Geetanjali Das

Andry Razakamanantsoa

Gontran Herrier

Dimitri Deneele

Compressive strength and microstructure evolution of lime-treated silty soil subjected to kneading action

Keywords: lime-treated soil, kneading compaction, Unconfined Compressive Strength, lime-dispersion, pozzolanic-reactions

Introduction

Efficient and effective management of natural resources such as soil is essential in any land development project. Thus, several chemical stabilisation processes associated with suitable implementation processes (compaction, soil-mixing, soil-chemical-mixing, etc.) are commonly practiced for improving the engineering properties of natural soil. Chemical stabilisation involves the use of inorganic or organic binders, such as slags [START_REF] Zhou | Application of granulated copper slag in massive concrete under saline soil environment[END_REF][START_REF] Poh | Soil stabilization using basic oxygen steel slag fines[END_REF][START_REF] Wild | Effects of partial substitution of lime with ground granulated blast furnace slag (GGBS) on the strength properties of lime-stabilised sulphatebearing clay soils[END_REF][START_REF] Bensaifi | Influence of crushed granulated blast furnace slag and calcined eggshell waste on mechanical properties of a compacted marl[END_REF], fly ashes produced from coal-burning [START_REF] Kolias | Stabilisation of clayey soils with high calcium fly ash and cement[END_REF][START_REF] Show | Reuse of incinerator fly ash in soft soil stabilization[END_REF], cement kiln dust [START_REF] Baghdadi | Soil modification by cement kiln dust[END_REF][START_REF] Miller | Influence of soil type on stabilization with cement kiln dust[END_REF], alkaline activator [START_REF] Wattez | Interactions between alkali-activated ground granulated blastfurnace slag and organic matter in soil stabilization/solidification[END_REF][START_REF] Cristelo | Effect of calcium content on soil stabilisation with alkaline activation[END_REF], lime [START_REF] Akula | Evaluating the Long-Term Durability of Lime Treatment in Hydraulic Structures: Case Study on the Friant-Kern Canal[END_REF][START_REF] Bicalho K V | Performance of an instrumented embankment constructed with lime-treated silty clay during four-years in the Northeast of France[END_REF][START_REF] Le Runigo | Performance of lime-treated silty soil under long-term hydraulic conditions[END_REF][START_REF] Makki-Szymkiewicz | Evolution of the properties of lime-treated silty soil in a small experimental embankment[END_REF][START_REF] Nguyen | Effect of freeze-thaw cycles on mechanical strength of lime-treated fine-grained soils[END_REF], etc.

The use of lime in the form of quicklime or hydrated lime is a widespread technique for such improvement. Lime is one of the most versatile [START_REF] Dowling | Lime in the limelight[END_REF], low-cost [START_REF] Inkham | Low-cost alkaline substances for the chemical stabilization of cadmium-contaminated soils[END_REF], and easily available chemicals. It was shown to be paramount in several applications using environmentally friendly techniques [START_REF] Dowling | Lime in the limelight[END_REF]. Soil treated by lime can be used repeatedly, which is another cost-effective measure [START_REF] Hopkins | Stockpiling Hydrated Lime-Soil Mixtures[END_REF]. Soil improvement by lime consists of two primary modification mechanisms: a) the instant cation exchange reactions and flocculation-agglomeration resulting in a reduction of soil plasticity and improvement of workability, and b) the long-term pozzolanic-reactions leading to the development of the cementitious compounds, thus increasing the soil strength [START_REF] Bell | Lime stabilization of clay minerals and soils[END_REF][START_REF] Diamond | Mechanisms of soil-lime stabilization[END_REF][START_REF] Little | Stabilization of pavement subgrades and base courses with lime[END_REF][START_REF] Rogers | Modification of clay soils using lime[END_REF]. Construction of earth structures such as earth embankments for roads, airports, or railway lines, hydraulic structures, pavement subgrades is successfully implemented because of improvement brought about by lime treatment [START_REF] Akula | Evaluating the Long-Term Durability of Lime Treatment in Hydraulic Structures: Case Study on the Friant-Kern Canal[END_REF][START_REF] Makki-Szymkiewicz | Evolution of the properties of lime-treated silty soil in a small experimental embankment[END_REF][START_REF] Little | Stabilization of pavement subgrades and base courses with lime[END_REF][START_REF] Das | Evaluation of the long-term effect of lime treatment on a silty soil embankment after seven years of atmospheric exposure: Mechanical, physicochemical, and microstructural studies[END_REF]. One of such examples of earthen hydraulic structures is the Friant-Kern Canal in California, United States. The bottom and blankets of several sections of the canal, initially built with heavy plastic clays, were renovated during the '70s using 4% quicklime by weight [START_REF] Herrier | Lime treated soil as an erosion-resistant material for hydraulic earthen structures[END_REF][START_REF] Knodel | Lime in canal and dam stabilization[END_REF]. Study up to more than 40 years after the renovation was conducted, which showed increased long-term strength, reduction in swelling potential, erosion resistance, thus indicating good geo-mechanical stability of the lime-treated structure [START_REF] Akula | Evaluating the Long-Term Durability of Lime Treatment in Hydraulic Structures: Case Study on the Friant-Kern Canal[END_REF]. Another study was recently reported by Das et al. [START_REF] Das | Evaluation of the long-term effect of lime treatment on a silty soil embankment after seven years of atmospheric exposure: Mechanical, physicochemical, and microstructural studies[END_REF], where the long-term effect of lime treatment on the mechanical, physicochemical, and microstructural evolution of an embankment constructed and atmospherically cured for 7 years were demonstrated. This study showed a significant evolution in average UCS of about 3.29 ± (0.45) MPa due to the development of cementitious compounds because of the long-term pozzolanic reactions. Thus, the durability and reliability of these structures throughout the service life are linked to the hydromechanical performance of the soil.

Several studies have reported that lime treatment increases the permeability of soil due to the flocculation of particles, which increases the inter-aggregates pore space [START_REF] Khattab | Etude multi-échelles d'un sol argileux plastique traité à la chaux[END_REF][START_REF] Rajasekaran | Permeability characteristics of lime treated marine clay[END_REF][START_REF] Nalbantoglu | Compressibility and hydraulic conductivity of a chemically treated expansive clay[END_REF]. Others have shown that permeability increases at the beginning of treatment and decreases with increased curing due to the evolution of cementitious compounds [START_REF] Locat | Mechanical and hydraulic behaviour of a soft inorganic clay treated with lime[END_REF][START_REF] Lasledj | Traitement des sols argileux à la chaux: processus physico-chimique et propriétés géotechniques[END_REF]. Makki-Szymkiewicz et al. [START_REF] Makki-Szymkiewicz | Evolution of the properties of lime-treated silty soil in a small experimental embankment[END_REF] conducted a permeability study on a 2.5% quicklime treated silty soil experimental embankment for about 6 months from the time of construction. The study reported a similar level of permeability values in the lime-treated and untreated soils over the 6 months of curing, concluding that conducting controlled mixing and compaction conditions during the construction improves the hydraulic performance of lime-treated soil. As a result, the same embankment, after 7 years from construction was also reported to have a uniform distribution of pH and water content by Das et al. [START_REF] Das | Evaluation of the long-term effect of lime treatment on a silty soil embankment after seven years of atmospheric exposure: Mechanical, physicochemical, and microstructural studies[END_REF], which led to a significant evolution in compressive strength. Thus, the long-term hydromechanical performance can be said to be associated with the compaction conditions implemented during the construction of such structures.

Compaction conditions, such as compaction procedure, energy, and compaction water content, were demonstrated to impact the hydromechanical improvement brought by lime treatment in the soil through several controlled laboratory investigations [START_REF] Le Runigo | Performance of lime-treated silty soil under long-term hydraulic conditions[END_REF][START_REF] Cuisinier | Microstructure and hydraulic conductivity of a compacted lime-treated soil[END_REF][START_REF] Deneele | Experimental assessment regarding leaching of lime-treated silt[END_REF][START_REF] Dhar | The strength and microstructural behavior of lime stabilized subgrade soil in road construction[END_REF][START_REF] Le Runigo | Impact of initial state on the fabric and permeability of a lime-treated silt under long-term leaching[END_REF][START_REF] Rosone | Microstructure and shear strength evolution of a limetreated clay for use in road construction[END_REF][START_REF] Cuisinier | Shear strength behaviour of compacted clayey soils percolated with an alkaline solution[END_REF][START_REF] Daniel | Water content-density criteria for compacted soil liners[END_REF][START_REF] Mitchell | Some Characteristics of the elastic and plastic deformation of clay on initial loading[END_REF][START_REF] Watabe | Influence of compaction conditions on pore-size distribution and saturated hydraulic conductivity of a glacial till[END_REF]. Le Runigo et al. [START_REF] Le Runigo | Impact of initial state on the fabric and permeability of a lime-treated silt under long-term leaching[END_REF] and Mitchell et al. [START_REF] Mitchell | Permeability of compacted clay[END_REF] stated that lime-treated soil compacted with different initial moisture contents and compaction energies show different magnitudes of initial permeability coefficient, k. A greater magnitude of k indicates a greater quantity of water percolation through the compacted soil. Water flow could enhance the dissolution of cementitious compounds, consequently decreasing the UCS of the lime-treated soil [START_REF] Le Runigo | Performance of lime-treated silty soil under long-term hydraulic conditions[END_REF][START_REF] Deneele | Experimental assessment regarding leaching of lime-treated silt[END_REF][START_REF] Le Runigo | Impact of initial state on the fabric and permeability of a lime-treated silt under long-term leaching[END_REF][START_REF] Cuisinier | Shear strength behaviour of compacted clayey soils percolated with an alkaline solution[END_REF]. This is because k of a soil depends on the pore size distribution (PSD) of the soil, which was shown to be a direct function of compaction conditions by several studies [START_REF] Ranaivomanana | Effects of cement treatment on microstructural, hydraulic, and mechanical properties of compacted soils: Characterization and modelling[END_REF][START_REF] Ranaivomanana | Permeability prediction of soils including degree of compaction and microstructure[END_REF][START_REF] González-López | Compaction effect on the compressive strength and durability of stabilized earth blocks[END_REF]. However, how well the above laboratory conditions represent the realistic field conditions is still an important question.

