Lucie Rolland
email: lucierolland24@gmail.com

Aurelie Amar-Hoffet

Vanessa Lubin

Lise Pr

Laura Miquel

Blandine Courbiere

Unexplained infertility: A French national survey of clinical practices

Keywords:

Introduction:

The aim of our study was to carry out a national survey of French practitioners to evaluate (i) their diagnostic criteria for making a diagnosis of unexplained infertility (UEI) and (ii) their management strategy when facing UEI. Materials and Method: An online questionnaire comprising ten multiple-choice questions was sent by mail to French reproductive practitioners in 80 fertility centres. Results: The response rate was 59.6% (195/327). Post coital testing was always or often prescribed by 14.8% of respondents (n = 36). Chlamydia trachomatis testing was never prescribed by 31.7% (n = 59) of them, 30.2% prescribed a pelvic MRI in cases of UEI and 18.4% (n = 33) always or often performed laparoscopy. For 87.6% (n = 169), advanced maternal age was always or often an indication of first-line IVF, with an average threshold of 37.4 years. For 68.6% (n = 129), diminished AMH was an indication for first-line IVF, with an average AMH threshold of 1.2 ng/ml. With respect to the management of UEI, we did not observe a consensus between the strategies of 2 to 6 intrauterine insemination cycles before IVF or IVF as the first-line treatment. Conclusion: There is no consensus in France on what tests should or should not be carried out to conclude UEI, and there is also no consensus on the management of UEI. UEI is one of the top 10 priorities for future infertility research. The diagnostic criteria must be standardized to enable the comparison of studies on this topic as well as to improve the translation of research into clinical practice.

Introduction

Unexplained infertility (UEI) is a diagnosis made by "exclusion" for couples who have not conceived and for whom standard investigations have not detected any male or female disease of the reproductive system. According to literature data, UEI affects approximately 15-50% of couples consulting for infertility [START_REF] Buckett | The management of unexplained infertility: an evidencebased guideline from the Canadian Fertility and Andrology Society[END_REF][START_REF]Evidence-based treatments for couples with unexplained infertility: a guideline[END_REF]. Based on the latest guidelines in the UK, Canada, USA and Cochrane [START_REF] Buckett | The management of unexplained infertility: an evidencebased guideline from the Canadian Fertility and Andrology Society[END_REF][START_REF]Evidence-based treatments for couples with unexplained infertility: a guideline[END_REF][START_REF]'s. Fertility: assessment and treatment for people with fertility problems[END_REF][START_REF] Wang | Interventions for unexplained infertility: a systematic review and network meta-analysis[END_REF], there is not a consensus in the literature regarding the methods for diagnosing UEI. For most authors, the definition of UEI is based only on the presence of normal ovulatory function, a normal semen analysis and at least one patent fallopian tube. Many factors are not taken into consideration for the definition of UEI, such as advanced maternal age (AMA), diminished ovarian reserve (DOR), body mass index, environmental exposure, tobacco, alcohol, or drug use, and sexual dysfunction. The management of couples consulting for UEI also varies from country to country. There is no consensus in France on what tests should or should not be carried out to conclude a diagnosis of UEI nor on the management of UEI. Indeed, there is a lack of clinical evidence to affirm the best management strategy between expectant attitude, IUI, and IVF +/-ICSI. Recently, UEI has been identified as one of the top 10 priorities for future infertility research [START_REF] Duffy | Top 10 priorities for future infertility research: an international consensus development studyyz[END_REF]. The diagnostic criteria must be standardized to enable a comparison of studies on this topic and to improve the translation of research into clinical practice.

The aim of our study was to carry out a national survey of French practitioners to evaluate (i) their diagnostic criteria for making a diagnosis of UEI and (ii) their management strategy for cases with a UEI diagnosis.

Materials and method

Population

We conducted a national survey via an online questionnaire among all practitioners certified in reproductive medicine who work in connection with assisted reproductive technology (ART) centres. For inclusion in the survey, practitioners must prescribe infertility tests in daily practice and perform at least intrauterine inseminations.

Online survey

The questionnaire was designed using an online survey software tool, Survey Monkey Ò (San Mateo, California, USA). All French clinics licenced by national authorities to perform assisted reproductive technologies (ARTs) were contacted via the head of their department. A link to the survey was emailed with an explanatory cover letter. The survey began in December 2020, with an email reminder sent out a week later. The last of the completed surveys was received in June 2021. The questionnaire comprised ten short multiple-choice questions with a section for additional comments (Annex 1). The total time to complete the survey was between 8 and 10 min. Before being sent to participants, the questionnaire was tested and revised by five fertility consultants. The questionnaire focused on the definition, investigation and medical management of UEI. To obtain a response rate, we asked each department head to whom a questionnaire was sent the number of physicians to whom he transmitted the questionnaire. The data were analysed using Excel software Ò (Microsoft). Only descriptive statistics were calculated. Continuous variables were summarized using the following statistics: n (sample size), mean and percentage.

Ethical considerations

Ethical consent was not required for this survey as the findings were kept anonymous, and there was a link to opt out of the survey.

Results

Of the 104 ART centres identified, 80 centres were contacted by email to obtain the email of the head of the department, and 24 centres were contacted by email or phone without response. Of the 80 centres that responded, 49 agreed to participate in the survey. The questionnaire was sent to 327 French practitioners, with a response rate of 59.6% (n = 195). Among the respondents, 18% (n = 35) were male; 82% (n = 160) were female. Question 2 explored the medical specialization of physicians. 57.4% (n = 112) of practitioners were obstetrician-gynecologist, 33.3% (n = 65) were medical gynecologist, 8.2% (n = 16) were endocrinologist and 1% (n = 2) were biologist. A total of 27.7% (n = 54) of participants were working in a private ART center, 54.4% (n = 106) were working in a university teaching hospital, 8.2% (n = 16) were working in a non-university public hospital and 9.7% (n = 19) had a mixed public and private activity. The average time of practice was 12.6 years, with a median of 10 years (min: 0.5 years, max: 45 years). Responses to questions 5, 6, 7, 8, and 9 are summarized in. Tables 1, 2, 3, 4, and 5, respectively. Responses to question 10 are summarized in Table . 6.

Which tests must be performed before concluding UEI?

The tests that are most prescribed by the responding practitioners before the diagnosis of UEI are AMH (87.6%, n = 169), ultrasound of antral follicles (95.9%, n = 187), hysterosalpingography (92.8%, n = 181) and spermogram (100%, n = 195) with total mobile sperm count (78.1%, n = 146) (Table . 1). Post coital testing is always or often prescribed by 14.8% (n = 26) of practitioners. Among respondents, 31.7% (n = 59) never prescribe chlamydia trachomatis testing, while hysterosalpingography is prescribed by 92.8% (n = 181). In our study, 30.2% (n = 59) of the practitioners prescribe an MRI in cases of UEI, and 18.4% (n = 33) of respondents always or often perform laparoscopy in cases of UEI.

Do environmental exposure factors rule out a UEI diagnosis?

In our study, 34.9% (n = 68) of respondents always or often exclude the diagnosis of UEI for women with a BMI > 30, and 57% (n = 110) discard the diagnosis of UEI for BMI > 35. A total of 43.1% (n = 74) of respondents always or often dismiss the diagnosis of UEI if women smoke more than 10 cigarettes per day (Table . 2). Should the management of UEI depend on maternal age?

Among respondents, 51.8% (n = 100) always or often exclude the diagnosis of UEI when the woman is over 38 years old (Table . 2). At the age of 28 and after one year of UEI, 63.8% (n = 120) of respondents always or often perform IUI (Table. 4). When the woman is 39 years old, IVF is always or often prescribed as the first-line treatment by 78.5% (n = 150) of the respondents (Table. 5).

For 87.6% (n = 169) of the practitioners, AMA is always or often an indication of first-line IVF, with an average threshold of 37.4 years (median: 38 years, min: 30 years, max: 41 years) (Table . 6).

Should the management of UEI depend on AMH?

For respondents, the management strategy also depends on AMH: 74.5% (n = 143) always or often dismiss a UEI diagnosis if AMH < 1 ng/ml (7.1 pmol/L) (Table . 2). For 68.6% (n = 129), a diminished or low AMH is an indication for a first-line IVF, with an average AMH threshold of 1.2 ng/ml (8.6 pmol/L) (median: 1 ng/ml (7.1 pmol/L), min: 0.5 ng/ml (3.6 pmol/L), max: 2 ng/ml (14.3 pmol/L)) (Table. 6).

UEI: expectant management, IUI or IVF first?

Out of 195 respondents to question 7 concerning the time of regular intercourse, 78% (n = 152) wait for at least one year of regular intercourse before initiating ART management when women are between 18 and 20 years of age. In contrast, 57.4% (n = 112) do not wait for more than 6 months for women between 36 and 39 years of age; this rate rose to 66.3% (n = 128) for patients over 40 (Table . 3).

For 74% (n = 142) of the practitioners, a long time to conceive is always or often an indication for IVF as a first intention, with an average delay of 2.75 years (median: 3 years, min: 1 year, max: 6 years) (Table .

6).

Concerning the number of IUIs performed before IVF, 96% (n = 185) of the practitioners responded that the failure of one or more IUIs is an indication for IVF as a second step, and the average number of attempts is 3.65 IUIs (median: 4 IUIs, min: 2 IUIs, max: 6 IUIs) (Table 10). UEI is an indication for first-line IVF for 5.8% (n = 11) of the physicians questioned (Table. 6).

Discussion

Need of consensus to define UEI diagnosis

UEI is a diagnosis of exclusion that depends on the investigations undertaken. For WHO, infertility is defined by the failure to achieve a pregnancy after 12 months or more of regular unprotected sexual intercourse. The diagnostic testing required to meet the definition of UEI that is most commonly described in the literature is based only on the presence of normal ovulatory function, a normal semen analysis and at least one patent Fallopian tube [START_REF] Buckett | The management of unexplained infertility: an evidencebased guideline from the Canadian Fertility and Andrology Society[END_REF][START_REF]Evidence-based treatments for couples with unexplained infertility: a guideline[END_REF][START_REF]'s. Fertility: assessment and treatment for people with fertility problems[END_REF][START_REF] Wang | Interventions for unexplained infertility: a systematic review and network meta-analysis[END_REF][START_REF] Ray | Unexplained infertility: an update and review of practice[END_REF]. For the American Fertility Society (AFS), UEI could be a diagnosis even in cases of mild endometriosis (AFS criteria I) or early-stage endometriosis [START_REF]Evidence-based treatments for couples with unexplained infertility: a guideline[END_REF][START_REF] Wang | Interventions for unexplained infertility: a systematic review and network meta-analysis[END_REF]. Before making a diagnosis of UEI, most French reproductive physicians who participated in this survey perform an evaluation of the ovarian reserve (by AMH and ultrasound of antral follicles), the tubal permeability and the sperm. More than 30% of physicians never prescribe chlamydia trachomatis testing, while hysterosalpingography is prescribed by more than 92% of practitioners. The major sequelae of C. trachomatis infection in women are tubal factor infertility and tubal ectopic pregnancy. A Chlamydia antibody test (CAT) could be interesting if anamnesis provides indications for tubal pathology, such as (recurrent) previous infections or lower abdominal surgery [7]. The NICE guidelines recommend screening for Chlamydia trachomatis before undergoing uterine instrumentation. Prophylactic antibiotics should be considered before uterine instrumentation if screening has not been carried out [START_REF]'s. Fertility: assessment and treatment for people with fertility problems[END_REF]. There is a lack of valid evidence on the attributable risk of post-infective tubal factor infertility after genital chlamydial infection [START_REF] Wallace | What is the excess risk of infertility in women after genital chlamydia infection? A systematic review of the evidence[END_REF][START_REF] Gottlieb | Screening and treatment to prevent sequelae in women with Chlamydia trachomatis genital infection: how much do we know?[END_REF].

