ∈ D(A), then [w, z] is a strong solution to (2.4)-(2.9);

(iii) Furthermore, if d = 0 or dM is globally Lipschitz continuous, then (2.4)-(2.9) is forward complete, i.e., T max = +∞ for all [w 0 , z 0 ] ∈ H × Z.

The proof of Theorem 3.1 is given in Section 4.1 and works around two technical difficulties (the possible unboundedness of C and the a priori non-monotonicity of the system with integral action) by an appropriate use of Hypothesis 2.2.

Sufficient conditions for output regulation

In view of the output regulation problem, we wish to investigate the existence of an attractive equilibrium. The following theorem is the main result of the paper.

Theorem 3.2 (Sufficient conditions for output regulation). Assume Hypotheses 2.1 and 2.2 are satisfied. Assume that

Range dM(0)B = Z, i.e., ∃λ > 0 | ∀z ∈ Z, B * dM(0) * z 2 U λ z 2 Z . (3.1)
If dM is globally (resp. locally) Lipschitz continuous, then there exist positive constants M , κ and r and a neighborhood N of the origin in H × Z such that for any [d, y ref ] in B H×Z (0, r) (resp. in {0} × B Z (0, r)), the following results hold.

• There exists an equilibrium point [w ⋆ , z ⋆ ] ∈ D(A) × Z in N of the closed-loop system (2.4)-(2.9) such that Cw ⋆ = y ref and for all [w 0 , z 0 ] in N , the corresponding solution [w, z] of (2.4)-(2.9) satisfies, for all t 0,

[w(t) -w ⋆ , z(t) -z ⋆ ] H×Z M exp(-κt) [w 0 -w ⋆ , z 0 -z ⋆ ] H×Z . (3.2) • Moreover, if B * dM(w) * z 2 U λ z 2 Z , ∀z ∈ Z, ∀w ∈ H, (3.3)
then for all [w 0 , z 0 ] in H × Z, the corresponding solution [w, z] of (2.4)-(2.9) satisfies:

[w(t), z(t)] → [w ⋆ , z ⋆ ] in H × Z as t → +∞. (3.4)
The proof of Theorem 3.2 is given in Section 4.2. Equation (3.2) guarantees the local exponential stability of the equilibrium point [w ⋆ , z ⋆ ], while (3.4) provides global asymptotic stability under the additional condition (3.3). This condition is stronger than a global version of the range condition (3.1), in the sense that the constant λ must be independent of w ∈ H. We shall show in the proof how this uniformity implies the global attractivity of the local exponential basin of attraction N .

Conditions (3.1) and (3.3) may be difficult to check in general. In Section 3.3, we show how to investigate them in the semilinear case. In the finite-dimensional context, (3.3) is equivalent to the incremental condition given in [GAAM21, Assumption 2]. In our work, it allows to obtain global asymptotic stability even in the presence of small perturbations. The local condition (3.1) is commonly used in the forwarding design (see, e.g., [START_REF] Astolfi | Integral action in output feedback for multiinput multi-output nonlinear systems[END_REF]) to obtain local exponential stability.

It is clear that if the operator C is continuous, i.e., C belongs to L(H, Z), then the output Cw(t) goes to y ref in Z whenever w(t) goes to w ⋆ in H. If C is unbounded, the question is more delicate and is discussed in Section 3.4.

The semilinear case

Consider now the case where A = A + F with D(A) = D(A), where -A is the infinitesimal generator of a strongly continuous semigroup {S t } of linear operators on H, and F is a nonlinear mapping satisfying F ∈ C 1 (H), dF locally Lipschitz continuous, (3.5) and without loss of generality, dF (0) = 0. Following [Paz12, CZ20], we shall say that A is semilinear. In that context, let us introduce the following set of assumptions.

Hypothesis 3.3. The operator A = A + F is semilinear and satisfies the following properties:

(i) C is A-bounded, i.e., there exist positive constants a and b such that

Cw Z a Aw H + b w H , ∀w ∈ D(A); (3.6) (ii) There exists α > 0 such that (Ah + dF (w)h, h) H α h 2 H , ∀w ∈ H, ∀h ∈ D(A). (3.7)
Note that (3.7) in Hypothesis 3.3 implies α-exponential stability of the linear semigroup {S t }; in particular, 0 is in the resolvent set of A. Remark also that Hypothesis 3.3 implies both Hypothesis 2.1 and maximal monotonicity of A. Indeed, writing F (w 1 ) -F (w 2 ) as an integral of dF along the line segment joining w 1 to w 2 , one can show that (3.7) implies (2.2). Then, standard results on Lipschitz perturbations of linear systems (see [Paz12, Section 6.1] or [CZ20, Chapter 11]) together with (2.2) yield that -A generates a strongly continuous semigroup of contractions (denoted by {T t }) on H, which in turn implies that A is maximal monotone by virtue of [START_REF] Komura | Differentiability of nonlinear semigroups[END_REF]Theorem 4].

Under Hypothesis 3.3, we show in the following theorem that (2.7) admits a solution that can be expressed in terms of C, A, and F . Theorem 3.4 (Existence of M in the semilinear case). Assume Hypothesis 3.3 is satisfied. Then there exists a map M ∈ C 1 (H, Z) satisfying (2.7) with dM locally Lipschitz continuous and given for all w ∈ H by

M(w) -C lim τ →+∞ τ 0 T t w dt = -CA -1 w + CA -1 +∞ 0 F (T t w) dt. (3.8) For all w, h ∈ H, dM(w)h = -CA -1 h + CA -1 +∞ 0 dF (T t w)dT t (w)h dt, (3.9)
where dT t (w) denotes the differential of T t at w, which exists. Moreover, if F and dF are globally Lipschitz continuous, then so is dM.

The proof of Theorem 3.4 is given in Section 4.3. Its purpose is to guarantee the existence of the control law (2.9) for a wide class of semilinear systems and provide easy conditions under which requirements of Theorem 3.2 are met. A notable consequence of (3.9) is that the coercivity condition (3.1) in Theorem 3.2 simply reads as

Range CA -1 B = Z, (3.10)
which, in the context of output regulation corresponds to a non-resonance condition between A and the zero dynamics of the integrator via the Schur complement (see, e.g., [START_REF] Isidori | Robust autonomous guidance: an internal model approach[END_REF][START_REF] Astolfi | Integral action in output feedback for multiinput multi-output nonlinear systems[END_REF]).

On the other hand, Theorem 3.4 also states that dM inherits the Lipschitz properties of F and dF . Now, regarding Hypothesis 3.3, (3.7) is roughly speaking a sufficient condition under which solutions to the uncontrolled w-equation linearized around a given trajectory (also called first variation equation) uniformly converge to that trajectory. Equation (3.7) is easily verified in (at least) two situations of interest.

• The nonlinearity F contributes to the contraction behavior of the w-dynamics. This is the case for example if A is coercive and F is monotone (i.e. such that (dF (w)h, h) H is nonnegative for all w, h in H). More generally, the reader may refer to [START_REF] Lohmiller | On contraction analysis for non-linear systems[END_REF] in finite dimension or [Tem12, Chapter V] for the contraction analysis of the system by means of its first variation equation.