On the field, the usual design-construct of an earthwork project is as follows: The proposed soil materials are initially subjected to a laboratory test to define the compaction parameters required for the design. During the construction, these compaction parameters are achieved by implementing the compaction procedure, which is often adjusted according to the type of soil. All soil types except for rocky soil are compacted by Smooth-wheeled roller [START_REF] Holtz | An introduction to geotechnical engineering[END_REF], while fine-grained soil, particularly clayey soil, is preferred to be compacted by Padfoot roller [START_REF] Kouassi | A new technique of kneading compaction in the laboratory[END_REF][START_REF] Williams | Compaction of Soils[END_REF][START_REF] Russ | Anisotropic properties of compacted silty clay[END_REF]. Lekarp et al. [START_REF] Lekarp | State of the art. II: Permanent strain response of unbound aggregates[END_REF] demonstrated that Smooth-wheeled roller develops two types of compaction mechanisms in the soil, i.e., generation of stress tensors at the point of contact of the roller and the soil and a continuous rotation of these stress tensors due to the cyclic passes of the roller wheel. However, Padfoot roller, in addition to the rolling action of the wheel, produces a kneading action by the pad foot present on the drum surface of the roller. Thus, penetration of the pads occurs in the soil during compaction [START_REF] Russ | Anisotropic properties of compacted silty clay[END_REF][START_REF] Lutenegger | Soils and Geotechnology in Construction[END_REF].

Though the effect of roller compaction was detailed by Lekarp et al. [START_REF] Lekarp | State of the art. II: Permanent strain response of unbound aggregates[END_REF], and recently its implementation was shown in le Vern et al. [START_REF] Le Vern | Experimental study of particle lift initiation on roller-compacted sand-clay mixtures[END_REF], the effect of kneading action has not been well investigated. Clegg [START_REF] Clegg | Kneading compaction[END_REF] urged the importance of implementing kneading compaction at a laboratory scale to produce realistic laboratory compacted fine-grained specimens by showing a similar generation of soil structure and residual interparticle stresses with the in-situ soil. Kouassi et al. [START_REF] Kouassi | A new technique of kneading compaction in the laboratory[END_REF] confirmed that soil properties, including the dry density and elastic stiffness obtained under kneading effect at a laboratory scale, were close to those obtained in the in-situ compacted soil. Cuisinier et al. [START_REF] Cuisinier | Microstructure and hydraulic conductivity of a compacted lime-treated soil[END_REF] and Herrier et al. [START_REF] Herrier | Lime treated soil as an erosion-resistant material for hydraulic earthen structures[END_REF] highlighted that the magnitude of k obtained from kneaded lime-treated silty soil was lower compared to the soil compacted statically. However, apart from these studies, the effect of kneading compaction on mechanical behaviour is less investigated. The consideration of compaction energy is not available in the previous studies, which is essential to analyse the soil mechanical behaviour. Moreover, since kneading compaction was shown to bring a difference in k, which correlates with microstructural characteristics, the contribution of the kneading effect to the mechanical behaviour must be evaluated.

In this context, the present study investigates how the kneading mechanism contributes to the mechanical performance and microstructural modifications of lime-treated soil at a laboratory scale. The first section of the study highlights the compressive strength of kneaded soil by comparing the same with soil compacted by a reference standard method. The second section describes the contribution of kneading compaction at microstructure levels. Finally, comments are made on how the generation of mesopores because of lime treatment contributes to the evolution of UCS.

Materials and Methodologies

Soil and Lime properties

The material used in this study is silty soil. Details regarding the geotechnical properties of the soil were obtained from the study reported by Makki-Szymkiewicz et al. [START_REF] Makki-Szymkiewicz | Evolution of the properties of lime-treated silty soil in a small experimental embankment[END_REF]. The soil was composed of 12% clay content, 82% silt fraction, and the content of soil passing through 80 µm sieve was 99.5%. The liquid limit and plasticity index of the soil was 31% and 11%, respectively. The Methylene value was 2.5 g/100 g. The mineralogy of the soil, obtained by X-ray diffraction, showed the presence of Illite, Kaolinite, and Chlorite as clay minerals along with Quartz and Feldspars [START_REF] Nguyen | Effect of freeze-thaw cycles on mechanical strength of lime-treated fine-grained soils[END_REF].

The quicklime (CaO) used for the treatment was supplied by a commercial supplier. The lime consists of 90.9% of available CaO and a reactivity (t60) of 3.3 min. The Lime Modification Optimum (LMO) of the silt, which defines the minimum lime content required to initiate the pozzolanic-reactions [START_REF] Cherian | A critical appraisal of the role of clay mineralogy in lime stabilization[END_REF], was determined by Eades and Grim test as per the ASTM D 6276-99a [START_REF]Standard Test Method for Using pH to Estimate the Soil-Lime Proportion Requirement for Soil Stabilization[END_REF]. The LMO was found to be 1% by weight of lime. Three different lime contents were used, lime content equal to LMO, 2.5%, and 4%.

Sample preparations

The maximum dry density, ρd(max), and OMC of the silt obtained as per the standard Proctor compaction test mentioned in ASTM D698-91 [START_REF] Astm A | D698-12e2[END_REF] are presented in Table . 1. Once the characteristics of compaction were deduced, the silt was then air-dried and sieved using 5 mm-sieve. Soil mixtures for both the kneading and static compaction were prepared at OMC (Table 1) and the wet of OMC, i.e., at WMC (= 1.1 × OMC) and stored in sealed plastic bags for about 24 hours to attain moisture content homogenization.

Thus, the initial compaction characteristics (Table 1) of the soil mixture subjected to both types of compactions were kept constant. The wet soil and the respective lime were then mixed and rested for 1 hour before compaction. This above process of soil preparation was as per the procedure mentioned in the French GTS Technical Guide for soil treatment [START_REF] Guide | Soil treatment with lime and/or hydraulic binders: Application to the Construction of fills and capping layers[END_REF], which is also a reference for in-situ construction of limetreated structures. Cylindrical specimens of dimensions having a length of 10cm and a diameter of 5cm were prepared by both kneading and static compactions. The static compaction involves compression of the specimens from top and bottom, as demonstrated by Holtz et al. [START_REF] Holtz | An introduction to geotechnical engineering[END_REF]. This was considered as the standard reference compaction method.

In this study, the kneading compaction was conducted by a laboratory-developed kneading tool, which was equipped with a dynamic load, as presented in Fig. 1. The application of the dynamic load was made successively with the rotation of the 3-kneading feet by an angle of 45° between 2 successive loadings. This procedure was demonstrated by Kouassi et al. [START_REF] Kouassi | A new technique of kneading compaction in the laboratory[END_REF]. The applied compaction energy was adjusted as recommended by ASTM D698-91 (Equation 1).

E = N×H×n×m×g V (1)
where, E is the compaction energy (kN-m/m³); N = 16, which is the number of blows per layer; H = 0.14 m, i.e., the height of the falling load; n = 5, the number of compacted layers; m = 1.042 kg, the mass of the falling load (kg), g = acceleration due to gravity (m/s²), V = 187 cm 3 i.e., the volume of the mould. After compaction, the bulk density of each specimen was evaluated. The average bulk density of all the kneading and statically compacted specimens was 20.0 kN/m 3 and 19.7 kN/m 3 , respectively. Thus, the average bulk density obtained using both compaction methods was similar. Hence, the precision of the kneading tool developed in the laboratory in achieving the targeted bulk density (i.e., obtained from the standard static tool) can be justified.

A total of 72 specimens, including duplicates for each soil configuration, were prepared for strength and microstructural investigations by both compaction methods (Table 2). Forty-eight of the total specimens were cured for 28-and 90-days at a laboratory temperature of about 20˚C, and the remaining were cured for 180 days at 40°C, i.e., under accelerated curing (Table 2).

After curing, specimens prepared for microstructural analysis were freeze-dried and then stored in vacuum bags to avoid atmosphere interactions until further analysis.

The following nomenclature is used for specimen's identification: lime content (1/2.5/4)-Static compaction method/Kneading compaction method (S/K)-compaction moisture content (OMC/WMC)curing time (28/90/180) days. For example, 1-K-OMC-28 means 1% lime-treated soil subjected to kneading compaction (K) at OMC and then cured for 28 days.

Laboratory tests

After curing, specimens were subjected to UCS test using a mechanical press with a load sensor at a constant displacement rate of 1mm/min.

Pore characterization was made by Mercury Intrusion Porosimetry (MIP) test and Barrett-Joiner-

Halenda pore (BJH) method [START_REF] Barrett | The determination of pore volume and area distributions in porous substances. I. Computations from nitrogen isotherms[END_REF] on freeze-dried specimens. The motive behind operating both these methods was to investigate the influence of lime treatment on pore structure modifications more elaboratively. MIP is generally used for macropore investigation [START_REF] Das | Evaluation of the long-term effect of lime treatment on a silty soil embankment after seven years of atmospheric exposure: Mechanical, physicochemical, and microstructural studies[END_REF][START_REF] Lei | Pore distribution characteristics of Long-dong loess in Northern Shanxi of China[END_REF][START_REF] Wang | Microstudy on soft foundations reinforcement with lime-fly ash piles[END_REF], while BJH for measuring mesopores and micropores [START_REF] Das | Evaluation of the long-term effect of lime treatment on a silty soil embankment after seven years of atmospheric exposure: Mechanical, physicochemical, and microstructural studies[END_REF][START_REF] Bin | Micropore structure of aggregates in treated soils[END_REF].