14.8% of French practitioners still prescribe post coital testing of cervical mucus. However, the routine use of post coital testing of cervical mucus in the investigation of fertility problems is no longer recommended in many guidelines because of the low predictive value for the pregnancy rate [START_REF]'s. Fertility: assessment and treatment for people with fertility problems[END_REF][START_REF]Coll ege National des Gyn ecologues et Obst etriciens Français[END_REF].

What pelvic exploration is required before reaching a UEI diagnosis?

In the UK, the Royal College of Gynaecology and Obstetrics recommends that women who are not known to have comorbidities (such as pelvic inflammatory disease, previous ectopic pregnancy or endometriosis) are screened for tubal occlusion using hysterosalpingography (HSG) because it is a reliable test for ruling out tubal occlusion, is less invasive and makes more efficient use of resources than laparoscopy [START_REF]'s. Fertility: assessment and treatment for people with fertility problems[END_REF]. There are still insufficient data for recommending hyfosy instead of hysterosalpingography [START_REF] Van Rijswijk | The FOAM study: is Hysterosalpingo foam sonography (HyFoSy) a costeffective alternative for hysterosalpingography (HSG) in assessing tubal patency in subfertile women? Study protocol for a randomized controlled trial[END_REF][START_REF] Exalto | Clinical Aspects of HyFoSy as Tubal Patency Test in Subfertility Workup[END_REF].

In our study, 18.4% of practitioners always or often perform laparoscopy to rule out subclinical endometriosis. In the absence of evidence for tubal or other pelvic pathologies (history of complicated appendicitis, pelvic surgery, pelvic inflammatory disease, clinical evidence of endometriosis, seropositivity for chlamydia and/or the presence of hydrosalpinx), laparoscopy is not systematically warranted before making a UEI diagnosis [START_REF] Buckett | The management of unexplained infertility: an evidencebased guideline from the Canadian Fertility and Andrology Society[END_REF][START_REF]'s. Fertility: assessment and treatment for people with fertility problems[END_REF][START_REF]Coll ege National des Gyn ecologues et Obst etriciens Français[END_REF]. In our study, 30.2% of the practitioners questioned prescribe an MRI in cases of UEI. Data are still insufficient to assess the value of MRI and laparoscopy in diagnosing UEI [START_REF] Badawy | Laparoscopy-or not-for management of unexplained infertility[END_REF].

Age-specific management of UEI

Age-related infertility is one of the top 10 research priorities for females and UEIs [START_REF] Duffy | Top 10 priorities for future infertility research: an international consensus development studyyz[END_REF]. Female age is a strong predictor of both natural and treatment-related live birth rates, with rates decreasing at age 35 years in ART [START_REF] Kamel | Management of the infertile couple: an evidence-based protocol[END_REF]; for spontaneous pregnancies these rates begin to decrease at age 30 [START_REF] Hunault | Prospective validation of two models predicting pregnancy leading to live birth among untreated subfertile couples[END_REF].

For 87.6% of the practitioners, maternal age is an indication of IVF as a first-line treatment, and this age is, on average, 37.4 years. However, IVF has not been shown to be the treatment for age-related infertility [START_REF] Xu | Cumulative live birth rates in more than 3,000 patients with poor ovarian response: a 15-year survey of final in vitro fertilization outcome[END_REF]. In several studies, neither IUI nor IVF showed superiority in cases of age-related reduction of ovarian reserve [START_REF] Freour | IVF conversion to IUI in poor responders: an observational study[END_REF][START_REF] Dosso | Impact of serum Anti-Mullerian Hormone levels on the results of assisted reproductive technologies. Single-center retrospective study from 2011 cycles (ICSI and bilateral tubal obstruction excluded)[END_REF][START_REF] Fujii | Conversion to IUI versus continuance with IVF in low responder patients: a systematic review[END_REF]. Some authors currently recommend a shorter period of expectant management for women who are 36 years or older [START_REF]'s. Fertility: assessment and treatment for people with fertility problems[END_REF]. UEI for more than 3 years would be a poor prognostic feature for future chance of pregnancy [START_REF] Isaksson | Present concept of unexplained infertility[END_REF].

Should AMH guide the management strategy for UEI?

We observed that 75% of practitioners consider an AMH rate <1 ng/ml to exclude the diagnosis of UEI. For 68.6% of participants, a reduction in AMH is an indication for first-line IVF, with an average AMH level of 1.2 ng/ml (8.6 pmol/L) in the UEI. Although AMH is considered the best marker of the ovarian reserve, the existing literature does not support the use of AMH as a marker of reproductive potential in the general population [START_REF] Dewailly | The physiology and clinical utility of anti-Mullerian hormone in women[END_REF][START_REF] Hawkins Bressler | Anti-M€ ullerian hormone as a predictor of reproductive potential[END_REF]. Several studies have reported a positive association of AMH levels with pregnancy rates after IVF treatment [START_REF] Tal | Antim€ ullerian hormone as predictor of implantation and clinical pregnancy after assisted conception: a systematic review and meta-analysis[END_REF]. Nevertheless, other studies reported a weak or nonsignificant association between AMH levels and a natural or medically assisted pregnancy outcome [START_REF] Alson | Anti-m€ ullerian hormone levels are associated with live birth rates in ART, but the predictive ability of anti-m€ ullerian hormone is modest[END_REF][START_REF] Tal | Antim€ ullerian hormone as a predictor of live birth following assisted reproduction: an analysis of 85,062 fresh and thawed cycles from the Society for Assisted Reproductive Technology Clinic Outcome Reporting System database for 2012-2013[END_REF]. Thus, if only AMH is decreased in the situation of apparent UEI, there is no evidence that AMH levels should guide management towards IVF or IUI.

Considering environmental exposures before concluding UEI

Studies defining the criteria to diagnose UEI generally do not consider the deleterious effects of environmental factors such as obesity and smoking.

In our study, 35.5% of respondents always or often exclude the diagnosis of UEI in women with a BMI > 30 kg/m 2 , and 58.6% discard the diagnosis of UEI for BMI > 35 kg/m 2 .

For a BMI above 29 kg/m 2 , every one point increase in BMI results in a 4% decrease in the chance of pregnancy (HR: 0.96; 95% CI: 0.91 to 0.99) [START_REF] Van Der Steeg | Obesity affects spontaneous pregnancy chances in subfertile, ovulatory women[END_REF]. Severe obesity (> 35 kg/m 2 ) reduces the chances of a live birth by a factor of 2 compared to women with BMI < 25. [START_REF]'s. Fertility: assessment and treatment for people with fertility problems[END_REF][START_REF]Coll ege National des Gyn ecologues et Obst etriciens Français[END_REF]. Similarly, for couples undergoing IVF, a decreased live birth rate following IVF was observed in obese (BMI ≥ 30 kg/m2) women compared with normal weight (BMI 18.5-24.9 kg/m 2 ) women [START_REF] Sermondade | Female obesity is negatively associated with live birth rate following IVF: a systematic review and meta-analysis[END_REF].

A total of 44.3% of respondents dismiss the diagnosis of UEI if women smoke more than 10 cigarettes per day. Tobacco is harmful to natural fertility with the consequence of prolonging the time for conception and reducing the chances of a live birth. This deleterious effect appears to be dose dependent [START_REF] Penzias | Smoking and infertility: a committee opinion[END_REF]. The environmental impact on fertility is not yet clearly defined, even if its implication is strongly suspected and increasingly studied [START_REF] S R | Lifestyle, and Female Infertility[END_REF]. In apparent UEI, screening for exposure to reprotoxic agents should be carefully performed.

What ART strategy for managing UEI?

For couples with UEI, Wang et al. concluded that there was insufficient evidence of differences in terms of live birth rates between expectant management and IVF/intracytoplasmic sperm injection (ICSI), ovarian stimulation (OS) with IUI or OS with sexual intercourse [START_REF] Wang | Interventions for unexplained infertility: a systematic review and network meta-analysis[END_REF]. The management of infertile couples should depend on the synthesis of multiple factors, including prognostic factors (age, duration of infertility, ovarian reserve), cost and side effects of treatments.

Clomiphene citrate or gonadotropins alone with timed intercourse should not be offered to couples with UEI. [START_REF]Evidence-based treatments for couples with unexplained infertility: a guideline[END_REF][START_REF]'s. Fertility: assessment and treatment for people with fertility problems[END_REF][START_REF] Wang | Interventions for unexplained infertility: a systematic review and network meta-analysis[END_REF]. The North American guidelines state that the best initial therapy for couples unsuccessful in achieving a pregnancy is 3 or 4 cycles of OS/IUI followed by IVF [START_REF] Buckett | The management of unexplained infertility: an evidencebased guideline from the Canadian Fertility and Andrology Society[END_REF][START_REF]Evidence-based treatments for couples with unexplained infertility: a guideline[END_REF]. For them, there is good evidence that immediate IVF in women >= 38 years of age may be associated with a higher pregnancy rate and shorter time to pregnancy compared to a strategy consisting of OS with IUI treatments prior to IVF [START_REF]Evidence-based treatments for couples with unexplained infertility: a guideline[END_REF]7] The NICE guidelines recommend first-line IVF after 2 years of regular sexual intercourse without pregnancy and to provide care without delay if maternal age is > 36 [START_REF]'s. Fertility: assessment and treatment for people with fertility problems[END_REF]. The Cochrane analysis concluded that there was no superiority in live birth rates between expectation, OS and IUI and IVF +/-ICSI. For couples with bad prognostic factors as determined according to the Hunault 2010 criteria (including age, duration of infertility, number of mobile spermatozoa and primary or secondary infertility) [START_REF] Hunault | Prospective validation of two models predicting pregnancy leading to live birth among untreated subfertile couples[END_REF], management by IUI or IVF is probably of value [START_REF] Wang | Interventions for unexplained infertility: a systematic review and network meta-analysis[END_REF]. However, in a cohort study of UEI with women aged ≤ 40, we previously reported that no clinical characteristics could identify favourable or unfavorable prognostic factors before starting ART [START_REF] Bosselut | Facteurs pronostiques des chances de naissance vivante en f econdation in vitro pour infertilit e inexpliqu ee : etude de cohorte[END_REF]. There is no reported difference in clinical pregnancy and live birth rates when comparing IVF with conventional fertilization to IVF with ICSI in the setting of UEI [START_REF] Buckett | The management of unexplained infertility: an evidencebased guideline from the Canadian Fertility and Andrology Society[END_REF][START_REF]Evidence-based treatments for couples with unexplained infertility: a guideline[END_REF][START_REF] Wang | Interventions for unexplained infertility: a systematic review and network meta-analysis[END_REF].

Limits

In our study, more than 50% of respondents work in a university teaching hospital, which is not representative of current practice and may lead to a bias.