• Variations of F are small with respect to linear dissipation brought by

A. More precisely, if F is K-Lipschitz continuous, then dF (•) L(H) is bounded by K, hence (3.7) is satisfied if (Ah, h) H β h 2 H for some constant β > K.
In the same spirit, under (3.7) and (3.10), by observing1 that for each w the integral map in (3.9) has operator norm bounded by K/α, one deduces from (3.9) that global uniform coercivity (3.3) holds whenever K < α.

Remark 3.5 (Linear case). If A is linear, i.e., F = 0, and 0 is in the resolvent set of A, then the solution M to (2.7) is explicitly and uniquely determined as M = -CA -1 . This choice corresponds to the linear forwarding approach followed by [MBA21, TJAMDSX19, MAA22].

The case of unbounded output

We conclude Section 3 with an informal discussion regarding the convergence of the output Cw(t) towards the reference when the operator C is unbounded. To get some insight on the situation, assume for a moment that the original w-system is linear. Then, the solution M to (2.7) provided by Theorem 3.4 is a bounded linear operator, and the closed-loop dynamics around the equilibrium are governed by a strongly continuous linear semigroup, which commutes with its generator. Recalling the notation from Section 3.3, it follows that for a strong solution [w, z] to (2.4)-(2.9), w(t)w ⋆ goes to 0 in D(A) endowed the graph norm, hence Cw(t) converges to y ref provided that C is A-bounded. In the nonlinear case, the argument breaks down.

Alternatively, one may look for weaker notions of convergence. In many applications, unbounded output operators of interest enjoy an admissibility property with respect to the uncontrolled dynamics. In the linear theory (see, e.g., [TW09, Section 4.3]), C is said to be A-admissible if C is A-bounded and there exist positive constants K and T such that

T 0 CS t w 0 2 Z dt K w 0 2 H , ∀w 0 ∈ D(A).
(3.11)

In the semilinear case, one can deduce from (3.11) that, first of all, the output Cw is well-defined in L 2 loc (0, +∞; Z) even for weak solutions [w, z] to the closed-loop equations (2.4)-(2.9), and secondly, that the output converges "in average" to the reference:

lim τ →+∞ T +τ τ Cw(t) -y ref 2 Z dt = 0 (3.12)
for any (including weak) solution [w, z] that converges to the equilibrium [w

⋆ , z ⋆ ] in H × Z.
When no semilinear structure is prescribed for the maximal monotone operator A governing the w-dynamics, we generalize (3.11) by assuming that for any w 0 i in D(A) and f i absolutely continuous with derivative in L2 (0, T ; H), the solution 2 w i to dw

i /dt + A(w) = f i with initial condition w i (0) = w 0 i , i ∈ {1, 2}, satisfies T 0 Cw 1 (t) -Cw 2 (t) 2 Z dt K w 1 (0) -w 2 (0) 2 H + K T 0 f 1 -f 2 2 H dt. (3.13)
In that case, the same conclusions hold for closed-loop solutions.

Proofs of the main results

This section is devoted to the proofs of the results of the

[w ⋆ , z ⋆ ] ∈ H × Z, any solution [w, z] of (2.4)-(2.9) converges towards some [w ⋆ , z ⋆ ] ∈ H ×Z if and only if [w, η] converges towards [w ⋆ , η ⋆ ],
where η = z -M(z) and η ⋆ = z ⋆ -M(w ⋆ ). Moreover, given a ball of H containing w ⋆ , by local Lipschtz continuity of M, there exist positive constants K 1 and K 2 such that

K 1 [w -w ⋆ , z -z ⋆ ] H×Z [w -w ⋆ , η -η ⋆ ] H×Z K 2 [w -w ⋆ , z -z ⋆ ] H×Z (4.3)
whenever w lies in that ball. Therefore, well-posedness, global attractivity (3.4), and local exponential stability (3.2), of (2.4)-(2.9) and (4.2) are actually equivalent. Finally, to alleviate notation when needed, we define

K(w) dM(w)B ∈ L(U, Z), ∀w ∈ H. (4.4)
4.1 Proof of Theorem 3.1 (well-posedness)

We start by proving that solutions to (4.2) exist at least on a finite time interval (Items (i) and (ii) of Theorem 3.1), and then we investigate forward completeness (Item (iii) of Theorem 3.1).

Step 1: Local well-posedness. First, we observe that (4.2) represents a locally Lipschitz perturbation of the following maximal monotone problem:

d dt [w, η] + [A(w), η] = 0. (4.5)
Indeed, by letting Recalling that z = η + M(w) and M ∈ C 1 (H, Z), we obtain Items (i) and (ii). Finally, if T max is finite, then the norm of [w(t), η(t)] must go to +∞ as t approaches T max .

F d,y ref [w, η] -BK(w) * η -d -η + K(w)K(w) * η + y ref + dM(w)d , ∀[w, η] ∈ H × Z, ( 4
Step 2: Sufficient conditions for forward completeness. Let us prove that if take d = 0 or dM is globally Lipschitz continuous, then T max = +∞ for any initial condition [w 0 , η 0 ] ∈ H × Z. Let ρ > 0 to be fixed (large enough) later on. Given a (strong) solution [w, η] with initial data [w 0 , η 0 ] ∈ D(A) × Z, we have

1 2 d dt w 2 H + ρ η 2 Z = -(A(w), w) H + (BK(w) * η, w) H + (d, w) H -ρ K(w) * η 2 U -ρ(y ref , η) Z -ρ(dM(w)d, η) Z (4.8)
holding a.e. on (0, T max ). Similarly as in our preliminary Lyapunov analysis (2.8)-(2.10), in order to deal with the term BK(w) * η, we use (2.2) together with Young's inequality and choose 2ρ

B 2 L(U,H) to obtain d dt w 2 H + ρ η 2 Z 2 w 2 H + 2ρ η 2 Z + ρ y ref 2 Z + d 2 H + ρ dM(w)d 2 Z (4.9)
If d = 0 or dM is globally Lipschitz continuous, the following inequality holds:

dM(w)d 2 Z K(1 + w 2 H ) d 2 H (4.10)
for some K > 0 independent of w or d. Combining (4.9) and (4.10) yields

d dt w 2 H + ρ η 2 Z K ′ w 2 H + ρ η 2 Z + K ′ a.e. (4.11)
for some K ′ > 0 independent of the initial data. As a strong solution to (4.2), [w, η] is absolutely continuous in H ×Z; therefore, we can deduce from (4.11) and Grönwall's inequality the following uniform estimate: for all t ∈ [0, T max ),

w(t) 2 H + ρ η(t) 2 Z exp(K ′ t){ w 0 2 H + ρ η 0 2 Z + 1}. (4.12)
We infer from (4.12) that the norm of [w, η] cannot blow up in finite time; thus, T max = +∞. Furthermore, by passing to the limit, we see that (4.12) is satisfied for weak solutions as well, which means that the same conclusion holds for any initial data in H × Z. This proves Item (iii) and ends the proof of Theorem 3.1.

