Thomas Radulesco
email: thomas.radulesco@ap-hm.fr

Julien Mancini

Martin Penicaud

Jean-Jacques Grob

Marie-Aleth Richard

Patrick Dessi

Nausicaa Malissen

Justin Michel

Cross-cultural adaptation into French and validation of the SCAR-Q questionnaire

INTRODUCTION

Scars are natural consequences of cicatrization following surgery, trauma or burn injury. It has been shown that poor aspect scarring can result in serious mental disorders or poor self-esteem [START_REF] Finlay | Validation of Sickness Impact Profile and Psoriasis Disability Index in Psoriasis[END_REF][START_REF] Zhang | The Correlation Between Quality of Life and Acceptability of Disability in Patients With Facial Burn Scars[END_REF].

Patient-reported outcome and experience measures (PROMs and PREMs) are now part of the quality assessment of the overall care of patients [START_REF] Baumhauer | Patient-Reported Outcomes -Are They Living Up to Their Potential[END_REF]. Klassen et al. recently developed a specific PRO instrument regarding scar evaluation [START_REF] Klassen | Development of a New Patientreported Outcome Instrument to Evaluate Treatments for Scars: The SCAR-Q[END_REF]. The SCAR-Q self-questionnaire is composed of three scales evaluating scar appearance (12 items), symptoms (12 items) and psychological impact (5 items). The SCAR-Q was designed to evaluate all scar types in children and adults. It was considered useful in research where appearance is an important outcome. Indeed when treatments aim to specifically improve the appearance of scars, asking patients what they think about how their scar looks seems a fundamental and practical measure [START_REF] Klassen | Development of a New Patientreported Outcome Instrument to Evaluate Treatments for Scars: The SCAR-Q[END_REF]. After rigorous development, a study among 731 patients validated the scale [START_REF] Ziolkowski | Psychometric Findings for the SCAR-Q Patient-Reported Outcome Measure Based on 731 Children and Adults with Surgical, Traumatic, and Burn Scars from Four Countries[END_REF]. Like most questionnaires designed to evaluate scarring from the point of view of patients, SCAR-Q was developed in English language [START_REF] Mundy | Patient-Reported Outcome Instruments for Surgical and Traumatic Scars: A Systematic Review of their Development, Content, and Psychometric Validation[END_REF][START_REF] Vercelli | How to assess postsurgical scars: A review of outcome measures[END_REF]. Unfortunately, such a questionnaire cannot be used in patients with another mother language. To our knowledge, no questionnaire regarding scarring evaluation by patients is available in French.

The objective of our study was to translate the SCAR-Q instrument into French [START_REF] Guillemin | Cross-cultural adaptation of healthrelated quality of life measures: literature review and proposed guidelines[END_REF][START_REF] Beaton | Guidelines for the process of cross-cultural adaptation of self-report measures[END_REF][START_REF] Chakka | Cross-cultural adaptations of health-related quality of life measures[END_REF][START_REF] Vasquez | Construct validity and responsiveness of the Colombian version of Skindex-29[END_REF] using International guidelines issued by ISPOR (International Society for Pharmacoeconomics and Outcomes Research) and by WHO (World Health Organization) [START_REF] Wild | Principles of Good Practice for the Translation and Cultural Adaptation Process for Patient-Reported Outcomes (PRO) Measures: report of the ISPOR Task Force for Translation and Cultural Adaptation[END_REF][START_REF]WHO | Process of translation and adaptation of instruments[END_REF], then to test the reliability and validity of the translated version.

MATERIALS AND METHODS

Ethical considerations

We obtained permission to use the SCAR-Q self-questionnaire from the original development team. All participating patients gave their written consent before joining this study, which was carried out in accordance with the Declaration of Helsinki (1983). We obtained an Ethical Committee Authorization (Authorization N° 2020-67, Assistance Publique des Hôpitaux de Marseille, Marseille, France).

Translation

ISPOR and WHO recommendations were used to carry out the translation process [START_REF] Wild | Principles of Good Practice for the Translation and Cultural Adaptation Process for Patient-Reported Outcomes (PRO) Measures: report of the ISPOR Task Force for Translation and Cultural Adaptation[END_REF][START_REF]WHO | Process of translation and adaptation of instruments[END_REF].

The translation process comprised 4 steps: 1/ forward translation 2/ back translation 3/ backtranslation review, and 4/ patient interviews. The translation process required 3 individuals who were fluent in both English and French. Two individuals whose mother tongue was French served as forward translators (English to French). One was a surgeon and the other one professional translator specialized in medical translations. Once the two forward translations were completed, cognitive debriefing was conducted to establish consensus to merge the 2 translations and produce a single version. The last professional translator, whose mother tongue was English and was fluent in French served as back translator. The back translator did not see or review the original English version of the SCAR-Q. Once the back translation was complete, the questionnaire was returned to the original development team and analyzed by an expert review panel including various 4 medical specialists and 2 paramedics. The development team provided feedback and instructions prior to conducting the cognitive debriefing interviews. The cognitive debriefing interviews engage ten patients in the target patient population to determine the quality of the translation. The final version was not aimed to provide a literal translation, but rather a conceptually equivalent translated version worded in language patients can understand easily. As suggested by the developer of the scale SCAR-Q scale scores were transformed into 0 (worst) to 100 (best) based on logits from Rasch measurement theory analysis [START_REF] Ziolkowski | Psychometric Findings for the SCAR-Q Patient-Reported Outcome Measure Based on 731 Children and Adults with Surgical, Traumatic, and Burn Scars from Four Countries[END_REF].

Population

Inclusion criteria were: French nationality and French as their first language; age ≥18 years; having a visible scar, whatever the location, size (centimeters), date or etiology. We did our best to cover the variability of location, size of scars and the demographic variability. Exclusion criteria were: illiterate patients or those unable to understand and respond to the survey, and persons with cognitive limitation.

The patients were recruited for the consultation of dermatology or surgery. The reason for consultation was not always the scar, but the presence of a visible scar made it possible to include the patient if he agreed and if the inclusion and exclusion criteria were met. The statistical power was calculated. Computing power and minimum sample size for RMSEA: with 200 patients and a RMSEA=0.065 we had a 95.7% power (with an alpha risk=5%) to show that this RMSEA was lower than 0.08. The first 200 patients that fully answered the questionnaire were included in the study. The majority of the patients approached to complete the survey participated; however, we do not have demographic information on persons with declined to participate.

Statistical validation

We used questionnaires completed by patients and stored in a secure database. All analyses were conducted using IBM SPSS Statistics 20.0 (IBM Inc., New York, USA) and the lavaan package for R.3.6.3 (R Foundation for Statistical Computing, Vienna, Austria). All tests were twosided and p-values <0.05 were considered significant.

Continuous and categorical variables were described using, respectively, means (± standard deviation) and counts (percentages). At scale level, floor and ceiling effects were considered to be present if more than 15% of respondents achieved, respectively, the lowest or the highest possible score, [START_REF] Terwee | Quality criteria were proposed for measurement properties of health status questionnaires[END_REF]. At item level, these effects were considered to be present if more than 95% of respondents answered in the lowest or highest response category [START_REF] Vet Hcw De | Measurement in Medicine: A Practical Guide[END_REF]. Items were considered redundant if the polychoric inter-item correlation was > 0.7 and irrelevant if < 0.2 [START_REF] Vet Hcw De | Measurement in Medicine: A Practical Guide[END_REF]. Reliability was assessed using ordinal Cronbach's alpha (α) [START_REF] Gadermann | Estimating ordinal reliability for Likert-type and ordinal item response data: A conceptual, empirical, and practical guide[END_REF] and considered satisfactory if ≥ 0.7 [START_REF] Bland | Cronbach's alpha[END_REF]. Structural validity was tested using confirmatory factor analysis (CFA) with the robust weighted least squares (WLSMV) estimator and Delta parameterization, based on the original 3-factor structure of the initial version. Model fit of a correlated 3-factor structure was examined using the root mean square error of approximation (RMSEA, good fit if <0.06, poor fit if ≥0.10, acceptable elsewhere), the comparative fit index (CFI) and the Tucker-Lewis index (CFI and TLI, good fit if >0.95, poor fit if <0.90, acceptable elsewhere) [START_REF]Cutoff criteria for fit indexes in covariance structure analysis: Conventional criteria versus new alternatives: Structural Equation Modeling[END_REF].

The Spearman rank coefficient (ρ) was used to assess correlations between subscales. Three hypotheses were identified based on previous findings to determine known group validity: higher SCAR-Q scores were excepted when the scars were bigger, more recent, and localized on the face [START_REF] Ziolkowski | Psychometric Findings for the SCAR-Q Patient-Reported Outcome Measure Based on 731 Children and Adults with Surgical, Traumatic, and Burn Scars from Four Countries[END_REF]. Univariate analyses were performed using ANOVAs.

To test the repeatability of this new scale, test-retest reliability was assessed in 10 patients (given the ICC estimated (≥0.94) we had an 87% power to detect an ICC higher than a lowest acceptable ICC of 0.6 with a two-sided test). We considered that no clinical changes would occur in one month given the long-time scars of the patients included (75% of scar older than 12 months, scars were more than 13 years old on average). Patients answered the questionnaire again under the same conditions 1 month later. Consistency between responses was evaluated using intra-class correlation coefficient. The closer the coefficient to 1, the higher the repeatability. This test was performed for each subscale.

Before initiating the study, all researchers were trained in patient interviewing. As all questionnaires were completed, there were no concerns regarding missing data.

RESULTS

Translation

Some differences were found between the two forward translations, regarding both instructions and items. For example, "How does your scar look?" was translated as "A quoi ressemble votre cicatrice?"by one translator (literally "what does your scar look like ?") and as "Quel est l'aspect de votre cicatrice?" (literally "what is the aspect of your scar?" by the other.

A "reconciled" version containing both instructions and items was submitted to the back translator. No major comments were made by the development team concerning the backtranslation. Patient interviews included ten patients (mean age = 37 years old, range 25-62) with different types of scars (burns, surgical or traumatic scars). This step entailed minor modifications of item wording. For example, for item n°6 of the SCAR Appearance scale the word "irrégulière" was added to the scar description.

Statistical validation

Two hundred patients completed the questionnaire. Population characteristics are reported in Table 1. We included 109 females (54.5%) and 91 males (46.5%). Mean age (SD) was 53.3 (18) years-old (min=18; max=84). Mean scar size was 6.8 (6.3) cm (min=1; max=35) and mean scar date was 156.7 (169) months (min=1; max=651).

Regarding item description, no floor or ceiling effect was found (max=85%) for all items.

Regarding scale description, we found a ceiling effect for all 3 subscales: in 34.0%, 33.0%, and 56.0% of patients for the appearance, symptoms, and psychological impact subscales respectively (Figure 1). Forty-one patients (20.5%) were not impaired at all in the three dimensions. Internal consistency was considered satisfactory with a Cronbach's alpha > 0.7 for all subscales (0.97, 0.90 and 0.97, respectively, for SCAR-Q Appearance, Symptom and Psychological impact scales). Several items from the Appearance and Psychosocial impact scales showed redundancy, with many inter-item correlations above 0.7. The CFA of the original structure displayed reasonable fit, with RMSEA=0.065 (90% Confidence Interval: 0.057-0.072), CFI=0.974, and TLI=0.972. Subscales were positively correlated but not strongly (0.45<rho<0.65, p<0.001).

Repeatability was tested on a randomized subgroup of 10 patients who did not differ from the other patients regarding their main characteristics and initial SCAR-Q scores. Intra-class correlation coefficients were 0.95 (95% Confidence Interval: 0.83-0.99), 0.94 (0.59-0.99), and 0.99 (0.94-1.00), for Appearance, Symptom and Psychological impact scales respectively.

DISCUSSION

Modern practitioners are committed to the diagnosis and treatment of patients and to addressing the patient's objectives and assessing patient satisfaction, as part of a comprehensive approach to care [START_REF] Tredget | Scar Management Following Burn Injury[END_REF]. Scar is a part of surgical outcome. Efforts to assess the patient's perspective will result in better therapeutic interventions to improve scar quality and acceptance [START_REF] Kim | Scar outcomes in dermatological surgery[END_REF]. For that purpose, an instrument like SCARQ must be available in other languages than English, for a much broader use. French alone is estimated to be spoken daily in Metropolitan France by 76 million native speakers and by 235 million fluent speakers worldwide.

Our translation process has already been used in other studies [START_REF] Poulsen | Danish translation and linguistic validation of the BODY-Q: a description of the process[END_REF][START_REF] Radulesco | French validation of the FACE-Q Rhinoplasty module[END_REF]. It ensures a translation of the idea or concept, more meaningful and accurate than a mere literal translation [START_REF] Lacasse | Health-related quality of life measurement: a readers' guide[END_REF]. ISPOR and WHO recommendations were chosen for the translation process, in order to obtain culturally-adapted French version of the SCAR-Q questionnaire. Combining forth and back translation methods, we reduced potential biases in the process. Two-way translation, combined with a panel expert review and patient interviews, guarantees a culturally and socially adapted version. No translation can perfectly match the original document because of conceptual differences due to diverging languages histories. However, backtranslation into the original language ensures a conceptually valid translation [START_REF] Lacasse | Health-related quality of life measurement: a readers' guide[END_REF]. By incorporating different medical specialties and paramedics in the panel of experts, we tried to make an easily understandable questionnaire. Actual patients were integrated into this process, to check whether the proposed version was also suitable for non-medical or non-paramedical users. To cover the variability of situations, we included patients with different types (keloid, contracture, hypertrophic) and locations (face and neck, chest, upper and lower members) of scars in order to be as representative as possible of target patients.