In addition, infertility management depends on access to care. For many centres access to IVF is more complex and requires a longer waiting period than the IUI. IUI cycles are often performed before validation of IVF management.

Conclusion

Our study showed heterogeneity in the diagnostic criteria and management of UEI in a sample of French reproductive physicians. Moreover, some clinical practices have no clear scientific basis, such as the orientation of management according to AMH level. There is a lack of clinical studies evaluating the value of additional tests such as MRI and laparoscopy in the diagnostic strategy. Moreover, there is still insufficient evidence to advise IVF or IUI rather than expectant management, and every practitioner often has an empirical strategy for UEI. Different countries have different recommendations for UEI. Thus, UEI would benefit from a diagnostic consensus so that all studies on management strategies could be comparable.

Table 1

 1 What tests do you prescribe before concluding -in case of normality -a diagnosis of unexplained infertility?.

		Yes, always	Yes often	Yes rarely	No never	n
	AMH	87.6% (n = 169)	8.3% (n = 16)	3.1% (n = 6)	1% (n = 2)	193
	Hysterosalpingography	92.8% (n = 181)	5.1% (n = 10)	2.1% (n = 4)	0% (n = 0)	195
	Spermogram	100% (n = 193)	0% (n = 0)	0% (n = 0)	0% (n = 0)	193
	Total mobil sperm count	78.1% (n = 146)	18.7% (n = 35)	1.6% (n = 3)	1.6% (n = 3)	187
	Ultrasound of antral follicles	95.9% (n = 187)	2.6% (n = 5)	1% (n = 2)	0.5% (n = 1)	195
	Chlamydia trachomatis Serology	8% (n = 14)	6.8% (n = 12)	11.4% (n = 20)	73.9% (n = 130)	176
	Post-coital testing	52.7% (n = 98)	15.6% (n = 29)	14.5% (n = 27)	17.2% (n = 32)	186
	Pelvic MRI	3.9% (n = 7)	26.3% (n = 47)	53.6% (n = 96)	16.2% (n = 29)	179
	Laparoscopy	1.1% (n = 2)	17.3% (n = 31)	49.7% (n = 89)	31.8% (n = 57)	179
	Sperm DNA Fragmentation Test	2.2% (n = 4)	5% (n = 9)	27.9% (n = 50)	64.8% (n = 116)	179

Table 2

 2 Would any of the following data make you rule out the diagnosis of unexplained infertility?.

	Yes, always	Yes often	Yes rarely	No never	Total

Table 3

 3 Depending on the woman's age, what period of regular sexual intercourse must transpire before actively managing unexplained infertility?.

	Time of regular sexual intercourse

Table 4 A

 4 couple with a 28-year-old woman has been referred for primary infertility for one year. The initial infertility assessment did not find any abnormalities (regular sexual intercourse, antral follicle count, hormonal assessment, hysterosalpingography, spermogram). What is your first-line medical practice?.

		Yes always	Yes often	Yes rarely	No never	Total
	Expectation and targeted sexual intercourse	10.2% (n = 19)	18,82% (n = 35)	29,57% (n = 55)	41.4% (n = 77)	186
	Ovarian stimulation	7% (n = 13)	23.8% (n = 44)	20% (n = 37)	49.2% (n = 91)	185
	Intrauterine insemination	20.7% (n = 39)	43.1% (n = 81)	17.6% (n = 33)	18.6% (n = 35)	188
	In vitro fertilization	2.7% (n = 5)	6.5% (n = 12)	21.5% (n = 40)	69.4% (n = 129)	186
	Invitro fertilizaton/ intracytoplasmic sperm injection	0.5% (n = 1)	1.6% (n = 3)	4.9% (n = 9)	92.9% (n = 171)	184

Table 5 A

 5 couple with a 39-year-old patient was referred for the first time for primary infertility for one year. The initial infertility assessment did not find any abnormalities (regular sexual intercourse, antral follicle count, hormonal assessment,hysterosalpingography, spermogram). What is your first-line medical practice?.

		Yes always	Yes often	Yes rarely	No never	Total
	Expectation and targeted sexual intercourse	1.6% (n = 3)	3.3% (n = 6)	12% (n = 22)	83.1% (n = 152)	152
	Ovarian stimulation	2.2% (n = 4)	8.8% (n = 16)	18.7% (n = 34)	70.3% (n = 128)	128
	Intrauterine insemination	5.4% (n = 10)	27.2% (n = 50)	30.4% (n = 56)	37% (n = 184)	68
	In vitro fertilization	30.4% (n = 58)	48.2% (n = 92)	8.4% (n = 16)	13.1% (n = 191)	25
	Invitro fertilizaton/ intracytoplasmic sperm injection	5.5% (n = 10)	16.6% (n = 30)	16% (n = 29)	61.9% (n = 112)	112

Table 6

 6 In your opnion, what are the indications of in vitro fertilization for couples with unexplained infertility?.

	Yes always	Yes often	Yes rarely	No never

Acknowledgement

The authors would like to thank all colleagues from participating ART centres. CH Aix-en-Provence, Cr eteil, Quatres-villes (Saint-Cloud), Saint-Denis, Saint-Nazaire, CHU de Amiens, Angers, Besançon, Bordeaux, Brest, Clermont-Ferrand, Grenoble, Lilles, Limoges, Lyon, Marseille, Montpellier, Nancy, Nantes, Nice, Nîmes, Paris (APHP Site Antoine ecl ere, Site Bichat, Site Tenon, Site Jean Verdier), Pointea-Pitre, Rennes, Rouen, Saint-Etienne, Strasbourg, Toulouse, Tours, Clinique Belharra (Bayonne), Clinique Bouchard-IMR (Marseille), Clinique Jules Verne (Nantes), Clinique La croix du Sud (Toulouse), Clinique Mutualiste de la Sagesse (Rennes), Clinique Natecia (Lyon), Clinique Procrealis (La Roche sur Yon), center Sainte Colette (Marseille), Clinique Saint-Georges (Nice), Clinique Saint-Michel (Toulon), Clinique Saint-Pierre (Perpignan), Clinique Saint-Roch (Montpellier), Clinique Val D'Ouest (Ecully), Institut Mutualiste Montsouris (Paris), Polyclinique Courlancy (Reims), Polyclinique Jean Villar (Bruges), Polyclinique Urbain V (Avignon).

Funding source

This research did not receive any specific grants from funding agencies in the public, commercial, or not-for-profit sectors.

Declarations of interest

None


Céline Pozniak

U Sfl

Paris

Mireille Copin

U Lnpl

Jean Toulouse

Giuseppina Jaurès

LLF, Turco

How specific are linguistic structures? Mathematical priming on relative clause attachment in French

The specificity of linguistic structures has been a central question in research related to questions on language universals and learnability. Scheepers et al. (2011) found that calculations like 76-(6+2)×2 vs. 76-6+2×2 respectively correspond to high and low attachment relative clause constructions. We report results from two production experiments further exploring the nature of mathematical priming on language. Contrary to Scheepers et al. (2011) and other previous studies, our results rather support relative clause attachment as association to thematic domains, as suggested for example by construal theory.

Introduction

The specificity of linguistic structures has been a central question in research related to questions on language universals and learnability. This question is still under debate (see Scheepers et al., 2019 for a review). Scheepers et al. (2011) suggested that calculations like 76-(6+2)×2 structurally resemble a high-attachment RC construction ([[the friend of a colleague] who lived in Spain]) whereas calculations like 76-6+2×2 are more similar to a low-attachment RC construction ([the friend of [a colleague who lived in Spain]]). This also applies for French as illustrated in Figure 1. 2011) found that solving calculations like 76-(6+2)×2 vs. 76-6+2×2 (Figure 1) influenced high versus low relative clause (RC) attachment preferences in subsequently presented English sentences that participants had to complete (e.g, The tourist guide mentioned the bells of the church that… ). Although mathematical priming effects have been found based on this method, replicability issues are regularly raised. Hedier et al. (2020) and Hedier (2020) found that the calculation corresponding to low attachment was not correct. Indeed, for the calculation 76-6+2×2, participants would rather start with 76-6, calculate 2×2, and then subtract the latter result from the first, leading to a different representation, as illustrated in Figure 2. Results from the first experiment suggest that priming may not be structural but based on associations to thematic domains. We test predictions for more adequate calculations in experiment 2 for high and low attachment primes.

2 Experiment 1 One variable was manipulated: prime (low vs. high). Primes consisted in calculations whose structure was equivalent to either that of a low attachment or of a high attachment of a RC. Items (N=12) were preceded by two structurally similar calculations, and consisted of structurally ambiguous target sentences containing complex NPs of the form NP1 of NP2, half of them singular-plural, half plural-singular, followed by the relativizer "qui" (Table 1). We used non-perceptual verbs in the main clause so as to avoid pseudo-RCs (Grillo & Costa, 2014), which have a different syntactic structure (Hedier et al. 2020 2019), participants were reminded before the experiment of the arithmetic operator-precedence rules. Then, they solved calculations, read and continued sentences in writing (see Table 1). Eighty native speakers of French (recruited on Prolific) participated. We present the results from monolingual participants who correctly answered the prime calculations in the critical trials (N=618 observations).

Results

As seen in Figure 3 (left), contrary to our expectations, Bayesian analyses showed that after a low calculation prime, continuations referred more to high attachment than low attachment (𝛽 ˆ=-0.53, CrI[-1.47,0.36], P(𝛽)>0=0.88). In post hoc analyses, taking number marking into account (Figure 3, right) we found an interaction between NP1 number and prime (𝛽 ˆ=2.69, CrI[0.30,5.19], P(𝛽)>0=0.99), meaning that there was a higher proportion of high attachment continuations with a singular NP1 than with a plural NP1, in the high prime condition. The preference for high attachment with singular NP1 persisted to a lower degree after low attachment calculations.

Discussion

Calculations did not prime the expected structure of RC attachment and, moreover, we found a strong interaction with number marking. Reanalyses of previous experiments with different materials and participants in French showed the same pattern regarding NP1 number which seems thus to be a robust effect. We suggest that the calculations primed RC construal to thematic domains (TD) as it has been suggested in Construal Theory (Frazier & Clifton, 1997, see also Keller, 1995, or Crysmann, 2005, for corresponding analyses in HPSG). According to construal theory, RCs (as other non-primary relations) are associated to a thematic domain in a first step (Figure 4). The antecedent of the RC is then chosen within the thematic domain based on a variety of factors (e.g. number). Priming of association to thematic domains could lead to the observed effects with our "high attachment" calculations priming a thematic domain above the complex NP (with a free choice of antecedents based on non-structural factors) and the "low attachment" calculations priming a thematic domain defined by NP1 (Table 2). We tested the robustness of thematic domain priming with the adequate calculations experiment 2. If shared representations triggering thematic domain priming do exist, the high attachment calculations should prime RC high attachment, and the low attachment calculations should prime RC low attachment (Table 2, Primes 2 and 3). 3 Experiment 2 3.1 Design & procedure Design and procedure are the same as Experiment 1 except that the calculations were different (primes 2 and 3 in table 1). We present the results from participants who correctly answered the prime calculations (N=506 observations).

