Florence Runyo

Marie Matignon

Etienne Audureau

William Vindrios

Anna Boueilh

Camille Gomart

Philippe Grimbert

Sébastien Gallien

Dr Giovanna Melica
email: giovanna.melica@aphp.fr

Infectious disease consultation is effective in boosting vaccine coverage in patients awaiting kidney transplantation: A French prospective study

Keywords: End stage renal disease GFR: glomerular filtration rate IF: Infectious Diseases IGRA: Interferon gamma release assays IQR: interquartile range LTBI: Latent tuberculosis infection MSM: men having sex with men MMR: measles, mumps, rubella SD: standard deviation TB: tuberculosis TDPaP: tetanus-diphtheria-polio-acellular pertussis VZV: varicella zoster virus

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Infectious disease consultation is effective in boosting vaccine coverage in patients awaiting kidney transplantation: a French prospective study

Florence Runyo [START_REF] Wolfe | Comparison of Mortality in All Patients on Dialysis, Patients on Dialysis Awaiting Transplantation, and Recipients of a First Cadaveric Transplant[END_REF][START_REF] Rose | Association of Kidney Transplantation with Survival in Patients with Long Dialysis Exposure[END_REF] , Marie Matignon 2, 3 , Etienne Audureau [START_REF] Sood | Kidney Transplant Outcomes After Primary, Repeat and Kidney After Nonrenal Solid Organ Transplantation: A Single-Center Experience[END_REF] , William Vindrios 1,5 , Anna Boueilh 2 , Camille Gomart [START_REF] Kinnunen | Secular Trends in Infection-Related Mortality after Kidney Transplantation[END_REF] , Philippe Grimbert 2,3 , Sébastien Gallien [START_REF] Wolfe | Comparison of Mortality in All Patients on Dialysis, Patients on Dialysis Awaiting Transplantation, and Recipients of a First Cadaveric Transplant[END_REF][START_REF]Special Issue: KDIGO Clinical Practice Guideline for the Care of Kidney Transplant Recipients[END_REF] , Giovanna Melica [START_REF] Wolfe | Comparison of Mortality in All Patients on Dialysis, Patients on Dialysis Awaiting Transplantation, and Recipients of a First Cadaveric Transplant[END_REF][START_REF]Special Issue: KDIGO Clinical Practice Guideline for the Care of Kidney Transplant Recipients[END_REF] 1: AP-HP (Assistance Publique-Hôpitaux de Paris), Infectious Diseases Department, Groupe Hospitalier Henri-Mondor/Albert Chenevier, Créteil, F-94010, France

Introduction

Kidney transplantation remains the best treatment for end-stage renal disease (ESRD) [START_REF] Wolfe | Comparison of Mortality in All Patients on Dialysis, Patients on Dialysis Awaiting Transplantation, and Recipients of a First Cadaveric Transplant[END_REF] , ensuring significant improvement in the quality of life of kidney failure patients, [START_REF] Rose | Association of Kidney Transplantation with Survival in Patients with Long Dialysis Exposure[END_REF] and fiveyear survival of transplant recipient's rate over 85% between 2012 and 2014 3 ˒ 4 Infections are the leading cause of non-cardiovascular mortality in kidney allograft recipients reaching 15 to 35% of deaths despite a functioning graft [START_REF]Special Issue: KDIGO Clinical Practice Guideline for the Care of Kidney Transplant Recipients[END_REF] ˒ 6 , which is significantly higher than the rate in the general population. Worldwide, preventive strategies employed before and after kidney transplantation have reduced infection-related mortality by half [START_REF] Kinnunen | Secular Trends in Infection-Related Mortality after Kidney Transplantation[END_REF] .

Before transplantation, anti-infection preventive strategies should include the treatment of potential latent tuberculosis infection (LTBI) 7 ˒ 8 ˒ 9 ˒ 10 , and an update of the national and local recommended pre-transplantation vaccines (French Guidelines"Immunizations for immunocompromised patients" from public health officials HCSP 2013) 11 ˒ 12 ˒ 13 . Vaccinepreventable infections have been associated with significant morbidity and mortality in solid organ transplant recipients, and the efficacy of post-transplantation vaccination seems to be lower [START_REF] Kotton | Vaccinations in Kidney Transplant Patients: Searching for Optimal Protection[END_REF] . Despite recommendations, the overall immunization rate against common vaccinepreventable infections remains low, as described in previous reports [START_REF] Lee | Low rates of vaccination in listed kidney transplant candidates[END_REF] . Interestingly, chronic kidney disease (CKD) patients showed better acceptability for pre-transplantation vaccinations if were offered during an infectious disease (ID) consultation or if vaccines were administered in dialysis centers. [START_REF] Kasper | Infectious diseases consult improves vaccination adherence in kidney transplant candidates[END_REF] Few European data are currently available on the subject. We performed a monocentric prospective study to evaluate pre-transplantation vaccination coverage, and to assess the impact of ID consultation on vaccination and on LTBI treatment rates.

Patients and Methods

Patients

We conducted a single center prospective descriptive cohort study, from January 2014 to December 2018, including all adult kidney allograft candidates listed on the French national registry in our center, and for whom an ID consultation was initiated in January 2014. All patients were informed about the use of their personal data for research. The study was approved by the local ethical committee (IRB#00003835).

Vaccinations

Information regarding types and dates of previous vaccinations against hepatitis A and B, Pneumococcus, Influenza, and tetanus-diphtheria-polio-acellular pertussis (TDPaP) was recorded. Serological status of hepatitis A and B, varicella-zoster virus (VZV), measles, and yellow fever in previously vaccinated patients were evaluated.

The catch up strategy for vaccinations based on the French national vaccination schedule and recommendations for immunocompromised patients/patients awaiting solid organ transplantation was proposed and customized as described in table 1 [START_REF] Mathew | Vaccination issues in patients with chronic kidney disease[END_REF] . For live attenuated vaccines, a temporary contraindication to transplantation of at least one month after being vaccinated was proposed.

Patients began the vaccine catch up schedule the day of the ID consultations and got vaccinated the same day by the outpatient nurse. Potential subsequent vaccine boosters were administered thereafter by the dialysis staff, and/or by the general practitioners.

We did not propose additional ID consultations for follow up.

Tuberculosis

A full past history on previous tuberculosis (TB) episodes or TB exposure was recorded. Pretransplantation chest X-rays and/or computed tomography (CT) scans were reviewed in all patients looking for active TB lesions or old healed TB lesions in asymptomatic never treated patients.

All patients included in the study underwent LTBI screening using Quantiferon-TB Gold-intube (QTF, Qiagen GmbH, Germany), an Interferon gamma release assays (IGRA), since it is documented as the most reliable test to diagnose LTBI in people with ESRD [START_REF] Rogerson | Tests for Latent Tuberculosis in People With ESRD: A Systematic Review[END_REF] . LTBI treatment was suggested to quantiferon positive patients and/or patients with radiological old healed TB lesions, not previously treated for tuberculosis, as follows: isoniazid alone at 3mg/kg for 9 months or combination of rifampicin (10mg/kg) and isoniazid (3mg/kg) for 3 months 8 ˒ 9 ˒ 10 . Furthermore, clinical (past history), biological, and radiological evaluation was performed to assess individual infectious risk leading to prophylaxis recommendations after transplant.

ID counseling enabled tailoring post-transplantation preventive strategies to patients' needs, such as the duration of cotrimoxazole prophylaxis in high risk Pneumocystis infections or, whenever needed, introducing another prophylaxis, e.g. posaconazole for previous invasive mold infections. We did not test patients for other potentially opportunistic pathogens like Strongyloides, Histoplasma or Leishmania.

Statistical analysis

Continuous variables were expressed in mean (± Standard Deviation [SD]) or median (+ Interquartile Range [IQR]). Categorical variables were expressed as percentages (%).

Comparisons between dialyzed and non-dialyzed patients and between patients previously vaccinated or not against influenza and pneumococcus were performed using Student-t test for non-categorical variables, and Fischer exact test for categorical ones. All analyses were performed on STATA V14.1 software.

Results

Patients' characteristics

From 2014 to 2018, 994 kidney transplantation candidates were listed on the French national registry, of whom 467 (47%) were evaluated by an infectious diseases specialist.

Explicitly, only half of the listed patients underwent ID consultation because of a resource issue at the beginnings, but the number of ID consultations increased over time reaching 147/250 (59%) patients in 2018 (Figure 1). Baseline characteristics of the study population are shown in Table 2. Our cohort included 302 (65%) men and 165 (35%) women (sex ratio 1.83), with 191 (41%) born out of Europe. Median age was 58 [46-66] years and 333 (71%) patients were on dialysis. Main causes of initial nephropathy included diabetes (N=116 (25%)) and immunological diseases (N=80 (18%)).

On the whole, 107(23%) patients were immunosuppressed: 15 (3%) patients were HIV positive, 10 (2%) were asplenic, 64 (14%) already had a transplant (kidney or others), and 18 (4%) were currently treated with immunosuppressive agents. For note, 252 patients (54%) planned to travel abroad after kidney transplantation.

Vaccination and immunity

Overall, immunization rates were low for TDPaP and Pneumococcus with 109 patients (24%) and 27 patients (6%) vaccinated before the ID consultation, respectively. Two hundred twenty-eight patients (51%) were vaccinated against Influenza in the previous year. For Pneumococcus, the 27 previously vaccinated patients had received the complete vaccine schedule, with one shot of 13-valent polysaccharide conjugate vaccine followed by 23-valent polysaccharide vaccine two months after.

Regarding hepatitis B, 252 (54%) patients were immunized and 84 (18%) had chronic hepatitis B or cured hepatitis. Sixteen patients of the cohort (3.6%) had past, treated hepatitis C virus infection. Regarding measles, 343/351 (98%) patients were IgG positives. Almost all patients tested for VZV IgG antibodies presented a past immunity (418/422, 99%).

Since June 2015, a live-attenuated shingle vaccine (Zostavax) has been proposed to people older than 65 years; in our cohort, 122/467 (26%) patients got Zostavax shots.

Concerning travel prophylactic vaccines, 68/368 patients (18%) were previously vaccinated against Yellow Fever, and 48/368 (13%) had Yellow Fever antibodies > 10IU. Three quarters of patients were already immunized against hepatitis A (224, 75%).

Patients insufficiently vaccinated were offered the catch up vaccination schedule directly after the ID consultation, as appropriate. Otherwise, vaccines were prescribed by the ID specialist and administered by the dialysis staffs or by the general practitioner. This strategy demonstrated a very high catch up vaccination rate. The exact proportions of shots administered by dialysis staff or by GPs were not quantified.

Vaccine acceptability was high with 465 (99%) patients who agreed to be vaccinated during the ID consultation. Only two patients refused.

Comparison according to patients' subgroups

We compared, in our cohort, patients treated or not with dialysis (table 4) to see whether dialyses patients received a better vaccination catch up strategy for recommended vaccinations than their counterparts. Age and gender were similar in both groups. There were more diabetic and sub-Saharan patients in the dialysis group (41.7% vs. 30.6%, P=0.027 for the former, and 23% vs. 13.6%, P = 0.003 for the latter).

For those vaccinated against Pneumococcus before the ID consultation, only 27/447 (6%) had received the current recommended vaccine schedule.

We compared previously vaccinated against Pneumococcus versus non-vaccinated patients.

Immunodepressive conditions were significantly associated with previous Pneumococcus vaccination (9/27, 33% vs 70/420, 17%, p=0.028; 6 asplenic condition, 3 HIV positive).