Proof of Theorem 3.2 (output regulation)

Our main result concerning the output regulation problem is demonstrated here, along with some auxiliary results.

Outline of the proof and intermediate results

The proof of Theorem 3.2 relies on a series of lemmas that are given below. Let us first give some insight on the main strategy.

1. Under the range condition (3.1), the operators K(w) * = B * dM(w) * involved in the ηequation (4.2b) enjoy a coercivity property that is uniform with respect to w, provided that w is small. This is shown in Lemma 4.1.

2. This allows to prove that there exists a (not necessarily invariant) region around the origin of H × Z where the dynamics generated by (4.2) are strictly contractive. This is stated in Lemma 4.2.

3. Another consequence of the local coercivity property is the existence of suitable attracting sets for the closed-loop dynamics (4.2), as demonstrated in Lemma 4.3. Furthermore, if the reference y ref and the perturbation d are sufficiently small, then those sets are contained in the contraction region of the previous step.

4. Finally, existence of a locally exponentially stable equilibrium for (4.2) is shown by using Banach fixed point arguments and properties of ω-limit sets associated with contraction semigroups.

5. Additionally, under the global coercivity condition (3.3), the attracting sets from Lemma 4.3 are in fact globally attractive, hence the equilibrium is also globally asymptotically stable.

Coming back to the [w, z]-coordinates and thus to the original closed-loop system (2.4)-(2.9) is then straightforward. In what follows, we shall be careful regarding the dependence of the various neighborhoods involved in the analysis with respect to each parameter. Before proceeding further, we introduce some additional notation. Recalling H×Z,ρ . In the sequel, we either take d = 0 or assume that dM is globally Lipschitz continuous. Therefore, by virtue of Theorem 3.1, solutions to the closed-loop equations (4.2) are well-defined for all positive time and any initial condition in H × Z. Recall from (3.1) in the statement of Theorem 3.2 that the following coercivity assumption is in force:

K(0) * z 2 U λ z 2 Z , ∀z ∈ Z. (4.14)
The first lemma is a consequence of (4.14) and continuity of dM.

Lemma 4.1 (Local coercivity).

There exist positive constants λ and r such that

K(w) * z 2 U λ z 2 Z , ∀z ∈ Z, ∀w ∈ B H (0, r). (4.15) Proof. Let w ∈ H and z ∈ Z. Recall that K(w) * z = B * dM(0) * z + B * [dM(w) -dM(0)] * z.
Using Cauchy-Schwarz and Young inequalities together with (4.14), we get

K(w) * z 2 U 2λ 3 z 2 H -2 B 2 L(U,H) [dM(w) -dM(0)] * 2 L(Z,U) z 2 Z . (4.16)
By continuity of dM at 0, we can choose r > 0 such that dM(w) -dM(0) 2

L(H,Z) λ 6 B 2 L(U,H)
, ∀w ∈ B H (0, r).

(4.17) Thus, by letting λ λ/3 > 0, we obtain the desired inequality (4.15).

The following constant appears in the next two lemmas:

κ min{α/4, λ/4}. ( 4

Lemma 4.2 (Local strong monotonicity).

There exists ρ 0 > 0 such that the following property holds: for all ρ ρ 0 , there exists a positive r 0,ρ r such that

( Ãd,y ref [w 1 , η 1 ] -Ãd,y ref [w 2 , η 2 ], [w 1 , η 1 ] -[w 2 , η 2 ]) H×Z,ρ κ [w 1 , η 1 ] -[w 2 , η 2 ] 2 H×Z,ρ (4.19) for all [w 1 , η 1 ], [w 2 , η 2 ] ∈ B H×Z (0, r 0,ρ ) ∩ D(A) × Z, d ∈ B H (0, r 0,ρ ) and y ref ∈ Z. Proof. Let [w 1 , η 2 ] and [w 2 , η 2 ] in D(A) × Z.
We write w w 1w 2 and η η 1η 2 . Then, for any ρ > 0,

( Ãd,y ref [w 1 , η 1 ] -Ãd,y ref [w 2 , η 2 ], [ w, η]) H×Z,ρ = (A(w 1 ) -A(w 2 ) -B[K(w 1 ) * η 1 -K(w 2 ) * η 2 ], w) H + ρ(K(w 1 )K(w 1 ) * η 1 -K(w 2 )K(w 2 ) * η 2 + [dM(w 1 ) -dM(w 2 )]d, η) Z . (4.20)
By adding and removing some terms, (4.20) can be rewritten as follows:

( Ãd,y ref [w 1 , η 1 ] -Ãd,y ref [w 2 , η 2 ], [ w, η]) H×Z,ρ = (A(w 1 ) -A(w 2 ), w) H + ρ(K(w 1 )K(w 1 ) * η, η) Z + ρ([K(w 1 )K(w 1 ) * -K(w 2 )K(w 2 ) * ]η 2 , η) Z -(BK(w 1 ) * η, w) H -(B[K(w 1 ) -K(w 2 )] * η 2 , w) H + ρ([dM(w 1 ) -dM(w 2 )]d, η) Z . (4.21)
Assume for a moment that w i H r/2, where r is given by Lemma 4.1. Therefore, w H r; and using (2.2) and (4.15) together with the Lipschtz continuity of K and K(•)K(•) * on B H (0, r), we infer from (4.21) that

( Ãd,y ref [w 1 , η 1 ] -Ãd,y ref [w 2 , η 2 ], [ w, η]) H×Z,ρ α w 2 H + ρ λ η 2 Z -ρK 1 w H η 2 Z η Z -K 2 η Z w H -K 3 η 2 Z w 2 H -ρK 4 w H d H η Z . (4.22)
where the K i are some positive constants independent of [w i , η i ], [d, y ref ] and ρ. Given ε > 0, we employ Cauchy-Schwarz and Young inequalities to obtain

( Ãd,y ref [w 1 , η 1 ] -Ãd,y ref [w 2 , η 2 ], [ w, η]) H×Z,ρ α w 2 H + ρ λ η 2 Z - ρ 2 {K 1 η 2 Z + K 4 d H }{ w 2 H + η 2 Z } - K 2 2ε η 2 Z - εK 2 2 w 2 H -K 3 η 2 Z w 2 H . (4.23)
Let ε = α/(2K 2 ) in (4.23) and define ρ 0 2K 2 /( λε). For all ρ ρ 0 ,

( Ãd,y ref [w 1 , η 1 ] -Ãd,y ref [w 2 , η 2 ], [ w, η]) H×Z,ρ α 2 w 2 H + ρ λ 2 η 2 Z - ρ 2 {K 1 η 2 Z + K 4 d H }{ w 2 H + η 2 Z } -K 3 η 2 Z w 2 H . (4.24)
Let ρ ρ 0 . We infer from (4.24) that there exists r 0,ρ r/2 such that

( Ãd,y ref [w 1 , η 1 ] -Ãd,y ref [w 2 , η 2 ], [ w, η]) H×Z,ρ α 4 w 2 H + ρ λ 4 η 2 Z (4.25)
as long as [w i , η i ] ∈ B H×Z (0, r 0,ρ ) and d ∈ B H (0, r 0,ρ ), which completes the proof.

