K Le Mercier
email: kevin.lemercier@univ-valenciennes.fr

M Watremez

E S Puchi-Cabrera

L Dubar

J D Guérin

L Fouilland-Paillé

Dynamic recrystallization behaviour of spheroidal graphite iron. Application to cutting operations

Keywords: SG iron, Hot cutting, Dynamic recrystallization, Finite element modelling

To increase the competitiveness of manufacturing processes, numerical approaches are unavoidable. Nevertheless, a precise knowledge of the thermo-mechanical behaviour of the materials is necessary to simulate accurately these processes. Previous experimental studies have provided a limited information concerning dynamic recrystallization of spheroidal graphite iron under hot cutting operations. The purpose of this paper is to develop a constitutive model able to describe accurately the occurrence of this phenomenon. Compression tests are carried out using a Gleeble 3500 thermo-mechanical simulator to determine the hot deformation behaviour of spheroidal graphite iron at high strains. Once the activation range of the dynamic recrystallization process is assessed, a constitutive model taking into account this phenomenon is developed and implemented in the Abaqus/Explicit software.

Finally, a specific cutting test and its finite element model are introduced. The ability of the numerical model to predict the occurrence of dynamic recrystallization is then compared to experimental observations.

Over the past few years, austempered ductile iron emerged for its application in several fields such as automotive and railway industries. This specific spheroidal graphite iron provides an efficient compromise between specific mechanical strength, fracture toughness and resistance to abrasive wear. Therefore, this material is intended to substitute forged steels for the weight reduction of numerous manufactured components [START_REF] Kovacs | Development of austempered ductile iron (ADI) for automobile crankshafts[END_REF]. To reach these enhanced mechanical properties, austempered ductile iron is obtained by a specific thermo-mechanical treatment (Figure 1). This consists in an austenitization of the cast iron in the temperature range 1123 -1223 K followed by quenching to an austempering temperature of 523 to 623 K causing the transformation of the austenite phase into ausferrite. A combined casting and forging process prior to this specific quenching is often performed to reduce manufacturing costs [START_REF] Meena | Drilling performance of green austempered ductile iron (ADI) grade produced by novel manufacturing technology[END_REF]. Also, with the aim of increasing the competitiveness, the removal of risers and feeder head is then performed at about 1273 K just after the casting operation. However, this stage can give rise to severe surface degradations under the cut surface, compromising the process viability.

A recent experimental investigation, performed by [START_REF] Fouilland | Experimental study of the brittle ductile transition in hot cutting of SG iron specimens[END_REF], The plastic flow stress evolution of a material which undergoes DRX is shown schematically in Figure 2. At stresses less than the critical stress for the onset of DRX (σ c ), the material undergoes both WH and DRV. However, once σ c is exceeded DRX will become operative and the three processes will occur simultaneously. As the strain applied to the material increases, the volume fraction recrystallized dynamically (X v ) will also increase In numerical cutting models, the [START_REF] Johnson | A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures[END_REF] constitutive formulation is generally implemented because of its simplicity and numerical robustness [START_REF] Limido | SPH method applied to high speed cutting modelling[END_REF]. However, this empirical law describes the flow stress as a function of the total strain applied to the material, which is not a valid state parameter, by means of a simple parametric power-law relationship. Such a description would be incompatible with the evolution of flow stress mentioned above. Therefore, the present paper deals with the development of a specific constitutive model which not only allows the DRX process to be considered, but also that expresses the flow stress in terms of valid state parameters. Thus, the first challenge is to determine the activation range in which DRX occurs, by means of hot compression tests. Then, the selected model is implemented in the finite element analysis of a specific cutting operation. Finally, the prediction of the numerical model concerning the occurrence of DRX is discussed in relation to the experimental observations.

Experimental techniques

Material

The material employed for the present study is an ASTM A536 100-70-03 iron similar to that employed by [START_REF] Fouilland | Experimental study of the brittle ductile transition in hot cutting of SG iron specimens[END_REF]. This spheroidal graphite iron (SGI) exhibits a pearlitic matrix at room temperature and a small amount of ferrite surrounding the graphite nodules, called bullseye ferrite (Figure 3). Its chemical composition is given in Table 1.

Page 7 of 35

Mechanical characterization

The experiments were performed on a Gleeble 3500 thermo-mechanical testing machine.

Compression specimens of 10 mm in diameter and 12 mm in length were tested under constant deformation conditions in a vacuum chamber. The samples were heated at 5 K.s -1 to the testing temperature and then held for one minute at the test temperature. A K-thermocouple was welded at the half height of the specimen to ensure the temperature measurement. The tests were conducted at mean effective strain rates of 0.5, 1 and 5 s -1 , at nominal temperatures of 1073, 1173 and 1273 K. At the end of the tests, the specimens were air cooled. At least two tests were conducted for each deformation condition.

Cutting operation

These experiments were conducted on an orthogonal cutting test bench. The SGI specimens replicated the feeder head obtained just after casting. These were cylindrical

MPa

Effective strain

1273 K -5 s -1 1273 K -1 s -1 1273 K -0.5 s -1 1073 K -5 s -1 1173 K -1 s -1 1173 K -5 s -1 1073 K -1 s -1

Experimental

Figure 5: Effective-stress-effective strain curves obtained at different temperatures and strain rates.

The samples tested at 5 s -1 were observed using optical microscopy. Figures 6a and6b show the optical micrographs of the samples deformed at 1073 K and 1173 K. These microstructures present a significant variation in their pearlitic matrix from the original state (Figure 3). Indeed, ferrite grains are more prominent and their size is finer. The microstructure of the sample deformed at 1273 K, exhibited in Figure 6c, has a pearlitic matrix and the crushed graphite nodules are surrounded by ferrite grains to a lesser extent.

A c c e p t e d M a n u s c r i p t

The predominant change is observed with the pearlite grains, which have a finer size than in the original state. Figure 8 shows the experimental cutting forces and also multiple pictures extracted from the recording of the cutting operation, whose main steps are described below.

1. The pin is in full contact with the tool and its base is deformed plastically, 2. A shear band is detected on the pin base, the cutting forces reach a maximum,

Constitutive model employed for the description of the flow stress curves

In the past few years, several constitutive relations including DRX effects have been developed to describe the behaviour of austenite under hot working conditions. Most of these constitutive models are based on the dependence of different variables on the Zener-Hollomon parameter [START_REF] Kim | Study on constitutive relation of AISI 4140 steel subject to large strain at elevated temperatures[END_REF][START_REF] Lin | Constitutive modeling for elevated temperature flow behavior of 42CrMo steel[END_REF][START_REF] Lurdos | Empirical and physically based flow rules relevant to high speed processing of 304L steel[END_REF]Wang et al., 2012).

Description of the constitutive model

In the present work, the description of the flow stress curves corresponding to the SGI samples deformed under hot-working conditions has been carried out on the basis of the models earlier advanced by Puchi-Cabrera et al. for structural steels deformed under hotworking conditions [2011; 2013a; 2013b; 2014a; 2014b; 2015]. Accordingly, the flow stress data is employed for determining the main stress parameters characteristic of the deformation of the material under these conditions, which include: yield stress, critical stress for the onset of DRX, actual or hypothetical saturation stress (depending on deformation conditions) and actual steady-state stress. Additionally, both the Avrami exponent and the time required to achieve 50% DRX are determined from the work-softening transients present in some of the stress-strain curves. However, the model can be simplified on the ba- Thus, the work-hardening (WH) and dynamic recovery (DRV) transient of each stressstrain curve is described by means of the evolution equation derived from the workhardening law advanced by [START_REF] Sah | Recrystallization during hot deformation of nickel[END_REF], which is expressed as:

dσ ε dε = µ(T ) A 1 - σ ε -σ y (T, ε) σ sat (T, ε) -σ y (T, ε) 2 σ sat (T, ε) -σ y (T, ε) σ ε -σ y (T, ε) (1)
In the above evolution equation, σ ε represents the current flow stress, σ y the yield stress, σ sat the hypothetical saturation stress, µ(T ) the temperature-dependent shear modulus, ε the effective strain rate, T the deformation temperature and A a material parameter that could either be a constant or a function of deformation conditions through the Zener-Hollomon parameter, which is defined as

Z = ε exp Q RT
where Q is an apparent activation energy for hot deformation and R is the universal gas constant. As shown in the forthcoming, the constant A in the above equation is computed from the experimental stress-strain data corresponding to each stress-strain curve determined under constant deformation conditions.

Regarding the temperature-dependent shear modulus, it can be confidently computed from the equation [START_REF] Kocks | Laws for work-hardening and low-temperature creep[END_REF]:

µ(T ) = 88884.6 -37.3T , MPa (2)
The two stress parameters present in eq.( 1) can also be expressed in terms

σ y (T, ε) = δ y sinh -1          ε exp Q RT B y     1/my      (3)
Whereas, for the hypothetical saturation stress:

σ sat (T, ε) = δ s sinh -1          ε exp Q RT B s     1/ms      (4)
In eqs. 3 and 4, δ y , B y and m y , as well as δ s , B s and m s represent material parameters and Q an apparent activation energy for hot deformation.

The actual steady-state stress (σ ss ), which as indicated above is considered to be equal to the critical stress for the onset of DRX (σ c ), can also be correlated with Z by means of the STG model, according to the following expression:

σ ss (T, ε) = δ ss sinh -1          ε exp Q RT B ss     1/mss      (5)
As in equation 4, δ ss , B ss and m ss represent material parameters.