Similarly, updated TDpaP and Influenza vaccination were higher in Pneumococcus vaccinated patients than non-vaccinated ones, 11/27 vs. 98/420 (41 vs. 23%, p=0.041) and 19/27 vs. 209/420 (70 vs. 50%, p=0.038), respectively. Patients who had not been dialyzed yet were significantly less vaccinated against HBV (62/134, 46%) than patients on dialysis (190/333, 57%, p< 0.0001). Non-dialyzed patients were more often up to date for the TDPaP vaccine (43/134, 33%) than dialyzed patients were (68/333, 20%, p=0.005). On the other hand, no difference between the two groups was observed for influenza and pneumococcus vaccination.

Diagnosis and treatment of LTBI

Thirty (6%) patients had a past history of treated tuberculosis.

Quantiferon result (QTF) was available in N= 439 (94%) patients, of whom 221 (47%) patients had already lived in a TB-endemic country for more than 3 months. QTF was positive in 75 (16%) patients, undetermined in 3 (1 %), and negative in 364 (82%).

Chest X-ray or CT-scan displayed old healed lesions in 26 (6%) patients with no signs of active illness. TB preventive treatment was prescribed in 77/439 (17%) patients: 69 patients with positive QTF and eight with negative QTF but presented never-treated radiological lesions.

We decided not to treat patients with a past history of a well-conducted TB treatment, with or without radiological sequelae, independently of their positive or negative QTF. Among treated patients, 61 patients (79%) were treated with a 9-month course of isoniazid alone, and 16 (21%) with a 3-month course of rifampicin and isoniazid. Only nine patients (11%) presented drug-related adverse events, mostly digestive symptoms (n=4) and moderate transaminases flare (n=2) (grade 1 of CTCAE criteria). One patient had an episode of generalized seizure attributed to isoniazid neurotoxicity and was the only patient to discontinue LTBI treatment in this cohort.

Discussion

Herein we present the first French cohort of patients listed for kidney allograft transplantation and who received ID consultation before transplantation as recommended by the national and international guidelines. 5 ˒ 8 ˒ 11 ˒ 12 ˒ 14 This collaborative project between nephrologists and ID specialists was prompted by the increasing number of kidney transplantations in our center and the need to ensure a structured preventive strategy.

Moreover, double-organ transplantation and second kidney transplantation are also increasing leading to more profound immunodepressive conditions.

In our center, from 2014 to 2018, the number of pre-transplantation infectious disease consultations increased over time reaching 59% of patients in 2018. The pre-transplantation period might be the perfect time for catch up preventive strategies, in particular vaccinations, since the immunogenicity status is much better in pre-than in posttransplantation where the immunosuppressive drugs reduce humoral and cellular responses adaptive immune responses, which predisposes them to infections and vaccine hyporesponsiveness; these effects increase with more advanced stages as well as with concomitant immunosuppressive treatment 2 ˒ 19 . Thus, it seems logical to vaccinate patients as early as possible in the course of their disease.

In this prospective observational study, we were able to show the low vaccination rates in kidney transplant candidates despite recommendations. According to patients' past history, immunization rates were extremely low for Pneumococcus and TDPaP (6% and 24% of patients, respectively) before ID consultation.

Our results are in line with that of previous studies 15 ˒ 16 ˒ 18 ˒ 20 . In particular, we found very low rates of pneumococcal vaccination (6%), which is even lower than results from recently published studies 15 ˒ 18 . If we consider comorbidities in listed kidney transplant patients, it is clear that for many reasons these patients should be vaccinated against Pneumococcal infections [START_REF] Vandecasteele | The ABC of pneumococcal infections and vaccination in patients with chronic kidney disease: Table 1[END_REF] . Of more and given that the incidence of invasive pneumococcal diseases after kidney transplantation is higher than that in the general population, Pneumococcal vaccination has been recommended for all adults with GFR below 30 ml/min since 2013 [START_REF] Vandecasteele | The ABC of pneumococcal infections and vaccination in patients with chronic kidney disease: Table 1[END_REF] .

For specific vaccines, the coverage was better in particular settings: hepatitis B for patients on dialysis, pneumococcus and influenza for previously immunodepressed patients, and influenza for older listed patients. Being previously vaccinated against pneumococcus was significantly associated with other immunosuppressive conditions as expected. Likewise, patients already vaccinated against Pneumococcus were also often vaccinated against seasonal influenza and TDPaP. It is likely that in severely immunocompromised patients preventive strategies were more strictly applied than in the other patients of the cohort.

For influenza, 51% of the patients were vaccinated in the previous year. Having received seasonal influenza vaccine was associated with older age and a better coverage for pneumococcus and TDPaP vaccination. Overall, aged patients were better vaccinated than younger ones.

Regarding patients on dialysis, we noticed that a better vaccination coverage was achieved in dialyzed patients only for hepatitis B, whereas for TDPaP and Pneumococcus vaccines dialyzed patients were insufficiently covered. Being vaccinated during the dialysis period could be an excellent choice for catch up vaccine strategies; it should be implemented, with a regular training for the dialysis staff whilst maintaining optimum networking with the ID consultation and transplantation centers. The use of an electronic immunization registry can be a practical tool to share information between centers on patients' history, immunization and serological results. Developing a vaccination checklist could augment the assessment of vaccines coverage, for example, in dialysis centers, and facilitate administrative work.

Of note, 54% of our patients planned to travel abroad after the transplantation, and 23% of the patients were already on immunosuppressive regimen before the transplantation. The issue of vaccination in immunosuppressed travelers and timing of live attenuated vaccines, e.g. for yellow fever, should be discussed. Such situations highlight the need for a specific pre-transplantation ID consultation tailored to patients' needs.

Interestingly, the acceptability of the ID consultation and of the vaccinations was very high, with only two patients refusing to be vaccinated before the graft. We described the high acceptability of proposed vaccinations but we neither quantified the number of administered vaccines nor reported the success of vaccination coverage in details because catch up schedules were often completed in dialysis centers or by the general practitioners.

Similarly, a study from Blanchard-Rohner et al. [START_REF] Blanchard-Rohner | Usefulness of a systematic approach at listing for vaccine prevention in solid organ transplant candidates[END_REF] showed that catch up vaccinations were well accepted in solid organ transplantation patients. Programmatic vaccination success was evaluated by pre-graft serology in transplanted patients and showed that catch up immunizations significantly increased patients' immunity. It has already been described in previous studies that patients with chronic diseases are in favor of preventive vaccination 19 ˒ 22 . Our finding contrasts with general public perception of vaccination. There is an overall lack of confidence in vaccination in several European countries, especially in France, with concerns regarding vaccines safety and efficacy, despite institutional reassuring messages [START_REF] Larson | The State of Vaccine Confidence 2016: Global Insights Through a 67-Country Survey[END_REF] .

The lack of confidence in vaccines safety may also concern the medical staffs, especially in kidney transplantation centers, generated by the potential risk of increasing donor specific antibodies and/or triggering graft rejection after vaccination. However, a recent metaanalysis on adverse immunologic events in solid organ transplantations did not show a risk of de novo donor specific antibodies or rejection after vaccination [START_REF] Mulley | Does vaccination in solid-organ transplant recipients result in adverse immunologic sequelae? A systematic review and meta-analysis[END_REF] . Furthermore, the American Society of Transplantation guidelines in 2013 stated that no evidence links clinical rejection episodes to vaccination [START_REF] Danziger-Isakov | AST Infectious Diseases Community of Practice. Vaccination in solid organ transplantation[END_REF] .

In alignment with the above mentioned, the acceptability of latent TB testing and treatment was excellent and no one refused treatment before the graft. Given the higher incidence and mortality of TB (up to 30%) in solid organ transplant recipients as compared with the general population 24 ˒ 25 , several guidelines recommend targeted isoniazid preventive therapy or isoniazid plus rifampicin for transplant candidates who were diagnosed with LTBI 25 ˒ 26 . In our cohort, the prevalence of LTBI infection was 16% and all patients in whom LTBI was diagnosed and who had never been treated for TB received a preventive therapy (with isoniazid alone for 79%). No case of active tuberculosis was discovered in the framework for latent TB screening. Additionally, tolerance was fairly good as only 11% of patients experienced adverse reactions and one had severe adverse event in form of generalized seizure caused by isoniazid. Even if some factors may limit the utility of post-transplantation TB preventive strategy (as limited accuracy of IGRAs or de novo acquisition of tuberculosis after the graft), pre-transplantation testing and treatment of LTBI is highly recommended.

Postponing LTBI treatment to the time of the transplantation is also possible, but could open the door for interactions and tolerance concerns generated by multiple co-medications that might be difficult to manage. This study is not without limitations. First, using vaccinations as an endpoint is a limit, because documentation may not always be an accurate reflection of previous immunizations. Having being vaccinated is often self-reported and documentation between health institutions might be unreliable to provide complete patients' records. Additionally, this study was conducted in a single center, in one kidney transplant population, and we were unable to report on post transplantation follow up and outcomes.

Conclusion

Despite specific vaccination recommendations for ESRD listed patients, rates of vaccination among renal transplant candidates remain suboptimal as described in our study.

Improvements can come from organizing ID consultation and implementing a collaborative network between ID specialists, transplantation, and dialysis centers. Given the optimal acceptability of patients to vaccine catch up schedules as well as to LTBI treatment, ID consultations might be the critical tool for promoting infection prevention strategies.

2 : 3 :

 23 AP-HP (Assistance Publique-Hôpitaux de Paris), Nephrology and Renal Transplantation Department, Institut Francilien de Recherche en Néphrologie et Transplantation (IFRNT), Groupe Hospitalier Henri-Mondor/Albert-Chenevier, Créteil, F-94010, France Université Paris-Est-Créteil, UPEC, DHU (Département Hospitalo-Universitaire) A-TVB, IMRB (Institut Mondor de Recherche Biomédicale) -EA 7376 CEpiA (Clinical Epidemiology And Ageing Unit), Créteil, F-94010, France 4 : AP-HP (Assistance Publique-Hôpitaux de Paris), Public Health Deparment, Groupe Hospitalier Henri-Mondor/Albert Chenevier, Créteil, F-94010, France 5 : INSERM U955, team 16, IMRB Créteil, F 94010 Créteil, France 6 : AP-HP (Assistance Publique-Hôpitaux de Paris), Microbiology Department, Groupe Hospitalier Henri-Mondor/Albert Chenevier, Créteil, F-94010, France

Figure 1 :

 1 Figure 1: Evolution of the number of ID consultations over time

Table 1 :

 1 recommended vaccines schedules for specific population, before kidney transplantation

	Specific populations	Vaccine strategies
	All patients	1 dose of Influenza tetravalent vaccine
		(inactivated)
	All patients	1 dose of 13vPCV followed by 1 dose of
		23vPPV
	All patients	3 double doses of HBV vaccine, or booster

Table 2 : Baseline characteristics of the study population

 2

	Age (years), (median )	55(61)
	Male gender, n (%)	302(64)
	Dialysis, n (%)	333(71)
	Primary diagnosis, leading to transplantation n (%)
	Diabetes	116(26)
	Hypertension	56(12)
	Genetic	55(12)
	Immunological	80(18)

Table 3 : vaccination rates and associated immunity

 3 Hepatitis B immunity: immunization history documented at least one vaccination dose, HB surface antibody quantitative result > 10 mIU/ml or the qualitative result was positive ** meningococcal conjugate vaccine (A, C, Y, W) and meningococcal B vaccine

	Vaccination rates/Past immunization	n/total	%
	Pneumococcal complete PCV 13 and PP23V 27/447	6
	PP23V alone	23/447	5
	TDPaP in past ten years	109/447	24
	Influenza	228/447	51
	Hepatitis B immunity*	252/467	54
	Haemophilus influenzae type B	11/447	2,5