Cronbach's alphas of the French version of SCAR-Q were high (0.97, 0.90 and 0.97 for SCAR-Q Appearance, Symptoms, and Psychological scales, respectively) thus demonstrating a high level of internal reliability. This is consistent Cronbach's alpha values of the original scale [START_REF] Klassen | Development of a New Patientreported Outcome Instrument to Evaluate Treatments for Scars: The SCAR-Q[END_REF], ie 0.96, 0.91, and 0.95, respectively. The ceiling effect found on each subscale was probably related to the fact that scars were not the main reason for consultation in our cohort, explaining the high number of high scores (Figure 1). Moreover, the majority of patients had scars for more than 1 year, thus probably increasing patients' acceptance and increasing the scores. This must be confirmed by multiple patients' evaluation at different time. As expected, subscales were positively correlated, but not too strongly (0.45<rho<0.65) which supports the idea that they measure distinct dimensions. The repeatability analysis attests to the efficacy of the questionnaire although generated by translation into French.

No translation process is perfect and conceptual differences may remain. However, the combination of two methodologies aimed to minimize bias and statistical validation provided acceptable results. Another study limit is that we do not have demographic information on persons who declined to participate. Generalizability of study finding may be limited due to the sample characteristics. The sample included a high percentage of persons who had their scars for surgery or trauma. Our sample, smaller than in Ziolkowski et al. study [START_REF] Ziolkowski | Psychometric Findings for the SCAR-Q Patient-Reported Outcome Measure Based on 731 Children and Adults with Surgical, Traumatic, and Burn Scars from Four Countries[END_REF], resulted in high ceiling effects across the 3 scales. Furthermore, our sample only included 2 patients with burn scars and our results may not be generalizable to patients with burns. While these finding demonstrate the initial validity of the French translation of the SCAR-Q, future studies are required to examine the reliability and validity in other samples, in particular with among persons with more recent and severe scars.

In further studies, the French SCAR-Q could also be used simultaneously with English version in a study including both English and French-speaking patients. This would allow examination of measurement equivalence.

CONCLUSION

The SCAR-Q questionnaire is a reference in the field of patient-reported outcomes regarding scar evaluation. The 4-step translation-backtranslation process made it possible to obtain a high-quality French version in line with the original document. This translated version is now usable in France.

Figure 1 .

 1 Figure 1. Bar graph showing scores of patients for all subscales. A ceiling effect was observed in

Table Test

 Test

		Upper	15	7.9%	0	0.0%
		extremity
		Breast	5	2.6%	1	10.0%
	Scar etiology Burn	2	-retest participant 1.1% 0	0.0%	0.72
		Surgical	No (n=190) 153 80.5%	Yes (n=10) 9	90.0%
		Traumatic	Mean 35	SD 18.4%	Mean 1	SD 10.0%	P-value
	Appearance Scale (0-100)	73.6	23.1	80.0	23.8	0.54
	Symptom Scale (0-100)	81.3	16.4	81.8	13.5	0.95
	Psychosocial Impact (0-	85.3	21.6	86.3	20.1
	100)						0.90
			N	%	N	%
	Sex	Female	104	54.7%	5	50.0%	0.99
		Male	86	45.3%	5	50.0%
	Age (y)	<=30	29	15.3%	1	10.0%	0.11
		31-60	92	48.4%	2	20.0%
		>60	69	36.3%	7	70.0%
	Scar size (cm) <=2	31	16.3%	1	10.0%	0.60
		3-10	130	68.4%	6	60.0%
		>10	29	15.3%	3	30.0%
	Scar date	<=3	25	13.2%	2	20.0%
	(months)						0.48
		4-12	23	12.1%	0	0.0%
		>12	142	74.7%	8	80.0%
	Scar location Abdomen	50	26.3%	3	30.0%	0.33
		Neck	16	8.4%	0	0.0%
		Back	7	3.7%	1	10.0%
		Face	70	36.8%	5	50.0%
		Lower	27	14.2%	0	0.0%
		extremity

Table 1 .

 1 Sample characteristics.

	Tables
	Characteristics	N (total = 200)	(%)
	Sex
	Female	109	54.5
	Male	91	46.6
	Age (years)	Min 18; max = 84; mean = 53.3 (18)
	30	30	15
	31-60	94	47
	> 60	76	38
	Scar size (cm)	Min = 1; Max = 35, mean = 6.8 (6.3)
	2 cm	32	16
	3-10 cm	136	68
	> 10 cm	32	16
	Scar date (months)	Min = 1; max = 651, mean 156.7 (169)
	3 months	27	13.5
	4 months -1 year	23	11.5
	> 1 year	150	75
	Scar etiology
	Surgical	162	81
	Traumatic	36	18
	Burn	2	1
	Scar location
	Face	75	37.5
	Abdomen	53	26.5
	Neck	16	8
	Back	8	4
	Lower extremity	27	13.5
	Upper extremity	15	7.5
	Breast	6	3

Table 1 .

 1 Sample characteristics.

Acknowledgments: none Declarations Funding (information that explains whether and by whom the research was supported) : none

Availability of data and material (data transparency): All data available

Code availability (software application or custom code) : Not applicable


Yacine Chitour
email: yacine.chitour@l2s.centralesupelec.fr

Ihab Haidar
email: ihab.haidar@ensea.fr

Paolo Mason
email: paolo.mason@l2s.centralesupelec.fr

Mario Sigalotti
email: mario.sigalotti@inria.fr

Upper and lower bounds for the maximal Lyapunov exponent of singularly perturbed linear switching systems

Keywords: Switching systems, Singular perturbation, Exponential stability, Maximal Lyapunov exponent, Differential inclusions

In this paper we consider the problem of determining the stability properties, and in particular assessing the exponential stability, of a singularly perturbed linear switching system. One of the challenges of this problem arises from the intricate interplay between the small parameter of singular perturbation and the rate of switching, as both tend to zero. Our approach consists in characterizing suitable auxiliary linear systems that provide lower and upper bounds for the asymptotics of the maximal Lyapunov exponent of the linear switching system as the parameter of the singular perturbation tends to zero.

Introduction

We consider in this paper a two-time-scales linear switching system, that is, a linear switching system for which some variables evolve on a much faster rate than the others. This class of systems appears in several industrial and engineering applications (see, e.g., [START_REF] Kimball | Singular perturbation theory for DC-DC converters and application to PFC converters[END_REF][START_REF] Malloci | Switched system modeling and robust steering control of the tail end phase in a hot strip mill[END_REF][START_REF] Wang | Extended dissipative control for singularly perturbed pdt switched systems and its application[END_REF]) where simplified models can be formulated by neglecting the effects of fast variables on the overall system. From control point of view, this allows to design a controller based on a reduced order model. However, the design based on a simplified model may not guarantee the stability of the overall system. To avoid this problem, a well-established framework developed in the mathematical and control community is that of singular perturbations [START_REF] Kokotović | Singular perturbation methods in control[END_REF]. The singular perturbation theory allows the separation between slow and fast variables where different controllers for different time-scale variables can be designed in order to lead the overall system to its desired performance.

In mathematical terms, we study the behavior of ẋ(t) = A(t)x(t) + B(t)y(t), ε ẏ(t) = C(t)x(t) + D(t)y(t),

where ε denotes a small positive parameter and A, B, C, D are matrix-valued signals undergoing arbitrary switching within a prescribed bounded range. We deal, therefore, with a 1-parameter family of linear switching systems Σ : ε → Σ ε . One of the main issues for such families of systems consists in understanding the time-asymptotic behavior of Σ ε as t → +∞ in the regime where ε is small. For instance, by saying that Σ is exponentially stable we refer to the fact that for every ε > 0 sufficiently small, the corresponding linear switching system Σ ε is exponentially stable. More refined notions, accounting for the uniform exponential behavior with respect to ε, are proposed in the paper (cf. Definition 4). Few stability criteria for singularly perturbed switching systems in the regime ε ∼ 0 have been obtained in the literature: among them, let us mention [START_REF] Malloci | Stabilization of continuous-time singularly perturbed switched systems[END_REF], where conditions are obtained based on the existence of a common quadratic Lyapunov function, [START_REF] Hachemi | Stability analysis of singularly perturbed switched linear systems[END_REF] characterizing the stability in dimension two based on the corresponding criteria in the nonsingularly-perturbed case [START_REF] Balde | A note on stability conditions for planar switched systems[END_REF][START_REF] Boscain | Stability of planar switched systems: the linear single input case[END_REF], and [START_REF] Alwan | Exponential stability of singularly perturbed switched systems with time delay[END_REF], where stability for time-delay singularly perturbed switching systems is based on dwell-time criteria.

A major mathematical difficulty relies on the fact that we are interested in characterizing a doubly asymptotic regime, where the order in which the limits are taken is crucial. Indeed, a limit as ε → 0 of the considered dynamics is well known to be given, through the Tikhonov decomposition, by ẋ(t) = A(t) -B(t)D(t) -1 C(t) x(t),

with y(t) = -D(t) -1 C(t)x(t). Heuristically, such a decomposition is obtained by saying that y tends instantaneously to -D(t) -1 C(t)x(t) (the equilibrium of the equation for y when x ≡ x(t)). However, as it has already been observed in the literature, the switching system (1) may be exponentially stable even when, for every ε > 0, Σ ε is unstable [START_REF] Malloci | Stabilization of continuous-time singularly perturbed switched systems[END_REF]. Notice that, in order to justify the Tikhonov decomposition, we are assuming here that each D(t) is a Hurwitz matrix. Actually, a standing assumption in this work is that the fast dynamics are exponentially stable, i.e., the trajectories of

ẏ(t) = D(t)y(t) (2)
converge to the origin with a uniform exponential rate. It should be noticed that this condition is necessary for the exponential stability of Σ ε in the regime ε ∼ 0, since if (2) admits trajectories diverging exponentially, then there is no hope for Σ ε to be exponentially stable for ε small (a precise mathematical statement of this fact is proved in Proposition 6). In the limit situation where (2) is stable but not exponentially stable, we can still have exponential stability of Σ ε for ε ∼ 0 [7, Section V.A], but this is a rather degenerate situation that we do not consider here. The goal of this paper is not only to give necessary or sufficient conditions ensuring that Σ has a certain time-asymptotic behavior in the regime ε ∼ 0, but also to prescribe upper and lower bounds on the limit as ε → 0 of the maximal Lyapunov exponent of Σ ε . We recall that the maximal Lyapunov exponent of a linear switching system is the largest asymptotic exponential rate as the time goes to +∞ among all trajectories of the system. Necessary or sufficient conditions for stability then follow as particular cases: indeed, a positive lower bound ensures that, for every ε small enough, system Σ ε is unstable, while a negative upper bound guarantees that Σ ε is exponentially stable for all ε in a right-neighborhood of zero.

The bounds on the limit as ε → 0 of the maximal Lyapunov exponent of Σ ε are obtained by identifying suitable auxiliary switching systems for the variable x (with a single time scale) that are either sub-or superapproximations of the asymptotic dynamics of Σ ε as ε → 0, in the following sense. A sub-approximation Σ of Σ is a system such that each of its trajectories can be approximated arbitrarily well, as ε → 0, by the x-component of a trajectory of Σ ε . Conversely, a super-approximation Σ of Σ is a system such that the x-component of each trajectory of Σ ε can be approximated arbitrarily well, as ε → 0, by a trajectory of Σ.

The first and rather natural choice for Σ is system (1) since it corresponds to the singular perturbation approach, see for instance [START_REF] Petar | Applications of singular perturbation techniques to control problems[END_REF]. The heuristic explanation is that, if switching occurs "slowly" with respect to the time-scale 1/ε, then the variable y(t) converges fast enough to -D(t) -1 C(t)x(t), and the transient phase is too short to affect the dynamics of x(t). As for a choice of super-approximation Σ, the idea is to consider the switching parameter as evolving on a time scale possibly much faster than 1/ε. Hence the dynamics of y can be seen, on a short time interval, as a switching system with affine vector fields y → C(t)x + D(t)y, where x is fixed. This kind of switching systems have been studied, for instance, in [START_REF] Della Rossa | Stability of switched affine systems: Arbitrary and dwell-time switching[END_REF][START_REF] Nilsson | Invariant sets of defocused switched systems[END_REF]. The exponential convergence of the solutions of (2) implies that the trajectories of such an affine switching system converge exponentially towards a compact set K(x) [START_REF] Della Rossa | Stability of switched affine systems: Arbitrary and dwell-time switching[END_REF]. A natural choice for Σ is then the differential inclusion

ẋ(t) ∈ A(t)x(t) + B(t)K(x(t)).
We show that, indeed, this system allows to identify an upper bound for the limit of the maximal Lyapunov exponent of Σ ε as ε → 0.