From the computational point of view, equation 1 is firstly integrated numerically. If the resulting value of σ ε is less than σ c (σ c = σ ss ), the flow stress is determined by σ ε (σ = σ ε ), otherwise, the flow stress should be computed from a second evolution equation, which includes the description of the work-softening transient associated to DRX. This second evolution law is expressed as:

dσ dε = µ(T ) A 1 - σ -σ y + ∆σX v σ sat -σ y 2 σ -σ y + ∆σX v σ sat -σ y - n Av ∆σ(1 -X v ) ln 2 ε t n Av 0.5 -t n Av 0.5 ln (1 -X v ) ln 2 1-1 n Av (6)
A c c e p t e d M a n u s c r i p t

Thus, the incremental change in the flow stress with strain, once DRX becomes operative, is observed to depend on σ, σ y , σ sat , σ ss , the dynamically recrystallized volume fraction, X v , the Avrami exponent, n Av , the time for 50% DRX, t 0.5 , µ(T ), ε, T and constant A.

Since plastic deformation of the material can occur under transient conditions involving arbitrary changes in temperature and strain rate, X v should also be computed from the Johnson-Mehl-Avrami-Kolmogorov (JMAK) equation expressed in differential form:

dX v dt = n Av (1 -X v ) ln 2 t n Av 0.5 -t n Av 0.5 ln (1 -X v ) ln 2 1-1 n Av (7)
However, in this case the change in the volume fraction recrystallized with time is also expressed in terms of the time required to achieved 50% DRX, t 0.5 , which can be conveniently computed by means of the simple parametric relationship proposed by Jonas et al.

(2009), expressed as:

t 0.5 = D ε exp Q RT -q exp Q DRX RT , s (8)
In the above equation, D represents a material constant weakly dependent on the austenitic grain size, whereas q and Q DRX represent a material parameter and the apparent activation energy for dynamic recrystallization, respectively. Thus, the constitutive description of the material is represented by equations 1 through 8. Clearly, two important features can be observed. Firstly, the flow stress is absolutely independent of the total strain (ε) applied to the material. Secondly, that given the determination of the flow stress from the numerical integration of two evolution laws, such a parameter can be readily evaluated and ( 6) of the constitutive formulation presented in the previous section. The accurate description of the experimental curves suggests that their individual modelling allows a precise and reliable identification of all the parameters of interest indicated above. Table 2 summarizes the value of the relevant parameters that were determined for each deformation condition. However, in order to formulate a global constitutive equation able to predict the flow stress of the material under arbitrary deformation conditions, the functional dependence of σ y , σ sat , σ ss , t 0.5 and A (if any) on ε, T should also be accurately established.

Previous studies conducted on 20MnCr5 steel deformed under hot-working conditions (Puchi-Cabrera et al., 2014a) indicated that a single activation energy value of approximately 283.3 kJ.mol -1 could be satisfactorily employed for the computation of the Zener-Hollomon parameter (Z), as well as for the corresponding description of σ y , σ sat , σ ss and t 0.5 as a function of deformation temperature and strain rate. Therefore, in the present work the same value will be employed for both purposes. Thus, Figure 13 illustrates the change in σ y , σ sat and σ ss as a function of Z, as well as their corresponding description according to the STG model (eqs. 3 through 5). Table 3

A c c e p t e d M a n u s c r i p t

As can be observed from Figure 13, the predicted change in each parameter with Z, indicated by the solid lines, describes quite satisfactorily the experimental data, which provides a reliable formulation for modelling purposes. An interesting feature that can be observed from this figure is that related to the behaviour exhibited by σ sat and σ ss .

The curve corresponding to the change in σ sat with Z, in the temperature and strain rate intervals explored in the present work, is always above that corresponding to the change in σ ss . However, as Z increases above a value of approximately 10 16 s -1 , both curves tend to approach each other, which suggests that DRX will occur to a lesser extent and therefore, that DRV will be the only operative dynamic restoration process. cutting operation. However, it also shows that as the temperature decreases and the strain rate increases, the extent to which DRX continues to occur is more limited.

Regarding the temperature and strain rate description of t 0.5 , Figure 14 clearly illustrates that the simple parametric relationship given in eq.( 8) constitutes a quite satisfactory approach for the computation of such a parameter under arbitrary deformation conditions.

1E-17 1E-16 1E-15 1E-14 1E-13 1E-12 1E-1 1 1E-10 1E+10 1E+11 1E+12 1E+13 1E+14 1E+15 1E+16 t 0.5 exp(-283300/RT) , s -1 283300 ε exp , s RT        -0.85 0.5 283300 Z = ε exp RT 283300 t = 0.0032 Z exp RT             
Figure 14: Evolution of the time required to achieve 50% DRX as function of Z.

As indicated on the plot, this relationship can be simplified further by assuming that the apparent activation energy for DRX has the same magnitude than that for hot deformation, which reduces the number of material parameters in the global constitutive formulation without compromising the accuracy of the model prediction. The values of the different constants involved in eq.( 8) are shown on the plot.

Another important consideration of the proposed constitutive formulation involves the temperature and strain rate dependence of constant A, as can be clearly observed from

A c c e p t e d M a n u s c r i p t

Table 2. Previous work conducted both on C-Mn, 20MnCr5 and Fe-Mn23-C0.6 steels [START_REF] Puchi-Cabrera | Constitutive description of a low C-Mn steel deformed under hot-working conditions[END_REF]2013a;2013b;2014a;2014b;[START_REF] Puchi-Cabrera | Constitutive description of FeMn23C0.6 steel deformed under hot-working conditions[END_REF] indicates that this constant in general does not exhibit any significant dependence on T and ε. However, in the present case it can be clearly observed that such a constant exhibits a significant dependence on deformation conditions, which should be taken into consideration into the global constitutive formulation. Thus, Figure 15 highlights the change in A as a function of Z, which clearly indicates that an increase in Zener-Hollomon parameter value leads to a significant and unexpected increase in the athermal work-hardening rate of the material, θ 0 = µ/A.

Finite element modelling of the cutting operation

Finite element model description

Finite element modelling and analysis of the cutting operation were performed with the Abaqus/Explicit software. Figure 20 shows the initial mesh of the workpiece and the cutting tool. The specimen was meshed using 8-nodes 3D solid elements (C3D8RT). A higher mesh density was applied to the pin, as compared to the rest of the model. In order to reduce the computing time of this analysis, only half of both the specimen and the tool were modelled. This results in 9710 elements. A symmetry condition on the z-axis was applied to both right faces of the workpiece and the tool. Appropriate boundary conditions were applied to constrain the bottom, front and left faces of the specimen. The tool, which is considered as a rigid surface, was constrained to move in the cutting direction, at a speed, which varies with time, as shown in Figure 7. No other displacements were allowed. The contact between the tool and the workpiece was modelled using a Coulomb's friction law. The friction coefficient has been identified by an inverse method and set to an almost constant value of 0.2. The workpiece was modelled as an elastic-plastic material A c c e p t e d M a n u s c r i p t with isotropic hardening.

Implementation of the constitutive model

Since the cutting operation is assumed to be performed at a mean constant temperature and strain rate, the integrated version of the constitutive formulation presented in section 4 was implemented using a VUHARD user subroutine (Figure 21). The variables to be defined in the subroutine are the flow stress, Σ, and its variations with respect to the total effective strain, the effective strain rate and the temperature.

ε r = 1 2 A µ(T ) (σ sat (T, ε) -σ y (T, ε)) ε c = -ε r ln 1 - σ ss (T, ε) -σ y (T, ε) σ sat (T, ε) -σ y (T, ε) 2 If ε < ε c σ ε = σ y (T, ε) + (σ sat (T, ε) -σ y (T, ε)) 1 -exp - ε ε r 1/2 Required variables Σ = σ ε ∂Σ ∂ε (T, ε) , ∂Σ ∂ ε (T,ε) , ∂Σ ∂T (ε, ε) Else t = ε -ε c ε X v = 1 -exp -ln(2) t t 0.5 n Av ∆σ = X v (σ sat (T, ε) -σ ss (T, ε)) Required variables Σ = σ ε -∆σ ∂Σ ∂ε (T, ε) , ∂Σ ∂ ε (T,ε) , ∂Σ ∂T (ε, ε)
A c c e p t e d M a n u s c r i p t

The volume fraction recrystallized, X v , was set as a solution-dependent variable within the subroutine. The critical strain for the onset of DRX (ε c ), the relaxation strain (ε r ) and the time during which DRX occurs (t) were introduced.

As mentioned in the previous section, the cutting operation is conducted at a mean temperature of 1273 K and a mean strain rate of approximately 200 s -1 . Furthermore, the high cutting speed allows no time for heat transfer between the tool and the workpiece material. Thus, the model was assumed to be adiabatic. According to [START_REF] Soo | 3D FE modelling of the cutting of Inconel 718[END_REF],

this assumption is generally used for the simulation of high-speed manufacturing processes.

The initial temperature of the test was applied to the specimen. No fracture criterion is introduced in this simulation as the emphasis is put on the beginning of the hot cutting operation corresponding to the first 4 ms. Since the constitutive law has been implemented in its integrated form, it assumes that during the cutting operation the mean strain rate remains constant and therefore, no effect of crack propagation has been taken into consideration concerning the evolution of the volume fraction recrystallized dynamically. In order to take into account changes in temperature and strain rate during the cutting operation, the constitutive law should be implemented in its differential formulation.

Page 29 of 35

A c c e p t e d M a n u s c r i p t

Simulation results

Figure 22 illustrates the comparison between the predicted forces before the crack initiation and the experimental forces. The normal force is predicted quite satisfactorily, whereas the cutting force is clearly overestimated between the steps 1 and 2. However at 4 ms, the prediction errors are respectively about 2.8 percent and 1.8 percent for the cutting and normal forces. The gap between the finite element model and the experimental results can be explained by the fact that the SGI specimen is not perfectly clamped in its insert. The relative velocity between the tool and the pin is then less than expected at the beginning of the test. This, results in a time lag between the predicted cutting force and the actual one.