The procedure of MIP test involves the evacuation of freeze-dried samples via heating inside a sealed penetrometer. Mercury was then progressively introduced into the samples through incremental hydraulic pressure. The volume of mercury intruded, and the applied pressure, p (MPa), was registered [START_REF] Romero | Microstructure investigation in unsaturated soils: a review with special attention to contribution of mercury intrusion porosimetry and environmental scanning electron microscopy[END_REF]. By measuring the pressure p required to be applied to force the mercury into a cylindrical pore of diameter, D, pore sizes can be obtained according to the Washburn equation (Equation 2) [START_REF] Romero | Microstructure investigation in unsaturated soils: a review with special attention to contribution of mercury intrusion porosimetry and environmental scanning electron microscopy[END_REF].

𝐷 = 4.γ.cos 𝜃 p (2)
D is the diameter of the entrance pore where mercury intrudes, γ is the surface tension of mercury, and 𝜃 represents contact angle.

The BJH method involves analysing pore structure from the isotherms obtained by Brunauer-Emmett-Teller (BET) test [START_REF] Brunauer | Adsorption of gases in multimolecular layers[END_REF], which uses nitrogen gas. Freeze-dried samples were evacuated or degassed at 50°C. Nitrogen gas at temperature, T of 77K, and pressure (p) lower than the equilibrium/saturation gas pressure (p0) was then injected. Pore structure is determined by using the Kelvin equation (Equation 3) [START_REF] Westermarck | Use of mercury porosimetry and nitrogen adsorption in characterisation of the pore structure of mannitol and microcrystalline cellulose powders, granules and tablets[END_REF].

𝑟 𝑘 = 2.𝑉 𝑚 .γ.cos 𝜃 R.T.ln ( 𝑝 𝑝 0 ) ( 3
)
where rk is the radius of curvature of the condensed gas inside the pore, γ is the surface tension, 𝑉 𝑚 is the molar gas volume of an ideal gas, 𝜃 is the contact angle, and R is the gas constant. In this study, the pore structure analysed by BJH was presented in the form of isotherm plots and cumulative mesopore volume.

The cumulative mesopore volume corresponding to the desorption branch of isotherm was used.

The discussion related to pore classifications was made as per the International Union of Pure and Applied Chemistry (IUPAC) [START_REF] Rouquerol | Recommendations for the characterization of porous solids (Technical Report)[END_REF], which classifies pores based on their pore-width as macropores (> 500 Å), mesopores (20-500 Å), and micropores (< 20 Å).

Calcium distribution evolution considering the Ca-mapping was shown to be a useful approach to observe the distribution of lime in lime-treated soil by Lemaire et al. [START_REF] Lemaire | Effects of lime and cement treatment on the physicochemical, microstructural and mechanical characteristics of a plastic silt[END_REF]. In this aspect, the µ-XRF images are recorded for lime-treated kneading and statically compacted specimens to assess the effect of compaction on the lime-dispersion. The µ-XRF analysis was performed on freeze-dried specimens, impregnated with polyester resin (LR White © ), and polished with ethanol. The device used equipped with a rhodium RX source and an EDS system (using an SDD detector). Chemical maps of 2 mm square sample sections were recorded.

The observed distribution of calcium was then quantitatively analysed by using imaging software NIS-Elements Basic Research 3.1.

Results

UCS of kneading compacted specimens

Fig. 2 presents the trend of strength evolution in the lime-treated kneading compacted specimens subjected to different curing periods and temperatures. The presentation was made in terms of the average stress-strain obtained from duplicates of each soil configuration. The UCS value, representing the peak of the stress-strain curve, is presented in Table 3. The UCS of the lime-treated kneading compacted soil increased with the increase in lime content and curing time, as seen in Table 3 and Fig. 2. This increase in UCS was significantly higher for the accelerated cured specimens after 180 days of curing compared to the increase in UCS from 28 to 90 days of curing at 20˚C.

Besides, the evolution of UCS was relatively higher in the WMC-compacted specimens than the corresponding OMC-compacted specimens treated with lime content higher than the LMO. The UCS measured was about 3% and 16% higher for 2.5% and 4% lime-treated specimens, respectively, for the 90days cured WMC-compacted specimens (Fig. 2d). Similarly, the UCS was about 50% and 70% higher for the 2.5% and 4% lime-treated accelerated cured specimens, respectively, compacted at WMC (Fig. 2f).

Comparison of UCS evolution

The UCS measured from the kneaded soil, were plotted against the respective values obtained from the standard statically compacted specimens in Fig. 3. Comparisons are made between specimens prepared at the same compaction water content, lime content and subjected to similar curing conditions. According to Fig. 3a-d, almost an equivalent UCS level was observed for all the 1% lime-treated compacted specimens, subjected to 28-and 90-days of curing at 20˚C. However, this UCS was about 30%

higher in kneaded specimens subjected to accelerated curing (Fig. 3e &f).

For the 2.5% lime-treated OMC-compacted specimens, the UCS level was similar for the 28-and 90-days cured specimens (Fig. 3a &c) and increased by about 45% in the accelerated kneaded specimen (Fig. 3e). For the corresponding WMC-compacted specimens, this UCS level remains almost the same after 28 days of curing (Fig. 3b); increased slightly in the kneaded soil after 90 days of curing (Fig. 3d), and increased by about 50% for the accelerated kneaded specimen (Fig. 3f).

At 4% lime treatment, all the kneaded specimens show about 6-40% greater UCS values.

Comparison of microstructural modifications

Pore size distribution by MIP

Fig. 4 presents the pore size modification brought by lime treatment in the untreated statically compacted specimens by MIP analysis. As expected, a significant decrease in the pores present at 10 5 and 10 4 Å and evolution of pores lower than 3000 Å was brought on lime additions. Such evolution was also reported by Cuisinier et al. [START_REF] Cuisinier | Microstructure and hydraulic conductivity of a compacted lime-treated soil[END_REF]. Besides, the evolution of pores lower than 3000 Å was enhanced with increased lime contents due to the increased formation of cementitious compounds [START_REF] Das | Evaluation of the long-term effect of lime treatment on a silty soil embankment after seven years of atmospheric exposure: Mechanical, physicochemical, and microstructural studies[END_REF][START_REF] Cuisinier | Microstructure and hydraulic conductivity of a compacted lime-treated soil[END_REF]. A small number of macropores of pore diameter 10 5 Å were observed in all three different limetreated kneaded specimens compacted at both OMC and WMC (Fig. 5a, b &e). On the other hand, for the statically compacted specimens, an increase in the intensity of macropores of pore diameter 10 5 Å was observed with the increased lime content (Fig. 5c, d &f).

Besides, almost a similar decrease in the presence of macropores of diameter 10 4 Å and increased intensity of pores of diameter lower than 3000 Å was observed in both types of compacted specimens, with increase lime content and curing time (Fig. 5a-f). The peak of the isotherms, as seen in Fig. 6a, represents the total nitrogen adsorption capacity of the soil [START_REF] Sotomayor | Characterization of micro/mesoporous materials by physisorption: concepts and case studies[END_REF]. In the analysis made with the present soil, the PSD analysed by the BJH method was found to be in the range of pore diameter of about 20-2000 Å. Thus, the higher is the peak of the isotherm, the greater is the presence of pores of diameter 20-2000 Å in the soil.

Fig. 6 shows that the peak of the isotherm rises with increased lime content. However, this rise was relatively more significant in the 4% lime-treated kneaded soil (Fig. 6a,b & e) than the corresponding rise in statically compacted soil (Fig. 6c, d &f).

The hysteresis developed in the isotherm was relatively distinctive in the 4% lime-treated 28 days cured kneaded soil (Fig. 6a &b). This hysteresis was demonstrated to be associated with the delay in capillary condensation and evaporation that occurs in the mesopores by McGregor et al. [START_REF] Mcgregor | Conditions affecting the moisture buffering measurement performed on compressed earth blocks[END_REF] and Collet et al. [START_REF] Collet | Porous structure and water vapour sorption of hempbased materials[END_REF]. Thus, an enhanced hysteresis indicates the presence of a greater volume of mesopores [START_REF] Zielinski | Physical characterization: surface area and porosity[END_REF]. The cumulative mesopore volume measured in the lime-treated compacted soil was in the range of mesopore diameter 25-250 Å. The variation of these mesopore volumes is presented in terms of the percentage of mesopore volume for a different interval of mesopore range (Fig. 7). It was observed that almost all the statically compacted soil exhibited a greater percentage of mesopore volume in the range of pore diameter of 100-250 Å. At the same time, the percentage of pore volume was relatively higher in the kneaded soil in the mesopore range of pore diameter lower than 100 Å, i.e., 25-75 Å.

µ-XRF images

Fig. 8 presents the distribution of calcium within the lime-treated soil matrix compacted at OMC and after 28 days of curing. The calcium distribution is represented by the blue region, while soil by the black region. The quantitative analysis of this calcium distribution is presented in Table 4.