Lemma 4.3 (Absorbing balls).

There exists ρ 1 > 0 such that the following property holds: for any ρ ρ 1 , there exist positive r 1,ρ and

K ρ such that, if [d, y ref ] ∈ B H×Z (0, r 1,ρ ), then the estimate [w(t), η(t)] 2 H×Z,ρ exp(-κt) [w 0 , η 0 ] 2 H×Z,ρ + K ρ {1 -exp(-κt)} [d,
-α w 2 H + 1 2ε B 2 L(U,H) K(w) * η 2 U + ε w 2 H + 1 2ε d 2 H -ρ K(w) * η 2 U + ρ 2µ y ref 2 Z + ρ 2µ dM(w)d 2 Z + ρµ η 2 Z . (4.30)
First, recall that we have either dM globally Lipschitz continuous or d = 0. Hence, there exists k > 0 independent of [w 0 , η 0 ] and [d,

y ref ] such that dM(w)d 2 Z k d 2 H
. By choosing ε = α/2 and µ = λ/4, we deduce from (4.30) the following differential inequality, valid on (0, 1):

1 2 d dt [w, η] 2 H×Z,ρ - α 2 w 2 H - ρ λ 4 η 2 Z + ρ 2µ y ref 2 Z + ρk 2µ + 1 2ε d 2 H (4.

Proof of Theorem 3.2

We can now prove the main result.

Step 1: Setting all neighborhoods. Pick ρ such that ρ max{ρ 0 , ρ 1 } as in Lemmas 4.2 and 4.3. In the sequel, [d, y ref ] is assumed to lie in the intersection of B H (0, r 0,ρ ) × Z and B H×Z (0, r 1,ρ ), so that the lemmas apply. Now that ρ is fixed, we will omit the dependence on ρ in further notation. Lemma 4.2 provides a neighborhood K of the origin in H × Z, which we will refer to as the contraction region, where (4.19) holds. On the other hand, according to (4.26) in Lemma 4.3, the set

V d,y ref B H×Z,ρ (0, K 1/2 ρ [d, y ref ] H×Z ) (4.34)
attracts all solutions to (4.2) originating from B H×Z (0, r 1,ρ ). By norm equivalence, any sufficiently small ball for the ρ-norm that is centered at the origin is contained in both K and B H×Z (0, r 1,ρ ). That being said, as a consequence of (4.26) and (4.34), there exists positive numbers δ and r such that, having let B B H×Z,ρ (0, δ), the following properties hold for any [d, y ref ] ∈ B H×Z (0, r):

• The closure of the corresponding attracting set V d,y ref is contained in B;

• Solutions to (4.2) with initial data in B remain in K.

If in addition we assume that (3.3) holds, i.e., (4.15) holds globally, then by Lemma 4.3,

• The set V d,y ref attracts all solutions to (4.2), whatever the initial data.

In what follows, we omit the dependence on [d, y ref ] in the notation and we denote by { Tt } the evolution semigroup associated with (4.2). Then, since solutions originating from (the nonempty open set) B remain in the contraction region K, we infer from (4.19) together with a density argument that

Tt [w 1 , η 1 ] -Tt [w 2 , η 2 ] H×Z,ρ exp(-κt) [w 1 , η 1 ] -[w 0 , η 0 ] H×Z,ρ (4.35)
for all t 0 and [w i , η i ] ∈ B, i ∈ {1, 2}.

Step 2: Existence of a fixed point. Pick an arbitrary [w 0 , η 0 ] ∈ B. By (4.35) and a usual contraction argument, we see that

{ Tn [w 0 , η 0 ]} n 0 is a Cauchy sequence in H × Z (4.36)
and converges to a fixed point [w ⋆ 0 , η ⋆ 0 ] of the nonlinear operator T1 . Now, consider the ω-limit set ω([w 0 , η 0 ]) of [w 0 , η 0 ] with respect to the evolution semigroup { Tt }. By the sequential characterization of ω-limit sets (see [Chu02, Lemma 2.1, p.19]), we observe that [w ⋆ 0 , η ⋆ 0 ] ∈ ω([w 0 , η 0 ]), which means that ω([w 0 , η 0 ]) is non-empty. Moreover, it is positively invariant by definition. Finally, since V d,y ref attracts all solutions originating from B, we must have

ω([w 0 , η 0 ]) ⊂ V d,y ref ⊂ B.
(4.37)

By following verbatim3 the proof of [DS73, Theorem 1], we obtain that for each t 0, Tt is an isometry on ω([w 0 , η 0 ]). On the other hand, for positive t, Tt is a strict contraction on ω([w 0 , η 0 ]); thus, ω([w 0 , η 0 ]) must be reduced to the singleton

{[w ⋆ 0 , η ⋆ 0 ]}. By invariance of the ω-limit set, [w ⋆ 0 , η ⋆ 0 ] is fixed by the semigroup { Tt }. Moreover, it follows from (4.35) that [w ⋆ 0 , η ⋆ 0 ] is the unique fixed point of { Tt } in B and is exponentially attractive in B. Thus, we write [w ⋆ , η ⋆ ] [w ⋆ 0 , η ⋆ 0 ].
Step 3: The fixed point lies in the domain. We now prove that w ⋆ ∈ D(A). Let ε > 0 sufficiently small and consider the ball for some t 0 0. In particular, A(w) H is bounded a.e. on (t 0 , +∞). On the other hand, w(t) converges to w ⋆ in H when t goes to +∞. Therefore, it follows from A being maximal monotone and [CP69, Lemma 2.3] that w ⋆ ∈ D(A).

C B H×Z,ρ ([w ⋆ , η ⋆ ], ε) ⊂ K. Since D(A) is dense in H and C has non-empty interior, we can pick some [w 0 , η 0 ] ∈ [D(A)×Z]∩C. Let [w(t), η(t)] Tt [w 0 , η 0 ]. As a strong solution to (4.2), [w, η] is differentiable in H × Z for a.
Step 4: Conclusion. Since [w ⋆ , η ⋆ ] belongs to D(A)×Z and is fixed by { Tt }, Ãd,y ref [w ⋆ , η ⋆ ] = 0. We come back to the original [w, z] coordinates by letting z

⋆ η ⋆ + M(w ⋆ ) and N {[w, η + M(w)], [w, η] ∈ B}. Then, [w ⋆ , z ⋆ ] belongs to [D(A) × Z] ∩ N
and is an equilibrium for (2.4)-(2.9); hence, Cw ⋆ = y ref .

Because M vanishes at 0 and is continuous, N is indeed a neighborhood of 0. Local exponential stability of [w ⋆ , z ⋆ ] with decay rate κ and bassin of attraction containing N follows from (4.35) and our prior remarks regarding the change of coordinates; the constant M in (2.3) comes from (4.3) and equivalence with the ρ-norm. Additionally, under the stronger condition (3.3), [w ⋆ , z ⋆ ] is globally asymptotically stable. The proof is now complete.