DRT

  Abbreviations DIC Digital image correlation DRV Dynamic recovery DRX Dynamic recrystallization JMAK Johnson-Mehl-Avrami-Kolmogorov model SGI Spheroidal graphite iron STG Sellars-Tegard-Garofalo model WH Work hardening Arabic symbols A Material parameter B s , B ss , B y Material parameters in the STG model, s -1 D Material parameter, s m s , m ss , m y Material parameters in the STG model n Av Avrami exponent Q Apparent activation energy for hot-working, kJ.mol -1 q Material parameter Q DRX Apparent activation energy for dynamic recrystallization, kJ.mol -1 R Universal gas constant, J.mol -1 .K -1 T Absolute temperature, K t Time during which DRX occurs, s t 0.5 Time for 50 percent recrystallization, s X v Volume fraction recrystallized Z Zener-Hollomon parameter, s -1 Greek symbols δ s , δ ss , δ y Material parameters in the STG model, MPa ε Total effective strain

Figure 1 :

 1 Figure 1: Heat treatment example for austempered ductile iron.

Figure 2 :

 2 Figure 2: Typical dynamic recrystallization hardening curve.

  effect of WH and DRV. As a consequence, a work softening transient will occur, leading to the presence of a peak stress (σ p ) on the flow stress curve. As the strain applied to the material continuous to increase, the balance among WH, DRV and DRX will lead to the achievement of a steady-state stress (σ ss ), whose magnitude is equal to σ c .

Figure 3 :

 3 Figure 3: Optical micrograph of the ASTM A536 100-70-03 iron in its original state (etched with saturated nitric acid).

  mm in height and 10 mm in diameter. Also, the fillet radius on the pin base was of 2.5 mm. At the beginning of the test, the specimens were heated in a furnace up to the required temperature. Then, they were clamped in a refractory insert bed, which prevented from heat losses. Finally, during the cutting operation, the high strength steel cutting tool moves against the cylinder, as shown in Figure4. This experimental device includes a piezoelectric sensor for measuring cutting loads. A high speed camera records the cutting operation at a frequency of 15000 frames per second with a resolution of 768 x 648 pixels. A speckle pattern covering the tool allows the determination of the effective cutting velocity by means of a digital image correlation (DIC). The tests were performed three times at a temperature of 1273 K with a tool rake angle of -10˚and an initial cutting speed of 1.2 m.s -1 . The choice of the negative rake angle was based on the results of the study conducted by[START_REF] Fouilland | Experimental study of the brittle ductile transition in hot cutting of SG iron specimens[END_REF], which revealed that such rake angle allows the observation of the brittle-ductile transition.

Figure 4 :

 4 Figure 4: Clamped specimen and cutting tool.

Figure 5

 5 Figure5illustrates the mean effective stress-effective strain curves obtained at different temperatures and strain rates. It was observed that the typical deviation of the flow stress values from the mean curve was about + or -2 MPa. The experimental stress-strain curves exhibit the same shape as that portrayed in Figure2, highlighting the occurrence of DRX during the compression tests.

Figure 6 :Figure 7 :

 67 Figure 6: Optical micrographs of the ASTM A536 100-70-03 iron deformed at 5 s -1 and different temperatures (etched with saturated nitric acid).

Figure 8 :Figure 9 :

 89 Figure 8: Axial and normal forces with the corresponding pictures recorded by the high speed camera.

  A c c e p t e d M a n u s c r i p t sis of the experimental results reported by Jonas et al. (2009) and Quelennec et al. (2011), who demonstrated that, for a broad range of steel grades, the steady state stress is equal to the critical stress for the nucleation of DRX.

  of T and ε by means of the well established Sellars-Tegart-Garofalo (STG) model [1966]. For the Page 15 of 35 A c c e p t e d M a n u s c r i p t yield stress, its functional dependence on T and ε is expressed as:

  when the deformation of the material occurs under transient deformation conditions, which are characteristic of actual industrial hot deformation processes. The experimental flow stress data determined at different deformation temperatures and strain rates constitutes the raw data for the rational computation of the different material parameters involved Identification of the different parameters involved in the constitutive modelThe precise determination of the different stress parameters involved in the constitutive description of material, as well as the time required to achieve 50% DRX, can be conducted by means of the individual modelling of each stress-strain curve determined under constant conditions of temperature and strain rate. Figures 10 through 12 illustrate the comparison of the experimental stress-strain curves and the predicted ones employing equations (1)

Figure 10 :

 10 Figure 10: Comparison of the experimental stress-strain curves and the constitutive formulation at 1073 K.

Figure 11 :Figure 12 :

 1112 Figure 11: Comparison of the experimental stress-strain curves and the constitutive formulation at 1173 K.

Figure 13 :

 13 Figure 13: σy, σsat and σss as a function of Z.

Figure 15 :Figure 16 :

 1516 Figure 15: A as a function of Z.

Figure 17 :

 17 Figure 17: Comparison between predicted and experimental stress-strain curves at 1173 K.

Figure 18 :

 18 Figure 18: Comparison between predicted and experimental stress-strain curves at 1273 K.

Figure 19 :

 19 Figure 19: Maximum relative error between the computed and predicted values of the flow stress.

Figure 20 :

 20 Figure 20: Isometric view of the initial mesh configuration.

Figure 21 :

 21 Figure 21: Algorithm defined in the VUHARD subroutine.

Figure 22 :

 22 Figure 22: Comparison between experimental and predicted forces.

Figure 23 :Figure 24 :

 2324 Figure23shows the von Mises stress (σ eq ) distribution within the specimen during the cutting operation. The fillet radius on the pin base is an area of stress concentration. The average value of von Mises stresses in the fillet radius at 4 ms is about 250 MPa. Under these deformation conditions, DRX is then operative as the effective stress associated to the WH and DRV curve is greater than the critical stress for the onset of DRX.

Table 1 :

 1 Chemical composition of the ASTM A536 100-70-03 iron.

	Element	C	Si	Mn	S	Cu	Ni	Cr	Mo	Mg
	Composition (wt%) 3.35 2.72 0.16 0.009 0.87 0.71 <0.03 0.21 0.043

Table 2 :

 2 Relevant parameters involved in the description of the individual stress-stain curves.

	T , K

  summarizes the value of the different material parameters involved.

	δ y , MPa	B y , s -1	m y δ s , MPa	B s , s -1	m s δ ss , MPa B ss , s -1	m ss
	19.0	1.77E+08	3	104.2	3.47E+10 4.96	88.3	1.13E+11 3.74

Table 3 :

 3 Materials parameters involved in description of σy, σsat and σss as a function of Z, according to the STG model.

A c c e p t e d M a n u s c r i p t of dynamic recrystallization has been determined. A typical cutting process has been modelled both from the experimental and numerical point of view. The numerical predictions agree with the experimental results and highlighted some explanations concerning the occurrence of dynamic recrystallization within the shear zone. Currently, further investigations are being carried out in order to validate the proposed constitutive description, by modelling other cutting configurations. Also, a fracture criterion is being characterized for the spheroidal graphite iron, in order to investigate the competition between the material fracture and the occurrence of DRX during the cutting operation.

Acknowledgements

The present research work has been supported by the ARTS Carnot Institute and was made possible through the collaboration between MSMP and LAMIH laboratories. The authors gratefully acknowledge the support of this institute. They also express their sincere


César Soto-Valero
email: cesarsotovalero@gmail.com

Johann Bourcier
email: johann.bourcier@irisa.fr

Benoit Baudry
email: baudry@kth.se

Detection and Analysis of Behavioral T-patterns in Debugging Activities

Keywords: Debugging interactions, developers' behavior, T-patterns analysis, empirical software engineering

A growing body of research in empirical software engineering applies recurrent patterns analysis in order to make sense of the developers' behavior during their interactions with IDEs. However, the exploration of hidden real-time structures of programming behavior remains a challenging task. In this paper, we investigate the presence of temporal behavioral patterns (T-patterns) in debugging activities using the THEME software. Our preliminary exploratory results show that debugging activities are strongly correlated with code editing, file handling, window interactions and other general types of programming activities. The validation of our T-patterns detection approach demonstrates that debugging activities are performed on the basis of repetitive and well-organized behavioral events. Furthermore, we identify a large set of T-patterns that associate debugging activities with build success, which corroborates the positive impact of debugging practices on software development.

INTRODUCTION

Debugging is a widely used practice in the software industry, which facilitates the comprehension and correction of software failures. When debugging, developers need to understand the pieces of the software system in order to successfully correct specific bugs. Modern Integrated Development Environments (IDEs) incorporate useful tools for facilitating the debugging process, allowing developers to focus only in their urgent needs during the fixing work. However, debugging is still a very challenging task that typically involves the interaction of complex activities through an intense reasoning workflow, demanding a considerable cost in time and effort [START_REF] Perscheid | Studying the advancement in debugging practice of professional software developers[END_REF].

Due to the complex and dynamic nature of the debugging process, the identification and analysis of repetitive patterns can benefit IDE designers, researchers, and developers. For example, IDE designers can build more effective tools to automate frequent debugging activities, suggesting related tasks, or designing more advanced code tools, thus improving the productivity of developers. Furthermore, researchers can better understand how debugging behavior is related to developers' productivity and code quality. Unfortunately, most of existing studies on debugging activities within IDEs do not consider the complex temporal structure of developers' behavior, thus including only information about a small subset of possible events in the form of data streams [START_REF] Parnin | Are Automated Debugging Techniques Actually Helping Programmers?[END_REF].

The detection of temporal behavioral patterns (T-patterns) is a relevant multivariate data analysis technique used in the discovery, analysis and description of temporal structures in behavior and interactions [START_REF] Magnusson | Discovering hidden time patterns in behavior: T-patterns and their detection[END_REF]. This technique allows to determine whether two or more behavioral events occur sequentially, within statistically significant time intervals.