Discussion

As expected, kneading compacted soil showed a rise in UCS with increased curing time and lime content (Fig. 2). The significant UCS evolution for the accelerated cured specimens was attributed to the acceleration of pozzolanic-reactions, as demonstrated by Lemaire et al. [START_REF] Lemaire | Effects of lime and cement treatment on the physicochemical, microstructural and mechanical characteristics of a plastic silt[END_REF] and Verbrugge et al. [START_REF] Verbrugge | Strength and microobservations on a lime treated silty soil, in: Road Materials and New Innovations in Pavement Engineering[END_REF] (Fig. 2e & f). At the same time, for soil treated with 2.5% and 4% lime, the UCS obtained was relatively higher for the WMC-compacted 90 days cured specimens when compared to the corresponding OMC-compacted specimens (Fig. 2c &d). This increase in UCS was more significant for all the accelerated cured WMCcompacted specimens (Fig. 2e &f). Thus, kneaded soil, treated at lime content higher than LMO, cured for a longer time, and under accelerated condition, shows enhanced UCS in WMC-compacted specimens when compared to OMC-compacted specimens. However, the literature has shown that lime-treated soil compacted at WMC, which is much higher than the OMC and cured for a shorter period, resulted in lower UCS than OMC-compacted soil due to the loss in soil grain-to-grain contact with increased water content [START_REF] Le Runigo | Performance of lime-treated silty soil under long-term hydraulic conditions[END_REF][START_REF] Ajayi | Effect of Lime Variation on the Moisture Content and Dry Density of Lateritic Soil in Ilorin[END_REF][START_REF] Consoli | Control factors for the long term compressive strength of lime treated sandy clay soil[END_REF][START_REF] Yin | Effects of initial water content on microstructure and mechanical properties of lean clay soil stabilized by compound calcium-based stabilizer[END_REF]. In the present case, the soil treated at WMC is only 1.1 times higher than OMC and cured for a longer time and under accelerated conditions. Thus, the generation of enhanced UCS in the WMCcompacted specimens indicates an appropriate availability of water in the soil matrix, which might have regulated a steady consumption of water by quicklime for the formation of cementitious compounds with increased curing time and under accelerated conditions. Thus, compacting lime-treated soil at WMC slightly higher than OMC is beneficial for long-term evolution of UCS.

On comparing the dispersion of calcium measured from the polished surface area of the compacted specimens in Fig. 8, it was observed that this dispersion was enhanced under kneading compaction. The percentage of the polished area covered by calcium was about 8% higher in the 2.5%, and 4% lime-treated OMC kneaded specimens compared to the corresponding statically compacted specimens (Table 4). The observed calcium can either be the calcium from the available lime or the cementitious compounds developed because of pozzolanic-reactions. This provides evidence that kneading action enhances limedispersion. This feature of enhanced lime dispersion, accompanied by water availability, would contribute to the enhanced pozzolanic-reactions, particularly in the long-term. Thus, kneading compaction acts in favour of pozzolanic-reaction, which consequently leads to better development of cementitious compounds.

This was confirmed by a relatively greater generation of mesopores of diameter 20-2000 Å as indicated by the enhanced isotherm peak, by the hysteresis developed (Fig. 6), and by the presence of a greater percentage of mesopore volume in the pore range 25-75 Å (Fig. 7). Increased development of cementitious compounds can increase the overall stiffness of the kneaded soil. This explains the relatively greater UCS obtained in all the 4% lime-treated kneaded soil (Fig. 3), in the 2.5% lime-treated 90 days cured kneaded soil, compacted at WMC (Fig. 3d), and in all the lime-treated accelerated cured kneaded soil (Fig. 3e &f).

Kneading effect was shown to bring more significant deformation in the natural soil aggregates compared to statically compacted specimens by Mitchell and McConnell [38]. Thus, upon kneading compaction of the lime-treated soil at OMC/WMC, a greater aggregate deformation accompanied by enhanced lime-dispersion occurred in comparison to the one that was statically compacted. Such a phenomenon lowered the availability of macropores of diameter 10 5 Å (Fig. 5a, b &e) and enhanced the mesopores evolutions (Fig. 6 & 7) in the kneaded soil. At the same time, macropores of diameter 10 5 Å increased with lime content in the statically compacted specimens (Fig. 5c, d &f), which was due to the formation of greater inter-aggregate pores at higher lime content, as reported by Tran et al. [START_REF] Tran | Effects of lime treatment on the microstructure and hydraulic conductivity of Héricourt clay[END_REF] and Wang et al. [START_REF] Wang | Temperature-accelerated strength development in stabilized marine soils as road construction materials[END_REF]. The same specimens after curing and on being subjected to UCS showed failure due to consumption of water for cementitious compounds formation. However, owing to the above-mentioned differences produced in soil structure due to the difference in the effect of compactions, kneaded soil showed a higher ductile failure characteristic during UCS test, as evidenced by the differences in the crack development observed between the lime-treated kneading and statically compacted specimens in Fig. 9.

The diagonal and vertical crack developed in the kneaded and statically compacted specimens (Fig. 9a & b) represents the shear and tensile crack, respectively, as reported by Kichou Z [START_REF] Kichou | A study on the effects of lime on the mechanical properties and behaviour of London clay[END_REF]. The present 2.5% lime-treated and WMC-compacted silty soil was used in the construction of an embankment using a vibrating padfoot roller that produces kneading action during compaction [START_REF] Makki-Szymkiewicz | Evolution of the properties of lime-treated silty soil in a small experimental embankment[END_REF][START_REF] Consoli | Control factors for the long term compressive strength of lime treated sandy clay soil[END_REF]. The process of construction was as per GTS Technical Guide for soil treatment [START_REF] Guide | Soil treatment with lime and/or hydraulic binders: Application to the Construction of fills and capping layers[END_REF]. After 7 years of atmospheric curing, the embankment was deconstructed, and the UCS of four core-sampled embankment's specimens, namely, T1-1 & T2-4 of length (l)/diameter(d) ratio of both 1 and 2 was evaluated, and the average UCS of these specimens was reported to be 3.29 (± 0.45) MPa [START_REF] Das | Evaluation of the long-term effect of lime treatment on a silty soil embankment after seven years of atmospheric exposure: Mechanical, physicochemical, and microstructural studies[END_REF]. The obtained in-situ UCS was also shown to be repeatable using the correction factor recommended by ASTM-C42-77 [START_REF]Standard method of obtaining and testing drilled cores and sawed beams of concrete[END_REF]. The average UCS of in-situ extracted samples was shown to be comparable with a 90-day laboratory accelerated-cured specimen of similar configuration and of the dimension of l/d=1. The UCS obtained from in-situ and laboratory accelerated cured specimen that has been reported by Das et al. [START_REF] Das | Evaluation of the long-term effect of lime treatment on a silty soil embankment after seven years of atmospheric exposure: Mechanical, physicochemical, and microstructural studies[END_REF] are compared with the trend of UCS obtained with the present accelerated kneaded soil in Fig. 11a &b. with the UCS of the corresponding 2.5% lime-treated accelerated cured laboratory kneaded soil, was found to be of a similar level for both dimensions. The difference in UCS observed between the laboratory 90and 180-days accelerated cured kneaded specimens was due to the difference in dimension which is in accordance with ASTM-C42-77 [START_REF]Standard method of obtaining and testing drilled cores and sawed beams of concrete[END_REF]. Besides, Das et al. [START_REF] Das | Evaluation of the long-term effect of lime treatment on a silty soil embankment after seven years of atmospheric exposure: Mechanical, physicochemical, and microstructural studies[END_REF] reported that the difference in UCS of the two in-situ cured core-sampled specimens was linked with the intensities of mesopores evolution by BJH. Fig. 12 shows that the present accelerated cured kneaded specimen exhibits intensities of mesopores greater than T2-4 and less than T1-1. Thus, the trend observed in the evolution of mesopores between the accelerated-cured and in-situ cured soil, both kneaded, corresponds well with the level of UCS measured. This confirms the fact that the soil cured under laboratory-accelerated condition and the soil sampled after 7 years of environmental exposure, both submitted to kneading action, exhibit a similar level of UCS. This shows evidence of the relevancy of the accelerated condition in the laboratory to access to the long-term UCS level similar to the sample submitted to field condition.

Additionally, Das et al. [START_REF] Das | Evaluation of the long-term effect of lime treatment on a silty soil embankment after seven years of atmospheric exposure: Mechanical, physicochemical, and microstructural studies[END_REF] reported a uniform distribution of pH above 11 and water content throughout the lime-treated embankment after 7 years of atmospheric curing. This evolution was due to the controlled mixing and compaction condition conducted during the construction of the embankment by Makki-Szymkiewicz et al. [START_REF] Makki-Szymkiewicz | Evolution of the properties of lime-treated silty soil in a small experimental embankment[END_REF] and Charles et al. [START_REF] Charles | An experimental full-scale hydraulic earthen structure in lime treated soil[END_REF], which again involves the kneading action due to the use of padfoot rollers. Considering the enhanced dispersive action created by kneading compaction in Fig. 8, a uniform evolution of lime and lime components might have resulted throughout the embankment, thus giving all pH values of the sampled specimens above 11 and a homogeneous water content.

Fig. 4 & 5 showed that lime treatment has led to the generation of pores lower than the pore diameter of 3000 Å, and such generation was shown to increase soil cohesion and enhance UCS by Verbrugge et al. [START_REF] Verbrugge | Strength and microobservations on a lime treated silty soil, in: Road Materials and New Innovations in Pavement Engineering[END_REF]. However, pores diameter lower than 3000 Å includes a part of macropores, total mesopores, and micropores as per IUPAC (1994). In the present soil, BJH measured the evolution of pores under the lime effect in the range 20-2000 Å (Fig. 6), which includes macropores and mesopores, and the cumulative pore volume in the range 25-250 Å that includes only mesopores (Fig. 7). Hence, to identify the contribution of different classes of pores to the UCS evolution, a regression equation was developed.