*

Table 4 : baseline characteristic and vaccination rates for dialyzed and not dialyzed patients

 4

	Dialyzed	Not Dialyzed	P value
	n= 333 (%)	n= 134 (%)

Acknowledgments: none

Disclosures

The authors of this manuscript have no conflict of interest to disclose as described by the

American Journal of Transplantation

Data availability statement: data will be available on demand to the corresponding author


Mathieu Bajodek

Alexandre Seuret

Frédéric Gouaisbaut

On the necessity of sufficient LMI conditions for time-delay systems arising from Legendre approximation

Keywords: Systems with time-delay, Infinite-dimensional systems, Stability analysis, Lyapunov functionals, LMI, Polynomial approximation

This work is dedicated to the stability analysis of time-delay systems with a single constant delay using the Lyapunov-Krasovskii theorem. This approach has been widely used in the literature and numerous sufficient conditions of stability have been proposed and expressed as linear matrix inequalities (LMI). The main criticism of the method that is often pointed out is that these LMI conditions are only sufficient, and there is a lack of information regarding the reduction of the conservatism. Recently, scalable methods have been investigated using Bessel-Legendre inequality or orthogonal polynomial-based inequalities. The interest of these methods relies on their hierarchical structure with a guarantee of reduction of the level of conservatism. However, the convergence is still an open question that will be answered for the first time in this paper. The objective is to prove that the stability of a time-delay system implies the feasibility of these scalable LMI, at a sufficiently large order of the Legendre polynomials. Moreover, the proposed contribution is even able to provide an analytic estimation of this order, giving rise to a necessary and sufficient LMI for the stability of time-delay systems.

Introduction

Time-delay systems represent a wide class of dynamical systems arising in many applications in electronics, biology, transport, etc... Their interest in automatic control is natural since they propose many challenging theoretical problems related to their intrinsic infinitedimensional nature [START_REF] Niculescu | Delay effects on stability: a robust control approach[END_REF][START_REF] Sipahi | Stability and stabilization of systems with time delay[END_REF]. In particular, their stability analysis has been at the heart of many research works for several decades [START_REF] Fridman | Introduction to Time-Delay Systems : analysis and control[END_REF][START_REF] Gu | Stability of Time-Delay Systems[END_REF][START_REF] Kharitonov | Time-Delay Systems: Lyapunov Functionals and Matrices[END_REF][START_REF] Richard | Time delay systems: an overview of some recent advances and open problems[END_REF][START_REF] Zhang | Overview of recent advances in stability of linear systems with time-varying delays[END_REF]] and many methods have been developed. Among them, the use of Lyapunov-Krasovskii theorem remains one of the most popular techniques because of its inherent robustness. In the context of linear time-invariant (LTI) delay systems, several conditions have been obtained over the last two decades by application of the Lyapunov-Krasovskii theorem leading generally to sum of square constraints [START_REF] Peet | A dual to Lyapunov's second method for linear systems with multiple delays and implementation using SOS[END_REF][START_REF] Peet | Positive forms and stability of linear time-delay systems[END_REF][START_REF] Valmorbida | Stability analysis for a class of partial differential equations via semidefinite programming[END_REF] or to linear matrix inequalities (LMI) [START_REF] Fridman | New Lyapunov-Krasovskii functionals for stability of linear retarded and neutral type systems[END_REF][START_REF] He | Delay-dependent robust stability criteria for uncertain neutral systems with mixed delays[END_REF][START_REF] Xu | A survey of linear matrix inequality techniques in stability analysis of delay systems[END_REF]. These conditions, combined with semidefinite convex optimization programs, only led to sufficient conditions. Hence, numerous works aimed at reducing the inherent conservatism and at recovering the necessity using, for instance, discretized functionals [START_REF] Gu | Complete Quadratic Lyapunov-Krasovskii Functional: Limitations, Computational Efficiency, and Convergence[END_REF][START_REF] Gu | Stability of Time-Delay Systems[END_REF] or delay partitioning methods [START_REF] Du | A delay-partitioning projection approach to stability analysis of continuous systems with multiple delay components[END_REF][START_REF] Gouaisbaut | Delay-dependent stability analysis of linear time-delay systems[END_REF]. Among them, an approach based on state extension has been proposed in [START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time-delay systems[END_REF]. The candidate Lyapunov-Krasovskii functional therein depends directly on the projections of the state of the delay system onto Legendre polynomials. This approach led to LMI stability conditions, which benefit from a particular hierarchical structure, arising from the use of the Bessel-Legendre inequality. Similar approaches based on orthogonal polynomials [START_REF] Lee | Bessel summation inequalities for stability analysis of discrete-time systems with time-varying delays[END_REF][START_REF] Park | Auxiliary functionbased integral inequalities for quadratic functions and their applications to time-delay systems[END_REF] have been also considered in the literature. The complexity of these LMI increases with the number of Legendre polynomials n taken under consideration but their conservatism drastically reduces as n increases, at least on examples. However, to the best of our knowledge, the proof of convergence of these LMI to necessary and sufficient LMI conditions of stability is still missing. Towards this direction, as for LTI finite-dimensional systems, there exists a converse Lyapunov-Krasovskii theorem for LTI systems with a constant delay. Indeed, it is possible to build a complete Lyapunov-Krasovskii functional for stable LTI delay systems, see for instance [START_REF] Kharitonov | Time-Delay Systems: Lyapunov Functionals and Matrices[END_REF] for a larger overview on the problem. While this method has been only seen as a theoretical contribu-tions, the authors of [START_REF] Medvedeva | Synthesis of Razumikhin and Lyapunov-Krasovskii approaches to stability analysis of time-delay systems[END_REF][START_REF] Medvedeva | Stability of neutral type delay systems: A joint Lyapunov-Krasovskii and Razumikhin approach[END_REF] paved the way to use the converse Lyapunov-Krasovskii theorem to derive not sufficient but necessary stability conditions for timedelay systems through approximation. Among them, the authors of [START_REF] Egorov | Necessary stability conditions for linear delay systems[END_REF] provided a necessary stability conditions for time-delay systems. This method was then extended to various classes of delay systems, see for instance [START_REF] Gomez | Necessary stability conditions for neutral type systems with a single delay[END_REF][START_REF] Gomez | Necessary stability conditions for neutral-type systems with multiple commensurate delays[END_REF]. More interestingly, their methods have led to necessary and sufficient stability conditions in [START_REF] Egorov | Necessary and sufficient stability conditions for linear systems with pointwise and distributed delays[END_REF][START_REF] Gomez | Necessary and sufficient stability condition by finite number of mathematical operations for time-delay systems of neutral type[END_REF]. The sufficiency part of their necessary condition has been obtained by estimating an upper bound of the approximation error due to the discretization process of the Lyapunov matrix. It is worth noting that these conditions are not LMI but only a test of positive definiteness of a given matrix, which makes their method efficient from computational reasons. However, even though the necessary condition of stability is simple to test, the sufficient condition of stability is still numerically complicated to verify, because the estimated order assessing the sufficiency can be very high.

The objective of the present paper follows the same spirit of deriving necessary and sufficient stability conditions of time-delay systems but with a different objective. Indeed, we aim at using the approach developed in [START_REF] Gomez | Necessary and sufficient stability condition by finite number of mathematical operations for time-delay systems of neutral type[END_REF][START_REF] Gu | Complete Quadratic Lyapunov-Krasovskii Functional: Limitations, Computational Efficiency, and Convergence[END_REF] to prove that the sufficient LMI conditions arising from the Bessel-Legendre inequality are asymptotically necessary, as the degree n of the Legendre polynomial to be considered increases. This corresponds to answer the following questions.

• If a time-delay system is stable, does there exist an order N * for which some LMI stability conditions are necessary satisfied? • Is it possible to estimate this order N * ?

The paper is organized as follows. Section 2 formulates the problem of stability analysis of LTI systems subject to a single point-wise constant delay. After presenting a brief summary of the properties of Legendre polynomials, sufficient scalable LMI stability conditions are proposed. Then, Section 3 proves the converse theorem of this first result, by ensuring the existence of an order N * for which the LMI stability conditions must be verified for stable time-delay systems. Section 4 goes beyond the existence of an order by providing an estimation of the order N * of the polynomial approximation for which the LMI stability conditions are guaranteed. Finally, the theoretical results are evaluated on academic examples leading to several discussions. The paper ends with a conclusion and several appendices gathering the technical proofs required to derive the main results.

Notations : Throughout the paper, N (N * ), R m×p and S m (S m + ) denote the sets of natural numbers (excluding zero), real matrices of size m × p and symmetric matrices of size m (positive definite), respectively. For any x in R, x stands for the least integer greater than or equal to x. For any square matrix M ∈ R m×m , M (p, q) denotes the entries of M located at the p th row and q th column, M denotes the transpose of M , H(M ) stands for M + M and inequality M 0 means that M ∈ S m + . Furthermore, for any M in S m , its minimal and maximal eigenvalues are denoted σ(M ) and σ(M ). The 2-norm of matrix M in R m×p is |M | = σ(M M ). The vector u = vec(M ) in R mp×1 collocates the columns of matrix M in R m×p and the inverse of this operation is denoted vec -1 and is such that vec -1 (vec(M )) = M . Moreover, notation ⊗ represents the Kronecker product so that, for any matrices (M

1 , M 2 ) in R m1×p1 ×R m2×p2 , M 1 ⊗M 2 = M1(1,1)M2 ... M1(1,p1)M2 . . . . . . . . . M1(m1,1)M2 ... M1(m1,p1)M2 in R m1m2×p1p2 . The set of square-integrable functions from (a, b) to R m×p is noted L 2 (a, b; R m×p ).
Let finally C pw (a, b; R m×p ) be the set of piecewise continuous functions with a finite number of discontinuity points.

2 Sufficient LMI stability conditions for timedelay systems

System data

Consider a LTI delay system given by

ẋ(t) = Ax(t) + A d x(t -h), ∀t ≥ 0, x(t) = ϕ(t) ∈ C pw (-h, 0; R nx ), ∀t ∈ [-h, 0], (1a) (1b)
where the single delay h > 0 and matrices A, A d in R nx×nx are constant and known. Without loss of generality, matrix A d is decomposed into a product BC as

A d = BC, with |C| = 1, (2)
with B, C T ∈ R nx×nz being full column rank matrices, with n z being the rank of matrix A d . Along the paper, the Shimanov notation x t (θ) = x(t + θ), for all (t, θ) in R + × [-h, 0] is adopted.

Definition 1 (GES)

The trivial solution x(t) ≡ 0 of system (1) is globally exponentially stable (GES), if there exist κ ≥ 1 and µ > 0 such that the solution to (1) generated by any initial condition ϕ ∈ C pw (-h, 0; R nx ), denoted as x(t, ϕ) verifies

|x(t, ϕ)| ≤ κe -µt sup θ∈[-h,0] |ϕ(θ)|, ∀t ≥ 0. ( 3
)
Several ways of assessing GES of LTI systems have been provided in the literature as mentioned in the introduction. Here, the contribution focuses on conditions arising from the application of the Lyapunov-Krasovskii theorem [START_REF] Gu | Stability of Time-Delay Systems[END_REF], adapted to the LTI case.