In this paper, we perform a T-patterns analysis to study debugging behavior. More specifically, we examine the relations of debugging events with other developers' activities. Through the analysis of the MSR 2018 Challenge Dataset, consisting of enriched event streams of developers' interactions on Visual Studio, we guide our work by the following research questions:

• RQ 1 : What developing events are the most correlated with debugging activities? • RQ 2 : Can we detect behavioral T-patterns in debugging activities? • RQ 3 : Is the analysis of T-patterns a suitable approach to show the effect of systematic debugging activities on software development?

We aim to answer these question by analyzing a set of 300 debugging sessions filtered from the MSR 2018 Challenge Dataset of event interactions. The objective of our analysis is twofold: [START_REF] Amann | FeedBaG: An interaction tracker for Visual Studio[END_REF] to provide the researchers with useful information concerning the application of T-patterns analysis in the study of developers' behavior; and (2) to present empirical evidence about the influence of debugging on software development.

Previous studies analyzed debugging behavior using patterns detection methods. For example, in the development of automated debugging techniques for IDE tools improvement [START_REF] Parnin | Are Automated Debugging Techniques Actually Helping Programmers?[END_REF]. However, to the best of our knowledge, this is the first attempt of using T-patterns analysis to investigate debugging session data.

DATA MANAGEMENT

The dataset for the 2018 MSR Challenge, released on March 2017 by the KaVE Project1 , contains over 11M enriched events that correspond to 15K hours of working time, originating from a diverse group of 81 developers [START_REF] Proksch | Enriched Event Streams: A General Dataset For Empirical Studies On In-IDE Activities Of Software Developers[END_REF]. The data was collected using FeedBaG, an interaction tracker for Visual Studio, which was designed with the purpose of capturing a large set of different in-IDE interactions during software developing in the shape of enriched event streams [START_REF] Amann | FeedBaG: An interaction tracker for Visual Studio[END_REF].

The THEME software2 supports the detection, visualization and analysis of T-patterns. It has been successfully applied in many different areas, from behavioral interaction between human subjects and animals to neural interactions within living brains [START_REF] Magnusson | Discovering hidden temporal patterns in behavior and interaction: T-pattern detection and analysis with THEME[END_REF]. Due to the data transferred by contributors is anonymous, we base our T-patterns analysis on the session Id that identifies developers' work during each calendar day. Our filtering routine removes duplicate events and generates individual session files with a structure appropriate for THEME. Date-time information of triggered events is converted to epoch-second values, which is an integer representing the number of elapsed seconds from 1970-01-01T00:00:00Z. Only sessions with debugging interactions where retained for further analysis. Our resulting dataset contains 300 sessions and more than 662K events. Figure 1 shows an example of the data inputs: the variable vs. value correspondence table with the debugging-related event types filtered ("vvt.vvt") and a data file of debugging interactions ("DebuggingSession.txt"). We are mostly interested in debugging events triggered using commands, such as "Debug.Start" or "Debug.StepInto", which represent the user's invocation of a direct debugging action in the IDE. We decide to keep other related event types that can bring additional information about the programmer's debugging behavior (e.g., "EditEvent", "TestEvent" or "BuildEvent"). To do so, we append onto each event type string its respective descriptor. For instance, we retain information about the amount of editing according to the size of changes made in the file (e.g., "Large" or "Short"), the result of tests (e.g., "Successful" or "Failed"), or the build result (e.g., "Successful" or "Unsuccessful").

Our analysis goes beyond the discovery of events' associations. We are more interested in explaining those connections in terms of developers' behaviour by means of T-patterns analysis. In the following, we perform the events analysis using THEME software. First, we show how interesting Tpatterns can be detected and visualized through the finegrained inspection of interactions in individual debugging sessions. Next, we aim to find general behavioral patterns that occur within statistical significance time thresholds for all the debugging sessions studied.

T-PATTERNS ANALYSIS

In this section, we summarize the main concepts regarding the detection and analysis of T-patterns [START_REF] Magnusson | Discovering hidden time patterns in behavior: T-patterns and their detection[END_REF]. Through the use of an active debugging session as case study, we illustrate the benefits of using THEME software as a tool for exploring hidden real-time structures of programming behaviour in IDEs. Our general approach consists of 3 phases: and evolution to deal with redundant detections, where partial and equivalent patterns are removed. THEME provides statistical validation features, global and per pattern, using randomization or Monte Carlo repeated [START_REF] Magnusson | Discovering hidden temporal patterns in behavior and interaction: T-pattern detection and analysis with THEME[END_REF].

T-patterns visualization. A T-pattern can be viewed as a hierarchical and self-similar pseudo fractal pattern, characterized by significant translation symmetry between their occurrences. Figure 2b shows the binary detection tree of a complex T-pattern of length 7 found in the debugging session of Figure 2a. The large vertical lines connecting event points indicate the occurrence time of the T-pattern. The node marked in green indicates an event that can be predicted from the earlier parts of the pattern (also called T-retrodictor).

GENERAL FINDINGS

We perform an exploratory data analysis to examine the association among events. We use the phi coefficient of correlation, a common measure for binary correlation, and the tidytext R package in order to visualize how often events appear together relative to how often they appear separately [START_REF] Silge | tidytext: Text Mining and Analysis Using Tidy Data Principles in R[END_REF]. Figure 3 shows the 10 developers' activities that we find more correlated with debugging (𝜑 > 0.5). From the figure, we observe that debugging activities are strongly correlated with code editing, window interactions, document saving, and activity events. In addition, we found that code completion, keyboard navigation and short code editing events are not directly correlated with debugging activities. Based on the observation of Figure 3, we derive the answer to the RQ 1 as follows:

Answer to RQ 1 : Debugging activities are more correlated with editing, file handling, window interactions and activity events than with other general commands or event types.

We are mostly interested in analyzing general patterns of events that occur within the debugging workflow. Such patterns allow for insights into the dynamic nature of developer's behavior while debugging software. Accordingly, all debugging sessions were ordered and concatenated in time to conform a single dataset for global analysis with THEME. Thus, the 300 debugging sessions were merged, resulting in a dataset with 263 different event types and more than 460K events' occurrences.

The following search parameters were fit in THEME via grid search: (a) detection algorithm = FREE; (b) minimum number of occurrences of pattern = 10; (c) significance level = 0.0005 (0.05% probability of any CI relationship to occur by chance); (d) maximum number of hierarchical search levels = 10; (e) exclusion of frequent event types occurring above the mean number of occurrences of ±1 standard deviations.

For the above parameters, more than of 12K of T-patterns were detected. We run the algorithm on 10 randomized versions of the data, using the same search parameters, to check if the set of detected T-patterns differentiate significantly from those obtained randomly. Figure 4 shows the comparison between the distributions of the detected patterns on the original data and the average number of patterns detected after the randomization procedure. The incidence of T-patterns in real data was significantly greater than in its randomized versions. Accordingly, it is clear that the T-patterns detected in the original dataset were not obtained by chance. This result demonstrates that debugging activities are organized on the basis of behavioral events, which occur sequentially and within significant constraints on the time intervals that separates them. Based on this result, we derive the answer to the RQ 2 as follows:

Answer to RQ 2 : The validation of the T-patterns detected using THEME provides meaningful evidence about the presence of behavioral patterns in debugging activities. Once T-patterns have been detected, the next challenge is to select relevant T-patterns for subsequent analysis. We are interested in study T-patterns that associate debugging activities with build results. To this end, we used the filters available in THEME, which allow to search for the presence of desired event types in patterns. We found a total of 735 T-patterns that directly associate debugging activities with successful builds, whereas only 67 T-patterns were found for unsuccessful builds. This result shows that, after a methodical sequence of debugging activities, generally the developers have much more chances to achieve successful builds.

Table 1 present a global comparison between the T-patterns found in debugging sessions that are directly related with successful and unsuccessful build results. From the table, we can see that T-patterns related to successful builds occurs more frequently and have a more complex structure, with higher values of patterns' length and duration. On the other hand, T-patterns associated with unsuccessful builds present a more simple structure, with a mean length value of nearly 2 events only and a duration that is almost five times smaller than T-patterns associated with successful builds. This result show that more complex debugging sessions (e.g., those in which developers utilize more specialized debugging tools or invert more time to complete) are more likely to pass the builds and correct software failures.

By analyzing the T-patterns of sessions with unsuccessful builds, we find that their contain mostly events that introduce minor changes in code (e.g., "Edit.Delete, "Edit.Paste"). We hypothesize that this type of debugging sessions were used to quick trace the effect of these changes. Table 1 also shows representative examples of T-patterns occurrences for both types of build results. Based on the T-patterns analysis performed, we derive the answer to the RQ 3 as follows:

Answer to RQ 3 : The quantitative analysis of detected T-patterns in debugging sessions shows that, in general, complex debugging activities achieve successful builds.

CONCLUSION

In this paper, we introduced T-patterns analysis as a useful approach to better understand developer's behavior during in-IDE activities. Through the analysis of 300 sessions with debugging interactions, the results obtained using the THEME software bring evidences about the presence of common Tpatterns during debugging. In particular, our analysis show a strong connection between debugging activities and successful builds. We believe that the study of the developers' activities using T-patterns analysis can advance the understanding about the complex behavioral mechanism that meddle during the process of software developing, which can benefit to both practitioners and IDE designers. In order to aid in future replication of our results, we make our THEME project, filtered dataset and R scripts publicly available online3 .