Proof of Theorem 3.4 (semilinear case)

We first give some auxiliary results valid under the hypotheses of Section 3.3, and then we prove Theorem 3.4.

Preliminaries

We endow D(A) with the graph norm. First, we claim that weak solutions (in the sense of nonlinear semigroup theory) to dw dt + A(w) = 0. ). Given w 0 in H, the next lemma states that the nonlinear operators T t are all differentiable at w 0 and their differentials coincide with the evolution family associated with (4.45). This is a classical fact for sufficiently smooth nonlinear dynamics; here, we give a proof that matches our particular set of hypotheses.

Lemma 4.5 (Differentiability of the semigroup). Each operator T t is Fréchet differentiable. Furthermore, for any t 0 and w 0 ∈ H, the differential dT t (w 0 ) is given by dT t (w 0 )h = v(t; w 0 , h) for all h ∈ H, (4.46)

where t → v(t; w 0 , h) is the unique mild solution to (4.45) with initial data h.

T t (w 0 + h) -T t w 0 = S t h - t 0 S t-s dF (T s w 0 ){T s (w 0 + h) -T s w 0 } ds - t 0 S t-s R(T s w 0 , T s (w 0 + h) -T s w 0 ) ds. (4.48)
Omitting the dependence on w 0 and h for the moment, we write v(t) v(t; w 0 , h) and R(t) T t (w 0 + h) -T t w 0v(t). Now, taking the difference between (4.48) and the integral identity satisfied by v as a mild solution to (4.45), one obtains

R(t) = t 0 S t-s dF (T s w 0 )R(s) ds - t 0 S t-s R(T s w 0 , T s (w 0 + h) -T s w 0 ) ds (4.49)
holding for all 0 t τ . Next, because dF is continuous on the compact set {T s w 0 , 0 s τ }, dF (T s w 0 ) L(H) is bounded by some m independent of h. Besides, S s L(H) 1 for all s 0. Therefore, it follows from (4.49) that for all 0 t τ ,

R(t) H m t 0 R(s) H ds + t 0 R(T s w 0 , T s (w 0 + h) -T s w 0 ) H ds.
(4.50) Using Grönwall's inequality in its integral form, we deduce from (4.50) that

R(τ ) H h H exp(mτ ) τ 0 R(T s w 0 , T s (w 0 + h) -T s w 0 ) H h H ds. (4.51)
To obtain the desired differentiability property, it suffices to show that the right-hand side of (4.51) converges to 0 as h H goes to 0. This is done using Lebegue's dominated convergence theorem. Let s ∈ [0, τ ]. By the contraction property of {T t }, for any nonzero h ∈ H such that T s (w 0 + h) -T s w 0 is nonzero, we have

R(T s w 0 , T s (w 0 + h) -T s w 0 ) H h H R(T s w 0 , T s (w 0 + h) -T s w 0 ) H T s (w 0 + h) -T s w 0 H , (4.52)
and if T s (w 0 + h) -T s w 0 = 0, then R(T s w 0 , T s (w 0 + h) -T s w 0 ) = 0. Either way, when h H → 0, T s (w 0 + h) -T s w 0 H → 0, and by definition of the residual term R, the right-hand side of (4.52) must converge to 0 as well. To conclude the proof, let us estimate the left-hand side of (4.52) uniformly with respect to h. We observe that the set of all points T s (w 0 + h), 0 s τ , h H 1, is contained in some open ball, on which F is K-Lipschitz continuous for some K > 0. Thus, using (4.47), one obtains that the left-hand side of (4.52) is smaller than 2K. First, T t w 1 and T t w 2 must remain in B H (0, R), where dF is, say, K R -Lipschitz continuous. Since dF (0) = 0, for all t 0 we have dF Step 3: Conclusion. At this point, it remains to check that our candidate M is a solution to (2.7). It is clear that M(0) = 0. Take w 0 in D(A) and consider the associated strong solution

(T t w 1 ) L(H) K R T t w 1 H K R w 1 H . ( 4
w t → T t w 0 ∈ C(R + , D(A)) ∩ C 1 (R + , H) to (4.40). Then, M(T t w 0 ) -M(w 0 ) t = C 1 t t 0 T s w 0 ds . (4.66)
As w is continuous in D(A), the term t -1 t 0 T s w 0 ds converges to w 0 in D(A) when t approaches 0. Thus, C being A-bounded, the difference quotient in (4.66) converges to Cw 0 in Z. On the other hand, since w ∈ C 1 (R + , H) and M ∈ C 1 (H, Z), by the chain rule, we have d dt [M(T t w 0 )] = -dM(T t w 0 )A(T t w 0 ), ∀t 0. (4.67)

Evaluating (4.67) at t = 0 yields dM(w 0 )A(w 0 ) + Cw 0 = 0 by uniqueness of the limit.

the consequence that for each bounded output operator C, M defined by M(w) -C +∞ 0

T t w dt is a solution to (2.7) with locally Lipchitz continuous differential.

Let w 0 1 and w 0 2 be taken in H, with w 0 i H

R.

In what follows, approximation arguments are omitted. Denote by w i the corresponding solution t → T t w 0 i to (5.10). Let h ∈ H and v i (t) dT t (w i )h, t 0. Then, by (5.11

), v v 1 -v 2 solves ∂ t v -∂ xx v + 3w 2 1 v 1 -3w 2 2 v 2 = 0 in (0, L) × (0, +∞), (5.12a) v(0, t) = v(L, t) = 0 for t ∈ (0, +∞), (5.12b)
with v(•, 0) = 0. Letting w w 1w 2 , we infer from (5.12a) that

∂ t v -∂ xx v + 3w 2 1 v + 3w{w 1 + w 2 }v 2 = 0 in (0, ) × (0, +∞).
(5.13) Let τ > 0 and Q τ Ω × (0, τ ). Multiplying (5.13) by v and integrating over

Q τ yields 1 2 Ω |v(•, τ )| 2 dx + Qτ |∂ x v| 2 + 3w 2 1 |v| 2 + 3{w 1 + w 2 }v 2 wv dx dt = 0, (5.14)
For a.e. (x, t) in Q τ , we have

|{w 1 (x, t) + w 2 (x, t)}v 2 (x, t)w(x, t)v(x, t)| K w 1 (t) H 1 0 (Ω) + w 2 (t) H 1 0 (Ω) v(t) H 1 0 (Ω) |w(x, t)v 2 (x, t)|, (5.15)
where K > 0 comes from the continuous embedding H 1 0 (Ω) ֒→ L ∞ (Ω) and does not depend on τ . Integrating (5.15) over Ω, we obtain that for a.e. t ∈ (0, τ ), (5.24) Equation (5.24) is valid for all h ∈ H and any w 0 i in the ball of (arbitrary) radius R in H. Thus, similarly as in the proof of Theorem 3.4, our first claim that w → Remark 5.1. example shows that the arguments used in Section 4.3 can be adapted beyond the semilinear context to derive existence and suitable properties of M and dM using specific features of the system, such as additional regularity and dissipativity in higher-order norms.