Theorem 1 (Lyapunov-Krasovskii theorem) If there exist positive scalars ε 1 , ε 2 and ε 3 and a continuous and differentiable functional V defined from C pw (-h, 0; R nx ) to R such that, for any ϕ in C pw (-h, 0; R nx ), the following inequalities hold

(i) ε 1 |ϕ(0)| 2 ≤ V(ϕ) ≤ ε 2 sup θ∈[-h,0] |ϕ(θ)| 2 and (ii) V(ϕ) ≤ -ε 3 |ϕ(0)| 2
, where V denotes here the derivative of V along the trajectories of (1), then, the trivial solution of system (1) is GES.

The main underlying idea of this theorem is to determine a positive definite functional V, such that its derivative with respect to time along the trajectories of the system (1) is negative definite. The main problem within the application of this theorem is to design a suitable functional and then to provide some conditions that guarantee its positive definiteness and the negative definiteness of its derivative. The derivation of stability conditions using the Lyapunov-Krasovskii theorem usually involves quite elaborate developments. To give an idea of the procedure involved in this approach and to provide a glimpse of its technical flavor, we present here some basics on the procedure to follow in order to derive asymptotic stability criteria for time-delay systems expressed in terms of LMI. The procedure follows three steps.

Step 1. To propose a candidate Lyapunov-Krasovskii functional V, based on the structure of the system.

Step 2. To compute the derivative of the functional along the trajectories of the system.

Step 3. To apply over-approximation technique to V and derive a stability condition expressed in terms of LMI.

Among the numerous methods employed in the literature, we will focus here on the method based on the application of the Bessel-Legendre integral inequality [START_REF] Seuret | Complete quadratic Lyapunov functionals using Bessel-Legendre inequality[END_REF][START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time-delay systems[END_REF], which is scalable with respect to the degree n of the Legendre polynomial to be considered. Interestingly, this method leads to LMI admitting a hierarchical structure with respect to n. Before stating this theorem, let us first recall the main definitions and some characteristics about these polynomials.

Definition of Legendre polynomials

Definition 2 The Legendre polynomials are defined over the normalized interval [0, 1] as

∀k ∈ N, l k (θ) = k j=0 ( k j )( k+j j )(θ -1) j , (4)
where ( k j ) stands for the binomial coefficients.

The Legendre polynomials have been widely used in the polynomial approximation theory [START_REF] Boyd | Chebyshev and Fourier Spectral Methods[END_REF] because they form an orthogonal sequence with respect to the inner product 1 0 φ (θ)ψ(θ)dθ, for any φ, ψ in L 2 (0, 1; R m ). Before going into the details, for any order n and m in N * and θ ∈ [0, 1], let us introduce the following notations:

m n (θ) = l 0 (θ) l 1 (θ) . . . l n-1 (θ) ⊗I m ∈ R nm×m , (5) I m n = I 1 n ⊗ I m ∈ R nm×nm , (6)
I 1 n (p, q) = 2p -1 if p = q, 0 otherwise, ∀(p, q) ∈ {1, . . . , n} 2 .
To ease the reading of the paper and where no confusion is possible, we will omit the upper script m = n z and only use notations n and I n .

The main motivation for employing these polynomials comes also from an efficient integral inequality, known as Bessel-Legendre inequality, which is stated here.

Lemma 1 (Bessel-Legendre inequality) Let z ∈ L 2 (-h, 0; R nz ) and S ∈ S nz + a positive definite matrix. The integral inequality

0 -h z (θ)Sz(θ)dθ ≥ 1 h ζ n (z)(I 1 n ⊗ S)ζ n (z), (7)
hold, for all n ∈ N * , where

ζ n (z) = 0 -h n θ+h h
z(θ)dθ. Moreover, if z is a polynomial function of degree n -1 over (-h, 0), the equality case holds

0 -h z (θ)Sz(θ)dθ = 1 h ζ n (z) T (I 1 n ⊗ S)ζ n (z). ( 8
)
Proof : The proof is derived from [29, Lemma 3] and is postponed to Appendix C.

Sufficient LMI conditions

We are now in position to state a stability theorem for system (1) based on the previous developments.

Theorem 2 If there exist an order n in N * and matrices (P n , R, S) in S nx+nnz × S nz + × S nz + such that the following LMI conditions hold

Φ + n = P n + 0 0 0 1 h (I 1 n ⊗ S) 0, (9a)
Φ - n = H(P n A n ) + Ψ(R, S) P n B n * -S ≺ 0, (9b)
where

Ψ(R, S) = C (hR + S)C 0 0 -1 h (I 1 n ⊗ R) , A n = A 0 n (1)C -1 h L nz n , B n = B -n (0) , n (1)
= In z . . . In z , n (0) = In z . . . (-1) n-1 In z ∈ R nnz , L nz n = L 1 n ⊗ I nz ∈ R nnz×nnz , L 1 n (p, q) = (2q -1)(1 -(-1) p+q ) if p ≥ q, 0 otherwise, ∀(p, q) ∈ {1, . . . , n} 2 ,
then, the trivial solution of system (1) is GES.

Proof : For a given integer n in N * , consider the functional defined as follows

V n (x t ) = x t (0) ζ n (Cx t ) P n x t (0) ζ n (Cx t ) (10)
+ 0 -h x t (θ)C ((θ + h)R + S)Cx t (θ)dθ,
where matrices P n , R and S are solution to the LMI stated in the theorem and where the augmented vector ζ n is defined as follows

ζ n (Cx t ) = 0 -h n (θ)Cx t (θ)dθ. ( 11
)
Applying the Bessel-Legendre inequality (7) yields

V n (x t ) ≥ x t (0) ζ n (Cx t ) Φ + n x t (0) ζ n (Cx t )
.

Therefore, if condition (9a) is verified, then there exists a sufficiently small

ε 1 > 0 such that V n (x t ) ≥ ε 1 |x t (0)| 2 . In addition, since V is quadratic with respect to x t , se- lecting ε 2 = σ(P n ) + h 2 2 σ(R) + hσ(S) ensures that in- equality V n (x t ) ≤ ε 2 sup θ∈[-h,0] |x t (θ)| 2 holds.
As in [START_REF] Seuret | Complete quadratic Lyapunov functionals using Bessel-Legendre inequality[END_REF],

computing the derivative of the functional along the trajectories of the system yields Vn (x t ) = 2 x t (0)

ζ n (Cx t ) P n ẋt (0) ζn (Cx t ) -x t (-h)C SCx t (-h) +x t (0)C (hR+S)Cx t (0)- 0 -h x t (θ)C RCx t (θ)dθ. (12)
Then, the dynamics (1a) and an integration by parts of ζn (Cx t ) provide an expression ẋt(0) ζn(Cxt) with respect to

x t (0), ζ n (Cx t ) and x t (-h). It ensures that ẋt (0) ζn (Cx t ) = A n x t (0) ζ n (Cx t ) + B n Cx t (-h).
Re-injecting this expression into [START_REF] Gu | Complete Quadratic Lyapunov-Krasovskii Functional: Limitations, Computational Efficiency, and Convergence[END_REF] and applying again the Bessel-Legendre inequality [START_REF] Fridman | Introduction to Time-Delay Systems : analysis and control[END_REF] yields

Vn (x t ) ≤ xt(0) ζn(Cxt) Cxt(-h) Φ - n xt(0) ζn(Cxt) Cxt(-h) .
Therefore, if condition (9b) is satisfied, then there exists a sufficiently small ε 3 such that Vn (x t ) ≤ -ε 3 |x t (0)| 2 , which concludes the proof by application of the Lyapunov-Krasovskii theorem.

Note that the proposed theorem has already been presented in [START_REF] Seuret | Complete quadratic Lyapunov functionals using Bessel-Legendre inequality[END_REF] or in [START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time-delay systems[END_REF]. As in [START_REF] Seuret | Complete quadratic Lyapunov functionals using Bessel-Legendre inequality[END_REF] or in [START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time-delay systems[END_REF], the previous stability condition is hierarchical with respect to n. This aspect is presented formally in the following lemma.

Lemma 2 If there exists an integer N in N * , for which a solution to LMI (9) exists, then there also exists a solution to the same problem for any integer n ≥ N .

Proof : The proof uses similar arguments to the ones provided in [29, Theorem 7]. A glimpse of the proof consists in introducing P n+1 = P n 0 0 0 , so that V n+1 (x t ) = V n (x t ) and so that one can exhibit a solution to the LMI problem at order n + 1 based on the solution at order n. The details of the proof are omitted but strongly relies on the structure of Φ + n and Φ - n .

To sum up the results presented so far, Theorem 2 presents scalable LMI conditions for the stability (GES) of LTI delay systems. These conditions is parameterized by n in N * , corresponding to the degree of the Legendre polynomials considered in the construction of the candidate Lyapunov-Krasovskii functional. Moreover, it is demonstrated that when n increases, the conservatism can only be reduced. It is however legitimate to wonder if a converse result can be proven. This direction refers to the possibility of assessing the satisfaction of these LMI for a sufficiently large n, when the system under consideration is known a priori to be GES for a given delay h. Apart the convergence, i.e. the existence of an order N * for which these LMI conditions are guaranteed, an interesting underlying question is to estimate analytically such an order. The next developments aim at providing a solution to both problems. More specifically, the next section presents a converse theorem with a proof of existence of N * , while the latter address the problem of estimating order N * .

Necessity of LMI stability conditions: Existence of an order

In this section, the objective is to prove the converse side of Theorem 2. In other words, we aim at proving that, if the trivial solution of system (1) is GES, then there exists an order N * in N * from which LMI conditions in [START_REF] Gomez | Necessary stability conditions for neutral-type systems with multiple commensurate delays[END_REF] hold. The developments go against the grain of usual approaches and are inspired by the converse Lyapunov-Krasovskii theorem [START_REF] Huang | Generalization of Liapunov's theorem in a linear delay system[END_REF][START_REF] Medvedeva | Synthesis of Razumikhin and Lyapunov-Krasovskii approaches to stability analysis of time-delay systems[END_REF], stated below.

Theorem 3 (Converse Lyapunov-Krasovskii theorem)

If the trivial solution of system (1) is GES, then there exist positive scalars ε 1 , ε 2 and ε 3 and a continuous and differentiable functional

V defined from C pw (-h, 0; R nx ) to R such that, for any ϕ in C pw (-h, 0; R nx ), inequalities (i) ε 1 |ϕ(0)| 2 ≤ V(ϕ) ≤ ε 2 sup θ∈[-h,0] |ϕ(θ)| 2 hold and (ii) V(ϕ) ≤ -ε 3 |ϕ(0)| 2 holds
, where V denotes here the derivative of V along the trajectories of (1), First, the so-called complete Lyapunov-Krasovskii functional is introduced. It fulfills the converse Lyapunov-Krasovskii theorem statement and serves as a target functional. Then, a link is established between such a complete functional and the functional used in the previous section with a particular structure. The two LMI conditions in ( 9) are consequently obtained. On one side, assuming that system ( 1) is GES, one ensures that (9a) holds, for any order n. On the other side, the time-derivative of the functional along the trajectories of system (1) becomes negative as the order increases so that (9b) holds, for sufficiently large orders. As a final step, the new converse theorem is stated and proven.

Step 1. To introduce the complete Lyapunov-Krasovskii functional which can be expressed analytically with respect to a Lyapunov matrix function.

Step 2. To make the link between the functional V n introduced in [START_REF] Gomez | Necessary and sufficient stability condition by finite number of mathematical operations for time-delay systems of neutral type[END_REF] and the Legendre approximation of the complete functional.

Step 3. To assess that the condition (9a) holds, assuming that system (1) is GES.

Step 4. To assess that the condition (9b) holds, for a sufficiently large order.

Step 5. To conclude on the necessity of LMI conditions given in Theorem 2.