As mentioned in the previous paragraph, the two choices of Σ and Σ just described correspond to signals switching at a rate much slower or much faster than 1/ε. We also consider a third regime, namely, the situation in which the switching occurs at rate exactly 1/ε. This corresponds to considering a signal t → (A(t), B(t), C(t), D(t)) =: σ(t) defined on an interval [0, T ] and all its reparameterizations σ ε : t → σ(t/ε) defined on [0, εT ]. The flow of Σ ε corresponding to σ ε , evaluated at time εT , has an effect of order O(ε) on the coordinate x (since the velocities are of order 1 and the length of the time interval is of order ε) and of order O(1) on the coordinate y. Suitable computations show that the evolution of x can actually be described as

x → x + εT Λ(T, σ)x + O(ε 2 ),
where the term Λ(T, σ) does not depend on the initial condition of the variable y. We then propose another choice of sub-approximation of Σ, denoted Σ, which coincides with the switching system having as possible modes all the matrices of the type Λ(T, σ). We provide an explicit expression for such matrices and we prove that, indeed, each trajectory of Σ can be approximated arbitrarily well, as ε → 0, by the x-component of a trajectory of Σ ε . Moreover, we show that the modes of Σ are also modes of Σ, implying that the maximal Lyapunov exponents of Σ is upper bounded by that of Σ. The main result of our paper, therefore, consists in providing an interval containing all limits points of the maximal Lyapunov exponents of Σ ε as ε goes to zero: the upper and lower bounds are given by the maximal Lyapunov exponents of Σ and Σ, respectively (see Theorem 9). The paper is organized as follows: In Section 2 we present precise definitions of the systems Σ, Σ, Σ, and of their maximal Lyapunov exponents. We also provide the statement of the main result, Theorem 9, whose proof is given in the reminder of the paper. In particular, in Section 3 we compare the maximal Lyapunov exponents of Σ and the asymptotic values as ε goes to zero of the maximal Lyapunov exponent of Σ ε (Proposition 11). The comparison with the maximal Lyapunov exponent of Σ is the subject of Section 4 (Proposition 14), while Section 5 discusses the comparison with the maximal Lyapunov exponent of Σ.

Notations

By R we denote the set of real numbers, by | • | the Euclidean norm of a real vector, and by • the induced matrix norm. We use R + to denote the set of non-negative real numbers and we write x to denote the evaluation of the floor function at a real number x. By M n (R) we denote the set of n × n real matrices. The n × n identity matrix is denoted by I n . We use ρ(M ) to denote the spectral radius of a matrix M ∈ M n (R), defined as the largest modulus among the eigenvalues of M . By B r (x) we denote the closed ball of radius r > 0 and center x ∈ R n . The Hausdorff distance between two nonempty subsets X and Y of R n is the quantity defined by 2 Problem statement and main results

Stability notions for singularly perturbed linear switching systems

Fix n, m ∈ N and a compact set of matrices M ⊂ M n+m (R). Let Σ = (Σ ε ) ε>0 be the family of linear switching systems

Σ ε : ẋ(t) = A(t)x(t) + B(t)y(t), ε ẏ(t) = C(t)x(t) + D(t)y(t),
where ε denotes a small positive parameter and

t → A(t) B(t) C(t) D(t)
is an arbitrary element of the set S M of measurable functions from R + to M. For a given ε > 0, the usual stability notions, recalled in the following definition, apply to the linear switching system Σ ε . Definition 1. Let d ∈ N and N be a bounded subset of M d (R). Consider the linear switching system

Σ N : ẋ(t) = N (t)x(t), N ∈ S N , (3)
and denote by Φ N (t, 0) the flow from time 0 to time t of Σ N associated with the switching signal N . Then Σ N is said to be 1. exponentially stable (ES, for short) if there exist C > 0 and δ > 0 such that

Φ N (t, 0) ≤ Ce -δt , ∀ t ≥ 0, ∀ N ∈ S N ;
2. exponentially unstable (EU, for short) if there exist C > 0, δ > 0, and a nonzero trajectory

t → x(t) of Σ N such that |x(t)| ≥ Ce δt |x(0)|, ∀t ≥ 0.
The maximal Lyapunov exponent of Σ N is defined as

λ(Σ N ) = lim sup t→+∞ 1 t sup N ∈S N log Φ N (t, 0) .
Remark 2. Notice that, for every t > 0 and every N ∈ S N , by Gelfand's formula,

ρ(Φ N (t, 0)) = lim k→+∞ Φ N (t, 0) k 1 k = lim k→+∞ Φ Ñ (kt, 0) 1 k
where Ñ denotes the t-periodic signal obtained by periodization of N | [0,t] . As a consequence,

ρ(Φ N (t, 0)) ≤ e tλ(Σ N ) . (4)
Actually, it is well known (see, e.g., [START_REF] Wirth | The generalized spectral radius and extremal norms[END_REF]) that an equivalent definition of the maximal Lyapunov exponent is

λ(Σ N ) = lim sup t→+∞ 1 t sup N ∈S N log ρ(Φ N (t, 0)). ( 5
)
Remark 3. It follows from the definitions that Σ N is ES if and only if λ(Σ N ) < 0. Also, by (5), we have that Σ N is EU if and only if λ(Σ N ) > 0. Moreover, the properties of exponential stability/instability and the value of λ(Σ N ) keep unchanged if we replace S N by the class of piecewise-constant functions from R + to N .

In the following we write Φ ε M (t, 0) to denote the flow from time 0 to time t of Σ ε associated with a signal M ∈ S M and we introduce the following stability notions for the 1-parameter family Σ of switching systems. Definition 4. We say that the 1-parameter family of linear switching systems Σ : ε → Σ ε is 1. ε-uniformly exponentially stable (ε-ES, for short) if there exist ε > 0, C > 0, and δ > 0 such that

Φ ε M (t, 0) ≤ Ce -δt , ∀t ≥ 0, ∀M ∈ S M , ∀ε ∈ (0, ε ); (6)
2. ε-uniformly exponentially unstable (ε-EU, for short) if there exist δ > 0, C > 0, and ε > 0 such that for every ε ∈ (0, ε ) there exist M ∈ S M and z 0 = 0 for which

|Φ ε M (t, 0)z 0 | ≥ Ce δt |z 0 |, ∀ t ≥ 0.
Remark 5. The ε-uniform exponential stability of Σ is stronger than the property of global uniform asymptotic stability introduced in [START_REF] Hachemi | Stability analysis of singularly perturbed switched linear systems[END_REF], since it not only guarantees that λ(Σ ε ) < 0 for every ε small enough, but also that the stability is uniform with respect to ε, i.e., λ(Σ ε ) is smaller than the negative number -δ, independent of ε small, and the constant C appearing in (6) is independent of ε small. Similar considerations concern the ε-uniform exponential instability.

Auxiliary switching systems and main result

Our stability analysis of Σ relies on the comparison with several auxiliary systems having a single time scale. Let us first discuss the system Σ D corresponding to the evolution of the fast variable y with x = 0. More precisely, let

M D = {D | ( A B C D
) ∈ M} and consider the linear switching system

Σ D : ẏ(t) = D(t)y(t), D ∈ S M D .
We first provide a useful result for the asymptotic behavior of Σ.

Proposition 6. With the notations above, it holds that

lim ε→0 ελ(Σ ε ) = max{0, λ(Σ D )}.
Proof. Consider the family (Σ Nε ) ε≥0 of linear switching systems, where, for every ε ≥ 0, the compact set of matrices N ε ⊂ M n+m (R) is defined by

N ε = εA εB C D | A B C D ∈ M .
Then one notices that, for every ε > 0, the time rescaling t → εt yields

φ Nε (t, 0) = φ ε M (εt, 0),
where M is an arbitrary signal in M and the signal N ε ∈ N ε is defined as

N ε (t) = εA(t) εB(t) C(t) D(t) , t ≥ 0.
This implies at once that, for every ε, ελ(Σ ε ) = λ(Σ Nε ). Next notice that the compact sets N ε converge to N 0 as ε goes to zero for the Hausdorff topology defined on the family of compact subsets of M n+m (R). One gets the conclusion using Lemma 3.5 and Equation ( 19) from [START_REF] Wirth | The generalized spectral radius and extremal norms[END_REF].

The above result yields, in the particular case where λ(Σ D ) > 0, that λ(Σ ε ) ≥ µ/ε for µ ∈ (0, λ(Σ D )) and ε > 0 small enough, hence Σ ε is unstable with trajectories diverging at an arbitrarily large exponential rate for ε small. This motivates the following working assumption for the rest of the paper.

Assumption 7. The switching system Σ D is ES, that is, λ(Σ D ) < 0.
In particular, all matrices in M D are Hurwitz (and then invertible). We also introduce the set

M := {A -BD -1 C | ( A B C D ) ∈ M} ⊂ M n (R),
which collects the modes of the switching system

Σ : ẋ(t) = A(t) -B(t)D(t) -1 C(t) x(t), ( A B C D ) ∈ S M .
As described in the introduction, we also consider as auxiliary system another switching system, denoted by Σ, with a larger set of modes than Σ, and which could be thought of as the slow dynamics corresponding to signals whose switching occurs at rate exactly 1/ε. In order to define Σ, let us associate with every T > 0 and every σ

= ( A B C D ) ∈ S M the matrices Λ 0 (T, σ) = T 0 Φ D (T, s)C(s) ds, Λ 1 (T, σ) = T 0 (A(s) + B(s)Λ 0 (s, σ)) ds, Λ 2 (T, σ) = T 0 B(s)Φ D (s, 0) ds, and
Λ(T, σ) = Λ 1 (T, σ) + Λ 2 (T, σ) (I m -Φ D (T, 0)) -1 Λ 0 (T, σ) T .
As proved in Lemma 12 given in Section 4, the set

M := {Λ(T, σ) | T > 0, σ ∈ S M } is bounded in M n (R).
The switching system Σ is then defined as

Σ : ẋ(t) = M (t)x(t), M ∈ S M. (7)
In the case of autonomous systems, i.e., in the absence of switching, singular perturbation theory [START_REF] Petar | Singular perturbations and order reduction in control theory -an overview[END_REF] guarantees that the exponential stability of Σ is completely characterized by the associated reduced dynamics Σ. In the switching case, however, it is well known that the stability of Σ is not sufficient to deduce the stability of the perturbed switching system Σ ε for ε ∼ 0 [START_REF] Malloci | Stabilization of continuous-time singularly perturbed switched systems[END_REF]. The main difference is that, fixed x ∈ R n , in the switching case the ω-limit set K(x) of the fast dynamics

Σ x : ẏ(t) = D(t)y(t) + C(t)x, ( A B C D ) ∈ S M , (8)
is in general not reduced to a single equilibrium point [START_REF] Della Rossa | Stability of switched affine systems: Arbitrary and dwell-time switching[END_REF][START_REF] Nilsson | Invariant sets of defocused switched systems[END_REF]. The precise definition of the set K(x) is the closure of the union of the ω-limit sets of all trajectories of Σ x starting from the initial condition y(0) = 0. We will prove in Proposition 17 and Lemma 19 that K(x) is compact and the set-valued map x → K(x) is homogeneous of degree one and globally Lipschitz continuous for the Hausdorff distance. Let us use the set-valued map K to introduce a last auxiliary system, which is not necessarily a switching system. Namely, let us set

M(x) := {Ax + By | ( A B C D ) ∈ M, y ∈ K(x)}, x ∈ R n , and consider the differential inclusion Σ : ẋ(t) ∈ M(x(t)).
Notice that M(x) is compact for every x ∈ R n , since both M and K(x) are compact. We recall that a solution to Σ (in the sense of Filippov) is an absolutely continuous map

R + t → x(t) ∈ R n such that ẋ(t) ∈ M (x(t))
for almost all t. The existence of solutions to Σ for every initial condition is a consequence of classical results on differential inclusions (see, e.g., [START_REF] Aubin | Differential inclusions: set-valued maps and viability theory[END_REF]Chapter 2]).

In analogy with Definition 1, we say that Σ is ES (for exponentially stable) if there exist C > 0 and δ > 0 such that every trajectory x(•) of Σ satisfies

|x(t)| ≤ Ce -δt |x(0)|, ∀t ≥ 0.
The notion of exponential instability is completely analogous to the one given for linear switching systems.

As for the Lyapunov exponent of Σ, it can be defined as

λ( Σ) := lim sup t→+∞ 1 t sup log |x(t)|,
where the sup is taken over all trajectories x(•) of Σ with |x(0)| = 1. Notice that the restriction to initial conditions with unit norm is justified by the fact that x → K(x), and hence x → M(x), is homogeneous of degree one. The Lyapunov exponent λ( Σ) satisfies the following property, which generalizes the corresponding one for switching systems recalled in Remark 3. The following theorem summarizes the main results obtained in this paper.

Theorem 9. Suppose that Assumption 7 holds. Then

λ( Σ) ≤ λ( Σ) ≤ lim inf ε→0 + λ(Σ ε ) ≤ lim sup ε→0 + λ(Σ ε ) ≤ λ( Σ). (9)
Moreover,

1. if system Σ is EU then system Σ is ε-EU; 2. If system Σ is ES then system Σ is ε-ES.

Comparison between the Lyapunov exponents of Σ and Σ

We prove in this section that λ( Σ) ≤ lim inf ε→0 + λ(Σ ε ) (Proposition 11). Strictly speaking, in view of Proposition 14 given below, this is not necessary for the proof of Theorem 9, but we prefer to provide the proof of this result since it illustrates, in a simplified framework, some of the ideas used in the proof of the inequality λ( Σ) ≤ lim inf ε→0 + λ(Σ ε ).