Figure 1 :

 1 Figure 1: Data input structure for THEME software.

  (1) visualization of debugging interactions in the form of T-data; (2) detection of T-patterns in debugging sessions; and (3) validation and analysis of the detected T-patterns. T-data. A T-data consists in a collection of one or more T-series, where each T-series represents the occurrence points 𝑝1, ..., 𝑝𝑖, ..., 𝑝𝑛 of a specific type of event during some observation interval [1, 𝑇 ]. Figure 2a shows an example of T-data coded from a debugging session with 166 squared data points (events occurrences), 25 T-series (event types), and a duration of 823 units. Each T-series in the Y-axis represents an event activity triggered in the IDE during the session, while the X-axis is the time in which each specific event was invoked. For the search parameters used, the blue squares represent detected T-patters, while the red ones did not. T-pattern. A T-pattern is composed of 𝑚 ordered components 𝑋1 . . . 𝑋𝑖 . . . 𝑋𝑚, any of which may be occurrence points or T-patterns, on a single dimension (time in this case), such that, over the occurrences of the pattern the distances 𝑋𝑖 𝑋𝑖+1, with 𝑖 . . . 𝑚 -1, varies within a significant small interval [𝑑1, 𝑑2]𝑖, called a critical interval (CI). Hence, a T-pattern 𝑄 can be expressed as: 𝑄 = 𝑋1[𝑑1, 𝑑2] 1 . . . 𝑋𝑖[𝑑1, 𝑑2]𝑖𝑋𝑖+1 . . . 𝑋𝑚-1[𝑑1, 𝑑2]𝑚-1𝑋𝑚 where 𝑚 is the length of 𝑄 and 𝑋𝑖[𝑑1, 𝑑2]𝑋𝑖+1 means that within all occurrences of the pattern in T-data, after an occurrence of 𝑋𝑖 at the instant 𝑡, there is a time window [𝑡 + 𝑑1, 𝑡 + 𝑑2]𝑖 within which 𝑋𝑖+1 will occur. Any T-pattern 𝑄 can be divided into at least one pair of shorter ones related by a corresponding CI: 𝑄 𝑙𝑒𝑓 𝑡 [𝑑1, 𝑑2]𝑄 𝑟𝑖𝑔ℎ𝑡 . Recursively, 𝑄 𝑙𝑒𝑓 𝑡 and 𝑄 𝑟𝑖𝑔ℎ𝑡 can thus each be split until the pattern 𝑋1 . . . 𝑋𝑚 is expressed as the 1 to 𝑚 terminals (occurrence points or event types) of a binary-tree. T-patterns detection. The T-patterns detection algorithm consists in a set of routines for CI detection, pattern construction and pattern completeness competition. The algorithm works bottom-up, level-by-level and uses competition

  (a) T-data representation. (b) T-pattern visualization.

Figure 2 :

 2 Figure 2: T-patterns analysis of a debugging session, both figures were created with THEME.

Figure 3 :

 3 Figure 3: Pairwise correlation between events related to debugging activities.

Figure 4 :

 4 Figure 4: Distribution of T-patterns lengths detected in real and randomized data.

Table 1 :

 1 Summary of T-patterns detected which reflect the relation of debugging activities with build results.

	Build Result Occurrence	Length	Duration	T-pattern Example
	Successful	735	4.87±0.72 580.09±232.51 (Debug.Start((Debug.StepOver Debug.StopDebugging)BuildEvent.Successful))
	Unsuccessful	67		120.71±35.91	(Debug.Start(Edit.Delete(DocumentEvent.Saved BuildEvent.Unsuccessful)))

Available at http://www.kave.cc/datasets

For more information see http://patternvision.com

https://github.com/cesarsotovalero/msr-challenge2018


Tonametl Sanchez
email: t.sanchez@inria.fr

Andrey Polyakov
email: andrey.polyakov@inria.fr

Jean-Pierre Richard
email: jean-pierre.richard@centralelille.fr

Denis Efimov
email: denis.efimov@inria.fr

J.-P Richard

A robust Sliding Mode Controller for a class of bilinear delayed systems

In this paper we propose a Sliding Mode Controller for a class of scalar bilinear systems with delay in both the input and the state. Such a class is considered since it has shown to be suitable for modelling and control of a class of turbulent flow systems. The stability and robustness analysis for the reaching phase in the controlled system are Lyapunov-based. However, since the sliding dynamics is infinite dimensional and described by an integral equation, we show that the stability and robustness analysis is simplified by using Volterra operator theory.

I. INTRODUCTION

Turbulent Flow Control is a fundamental problem in several areas of science and technology, and improvements to address such a problem can produce a very favourable effect in, for example, costs reduction, energy consumption, and environmental impact [START_REF] Brunton | Closed-Loop Turbulence Control: Progress and Challenges[END_REF]. Unfortunately, in general, model based control techniques find several obstacles to be applied to the problem of Flow Control. One of the main difficulties is that the par excellence model for flow is the set of Navier-Stokes equations that is very complicated for simulation and control design [START_REF] Brunton | Closed-Loop Turbulence Control: Progress and Challenges[END_REF]. On the other hand, when the model is very simple it is hard to represent adequately the behaviour of the physical flow. In [1] the authors say that the remaining missing ingredient for turning flow control into a practical tool is control algorithms with provable performance guarantees. Hence, adequate models (a trade-off between simplicity, efficiency, and accuracy) and algorithms for Flow Control are required.

In [START_REF] Feingesicht | A bilinear input-output model with state-dependent delay for separated flow control[END_REF] a model for a flow system was proposed, such a model consists in a bilinear differential equation with delays in the input and in the state. An attractive feature of the model is that, according to the experimental results, with some few parameters the model reproduces the behaviour of the physical flow with a good precision. The justification for using such a kind of equations as models for flow systems was presented in [START_REF] Feingesicht | Nonlinear active control of turbulent separated flows: Theory and experiments[END_REF]. We reproduce that reasoning as a motivational example in Section II.

For a particular case of the model introduced in [START_REF] Feingesicht | A bilinear input-output model with state-dependent delay for separated flow control[END_REF], a sliding mode controller was proposed in [START_REF] Feingesicht | SISO model-based control of separated flows: Sliding mode and optimal control approaches[END_REF]. That control technique was chosen due to the switching features of the actuators. A good experimental performance was obtained with such a controller1 . Hence, it is worth to continue with the study of the class of bilinear delayed systems and to develop general schemes for analysis and control design. In this paper we design a Sliding Mode Controller for a subclass of such systems by following the idea proposed in [START_REF] Feingesicht | SISO model-based control of separated flows: Sliding mode and optimal control approaches[END_REF]. Nonetheless, the result of this paper differs from [START_REF] Feingesicht | SISO model-based control of separated flows: Sliding mode and optimal control approaches[END_REF] in the following points.

• The dynamics on the sliding surface is infinite dimensional and is described by an integral equation. In [START_REF] Feingesicht | SISO model-based control of separated flows: Sliding mode and optimal control approaches[END_REF] the asymptotic stability of the sliding motion was analysed in the frequency domain. In this paper we propose, as one of the main contributions, to analyse the stability properties of such dynamics by considering it as a Volterra integral equation. This allows us to simplify the analysis and to give simple conditions to guarantee asymptotic stability of the solutions. Hence we avoid the necessity of making a frequency domain analysis to determine the stability properties of an infinite dimensional system. • The analysis of the reaching phase is Lyapunov-based, this is important because it is not only useful to establish stability properties, but also robustness, whose analysis is performed applying Volterra operator theory. • Although the systems considered in this paper and those in [START_REF] Feingesicht | SISO model-based control of separated flows: Sliding mode and optimal control approaches[END_REF] are similar, the assumptions on the parameters are different. This allows us to enlarge the class of systems which can be considered for the application of the proposed methodology. Paper organization: In Section III a brief description of the control problem is given. Some properties of the system's solutions are studied in Section IV. The design and analysis of the proposed controller are explained in Section V. A robustness analysis is given in Section VI. A numerical example is shown in Section VII. Some final remarks are stated in Section VIII.

Notation: R denotes the set of real numbers. For any a ∈ R, R ≥a denotes the set {x ∈ R : x ≥ a}, and analogously for R >a . For any p ∈ R ≥1 , L p (J) denotes the set of measurable functions x : J ⊂ R → R with finite norm x L p (J) = J |x(s)| p ds 1 p , and L ∞ (J) denotes the set of measurable functions with finite norm x L ∞ (J) = ess sup t∈J |x(t)|.

II. MOTIVATIONAL EXAMPLE

In this section we reproduce the example given in [START_REF] Feingesicht | Nonlinear active control of turbulent separated flows: Theory and experiments[END_REF] on how a bilinear delayed differential equation can be obtained as a model for a flow system.

A unidimensional approximation to the Navier-Stokes equations is the Burgers' equation given by

∂v(t, x) ∂t + v(t, x) ∂v(t, x) ∂x = ν ∂ 2 v(t, x) ∂x 2 , (1)
where v : R2 → R is the flow velocity field, x ∈ R is the spatial coordinate, and ν ∈ R ≥0 is the kinematic viscosity. Assume that x ∈ [0, F ] for some F ∈ R >0 . Suppose that v(t, x) = v(x -ct), i.e. the solution of (1) is a travelling wave with velocity c ∈ R ≥0 , it has been proven that (1) admits this kind of solutions [START_REF] Debnath | Nonlinear Partial Differential Equations for Scientists and Engineers[END_REF]. A model approximation of (1) can be obtained by discretizing (1) in the spatial coordinate. Here, we use central finite differences, for the spatial derivatives, with a mesh of three points (and step of h = F/2). Thus

∂v(t, F/2) ∂t + v(t,F/2) F [v(t, F ) -v(t, 0)] = 4ν F 2 [v(t, F ) -2v(t, F/2) + v(t, 0)] . ( 2
)
Since v is assumed to be a travelling wave, it has a periodic pattern in space and time. In particular, note that v(t,

F/2) = v(F/2 -ct) = v(t + F/(2c), F ) = v(t -F/(2c), 0). Now, define y(t) = v(t, F ) and u(t) = v(t, 0), thus, (2)
can be rewritten as

ẏ(t) = -1 F y(t -ς)u(t -2ς) + 1 F y(t)u(t -ς)+ 4ν F 2 [y(t -ς) -2y(t) + u(t -ς)]
, where ς = F/2c. Hence, in [START_REF] Feingesicht | Nonlinear active control of turbulent separated flows: Theory and experiments[END_REF], [START_REF] Feingesicht | A bilinear input-output model with state-dependent delay for separated flow control[END_REF], [START_REF] Feingesicht | SISO model-based control of separated flows: Sliding mode and optimal control approaches[END_REF] the authors propose a more general model for separated flow control:

ẏ = N1 i=1 a i y(t -τ i ) + N2 i=1 N3 k=1 āk y(t -τk ) + b j × u(t -ς k ) . (3)
Observe that this approximating model still recovers two main features of the original flow model: first, it is nonlinear; and second, it is infinite dimensional.