The section is decomposed in five subsections respectively based on the steps listed above.

The complete Lyapunov-Krasovskii functional

In this subsection, based on [START_REF] Kharitonov | Time-Delay Systems: Lyapunov Functionals and Matrices[END_REF], we introduce the socalled complete Lyapunov-Krasovskii functional, which will be used in conjonction with Theorem 3.

Definition 3 For any

(W 1 , W 2 , W 3 ) in S nx + × S nz + × S nz + , the complete Lyapunov-Krasovskii functional V is V(x t ) = x t (0)U (0)x t (0)+2x t (0) 0 -h U (θ+h)BCx t (θ)dθ + 0 -h 0 -h x t (θ 1 )(BC) U (θ 2 -θ 1 )BCx t (θ 2 )dθ 1 dθ 2 + 0 -h x t (θ)C ((θ + h)W 2 + W 3 )Cx t (θ)dθ, ( 13
)
where U is the Lyapunov matrix defined from [-h, h] to R nx×nx as the solution of the following matrix delayed differential equation

     U (θ) = -U (θ)A -U (θ + h)A d , ∀θ ≤ 0, A U (0)+A d U (h)+U (0)A+U (-h)A d = -W, U (θ) = U (-θ), ∀θ > 0, (14)
where [START_REF] Kharitonov | Time-Delay Systems: Lyapunov Functionals and Matrices[END_REF]Section 2.10] provides an analytic expression of U given by

W = W 1 + C hW 2 + W 3 C. (15) Interestingly, using vec(M 1 M 2 M 3 ) = (M 3 ⊗M 2 )vec(M 2 ), for any matrices M 1 , M 2 , M 3 in R nx×nx , a technique de- veloped in
U (θ) = vec -1 [ I n 2 x 0 ]e θM N -1 0 -vec(W ) if θ ≤ 0, U (-θ) if θ > 0, ( 16
)
where W is given in [START_REF] Huang | Generalization of Liapunov's theorem in a linear delay system[END_REF] and

M = -A ⊗In x -A d ⊗In x In x ⊗A d In x ⊗A , (17)
N = I n 2 x 0 A ⊗In x +In x ⊗A A d ⊗In x + 0 -I n 2 x In x ⊗A d 0 e -hM .
It is well-known that the complete functional V satisfies the following lemma.

Lemma 3 For any (W 1 , W 2 , W 3 ) in S nx + × S nz + × S nz + , the time-derivative along the trajectories of system (1) of the complete Lyapunov-Krasovskii functional V given by (13) yields

V(x t ) = - 1 h 0 -h xt(0) hCxt(θ) Cxt(-h) W1 0 0 * 1 h W2 0 * * W3 xt(0) hCxt(θ) Cxt(-h) . ( 18
)
Proof : The proof is provided in [START_REF] Kharitonov | Time-Delay Systems: Lyapunov Functionals and Matrices[END_REF]Theorem 2.11].

Remark 1 For any symmetric positive definite matrices (W 1 , W 2 , W 3 ) and by assuming that there is no characteristic root of (1) such that its opposite is also characteristic root, it is worth mentioning that this functional exists and is the unique solution of [START_REF] Kharitonov | Time-Delay Systems: Lyapunov Functionals and Matrices[END_REF]. For more details, one can refer to [18, Theorems 2.8 and 2.10].

It is also well-known that the complete funtional V is positive definite if the trivial solution of system ( 1) is GES as demonstrated in [START_REF] Gu | Stability of Time-Delay Systems[END_REF]Theorem 5.19]. In light of paper purpose, a variation of this result is proposed, which will corresponds exactly to the present framework.

Lemma 4 Consider (W 1 , W 2 , W 3 ) in S nx + × S nz + × S nz + .
If the trivial solution of system (1) is GES, there exists a positive scalar ε > 0 such that the complete functional V given by (13) satisfies

V(ϕ) - 0 -h (θ + h)ϕ (θ)C W 2 Cϕ(θ)dθ ≥ ε |ϕ(0)| 2 + 0 -h |Cϕ(θ)| 2 dθ , (19)
for any ϕ in C pw (-h, 0; R nx ).

Proof : The proof is postponed to Appendix D.

Lastly, the Lyapunov matrix U introduced in Definition 3 satisfies several properties. The first one is related to its continuity. The second property concerns its derivative U , which is only continuous over (0, h] and [-h, 0) and has a discontinuity at 0, which is measured as follows.

Lemma 5 The Lyapunov matrix U given by ( 16) verifies the following properties.

(i) The Lyapunov matrix U is continuous on [-h, h] and

U (0) = U (0).
(ii) The Lyapunov matrix U is infinitely differentiable on (0, h] and [-h, 0). Its derivative has a discontinuity only at 0 so that

∆U (0) := lim →0 (U ( ) -U (-)) = W.
Proof : The two items are proved in [18, Lemma 2.4 and Lemma 2.6], respectively.

These regularity conditions satisfied by the Lyapunov matrix have been used for H 2 [START_REF] Jarlebring | Characterizing and computation H 2 norm of time delay systems by solving the delay Lyapunov equation[END_REF] or H ∞ [START_REF] Kharitonov | Lyapunov-Krasovskii approach to the robust stability of time-delay systems[END_REF] analysis. They will also be at the heart of the derivation of the convergence results.

To summarize, this subsection emphasized the structure of the complete functional whose kernels are defined by the Lyapunov matrix U . This particular structure has the benefits of ensuring by construction that its time-derivative along the trajectories of the system verifies [START_REF] Kharitonov | Time-Delay Systems: Lyapunov Functionals and Matrices[END_REF] and that the positive definiteness is ensured for GES systems.

Construction of matrices P n , R and S

In this subsection, the objective is to understand how to relate the functional defined in ( 10) and the complete one in [START_REF] Gu | Stability of Time-Delay Systems[END_REF], for a particular structure of matrices (P n , R, S).

To do so, the main idea is to exploit the terms of ( 13) that are expressed in U (θ+h)B and B U (θ 2 -θ 1 )B. The Legendre polynomial approximation of these functions at any order n in N * writes

U (θ + h)B = U 1,n n θ+h h + Ũ1,n (θ), ∀θ ∈ [-h, 0], B U (θ)B = U 2,n n θ+h 2h + Ũ2,n (θ), ∀θ ∈ [-h, h]. (20)
In this decomposition, the constant matrices U 1,n and U 2,n have been selected as the orthogonal projection of U (θ + h)B and B U (θ)B, respectively, on the n first Legendre polynomials n . Their expressions are given by

U 1,n = 1 h 0 -h U (θ + h)B n θ + h h dθ I n , U 2,n = 1 2h h -h B U (θ)B n θ + h 2h dθ I n . (21)
Functions Ũ1,n (θ) and Ũ2,n (θ) can be interpreted as the approximation errors of the orthogonal projections, and verify

0 -h Ũ1,n (θ) n θ+h h dθ = 0 -h U (θ+h)B n θ+h h dθ =U1,n(In/h) -1 -U 1,n 0 -h n θ+h h n θ+h h dθ =(In/h) -1 = 0.
Similarly, the same calculations ensure that error Ũ2,n is orthogonal to the n first Legendre polynomials considered over [-h, h], i.e.

h -h Ũ2,n (θ) n θ + h 2h dθ = 0.
The next developments aim at demonstrating the uniform convergence of the polynomial approximation. Following the theory of polynomial approximation (see for instance [START_REF] Boyd | Chebyshev and Fourier Spectral Methods[END_REF] and references therein), it results that Ũ1,n and Ũ2,n given in [START_REF] Medvedeva | Synthesis of Razumikhin and Lyapunov-Krasovskii approaches to stability analysis of time-delay systems[END_REF] converge to zero in the sense of the L 2 norm. This is actually a by-product of Bessel-Legendre inequality. This implies that the approximation errors converges to zero almost everywhere on their domain of definition, as n tends to infinity. Nevertheless, uniform convergence properties can even be obtained using the regularity of the Lyapunov matrix.

Ũ1,n (θ) ≤ ū1,n := 1 √ n -3 |W | , (22)
where

1 = √ n x π 2 3 2 e h|M| M 2 N -1 h 2 |B| , (23)
and matrices M, N and W are given by [START_REF] Kharitonov | Lyapunov-Krasovskii approach to the robust stability of time-delay systems[END_REF].

(ii) The approximated Legendre function 1 h U 1,n n θ+h h
converges uniformly to U (θ + h)B, when n tends to infinity, on the interval [-h, 0). More precisely, for any n ≥ 6, the following inequality holds

sup θ∈[-h,0) Ũ 1,n (θ) ≤ ū2,n := 2 √ n -5 |W | , (24)
where

2 = 1 2 √ n x π 2 3 2 e h|M| M 4 N -1 h 3 |B| , (25)
and matrices M, N and W are given by [START_REF] Kharitonov | Lyapunov-Krasovskii approach to the robust stability of time-delay systems[END_REF]. (iii) The approximated Legendre function U 2,n n ( θ+h 2h ) converges uniformly to B U (θ)B, when n tends to infinity, on the closed interval [-h, h]. More precisely, for any n ≥ 4, the following inequality holds

sup θ∈[-h,h] Ũ2,n (θ) ≤ ū3,n := 3 √ n -3 |W | , (26)
where

3 = √ 2π 1 + √ n x πhe h|M| M 2 N -1 h |B| 2 , ( 27
)
and matrices M, N and W are given by [START_REF] Kharitonov | Lyapunov-Krasovskii approach to the robust stability of time-delay systems[END_REF].

Proof : The proof follows the arguments provided in [32, Theorem 2.5] and is based on the regularity properties of the Lyapunov matrix U highlighted in Lemma 5. The proof is postponed to Appendix E, for the sake of readability.

The previous lemma provides uniform upper bounds on the error done by three approximations related to the Lyapunov matrix U . These uniform convergence results are only presented as the polynomial approximations of U but addresses more general problems of functional analysis and approximation of continuous functions.

All these upper bounds depend explicitly on the arbitrary symmetric positive definite matrices (W 1 , W 2 , W 3 ) through the term |W |.

As a final result, Lemma 6 implies the following corollary.

Corollary 1 For any positive scalar η > 0, there exists an integer N * η such that

max (ū 1,n , ū2,n , ū3,n ) ≤ η, ∀n ≥ N * η . (28)
Proof : Since the upper bounds of approximation errors tend to 0 as n tends to infinity, the existence of such N * η for any η > 0 is guaranteed.

We are now in position to construct matrices (P n , R, S) for given matrices (W 1 , W 2 , W 3 ). The guiding principle is to mimic the complete Lyapunov-Krasovskii functional, replacing U (θ +h)B and B U (θ 2 -θ 1 )B by their Legendre polynomial approximation given by ( 20) in order to benefit from its properties as formulated in the following lemma.