Results & Conclusion

As seen in Figure 5, continuations referred more to high attachment than low attachment after a high attachment prime (𝛽 ˆ=0.57, CrI[-0.6,1.7], P(𝛽)>0=0.84). Our results support the hypothesis of shared representations between mathematics and language. However, it seems that structural priming may not be at stake here, but rather thematic domain priming, suggesting that relative clauses as non-primary relations are associated to a thematic domain. This would explain replicability issues from previous studies, especially in French since the calculations were rather structural and not linked to thematic domain (see Hedier et al., 2020 for the calculations). A better analysis of the number effect, especially regarding the semantic particularity of singulars, is also necessary to understand the nature of priming. Finally, our results support relative clause attachment as association to thematic domains, as suggested by construal theory.

Figure 1 .

 1 Figure 1. Calculations and relative clause attachment

Figure 2 .

 2 Figure 2. Real representation for 76-6+2×2 (Hedier, 2020)

Figure 3 .

 3 Figure 3. Results from Experiment 1

Figure

  Figure 4. Association to thematic domains

Figure 5 .

 5 Figure 5. Results from Experiment 2

Table 1 .

 1 ). Example of two items in all conditions Prediction If shared structural representations between mathematics and language exist that trigger priming(Scheepers et al., 2011), high attachment calculations should prime RC high attachment, and low attachment calculations should prime RC low attachment.

	Item NP1	Prime	Sentence
	1	Plural	High 76-(6+2)×2= Marc a cherché les secrétaires du directeur qui…
			Low 76-(6+2×2)=	Mark sought the personal assistants of the general manager that…
	2	Singular High 8 -(6 -2) / 2 = Philippe a épousé l'éditrice des journalistes qui…
				Philip married the editor of the journalists that…
			Low 8 -(6 -2 / 2) =
	2.1

2.2 Procedure Inspired

  by Scheepers et al. (

Selected references


Andrey Polyakov

Denis Efimov

Xubin Ping

On consistent discretization of a homogeneous 2-SM controller

The paper presents a discretization scheme for a homogeneous second order sliding mode (2-SM) controller, which preserves the finite-time stability of the closed-loop system even in the case the sampled-time implementation of the control. Numerical simulations confirm the efficiency of the proposed scheme for chattering reduction.

I. INTRODUCTION

An asymptotically stable homogeneous system is finitetime stable in the case of negative homogeneity degree and nearly fixed-time stable in the case of the positive homogeneity degree (see, e.g. [START_REF] Nakamura | Smooth Lyapunov functions for homogeneous differential inclusions[END_REF], [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF]). However, the finite/fixed-time stability is a fragile property in the sense that an improper discretization of a finite-time or a fixedtime stable ordinary differential equation (ODE) results in chattering [START_REF] Acary | Implicit euler numerical scheme and chattering-free implementation of sliding mode systems[END_REF], [START_REF] Levant | Chattering analysis[END_REF] or even in a finite-time blow up [START_REF]On fixed and finite time stability in sliding mode control[END_REF]. That is why the discretization issues are very important for practical implementation of finite/fixed-time sliding mode (SM) control/estimation algorithms [START_REF] Acary | Implicit euler numerical scheme and chattering-free implementation of sliding mode systems[END_REF], [START_REF] Kikuuwe | Proxybased sliding mode control: A safer extension of pid position control[END_REF], [START_REF] Livne | Proper discretization of homogeneous differentiators[END_REF], [START_REF] Huber | Lyapunov stability analysis of the implicit discrete-time twisting control algorithm[END_REF], [START_REF] Koch | Discrete-time implementation of homogeneous differentiators[END_REF], [START_REF] Brogliato | Digital implementation of slidingmode control via the implicit method: A tutorial[END_REF], [START_REF] Michel | A noise less-sensing semi-implicit discretization of a homogeneous differentiator : principle and application[END_REF], [START_REF] Hanan | Low-chattering discretization of sliding mode control[END_REF].

The concept of consistent discertization introduced in [START_REF] Polyakov | Consistent discretization of finite-time and fixed-time stable systems[END_REF] postulates that stability properties of a continuous-time system are preserved in its discrete-time counterpart (approximation). Consistent discretizations for stable generalized homogeneous ODEs were developed in [START_REF] Polyakov | Consistent discretization of finite-time and fixed-time stable systems[END_REF], [START_REF] Sanchez | Lyapunov-based consistent discretisation of stable homogeneous systems[END_REF] based on Lyapunov function theory. Some schemes with state dependent discretization step were given in [START_REF] Efimov | Discretization of homogeneous systems using euler method with a state-dependent step[END_REF]. Being efficient for numerical simulations, the mentioned schemes do not allow a consistent discretization (sampled-time implementation) of finite-time controllers in the general case. To the best of authors' knowledge, such implementations are developed only for the conventional (first order) SM algorithms [START_REF] Acary | Implicit euler numerical scheme and chattering-free implementation of sliding mode systems[END_REF], [START_REF] Huber | Lyapunov stability and performance analysis of the implicit discrete sliding mode control[END_REF] as well as for the twisting algorithm [START_REF] Huber | Lyapunov stability analysis of the implicit discrete-time twisting control algorithm[END_REF] and supertwisting algorithm [START_REF] Brogliato | The implicit discretization of the super-twisting sliding-mode control algorithm[END_REF] based on the implicit method. This paper presents a consistent discretization for a homogeneous 2-SM controller studied in [START_REF] Polyakov | Finite-time and fixedtime stabilization: Implicit Lyapunov function approach[END_REF]. The latter controller is a version of the so-called quasi-continuous SM algorithm [START_REF] Levant | Quasi-continuous high-order sliding-mode controllers[END_REF], which has a discontinuity only in the origin. It is shown that the sampled-time implementation of the controller according to the developed scheme preserves the finite-time stability property of the original closed-loop continuous-time system in the disturbance-free case. Numerical simulations show the *This work was partially supported by the National Natural Science Foundation of China under Grant 62050410352 and by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898.

1 Andrey Polyakov and Denis Efimov are with Univ. Lille, Inria, CNRS, Centale Lille, France, e-mail: andrey.polyakov(denis.efimov)@inria.fr, and with with ITMO University, Saint-Petersburg, Russia.

2 Xubin Ping (the corresponding author) is with Xidian University, Xi'an, Shaanxi, China, e-mail:pingxubin@126.com. efficiency of this scheme for a chattering reduction even in the case of matched perturbations.

Notation: N is the set of natural numbers; R is the field of real numbers; C is the field of complex numbers; 0 is the zero of a vector space (e.g., the zero vector in R n or the zero matrix in R n×m ) ; I n ∈ R n×n is the identity matrix; P 0 denotes positive definiteness of a matrix P = P ∈ R n×n ; λ max (P ) is a maximum eigenvalue of a symmetric matrix P ; x =

√

x P x with P 0 denotes the weighted Euclidean norm in R n ; K denotes a class of strictly increasing positive definite functions [0, +∞) → [0, +∞); a function σ ∈ K of the class K ∞ if σ(s) → +∞ as s → +∞; the set of continuous maps X → Y between open connected sets X and Y of finite-dimensional vectors spaces is denoted by C(X, Y ).

II. PROBLEM STATEMENT

Let us consider a linear control system

ẋ(t) = Ax(t) + Bu(t), t > 0, x(0) = x 0 ∈ R n , (1)
where x(t) ∈ R n is the system state, u(t) ∈ R m is control input and A ∈ R n×n , B ∈ R n×m are known matrices.

Recall [START_REF] Bhat | Finite time stability of continuous autonomous systems[END_REF] that a system ẋ = f (t, x), t > 0, x(0) = x 0 is globally uniformly finite-time stable if it is Lyapunov stable and there exists a locally bounded function T : R n → [0, +∞) such that any trajectory of the system vanishes to zero in a finite time:

x(t, x 0 ) = 0, ∀t ≥ T (x 0 ), ∀x 0 ∈ R n .
Definition 1: Let the system (1) with a feedback u ∈ C(R n \{0}, R m ) be globally uniformly finite-time stable. A family of functions ũh : R n → R m parameterized by a scalar h > 0 is said to be a consistent discretization of u if

• Finite-time stability: the closed-loop system (1)

u(t) = ũh (x(t i )), ∀t ∈ [t i , t i+1 ), t i = ih, i = 0, 1, ... (2
) is globally uniformly finite-time stable;

• Approximation: ∀r 1 > 0, ∀r 2 > r 1 , ∃ω r1,r2 ∈ K : sup r1≤ x ≤r2 ũh (x) -u(x) ≤ ω r1,r2 (h). (3)
The aim of this paper is to develop a consistent (in the sense of the above definition) discretization for a homogeneous 2-SM controller [START_REF] Polyakov | Finite-time and fixedtime stabilization: Implicit Lyapunov function approach[END_REF]. In this case, A and B satisfy the following assumption.

Assumption 1:

A = [ 0 1 0 0 ] and B = [ 0 1 ] .

III. PRELIMINARIES: HOMOGENEOUS SYSTEMS

A. Linear dilation and homogeneous norm

In this paper we deal only with the so-called linear geometric dilation [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]Chapter 6] in R n given by

d(s) = e sG d = ∞ i=0 (sG d ) i i! , s ∈ R, (4)
where G d ∈ R n×n is an anti-Hurwitz matrix1 known as the generator of linear dilation. The latter guarantees that d satisfies the limit property ( d(s)x → 0 as s → -∞ and d(s)x → +∞ as s → +∞) required for a group to be a dilation in R n (see, e.g., [START_REF] Kawski | Families of dilations and asymptotic stability[END_REF]). The dilation introduces an alternative norm topology in R n by means the so-called canonical homogeneous norm. Definition 2: [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF] The functional • d : R n → R + given by x d = 0 for x = 0 and

x d = e sx , where s x ∈ R : d(-s x )x = 1, x = 0 (5)
is called the canonical homogeneous norm in R n , where d is a monotone dilation 2 .

Notice that x = 1 (resp. x ≤ 1) is equivalent to

x d = 1 (resp. x d ≤ 1). For the uniform dilation d(s) = e s I n , s ∈ R we have • = • d .
Theorem 1: [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF] If d is a monotone dilation, the canonical homogeneous norm is continuous on R n , locally Lipschitz continuous on R n \{0} and there exist σ, σ ∈ K ∞ such that

σ( x ) ≤ x d ≤ σ( x ), ∀x ∈ R n . Moreover, • d is differentiable on R n \{0} provided that
• is differentiable on R n \{0}. For x =

√

x P x with the symmetric matrix P ∈ R n×n satisfying P 0, P G d + G d P 0 we have

∂ x d ∂x = x d x d (-ln x d )P d(-ln x d ) x d (-ln x d )P G d d(-ln x d )x , ∀x = 0. (6)
Below the canonical homogeneous norm is used as a Lyapunov function for analysis and design of a homogeneous SMC.