The proposed equation (Equation 4 This indicates how lime treatment brings greater development of mesopores due to the cementitious bonding formed because of pozzolanic-reactions and how this contributes towards the strength evolution.

Conclusions

The effect of the kneading mechanism on the UCS evolution and microstructural properties of limetreated silty soil was investigated. Based on the studies, the following findings are derived:

1. Lime-treated kneaded soil, particularly compacted at WMC, prepared at lime content higher than LMO and subjected to longer and accelerated curing, showed enhanced evolution of UCS. With appropriate availability of water in the soil matrix, a steady consumption of water by quicklime was regulated for the formation of cementitious bonding with increased curing time and under accelerated conditions.

2. The kneading action caused relatively greater deformation of large aggregates and decreased the macropores of diameter 10 5 Å, compared to the corresponding statically compacted soil.

3. Kneading compaction enhanced better dispersion of lime during compaction. When accompanied by the available water, such configuration works in favour of long-term pozzolanic-reactions. This resulted in up to about 50% greater UCS in the kneaded soil than the statically compacted soil prepared at lime content greater than LMO, compacted at WMC, and subjected to longer and accelerated curing.

4. The UCS obtained from the laboratory accelerated kneaded soil was of a similar level as the average UCS obtained from in-situ specimens sampled from the 7-year atmospherically cured embankment. The mesopores generation showed a positive trend with respect to the evolution of UCS between the laboratoryaccelerated cured, and in-situ cured sampled soil.

5. Increased lime content and curing time resulted in increased isotherm peak, which indicated the greater generation of pores in the range of pore diameter 20-2000 Å. Among the pore ranges produced under the lime effect, the contribution of mesopores in the pore range 25-250 Å towards the evolution of UCS was shown to be most significant, as highlighted by a regression equation relating UCS with pore size. Thus, the above findings are part of a holistic investigation into the mechanical behaviour and microstructural modification of lime-treated silty soil subjected to kneading action. The kneading action caused compressive strength gain, particularly in the long-term and for WMC-compacted soil. These results were evidenced by UCS and microstructure evaluated from 7 years in-situ cured core-sampled soil. Further studies are required to be conducted to understand the hydraulic characteristics of lime-treated soil subjected to the kneading effect.

Fig. 1 .

 1 Fig. 1. Laboratory Kneading Compaction tool

Fig. 2 .

 2 Fig. 2. Strength evolution of 1%, 2.5% and 4% lime-treated kneading compacted specimens at OMC-28 days (a), WMC-28 days (b), OMC-90 days (c), WMC-90 days (d), OMC-180 days (e), and WMC-180 days (f)

Fig. 3 .

 3 Fig. 3. UCS measured from 1%, 2.5% and 4% lime-treated specimens subjected to static and kneading compactions at OMC-28 days (a), WMC-28 days (b), OMC-90 days (c), WMC-90 days (d), OMC-180 days (e), and WMC-180 days (f)

Fig. 4 :

 4 Fig. 4: Evolution of pore structure after addition of 1%, 2.5%, and 4% lime in the untreated statically compacted specimens at OMC and after 28 days of curing

Fig. 5 .

 5 Fig. 5. Analysis of pore structure by MIP for 1%, 2.5% and 4% lime-treated specimens compacted by kneading and static compaction at OMC, and WMC and after 28 days (a-d) and 90 days (e, f) of curing at 20˚C

Fig. 6 &

 6 Fig. 6 & 7 presents the pore structure analysis by the BJH method in terms of the generation of isotherms and cumulative pore volume evolutions, respectively, in the lime-treated kneading and statically compacted specimens.

Fig. 6 .

 6 Fig. 6. Evolution of isotherms in 1%, 2.5%, and 4% lime-treated OMC, and WMC kneading and statically compacted specimens after 28 days (a-d) and 90 days (e, f) of curing at 20˚C

Fig. 7 .

 7 Fig. 7. Percentage of cumulative mesopores volume measured in 1%, 2.5%, and 4% lime-treated OMC, and WMC kneading and statically compacted specimens after 28 days (a-f) and 90 days (g-i) of curing at 20˚C

Fig. 8 .

 8 Fig. 8. µ-XRF image highlighting calcium distributions in 1%, 2.5%, and 4% lime-treated static (a, b, & c) and kneading compacted (d, e, & f) specimens prepared at OMC after 28 days of curing time at 20˚C.

Fig. 9 .

 9 Fig. 9. Diagonal Crack in kneading compacted (a), and vertical crack in statically compacted (b) specimens after UCS test at 1% lime treatment, compacted at OMC and after 28 days curing

Fig. 10 .

 10 Fig. 10. Schematic diagram showing the difference in the effect of kneading and static compaction on aggregates deformation and lime-dispersion

Fig. 11

 11 Fig. 11 Comparative evolution of UCS in in-situ core sampled and laboratory-accelerated cured soil of dimension of l/d ratio of 1 (a) and 2 (b).

Fig.

  Fig. 11a & b presents the comparative trend in UCS evolution in the in-situ and laboratory kneaded specimen of dimension l/d=1 and l/d=2, respectively. The average of the in-situ UCS values, if compared

Fig. 12 .

 12 Fig. 12. Comparison of PSD obtained by BJH in 2.5% lime-treated kneaded specimens obtained between laboratory accelerated cured specimen and 7 years in-situ atmospherically cured specimens

  ) considers compacted specimens cured at 20˚C. The input data of the proposed equation involves the isotherm peak which indicates the total volume of nitrogen injected in pores of diameter 20-2000 Å, and the cumulative pore volume measured in the mesopore range 25-250 Å. The input data is provided in Table 5. UCS = -0.034 + 0.015 × (pores of diameter 20-2000 Å) + 27.7 × (mesopores of diameter 25-250 Å)

Table 1

 1 Maximum dry density and OMC of untreated and lime-treated silty soil using Standard Proctor test

	Soil	ρd(max)	OMC
		(kN/m 3 )	(%)
	Untreated silty soil	18.4	14.3
	Silty soil treated with 1% lime	17.4	17.6
	Silty soil treated with 2.5% lime	17.1	18.5
	Silty soil treated with 4% lime	17.0	18.7

Table 2

 2 Types and number of specimens prepared by static-and kneading-compactions

						Number of
	Compaction modes	Curing conditions	Compaction moisture content	Lime contents	specimens (including
						duplicates)
		Curing time	Curing
		(days)	temperature (°C)
				OMC		6
		28	20	WMC	1%, 2.5%, 4%	6
	Static	90		OMC WMC	1%, 2.5%, 4%	6 6
		180	40	OMC WMC	1%, 2.5%, 4%	6 6
	Kneading	28 90	20	OMC WMC OMC WMC	1%, 2.5%, 4% 1%, 2.5%, 4%	6 6 6 6
		180	40	OMC WMC	1%, 2.5%, 4%	6 6

Table 3

 3 UCS measured in lime-treated kneaded soil subjected to different curing time and temperatures

	OMC-compacted	WMC-compacted
	specimens	specimens

Table 4

 4 Percentage distribution of Calcium (Ca) within the polished surface area of the kneading-and statically-compacted specimens

	18

Table 4

 4 highlights that, for a constant surface area, the percentage of calcium distribution was about 4 to 8% higher in the lime-treated kneaded specimens than the corresponding statically compacted specimens.

Table 5

 5 Isotherm peak and cumulative pore volume measured in lime-treated soil

			1 Vcum				Vcum
	Samples name	Isotherm peak (cm³/g)	in pore range (25-250 Å)	UCS (MPa)	Samples name	Isotherm peak (cm³/g)	in pore range (25-250 Å)	UCS (MPa)
			(cm³/g)				(cm³/g)
	1-K-OMC-28	27.3	0.008	0.60	1-S-OMC-28	26.0	0.011	0.70
	2.5-K-OMC-28	37.6	0.015	0.86	2.5-S-OMC-28	29.3	0.013	0.86
	4-K-OMC-28	39.3	0.018	1.00	4-S-OMC-28	31.3	0.014	0.90
	1-K-WMC-28	26.6	0.010	0.60	1-S-WMC-28	27.0	0.010	0.64
	2.5-K-WMC-28	32.2	0.012	0.68	2.5-S-WMC-28	32.0	0.012	0.77
	4-K-WMC-28	38.9	0.014	0.96	4-S-WMC-28	34.4	0.012	0.87
	1-K-OMC-90	31.4	0.011	0.75	1-S-OMC-90	28.8	0.010	0.77
	2.5-K-OMC-90	39.3	0.015	0.98	2.5-S-OMC-90	42.5	0.016	1.00
	4-K-OMC-90	46.8	0.020	1.23	4-S-OMC-90	43.0	0.019	1.00
	1-K-WMC-90	29.0	0.009	0.60	1-S-WMC-90	28.8	0.011	0.72
	2.5-K-WMC-90	41.4	0.017	1.03	2.5-S-WMC-90	39.3	0.016	0.91
	4-K-WMC-90	47.7	0.020	1.43	4-S-WMC-90	43.8	0.018	1.23

Vcum: Cumulative pore volume Equation (4) demonstrates accurately (R

= 0.90) that the mesopore range 25-250 Å makes the maximum contribution to the rise in UCS in the range of pores lower than 2000 Å developed under the lime effect.