Lemma 7 For any [START_REF] Gomez | Necessary and sufficient stability condition by finite number of mathematical operations for time-delay systems of neutral type[END_REF] with matrices

(W 1 , W 2 , W 3 ) in S nx + × S nz + × S nz + , define the functional V n by
P n = U (0) U 1,n * T n , R = W 2 , S = W 3 , T n = 0 0 -h -h I n h n θ 1 +h h U 2,n (θ 2 -θ 1 ) n θ 2 +h h I n h dθ 1 dθ 2 , U 2,n (θ) = U 2,n n θ + h 2h , (29)
where U 1,n , U 2,n are given in [START_REF] Medvedeva | Stability of neutral type delay systems: A joint Lyapunov-Krasovskii and Razumikhin approach[END_REF]. The time-derivative of this functional along the trajectories of system (1) yields

Vn (x t ) = - 1 h 0 -h xt(0) hCxt(θ) Cxt(-h) Ψ n (θ) xt(0) hCxt(θ) Cxt(-h) , (30)
where matrix

Ψ n (θ) := W1+H( Ũ1,n(0)C) Ψ 1 n (θ) -Ũ1,n(-h) * 1 h W2 Ψ 2 n (θ) * * W3 0, ( 31
)
is defined for any n in N * and for all θ in [-h, 0] with

Ψ 1 n (θ) = A Ũ1,n (θ) -Ũ 1,n (θ) + C Ũ2,n (θ), Ψ 2 n (θ) = B Ũ 1,n (θ) -Ũ 2,n (θ + h),
and with Ũ1,n , Ũ 1,n and Ũ2,n being the approximation errors of the Lyapunov matrix U of system (1).

Proof : To begin with, re-injecting the expression of U (θ +h)B and B U (θ)B and using their approximation given by (20) into the complete functional V leads to

V(x t ) = x t (0)U (0)x t (0) +2x t (0)U 1,n 0 -h n θ+h h Cx t (θ)dθ + 0 -h 0 -h x t (θ 1 )C U 2,n n θ 2 -θ 1 +h 2h Cx t (θ 2 )dθ 1 dθ 2 + 0 -h x t (θ)C (W 2 + (θ + h)W 3 )Cx t (θ)dθ +2x t 0 -h Ũ1,n (θ)Cx t (θ)dθ + 0 -h 0 -h x t (θ 1 )C Ũ2,n (θ 2 -θ 1 )Cx t (θ 2 )dθ 1 dθ 2 . ( 32
) Since U 2,n (θ 2 -θ 1 ) = U 2,n n θ2-θ1+h 2h
belongs to R nz×nz and is a polynomial function of degree n -1 in both θ 1 and θ 2 , it can be decomposed using the basis of Legendre polynomials n θ1+h h

and n θ2+h h

. Thanks to the orthogonality of Legendre polynomials, this decomposition writes

U 2,n n θ 2 -θ 1 +h 2h = n θ 1 +h h T n n θ 2 +h h ,
where matrix T n is the symmetric matrix given in [START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time-delay systems[END_REF]. Note that the symmetry of T n is ensured by the symmetry of Ũ2,n highlighted in Property 2 that has been postponed in Appendix B in order to ease the reading. Hence, using the same augmented vector as in the proof of Theorem 2, i.e.

ζ n (Cx t ) = 0 -h n θ+h h
Cx t (θ)dθ, functional V reduces to the following expression

V(x t ) = x t (0) ζ n (Cx t ) U (0) U 1,n * T n x t (0) ζ n (Cx t ) + 0 -h x t (θ)C (W 2 + (θ + h)W 3 )Cx t (θ)dθ +2x t (0) 0 -h Ũ1,n (θ)Cx t (θ)dθ + 0 -h 0 -h x t (θ 1 )C Ũ2,n (θ 2 -θ 1 )Cx t (θ 2 )dθ 1 dθ 2 .
(33) Therefore, by selecting P n , R and S as in [START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time-delay systems[END_REF], functional V n in ( 10) is retrieved and we obtain

V n (x t ) = V(x t )-2x t (0) 0 -h Ũ1,n (θ)Cx t (θ)dθ - 0 -h 0 -h x t (θ 1 )C Ũ2,n (θ 2 -θ 1 )Cx t (θ 2 )dθ 1 dθ 2 .

Differentiating the previous expression leads to Vn

(x t ) = V(x t ) -2 ẋ t (0) 0 -h Ũ1,n (θ)Cx t (θ)dθ -2x t (0) 0 -h Ũ1,n (θ)C ẋt (θ)dθ - 0 -h 0 -h ẋ t (θ 1 )C Ũ2,n (θ 2 -θ 1 )Cx t (θ 2 )dθ 1 dθ 2 . - 0 -h 0 -h x t (θ 1 )C Ũ2,n (θ 2 -θ 1 )C ẋt (θ 2 )dθ 1 dθ 2 .
Since V is the complete functional, its derivative is expressed using W 1 , W 2 , W 3 . Several integrations by parts lead to the following expression of Vn

Vn (x t ) = - 1 h 0 -h xt(0) hCxt(θ) Cxt(-h) W1 0 0 * 1 h W2 0 * * W3 xt(0) hCxt(θ) Cxt(-h) -2(Ax t (0) + BCx t (-h)) 0 -h Ũ1,n (θ)Cx t (θ)dθ -2x t (0) Ũ1,n (0)Cx t (0) -Ũ1,n (-h)Cx t (-h) +2x t (0) 0 -h Ũ 1,n (θ)Cx t (θ)dθ -x t (0) 0 -h C ( Ũ2,n (θ) + Ũ 2,n (-θ)) =2 Ũ2,n(θ) Cx t (θ)dθ +x t (-h) 0 -h C ( Ũ2,n (θ+h)+ Ũ 2,n (-h-θ)) =2 Ũ2,n(θ+h) Cx t (θ)dθ + 0 -h 0 -h x t (θ 1 )C Ũ 2,n (θ 2 -θ 1 )Cx t (θ 2 )dθ 1 dθ 2 , - 0 -h 0 -h x t (θ 1 )C Ũ 2,n (θ 2 -θ 1 )Cx t (θ 2 )dθ 1 dθ 2 .
We first notice that the two last terms of the previous expression are opposite and thus sum up to zero. Moreover, Property 2 given in Appendix B helps reducing the expression of the terms that depend on Ũ2,n . Re-ordering the previous expression yields

Vn (x t ) = - 1 h 0 -h xt(0) hCxt(θ) Cxt(-h) Ψ n (θ) xt(0) hCxt(θ) Cxt(-h)
dθ, [START_REF] Zhang | Overview of recent advances in stability of linear systems with time-varying delays[END_REF] where Ψ n is given in [START_REF] Valmorbida | Stability analysis for a class of partial differential equations via semidefinite programming[END_REF] and concludes the proof.

To summarize the previous developments, a candidate functional V n has been built to fulfill the structure given in [START_REF] Gomez | Necessary and sufficient stability condition by finite number of mathematical operations for time-delay systems of neutral type[END_REF] and to follow the time-derivative of the complete functional that is defined for any symmetric positive definite matrices (W 1 , W 2 , W 3 ).

Necessary condition for LMI condition (9a)

As an extension to Lemma 4, the next developments aim at understanding how the positive definiteness condition on the complete functional V can be transferred to the candidate functional V n and then to LMI (9a) at any order. This is formulated in the next lemma.

Lemma 8 If the trivial solution of system (1) is GES, then condition Φ + n 0 holds with matrices (P n , S, R) given by (29) for all n in N * .

Proof : Assume that system (1) is GES. According to Lemma 4, the complete functional V in (13) verifies inequality [START_REF] Lee | Bessel summation inequalities for stability analysis of discrete-time systems with time-varying delays[END_REF] for a sufficiently small scalar ε > 0 for all ϕ in C pw (-h, 0; R nx ). In particular, consider ϕ as a function expressed as follows

ϕ(θ) = δ 0 (θ)I nx 1 h C † f n (θ)I n δ -h (θ)C † ϕ0 ϕn ϕ h , (35)
with

C † = C (CC ) -1 in R nx×nz is the right pseudo- inverse of C and with f n (θ) = n (θ) -n (1)δ 0 (θ) -n (0)δ -h (θ).
In this formulation, vector

ϕ0 ϕn ϕ h in R nx+nz(n+1
) is arbitrary and δ θ0 is zero everywhere except at θ 0 , where it equals 1, i.e. δ θ0 (θ) = 1 for θ = θ 0 , 0 otherwise.

Note that such a function ϕ has been selected so that ϕ at the boundary of [-h, 0] is given by ϕ(0) = ϕ 0 , ϕ(-h) = C † ϕ h , and in the interval (-h, 0) is given by ϕ(θ) = 1 h C † n (θ)I n ϕ n , for all θ in (-h, 0)., which is a polynomial function of θ of degree n -1. Re-injecting the particular function ϕ into the definition of V in [START_REF] Gu | Stability of Time-Delay Systems[END_REF], the orthogonality of the Legendre polynomials decomposition [START_REF] Medvedeva | Synthesis of Razumikhin and Lyapunov-Krasovskii approaches to stability analysis of time-delay systems[END_REF] ensures that

V(ϕ) = ϕ 0 U (0)ϕ 0 +2ϕ 0 0 -h U (θ+h)B n θ+h h dθ I n h U1,n ϕ n +ϕ n I n h 0 0 -h -h n θ 1 +h h B U (θ 2 -θ 1 )B n θ 2 +h h dθ 1 dθ 2 I n h Tn ϕ n + 0 -h x t (θ)C (W 2 + (θ + h)W 3 )Cx t (θ)dθ = V n (ϕ),
(36) where we recognize the functional V n built with matrices (P n , S, R) in [START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time-delay systems[END_REF]. Hence, applying the Bessel-Legendre equality case [START_REF] Gomez | Necessary stability conditions for neutral type systems with a single delay[END_REF] in Lemma 1 leads to

V(ϕ)- 0 -h ϕ (θ)C (θ+h)W 2 Cϕ(θ)dθ = [ ϕ0 ϕn ] Φ + n [ ϕ0 ϕn ].
(37) Moreover, re-injecting (35) into the inequality [START_REF] Lee | Bessel summation inequalities for stability analysis of discrete-time systems with time-varying delays[END_REF] yields

V(ϕ)- 0 -h ϕ (θ)C (θ+h)W 2 Cϕ(θ)dθ ≥ ε |ϕ 0 | 2 +ϕ n I n ϕ n ≥ ε | ϕ0 ϕn | 2 . ( 38
)
Altogether, (37) and ( 38) ensure

[ ϕ0 ϕn ] Φ + n [ ϕ0 ϕn ] ≥ ε | ϕ0 ϕn |
2 , for any vector [ ϕ0 ϕn ] in R nx+nnz , which guarantees the positive definiteness of matrix Φ + n .

Existence of an order for having LMI condition (9b)

As an extension to Lemma 7, the next developments use convergence arguments to asymptotically ensure the negative definiteness of the time-derivative of V n along the trajectories of system (1). The following lemma guarantees the satisfaction of LMI (9b), for sufficiently large orders.

Lemma 9 There exists an order N * in N * such that LMI condition Φ - n ≺ 0 holds with matrices (P n , S, R) given by ( 29) for all n ≥ N * .