B. Homogeneous functions and vector fields

Definition 3: [23] A vector field f : R n → R n (resp. a function h : R n → R) is said to be d-homogeneous of degree µ ∈ R if f (d(s)x) = e µs d(s)f (x) (resp. h(d(s)x) = e µs h(x)), for all x ∈ R n , s ∈ R.
Homogeneity of a function (operator) is inherited by any other object induced by this function. For example, the Euler homogeneous function theorem implies that the derivative of the homogeneous function is homogeneous as well. If a vector field f is d-homogeneous of degree µ then solutions of the ODE ẋ = f (x) [START_REF] Kikuuwe | Proxybased sliding mode control: A safer extension of pid position control[END_REF] are symmetric [START_REF] Kawski | Families of dilations and asymptotic stability[END_REF]:

x(t, d(s)x 0 ) = d(s)
x(e µs t, x 0 ), where x(t, z) denotes a solution of [START_REF] Kikuuwe | Proxybased sliding mode control: A safer extension of pid position control[END_REF] with the initial condition x(0) = z. Example 1: [START_REF] Zimenko | Robust feedback stabilization of linear mimo systems using generalized homogenization[END_REF] The linear vector field

x → Ax, A ∈ R n×n is d-homogeneous of the degree µ = 0 ⇔ A is nilpotent ⇔ AG d = (µI n + G d )A.
The homogeneity degree specifies the convergence rate. Theorem 2: [START_REF] Bhat | Geometric homogeneity with applications to finite-time stability[END_REF] Let the vector field f : R n → R n be continuous and d-homogeneous of a degree µ ∈ R. If the system ( 7) is asymptotically stable then it is globally finitetime stable for µ < 0.

The homogeneous control systems are robust (input-tostate stable) with respect to a rather large class of perturbations [START_REF] Hong | H∞ control, stabilization, and input-output stability of nonlinear systems with homogeneous properties[END_REF], [START_REF] Andrieu | Homogeneous Approximation, Recursive Observer Design, and Output Feedback[END_REF].

C. Homogeneous stabilization of linear plant

The following theorem combines the results of [START_REF] Polyakov | Finite-time and fixedtime stabilization: Implicit Lyapunov function approach[END_REF], [START_REF] Polyakov | Generalized Homogeneity in Systems and Control[END_REF]. Theorem 3: Let the pair {A, B} be controllable. Then 1) the linear algebraic equation

AG 0 -G 0 A + BY 0 = A, G 0 B = 0 (8)
has a solution with respect to the pair

Y 0 ∈ R m×n , G 0 ∈ R n×n , and G 0 -I n is invertible, G d = I n +µG 0 is anti-Hurwitz for any µ ∈ [-1, 1/k], where k is minimal natural number such that rank[B, AB, ..., A k-1 B] = n, the matrix A 0 = A + BY 0 (G 0 -I n ) -1 satisfies A 0 G d = (G d + µI n )A 0 , G d B = B; (9)
2) the system of linear matrix inequalities (LMIs) with the linear matrix equation

A 0 X +XA 0 +BY +Y B +ρ(G d X +XG d ) = 0, G d X + XG d 0, X = X 0 ( 10
)
has a solution X ∈ R n×n , Y ∈ R m×n for any fixed ρ > 0; 3) the canonical homogeneous norm • d induced by the weighted Euclidean norm x = √ x P x with P = X -1 is a Lyapunov function of the system (1) with

u(x) = K 0 x + x 1+µ d Kd(-ln x d )x, (11)
K 0 = Y 0 (G 0 -I n ) -1 , K = Y X -1 , ( 12
)
where d is a dilation generated by G d ; moreover,

d dt x d = -ρ x 1+µ d , x = 0; (13)
4) the feedback law u given by ( 11) is continuously differentiable on R n \{0}, u is continuous at zero if µ > -1 and u is discontinuous at zero but locally bounded if µ = -1; 5) the closed-loop system (1), ( 11) is d-homogeneous of the degree µ. Obviously, the closed-loop system (1),( 11) is finite-time stable if µ < 0 and it is nearly fixed-time stable if µ > 0. For µ = 0 the control [START_REF] Brogliato | Digital implementation of slidingmode control via the implicit method: A tutorial[END_REF] becomes the linear exponentially stabilizing feedback u = K 0 x + Kx.

Remark 1: For µ = -1 the control ( 11) is discontinuous at x = 0. If, additionally, K 0 = 0 then u is uniformly bounded

u u = x d (-ln x d )K Kd(-ln x d )x ≤ u 2 max := λ max (P -1/2 K KP 1/2 ).
The control u is a high order sliding mode control, which rejects matched bounded uncertainties (see, [START_REF] Polyakov | Finite-time and fixedtime stabilization: Implicit Lyapunov function approach[END_REF] for more details). Notice also that for µ = -1 and ρ = 1 the canonical homogeneous norm is the settling-time function of the closed-loop system. Such a control law (under some variation and/or simplifications) is well-known in the literature as a solution to a finite-time stabilization problem for linear plants [START_REF] Korobov | A solution of the synthesis problem using controlability function[END_REF], [START_REF] Praly | Generalized weighted homogeneity and state dependent time scale for linear controllable systems[END_REF], [START_REF] Polyakov | Finite-time and fixedtime stabilization: Implicit Lyapunov function approach[END_REF]. Using the topological equivalence of any stable d-homogeneous system to a standard homogeneous one we prove the following corollary which allows the explicit formula for solutions of the closed-loop system (1), ( 11) to be presented.

Corollary 1: Under conditions of Theorem 3 with µ = 0, the solution of the closed-loop system (1), ( 11) is unique and satisfies the identity

x(t + τ ) = Q τ ( x(t) d )x(t), (14)
where τ, t ≥ 0 and Q τ (0) = 0,

Q τ (r) = e G d ln r Qτ (r)e -G d ln r if 1 r µ >-µρτ, 0 if 1 r µ ≤-µρτ, r > 0, (15)
Qτ (r) = e G d µ ln 1 1+µρτ r µ e (A+B(K 0 +K)+ρG d ) ln(1+µρτ r µ ) ρµ .
Moreover, for x = 0 one holds

d dτ Q τ ( x d )x = AQ τ ( x d )x + Bu(Q τ ( x d )x). (16) Proof. Denoting y = x d d(-ln x d )x, we derive y = x d d(-ln x d )x = x d and ẏ = d x d dt (I n -G d )d(-ln x d )x + x d d(-ln x d ) ẋ =ρ y µ (G d -I n )y + x d d(-ln x d )(A 0 x+ x 1+µ d BKd(-ln x d )x). Since d(s)A 0 = e -µs A 0 d(s) and d(s)B = e s B for all s ∈ R then ẏ = y µ (A 0 + BK + ρ(G d -I n ))y.
In this case, using [START_REF] Koch | Discrete-time implementation of homogeneous differentiators[END_REF] we conclude

d dt y(t) = y P ẏ(t) y(t) = y µ-1 y P (A 0 +BK +ρ(G d -I n )) y = y µ-1 y {P (A0+BK+ρG d )+(A0+BK+ρG d ) P}y 2 -ρy P y = -ρ y(t) µ+1 and y(t + τ ) -µ = y(t) -µ + µρτ,
for y(t) -µ + µρτ ≥ 0. Obviously, y(t + τ ) = 0 if y(t) -µ + µρτ ≤ 0. The latter corresponds to the negative homogeneity degree µ < 0 and the finite-time stability of the closed-loop system. Hence, denoting

K lin = K 0 + K we obtain y(t + τ ) =e (A+BK lin +ρ(G d -In)) τ 0 y(t+σ) µ dσ y(t) =e (A+BK lin +ρ(G d -In)) τ 0 1 y(t) -µ +µρσ dσ y(t) =e (A+BK lin +ρ(G d -In)) 1 µρ ln y(t) -µ +µρτ y(t) -µ y(t)
=e (A+BK lin +ρ(G d -In)) 1 µρ ln(1+µρτ y(t) µ ) y(t). Returning to the original coordinates we derive

x(t + τ ) =d(ln y(t + τ ) y(t+τ ) y(t+τ ) =d(ln( y(t) -µ + µρτ ) 1 -µ ) • e (A+BK lin +ρ(G d -In)) 1 ρµ ln(1+µρτ y(t) µ ) • y(t) ( y(t) -µ +µρτ ) 1 -µ =d(ln( x(t) -µ d + µρτ ) 1 -µ ) • e (A+BK lin +ρG d ) 1 ρµ ln(1+µρτ x(t) µ d ) • d(-ln x(t) d )x(t) =Q τ ( x(t) d )x(t)
for all t ≥ 0 and all τ ≥ 0. The identity ( 16) follows immediately from the fact that τ → x(τ ) is a solution of (1), [START_REF] Brogliato | Digital implementation of slidingmode control via the implicit method: A tutorial[END_REF]. The latter corollary presents an important result for a class of nonlinear homogeneous system under consideration, since this is a very rare case that a solution of an essentially nonlinear system in R n can be found explicitly (see the formula ( 14)). Such a property may be very useful for various analysis. In this paper, we design a consistent discretization for the homogeneous control [START_REF] Brogliato | Digital implementation of slidingmode control via the implicit method: A tutorial[END_REF] for the double integrator. In such a particular case, the control (11) may be defined analytically (with an explicit representation of the canonical homogeneous norm by means of Ferrari formulas [START_REF] Polyakov | A new homogeneous quasi-continuous second order sliding mode control[END_REF]).

Remark 2: For µ = -1 under Assumption 1 we have

K 0 = 0, A 0 = A, G d = 2 0 0 1 .
In this case, for ρ = 1 the condition [START_REF] Koch | Discrete-time implementation of homogeneous differentiators[END_REF] with

X = x 11 x 12 x 12 x 22 , ∈ R 2×2 and Y = [y 1 , y 2 ] ∈ R 1×2 becomes x 11 > 0, x 12 = -2x 11 , x 22 > 9 2 x 11 , y 1 = -3x 12 -x 22 , y 2 = -x 22 .
(17) and the control [START_REF] Brogliato | Digital implementation of slidingmode control via the implicit method: A tutorial[END_REF] has the form

u(x) = k 1 x -2 d x 1 +k 2 x -1 d x 2 , K = [ k1 k2 ] = Y P, (18)
where x = [x 1 , x 2 ] ∈ R 2 , P = p 11 p 12 p 12 p 22 = X -1 and the canonical homogeneous norm x d is induced by the norm x = √ x P x. From (5) we derive

x x -1+µ d 0 0 x -1 d [ p11 p12 p12 p22 ] x -1+µ d 0 0 x -1 d x = 1.
Therefore, x d = r for µ = -1, where r ∈ R is a unique positive root of the following quartic equation

r 4 = ar 2 +br+c, a = p 22 x 2 2 , b = 2p 12 x 1 x 2 , c = p 11 x 2 1 , (19)
which can be solved using Ferrari formulas.

IV. CONSISTENT DISCRETIZATION OF HOMOGENEOUS CONTROL FOR DOUBLE INTEGRATOR

Let us consider the discrete-time version of the system (1) with the sample-time control implementation. Obviously, if

u(t) = u k := u(t k ) for t ∈ [t k , t k+1 ) then x k+1 = A h x k + B h u(t k ), k = 0, 1, ..., (20)
where

x k = x(t k ), t k = kh, A h = e hA and B h = h 0 e sA ds B.
The consistent discretizations of twisting and super-twisting algorithms presented in [START_REF] Huber | Lyapunov stability analysis of the implicit discrete-time twisting control algorithm[END_REF] and [START_REF] Brogliato | The implicit discretization of the super-twisting sliding-mode control algorithm[END_REF] are based on the socalled implicit scheme. The application of the same approach to the system (1), [START_REF] Brogliato | Digital implementation of slidingmode control via the implicit method: A tutorial[END_REF] gives the generalized equation

x k+1 = Ax k + Bu k , u k ∈ R(x k+1 ) := u(x k+1 ) if x k+1 = 0, [ -u max , u max ] if x k+1 = 0,
which is difficult to solve since R is not a maximal monotone operator. This equation may not have a solution without a relaxation (e.g., a Yosida approximation [START_REF] Miranda-Villatoro | Set-valued sliding-mode control of uncertain linear systems: Continuous and discrete-time analysis[END_REF]). In this case the finite-time stability of the discrete-time system may not be guaranteed. To overcome this difficulty, we introduce an alternative technique inspired by Corollary 1.