The following lemma recalls a rather classical relation between Lyapunov exponents of linear switching systems that will be useful in the sequel. We provide its proof for completeness.

Lemma 10. Consider a linear switching system Σ N as in (3). Then, for every

µ ∈ R, λ(Σ N +µIn ) = µ + λ(Σ N ). Proof. Let µ ∈ R. Observe that t → x(t) is a trajectory of Σ N if and only if t → e µt x(t) is a trajectory of Σ N +µIn . By consequence, λ(Σ N +µIn ) = lim sup t→+∞ 1 t sup N ∈S N +µIn log Φ N (t, 0) = lim sup t→+∞ 1 t sup N ∈S N log e µt Φ N (t, 0) = µ + λ(Σ N ),
concluding the proof.

Proposition 11. Suppose that Assumption 7 holds. Then λ( Σ) ≤ lim inf ε→0 + λ(Σ ε ).

Proof. For µ ∈ R and ε > 0 consider the switching systems

Σ ε,µ : ẋ(t) = A µ (t)x(t) + B(t)y(t), ε ẏ(t) = C(t)x(t) + D µ (t)y(t), (10)
and

Σµ : ẋ(t) = A µ (t) -B(t)D -1 (t)C(t) x(t),
where

( A B C D ) ∈ S M , A µ (t) = A(t) + µI n , and D µ (t) = D(t) + εµI m .
It follows from Lemma 10 that

λ(Σ ε,µ ) = λ(Σ ε ) + µ and λ( Σµ ) = λ( Σ) + µ.
Fix, for now, µ ∈ R, T > 0, and a piecewise-constant switching signal σ = ( A B C D ) ∈ S M . Denote by t 1 < • • • < t N the switching instants of σ within the interval [0, T ] and set t 0 = 0, t N +1 = T . Following a classical approach (see e.g. [START_REF] Petar | A Riccati equation for block-diagonalization of ill-conditioned systems[END_REF]), we introduce the variable

z(t) = y(t) + D -1 µ (t)C(t)x(t) + εP ε (t)x(t), (11)
where P ε is constant on each interval [t k , t k+1 ), k = 0, . . . , N , and is chosen in such a way that Σ ε,µ is equivalently represented on each interval [t k , t k+1 ) as

ẋ(t) = (M µ (t) -εB(t)P ε (t)) x(t) + B(t)z(t), (12)
ε ż(t) = (D µ (t) + εQ ε (t)B(t)) z(t), (13)
where

M µ = A µ -BD -1 µ C, Q ε = D -1
µ C + εP ε , and P ε (t) is upper bounded uniformly with respect to t ∈ [0, T ] and ε small enough. System ( 12)-( 13) is discontinuous at the instants of switching, since the variable z depends on σ. Recall that Φ ε -1 Dµ denotes the flow associated with 1 ε D µ . Thanks to Assumption 7, there exist c, α > 0 such that for ε small enough and for every 0 < s < t

Φ ε -1 Dµ (t, s) ≤ ce -α ε (t-s) . (14)
Observe that there exists a positive constant K independent of ε (but possibly depending on T, M, µ) such that for every (x 0 , y 0 ) ∈ R n × R m and every ε > 0 small enough one has

|(x(t), z(t))| ≤ K|(x 0 , y 0 )|, t ∈ [0, T ], (15)
where (x(•), y(•)) is the trajectory of Σ ε,µ associated with σ and the initial condition (x 0 , y 0 ) and z(•) is given by [START_REF] Petar | A Riccati equation for block-diagonalization of ill-conditioned systems[END_REF]. By a slight abuse of notation, in what follows we still use K to denote possibly larger constants independent of ε. For t ∈ [t k , t k+1 ), by applying the variation of constant formula to (13) and using [START_REF] Petar | A Riccati equation for block-diagonalization of ill-conditioned systems[END_REF] for t = t k , we have

z(t) = Φ ε -1 Dµ (t, t k )z(t k ) + t t k Φ ε -1 Dµ (t, s)Q ε (s)B(s)z(s)ds.
As an immediate consequence of ( 14) and ( 15), one gets that

|z(t)| ≤ (e -α ε (t-t k ) + ε)K|(x 0 , y 0 )|, ∀t ∈ [t k , t k+1 ). ( 16
)
By applying the variation of constant formula to [START_REF] Petar | Applications of singular perturbation techniques to control problems[END_REF], one deduces that, for every t ∈ [0, T ],

x(t) -x(t) = t 0 Φ Mµ (t, s)B(s) D -1 (s) -D -1 µ (s) C(s)x(s)ds -ε t 0 Φ Mµ (t, s)B(s)P ε (s)x(s)ds + t 0 Φ Mµ (t, s)B(s)z(s)ds,
where Mµ := A µ -BD -1 C. By the definition of D µ we have that

D -1 (t) -D -1 µ (t) ≤ εK, ∀ t ∈ [0, T ]. (17)
Hence, using estimates ( 15) and ( 16), one deduces that

|x(t) -x(t)| ≤ εK|(x 0 , y 0 )|. ( 18
)
From [START_REF] Petar | A Riccati equation for block-diagonalization of ill-conditioned systems[END_REF] together with ( 16), [START_REF] Wang | Extended dissipative control for singularly perturbed pdt switched systems and its application[END_REF], and ( 18) we obtain that

|y(T ) + D -1 N C N x(T )| ≤ |z(T )| + K|x(T ) -x(T )| + εK|(x 0 , y 0 )| ≤ (e -α ε (T -t N ) + ε)K|(x 0 , y 0 )| ≤ εK|(x 0 , y 0 )| ( 19
)
for ε sufficiently small, where (

A N B N C N D N ) is the value of σ on [t N , T ]. If Φ ε
σ,µ denotes the flow of (10) associated with σ, one deduces from ( 18) and ( 19) that

Φ ε σ,µ (T, 0) - Φ Mµ (T, 0) 0 -D -1 N C N Φ Mµ (T, 0) 0 ≤ εK.
Let µ be any constant such that λ( Σ) + µ = λ( Σµ ) > 0, and choose the time T > 0 and the piecewiseconstant switching signal σ so that the spectral radius of Φ Mµ (T, 0) is larger than one (this is possible thanks to ( 5)). By continuity of the spectral radius, it follows that ρ(Φ ε σ,µ (T, 0)) is also larger than one for ε small enough. Then one deduces from (4) that λ(Σ ε ) + µ = λ(Σ ε,µ ) > 0. The conclusion follows by arbitrariness of µ > -λ( Σ).

Comparison between the Lyapunov exponents of Σ and Σ

We have the following preliminary results.

Lemma 12. Under Assumption 7 the set M is bounded.

Proof. By Assumption 7 we have that

Φ D (t 2 , t 1 ) ≤ ce -α(t2-t1) (20)
for every t 2 ≥ t 1 , for some constants α > 0 and c ≥ 1 independent of the switching law D ∈ S M D . As a consequence of (20) and the boundedness of M, we have Λ 0 (T, σ) ≤ C 1 min{1, T } and Λ 2 (T, σ) ≤ C 1 min{1, T } for some C 1 > 0, and Λ 1 (T, σ) ≤ C 2 T for some C 2 > 0. Moreover since ρ(Φ D (T, 0)) < 1 for every T > 0 and D ∈ S M D according to (4), (I m -Φ D (T, 0)) -1 is well defined and expandable as an absolutely convergent power series in Φ D (T, 0). Letting T = log(2c) α so that Φ D (τ, 0) ≤ 1 2 for every τ ≥ T , we can write

(I m -Φ D (T, 0)) -1 = k≥0 Φ D (T, 0) k = T T h=0 Φ D (T, 0) h + Φ D (T, 0) T T +1   T T h=0 Φ D (T, 0) h   + Φ D (T, 0) 2( T T +1)   T T h=0 Φ D (T, 0) h   + . . . =   k≥0 Φ D (T, 0) ( T T +1)k     T T h=0 Φ D (T, 0) h   .
As Φ D (T, 0) h corresponds to the flow of Σ D for a T -periodic signal at time hT we have Φ D (T, 0) h ≤ c by (20). Hence

T T h=0 Φ D (T, 0) h ≤ 1 + T T c ≤ c + c T T .
Moreover, as

( T T + 1)T ≥ T , k≥0 Φ D (T, 0) ( T T +1)k ≤ k≥0 Φ D (T, 0) ( T T +1) k ≤ k≥0 1 2 k = 2.
Summing up

Λ(T, σ) ≤ C 2 + C 2 1 min{1, T } 2 (2c + 2c T /T ) T = C 2 + 2cC 2 1 min 1 T + T T 2 , T + T ≤ C 2 + 2cC 2 1 (1 + T ),
concluding the proof of the lemma.

Lemma 13. Let T > 0, σ = ( A B C D ) ∈ S M , and µ ∈ R. For every ε > 0 denote by M (ε) the flow at time εT of Σ ε,µ defined in [START_REF] Kokotović | Singular perturbation methods in control[END_REF] and corresponding to the signal σ(•/ε). Then there exists P (ε) of the form

P (ε) = I n 0 Q(ε) I m (21)
such that

P (ε) -1 M (ε)P (ε) = I n + εT (Λ(T, σ) + µI n ) + O(ε 2 ) O(ε) 0 Φ D (T, 0) + O(ε) , (22)
where

Q(ε) = I m -Φ D (T, 0) -1 Λ 0 (T, σ) + O(ε) (23)
and the functions O(ε k ) are such that O(ε k ) ≤ Cε k for ε small, for some positive constant C independent of ε and σ.

Proof. We first prove that ε → M (ε) admits a first order expansion

M (ε) = M 0 + εM 1 + O(ε 2 ), (24)
for some matrices M 0 , M 1 to be computed. To see that, we first apply the time rescaling τ = t/ε and we have that M (ε) is equal to Φ N0+εN1 (T, 0), where the signals N 0 and N 1 are defined as

N 0 (τ ) = 0 0 C(τ ) D(τ ) , N 1 (τ ) = A(τ ) + µI n B(τ ) 0 µI m .
By the variation of constant formula, one has

Φ N0+εN1 (T, 0) = Φ N0 (T, 0) + ε T 0 Φ N0 (T, τ )N 1 (τ )Φ N0+εN1 (τ, 0) dτ.
One deduces that (24) holds true with

M 0 = Φ N0 (T, 0) and M 1 = T 0 Φ N0 (T, τ )N 1 (τ )Φ N0 (τ, 0) dτ.
It is easy to get that

M 0 = I n 0 Λ 0 (T, σ) Φ D (T, 0)
, and

M 1 = Λ 1 (T, σ) + T µI n Λ 2 (T, σ) Λ 3 (T, σ) Λ 4 (T, σ) , (25)
where

Λ 3 (T, σ) = Λ 0 (T, σ) (Λ 1 (T, σ) + T µ) - T 0 Φ D (T, τ )Λ 0 (τ, σ)(A(τ ) + B(τ )Λ 0 (τ, σ))dτ and Λ 4 (T, σ) = µT Φ D (T, 0) + Λ 0 (T, σ)Λ 2 (T, σ) - T 0 Φ D (T, τ )Λ 0 (τ, σ)B(τ )Φ D (τ, 0)dτ.
According to (24) and ( 25), it follows that (22) holds true with P (ε) and Q(ε) as in ( 21) and ( 23). This concludes the proof of the lemma.

We can now prove the main result of this section.

Proposition 14. Suppose that Assumption 7 holds. Then

λ( Σ) ≤ λ( Σ) ≤ lim inf ε→0 + λ(Σ ε ). Moreover, if Σ is EU then Σ is ε-EU.
Proof. We first prove the inequality λ( Σ) ≤ λ( Σ) by showing that M ⊂ M. To see that, let us check that A -BD -1 C belongs to M for every M = ( A B C D ) in M. Indeed, letting T > 0 and σ ∈ S M constantly equal to M on [0, T ], it holds Λ(T, σ) = A -BD -1 C.

Assume that µ is chosen so that λ( Σ) + µ > 0.

Then, according to [START_REF] Boscain | Stability of planar switched systems: the linear single input case[END_REF], there exist ∈ N, matrices Λ(T 1 , σ 1 ), . . . , Λ(T , σ ) ∈ M, and positive times t 1 , . . . , t so that ρ e t (Λ(T ,σ )+µIn) . . . e t1(Λ(T1,σ1)+µIn) > 1.