Proof : As a first step to the proof, let emphasize the structure of matrix Ψ n in [START_REF] Valmorbida | Stability analysis for a class of partial differential equations via semidefinite programming[END_REF]. Such a matrix can be decomposed as the sum of a block diagonal positive definite matrices that is independent of n and of a matrix whose entries are all expressed using the approximation errors Ũn and Ũ n , which can be made uniformly arbitrarily small in light of convergence properties in Lemma 6. Therefore, we will use the following equivalence results. For any matrix X in R p×q such that X X |X| 2 I q , then inequality |X|Ip X X |X|Iq 0 holds. Using this inequality, the following lower bounds of Ψ n is derived

Ψ n (θ) µ1,nIn x 0 0 0 µ2,nIn z 0 0 0 µ3,nIn z , with µ 1,n = σ(W 1 )-(1+|A|+2|C|)ū 1,n -ū2,n -|C|ū 3,n , µ 2,n = 1 h σ(W 2 )-(|A|+|B|)ū 1,n -ū2,n -(1+|C|)ū 3,n , µ 3,n = σ(W 3 )-(1+|B|)ū 1,n -ū3,n , (39
) where ū1,n , ū2,n and ū3,n are the upper bounds of the approximation errors of the polynomial approximation given by ( 22), ( 24) and ( 26), respectively. Then, Corollary 1 ensures that all the negative terms can be made arbitrarily small as n increases, while the first terms are positive and independent of n. In particular, there exist N * in N * and a positive scalar ε = min(µ 1,n , hµ 2,n , µ 3,n ) > 0 such that the functional V n associated to matrices (P n , S, R) given by [START_REF] Gomez | Necessary and sufficient stability condition by finite number of mathematical operations for time-delay systems of neutral type[END_REF] satisfies

Vn (ϕ) ≤ -ε |ϕ(0)| 2 + 0 -h |Cϕ(θ)| 2 dθ+|Cϕ(-h)| 2 ,
(40) for all ϕ in C pw (-h, 0; R nx ) and n ≥ N * . As a second step, we show that this inequality leads to the satisfaction of LMI condition (9b). Indeed, consider ϕ as the function expressed in (35), which is a polynomial of degree n -1 over the interval (-h, 0). Re-injecting expression (35) into the definition the time-derivative of the functional V n in [START_REF] Gu | Complete Quadratic Lyapunov-Krasovskii Functional: Limitations, Computational Efficiency, and Convergence[END_REF] yields

Vn (ϕ) = ϕ0 ϕn ϕ h Φ - n ϕ0 ϕn ϕ h , ( 41
)
where we have used the Bessel-Legendre equality case (8) of Lemma 1, since ϕ is a polynomial of degree n -1 over (-h, 0). Similarly, re-injecting (35) into the right-hand part of inequality (40) leads to

Vn (ϕ) ≤ -ε |ϕ 0 | 2 +ϕ n I n ϕ n +|ϕ h | 2 ≤ -ε ϕ0 ϕn ϕ h 2 . (42)
Hence, the following inequality holds

ϕ0 ϕn ϕ h Φ - n ϕ0 ϕn ϕ h ≤ -ε ϕ0 ϕn ϕ h 2 , ∀ ϕ0 ϕn ϕ h ∈ R nx+(n+1)nz ,
which ensures that matrix Φ - n is negative definite and concludes the proof.

Necessary LMI conditions

The final step consists in demonstrating that the previous corollaries impose the satisfaction of the LMI in Theorem 2, which corresponds to the following statement.

Theorem 4 If the trivial solution of system (1) is GES, then then there exist an order N * in N * and matrices

(P N * , R, S) in S nx+N * nz × S nz + × S nz + such that inequal- ities Φ + N * 0 and Φ - N * ≺ 0 in (9) hold.
Proof : Consider the functional V n given by ( 10) associated to matrices (P n , S, R) in S nx+nnz × S nz + × S nz + given by [START_REF] Seuret | Hierarchy of LMI conditions for the stability analysis of time-delay systems[END_REF]. Recall that these matrices have been built for given matrices (W 1 , W 2 , W 3 ) in S nx + × S nz + × S nz + with the idea to propose an approximated reconstruction of the complete functional V. We have then shown in Lemma 8 that having system (1) GES guarantees that Φ + n 0 holds, for any order n in N * . Applying Lemma 9, we have also ensured the existence of an order N * such that Φ - N * ≺ 0 holds. These two results close the proof.

To sum up, this section has been dedicated to the proof of the converse side of Theorem 2. The next section aims at providing an estimation of the order N * for which this converse statement holds.

4 Necessity of LMI stability conditions: Estimation of the order N *

This section aims at providing an estimation of the order N * for which the sufficient LMI conditions of Theorem 2 are necessarily true if the system is assumed to be GES.

Theorem 5 If the trivial solution of system (1) is GES, then there exist matrices

(P N * , R, S) in S nx+N * nz × S nz + × S nz + such that inequalities Φ + N * 0 and Φ - N * ≺ 0 in (9) hold with N * = 5+ ( 4 1+h 2 +|A|+|B|) 1 + 2 +2 3 2 1+h 2 2 ,
(43) where parameters 1 , 2 , 3 are defined by ( 23), ( 25), ( 27), but are recalled here for consistency

1 = √ n x π 2 3 2 e h|M| M 2 N -1 h 2 |B| , 2 = 1 2 √ n x π 2 3 2 e h|M| M 4 N -1 h 3 |B| , 3 = √ 2π(1 + √ n x πhe h|M| M 2 N -1 )h |B| 2 .
Proof : The key step for the estimation of N * appears in the first part of the proof of Lemma 9, more particularly in equations (39), that are recalled here for the sake of readability (with |C| = 1),

0 < µ 1,n = σ(W 1 )-(3+|A|)ū 1,n -ū2,n -ū3,n , 0 < µ 2,n = 1 h σ(W 2 )-(|A|+|B|)ū 1,n -ū2,n -2ū 3,n , 0 < µ 3,n = σ(W 3 )-(1+|B|)ū 1,n -ū3,n .
(44) As matrices (W 1 , W 2 , W 3 ) are arbitrary, it is possible to select for all λ > 0

W 1 = λη 1 I nx , hW 2 = λη 2 I nz , W 3 = λη 3 I nz , (45)
where η 1 , η 2 and η 3 > 0 are positive scalars such that η 1 + η 2 + η 3 = 1. This selection makes that

|W | = |W 1 + hW 2 + W 3 | = λ.
Therefore, the upper bounds of the approximation errors given in ( 22), ( 24) and ( 26) verify for all n ≥ 6,

ū1,n = 1 √ n -3 λ ≤ 1 √ n -5 λ, ū2,n = 2 √ n -5 λ, ū3,n = 3 √ n -3 λ ≤ 3 √ n -5 λ.
The objective is now to evaluate the necessary condition obtained in the previous section, more particularly the ones arising from inequalities (44). We select then η 1 , η 2 and η 3 as follows

η 1 ≥ 1 (3 + |A|) + 2 + 3 √ n -5 , η 2 ≥ 1 (|A| + |B|) + 2 + 2 3 √ n -5 h 2 , η 3 ≥ 1 (1 + |B|) + 3 √ n -5 .
Recalling that η 1 + η 3 + η 3 = 1, we obtain

4 1 + (|A|+|B|) 1 + 2 +2 3 (1+h 2 ) √ n -5 ≤ 1,
which yields condition (43).

Remark 2 It is worth noticing that the minimal order N * given by (43) is an over estimate of the necessary order. Indeed, to get such estimate, several over-bounding approximations have been performed, leading to a conservative estimation. Nevertheless, this is a first solution to the problem of necessary and sufficient LMI conditions for the stability of time-delay systems. Several improvements to get a more accurate estimation is let to future works. One of the possible improvements would consider, for instance, more accurate estimation of 1 and 2 which would greatly reduce the estimation by using the fact that U is infinitely continuous on [-h, 0). Another possibility to improve our estimation of N * is to compute the minimal order directly from inequality (31).

Numerical results

The numerical application of Theorems 2 and 5 is commented and illustrated on the following academic examples corresponding to (1) with

Example 1 A = 1 and A d = -2. Example 2 A = [ 0 0 0 0 ] and A d = -1 0.2 -0.1 0 . Example 3 A = 0 0 1 0 0 0 0 1 -10-λ 10 0 0 5 -15 0 -1 4
and A d = 0 0 0 0 0 0 0 0 λ 0 0 0 0 0 0 0 . First, we recall the efficiency of the sufficient stability condition of Theorem 2. For Examples 1 and 2, Table 1 reports the maximal allowable delay, for which LMI (9) at order n = 1, 2, 3 are satisfied. Figure 1a shows the stability region for Example 3 achieved by the same condition for several orders. One can see that in this table and in this figure the efficiency of Theorem 2 even for very low orders n to provide an inner approximation of the stability regions.

Thanks to Theorem 5, these LMI conditions of stability are proven to be also necessary. Therefore, the inner approximation mentioned above is supposed to converge toward the expected regions of stability as the order n increases. Table 2 reports the estimated order N * for which the LMI conditions become necessary. Similarly, Figure 1b shows the values of N * given by (43) for various pairs of (λ, h) in [0.1, 10] × [0, 3]. Clearly the values of N * computed here are too large to propose tractable test of instability in comparison to [START_REF] Gomez | Necessary and sufficient stability condition by finite number of mathematical operations for time-delay systems of neutral type[END_REF]. Nonetheless, Theorem 5 provides a theoretical estimation of N * , from which these sufficient LMI conditions become necessary.

This estimation is a by-product of our main result that has not been optimized in this paper. Some guidelines to improve this estimation have been suggested in Remark 2.

Interestingly Figure 1b shows that increasing both λ and h makes that N * also increases very fast and reach very large values. Indeed, formula (43) shows that the order N * grows as the delay h or the norm |B| increase, respectively in h 8 e 2h|M | and |B| 4 . It is also worth noticing that when the parameter λ and h are getting closer to the black lines, the estimation N * increases even more faster. These black lines correspond to the situation where some roots of the time-delay system cross the imaginary axis. This makes sense because if characteristic roots are approaching the imaginary axis, then matrix N tends to a singular matrix and N -1 tends to infinity. Hence, the upper bounds 1 , 2 , 3 also tend to infinity.

All together, this ascertainment can be correlated with Figure 1a and may explain why some stable regions are difficult to reach with LMI (9), especially for low orders.

Conclusions

This paper studied the convergence of sufficient LMI conditions for the stability analysis of time-delay systems, based on the Bessel-Legendre inequalities. While the framework of Bessel-Legendre already showed their relevance regarding the hierarchical structure of these LMI, i.e. increasing their order can only reduce their conservatism, the main contribution of this paper demonstrates that this framework also offers an asymptotically necessary condition of stability for time-delay systems. In other words, it is now proven that if a time-delay system is stable, then there exists an order N * such that these LMI conditions are verified at least at this order. A numerical estimation of this necessary order has also been provided, ensuring that if these LMI conditions are not verified at this order, then the system is proven to be unstable. To summarize, these LMI conditions arising from the Bessel-Legendre inequality are sufficient and asymptotically necessary.

The estimation of the order N * provided in this paper can be numerically very large, even on some simple examples. This can be seen as a major drawback of the contribution. Nevertheless, the main theoretical achievement of the paper remains the proof of convergence of these LMI conditions. Providing more accurate estimations is let to future direction of research. A generalization to other approximation or discretization methods or the investigation of combined procedures could also be considered.

A Preliminaries on Legendre Polynomials for the technical proofs

In this section, several properties of Legendre polynomials are presented to help in the technical developments of this paper.

B Symmetric property

In the core of the paper, we have used the following property satisfied by the Legendre approximated error Ũ2,n of function B U B on the interval [-h, h].

Property 2

The approximation error Ũ2,n given by (20) and recalled below

Ũ2,n (θ) = B U (θ)B -U 2,n n θ + h 2h , ∀θ ∈ [-h, h], verifies Ũ2,n (θ) = Ũ 2,n (-θ), for all θ ∈ [-h, 0].
Proof : From the definitions of Ũ2,n and U 2,n , we have

Ũ 2,n (-θ) = B U (-θ)B - 1 2h h -h n h-θ 2h I n n h+θ 1 2h B U (θ 1 )Bdθ 1 .
Then, property (A.3) of the Legendre polynomials and since n h-θ1 2h

I n n h+θ 2h
is an n z × n z matrix that is proportional to I nz , it commutes with B U (θ 1 )B, so that the previous expression writes

Ũ 2,n (-θ) = B U (-θ)B - 1 2h h -h B U (θ 1 )B n h-θ 1 2h I n n h+θ 2h dθ 1 .
Recalling that U (-θ) = U (θ) holds for all θ in [-h, h] and performing the change of variable θ 2 = -θ 1 , the previous expression becomes

Ũ 2,n (-θ) = B U (θ)B - 1 2h h -h B U (θ 2 )B n h+θ 2 2h dθ 2 I n n h+θ 2h ,
which is the definition of Ũ2,n (θ).