Using [START_REF] Levant | Quasi-continuous high-order sliding-mode controllers[END_REF] we derive

x k+2 = B h u(t k+1 ) + A h B h u(t k ) + A 2 h x k . (21)
For the double integrator (see Assumption 1) we have

A h = I 2 + hA, B h = hI 2 + h 2 2 A B
and, obviously, the matrix

W h = [B h , A h B h ] (22)
is invertible. Since by Corollary 1 for any solution of the closed-loop continuous-time system (1) with the homogeneous control [START_REF] Brogliato | Digital implementation of slidingmode control via the implicit method: A tutorial[END_REF] we have

x(t + 2h) = Q 2h ( x(t) d )x(t) then to guarantee x k+2 = Q 2h ( x k d )x k we may select u(t k+1 ) u(t k ) = W -1 h Q 2h ( x k d ) -A 2 h x k , (23)
where Q τ is given by [START_REF] Sanchez | Lyapunov-based consistent discretisation of stable homogeneous systems[END_REF]. By construction, the discretetime system (20), [START_REF] Kawski | Families of dilations and asymptotic stability[END_REF] tracks the states of the closed-loop continuous-time system (1), [START_REF] Brogliato | Digital implementation of slidingmode control via the implicit method: A tutorial[END_REF] at discrete time instances t 2k . This means that the discretization [START_REF] Kawski | Families of dilations and asymptotic stability[END_REF] preserves the convergence rate of the continuous time system. However, the control ( 23) is not a conventional feedback law, since u(t k+1 ) depends on x k = x(t k ) but not on x k+1 . Inspired by [START_REF] Kawski | Families of dilations and asymptotic stability[END_REF], we consider the control law:

ũh (x k ) = Kh ( x k d )x k , Kh (•) = [0, 1]W -1 h Q 2h (•)-A 2 h , (24)
which selects only u(t k ) by the formula (23) leaving u(t k+1 ) to be defined at the next sampling instant. To show that the static nonlinear feedback ( 24) is a consistent discretization of the finite-time stabilizing feedback [START_REF] Brogliato | Digital implementation of slidingmode control via the implicit method: A tutorial[END_REF], the discrete-time closed-loop system (20), [START_REF] Zimenko | Robust feedback stabilization of linear mimo systems using generalized homogenization[END_REF] has to be globally finite-time stable for µ < 0 and the feedback (24) must tend to the control (11) as h → 0 + . Theorem 4: For n = 2, µ = -1, ρ = 1 the control u defined by [START_REF] Brogliato | Digital implementation of slidingmode control via the implicit method: A tutorial[END_REF] under conditions of Theorem 3 has a consistent discretiation ũh given by [START_REF] Zimenko | Robust feedback stabilization of linear mimo systems using generalized homogenization[END_REF] provided that

F ( h)X -1 F ( h) < X -1 , ∀ h ∈ (0, 1/2] (25)
where

F ( h) =I 2 + hA + (I 2 + h 2 A)B 1 -h 2 • • e G d ln(1-2 h) e -(A+BK+G d ) ln(1-2 h) -I-2 hA h . (26)
Proof. 1) First, let us show that ũh → u as h → 0 + . Since

A h = [ 1 h 0 1 ], B h = h 2 2 h , W h = h 2 2 3h 2 2 h h , W -1 h = -1 h 2 3 2h 1 h 2 -1 2h then Kh (r) = 1 h 2 -1 2h (Q 2h (r) -A 2 h ).
Denoting s h = ln(1 -2h/r) -1 and s = ln r with r = x d > 0 we derive

Q 2h (r) = d(s) Q2h (r)d(-s),
where

Q2h (r) = I 2 -s h G d + s 2 h 2 G 2 d + O(s 3 h ) • • I 2 +(A+BK +G d )s h + (A+BK+G d ) 2 s 2 h 2 +O(s 3 h ) =I 2 + (A + BK)s h -s 2 h G d (A + BK) - s 2 h 2 G 2 d + s 2 h 2 (A + BK + G d ) 2 + O(s 3 h ).
Using Ad(s) = e -s d(s)A we derive

Q 2h (r) -A 2 h = d(s) Q2h (r)d(-s) -I 2 -2hA = = d(s) Q2h (r) -I 2 -2h/rA d(-s) = O(s 3 h )+ d(s) A(rs h -2h) r + 2BKs h +((A+BK+G d ) 2 -2G d (A+BK)-G 2 d )s 2 h 2 d(-s)
Taking into account

s h = ln 1 -2h r -1 = -ln 1 -2h r = 2h r + (2h/r) 2 2 + O(h 3 )
we derive

Kh (r) = O(h) + [ 1 h 2 -1 2h ] 2hBK r + ( 2h r ) 2 [ 1 h 2 -1 2h ]• •d(s) A 2 + ((A+BK+Gd) 2 -2G d (A+BK)-G 2 d ) 2 d(-s) = O(h)+ 2[ 1 -h 2 ]d(s)(A-2Gd(A+BK)-G 2 d +(A+BK+G d ) 2 )d(-s)-r 2 Kd(-s) r 2
.

Since

[ 1 0 ] B = 0, [ 1 0 ] G d = [ 2 0 ], [ 1 0 ] d(s) = r 2 [ 1 0 ] then Kh (r) = 2 [ 1 0 ] A-2G d A-G 2 d +(A+G d ) 2 +ABK d(-s)+ +O(h) -Kd(-s) = 2 [ 1 0 ] (A-2G d A+AG d +G d A) d(-s)+Kd(-s) + O(h) = Kd(-ln r) + O(h),
where the identity ( 9) is utilized on the last step. Hence, ũ → u as h → 0 + uniformly on compacts from R 2 \{0}.

2) Finite-time stability. a) Let us denote

z(h,x)=x+hAx+(hB+ h 2 2 AB) 1 h 2 - 1 2h • d(ln x d )e G d ln x d -2h x d e -(A+BK+G d )ln x d -2h x d d(-ln x d )-I2-2hA x.
The function z : R × R n → R n is well defined on the set

D = {h > 0, x ∈ R n : x d > 2h}.
Obviously, the closed-loop system ( 20), ( 24) can be rewritten as

x k+1 = z(h, x k ) as long as x k / ∈ Ω h , Ω h = {x : x d ≤ 2h}.

Notice that

d(s)x+hd(s)Ax+d(s)(hI2+ h 2 2 A)B 1 (e s h) 2 - 1 2e s h • •(d(s)d(ln x d )e G d ln(1-2h/ x d ) • e -(A+BK+G d ) ln(1-2h/ x d ) d(-ln x d )-d(s)-2hd(s)A)x= d(s)z(h,x), ∀s∈R,∀x∈R 2 \{0}.
Recall that • d is a Lyapunov function of the closed-loop continuous time system (1), [START_REF] Brogliato | Digital implementation of slidingmode control via the implicit method: A tutorial[END_REF] such that

x(t + h) d = x(t) d -h.
Since ũ → u as h → 0 uniformly on compacts from R 2 \{0} then away from the origin a solution of the sampled-time (discrete-time) system uniformly (on compact intervals of time) converge to a solution of the continuous-time system, so there exist a natural number n 0 ≥ 2 and ε 0 > 0 such that

z(h, x) d ≤ -ε 0 h+ x d , ∀x : x d = 1, ∀h ∈ (0, 1/2 n0 ]. In this case, if x d > 1 then h/ x d < h and for s = -ln x d we have d(s)x d = 1, z(h, x) d = d(-s)z(e s h, d(s)x) d = e -s z(h/ x d , d(s)x) d ≤ e -s ( d(s)x d -ε 0 h/ x d ) = x d -ε 0 h. Therefore, we have z(h, x) d ≤ x d -ε 0 h, ∀x : x d ≥ 1, ∀h ∈ (0, 1/2 n0 ].
(28) b) On the one hand, since the first identity from [START_REF] Koch | Discrete-time implementation of homogeneous differentiators[END_REF] guarantees that

X -1 2 (A + BK + G d )X 1 2 + X 1 2 (A + BK + G d ) X -1 2 = 0 then the matrix X -1/2 (A + BK + G d )X 1/2 is skew- symmetric and e -X -1/2 (A+BK+G d )X 1/2 φ = R(φ), ∀φ ∈ R is a rotation matrix R R = I 2 , so lim sup φ→±∞ R(φ) < +∞. Since e G d ln(1-2 h) = (1 -2 h) 2 0 0 1 -2 h , h ∈ [0, 1/2] then e G d ln(1-2 h) e -(A+BK+G d ) ln(1-2 h) → 0 as h → 1/2.
Hence, the function z can be continuously prolonged to the set {h > 0, x ∈ R n :

x d ≤ 2h} as follows z(h, x) := I 2 +hA+(B + h 2 AB) 1 h - 1 2 (-I 2 -2hA) x.
Since any continuous function on a compact reaches the maximal value on this compact then there exists x * :

x * d = 1 and h * ∈ [1/2 n0 , 1/2] such that max h∈[1/2 n 0 ,1/2], x d =1 z(h, x) d = z(h * , x * ) d = c ≥ 0.
On the other hand, the matrix inequality [START_REF] Hong | H∞ control, stabilization, and input-output stability of nonlinear systems with homogeneous properties[END_REF] implies that

z (h, x)X -1 z(h, x) < x X -1 x = 1, ∀x : x d = 1, ∀h ∈ 1 2 n 0 , 1 2 . or, equivalently, z(h, x) d < x d = 1, ∀x : x d = 1, ∀h ∈ 1 2 n 0 , 1 2 . Therefore, we derive z(h, x) d ≤ c = z(h * , x * ) d < 1 or equivalently, z(h, x) d ≤ 1 -ε, ∀x : x d = 1, ∀h ∈ 1 2 n 0 , 1 2 where ε = 1 -c > 0. If x d ∈ [1/2 n0-1 , 1] then for s = -ln x d we have e s /2 n0 ∈ [1/2 n0 , 1/2] and z(1/2 n0 , x) d = d(-s)z(e s /2 n0 , d(s)x) d = e -s z(e s /2 n0 , d(s)x) d ≤ x d (1 -ε) ≤ x d -ε/2 n0-1
for all ∀x :

x d ∈ [1/2 n0-1 , 1].
Taking into account the inequality (28) we derive

z(h 0 , x) d ≤ x d -εh 0 , ∀x : x d ≥ 2h 0 , where h 0 = 1/2 n0 , ε = min{ε, ε 0 }.
This property implies a global finite-time convergence to the set Ω h0 .

c) Let us show that the set Ω h0 is invariant. Notice that for x ∈ Ω h0 we have Q 2h0 ( x d ) = 0 and

z(h 0 , x) = I 2 +h 0 A+(B + h0 2 AB) [ 1 h 0 -1 2 ] (-I 2 -2h 0 A) x. Since x d ≤ 2h 0 is equivalent to x d(-ln(2h 0 ))X -1 d(-ln(2h 0 ))x ≤ 1, then to prove that x d ≤ 2h 0 ⇒ z d ≤ 2h 0 it is sufficient to fulfill the inequality z (h 0 , x)d(-ln(2h 0 ))X -1 d(-ln(2h 0 ))z(h 0 , x) ≤ x d(-ln(2h 0 ))X -1 d(-ln(2h 0 ))x,
The latter follows from [START_REF] Hong | H∞ control, stabilization, and input-output stability of nonlinear systems with homogeneous properties[END_REF] since

d(-ln(2h 0 ))z(h 0 , x) = d(-ln(2h 0 )) I 2 +h 0 A+(B + h0 2 AB) [ 1 h 0 -1 2 ] (-I -2h 0 A) x = F (0.5) d(-ln(2h))x.
Therefore, Ω h0 is a finite-time stable invariant set of the system (20), [START_REF] Zimenko | Robust feedback stabilization of linear mimo systems using generalized homogenization[END_REF].