Acknowledgements

This work was financially supported by Association Nationale de la Recherche et de la Technologie with grant N˚2018/0219 and Lhoist Southern Europe with grant N˚RP2-E18114. The authors are very thankful to the research team of Université Gustave Eiffel and Lhoist Nivelles for their great support in performing laboratory experiments and technical supports.


domain: math spi

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

Jean Auriol

Yann Le

Yann Le Gorrec
email: legorrec@femto-st.fr

Distributed Damping Assignment for a Wave Equation in the Port-Hamiltonian Framework

Keywords: infinite dimensional systems, Port-Hamiltonian Systems, backstepping methodology

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

The interaction of physical systems with their environment can be seen as a power flow through interaction ports. From this observation, [START_REF] Maschke | Port-controlled hamiltonian systems: modelling origins and systemtheoretic properties[END_REF] introduced three decades ago the notion of Port-Hamiltonian systems (PHS), extending the Hamiltonian formulation to open physical systems. This formalism has then been extended to distributed parameter systems in (van der Schaft and [START_REF] Van Der Schaft | Hamiltonian formulation of distributed-parameter systems with boundary energy flow[END_REF], and has been since widely used to model and control systems described by Partial Differential Equations (PDEs) (Le [START_REF] Gorrec | A semigroup approach to port hamiltonian systems associated with linear skew symmetric operator[END_REF][START_REF] Gorrec | Dirac structures and boundary control systems associated with skew-symmetric differential operators[END_REF][START_REF] Villegas | A port-hamiltonian approach to distributed parameter systems[END_REF][START_REF] Jacob | Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces[END_REF].

Since this framework takes into account physical properties (dissipation, interconnections), it can be used to design boundary controllers efficiently exploiting the physical properties of the system. For instance, Passivity-based control (PBC) design, initially proposed for finite dimensional non linear systems [START_REF] Ortega | Control by interconnection and standard passivity-based control of port-hamiltonian systems[END_REF], consists in using structural invariants and damping injection to modify the closed-loop properties of the system. This control design technique has been extended to boundary controlled PHS in [START_REF] Macchelli | On the synthesis of boundary control laws for distributed port-hamiltonian systems[END_REF]. In this case it has been shown that the closed-loop properties can only be partially modified using the available information at the boundary.

In this paper, we consider a one-dimensional wave-like equation with varying parameters and possible indefinite (negative, positive or null) damping characterizing the motion of a string. This system is clamped at one end and actuated at the other end. Depending on the damping values, this system can be either exponentially stable or unstable in open-loop [START_REF] Cox | The rate at which energy decays in a damped string[END_REF][START_REF] Freitas | Stability results for the wave equation with indefinite damping[END_REF]. The controller is generally used to damp out the oscillations or to improve the closed-loop performance. Our objective is to design a dynamic boundary feedback such that the closed loop system is equivalent to a target system known to have satisfying stability properties.

The backstepping methodology has been proved to be an efficient method for stabilizing hyperbolic systems [START_REF] Krstic | Control of a tip-force destabilized shear beam by observer-based boundary feedback[END_REF][START_REF] Krstic | Backstepping boundary controllers and observers for the slender timoshenko beam: Part i -design[END_REF]. The main idea is to map the original system into an exponentially stable target system using a boundedly invertible integral transform. However, the adequate target system is obtained on a case by case basis, sometimes facing difficulties. When possible, the simplest exponentially stable system is chosen, without paying attention to the intrinsic properties of the original system.

Since the PHS framework naturally emphasizes the physical properties of the system we aim at showing in this paper that it can be used against this main drawback of the backstepping approach. Specifically for linear hyperbolic PDEs, the Port-Hamiltonian formalism has been efficiently used to prove the existence of solutions or determine adequate boundary conditions guaranteeing the well-posedness of the associated boundary control system [START_REF] Zwart | Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain[END_REF]. It also helps inferring observability or controllability properties [START_REF] Jacob | Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces[END_REF]. Therefore, it could be of great interest to investigate further on developing state feedback boundary controllers using the backstepping approach and taking advantages of this formalism. In this respect in [START_REF] Ramirez | On backstepping boundary control for a class of linear port-hamiltonian systems[END_REF], the authors considered a general class of linear PHS, and used a coordinate transformation to derive an invertible mapping with a target system containing in-domain dissipative terms and homogeneous boundary conditions. However, in the case of an initial system which already contains a dissipative term, and with spatially varying parameters, a similar multiplicative operator cannot be used.

In this paper, we propose an innovative approach taking advantage of the backstepping methodology and the Port-Hamiltonian framework for a simple toy-system. We choose an exponentially stable target system based on energy considerations. We propose a new form of integral transform to map the initial system to this target system. We prove that such a transformation exists and is invertible and bounded to guarantee that the resulting control law imposes the desired dynamics to the initial plant.

The organization of the paper is as follows. First, in Section 2, we present the system under consideration and the control objective. Then, in Section 3, we present the stepby-step approach and the backstepping transform used. Finally, the performance of the controller is illustrated in Section 4. Some concluding remarks and perspectives end this paper (Section 5).

Notations

We denote C 1 ([0, 1]) the space of real differentiable functions defined on [0, 1] with a continuous derivative. We denote C 1 ([0, 1]) + ⊂ C 1 ([0, 1]), the subset of C 1 ([0, 1]) that contains positive functions (i.e. functions that are in C 1 ([0, 1]) and that are positive for all x ∈ [0, 1]). Let H ∈ C 1 ([0, 1]; R 2×2 ) + , a positive diagonal matrix. We denote χ . = L 2 ([0, 1]; R 2 ) the Hilbert-space equipped with the inner-product < u, v > χ = 1 2 1 0 u(x) T H(x)v(x)dx, and we denote ∥u∥ χ the associated norm (which is equivalent to the standard L 2 -norm). We denote the lower triangular part of the unit square as

T -= {(x, y) ∈ [0, 1] 2 | 0 ≤ y ≤ x}.
For any space B, Id B corresponds to the identity operator for the space. When there is no ambiguity, the subscript may be omitted. Similarly, the time and/or space dependency may be omitted.

SYSTEM UNDER CONSIDERATION

Vibrating string model

Let us consider a vibrating string clamped at one end and actuated at the other. Such a system is schematically pictured in Figure 2.1. We denote w(x, t) the vertical position of the string at point x and time t > 0. The spatial domain is normalized so that x ∈ [0, 1], the left end (that is clamped) corresponds to x = 0, while the actuator is located at x = 1. For all x ∈ [0, 1] and all t > 0, the state w(x, t) satisfies ) is positive, the results of this paper apply to the more general case where κ(x) can be positive, negative (antidamping) or equal to zero. At the fixed end of the string, no movement is allowed such that ∂w ∂t | x=0 (t) = 0. We exert a force u(t) (control input) on the opposite end, i.e.

ρ(x) ∂ 2 w ∂t 2 (x, t) = ∂ ∂x E(x) ∂w ∂x (x, t) -κ(x) ∂w ∂t (x, t), (1)
E(1) ∂w ∂x | x=1 (t) = u(t).
Initially, the position of the string is given by w(x, 0) = w 0 (x) ∈ C 1 ([0, 1]). The energy of this system is defined by

E(t) = 1 2 1 0 ρ(x) ∂w ∂t (x, t) 2 + E(x) ∂w ∂x (x, t) 2 dx.
(2) Using the dynamics of the string (1) and the specific boundary conditions, we can integrate by parts, and obtain the derivative of the energy

dE dt (t) = ∂w ∂t x=1 u(t) - 1 0 κ(x) ∂w ∂t 2 dx. (3)
Thus, the boundary control can be used to guarantee that the energy of the system is strictly decreasing.

Port-Hamiltonian formulation

To rewrite the model as a Port-Hamiltonian system, we introduce the Hamiltonian density matrix H(x) = diag(E(x), 1 ρ(x) ) and the following state variables

X = (X 1 , X 2 ) with      X 1 (x, t) = ∂w ∂x (x, t) : strain. X 2 (x, t) = ρ(x) ∂w ∂t (x, t) : momentum density. (4)
In this formalism, the dynamics of the damped string (1) rewrite

∂ ∂t X 1 X 2 =    0 ∂ ∂x 1 ρ(x) • ∂ ∂x (E(x)•) -c(x)    X 1 X 2 , (5)
where c(x) = κ(x) ρ(x) . Following the approach proposed in [START_REF] Jacob | Linear Port-Hamiltonian Systems on Infinite-dimensional Spaces[END_REF]

, Equation (5) rewrites ∂X ∂t (x, t) = P 1 ∂ ∂x (H(x)X(x, t)) + G 0 (H(x)X(x, t)), with P 1 = 0 1 1 0 , G 0 = 0 0 0 -c(x)ρ(x)
. The boundary effort e ∂ and boundary flow f ∂ are given by

     e ∂ = 1 √ 2 ((HX)(1) + (HX)(0)) f ∂ = 1 √ 2 (P 1 (HX)(1) -P 1 (HX)(0)) (6)
The boundary conditions rewrite

W B f δ e δ = u(t) 0 , with W B = 1 √ 2 0 1 1 0 1 0 0 -1 . (7)
We define the operator A .

= P 1 ∂ ∂x H + G 0 H, with domain D(A) = {X ∈ χ| HX ∈ H 1 ([0, 1]; R 2 ), W B f δ e δ = 0}.
Then, A is the infinitesimal generator of a C 0 -semigroup on χ. For any control input u ∈ C 2 ([0, T ], R), for any initial conditions given by X 1 (x, 0) = dw0 dx , X 2 (x, 0) = ρ(x) ∂w ∂t | t=0 (x) satisfying the corresponding compatibility conditions, there exists a unique classical solution of ( 5)-( 7) (Jacob, 2012, Lemma 13.2.1). Note that in this formalism, the energy of the system (or Hamiltonian) rewrites

E(t) = ∥X∥ 2 χ .