(26)

Let ε be sufficiently small so that εT k < t k for every k = 1, . . . , , and denote by

N k = t k εT k the number of intervals of length εT k contained in [0, t k ].
Then, for every k = 1, . . . , , consider the flow M k (ε) of system Σ ε,µ (cf. ( 10)) corresponding to the signal σ k (•/ε), evaluated at time εT k . Thanks to Lemma 13, there exists P k (ε) given by P k (ε) =

I n 0 Q k (ε) I m such that T k (ε) = P k (ε) -1 M k (ε)P k (ε) satisfies T k (ε) = I n + εT k (Λ(T k , σ k ) + µI n ) + O(ε 2 ) O(ε) 0 Φ D (T k , 0) + O(ε) .
We repeat N k times σ k (•/ε) to get a signal on [0, εT k N k ] and the corresponding flow of Σ ε,µ at time εT k N k is given by

M k (ε) N k = P k (ε)T k (ε) N k P k (ε) -1 . ( 27
)
We claim that

T k (ε) N k = e t k (Λ(T k ,σ k )+µIn) + O(ε) O(ε) 0 O(ε) . ( 28
)
This follows from the general formula

A 11 A 12 0 A 22 N = A N 11 W 0 A N 22 , W = N -1 j=0 A j 11 A 12 A N -j-1 22 , applied to N = N k , A 11 = I + B 11 /N k + O(ε 2 ) = e B 11 N k + O(ε 2 ) with B 11 = t k (Λ(T k , σ k ) + µI n ), and
A 22 = Φ D (T k , 0) + O(ε).
Then, for ε small enough one gets that

A j 11 ≤ 1 + 2 B 11 j j ≤ e 2 B11 , 1 ≤ j ≤ N k , (29)
A N k 11 = e B11 + O(ε).
Moreover, since ρ(A 22 ) < 1 for ε small, it follows that

A j 22 ≤ Kλ j , 1 ≤ j ≤ N k , (30)
for some K > 0 and λ ∈ (0, 1) independent of ε small enough and k ∈ {1, . . . }. In particular,

A N k 22 ≤ Kλ t k εT k -1 = O(ε).
Using now (29), (30), and recalling that K can be taken so that A 12 ≤ Kε, one deduces that

W ≤ N k -1 j=0 A j 11 A 12 A N k -j-1 22 ≤ K 2 e 2 B11 ε N k -1 j=0 λ N k -1-j = Cε,
for some C > 0 independent of ε small enough and k ∈ {1, . . . }. This concludes the proof of (28). We next use ( 23) and ( 28) in ( 27) to deduce that

M k (ε) N k = e t k (Λ(T k ,σ k )+µIn) 0 r k 0 + O(ε), (31)
where

r k = I m -Φ Dσ k (T k , 0) -1 Λ 0 (T k , σ k )e t k (Λ(T k ,σ k )+µIn) .
Set t ε = ε k=1 N k T k and notice that t ε tends to t 1 + • • • + t as ε tends to zero. We concatenate the N k times repetitions of σ k (•/ε) for k = 1, . . . , to get a signal on [0, t ε ] and the corresponding flow of Σ ε,µ at time t ε is given by the matrix product

Υ ε = M (ε) N M -1 (ε) N -1 • • • M 1 (ε) N1 .

Using (31), one deduces that

Υ ε = e t (Λ(T ,σ )+µIn) • • • e t1(Λ(T1,σ1)+µIn) 0 r 0 + O(ε),
where the matrix r does not depend on ε.

Using (26), one gets that ρ(Υ ε ) > 1 for ε small enough, yielding that

λ(Σ ε ) + µ = λ(Σ ε,µ ) > 0.
In particular lim inf ε→0 + λ(Σ ε ) + µ ≥ 0. By letting µ tend to -λ( Σ) from above, one concludes that λ( Σ) ≤ lim inf ε→0 + λ(Σ ε ) as desired.

We are left to show that if Σ is EU then Σ is ε-EU. For this purpose, we take µ = 0 in the previous calculations and we observe that the flow Υ ε,t of Σ ε at time t ∈ [0, t ε ] corresponding to the signal defined above satisfies

Υ ε,t = e (t-k-1 h=1 t h )Λ(T k ,σ k ) k-1 =1 e t Λ(T ,σ ) + O(ε) O(ε) r(ε, t) q(ε, t) , ( 32
)
whenever t ∈ [ k-1 h=1 t h , k h=1 t h ],
for some matrix functions r, q, where the terms O(ε) are uniform with respect to t ∈ [0, t ε ]. The matrix Υ ε,tε converges, as ε goes to zero, to

Ῡ = e t Λ(T ,σ ) • • • e t1Λ(T1,σ1) 0 r 0 ,
for some matrix r. For ε small enough we construct a trajectory

z ε (t) = (x ε (t), y ε (t)) of Σ ε satisfying |z ε (t)| ≥ Ĉe λt |z ε (0)| > 0 for every t ≥ 0, with λ ∈ (0, λ( Σ)
) and Ĉ > 0 independent of ε. Let π be the sum of the projectors on the generalized eigenspaces associated with the eigenvalues of Ῡ of modulus ρ(e t Λ(T ,σ ) • • • e t1Λ(T1,σ1) ) > 1. Since Υ ε,tε converges to Ῡ as ε goes to zero, by classical results (see [8, Theorem 5.1, Chapter II]) there exists π ε , a sum of projectors on generalized eigenspaces of Υ ε,tε , satisfying lim ε→0 π ε = π, and the corresponding eigenvalues also converge. Let v ε be a possibly complex eigenvector of the restriction of Υ ε,tε to the image of π ε , associated with an eigenvalue α ε . If α ε is real then v ε can be taken real as well, otherwise we assume without loss of generality that

v ε satisfies |Re(v ε )| = min θ∈R |Re(e iθ v ε )|. In particular |Re(βv ε )| ≥ |β||Re(v ε )| for every β ∈ C. Note that, for every positive integer k, one has that (Υ ε,tε ) k v ε = α k ε v ε and (Υ ε,tε ) k vε = ᾱk ε vε which implies (Υ ε,tε ) k Re(v ε ) = Re(α k ε v ε ). Consider the trajectory z ε (t) = (x ε (t), y ε (t)) of Σ ε obtained
applying the flow Υ ε,t to the initial condition z ε (0) = Re(v ε ) and repeating periodically after time t ε . Letting Π x be the projection of a vector of R n+m onto its first n components, it is easy to see that |Π x v| ≥ C|v| for every v in the image of Ῡ, where

C = (1 + r 2 (e t Λ(T ,σ ) • • • e t1Λ(T1,σ1) ) -1 2 ) -1/2 .
Hence, for every nonnegative integer h and ε small enough,

|x ε (ht ε )| = |Π x z ε (ht ε )| = |Π x π ε z ε (ht ε )| ≥ |Π x πz ε (ht ε )| -Π x π -π ε |z ε (ht ε )| ≥ |Π x πz ε (ht ε )| -π -π ε |z ε (ht ε )| ≥ C|πz ε (ht ε )| -π -π ε |z ε (ht ε )| ≥ C|z ε (ht ε )| -(1 + C) π -π ε |z ε (ht ε )| ≥ C 2 |z ε (ht ε )|. ( 33
) As z ε (ht ε ) = Re(α h ε v ε ) we also have |x ε (ht ε )| ≥ C 2 |α ε | h |z ε (0)|. (34)
By (32), (33), and setting κ = max Λ∈ M Λ it follows that

|x ε (ht ε + τ )| ≥ |x ε (ht ε )|e -κτ -|z ε (ht ε )|O(ε) ≥ |x ε (ht ε )|e -κτ - 2 C |x ε (ht ε )|O(ε) ≥ 1 2 |x ε (ht ε )|e -κτ (35)
for every τ ∈ [0, t ε ]. Take ε small enough in such a way that 1 < α ≤ |α ε | ≤ ᾱ for some positive constants α, ᾱ, and t ε ≤ t, where t = 2 h=1 t h . By (34) and (35) we thus get for every t ≥ 0

|z ε (t)| ≥ |x ε (t)| ≥ C 4 e -κtε |α ε | t tε |z ε (0)| ≥ Ce -κtε 4|α ε | e log |αε| tε t |z ε (0)| ≥ Ĉe λt |z ε (0)|,
where Ĉ = Ce -κt 4 ᾱ and λ = log α t , which shows that Σ is ε-EU.

Example 15 ( Σ gives sharper bounds than Σ). Consider system Σ with

M = M 1 = -1 1 0 -0.1 , M 2 = -3 0 2 -0.1 .
The stability of singularly perturbed planar switching systems is completely characterized in [7, Theorem 2] through some necessary and sufficient conditions. Based on this characterization (cf., in particular, Item (SP5) in [7, Theorem 2]) the condition

Γ(M 1 , M 2 ) := 1 2 (tr(M 1 )tr(M 2 ) -tr(M 1 M 2 )) < -det(M 1 ) det(M 2 ) (36)
implies that Σ ε is EU for all ε > 0. Condition (36) is satisfied in the case of this example with Γ(M 1 , M 2 ) = -0.8 and det(M 1 M 2 ) = 0.03. Look now at systems Σ and Σ. We have M = {-1, -3} and then the associated system Σ is ES. Concerning system Σ, let us consider the switching signal

σ(t) = α(t)M 1 + (1 -α(t))M 2
associated with the 2-periodic function

α(t) = 1 t ∈ [0, 1], 0 t ∈ [1, 2],
and take T = 2. For this choice of σ and T one can easily verify that

Λ(T, σ) = -2 + 100(1 -e -0.2 ) -1 (1 -e -0.1 ) 2 > 0.
Then Σ is EU, as illustrated in Figure 1. 5 Comparison between the Lyapunov exponents of Σ and Σ

Definition and structural properties of the differential inclusion Σ

The following lemma studies the ω-limit set of the dynamics Σ x given by [START_REF] Kato | Perturbation theory for linear operators[END_REF].

Lemma 16. Let x ∈ R n . Then, for every y 0 ∈ R m and σ ∈ S M , the ω-limit set ω x σ of the trajectory of Σ x associated with σ and starting at y 0 does not depend on the initial condition y 0 .

Proof. Let y 0 , y 1 ∈ R m and let y 0 (•), y 1 (•) be the trajectories of Σ x associated with σ and starting from y 0 and y 1 , respectively. Setting z(•) = y 0 (•) -y 1 (•), one has ż(t) = D(t)z(t), and thanks to Assumption 7, there exist c, δ > 0 such that

|y 0 (t) -y 1 (t)| ≤ ce -δt |y 0 -y 1 |, ∀ t ≥ 0. ( 37
)
By definition of an ω-limit set, one deduces at once that ω x σ does not depend on y 0 .

Let us introduce the set valued-map K : R n R m defined by K(x) = σ ω x σ . We have the following proposition.

Proposition 17. For each x ∈ R n the set K(x) is compact and forward invariant for the dynamics of Σ x , and there exist c, δ > 0 such that

d(y 0 (t), K(x)) ≤ ce -δt d(y 0 , K(x)), ∀t ≥ 0, y 0 ∈ R m , (38)
for all σ ∈ S M , where y 0 (•) is the trajectory of Σ x associated with σ and starting from y 0 .

Proof. Let c and δ be as in (37). The set K(x) is closed by definition and its boundedness follows from the fact that for every σ ∈ S M the corresponding solution y(•) of Σ x with initial condition y(0) = 0 satisfies

|y(t)| = t 0 Φ D (t, s)C(s)x ds ≤ c|x| max M ∈M M δ ,
for t ≥ 0. Hence K(x) is compact. Let now ȳ ∈ K(x), i.e., ȳ = lim k→+∞ y k , where y k ∈ ω x σ k , σ k ∈ S M . For t > 0 and σ, denote by Φ x,σ the flow of Σ x associated with the signal σ and let us prove that ỹ := Φ x,σ ( t, 0)ȳ is in K(x). For k ≥ 1, let r k = |ỹ -Φ x,σ ( t, 0)y k )| and notice that lim k→+∞ r k = 0. Moreover, for k ≥ 1, there exists ρ k > 0 such that Φ x,σ ( t, 0)B ρ k (y k ) ⊂ B 2r k (ỹ). We next define recursively the sequence (z k ) k≥0 with z 0 ∈ R m by setting

z k+1 = Φ x,σ ( t, 0)Φ x,σ k+1 (t k , 0)z k , k ≥ 0,
where the sequence of times t k is chosen so that Φ x,σ k+1 (t k , 0)z k is in B ρ k+1 (y k+1 ). This is possible since

y k+1 ∈ ω x σ k+1 for k ≥ 0. By construction, z k is in B 2r k (ỹ) for every k ≥ 1. Moreover, z k = Φ x,σ (τ k , 0)z 0 where τ k → ∞ and σ is constructed by repeatedly concatenating σ k | [0,t k ] and σ| [0, t] . Then ỹ is in ω x σ ⊂ K(x).
Finally, using (37) with y 1 ∈ K(x) and the forward invariance of K(x) for the dynamics of Σ x , we have

d(y 0 (t), K(x)) ≤ |y 0 (t) -y 1 (t)| ≤ ce -δt |y 0 -y 1 |, ∀ t ≥ 0,
and one gets (38) by arbitrariness of y 1 ∈ K(x).

Remark 18. The contents of the above proposition are essentially contained in the preprint [START_REF] Della Rossa | Stability of switched affine systems: Arbitrary and dwell-time switching[END_REF] (Theorem 1 and Proposition 2), where the authors study general switching affine systems and the role of K(x) is played by the set K ∞ . Lemma 19. The set-valued map K is globally Lipschitz continuous for the Hausdorff distance and homogeneous of degree one.