III. PROBLEM STATEMENT

Consider the system

ẋ(t) = a 1 x(t -τ 1 ) -a 2 x(t -τ 2 ) + [c 1 x(t -τ1 ) -c 2 x(t -τ2 ) + b] u(t -ς) , (4)
where a 1 , a 2 , c 1 , c 2 , b, τ 1 , τ 2 , τ1 , τ2 ∈ R ≥0 . We assume that all the delays are bounded and constant. We also assume that the initial conditions of (4) are x(t) = 0 for all t < 0 and x(0) = x 0 for some x 0 ≥ 0.

The control objective is to drive the state of the system to a constant reference x * ∈ R >0 . Such an objective must be achieved under the following general restrictions:

• Since the equation is used to model a positive physical system, some conditions on the model parameters have to be given to guarantee that the solutions of (4) can only take nonnegative values. • Due to the physical nature of the on/off actuator, the control input is restricted to take values from the set {0, 1}.

IV. SYSTEM'S PROPERTIES

As stated in Section III we require some features of the solutions of (4) to guarantee that it constitutes a suitable model for the physical system. In this section we study the conditions on the parameters of ( 4) that guarantee nonnegativeness and boundedness of the solutions. Of course, existence and uniqueness of solutions must be guaranteed. To this aim we rewrite (4) as

ẋ = a 1 x(t -τ 1 ) + c 1 u(t -ς)x(t -τ1 ) -a 2 x(t -τ 2 )- c 2 u(t -ς)x(t -τ2 ) + bu(t -ς) , (5)
that can be seen as a linear delayed system with timevarying coefficients. The term bu(t -ς) is considered as the input. In a first time, we will consider the general case u : R ≥0 → R and, then, restrict it to u : R ≥0 → {0, 1}.

A locally absolutely continuous function that satisfies [START_REF] Feingesicht | Nonlinear active control of turbulent separated flows: Theory and experiments[END_REF], for almost all t ∈ [0, ∞), and its initial conditions for all t ≤ 0 is called a solution of ( 5) [START_REF] Agarwal | Nonoscillation Theory of Functional Differential Equations with Applications[END_REF]. Hence, if in addition to the assumptions in the previous section we assume that u : [0, ∞) → R is a Lebesgue-measurable locally essentially bounded function, then the solution of ( 5) exists and it is unique, see Appendix A. Such a definition of solution is adequate for the analysis made in this section, however, for the closed-loop behaviour analysis, we will also consider another framework, see Remark 1 in Section V.

A. Nonnegative solutions

We have said that the model has to be guaranteed to provide nonnegative solutions. Thus, we first search for some conditions that guarantee that the solutions of (5) are nonoscillatory 2 . Consider (5) and define

P (t) = a 1 +c 1 u(t- ς) and N (t) = a 2 + c 2 u(t -ς).
Lemma 1 ([2], Corollary 3.13): Consider (5) with b = 0. If min(τ 2 , τ2 ) ≥ max(τ 1 , τ1 ), N (t) ≥ P (t) for all t ≥ t 0 , and there exists λ ∈ (0, 1) such that

lim t→∞ sup t-min(τ1,τ1) t-max(τ2,τ2) (N (s) -λP (s)) ds < ln(1/λ) e , lim t→∞ sup t t-max(τ2,τ2) (N (s) -λP (s)) ds < 1 e ,
then, the fundamental solution of ( 5) is such that X(t, s) > 0, t ≥ s ≥ t 0 , and ( 5) has an eventually positive solution with an eventually nonpositive derivative. Now, having nonoscillation conditions for (5) we can state the following.

Corollary 1: Consider (5) with b ≥ 0. Suppose that the assumptions of Lemma 1 hold. Assume that x(t) = 0, u(t) = 0 for all t < 0 and x(0) = x 0 for some x 0 ≥ 0. If u(t) ≥ 0 for all t ≥ 0, then x(t) ≥ 0 for all t ≥ 0. The proof is straightforward through the solution representation by using the fundamental function, see Lemma 3. Note that, in particular, the integral conditions of Lemma 1 are satisfied if

a 2 - 1 e (a 1 + c 1 ) max(τ 2 , τ2 ) < 1 e .
Although this is only sufficient, it constitutes a simple formula to verify the integral conditions of Lemma 1.

B. Boundedness of solutions

Observe that the nonoscillation conditions of Lemma 1 also guarantee the boundedness of the system's trajectories for b = 0. For the case b = 0 we have the following result.

Lemma 2: Consider (4) with its parameters satisfying Lemma 1, and with the initial conditions

x(t) = 0, u(t) = 0 for all t ≤ 0. If b = 0, N (t) -P (t) ≥ α , ∀ t ≥ 0 , (6)
for a strictly positive α, and u(t) = 1 ∀t ≥ 0, then the solution of ( 5) is such that x(t) ≤ x for all t ≥ 0 and

lim t→∞ x(t) = x , x = b a 2 + c 2 -a 1 -c 1 . (7)
Proof: According to Lemma 1, if b = 0, then we can ensure that there exists t 1 such that x(t) > 0 and ẋ(t) ≤ 0 for all t ≥ t 1 . Hence, there exists t 2 ≥ t 1 such that for all

t ≥ t 2 ẋ≤a 1 x(t -max(τ 1 , τ1 )) + c 1 u(t -ς)x(t -max(τ 1 , τ1 )) -a 2 x(t -min(τ 2 , τ2 )) -c 2 u(t -ς)x(t -min(τ 2 , τ2 )) ≤P (t)x(t -max(τ 1 , τ1 )) -N (t)x(t -min(τ 2 , τ2 )) ,
thus, since N (t) -P (t) ≥ α, we can ensure that lim t→∞ x(t) = 0, see e.g. [START_REF] Agarwal | Nonoscillation Theory of Functional Differential Equations with Applications[END_REF]Theorem 3.4]. Now, for the particular case u(t) = 1 and b = 0, (4) is time-invariant and the asymptotic behaviour of x(t) guarantees that x = 0 is asymptotically stable, therefore, it is exponentially stable and its fundamental solution X(t, s) is exponentially bounded (see e.g. [START_REF] Györi | Oscillation Theory of Delay Differential Equations With Applications[END_REF], [START_REF] Fridman | Introduction to Time-Delay Systems[END_REF]). Hence, for the case b = 0, u(t) = 1, the solution of (4) can be expressed as (see Lemma 3 in Appendix A)

x(t) = X(t, t 0 )x(0) + t t0 X(t, s)b ds .
Since X(t, s) decreases exponentially in t, x(t) is bounded, moreover, x(t) increases monotonically due to the input term. Thus lim t→∞ x(t) exists and it is some constant x, therefore, lim t→∞ ẋ(t) = 0 = -(a 2 + c 2 -a 1 -c 1 )x + b. This equality gives the limit value [START_REF] Feingesicht | A bilinear input-output model with state-dependent delay for separated flow control[END_REF].

V. SLIDING MODE CONTROLLER

In this section we present the Sliding Mode Controller for (4), but first, define k : R → R given by

k(r) = k a1 (r) -k a2 (r) + k c1 (r) , (8)
where

k a1 (r) = a 1 , r ∈ [min(ς, τ 1 ), max(ς, τ 1 )], 0, r / ∈ [min(ς, τ 1 ), max(ς, τ 1 )], k a2 (r) = a 2 , r ∈ [ς, τ 2 ], 0, r / ∈ [ς, τ 2 ], k c1 (r) = c 1 , r ∈ [ς, τ1 ], 0, r / ∈ [ς, τ1 ].
Theorem 1: If system (4) satisfies the conditions of Lemma 1, the condition (6), ς ≤ τ1 , and

τ2 min(ς,τ1) |k(r)| dr < 1 , (9)
then, for any x * ∈ (0, x) (where x is given by ( 7)), the solution of the closed loop of (4) with the controller

u(t) = 1 2 (1 -sign(σ 0 (t) -σ * )) , (10)
σ 0 (t) = x(t) + a 1 t t-τ1
x(s) ds -a 2 t t-τ2

x(s) ds+

c 1 t t-τ1+ς x(s) ds + t t-ς [c 1 x(s -τ1 + ς)- c 2 x(s -τ2 + ς) + b] u(s) ds , (11)
where

σ * = x * [1 -a 2 (τ 2 -ς) + a 1 (τ 1 -ς) + c 1 (τ 1 -ς)],
establishes a sliding motion in finite-time on the surface σ 0 (t) = σ * , and the sliding motion converges exponentially to x * . The design procedure is explained through the proof of the theorem given in the following sections. Note that for implementation, the following equivalent formula can also be used

σ 0 (t) = x(t) + t 0 {(a 1 + c 1 -a 2 )x(s) -a 1 x(s -τ 1 )+ a 2 x(s -τ 2 ) -c 1 x(s -τ1 + ς)(1 -u(s))+ [-c 2 x(s -τ2 + ς) + b] -[c 1 x(s -τ1 )- c 2 x(s -τ2 ) + b] u(s -ς)} ds .