Control objective

In the power balance (2), the actuation at the boundary of the spatial domain impacts the change of internal power.

In the case κ > 0, the system is already stable as its energy is strictly decreasing in open-loop. However, we can use the control input u(t) to fasten its stabilization. In case of wave equation stemming from the linearization of an unstable system (κ < 0), the control input can be used for stabilization purposes.

As mentioned in the introduction, we already have examples of boundary feedback controllers stabilizing system (5)-( 7). In this paper, we do not simply want to stabilize the system but instead impose a specific decay rate to the energy of the system E, using a distributed damping assignment. More precisely, we want to design a control law u(t) making the dynamics of X equivalent to the dynamics of X = ( X1 , X2 ) satisfying

∂ ∂t X1 X2 =    0 ∂ ∂x 1 ρ(x) • ∂ ∂x (E(x)•) -K    X1 X2 , (8)
with boundary conditions X1 (1, t) = 0, X2 (0, t) = 0.

(9) This system is well-posed for any smooth initial conditions X1 (x, 0) = X10 (x), X2 (x, 0) = X20 (x) and exponentially stable. We define the new damping coefficient K > 0. For sake of simplicity, we considered here that the coefficient K > 0 is constant. However, the proposed approach could be extended for spatially-varying damping term K(x). In closed-loop the energy decreases proportionally to it

d Ē dt = -K 1 0 X2 (x, t) ρ(x) 2 dx. (10)

CONTROLLER DESIGN

To find the full-state feedback controller u(t) satisfying the control objective, we propose a specific integral transform of the form T : C + x 0 N dx mapping system ( 5)-( 7) to ( 8)-( 9). We prove that this transform is boundedly invertible.

It is an extension of the classical backstepping transform to Port Hamiltonian systems. Indeed, we aim to map the original system (5)-( 7) into the target system (8)-( 9). Although this transformation can be obtained directly, it can be decomposed into several classical elementary transforms (exponential changes of variables and classical backstepping coordinates changes). To facilitate the generalization of the proposed approach to other systems, we first present this more intuitive step-by-step approach.

Insights on the transform

In this section, we briefly present the methodology we used to find the form of the adequate transform T . Due to space restrictions, we only give here the main ideas. The global integral transform will be presented in the next section. To simplify the notations, we introduce the following functions λ(x) = E(x)/ρ(x), γ(x) = E(x)ρ(x) and

δ(x) = ρ ′ ρ + E ′ E (x), δ 3 1 (x) = 3 ρ ′ ρ -E ′ E (x), δ 1 3 (x) = ρ ′ ρ -3 E ′ E (x).
First change of variables: Riemann coordinates. First, we rewrite system ( 5)-( 7) as two coupled transport equations using an invertible change of variables (Riemann coordinates). Indeed, since the matrix A(x) = 0 1/ρ(x) E(x) 0 admits two eigenvalues ±λ(x), there ex-

ists P(x) ∈ C 1 ([0, 1], R 2×2 ) invertible, such that A(x) = P(x)Λ(x)P(x) -1 , with Λ(x) = diag(-λ(x), λ(x)). The new state variables ξ + ξ -= P -1 (x)
X 1 X 2 satisfy two heterodirectional hyperbolic PDEs with in-domain spatially varying coupling terms ∂ ∂t

ξ + ξ -= Λ(x) ∂ ∂x ξ + ξ -+ σ ++ σ +- σ -+ σ -- ξ + ξ -, (11) with boundary conditions ξ + (0, t) = q 0 ξ -(0, t), (12) ξ -(1, t) = r 1 ξ + (1, t) + u 1 (t), (13)
where

q 0 = 1 γ(0) , r 1 = -γ(1) u 1 (t) = √ 2 λ(1) u(t). The in- domain couplings σ •• ∈ C 0 ([0, 1]
) are continuous functions depending on the system parameters. They are defined by

                       σ ++ (x) = 1 2 -c(x) + λ 2 δ 1 3 (x) , σ +-(x) = 1 2ρ(x) c(x) λ(x) - 1 2 δ(x) , σ -+ (x) = E(x) 2 c(x) λ(x) + 1 2 δ(x) , σ --(x) = 1 2 -c(x) - λ 2 δ 3 1 (x) .
Inspired by the constructive backstepping approach [START_REF] Krstic | Boundary control of PDEs. A course on backstepping designs[END_REF], we use a Volterra integral transform to move the indomain couplings to the actuated boundary.

Exponential change of variables

Before doing so, we apply an exponential change of variables to suppress the diagonal coupling terms. Define the new set of variables ξ+ (x, t) ξ-(x, t) = diag(f (x)e Ic(x) , g(x)e -Ic(x) ) f (x) σ -+ (x). They rewrite as

ξ + (x, t) ξ -(x, t) , with
σ+ (x) = 1 2γ(0) e 2Ic(x) (c - λ 2 δ(x)), σ-(x) = γ(0) 2 e -2Ic(x) (c + λ 2 δ(x)),
The boundary conditions are given by ξ+ (0, t)

= q 0 ξ-(0, t), ξ-(1, t) = r1 ξ+ (1, t) + Ū (t), ( 15
)
where r1 = -γ(0)e -2Ic(1) , Ū (t) = g(1)e -Ic(1) u 1 (t).

Volterra integral transform

Next, we use a classical invertible Volterra integral transform of the second kind Id +

x 0 K to replace the in-domain coupling terms by adequate terms that correspond to the ones we would have obtained performing the change of variables P -1 and the exponential change of coordinates on the system (8). More precisely, we define the new state variables as

α + α -= ξ+ ξ-- x 0 K ++ K +- K -+ K --(x, y) ξ+ ξ-(y)dy,
with kernels K ±∓ defined on T -. Following the backstepping methodology [START_REF] Vazquez | Backstepping boundary stabilization and state estimation of a 2× 2 linear hyperbolic system[END_REF], one can easily show that there exists a unique set of kernels such that the new variables verify ∂ ∂t

α + α -= Λ(x) ∂ ∂x α + α -+ 0 b + b -0 (x) α + α -, (16) with the boundary conditions α + (0, t) = q 0 α -(0, t), α -(1, t) = a 1 α + (1, t). (17)
The two continuous functions b ± are defined on [0, 1] by

b + (x) = 1 2γ(0) K - λ(x) 2 δ(x) e 2I K (x) , b -(x) = γ(0) 2 K + λ(x) 2 δ(x) e -2I K (x) , with I K (x) = 1 2 x 0 K λ(s)
ds, and a 1 = -γ(0)e -2I K (1) . The control input is given by Ū (t) =(a 1 -r1 ) ξ+ (1, t)

+ 1 0 (K -+ (1, y) -a 1 K ++ (1, y)) ξ+ (y) + (K --(1, y) -a 1 K +-(1, y))
ξ-(y)dy. Note that system ( 16)-(17) corresponds to system ( 14)-( 15) where the parameter c has been replaced by K. Using the invertibility of the different transformations, it is straightforward to express Ū (t) as a function of the original states X i .

Second exponential transform and inverse change of variables. Finally, we define another exponential transform by

ᾱ+ ᾱ-= f (x) -1 e -I K (x) 0 0 g(x) -1 e I K (x) α + α -. (18)
We can show that P -1 (x) ᾱ+ ᾱ-corresponds to the expected target system ( 8)-( 9) in the Port Hamiltonian formalism. This step-by-step strategy is illustrated in Fig. 2.

Fig. 2. Presentation of the two control strategies

Overall transform

We now present a specific integral transform that directly maps the original system (5) into the target system (8). It is obtained as a composition of all the transformations presented in the last section.

Definition Let us introduce the function C defined by

C : [0, 1] → C 1 ([0, 1]; R 2×2 ) x → cosh (I(x)) -1/γ(x) sinh (I(x)) -γ(x) sinh (I(x)) cosh (I(x))
where the function I is defined by I :

x → I c (x)-I K (x) ∈ C 1 ([0, 1]).
Note that for all x ∈ [0, 1], det(C(x)) = 1 such that C(x) admits a unique inverse C -1 (x). We then define the invertible operator

C : χ → χ X → C × X.
We finally introduce the integral operator T : χ → χ,

T (X)(x) = C(X)(x) + x 0 N (x, y)X(y)dy, ( 19
)
where N is a bounded function defined on T -.

Invertibility of T

Our objective is to find the appropriate full-state feedback controller u(t) mapping ( 5)-( 7) to ( 8)-( 9) using T defined by ( 19), i.e X1 X2 = T X 1 X 2 . To guarantee that both systems share the same stability properties, the proposed transformation must be boundedly invertible. This is straightforward as T is a composition of the invertible transform C with an invertible Volterra integral transform of the second kind [START_REF] Yoshida | Lectures on differential and integral equations[END_REF]. The inverse transform verifies

T -1 u(x) v(x) = C -1 u(x) v(x) - x 0 N (x, y) u(y) v(y) dy, (20)
where N is a bounded function defined on T -by N (x, y) = C -1 (x)N (x, y)C -1 (y) -x y N (x, s) N (s, y)ds).