Proof. Notice that, given x ∈ R n and a nonzero λ ∈ R, y(•) is a trajectory of Σ x if and only if λy(•) is a trajectory of Σ λx . We deduce that z ∈ ω x σ if and only if λz ∈ ω λx σ and hence that K(λx) = λK(x). As for the Lipschitz continuity of K, let x 1 , x 2 ∈ R n . It is easy to deduce from the variation of constant formula that there exists L K > 0 independent of x 1 , x 2 such that

|y 1 (t) -y 2 (t)| ≤ L K |x 1 -x 2 |, ∀ t ≥ 0, ∀ σ ∈ S M ,
where y 1 (•) and y 2 (•) are the trajectories of Σ x1 and Σ x2 , respectively, associated with σ and starting from the same initial condition y 0 . By consequence, we have

max{d(y 1 , ω x2 σ ), d(y 2 , ω x1 σ )} ≤ L K |x 1 -x 2 |, for y 1 ∈ ω x1
σ and y 2 ∈ ω x2 σ , where ω x1 σ and ω x2 σ are the ω-limit sets of Σ x1 and Σ x2 , respectively, which, thanks to Lemma 16, do not depend on the initial condition y 0 . From the previous inequality we obtain that, max{d(y

1 , ∪ σ ω x2 σ ), d(y 2 , ∪ σ ω x1 σ )} ≤ L K |x 1 -x 2 |, for y 1 ∈ ∪ σ ω x1 σ and y 2 ∈ ∪ σ ω x2 σ . By a standard density argument, max{d(y 1 , K(x 2 )), d(y 2 , K(x 1 ))} ≤ L K |x 1 -x 2 |,
for y 1 ∈ K(x 1 ) and y 2 ∈ K(x 2 ), from which we obtain, by the arbitrariness of y 1 and y 2 , that

d H (K(x 1 ), K(x 2 )) ≤ L K |x 1 -x 2 |,
where we recall that d H denotes the Hausdorff distance in R m .

Asymptotic estimates by converse Lyapunov arguments

The argument provided below bears similarities with proofs given in [START_REF] Watbled | On singular perturbations for differential inclusions on the infinite interval[END_REF], where more general dynamics are considered.

We consider the µ-shifted differential inclusion

Σµ : ẋ ∈ {A µ x + By | M = ( A B C D ) ∈ M, y ∈ K(x)},
where A µ = A + µI n with µ ∈ R. By homogeneity of K(•) it follows that for every solution x(•) of Σ the trajectory t → xµ (t) = e µt x(t) is a solution of Σµ . As a consequence λ( Σµ ) = λ( Σ) + µ. Hence, recalling that λ(Σ ε,µ ) = λ(Σ ε ) + µ, in order to prove the right-hand side of inequality [START_REF] Kimball | Singular perturbation theory for DC-DC converters and application to PFC converters[END_REF] it is enough to show that, under Assumption 7, λ( Σµ ) < 0 implies λ(Σ ε,µ ) < 0 for ε small enough. In order to prove the latter statement we will construct a common Lyapunov function for the systems Σ ε,µ as the sum of Lyapunov functions for Σµ and for Σ x defined next. Assume that λ( Σµ ) < 0. Then, by Lemma 8, |x(t)| ≤ ce -γt |x(0)| for some c ≥ 1 and γ > 0, for every trajectory x(•) of Σµ . Define

V 1 (x) = sup x(•),t∈[0, t ] e γt |x(t)| 2 , (39)
where the supremum is computed among all trajectories x(•) of Σµ starting from x ∈ R n , and t = log(c) 2γ . Furthermore, let δ ∈ (0, |λ(Σ D )|) and c ≥ 1 be such that |y(t)| ≤ ce -δt |y(0)| for every trajectory y(•) of Σ D . By Proposition 17 one has

d(y x (t), K(x)) ≤ ce -δt d(y x (0), K(x))
for every trajectory y x of Σ x , for every x ∈ R n and t > 0. Define

V 2 (x, y) = sup yx(•),t∈[0, t ] e δt d(y x (t), K(x)) 2 , (40)
where the supremum is computed among all trajectories y x (•) of Σ x starting from y ∈ R m and t = log(c) 2δ . In the next lemma, we summarize the main properties of V 1 and V 2 .

Lemma 20. The positive definite functions V 1 and V 2 introduced in (39) and (40) are homogeneous of degree two and locally Lipschitz continuous. Moreover, V 1 and V 2 are nonincreasing along every trajectory of Σµ and Σ x respectively and satisfy the following estimates:

V 1 (x(t)) ≤ e -γt V 1 (x(0)), (41)
V 2 (x, ȳx (t)) ≤ e -δt V 2 (x, ȳx (0)), (42)
where t ≥ 0, x ∈ R n , x(•) is an arbitrary trajectory of Σµ , and ȳx (•) is an arbitrary trajectory of Σ x .

Proof. It is clear that both V 1 and V 2 are homogeneous of degree two. We give the proof of the remaining properties for V 2 , the corresponding arguments for V 1 being completely analogous.

Let us next show that V 2 is locally Lipschitz continuous. For every bounded set B ⊂ R m there exists a compact set of R m containing every trajectory of Σ x starting from B. Take two points y 1 , y 2 in B and consider the trajectories y 1

x , y 2

x of Σ x corresponding to the same switching law starting respectively from y 1 and y 2 . Then |y 2

x (t) -y 1 x (t)| ≤ ce -δt |y 2 -y 1 | for every t ≥ 0. We deduce that the function y → e δt d(y x (t), K(x)) 2 , where y x is the trajectory starting from y ∈ B corresponding to a fixed switching law σ ∈ S M , is Lipschitz continuous, and the Lipschitz constant does not depend on t nor on the switching law. Since the supremum among a family of uniformly Lipschitz continuous functions is Lipschitz continuous, we deduce that V 2 is locally Lipschitz continuous in the variable y. Similarly, local Lipschitz continuity of the map x → e δt d(y x (t), K(x)) 2 for a fixed switching law follows from the fact that y x (•) is affine with respect to the variable x and K(•) is Lipschitz continuous. Furthermore, the corresponding Lipschitz constant (locally) does not depend on t nor on the switching law.

Consider now a trajectory ȳx (•) of Σ x and let us prove that V 2 (x, •) is nonincreasing along it. For h > 0 one has = e -δh V 2 (x, ȳx (0)), where we have used the fact that e δt d(y x (t), K(x)) 2 ≤ c2 e -δt d(y x (0), K(x)) 2 < d(y x (0), K(x)) 2

for every t > t along every trajectory of Σ x .

Based on the above construction of V 1 and V 2 , we next show the existence of a common Lyapunov function allowing us to prove that (Σ ε,µ ) ε>0 is ε-ES.

Proposition 21. There exists χ > 0, 0 < α -< α + , η > 0, and ε * > 0 such that, setting V = V 1 + χV 2 , one has

α -|(x, y)| 2 ≤ V (x, y) ≤ α + |(x, y)| 2 , ∀x ∈ R n , ∀y ∈ R m , (43)
V (x(t), y(t)) ≤ V (x(0), y(0))e -ηt , ∀t ≥ 0,

where (44) holds true for every solution (x(•), y(•)) of Σ ε,µ for ε < ε * . As a consequence lim sup ε→0 + λ(Σ ε ) ≤ λ( Σ) and if λ( Σ) < 0 then Σ is ε-ES. The desired inequality holds true with

α -= min 1 1 + 4L 2 K , χ 4
.

Consider now a trajectory (x(•), y(•)) of Σ ε,µ corresponding to the switching law σ. In order to prove (44) we first estimate the difference V (x(εh), y(εh)) -V (x(0), y(0)) for small h. Note that as it follows from the fact that the Lipschitz constant of V 1 on a ball of radius r is of order r. By using (41), V (x(t + τ ), y(t + τ )) -V (x(t), y(t)) τ ≤ -γ 2 V (x(t), y(t)).

V 1 (x(εh)) -V 1 (x(0)) = V 1 (x(εh)) -V 1 (x(εh)) + V 1 (x(εh)) -V 1 (x(0)) ≤ V 1 (x( 0
Since V (x(•), y(•)) is absolutely continuous we deduce that d dt V (x(t), y(t)) ≤ -γ 2 V (x(t), y(t)), a.e. t ≥ 0, and V (x(t), y(t)) ≤ V (x(0), y(0))e -γ 2 t , ∀ t ≥ 0, concluding the proof.

d

  H (X, Y ) = max sup x∈X d(x, Y ), sup y∈Y d(y, X) , where d(x, Y ) = inf y∈Y |x -y| and d(y, X) = inf x∈X |x -y|. Given a subset N of M n (R), we denote by S N the set of all measurable functions from R + to N .

Lemma 8 .

 8 System Σ is ES if and only if λ( Σ) < 0. Proof. The direct implication is trivial. On the other hand, by definition of λ( Σ), if the latter is negative then for every δ ∈ (0, |λ( Σ)|) there exists T > 0 such that |x(t)| ≤ e -δt |x(0)| for every solution x(•) of the differential inclusion and t ≥ T . Furthermore, | ẋ(t)| ≤ Ĉ|x(t)| for some Ĉ > 0, because of the compactness, continuity, and homogeneity of the map M. Hence |x(t)| ≤ e Ĉt |x(0)| ≤ Ke -δt |x(0)| on [0, T ], where K = max t∈[0,T ] e ( Ĉ+δ)t = e ( Ĉ+δ)T . We conclude that |x(t)| ≤ Ke -δt |x(0)| for every t > 0.

Figure 1 :

 1 Figure 1: Fast and slow variables evolution of system Σ ε with signal σ and ε = 0.1, starting from (x 0 , y 0 ) = (1, 1).

V 2 2 =

 22 (x, ȳx (h)) = sup yx(•), yx| [0,h] =ȳx| [0,h] t∈[h, t+h] e δ(t-h) d(y x (t), K(x)) 2 ≤ sup yx(•), yx(0)=ȳx(0) t∈[h, t+h] e δ(t-h) d(y x (t), K(x)) 2 ≤ sup yx(•), yx(0)=ȳx(0) t∈[0, t+h] e δ(t-h) d(y x (t), K(x)) e -δh sup yx(•), yx(0)=ȳx(0) t∈[0, t]e δt d(y x (t), K(x)) 2

Proof. 2 andV 2 χ 4 |y| 2 , 1 1 + 4L 2 K

 22212 The right inequality in (43) follows from the boundsV 1 (x) ≤ c 2 |x| (x, y) ≤ c2 d(y, K(x)) 2 ≤ c2 (|y| + d(0, K(x))) 2 ≤ c2 (|y| + L K |x|) 2 ,where L K is the Lipschitz constant for K(•). Concerning the left inequality in (43), note that d(y, K(x)) ≥ |y| -L K |x|, where L K is the Lipschitz constant for K(•). Then, either |y| > 2L K |x|, in which case d(y, K(x)) > 1 2 |y| and V (x, y) ≥ |x| 2 + χd(y, K(x)) 2 > |x| 2 + or |y| ≤ 2L K |x|, in which case V (x, y) ≥ |x| 2 ≥ |(x, y)| 2 .

  |x(εh) -x(0)| = ε|(x(0), y(0))| O(h), |y(εh) -y(0)| = |(x(0), y(0))| O(h),where O(h) denotes a function bounded in absolute value by Ch, where the constant C does not depend on x(0), y(0), h, nor ε, for ε, h in a small right-neighborhood of zero. Knowing thatt → d( ẋ(t), A µ (t)x(t) + B(t)K(x(t))) ≤ B d(y(t), K(x(t)))is integrable, we deduce from Theorem 10.4.1 in[START_REF] Aubin | Set-Valued Analysis[END_REF] that there exists a solution x(•) of Σµ such that x(0) = x(0) and|x(εh) -x(εh)| ≤ B e ( Aµ +L K B )εh εh 0 d(y(s), K(x(s)))ds ≤ C 1 εh d(y(0), K(x(0))) + ε|(x(0), y(0))| O(h 2 ),for some C 1 > 0, where we have used the fact thatd(p, P ) ≤ |p -q| + d(q, Q) + d H (Q, P ), for p, q ∈ R m and P, Q ⊂ R m . Hence V 1 (x(εh)) -V 1 (x(h)) ≤ε|(x(0), y(0))| d(y(0), K(x(0))) O(h) + ε|(x(0), y(0))| 2 O(h 2 ),

  ))(e -γεh -1) + ε|(x(0), y(0))| 2 O(h 2 ) + ε|(x(0), y(0))| d(y(0), K(x(0))) O(h).

Moreover, V 2

 2 (x(εh), y(εh)) -V 2 (x(0), y(0)) ≤ |V 2 (x(εh), y(εh)) -V 2 (x(0), y(εh))| + |V 2 (x(0), y(εh)) -V 2 (x(0), ỹ(εh))| + V 2 (x(0), ỹ(εh)) -V 2 (x(0), y(0)),and ε is small enough. By a time-shift we obtain, for t ≥ 0, lim sup τ →0 +

where ỹ(•/ε) is the solution of Σ x(0) starting at y(0) and corresponding to the signal σ(•/ε). The first term is of order ε|(x(0), y(0))| 2 O(h), while, by (42),

from which one gets that |y(εh) -ỹ(εh)| = ε|(x(0), y(0

Summing up,

where C 2 , C 3 , C 4 do not depend on x(0), y(0), h, nor ε, for h, ε in a small right-neighborhood of zero. We have lim sup

We write the right-hand side as the sum of the three terms

where the inequalities in W 2 , W 3 are obtained assuming ε ≤ 2δ γ . For any given χ > 0, it is easy to see that

we get |(x(0), y(0))| 2 ≤ (1 + 2L 2 K )|x(0)| 2 + 2d(y(0), K(x(0))) 2 so that W 3 ≤ 0, provided that χ is chosen so that


Mohammad Ghalei

Parastoo Mahdi Khoshouei

Dr Johan Vandenborre
email: johan.vandenborre@subatech.in2p3.fr

François Guérard

Guillaume Blain

Mojtaba Zarei

Ferid Haddad

Massoud Fattahi

François Guerard

How radiolysis impacts astatine speciation?