A. Sliding variable

Let us, from [START_REF] Gripenberg | Volterra Integral and Functional Equations[END_REF], define the sliding variable as σ(t) = σ 0 (t) -σ * . The time derivative of σ is

σ(t) = -(a 2 -a 1 -c 1 )x(t) -c 1 x(t -τ1 + ς)+ [c 1 x(t -τ1 + ς) -c 2 x(t -τ2 + ς) + b] u(t).( 12
)
Observe that σ 0 is acting as a kind of predictor since it allows us to have u without delay in [START_REF] Kiong | Comments on "robust stabilization of uncertain input-delay systems by sliding mode control with delay compensation[END_REF]. Now, let us verify that the trajectories of (4) in closed loop with [START_REF] Gripenberg | Volterra Integral and Functional Equations[END_REF] reach and remain on the sliding surface σ = 0 in finite-time. To this end, we substitute [START_REF] Gripenberg | Volterra Integral and Functional Equations[END_REF] in [START_REF] Kiong | Comments on "robust stabilization of uncertain input-delay systems by sliding mode control with delay compensation[END_REF] to obtain the differential equation

σ(t) = -1 2 g 1 (t) sign(σ(t)) + g 2 (t) , (13)
where

g 1 (t) = c 1 x(t -τ1 + ς) -c 2 x(t -τ2 + ς) + b , g 2 (t) = g 1 (t)/2 + (a 1 + c 1 -a 2 )x(t)- (14) c 1 x(t -τ1 + ς) .
Before we proceed to prove the establishment of a sliding regime on σ = 0, we have to guarantee the existence of solutions of [START_REF] Polyakov | Minimization of disturbances effects in time delay predictor-based sliding mode control systems[END_REF].

Remark 1: Note that (13) can be seen as a nonautonomous differential equation with discontinuous right-hand side, therefore, we can use the definition of solutions given by Filippov in [8, p. 50]3 . But, g 1 and g 2 in (13) depend on x, and it is the solution of the functional differential equation ( 4) that in turn depends on σ through the input u. However, if we study recursively the system (4), [START_REF] Polyakov | Minimization of disturbances effects in time delay predictor-based sliding mode control systems[END_REF], on the intervals [nς, (n + 1)ς), n = 0, 1, 2..., we can see that the Filippov approach still works. Indeed, from the assumptions of the initial conditions for (4), u(t) = 0 for t ∈ [0, ς), hence, in such an interval the solutions of (4) do not depend on σ. Therefore, in the same interval, (13) can be seen as a simple differential equation with discontinuous right-hand side. Now, the solutions of (4) are not affected for the values of σ(t) for any t ∈ [ς, 2ς), thus, in such interval, ( 13) is a simple differential equation with discontinuous right-hand side, and so forth.

Consider the Lyapunov function candidate V (σ) = 1 2 σ 2 , whose derivative along ( 13) is given by

V = σ(g 2 (t) -1 2 g 1 (t) sign(σ)) , = -1 2 (g 1 (t) -2g 2 (t) sign(σ))|σ| .
Hence, V is a Lyapunov function for (13) if g 1 (t) -2g 2 (t) sign(σ) ≥ 0. Let us start with the case σ > 0. In this case we have g 1 (t)-2g 2 (t) = (a 2 -a 1 -c 1 )x(t)+c 1 x(t-τ 1 + ς). Therefore, since the solutions of (4) are guaranteed to be nonnegative, g 1 (t) -2g 2 (t) ≥ 0. For the case σ < 0 we have

g 1 (t) -2g 2 (t) = 2(b -(a 2 -a 1 -c 1 )x(t) -c 2 x(t -τ2 + ς)).
Note that since σ(0) < σ * then x(0) < σ * , and we know for this case that x(t) is bounded from above by x. This clearly implies that b -(a 2 -a 1 -c 1 )x(t) -c 2 x(t -τ2 + ς) ≥ 0.

Up to now, we have proven that σ = 0 is Lyapunovstable, however, to guarantee finite-time convergence of σ(t) to the origin, we have to verify that g 1 (t) -2g 2 (t) sign(σ) is bounded from below by a strictly positive constant. The condition σ > 0 implies that x 0 > σ * . If x(t) is increasing it is convenient for the analysis, however, the critic situation is when x(t) is decreasing and x(t) < x. Note that, in such a case, u = 0 necessarily. Now, suppose that for some t 1 we have x(t 1 ) = x * , then

σ(t 1 ) = -a 2 t t-τ2 x(s) ds + a 1 t t-τ1 x(s) ds+ c 1 t t-τ1+ς x(s) ds -a 2 (τ 2 -ς)+ a 1 (τ 1 -ς) + c 1 (τ 1 -ς) ,
which is clearly negative. Hence, we can guarantee that, for the case σ > 0, x(t) is bounded from below by x * , and therefore, g 1 (t) -2g 2 (t) ≥ (a 2 -a 1 )x * . Now, for practical purposes, let us define

S(t) = x(t) -a 2 t-ς t-τ2 x(s) ds+ a 1 t-ς t-τ1 x(s) ds + c 1 t-ς t-τ1
x(s) ds . [START_REF] Utkin | Sliding Modes in Control and Optimization[END_REF] Observe that the sliding variable σ can be rewritten as σ(t) = S(t) -σ * + R(t), where

R(t) = - t t-ς [(a 2 -a 1 -c 1 )x(s) + c 1 x(s -τ1 + ς)] ds+ t t-ς [c 1 x(s -τ1 + ς) -c 2 x(s -τ2 + ς) + b] u(s) ds .
Now, we want to prove that b -(a 2 -a 1 -c 1 )x(t)c 2 x(t -τ2 + ς) is strictly positive when σ < 0. In this case the critic situation is when x is monotonically increasing. This happens only if u = 1. Note that in such situation

σ(t) = S(t) -σ * + t t-ς [b -(a 2 -a 1 -c 1 )x(s)- c 2 x(s -τ2 + ς)] ds ,
where the term

t t-ς [b-(a 2 -a 1 -c 1 )x(s)-c 2 x(s-τ 2 +ς)] ds is strictly positive. Note also that S(t) -σ * ≥ x(t) -x * .
Hence, if for some t 1 we have that x(t 1 ) = x * then σ(t 1 ) ≥ 0. Thus, we can conclude that b-(a 2 -a 1 -c 1 )x(t)-c 2 x(t-τ2 + ς) is bounded from below by b -(a 2 + c 2 -a 1 -c 1 )x * when σ < 0. Therefore, we have proven that the sliding mode is established in finite-time.

B. Sliding dynamics

To obtain the dynamics on the sliding surface σ = 0, we use the Equivalent Control method [START_REF] Utkin | Sliding Modes in Control and Optimization[END_REF], see also [START_REF] Utkin | Sliding Mode Control in Electro-Mechanical Systems[END_REF], [START_REF] Fedorovich | Differential equations with discontinuous right-hand side[END_REF], [START_REF] Polyakov | Stability notions and lyapunov functions for sliding mode control systems[END_REF]. To compute the equivalent control, we make σ(t) = 0 and obtain that

[c 1 x(t -τ1 + ς) -c 2 x(t -τ2 + ς) + b] u(t) = -(a 1 + c 1 -a 2 )x(t) -c 1 x(t -τ1 + ς) .
By substituting this expression in the equation for σ(t) = 0 we obtain that the sliding dynamics is given by the integral equation

S(t) -σ * = 0 , ( 16
)
where S is given by [START_REF] Utkin | Sliding Modes in Control and Optimization[END_REF]. Hence, our objective is to prove that the solution x(t) of ( 16) converges exponentially to x * .

Here we are going to use the results provided in Appendix B. First, let us rewrite ( 16) in a more suitable way. Define the change of variable z(t) = x(t) -x * , thus, from the dynamics on the sliding surface, we obtain the following integral equation

z(t) -a 2 t-ς t-τ2 z(s) ds + a 1 t-ς t-τ1 z(s) ds+ c 1 t-ς t-τ1
z(s) ds = 0 .

Note that this equation can be rewritten as follows

z(t) + t t * k(t -s)z(s) ds = f (t) , t ≥ t * , (17)
where t * is the reaching time to the sliding surface (i.e. the minimum t such that σ(t) = 0), k is given by ( 8) replacing the parameter r by t -s, and

f (t) = - t * t * -τ2 k(t -s)φ(s) ds , φ(t) = z(t) , ∀t ≤ t * .
Observe that (17) is a Volterra integral equation of the second type and the kernel k of the integral is a convolution kernel. Now, we can state directly the following result.

Theorem 2: If k : R ≥t * × R ≥t * → R is a measurable kernel with ||k|| L p (R ≥t * ) < 1, then for any f ∈ L 1 (R ≥t * )
there exists a unique solution of (17) and it is such that z ∈ L 1 (R ≥t * ). Moreover, z(t) → 0 exponentially as t → ∞.

Proof: First we claim that f is in L 1 (R ≥t * ), see the verification in Appendix C. According to Lemma 5, the assumptions in Theorem 1 guarantee that ||k|| L p (R ≥t * ) < 1. Thus, we can use Lemma 4 to guarantee existence and uniqueness of solutions of (17). Now, Lemmas 6 and 7 guarantee the exponential stability of z = 0.

Since z = 0 is exponentially stable, x(t) → x * exponentially on the sliding surface.