Ω |{w 1 (x, t) + w 2 (x, t)}v 2 (x, t)w(x, t)v(x, t)| dx K w 1 (t) H 1 0 (Ω) + w 2 (t) H 1 0 (Ω) v(t) H 1 0 (Ω) w(t) H v 2 (t

Conclusions

We finish our article with some comments on our results. We wish to point out two notable byproducts of our approach in terms of robustness.

In order to take into account nonlinear behavior of actuators (e.g., saturation) in the feedback loop, one may investigate stability properties of (2.4)-(2.9) when a nonlinearity g is applied to the input u. This is also relevant in applications where the control signal must satisfy some prescribed bound in norm (see e.g. [START_REF] Marx | Forwarding techniques for the global stabilization of dissipative infinite-dimensional systems coupled with an ODE[END_REF]). Consider a Lipschitz continuous map g that vanishes at 0 and is strongly monotone in some neighborhood of 0. Then, the local (strict) contraction property of the [w, η]-dynamics is preserved. This leaves room for a possible adaptation of Theorem 3.2 in the case of saturated or a priori bounded control.

We also believe that our framework provides tools to analyse the behavior of the closed-loop (2.4)-(2.9) under certain time-varying disturbances. Indeed, given [d 0 , y ref ] as in Theorem 3.2, to which we associate an equilibrium [w ⋆ , z ⋆ ], consider a disturbance of the form d(t) = d 0 + d 1 (t) with d 1 small in L 2 (0, +∞; H). It can then be deduced from (4.19) that the system (2.4)-(2.9) is incrementally input-to-state stable in a neighborhood of [w ⋆ , z ⋆ ], allowing us to quantify the deviation from equilibrium due to the exogenous signal d 1 in terms of its L 2 -norm.

Finally, putting aside the problem of output regulation and following [MBA21, MAA22], we might be interested in stabilizing the cascade composed of the (actuated) w-subsystem and a more general z-subsystem governed by dz/dt = Sz + Cw, where S is a skew-adjoint operator on Z. This would require to investigate a nonlinear Sylvester equation of the form dM(w)A(w) + SM(w) Cw = 0, ∀w ∈ D(A).

(6.1)

Under the condition that a sufficiently regular solution M to (6.1) exists, the Lyapunov analysis performed in Section 2.2 remains valid, which is a good starting point for analysing stability of the new closed-loop with control law given by (2.9).

  (4.6) and (4.7), given [d, y ref ], we denote by Ãd,y ref the nonlinear operator [A, id] + F d,y ref defined on D(A) × Z and associated with the closed-loop system (4.2) in the [w, η]-coordinates. Also, for ρ > 0, we denote by • H×Z,ρ the Hilbert norm on H × Z given by [w, η] 2 H×Z,ρ w 2 H + ρ η 2 Z , ∀[w, η] ∈ H × Z. (4.13) All these norms are equivalent. We denote by (•, •) H×Z,ρ the associated scalar product and by B H×Z,ρ ([w, η], r) the open ball of radius r > 0 centered at [w, η] ∈ H ×Z with respect to the norm • H×Z,ρ . Bearing in mind our Lyapunov analysis in the [w, z]-coordinates in Section 2.2, the ρ-norm is connected to the Lyapunov function V by V (w, z) = (1/2) [w, z -M(w) 2

  .18) Those concern the contraction property of the dynamics governed by (4.2) around the origin and the existence of attractive sets depending on [d, y ref ].

  e. time and d dt [w, η] + Ãd,y ref [w, η] = 0 a.e. (4.38) Besides, we infer from (4.35) that C is positively invariant so that { Tt } restricted to C is still a well-defined contraction semigroup. Thus, we can apply [CP69, Theorem 1.4] to obtain Ãd,y ref [w, η] H×Z,ρ Ãd,y ref [w(t 0 ), η(t 0 )] H×Z,ρ a.e. on (t 0 , +∞) (4.39)

0 S→+∞ τ 0 TF 0 T 0 F

 0000 (4.40) coincide with mild solutions (in the sense of perturbation of linear equation) -see, e.g.,[START_REF] Curtain | Introduction to infinite-dimensional systems theory[END_REF] Theorem 11.1.5]. In other words, solutions t → T t w 0 to (4.40) with initial data w 0 ∈ H are characterized byT t w 0 = S t w 0 -t t-s F (T s w 0 ) ds, ∀t 0.(4.41)Strong solutions w to (4.40) enjoy the regularity w∈ C 1 (R + , H) ∩ C(R + , D(A)). Lemma 4.4given below allows us to circumvent the possible unboundedness of the output operator C and, together with the A-boundeness of C, guarantees that M given by the limit in (3.8) is a welldefined nonlinear mapping on the whole space H.Lemma 4.4 (Integral formula). For any w 0 ∈ H,lim τ t w 0 dt = A -1 w 0 -A -1 +∞ 0 (T t w 0 ) dt. (4.42)In particular, lim τ →+∞τ 0 T t (w 0 ) dt belongs to D(A). Proof. Let τ > 0 and w 0 ∈ D(A). Then, t → T t w 0 ∈ C 1 (R + , H) ∩ C(R + , D(A)) solves (4.40) in a classical sense. After applying the bounded operator A -1 to the differential equation, one obtains d dt [A -1 T t w 0 ] + T t w 0 + A -1 F (T t w 0 ) = 0, ∀t 0. (4.43) Integrating (4.43) over (0, τ ) yields τ t w 0 dt = A -1 w 0 -A -1 T τ w 0 -A -1 τ (T t w 0 ) dt. (4.44)

Proof. Let τ 0 and w 0

 0 ∈ H. It is clear that the mapping h → v(τ ; w 0 , h) is linear; it is also continuous by [Paz12, Theorem 1.2, p.184]. First, since F is differentiable, the following Taylor formula holds: F (a + b) -F (a) = dF (a)b + R(a, b), ∀a, b ∈ H, (4.47) with R(a, b) = o( b H ) when b H → 0 for fixed a ∈ H. Now, take a nonzero h ∈ H. Combining the variation of the constant formula (4.41) with (4.47) leads to

  +∞ 0 dT t (w) dt is a locally Lipschitz continuous map of H into L(H) follows. Let B : U → H and C : H → Z be a bounded input and output operators. With our choice of M, the "non-resonance condition" reads as Range CA -1 B = Z, where A is -∂ xx with zero Dirichlet boundary conditions. As in Section 5.1, we choose a non-empty open subset O of Ω and let Bu = u1 O , u ∈ R. By virtue of Theorem 3.2 (in the case that dM is only locally Lipschitz continuous, which requires d = 0), this localized scalar control achieves local regulation of the nonlinear reaction-diffusion equation (5.10) with one-dimensional bounded output maps such as the mean-value Cw = Ω w.