Keywords: Astatine Chemistry, Chloroform, Radiolysis, Hydrogen Chloride, G-yield

This work deals with the impact of chloroform radiolysis on astatine chemistry. Astatine-211 is a promising radionuclide for the targeted alpha therapy of cancers. Yet, the production of 211 At based radiopharmaceuticals is often complicated by radiolysis due to its own decay, leading to solvent degradation and then impacting astatine chemical state and radiolabeling efficiency. A better understanding of the radiolysis phenomenon is thus necessary. Two experiments are performed: (1) pH measurement during the At extraction in chloroform medium, (2) pH determination calculated from the HCl chemical yield which is obtained during the radiolysis of chloroform medium by γ-ray.

In both cases, the pH values are determined at 2.0 ± 0.5. Then, it is possible to predict the acidic conditions during the 211 At production by HCl radiolytic yield determination. This work demonstrated that the radiolysis induced by At during its production in CHCl3 medium should be taken into account as it has a significant effect on acid-basic conditions.

The work helps understanding the mechanisms of production of a major impurity generated from 211 At radiolysis in chloroform which alters its radiochemistry, allowing for the identification of solutions to counteract these unwanted effects.

INTRODUCTION

Astatine-211 is a promising alpha-emitter radionuclide with a high potential for applications in targeted alpha therapy of cancers [START_REF] Guérard | Production of [211At]-Astatinated Radiopharmaceuticals and Applications in Targeted α-Particle Therapy[END_REF]. Its intermediate half-life (7.2 h) in comparison with other alpha emitters makes it potentially compatible with various carrier compounds, from small molecules and peptides to large proteins such as antibodies. Radiolabeling chemistry which is employed for 211 At in most cases is similar to other halogens, in particularly to radioiodine. Various strategies have been developed recently which provided biologically relevant molecules labelled with 211 At with high radiochemical yields [START_REF] Navarro | Prosthetic groups for radioiodination and astatination of peptides and proteins: A comparative study of five potential bioorthogonal labeling strategies[END_REF][START_REF] Reilly | Rapid Cu-Catalyzed [211At]Astatination and []Iodination of Boronic Esters at Room Temperature[END_REF][START_REF] Watanabe | A convenient and reproducible method for the synthesis of astatinated 4-[211At]astato-lphenylalanine via electrophilic desilylation[END_REF]). Yet, difficulties to obtain robust labelling procedures have been reported when using certain solvents particularly chloroform. Lack of robustness of radiolabeling approaches performed in chloroform, which is clearly evident at high radioactivity , has been attributed to solvent radiolysis, induced by high energy α-particles (5,9-7,4 MeV), which in turn comes from 211 At decay. Products of chloroform decomposition may be responsible for the reduction in RCYs due to their reactivity with the reaction precursors (Pozzi and Zalutsky, 2005b) and/or alteration of astatine species necessary for the radiolabelling reaction [START_REF] Aneheim | Towards elucidating the radiochemistry of astatine -Behavior in chloroform[END_REF]. Although other solvents (e.g. alcohols or acetonitrile) are now preferred for radiolabeling (in order to reduce issues related to radiolysis), chloroform remains an interesting solvent for astatine recovery after separation from its irradiated bismuth target. The main reason being that it is the only known solvent allowing evaporation to dryness without co-evaporation of astatine activity. This properties is highly interesting since it allows the fractionation of a batch of astatine into several samples that can be evaporated to dryness and then dissolve in any solvent adapted for each experiment [START_REF] Aneheim | Towards elucidating the radiochemistry of astatine -Behavior in chloroform[END_REF].

Thus, the study of chloroform radiolysis appears essential to better counteract issues associated with the use of this solvent.

A large number of reports have previously focused on chloroform radiolysis in various contexts and experimental conditions [START_REF] Abadie | Radiolysis of liquid chloroform in an oxygen free atmosphere[END_REF][START_REF] Bibler | Radiolysis of Chloroform in the Intense Radiation Pulse from a Nuclear Explosion[END_REF][START_REF] Chen | Radiolysis of Chloroform and Carbon Tetrachloride[END_REF][START_REF] Johnston | Effects of Temperature and Added Hexachloroethane on the Radiolyses of Carbon Tetrachloride and Chloroform[END_REF][START_REF] Ottolenghi | The Radiation Chemistry of Chloroform[END_REF][START_REF] Schulte | Chemical Effects Produced in Chloroform by γ-Rays1[END_REF][START_REF] Werner | Kinetics of the γ-Ray-Induced Decomposition of Chloroform1[END_REF]. These authors suggested that hydrogen chloride is the main product of chloroform radiolytic degradation. However, all these publications (from the 50's to the 80's) gave equivocal radiolytic yield (G-yield) of generated hydrogen chloride because of differences in the experimental conditions. Either chloroform solutions purity were different which implies discrepancies in G-yield of HCl [START_REF] Abadie | Radiolysis of liquid chloroform in an oxygen free atmosphere[END_REF][START_REF] Chen | Radiolysis of Chloroform and Carbon Tetrachloride[END_REF][START_REF] Johnston | Effects of Temperature and Added Hexachloroethane on the Radiolyses of Carbon Tetrachloride and Chloroform[END_REF][START_REF] Ottolenghi | The Radiation Chemistry of Chloroform[END_REF] or experimental conditions, for example various temperature, pressure or radiation dose rate [START_REF] Bibler | Radiolysis of Chloroform in the Intense Radiation Pulse from a Nuclear Explosion[END_REF][START_REF] Johnston | Effects of Temperature and Added Hexachloroethane on the Radiolyses of Carbon Tetrachloride and Chloroform[END_REF][START_REF] Schulte | Chemical Effects Produced in Chloroform by γ-Rays1[END_REF][START_REF] Werner | Kinetics of the γ-Ray-Induced Decomposition of Chloroform1[END_REF]. For these reasons, we planned to revisit the experimental measurements of HCl yield during γ-ray experiments, more easy to be measured and close to the α ones, in specifically well-defined conditions in order to establish the potential quantities of HCl in routine production of astatine-211 solution during radiolabeling and production of [ 211 At]-astatinated radiopharmaceuticals.

MATERIALS AND METHODS

Irradiation experiments

γ-Ray irradiations were performed on an gamma irradiator located at the ARRONAX cyclotron facility (Nantes -France). Irradiation cells made of PEEK (polyether ether ketone) with an internal volume of 42 ml were mounted with a rotulex 19/9 glass tube. These cells were gas-tightened using a screwed joint in a glass-metallic valve. Specificity of each experiment is discussed thereafter. Irradiation of different 20 ml-samples of chloroform was performed from 1 to 7 days under aerated or oxygen free atmospheres under normal pressure (760 mmHg).

γ-Ray irradiation γ-Ray irradiation was performed with a GSM D1 (Gamma-Service Medical) irradiator containing a 123 TBq 137 Cs source. This radionuclide decays to 137m Ba, which delivers 661.7 keV γ-Ray. A dose cartography performed by Fricke dosimetry [START_REF] Fricke | Chemical Dosimetry, Radiation Dosimetry[END_REF] inside gamma irradiation chamber, indicated that the average deposited dose inside the aqueous samples was between 7 and 9 Gy.min -1 . As γ-Ray is a penetrating radiation, the dose can be considered as homogeneously deposited inside the whole sample.

Analytical procedures

Solutions described in this section were prepared with commercial chemical products as received with no further purification. All reactants were analytical grade and the aqueous samples were prepared with ultrapure (MilliQ) water.

HCl analysis

To quantify HCl, produced by the irradiation of chloroform, it is necessary to extract HCl molecules from chloroform bywater. Extraction experiments were performed to obtain the percentage of HCl extracted from chloroform samples by water. Using 4 different concentrations of HCl in 20 ml of chloroform, samples prepared ahead of time (0.1, 0.2, 0.5 and 1 mol.l -1 by addition of 100 µl of aqueous HCl solution) and extraction protocol was performed with a large volume of Ultrapure water (100 ml). After separation of layers, titration of aqueous layer was performed using 0.1 mol.l -1 NaOH to determine the percentage of HCl extracted. Similarly, extraction and determination of HCl from irradiated samples of chloroform was done using a decanter followed by titration of acid with NaOH (Figure 1).

The results show on average 75 ± 8 % of HCl from the HCl-chloroform samples was extractable from chloroform to the aqueous phase. This ratio determined by the extraction step that used in this work to establish the realistic HCl concentration obtained during the radiolysis experiments. Therefore, we consider these uncertainties for all data plotted into the figures.

Radiolytic Yield Definition

The radiolytic yield is defined as the amount of a species that is formed or disappeared per unit of deposited energy. It is expressed in the international system in µmol. J -1 and is calculated at a time t after transition of the ionizing irradiation according to the following equation:

• • (•) = [••] ••
where [Xt] is the concentration of the species X at time of t (mol.l -1 ), ρ is the density of the irradiated solution (kg.l -1 ) and D the absorbed dose (J.kg -1 ).

Thus, for the determination of the radiolytic yield of HCl during the irradiation of chloroform, the equation is as following:

• (!"#) = [!"#] • $%$& ' . • In which • $%$& ' is equal to 1.48 kg.l -1 .

Production of astatine-211 solutions in chloroform

211 At was produced at Arronax cyclotron facility using the 209 Bi(α,2n) 211 At reaction and recovered from the irradiated target in chloroform using a dry-distillation protocol as reported previously [START_REF] Guérard | Bifunctional aryliodonium salts for highly efficient radioiodination and astatination of antibodies[END_REF]. The radioactive sample was isolated and shipped to CRCINA laboratory where radiolabelling chemistry is performed. On average there is 300 to 400 MBq activity in a total volume of about 600 µl of chloroform with a contact time of about 2 hours between the end of distillation and the HCl extraction experiments, which corresponds to a dose range of 2.3 to 3.0 kGy received by chloroform solution. Before radiolabeling, and approximately 2 h after recovering of 211 At activity in chloroform, the chloroform 211 At solution was washed with 200 µl of water (TraceSelect Grade, Fluka). After decantation, both layers were separated. The 211 At activity found in the aqueous layer was negligible, nearly all activity being found in the chloroform layer.

The pH of the aqueous layer was measured by mean of a pH indicator strip (Merck, pH range 0-6, pH gradation = 0.5). All assays indicated a pH of 1.5 ± 0.5 (n = 4). pH indicator was used to compare it with HCl titration result not as the tool for HCl analysis.

RESULTS AND DISCUSSION

In this study, we investigated the amounts of HCl produced through irradiation of chloroform by γ radiations in two different conditions (presence or absence of oxygen) at low doses of irradiation. The latter one was the dose rate corresponding to the real conditions of exposure of chloroform when performing [ 211 At]-astatinated radiopharmaceuticals for pre-clinical development.

Determination of initial G-values of HCl during irradiation in the aerated and oxygen free atmospheres

Figure 2 presents HCl production as function of dose in aerated and oxygen free atmospheres for CHCl3 irradiated by γ-ray.

According to the data derived from this figure, G-value for HCl in the dose range from 0 to 2580 Gy in the presence of air, is equal to 1.66 ± 0.24 µmol.J -1 whereas it is only 1.14 ± 0.17 µmol.J -1 in the presence of Ar. We can see a clear difference between the radiolytic yields of HCl production that can be explained by looking the possible reactions in both conditions.

For the Chloroform irradiation mechanisms, two different behaviors are possible:

(i) One of them, when air is present above the sample, are the chemical reactions of Chloroform with Oxygen in order to produce molecules of HCl, Cl3COOH and COCl2 which have been proposed in the primarily steps of chloroform irradiation [START_REF] Schulte | Chemical Effects Produced in Chloroform by γ-Rays1[END_REF]:

CHCl3 + O2 → Cl3COOH (1) 2CHCl3 + O2 → 2COCl2 + 2HCl (2)
However, reaction 2 can take place even without ionizing radiation, but it takes too long (some days) [START_REF] Clover | The auto-oxidation of chloroform[END_REF]. As a result, when measuring the radiolytic yield of HCl in presence of air, we consider the HCl concentration zero at no absorbed radiation dose.

HCl is produced through reaction (2) while Cl3COOH is produced through reaction ( 1).

(ii) The second one, in a free-oxygen environment, is the direct radiolysis of the CHCl3 molecules to produce HCl via H° and Cl° radicals production [START_REF] Chen | Radiolysis of Chloroform and Carbon Tetrachloride[END_REF]:

CHCl3 → CHCl2 ° + Cl°. (3) CHCl3 + Cl °→ CCl3 ° + HCl (4) or CHCl3 → CCl3 ° + H°. ( 5
)
CHCl3 + H °→ CHCl2 ° + HCl (6)

Radiolytic products such as dihydrogen (H2) and dichloride (Cl2) gases result from recombination of H ° and Cl ° radicals, respectively [START_REF] Lainé | Reaction mechanisms in swelling clays under ionizing radiation: influence of the water amount and of the nature of the clay mineral[END_REF][START_REF] Mostafa | Chlorine species evolution during electrochlorination on boron-doped diamond anodes: In-situ electrogeneration of Cl2, Cl2O and ClO2[END_REF].