VI. ROBUSTNESS

In this section we analyse the robustness of the closed loop of (4) with [START_REF] Gripenberg | Volterra Integral and Functional Equations[END_REF], [START_REF] Györi | Oscillation Theory of Delay Differential Equations With Applications[END_REF]. For this, consider the system

ẋ(t) = a 1 x(t -τ 1 ) -a 2 x(t -τ 2 ) + [c 1 x(t -τ1 )- c 2 x(t -τ2 ) + b] u(t -ς) + δ(t) , (18)
where δ : R → R is an external disturbance. We assume that δ L ∞ (R ≥0 ) = ∆ for some finite ∆ ∈ R ≥0 . Considering (18), the time derivative of the sliding variable σ is

σ(t) = -1 2 g 1 (t) sign(σ(t)) + g 2 (t) + δ(t) , (19)
where g 1 and g 2 are given by [START_REF] Polyakov | Stability notions and lyapunov functions for sliding mode control systems[END_REF]. Consider V (σ) = 1 2 σ 2 as a Lyapunov function candidate for (19). The derivative of V along ( 13) is given by

V ≤ -1 2 [g 1 (t) -2g 2 (t) sign(σ) -|δ(t)|] |σ| .
In Section V we proved that there exists a strictly positive such that g 1 (t) -2g 2 (t) sign(σ) ≥ for all t along the reaching phase, thus if ∆ < , then V ≤ -1 2 [ -∆|] |σ| and the sliding regime is established in finite-time. Nevertheless, since the sliding variable contains delayed terms of the control, the establishment of the sliding mode does not guarantee the complete disturbance rejection, see e.g. [START_REF] Kiong | Comments on "robust stabilization of uncertain input-delay systems by sliding mode control with delay compensation[END_REF], [START_REF] Polyakov | Minimization of disturbances effects in time delay predictor-based sliding mode control systems[END_REF]. Thus, let us analyse the behaviour of the sliding motion in the presence of the disturbance δ. By using again the equivalent control method (by taking into account the disturbance) we obtain the sliding dynamics S(t) -σ * -δ(t) = 0. If we use again the change of variable z(t) = x(t) -x * , then the sliding dynamics can be rewritten as

z(t) + t t * k(t -s)z(s) ds = f (t) + δ(t) , t ≥ t * , (20)
or equivalently z(t) + (k z)(t) = f (t) + δ(t). We have proven that the solution of (20) is given by

z(t) = [f + δ](t) -(r [f + δ])(t) ,
where r is a convolution operator of type L 1 (R ≥t * ), see Theorem 2 and Appendix B. Now, since f is bounded (see

Appendix C) and δ ∈ L ∞ (R ≥0 ) then [f + δ] ∈ L ∞ (R ≥t * ).
Hence, according to Lemma 8, we can ensure that z ∈ L ∞ (R ≥0 ).

VII. NUMERICAL EXAMPLE

Consider (4) with the parameters a 1 = 0.2, a 2 = 1, c 1 = 0.1, c 2 = 0.4, b = 1, τ 1 = 0.05, τ 2 = 0.11, τ1 = 0.07, τ2 = 0.09. The values of these parameters were chosen in the same order as those obtained in [START_REF] Feingesicht | SISO model-based control of separated flows: Sliding mode and optimal control approaches[END_REF]. Of course, they satisfy all the conditions of Theorem 1. The simulations were made with Matlab by using an Explicit Euler integration method with a step of 1ms. In Fig. 1 we can observe the system's state for a simulation with initial condition x 0 = 0 in the nominal case. Fig. 2 shows the control signal. In Fig. 3 we can see a simulation considering a disturbance δ(t) = sin(10t)/10. Note in Fig. 4 that, for this example, the amplitude in steady state is less than the amplitude of the disturbance.

VIII. CONCLUSIONS

We proposed a Sliding Mode Controller for a class of scalar bilinear systems with delays. We have shown that the combination of Lyapunov function and Volterra operator theory provides a very useful tool to study the stability and robustness properties of the proposed control scheme. Naturally, a future direction in this research is to try to extend the control scheme to higher order systems.

APPENDIX

A. Solutions of delayed differential equations

The theory recalled in this section was taken from [START_REF] Agarwal | Nonoscillation Theory of Functional Differential Equations with Applications[END_REF], see also [START_REF] Györi | Oscillation Theory of Delay Differential Equations With Applications[END_REF] and [START_REF] Fridman | Introduction to Time-Delay Systems[END_REF]. Consider the system

ẋ = N i=1 a i (t)x(t -τ i ) , (21)
where each τ i ∈ R ≥0 , and each a i is a Lebesgue-measurable locally essentially bounded function. Definition 1: The function X(t, s) that satisfies, for each s ≥ 0, the problem

ẋ = N i=1 a i (t)x(t -τ i ) ,
x(t) = 0 for t < s, x(s) = 1, is called the fundamental function of (21).

It is assumed that X(t, s) = 0 when 0 ≤ t < s. Now consider the system

ẋ = N i=1 a i (t)x(t -τ i ) + f (t) , (22)
with initial conditions x(t) = 0 for all t < 0 and x(0) = x 0 for some x 0 ∈ R. Lemma 3: Assume that a i , τ i are as above and f is a Lebesgue-measurable locally essentially bounded function, then there exists a unique solution of ( 22) and it can be written as

x(t) = X(t, 0)x 0 + t 0 X(t, s)f (s) ds .

B. Volterra equations

Most of the results recalled in this section was taken from [START_REF] Gripenberg | Volterra Integral and Functional Equations[END_REF]. For z : R → R consider the integral equation

z(t) + t t * k(t, s)z(s) ds = f (t) , t ≥ t * . ( 23
)
Define the map t → t t * k(t, s)z(s) ds as k z. Hence, we rewrite (23) as

z(t) + (k z)(t) = f (t) . (24)
A function r is called a resolvent of (24

) if z(t) = f (t) - (r f )(t). We say that the kernel k : J × J → R is of type L p on the interval J if k L p (J) < ∞, where k L p (J) = sup g1 L p (J) ≤1 g2 L p (J) ≤1 J J |g 1 (t)k(t, s)g 2 (s)| p ds dt.
The question about the existence and uniqueness of solutions of (24) is answered by the following lemma.

Lemma 4 ([10], Theorem 9-3.6): If k is a kernel of type L P on J that has a resolvent r of type L P on J, and if f ∈ L P (J), then (24) has a unique solution z ∈ L P (J), and such solution is given by z

(t) = f (t) -(r f )(t).
Now, we have two problems, verify if k is a kernel of type L P and if it has a resolvent r of type L P . For the particular case of p = 1 we have the following lemma.

Lemma 5 ([10], Proposition 9-2.7): Let k : J × J → R be a measurable kernel. k is of type L 1 on J if and only if N (k) = ess sup s∈J J |k(t, s)| dt < ∞. Moreover N (k) = k L p (J) . And finally. Lemma 6 ([10], Corollary 9-3.10): If k is a kernel of type L P on J and k L p (J) < 1, then k has a resolvent r of type L P on J. Now we can guarantee some asymptotic behaviour of z(t) according to the asymptotic behaviour of f (t) for a Volterra kernel k. Nonetheless, if such a kernel is of convolution kind, i.e. k(t, s) = k(t -s), we can say something else. For the following lemma let us denote the Laplace transform of k(t) as K(z), z ∈ C.

Lemma 7 ([10], Theorem 2-4.1): Let k be a Volterra kernel of convolution kind and L 1 type on R ≥0 . Then the resolvent r is of type L 1 on R ≥0 if and only if det(I + K(z)) = 0 for all z ∈ C such that Re{z} ≥ 0.

To finalise this section we recall the following lemma that is useful for the robustness analysis. Here we verify that f is in L 1 . First note that the integral in f restricts to t * -τ 2 ≤ s ≤ t * , therefore, the argument of k(t -s) is restricted to t -t * ≤ t -s ≤ t -t * + τ 2 . Recall that k(t-s) is different from zero only in the interval [min(ς, τ 1 ), τ 2 ]. Hence, under the integral in f , k(t -s) can be different from zero only for t * + min(ς, τ 1 ) -τ 2 ≤ t ≤ t * + τ 2 . Thus

||f (t)|| L 1 (R ≥0 ) = ∞ 0 |f (t)| dt = t * +τ2
t * +min(ς,τ1)-τ2 |f (t)| dt . Now, since φ(t) = x(t) -x * and x(t) was guaranteed to be bounded then there exists a finite φ * ∈ R ≥0 such that |φ(t)| ≤ φ * for all t ∈ [t * -τ 2 , t * ]. Note that also k is bounded by some finite k * , thus

f (t) L 1 (R ≥0 ) ≤ t * +τ2 t * +min(ς,τ1)-τ2 t * t * -τ2
k * φ * ds dt , therefore f (t) L 1 (R ≥0 ) ≤ k * φ * τ 2 (2τ 2 -min(ς, τ 1 )) .

Fig. 1 .Fig. 2 .

 12 Fig. 1. State of the system in the nominal case.

Fig. 3 .Fig. 4 .

 34 Fig. 3. State of the system in presence of disturbance.

Lemma 8 (

 8 [START_REF] Gripenberg | Volterra Integral and Functional Equations[END_REF], Theorem 2-2.2): Let r be a convolution Volterra kernel of type L 1 (R ≥0 ), and let b ∈ L P (R ≥0 ) for some p ∈ [1, ∞]. Then r b ∈ L P (R ≥0 ), andr b L P (R ≥0 ) ≤ r L 1 (R ≥0 ) b L p (R ≥0 ) . C. Function f is in L 1

A video with some experiments, reported in[START_REF] Feingesicht | SISO model-based control of separated flows: Sliding mode and optimal control approaches[END_REF], can be seen at https://www.youtube.com/watch?v=b5NnAV2qeno.

For the definition of a nonoscillatory solution see e.g.[START_REF] Györi | Oscillation Theory of Delay Differential Equations With Applications[END_REF],[START_REF] Agarwal | Nonoscillation Theory of Functional Differential Equations with Applications[END_REF].

For the particular case of (13), the three methods given in[8, p. 50-56] to construct the differential inclusion coincide, see also[START_REF] Polyakov | Stability notions and lyapunov functions for sliding mode control systems[END_REF].