  The nonlinear map F d,y ref is locally Lipschitz continuous on H × Z. Besides, since A is maximal monotone on H, so is the mapping [w, η] → [A(w), η] on H × Z (with dense domain D(A) × Z). Furthermore, [A(0), 0] = 0. Thus, it follows from [CEL02, Theorem 7.2] that for each initial condition [w 0 , η 0 ] ∈ H × Z, there exists a unique maximal weak solution [w, η] to (4.2) defined on [0, T max ), with T max ∈ (0, +∞]. Moreover, if w 0 ∈ D(A), [w, η] is actually a strong solution.

	we can rewrite (4.2) as follows:
	d dt	[w, η] + [A(w), η] + F d,y ref [w, η] = 0.	(4.7)
			.6)

  y ref ] 2

						in the
	proof of Theorem 3.1, using (4.8) we obtain
			d dt	[w, η] 2 H×Z,ρ	k ρ { [w, η] 2 H×Z,ρ + y ref	2 Z + d 2 H }	(4.28)
	for some k 0, 1) we have
	1 2	d dt	[w, η] 2 H×Z,ρ

H×Z

(4.26) holds for any solution [w, η] to (4.2) with initial data [w 0 , η 0 ] in B H×Z (0, r 1,ρ ). Furthermore, if (4.15) holds globally, then (4.26) is true for all initial data in H × Z.

Proof. All formal computations performed below are justified by considering appropriate sequence of strong solutions as provided by Theorem 3.1 and then passing to the limit at the very end. First, let

ρ 1 B 2 L(U,H) max{1, 2α

-1 }. (4.27) Let ρ ρ 1 be fixed and let [w, η] be a solution to (4.2) with initial condition [w 0 , η 0 ]. As ρ > 0 independent of [w 0 , η 0 ] and [d, y ref ]. It follows from (4.28) that [w(t), η(t)] 2 H×Z,ρ exp(k ρ ) [w 0 , η 0 ] 2 H×Z,ρ + exp(k ρ ){ y ref 2 Z + d 2 H } (4.29) for all t ∈ [0, 1]. As a consequence of (4.29), there exists a positive constant r ρ such that max{ w 0 H , η 0 Z , d H , y ref Z } r ρ implies w(t) H r for all t ∈ [0, 1], where r is defined in Lemma 4.1. Therefore, for such initial data, we can use (4.15) in (4.8) to refine our previous estimate: for all positive ε and µ, on (

  ∈ [0, 1], where κ is defined in (4.18) and K ρ 1 is some constant independent of [w 0 , η 0 ] and [d, y ref ]. Next, by norm equivalence, there exists r 1,ρ > 0 such that the following implication holds: if [w 0 , η 0 ] and [d, y ref ] are in B H×Z (0, r 1,ρ ), then [w 0 , η 0 ] H×Z,ρ 2 -1/2 r ρ , and [d, y ref ] H×Z (2K ρ ) -1/2 r ρ . H×Z,ρ r ρ , and so on. The conclusion readily follows by induction. Moreover, if it is assumed that (4.15) holds for all w ∈ H, then (4.31) is valid on (0, +∞) whatever the initial condition, so that (4.29) immediately holds for all t 0. In this case, no additional condition on [d, y ref ] is required.

	Applying Grönwall's inequality to (4.31) yields
	[w(t), η(t)] 2 H×Z,ρ
		exp(-κt) [w 0 , η 0 ] 2 H×Z,ρ + K ρ (1 -exp(-κt)) [d, y ref ] 2 H×Z	(4.32)
	for all t (4.33)
	Now, we claim that the estimate (4.32) remains valid for all t	0. Indeed, (4.32) shows that
	[w(1), η(1)] H×Z,ρ	r ρ . Therefore, by definition of r ρ , we infer from the estimate (4.29)
	applied to the initial data [w(1), η(1)] that w(t) H	r for all t ∈ [1, 2]. As a consequence, the
	differential inequality (4.31) is valid on (0, 2); hence, (4.32) holds on [0, 2], with in particular
	[w(2), η(2)
			31)

  Because the map w 0 → (t → T t w 0 ) is continuous from H to C([0, τ ], H), a density argument shows that (4.44) is actually valid for all w 0 ∈ H. To obtain (4.42), it then suffices to let τ → +∞, having in mind that T τ w 0 → 0 in H and the integral at the right-hand side of (4.44) is absolutely convergent.

	Now, let us come back to the uncontrolled w-equation (4.40), which we linearize around a
	given trajectory {T t w 0 , t 0}:
	dv dt	+ Av + dF (T t w 0 )v = 0.	(4.45)
	Equation (4.45) is linear but non-autonomous in general. Given h ∈ H, (4.45) possesses a unique
	mild solution t → v(t) = v(t; w 0 , h) satisfying v(0) = h and v ∈ C(R + , H) (see [Paz12, Theorem
	1.2, p.184]

  .59) Now, let us estimate dT t (w 1 ) -dT t (w 2 ) in operator norm. Pick h ∈ H. In what follows, we denote by v(t) the difference v(t) dT t (w 1 )h -dT t (w 2 )h. Then, v(0) = 0 and v is a (mild) solution to the following non-automonous equation:

				dv dt	+ Av + [dF (T t w 1 )dT t (w 1 ) -dF (T t w 2 )dT t (w 2 )]h = 0,	(4.60)
	which can be rewritten as
			dv dt	+ Av + dF (T t w 1 )v + [dF (T t w 1 ) -dF (T t w 2 )]dT t (w 2 )h = 0.	(4.61)
	Justifications for the formal computations performed below are similar to those in the proof of
	Lemma 4.6; they are omitted here. Taking the scalar product in H of (4.61) with v and using
	(3.7) along with Cauchy-Schwarz and Young inequalities leads to
	1 2	d dt	v 2 H	-	α 2	v 2 H +	1 2α	[dF (T t w 1 ) -dF (T t w 2 )]dT t (w 2 ) 2 L(H) h 2 H .	(4.62)
	Since v(0) = 0, we deduce from (4.62) multiplied by exp(αt), (4.53) and (2.3) that
					v(t) 2 H	K 2 R exp(-αt) 3α 2	w 1 -w 2	2 H h 2 H , ∀t 0.	(4.63)
	We infer from (4.59) and (4.63) that
		dF (T

t w 1 )[dT t (w 1 ) -dT t (w 2 )] L(H) mRK 2 R exp(-αt/2) w 1w 2 H (4.64)

for all t 0, where m is some constant independent of R; on the other hand, coming back to the second term of (4.58), we also have

[dF (T t w 1 ) -dF (T t w 2 )]dT t (w 2 ) L(H) K R exp(-2αt) w 1w 2 H . (4.65)

Thus, the desired local Lipschitz continuity is obtained by applying the triangular inequality to (4.58) and integrating (4.64) and (4.65) over (0, +∞). Furthermore, if we assume that both F and dF are globally Lipschitz continuous, then for some K F we can choose K R = K F independent of R and replace (4.59) with dF (T t w 1 ) L(H) K F , thereby proving global Lipschitz continuity of the differential dM.

See Lemma 4.6 below for the bound K/α.

Existence and uniqueness of a strong solution is guaranteed by [Sho13, Chapter 4, Theorem 4.1]. In particular, each w i is absolutely continuous; hence A(w i ) is measurable, and so is Cw i by (2.7).