This implies that HCl and radiolytic gases compete for formation reactions, which needs to be measured in further studies.

In addition, recent studies by using pulse radiolysis shown that if surrounded chloroform molecules participate with the CHCl +° the reaction ( 7) and ( 8) are more favorable [START_REF] Bird | Pushing the limits of the electrochemical window with pulse radiolysis in chloroform[END_REF]:

CHCl +° + CHCl3 → CCl3 + + CHCl2 ° + HCl (7)
CHCl +° + CHCl3 → CHCl2 + + CCl3 ° + HCl (8)

At low doses, typically until 3000 Gy, in the first steps of chloroform irradiation, Cl3COOH

(see reaction (1)) has been considered as a major product which produces quantitative amount of peroxide before its reduction [START_REF] Schulte | Chemical Effects Produced in Chloroform by γ-Rays1[END_REF]. According to our results at low doses (< 500 Gy) a difference is observed for the radiolytic yield of acids with and without air. This difference was increased when the dose was increased which means that reactions 2 and 4 are occurring in parallel. It can be concluded that the higher the dose, the more important the effect of oxygen.

range (< 3 kGy) G-value = 1.66 µmol.J -1 while at high range (< 93 kGy) G-value = 0.66 µmol.J -1 . It should be noted that the G-value at higher doses was calculated from obtained data in dose range of 3-93 kGy (cf figure 3).

That is the reason why, we have chosen to use the same dose range (< 3 kGy), determined in the experimental section, for the radiolysis and the 211 At extractions experiments in order to determine the realistic quantity of HCl produced by radiolysis which comes from 211 At.

Therefore, in the next part, we compare the HCl production for the real dose deposited in chloroform solution defined in the 211 At extraction (between 2.3 and 3 kGy), which corresponds to a contact time between At and chloroform solutions of 2 hours, with our γ irradiation conditions in a close range of dose rate in the both cases (27 Gy.min -1 for 211 At extraction vs. 10 Gy.min -1 in the radiolysis ones). However, the possibility of differences in HCl production between γ and α radiolysis may be considered in the context of radiolysis arising from astatine-211. Nonetheless, according to reactions (3-6), HCl production depends on the radicals produced by radiolysis (H°, Cl°). As the linear energy transfer (LET) in α radiation is higher than γ one, therefore the produced radicals are combined faster in spurs therefore, the molecular yield is higher than radical yield in α radiation [START_REF] Appleby | Radical and molecular yields in water irradiated by g-rays and heavy ions[END_REF]. That is the main difference between the two types of radiation. Consequently, by considering this difference, the HCl production is, in theory, higher in the case of γ-ray radiolysis and the G-value determined in this work is overestimated in comparison with the one actually occurring during exposition of chloroform to 211 At. In order to determine this overestimation, we have measured the HCl production by extraction from batches of 211 At in chloroform.

Impact/Consequences for Astatine Extraction/Complexation

Being a radioelement, with only short half-life radioisotopes, astatine chemistry is extremely complex to study. Most work with astatine has been performed with 211 At (t1/2 = 7.2 h), the most stable isotope, 210 At, exhibiting a t1/2 of only 8.1 h being avoided for radioprotection concerns. Additionally, the weak amounts produced by accelerators (few pictograms at best), make it nearly impossible to use conventional spectroscopic techniques for precise determination of chemical properties of astatinated species. A major issue in studying astatine chemistry is the evolution over time of the astatinated species in solution, including their oxidation state, which affects the reproducibility of radiolabelling procedures. This phenomenon has been attributed to radiolysis of the solvent, which generates degradation products that may then react with astatine and modify its chemical nature. In the case of chloroform, previous reports have suggested that radicals formed during radiolysis could impact radiolabelling procedures (eg Cl ° that may react with precursors instead of astatine and then decrease radiochemical yields (Pozzi and Zalutsky, 2005a), or that recombination of radicals can form species suspected to react with astatine, or adducts affecting its behavior (volatility, extractability, solubility) [START_REF] Aneheim | Towards elucidating the radiochemistry of astatine -Behavior in chloroform[END_REF].

Here we focused our attention on HCl production upon chloroform radiolysis since acidity of the media appears as an essential parameter governing the astatine species in solution as shown in recent reports coupling experimental data with computer simulations to construct the Pourbaix diagram of astatine in water [START_REF] Champion | Investigation of Astatine(III) Hydrolyzed Species: Experiments and Relativistic Calculations[END_REF][START_REF] Sergentu | Advances on the Determination of the Astatine Pourbaix Diagram: Predomination of AtO(OH)2-over At-in Basic Conditions[END_REF], and that may impact its use for analysis and radiolabeling chemistry. It is assumed that in acidic and oxidative medium, three species coexist (At + , AtO + , and AtOOH), the predominant species being in part determined by the pH of the solution.

Considering this, and since concentration of HCl varies over time in chloroform it is clear that experiments conducted with astatine isolated in chloroform are likely to lack difference between γ and α irradiations and the production of other radiolytic species, which contribute to the acido-basic conditions and must be characterized such as

Cl3COOH. Experiment performed in further studies would be the α irradiation of chloroform solutions using a helium beam to measure the G-yield in these specific conditions and compare them with astatine irradiations (α in the both cases) and the analysis by Infrared Spectrometry of other radiolytic species produced during chloroform irradiation.

Our results may also explain why different oxidation states of 211 At exist in chloroform at the same time [START_REF] Aneheim | Towards elucidating the radiochemistry of astatine -Behavior in chloroform[END_REF]. It is helpful to measure the radiolytic yield of H2 and Cl2 in order to better understand the competition of HCl formation reaction with those gases, as H° and Cl° radicals are involved in these reactions. Moreover, based on this experiment, we would then recommend washing astatine solution with water in order to improve consistency before using for radiolabeling.

Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐ The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

• Parastoo Mahdi Khoshouei and Guillaume Blain performed the experiments.

• Johan Vandenborre and Mohammad Ghalei wrote the manuscript with contribution from all co-authors. • All co-authors contributed to the discussion and interpretation of the data.

  consistency and reproducibility due to the dose deposited in stock solutions when activity in solution and time before radiolabeling and use vary. The solution to this problem is simple to set up, by adding an extraction step in water before use for radiolabeling.Extraction experiments discussed above showed that most of HCl could be extracted from chloroform to water, which was confirmed when applied to 211 At stock solutions. This additional step in the purification process of 211 At is time-consuming (< 10 min) and generates negligible loss of activity due to the extremely low transfer of astatine to aqueous layer. From the G-yield value determined above (1.66 µmol.J -1 in the dose range studied), we can estimate the HCl concentration for the dose applied to the chloroform corresponding to the time between the isolation of astatine in chloroform, to the extraction step (between 2.3 and 3 kGy). The calculated range of HCl concentration is between 3.8 and 5 mmol.L -1 (i.e. pH = 2.3 ± 0.1) close to the real pH value measured in the Astatine extraction experiments (= 1.5 ± 0.5). The difference can be attributed to the G-values

Figure 1 :Figure 2 :

 12 Figure 1 : Amount of HCl extracted in water from chloroform Figure 2 : Experimental G-yield values of HCl), produced during irradiation of CHCl3 in aerated (red spots) and oxygen free (blue ones) atmospheres by 137 Cs-source irradiator Eγ = 0.667 MeV, DR = 7-9 Gy.min -1 , Total Dose = 2580 Gy. The obtained data are shown

Figure 3 :

 3 Figure 3 : HCl concentration, produced during irradiation of CHCl3 in aerated atmosphere by 137 Cs-source irradiator Eγ = 0.667 MeV, DR = 7-9 Gy.min -1 , Total Dose = 93300 Gy.

Figure 2 :

 2 Figure 2 : Experimental G-yield values of HCl), produced during irradiation of CHCl3 in aerated (red spots) and oxygen free (blue ones) atmospheres by 137 Cs-source irradiator Eγ = 0.667 MeV, DR = 7-9 Gy.min -1 , Total Dose = 2580 Gy. The obtained data are

Figure 3 :

 3 Figure 3 : HCl concentration, produced during irradiation of CHCl3 in aerated atmosphere by 137 Cs-source irradiator Eγ = 0.667 MeV, DR = 7-9 Gy.min -1 , Total Dose = 93300 Gy.

TABLE and FIGURESTable 1 :

 and1 G-yield values of HCl from chloroform irradiation in literature and this work

		0.02
		0.018
	Extracted HCl(mol)	0.016 0.006 0.008 0.01 0.012 0.014		y = 0.7551x R² = 0.9915
		0.004
		0.002
		0
		0	0.005	0.01	0.015	0.02	0.025
			Initial HCl in Chloroform(mol)
	Figure 1 : Amount of HCl extracted in water from chloroform
		0.008
		0.007
	[HCl]/(mol.l -1 )	0.006 0.003 0.004 0.005		y = 2.45E-06x R² = 9.91E-01
		0.002				y = 1.69E-06x
		0.001				R² = 9.74E-01
		0
		0	500	1000	1500	2000	2500	3000
					Dose(Gy)
	Dose(kGy)	•. !	#$ With Ar	%. & -((. ) $& -$( With the air	(& -%$	*#. #
	G(HCl)	0.45	1.18	0.59		1.23	0.55	0.82
	(µmol.J -1 )
	Ref.		(Abadie,	(Chen	(Ottolenghi	(Johnston	(Werner	This Work
			1982)	et al.,	and Stein,	et al.,	and
				1960)	1961)		1961)	Firestone
								, 1965)

ACKNOWLEDGEMENTS

This work has been supported in part by a grant from the French National Agency for Research called "Investissements d''Avenir", IRON Labex no. ANR-11-LABX-0018-01 and Equipex ArronaxPlus n°ANR-11-EQPX-0004 and ISITE NExT n°ANR-16-IDEX-0007. The authors would like to express their gratitude for the ARRONAX facility teams for their technical support.

Irradiation of chloroform at higher doses

The results obtained in this study was over a wide range of doses, from 0 to 93 kGy, (see Figure 3). This dose range corresponds to irradiation times of chloroform in our gammairradiator varying from 0 to 10 days.

Based on the results reported in Figure 3 and using the reactions (2-6), the radiolytic yield of HCl production in the dose range of 0-93294 Gy has been determined as 0.82 ± 0.11 µmol.J -1 . Moreover, as already suggested in the previous part, two behaviors append during the HCl formation by chloroform irradiation: (i) in presence and absent of oxygen (ii) in the high dose range (> 3 kGy) the radicals H° and Cl° production by the direct radiolysis of CHCl3 (reactions (3-6)).

The mechanism of abstraction of H and Cl from CHCl3 during dose enhancement, suggested in this work, is consistent with literature [START_REF] Chen | Radiolysis of Chloroform and Carbon Tetrachloride[END_REF]. Authors have proposed that the abstraction of H and Cl atoms from CHCl3 molecules, by Cl ° and H ° radicals, can be considered as the major reactions producing HCl during the irradiation in the absence of oxygen by γ-rays [START_REF] Chen | Radiolysis of Chloroform and Carbon Tetrachloride[END_REF].

Most of the reported results regarding irradiation of chloroform in the literature have been obtained in O2 and water-free situations as in our conditions. However, the literature shows a large discrepancy (see Table 1).

All authors have underlined the discrepancy of the G-values (from 0.45 to 1.23 µmol.J -1 ).

Then, we can explain the difference by temperature, pressure conditions and HCl production behavior due to the impact of oxygen as already suggested. In addition, the range of dose studied in previous works may explain this discrepancy. For example in our study, a neat difference of G-value is observed as a function of the dose range: for low

CONCLUSION

The gamma radiolysis of chloroform using a 137 Cs source was investigated in this study in order to determine the amount of hydrogen chloride (HCl) produced as a major product of chloroform radiolysis. Radiation doses ranged from 0 to 93 kGy, which had an impact on the chemistry of Astatine. According to our finding, during the accumulation of the dose, the amount of HCl acid follows a linear behavior at low dose with a G-value of (1.66 µmol.J -1 ) and a linear behavior at high dose with a G-value of (0.82 µmol.J -1 ). This different behavior at low and high dose may explain the large discrepancy found in the literature as a function of the dose range studied. Moreover, in the low dose range (< 3kGy), the trichlorohydroperoxymethane (Cl3COOH) production by chloroform reaction with O2, must be taken into account for the mechanisms of HCl production.

These experimental results contribute in understanding of chloroform radiolysis mechanisms induced by astatine radioactivity during its extraction/complexation processes. In fact, with the G-value determined in this work we calculated the theoretical pH value in the Chloroform/Astatine solution (pH = 2.3) close to the one measured in the real Astatine complexation/extraction process (pH= 1.5). Then, as a conclusion, the radiolysis mechanism which can explain the acidic pH in the chloroform/astatine solution must be taken into account for this kind of radiochemical experiments as already suggested but not, until now, demonstrated in the literature [START_REF] Aneheim | Towards elucidating the radiochemistry of astatine -Behavior in chloroform[END_REF]Pozzi andZalutsky, 2005a, 2017). Moreover, synthesis of 211 At radiopharmaceuticals such as SAB (N-succinimidyl 3-[ 211 At]-astatobenzoate) suffers from chemical decomposition (Pozzi and Zalutsky, 2005a) and formation of HCl by radiolysis may explain this problem.