Since x k+1 = z(h, x k ) then from the identity ( 27) we derive the symmetry of solutions of the system with respect to simultaneous the scaling of the sampling time h → e s h and the initial x 0 → d(s)x 0 (see, e.g., [START_REF] Efimov | Realization and discretization of asymptotically stable homogeneous systems[END_REF] for more details). Therefore, for any h > 0 the set Ω h = {x : x d ≤ 2h} is a finite-time stable invariant set of the system (20), [START_REF] Zimenko | Robust feedback stabilization of linear mimo systems using generalized homogenization[END_REF] and for any x 0 ∈ R 2 there exists k * such that x k ∈ Ω h , ∀k ≥ k * . d) Since for x k ∈ Ω h we have

x k+1 = F (h)x k
where

F (h) = I 2 + hA + (I 2 + h 2 A)B 1 h -1 2 (-I 2 -2hA) = 1 2 h 4 -1 h -1 2
then to complete the proof we just need to show that the matrix F (h) is Schur stable and nilpotent (to have finitetime convergence). The required conclusion follows from the fact that the matrix F (h) has only zero eigenvalues for all h > 0. Therefore, we have shown that the system (20), ( 24) is finite-time stable for any h > 0.

To design a consistent discretization for [START_REF] Brogliato | Digital implementation of slidingmode control via the implicit method: A tutorial[END_REF], the matrix inequality [START_REF] Koch | Discrete-time implementation of homogeneous differentiators[END_REF] has to be fulfilled together with [START_REF] Hong | H∞ control, stabilization, and input-output stability of nonlinear systems with homogeneous properties[END_REF].

Remark 3 (On feasibility of matrix inequalities): If n = 2, ρ = 1, µ = -1 then the matrix

X = x 22 1 32 -1 16 -1 16 1
, x 22 > 0 satisfy both conditions [START_REF] Koch | Discrete-time implementation of homogeneous differentiators[END_REF] and [START_REF] Hong | H∞ control, stabilization, and input-output stability of nonlinear systems with homogeneous properties[END_REF]. Indeed, in this case, we have X -1 = 256 7x 22 Let us consider the matrix-valued function W : (0, 1/2] → R 2 given by

W ( h) = 1 1 16 1 16 1 32 -F ( h) 1 1 16 1 16 1 32 F ( h).
The minimal eigenvalue of W ( h) is always positive for all h ∈ (0, 1/2] (see, Fig. 1).

The latter means

X -1 -F ( h) X -1 F ( h) 0, ∀ h ∈ (0, 1/2].
Notice, despite that the gain K is fixed the settling time of the closed-loop system can be tuned by selection of x 22 . Indeed, the settling time of the closed-loop system (1), [START_REF] Brogliato | Digital implementation of slidingmode control via the implicit method: A tutorial[END_REF] with n = 2, µ = -1, ρ = 1 is given by the canonical homogeneous norm • d which depends on X. It is easy to see that larger x 22 faster the convergence of the system.

Remark 4 (Practical implementation of sampled control): The sampled control [START_REF] Zimenko | Robust feedback stabilization of linear mimo systems using generalized homogenization[END_REF] is explicit for µ = -1. Indeed, the canonical homogeneous norm x d is defined explicitly (see Remark 2). The matrix Q 2h ( x d ) has the explicit representation as well since for µ

= -1, ρ = 1 one holds e - G d µ ln 1+µρ2h x(t) µ d x(t) µ d = φ 2 0 0 φ , φ = x d -2h, e -G d ln x(t) d = 1/ x 2 d 0 0 1/ x d
and there exists J ∈ R 2×2 (see, the proof of Theorem 4) such that e (A+B(K 0 +K)+ρG d ) ln(1+µρ2h x(t) µ d ) ρµ = J -1 cos(ln φ) sin(ln φ)

-sin(ln φ) cos(ln φ) J.

Hence, for x d > 2h we, obviously, have

Q 2h ( x d ) = φ 0 0 1 J -1 φ cos(ln φ) φ sin(ln φ) -φ sin(ln φ) φ cos(ln φ) J 1 x 2 d 0 0 1 x d , with φ = x d -2h, so Q 2h → 0 as φ → 0 + . V. NUMERICAL EXAMPLE
Let us consider the control law u given by the formula [START_REF] Brogliato | Digital implementation of slidingmode control via the implicit method: A tutorial[END_REF] and designed according to Remarks 2 and 3 for n = 2, µ = -1, ρ = 1 and x 22 = 1. The sampling period is h = 0.1. The initial condition is x 0 = [ 2 1 ]. The results of the consistent discretization u h of the control u are depicted in Fig. 3. For comparison purposes, we present the simulation results (see Fig. 2 ) of the explicit discretization

u(t) = u(x(t i )), t ∈ [t i , t i+1 ]
of the homogeneous 2SM control [START_REF] Brogliato | Digital implementation of slidingmode control via the implicit method: A tutorial[END_REF].

The numerical simulations confirm our theoretical conclusion then the consistent discretization of the homogeneous SM control guarantees the finite time stability of the continuous-time system even in the case of the sampledtime implementation of the control law. The settling-time obtained by the simulation (3.6 sec) is close to the theoretical settling-time of the continuous-time system (T (x 0 ) = x 0 d ≈ 3.7442 sec). Computations for various sampling periods show that the error of the settling-time estimate is O(h). The system with the explicitly discretized control has a large chattering (see Fig. 2) even in the disturbancefree case. The consistent discretization scheme allows a chattering reduction even in the case of perturbations. To show this the system with the matched additive perturbation d(t) = B sin(t) has been simulated as well. Fig. 4 presents the simulation results for the perturbed system with the consistently discretized controller. The zero set-point is well tracked even in the case of matched perturbation. It is worth stressing that the control does not use any knowledge about the perturbation.

VI. CONCLUSIONS

The paper introduces a new concept of a digital (sampledtime) implementation of the control law, which is called a "consistent discretization of a control". It consists in a digital (sampled) implementation of the control law in such a way that the closed-loop system with a sampled control implementation preserves convergence rate of the system with the original (continuous-time) control law. In this paper, we present such a discretization for a homogeneous finite-time 2-SM controller. Numerical simulations confirm essential improvement of the control precision (comparing with conventional explicit scheme). Development of consistent discretization schemes for other popular finite-time and fixed-time controllers seems an interesting direction for the further research. A rigorous robustness analysis of the consistently discretized control is an important problem as well.

  Ad(s) = e -s d(s)A, d(s)B = e s B, d(s)x d = e s x d and d(ln d(s)x d ) = d(ln x d )d(-s). Hence, we have z(e s h, d(s)x) = d(s)z(h, x), ∀s ∈ R, ∀(h, x) ∈ D. (27) Indeed, z(e s h,d(s)x)=d(s)x+e s hAd(s)x+(e s hB+ e 2s h 2 2 AB)[ 1 e 2s h 2 -1 2e s h ]• d(ln d(s)x d )e G d ln 1-2e s h d(s)x d •e -(A+BK+G d ) ln 1-2he s d(s)x d d(ln d(s)x d )-I2-2e s hA d(s)x=

Fig. 1 :

 1 Fig. 1: The minimal eigenvalue of the matrix W ( h)

Fig. 2 :

 2 Fig.2: Evolution of the system with the explicitly discretized control (the disturbance-free case).

Fig. 3 :

 3 Fig.3: Evolution of the system with the consistently discretized control (the disturbance-free case).

Fig. 4 :

 4 Fig.4: Evolution of the system with the consistently discretized control (the perturbed case).

A matrix G d ∈ R n×n is called aniti-Hurwitz if -G d is Hurwitz.

A dilation in R n is monotone if and only if for any x ∈ R n the function s → d(s)x , s ∈ R is monotone.


I Janowska
email: janowskai@unistra.fr

Evaporation induced self-assembling of few layer graphene into fractal-like conductive macro-network: the way to reduce percolation threshold

Transcription of nanoproperties to others dimensions in efficient and simple way by appropriate design of devices is a challenge; and conductive nanocarbon systems are concerned. Here, the evaporation induced self-assembling method is proposed to form branched, fractal-like "2D" conductive macrostructures from (nano) micro-few layer graphene flakes. The self-assembled conductive graphene networks reveal a reduction of percolation threshold compared to random arrangement (reduction of matter for a given surface and conduction at lower surface coverage), and becomes a promising matrix for conductive (transparent) films.

 The micro(nano-)graphenes originated from top down approach synthesis (exfoliation of graphite based materials) exhibit usually lower conductivity and their