Kernel equations

Our objective is to map ( 5)-( 7) to ( 8)-( 9). Differentiating Equation ( 19) with respect to time and space and integrating by parts, we obtain the following set of kernel equations:

1 ρ(x) N 21 x + E(y)N 12 y = ρ ′ (x) ρ(x) 2 N 21 (x, y), (21)
1 ρ(y) N 11 y + 1 ρ(x) N 22 x = ρ ′ (x) ρ(x) 2 N 22 (x, y) -c(y)N 12 (x, y), E(x)N 11 x + E(y)N 22 y = KN 21 (x, y) -E ′ (x)N 11 (x, y), E(x)N 12 x + 1 ρ(y) N 21 y = -E ′ (x)N 12 (x, y) + (K -c(y))N 22 (x, y), (22) with the boundary conditions N 12 (x, 0) = 0, N 22 (x, 0) = 0, (23)
N 11 (x, x) = N 22 (x, x) - 1 2λ(x) (c(x) + K) sinh α(x), N 21 (x, x) = ρE N 12 (x, x) + 1 2E (c(x) -K) cosh α(x) - 1 2λρ E ′ E + ρ ′ ρ sinh α(x) . (24)
We have the following theorem Theorem 1. The system (21)-( 22) with boundary conditions ( 23)-( 24) admits a unique continuous solution N defined on T -.

Proof. To prove the well-posedness of the kernel equations, we rewrite ( 21)-( 24) using the formalism of (Di Meglio et al., 2018, Theorem 3.2). Define a new set of kernels

K i , i ∈ 1, 4 on T -by     K 1 K 2 K 3 K 4     =          1 0 0 γ(y) γ(x) - γ(x) γ(y) 0 0 1 0 -γ(y)γ(x) 1 0 0 1 1 γ(y)γ(x) 0              N 11 N 12 N 21 N 22     . (25)
The kernels K i satisfy

ϵ i (x)∂ x K i + ν i (y)∂ y K i = 4 j=1 Σ ij (x, y)K j (x, y), (26) with ϵ(x) = (1 1 1 1) T λ(x), ν(y) = (1 -1 -1 1)
T λ(y), and the continuous in-domain coupling terms are defined by (27). Each kernel K i has boundary conditions on Ω i ⊂ ∂T -of the form

K i | Ωi = f i + 4 j=1 Γ ij K j | Ωi (28)
with

Ω 1 = Ω 4 = {(x, y) ∈ T -|y = 0}, Ω 2 = Ω 3 = {(x, y) ∈ T -|y = x}, with Γ 12 (x) = -γ(0) γ(x) , Γ 43 (x) = 1 γ(0)γ(x) , f 2 (x) = c(x)+K 2λ(x) sinh(α(x)), f 3 (x) = ρ(x)
2 ((c -K) cosh(α(x))λ(x)δ(x) sinh(α(x), the other coupling terms being equal to zero. We show that the assumptions of (Di Meglio et al., 2018, Theorem 3.2) are verified. Thus, the system ( 26)-( 28) admits a unique bounded solution K. The regularity of the solution derives from the one of the couplings.

Controller design

We can now design the stabilizing control law. We have the following theorem. Theorem 2. The initial system ( 5)-( 7) with the control law

u(t) = γ(1) tanh(I(1))X 2 (1, t) (29)
-E(1) cosh(I(1))

1 0 N 11 (1, y)X 1 (y, t) + N 12 (1, y)X 2 (y, t)dy
has the same dynamics as ( 8)-( 9).

Proof. The proof is a direct application of the backstepping methodology. ■

We therefore obtain a full-state feedback controller for the initial system, that stabilizes it with the desired damping coefficient.

Comment on robustness It is worth noticing that the control law (29) involves the pointwise term γ(1) tanh(I(1)) X 2 (1, t). This corresponds to the cancellation of the reflection term (a 1 -r1 ) ξ+ (1, t) to obtain the target system ( 16) in the step by step approach. As shown in [START_REF] Auriol | An explicit mapping from linear first order hyperbolic PDEs to difference systems[END_REF], this may have major consequences regarding the robustness margins of the closed-loop system since the corresponding feedback law is not strictly proper. To avoid any robustness issue, we must have |q 0 (a 1 -r1 )|+ |q 0 r1 | < 1 which implies the condition e -2Ic(1) + |e -2I K (1) -e -2Ic(1) | < 1.

(30) Note that (30) is never satisfied in the case κ ≤ 0. Indeed, the terms neglected in the linearization can effect positively the robustness of the real system.

SIMULATION RESULTS

In this section, we illustrate the performance of our controller in simulations. Consider the wave equation ( 1) with constant coefficients ρ = 936kg.m -3 , E = 4.14GPa. The initial string position is w 0 (x) = 0.1x. We simulated system ( 14)-( 15) on a time horizon of 20s using a Godounov Scheme (LeVeque, 2002) (CF L = 1). The space domain [0, 1] is discretized with a mesh of 200 points. Beforehand, the kernels K ±∓ are computed offline using a fixed-point algorithm. The control input is computed at each time step using (29). The integral term is approximated using trapz.

We first consider the undamped case, i.e. c = 0. Note that in this case the robustness property is not satisfied, and the purpose is simply to illustrate the performance of our control strategy. As illustrated in Figure 3, the energy of the open-loop system is constant, while the energy of the closed-loop system decays to zero with the same rate as the one of the target system (dotted black line). From

               Σ 1j (x, y) = 1 4 ((λδ 1 3 )(x) + (λδ)(y)) 1 4 ( γ(y)δ(x) ρ(x) + E(y)δ(y) γ(x) ) K + c(y) 2γ(x) γ(y)(K -c(y)) 2 Σ 2j (x, y) = 1 4 ( E(x)δ(x) γ(y) - γ(x]δ(y) ρ(y) ) 1 4 ((λδ 3 1 )(x) + (λδ)(y)) c(y) -K 2γ(y) - γ(x)(K + c(y)) 2 Σ 3j (x, y) = γ(x)(c(y) -K) 2 γ(y)(c(y) -K) 2 1 4 ((λδ 3 1 )(x) -(λδ)(y)) 1 4 (E(y)(γδ)(x) + E(x)(γδ)(y)) Σ 4j (x, y) = K -c(y) 2γ(y) K -c(y) 2γ(x) 1 4 ( δ(x) ρ(x)γ(y) - δ(y) ρ(y)γ(x) ) 1 4 ((λδ 1 3 )(x) -(λδ)(y)) (27)
Fig. 3. Evolution of the energy E(t).

the values of ( ξ+ , ξ-), we can numerically compute the evolution of the displacement w(x, t) along the string. We see in Figure 4 (bottom) that the oscillations naturally present in open-loop (top) are substantially damped. We Second, we consider the case c = 0.1 and we choose as target system ( 8)-( 9) with K = 5c > 0. We compare the proposed control law with a simple boundary feedback X 1 (t, 1) = -0.1X 2 (t, 1). We represent the evolution of

CONCLUSION

In this paper, we presented an innovative state-feedback boundary controller inspired by the backstepping methodology that stabilizes a wave equation with in-domain damping with predefined closed-loop properties. We used the Port-Hamiltonian formalism to determine a target system of strictly decreasing energy, whose behavior matches the one of the original system. This combined approach was proposed on a simple 'toy-system' and is a first step towards generalization to more complex PHS. The controller was designed such that the closed-loop initial system can be mapped to this target system with a new form of boundedly invertible transform. Its implementation requires the knowledge of the state on the entire string. This can be done using an observer, similarly to the ones proposed in [START_REF] Guo | The strong stabilization of a one-dimensional wave equation by non-collocated dynamic boundary feedback control[END_REF][START_REF] Smyshlyaev | Boundary stabilization of a 1-d wave equation with in-domain antidamping[END_REF].

We would like to take advantage of the intrinsic modularity of the Port-Hamiltonian framework to adapt this approach to larger networks of interconnected systems. We believe that Port-Hamiltonian formalism can be of great use to get intuition on adequate target systems in the backstepping approach. Indeed, adjusting energy-shaping techniques to design amenable target systems could improve the performance of the closed-loop system while reducing the control effort (since the target system would take advantage of the natural dissipation of the system). In future works, we will consider systems of higher dimension (as Timoshenko beams) and interconnected systems for which the backstepping approach has already allowed the design of stabilizing control laws [START_REF] Redaud | Outputfeedback control of an underactuated network of interconnected hyperbolic PDE-ODE systems[END_REF]. We could also extend our result to introduce adaptive time-varying damping with k(t), which would require timedependent integral transforms.

Fig. 1 .

 1 Fig. 1. Schematic representation of the system with ρ(x), E(x) ∈ C 1 ([0, 1]) + being the mass density and the Young's modulus. They may vary along the string. Even if in the considered case κ(x) ∈ C 0 ([0, 1]) is positive, the results of this paper apply to the more general case where κ(x) can be positive, negative (antidamping) or equal to zero. At the fixed end of the string, no movement is allowed such that ∂w ∂t | x=0 (t) = 0. We exert a force u(t) (control input) on the opposite end, i.e. E(1) ∂w ∂x | x=1 (t) = u(t). Initially, the position of the string is given by w(x, 0) = w 0 (x) ∈ C 1 ([0, 1]). The energy of this system is defined by

  σ± ∈ C 0 ([0, 1]) defined by σ+ (x) = e 2Ic(x) f (x) g(x) σ +-(x) and σ-(x) = e -2Ic(x) g(x)

Fig. 4 .

 4 Fig. 4. Evolution of the displacement w(x, t).

Fig. 6 .

 6 Fig. 6. Evolution of the energy E(t). the energy for both the open-loop and closed-loop systems with the two different controllers on Figure 6. As expected, the energy decays faster with the control input we propose (blue). One can verify that the control effort goes to zero.

Fig. 7 .

 7 Fig. 7. Displacement w(0.5, t) in open-loop (red), and closed-loop for the proposed controller (blue) and simple boundary feedback (black).