The only difference with [DS73, Theorem 1] is that { Tt} is a contraction only on a region (containing the ωlimit set) that is not a priori positively invariant. However, in order to obtain the isometry property, contraction is only needed on the points of the ω-limit set.

Consider for instance the image by CA -1 B of [0, 1 I ] where I is some segment contained in O.

Acknowledgments

The authors would like to thank Vincent Andrieu, Daniele Astolfi and Christophe Prieur for many fruitful discussions.

This work was partially funded by the French Grants ANR ODISSE (ANR-19-CE48-0004-01) and MIAI@Grenoble Alpes (ANR-19-P3IA-0003).

We continue by establishing exponential decay of solutions to (4.45).

Lemma 4.6 (Stability of the linearized equation). Let w 0 ∈ H. For any h ∈ H, the solution t → v(t; w 0 , h) to (4.45) with initial data h satisfies v(t; w 0 , h) H exp(-αt) h H , ∀t 0.

(4.53)

Proof. Let w 0 and h in H. Let τ 0 and f t → dF (T t w 0 )v(t) ∈ C([0, τ ], H). Then, pick sequences {h n } ⊂ D(A) and absolutely continuous {f n } such that h n → h in H and f n → f in L 2 (0, τ ; H). For each n, there exists a unique strong solution v n to dv n /dt + Av n + f n (t) = 0 satisfying the initial condition v n (0) = h n . Furthermore,

We deduce from (4.55) that for all 0 t τ ,

As n goes to +∞, the integral term in (4.56) tends to 0 and we obtain the desired result by passing to the limit.

Proof of Theorem 3.4

Now that we have established that all objects in the statement of the theorem are well-defined, we can give the proof of the result.

when h H goes to 0. We use Lebesgue's dominated convergence theorem. Since F and T t are differentiable, by the chain rule, the integrand in (4.57) converges to 0 pointwise. Let us now find some integrable dominating function. As in the proof of Lemma 4.5, we can find some open ball where F is K-Lipschitz and which contains the set of all T t (w 0 + h), h H 1, t 0. Thus, it follows from (2.3) and (4.53) that the integrand in (4.57) is dominated by t → 2K exp(-αt), which is integrable.

Step 2: Lipschitz continuity of the differential. Pick two elements w 1 and w 2 in H. We write R max{ w 1 H , w 2 H }. Subsequent estimates are motivated by the following decomposition:

Examples

We provide three examples for which our previous results apply. In what follows, we use standard notation for real-valued Lebesgue and Sobolev spaces.

Sine-Gordon equation

Let ξ, γ, and L be positive constants. Let O be a non-empty open subset of Ω (0, L). Consider the following damped sine-Gordon equation with control acting on O and homogeneous Dirichlet boundary conditions:

(5.1b)

In this example inspired by [Tem12, Chapter IV], (5.1) may represent the voltage dynamics of the continuous limit case for coupled Josephson junctions, with the control u being proportional to the applied current. The uncontrolled dynamics generated by (5.1) are well-posed on the energy space H H 1 0 (Ω) × L 2 (Ω). Using the state variable w = [θ, ∂ t θ], we can recast (5.1) into a semilinear evolution problem on H as in Section 3.3. Letting D(A) [H 2 (Ω) × H 1 0 (Ω)] ∩ H and U R, we define the unbounded linear operator A, the input operator B, and the nonlinear mapping F by

)

)

As an output, consider the Neumann trace at, say, x = 0:

which is modeled by an unbounded (but A-bounded) operator. That A -1 exists in L(H) can be proved using Riesz representation theorem in H 1 0 (Ω). Besides, F is in C 1 (H) and both F and dF are globally Lipschitz continuous , dF being given by

(5.4)

We equip H with a scalar product that is equivalent to the usual one:

where ε min{ξ/4, λ 1 /(2ξ)}, with λ 1 being the optimal Poincaré inequality constant. Then, after some computations similar to [Tem12, Section IV.1.2], we obtain

Therefore, Hypothesis 3.3 is satisfied as long as γ < ε/(2λ 1 ). In that case, Theorem 3.4 provides a suitable solution M to (2.7) with dM globally Lipschitz continuous, upon which a forwarding control law can be built for the output regulation problem. Since the range of CA -1 B is non-zero 4 and hence R, Theorem 3.2 guarantees the existence of a locally exponentially stable equilibrium for the closed-loop system with small reference and disturbance. Furthermore, following the discussion subsequent to Theorem 3.2, the global coercivity condition (3.3) holds whenever ε/2(1 + λ 1 ) > γ, in which case the theorem provides a globally asymptotically stable equilibrium.

A pre-stabilized Wilson-Cowan equation

The following example is inspired by the study of neural fields (see for instance [START_REF] Boscain | A bioinspired geometric model for sound reconstruction[END_REF]). Let Ω be a bounded domain in R n and O be an open subset of Ω. Given a positive gain α, a kernel k ∈ L ∞ (Ω × Ω), and a smooth scalar nonlinearity s that has bounded derivative and vanishes at 0, consider the following non-local evolution equation:

(5.7)

We look at the (vector-valued) output given by

(5.8)

Having set H L 2 (Ω), we define two mappings K and F on H by [Kw](x) s ′ (0) Ω k(x, ν)w(ν) dν and [F (w)](x) Ω k(x, ν){s(w(ν))s ′ (0)w(ν)} dν for all w ∈ H. Then, K is a bounded linear operator, and F is continuously differentiable with F and dF globally Lipschitz continuous. We also let A αid + K and Bu 1 O u with U L 2 (O). As an integral operator, K is compact; thus, 0 lies in the resolvent set of A except for a bounded and countable set of values for α. Here, A is continuous; therefore, we can choose Z Range A -1 B, which as a closed subspace of H is a Hilbert space as well. By letting C id, the condition (3.10) is automatically satisfied. Furthermore, Z = H if and only if O = Ω. As for condition (3.7), we have

where M k,s Ω×Ω |k(x, ν)s ′ (ν)| 2 dx dν. Hypothesis 3.3 is satisfied whenever α > M k,s , while global uniform coercivity (3.3) holds provided that α > 2M k,s .

A reaction-diffusion equation

Let Ω (0, L), L > 0. Consider the following reaction-diffusion equation with homogeneous Dirichlet boundary conditions:

w(0, t) = w(L, t) = 0 for t ∈ (0, +∞).

(5.10b)

To (5.10) we can associate a contraction semigroup {T t } on H L 2 (Ω) with generator -A satisfying Hypothesis 2.1. Note that unlike the previous examples, this problem does not enter the realm of application of Section 3.3 as the (pointwise) function w → w 3 cannot be modelled by a nonlinear map F : H → H. Nevertheless, it can be proved that {T t } is Fréchet differentiable [Tem12, Section VI.2.1], the first variation equation around a trajectory w of (5.10) being given by

in (0, L) × (0, +∞), (5.11a) v(0, t) = v(L, t) = 0 for t ∈ (0, +∞).

(5.11b)

We start by proving that w → +∞ 0 dT t (w) dt is a locally Lipschitz continuous map of H into L(H). Following the arguments given at the very end of the proof of Theorem 3.4, this will have