(transparent) conductive films formed by spin coating, hot-spray or other techniques show few-few-dozen kΩ sheet resistance. [6][7][8][9][10] Varied transparency/conductivity property of the films is required depend on the final (opto)electronic application (e-readers, electrochromic windows, touch sensors, photovoltaic, etc…); 11 this relationship can be modified by films' thickness within a given matter. A variant is low surface coverage e.g. formation of graphenecarbon nanotubes hybrids, graphene mesh electrodes. [12][13][14][15] At present, we propose the evaporation induced self-assembling method as an alternative way to reach highly conductive graphene (few layer graphene) network at macroscale. The approach leads to branched, fractal-like patterns. In principle, these autoscalable structures achieve the goal of nano micro  macro scale multiplication. Apart from branched geometries of trees or river beds widely present in nature, the fractals approach is also present in time or market analysis. [START_REF] Mandelbrot | The Fractal Geometry of Nature[END_REF][START_REF] Manna | [END_REF][START_REF] Vrobel | Fractal Time: Why a Watched Kettle Never Boils[END_REF][START_REF] Peters | Fractal Market Analysis: Applying Chaos Theory to Investment and Economics[END_REF] The formation of self-assembled structures from dried nanofluids is not entirely understood and final morphologies depend on many factors such as solvent, evaporation conditions, particle size, identity, concentration and thermodynamic state, which affect chemical potential, nanoparticles mobility and attractions (repulsions) between particles, particles and substrate, particles and solvent. [START_REF] Vancea | [END_REF][21][22] Initially it was described by hydrodynamic instabilities and solute transports phenomena. 23,24 Further, the formation of branched structure could be predicted with Kinetic Monte Carlo (KMC) dynamics and "coarse-grained" two dimensional microscopic lattice gas model based on uniformity of solvent dynamics and the fluctuation of nanoparticles boundaries following evaporation. 25 Following investigations introduced additional instability at the macroscopic solvent-airsubstrate contact line in accordance with Marangoni effect. [START_REF] Vancea | [END_REF]21 The recently reported "open" domain simulations consider also particles aggregation during evaporation process, assisted by shrinking of drying fronts from the edges toward the center. 26 On the other hand, the aggregation of particles appeared in general as a central problem in the applied colloid science and was described by diffusion-limited aggregation (DLA) and cluster cluster aggregation (CA) mechanisms based on Monte Carlo simulations. [27][28][29][START_REF] Hentschel | The structure and fractal dimension of cluster-cluster aggregates 117-120[END_REF][START_REF] Schaefer | Aggregation of colloidal silica in Kinetics of Aggregation and Gelation[END_REF] The aggregation process limited by diffusion of particles to the aggregate surface is similar to precipitation of species from a supersaturated matrix or to a crystal grows from a supercooled melt, 29 while DLA model produces complicated shapes such as dendrits and clusters with fractal dimensionality. Curiously, micro-dendritic and fractals mono and polycrystals were observed for CVD grown graphene islands on copper and gold foils which suggested diffusion-limited process; [START_REF] Nie | [END_REF][33][34] the fractal etching of graphene on liquid copper surface was reported as well. 35 Recently Wang et al. reported on the self-assembly of DNA and biofunctional polymer on graphene nanorribons, where non-covalent interactions (van der Waals, charge transfer) permitted an adhesion of graphene to SiO 2 surface and assembling of the macromolecules on graphene. 36,37 While a formation of branched structures is more elucidated, their impact on percolation phenomenon in the conductive films is rarely studied. The germinal theoretical simulations show that reduction of percolation threshold can occur in self-assembled patterns compared to randomly assembled structures. 38,39 Two types of few layer graphene (FLG) flakes originated from different synthesis methods are here considered. The FLG obtained by mechanical ablation of pencil lead has an average lateral size of 2.5 µm and multi step structures (the lateral size of sheets differs within a given flake, (FLG-Abl). 40,41 The second FLG was extracted from expanded graphite (FLG-EG) under µ-waves irradiations, it exhibits higher 2D aspect compared to FLG-Abl i.e. few up to dozen of µm lateral sheets size with similar average sheets number ( 6 and up to 20). Both FLGs have similar atomic oxygen percentage (5%, XPS), while higher conductivity was measured for FLG-EG. The minimal resistance measured for FLG-EG individual flakes is 500Ω, while for FLG-Abl flakes a min. resistance was of 1.6 kΩ after annealing at 800°-900°C. 40 Three different solvents are used in the evaporation process: chloroform, toluene and ethanol. These are the solvents having an ability to significantly disperse graphene, which aimed to avoid an agglomeration of the flakes prior to evaporation processes. A diffusion rate was additionally influenced by evaporation time; the samples are left to dry in open or closed systems. The ethanol suspension of FLG-Abl was drop casted on flat glass plate and left as such for drying (f-1), two toluene suspension of FLG-Abl were left in the closed glass vessel with two different concentrations (f-2 higher, f-3 lower); and chloroform suspension of FLG-EG was drop casted on flat glass plate and left for evaporation in closed vessel.

Once the evaporation processes finished, all investigated FLGs formed well-organized continuous branched patterns (f-1, f-2 in Fig. 1, f-3 in Fig. 2 and f-4 in Fig. 3), which in majority can be detected by eyes; and mostly with fractal characteristics.

Figure1.Optical photos of self-assembled patterns formed via evaporation: f-1 from FLGpencil in ethanol in open system on flat glass plate, f-2 from FLG-pencil in toluene in closed system on bossy round substrate, f-4 from FLG-EG in chloroform in closed system on glass plate. 2), which is in agreement with higher 2D aspect (lateral size/thickness) of its components. In the patterns formed from FLG-Abl, the flakes have tendency to lift from the surface and stack each other at higher angle from the surface, which is due to their lower 2D feature and/or multi step structure. 41 In traditional approach they assimilate rather "particles" than "sheets" behavior. The observed densification of matter in a center of f-1, f-2, and f-3 is in accordance with "open" domain simulations, where an amount of flakes are pulled toward the center after shrinking of drying front. 26 F-4 network exhibits more homogenous overall density and "order" of branches, where main branches initiate from one side of the substrate plate. The main branches have seeds in a center of plate in fractals f-1, and f-3 (fig. 1, fig. 3) and their density as well as a density of sub-branches decrease with increase of substrate radius from that center. In f-2 (fig. 1), where a round, bossy surface and long time evaporation was applied, the overall pattern consists of three connected circular networks of trees which keep the same morphology but their size and (sub-)branches densities change periodically with the substrate radius (density decreases and size of substructures increases with the radius increase). This circular symmetry suggests a uniform evaporation and density of the FLG particles on the overall surface (number of flakes/substrate area), while the substrate center remains FLG-free and the seeds of trees are arranged at three different radius circular lines due to the bossy form of the substrate.

1 cm f-1 1 mm f-2 f-2 f-4 1 cm
The AFM topography profiles performed for f-4 (fig. 4) indicate inhomogeneous thickness of the network, while the highest detected points ( 400 nm) are linked to lifted flakes extremities, the flakes which are localized on the top but not at the edges of patterns (see also The fractal dimension (D f ) can be defined as logN(l)/logl, where "N(l)" is a number of boxes and "l" is a length of the boxes. D f was estimated for f-3 and f-4 by box counting method via linear fitting of the logarithmic relation with the "ImageJ" software. [START_REF] Mandelbrot | Les objects fractals: forme, hazard et dimension[END_REF] Prior to the counting the chosen SEM micrographs were converted into binary images, which are next divided by squares (boxes) with varied size from 2 to 64 pixels. Fig. 5 demonstrates the representative binary images of f-3 and f-4 with corresponding logN(l)/logl plots. The D f , a slope of the plots are 1.73  0.02 for both networks and suggests diffusion limited aggregation mechanism (DLA). 27,28 For few images D f goes up to higher value, 1.85  0.02, which can be related to cross-linking between the branches. The charge transport properties of f-1, f-3 and f-4 networks were investigated by four point probe (FPP) method with distance fixed probes, by injection of current in the range up to 10 - 2 A. The experimental values were multiplied by geometrical correction factor according to the method protocol of FPP method for thin flms. [START_REF] Smit | [END_REF] The measurements in addition were performed after annealing of the samples at 400°C in He for 4h, which aimed to removed adsorbed The main finding from the charge transport measurements is that the percolation threshold is strongly reduced in the branched, fractal-like networks compared to a random arrangement, which confirms preliminary simulation results for self-assembled gold nanoparticles. 36 The performed earlier investigations considering FLG-Abl flakes shows, that when they are randomly deposited on glass substrate by hot spray techniques, a charge transport performance is much lower. The minimal resistance of the formed film and at minimal coverage, after annealing at 800°-900°C, was 760  by Hall Effect technique, while the FPP measurements showed 85 k and 15 k before and after annealing respectively. Here, obtained by the FPP Rs for assemblies are one, two orders of magnitude lower, moreover the conductive film with random arrangement could not be achieved for a transparency higher than 32% (high coverage). 44 It is however difficult to determine the exact impact degree of self-assembly on conductivity improvement due to the undetermined graphene-graphene junction and local thickness variation. One thing is certain; the presented self assembly allows to occur a percolation at lower FLG concentration for a given surface, while the low coverage of substrate surface can increase additionally a transparency of the film. To improve a transparency initial concentration of the flakes needs to be also much lowered to avoid excessive stacking in "z" direction. High conductivity and related continuity of the fractal networks are in addition reflected by SEM micrographs quality. A very low charging effect of the assemblies occurs during the analysis despite the large and thick insulator substrates of which significant surface area is FLG free. Beside the observed black FLG conductive paths, separated and charged FLG flakes can be punctually identified on network-free areas (flashes, fig. 2 and3).

The ohmic behavior of the fractal networks and such low sheet resistance confirm first: a high quality of the flakes (diffusive motion of electrons is limited only by scattering at functional groups and lattice defects); 45 second: an overlapping of the flakes (HR-SEM, fig. 2 and 3) and formation of bilayer boundary region via van der Waals forces, which allows electron flow to overcome edges defect by connection of sp 2 carbons from overlapped flakes (contrary to direct atomic bonding with discrete atomic domain boundary), 46 third and principal: the self-assembled branched network strongly helps the percolation to occur compared to random assembly. The latter is of huge importance in view of (opto)electronic applications, where continuous branched networks constitute a path for charge transport, and when assisted by low coverage of substrate increases a transparency of the film similarly to graphene-carbon nanotubes hybrids or graphene mesh approach. [12][13][14][15] Moreover, according to "open" domain simulations branched structures can be reached at low surface coverage, only 10% . 26 In conclusion, it should be underline that the presented evaporation conditions were not optimized with respect to (transparent) conductive films, and mainly aimed to demonstrate a possibility to form macroscopic self-assembled conductive graphene networks from (micro)nanoscopic matter with reduced percolation threshold (lower matter concentration is sufficient to reach the percolation for a given surface compare to random arrangement), via easy, cost-effective and substrate-size unlimited method. Several parameters need to be set in order to fully understand the graphene fractals-like self-assembly formation and to control conductivity and (or) transparency property in most important manner. The thickness of the networks can be optimized by the use of lower initial concentration and subsequent higher dispersion of flakes containing lower sheets number; and adapted evaporation conditions. The appropriate choice of the conditions would modify a conductivity/transparency ratio withtin a given matter.

Figure 2 .

 2 Figure 2. SEM micrographs of self-assembled pattern of FLG-pencil formed via evaporation of toluene in closed system (f-3).

Figure 3 .

 3 Figure 3. SEM micrographs of self-assembled pattern of expanded graphite originated few layer graphene formed via evaporation of chloroform in closed system on glass plate (f-4).

fig. 3 ,

 3 fig.3, HR images). The flakes at network edges making interface with glass lie flat confirming significant adhesion interactions (thickness of few nanometers is detected at the edges, fig.4right, see also fig.3HR images), while the top curved flakes suggests an excess of the flakes, which is linked to excessive initial concentration.

Figure 4 .

 4 Figure 4. The AFM image with two profiles performed at the edges of patterns (right), and corresponding AFM tomography profile.

Figure 5 .

 5 Figure 5. Binary images of f-3 and f-4 and corresponding logN(l)/logl plots.

  impurities and solvents residues. The networks exhibit ohmic behaviors; representative I(V) curves with corresponding calculated resistances are presented in fig.6. The resistance values before annealing are 15.4 k, 2.4 k for f-1 and f-3 respectively (high density center omitted), and 4.4 k and 541  for f-4 (at lower and higher density). The resistance values significantly drop after annealing treatment an order of magnitude to 2.3 k, 130  for f-1 and f-3 respectively and to 97  and 48  for f-4. The relatively lower resistance of f-4 (48 , 97  depending on local thickness) compared to FLG-Abl are in agreement with higher conductivity of individual FLG-EG components, however other factors such as flakes identity, different evaporation conditions and related pattern morphology can play an additional role.

Figure 6 .

 6 Figure 6. Representative I(V) curves obtained for f-1, f-3 and f-4 by four point method with corresponding resistance values.

Acknowledgements

I would like to thank "Conectus" program (Alsace) for financial support; Yue Feng Liu and Azhar A. Pirzado for help during SEM microscopy analysis and four point probe measurements.