C Proof of Lemma 1

Proof : Consider a function z in L 2 (-h, 0; R nz ), a matrix S in S nz and h > 0. Define the function zn by

zn (θ) = z(θ) - 1 h n θ + h h I n ζ n (z), (C.1) where ζ n (z) = 0 -h n θ1+h h z(θ 1 )dθ 1 .
The function zn in L 2 (-h, 0; R nz ) represents the Legendre approximation error of function z at order n. The quantity 0 -h z n (θ)S zn (θ)dθ exists and the orthogonal property of the Legendre polynomials (6) yields

0 -h z n (θ)S zn (θ)dθ = 0 -h z (θ)Sz(θ)dθ - 1 h ζ n (z)(I 1 n ⊗S)ζ n (z),
Clearly, if z is a polynomial of order n-1, then zn (θ) = 0 for all θ ∈ (-h, 0) so that 0 -h z n (θ)S zn (θ)dθ = 0 , which ensures that the equality (8) holds. Otherwise, recalling that 0 -h z n (θ)S zn (θ)dθ > 0 since S 0, inequality [START_REF] Fridman | Introduction to Time-Delay Systems : analysis and control[END_REF] holds and concludes the proof.

D Proof of Lemma 4

Proof : First, let us introduce the functional

W(ϕ) =V(ϕ) - 0 -h ϕ (θ)C (θ + h)W 2 Cϕ(θ)dθ -ε 0 -h ϕ(0) Cϕ(θ) ϕ(0) Cϕ(θ) dθ, (D.1)
where V is the complete Lyapunov-Krasovskii functional given by ( 13). According to Lemma 3, the timederivative of W along the trajectories of system (1) writes

Ẇ(x t ) = -xt(0) Cxt(-h) W1+C hW2C 0 * W3 ϕ(0) Cxt(-h) +ε xt(0) Cxt(-h) H(A)+C C B * -In z xt(0)
Cxt(-h) . (D.2) Clearly, there exists a sufficiently small ε > 0 such that Ẇ(x t ) ≤ 0. Then, integrating (D.2) from 0 to ∞ and assuming that system (1) is GES yields W(x 0 ) ≥ 0, for any initial conditions x 0 = ϕ in C pw (-h, 0; R nz ), which concludes the proof.

E Proof of Lemma 6

The next properties reflect the fact that U has continuous and bounded second and fourth order derivatives on [-h, 0) (and (0, h]). Their bound are provided therein.

Property 3

The Lyapunov matrix U associated to W defined in [START_REF] Jarlebring | Characterizing and computation H 2 norm of time delay systems by solving the delay Lyapunov equation[END_REF]

satisfies sup θ∈[-h,0) U (2) (θ) = sup θ∈(0,h] U (2) (θ) ≤ ρ |W | , (E.1) sup θ∈(0,h] U (4) (θ) ≤ ρ |W |, (E.2)
with parameters ρ, ρ given by

ρ = √ n x e h|M| M 2 N -1 , ρ = √ n x e h|M| M 4 N -1 , (E.
3) and matrices M, N given by [START_REF] Kharitonov | Lyapunov-Krasovskii approach to the robust stability of time-delay systems[END_REF].

Proof : Thanks to the equivalence of matrix norms, inequalities

M = |M | ≤ |vec(M )| ≤ √ p |M | hold, for
any square M of dimension p. Then, for all θ in (0, h], we have

U (k) (θ) ≤ [ I n 2 x 0 ]e θM M k N -1 -vec(W ) 0 , ≤ e θM M k N -1 |vec(W )| , ≤ √ n x e θM M k N -1 |W | ,
where U (k) stands for the k th derivatives of function U . Moreover, recalling the definition of exponential matrices, i.e. e θM = ∞ k=0

(θM) k k! , an upper bound of |e θM | is obtained as follows e θM ≤ ∞ k=0 (θM) k k! ≤ ∞ k=0 |θ| k |M| k k! ≤ ∞ k=0 h k |M| k k! = e h|M| ,
which yields the results (E.1) and (E.2).

E.1 Proof of item (i) of Lemma 6

Proof : The objective of the proof is to provide an upper bound of the norm of the approximation error Ũ1,n , which depends explicitly on order n. To do so, let us first rewrite the expression of this error as follows

Ũ1,n (θ) = U (h+θ)B - n-1 k=0 U k 1 l k θ + h h , with U k 1 being the projection of U (θ + h)B onto the k th Legendre polynomial, that is U k 1 = 2k + 1 h 0 -h U (θ + h)Bl k θ + h h dθ, = (2k + 1) 1 0 U (hθ)l k (θ)dθ. (E.4)
Using the differentiation rule (A.6), the previous expression can be rewritten as

U k 1 = 1 2 1 0 U (hθ)B l k+1 (θ) -l k-1 (θ) dθ. (E.5)
Then, an integration by parts yields

U k 1 = h 2 1 0 U (hθ)B (l k-1 (θ) -l k+1 (θ)) dθ, (E.6)
where we have used l k+1 (0) = l k-1 (0) and l k+1 (1) = l k-1 (1) (see (A.2)), which cancels the first terms of the integration by parts. Repeating this operation, we get = π, the following upper bound is obtained

U k 1 = h 2 4(2k-1)
U k 1 ≤ ρ( π 2 ) 3 2 h 2 |B| |W | 2(k -2) 3 2
, ∀k ≥ 3.

Applying now (A.4) ensures that for any integer N ≥ n and for all θ in [-h, 0]

N k=n U k 1 l k h+θ h ≤ N k=n U k 1 ≤ N k=n 1 |W | 2(k -2) 3 2
,

denoting 1 = ρ( π 2 )
3 2 h 2 |B|. Finally, using an integral over estimation of the sum, we obtain

N k=n U k 1 l k θ+h h ≤ 1 |W | 1 √ n -3 - 1 √ N -3 .
We conclude that the approximation error Ũ1,n (θ) =

∞ k=n U k 1 l k θ+h h
is uniformly bounded as in [START_REF] Niculescu | Delay effects on stability: a robust control approach[END_REF] and converges also uniformly to zero as n tends to infinity.

E.2 Proof of item (ii) of Lemma 6

Proof : The objective of the proof is to demonstrate the uniform convergence towards zero of where α k,i are positive coefficients whose expression is omitted for simplicity but which verify |α k,i | ≤ 1. Therefore, an upper bound of the norm of U k 1 can be derived using property (A.5) on the 2 4 terms, yielding

U k 1 ≤ 1 2 π 2 h 4 |B| (2k-5)(2k-3)(2k-1) 1 0 U (4) (hθ) (k-4)θ(1 -θ) dθ.
Using (E.2), (2k -5) ≤ 2(k -4) and

1 0 dθ √ θ(1-θ) = π,
the following upper bound is obtained .

U k 1 ≤ ρ ( π 2 ) 3 2 h 4 |B| |W | 2(k -4)
Finally, an integral over estimation of the sum leads to

1 h N k=n U k 1 l k θ+h h ≤ 2 |W | 1 √ n-5 - 1 √ N -5 .
We conclude that the approximation error Ũ 1,n (θ) = d dθ

∞ k=n U k 1 l k θ + h h
is uniformly bounded as in [START_REF] Peet | A dual to Lyapunov's second method for linear systems with multiple delays and implementation using SOS[END_REF] and converges to zero as n tends to infinity. Using the differentiation rule (E.5) and integration by parts (E.6) on both intervals (0, 1 2 ) and ( 1 2 , 1), we obtain

U k 2 = h 1 0
B U h(2θ -1) B(l k-1 (θ) -l k+1 (θ)) dθ, since l k+1 (θ) = l k-1 (θ) for θ ∈ {0, 1} is ensured by (A.2) and since the continuity of U h(2θ -1) at θ = 1 2 is ensured by Property 5 (i). Then, repeating this operation,

  (a) Sufficiency: stability areas in the plan (λ, h) given by (9) for n ∈ {1, 2, 3, 4, 5}.(b) Necessity: order N * in the plan (λ, h) given by (43).

Fig. 1 .

 1 Fig. 1. Example 3.

1 0UU

 1 (hθ)B (l k-2 (θ) -l k (θ)) dθ -(hθ)B (l k (θ) -l k+2 (θ)) dθ.Then, Property (A.5) ensures that

Ũ 1 U k 1 U ( 4 ) 1 0U ( 4 )U ( 4 ) 1 0U ( 4 ) 1 0U ( 4 ) 1 0U ( 4 ) 1 0U ( 4 ) 1 0U ( 4 ) 1 0U ( 4 ) 1 0U ( 4 )U ( 4 ) 1 0U ( 4 ) 1 0U ( 4 ) 1 0U ( 4 ) 1 0U ( 4 )

 11414414141414141414414141414 given by (E.4). Here, we repeat the pro-14 cess (E.5)-(E.6) four times successively to obtain (hθ)Bl k-4 (θ)dθh 4 2 4 (2k-5)(2k-3)(2k-1) (hθ)Bl k-2 (θ)dθ -(hθ)Bl k (θ)dθ h 4 2 4 (2k-1) 2 (2k+1) (hθ)Bl k-2 (θ)dθ + h 4 2 4 (2k-1) 2 (2k+1) (hθ)Bl k (θ)dθ h 4 2 4 (2k-1)(2k+1)(2k+3) (hθ)Bl k-2 (θ)dθ + h 4 2 4 (2k-1)(2k+1)(2k+3) (hθ)Bl k (θ)dθ + h 4 2 4 (2k-1)(2k+1)(2k+3) (hθ)Bl k (θ)dθ h 4 2 4 (2k-1)(2k+1)(2k+3) (hθ)Bl k+2 (θ)dθ + h 4 2 4 (2k+1)(2k+3) 2 (hθ)Bl k (θ)dθ -(hθ)Bl k+2 (θ)dθ + h 4 2 4 (2k+3) 2 (2k+5) (hθ)Bl k (θ)dθ h 4 2 4 (2k+3) 2 (2k+5) (hθ)Bl k+2 (θ)dθ h 4 2 4 (2k+3)(2k+5)(2k+7) (hθ)Bl k+2 (θ)dθ + h 424 (2k+3)(2k+5)(2k+7) (hθ)Bl k+4 (θ)dθ, i l k-4+2i (θ) dθ (2k-5)(2k-3)(2k-1) ,

3 2 3 2 h 3

 333 (2k-3)(2k-1), ∀k ≥ 5. (E.7)We denote 2 = 1 2 ρ ( π 2 ) |B| and we use this upper bound to obtain the result. The application of (A.7) and (E.7) gives, for any integer N ≥ n ≥ 5 and for all θ in [-h, 0

E. 3 U k 2 - 1 )

 321 Proof of item (iii) of Lemma 6 Proof : The objective of the proof is to demonstrate the uniform convergence towards zero of Ũ2,n (θ) = B U (θ)B -U 2,n n θ+h 2h , = B U (θ)Bthe k-th coefficient of matrix U 2,n given by Bl k (θ)dθ. (E.8)

  Lemma 6 Consider the Lyapunov matrix U for the time-delay system (1) defined for any (W 1 , W 2 , W 3 ) in S nx + × S nz + × S nz + . The following statements hold. (i) The approximated Legendre function U 1,n n θ+h h converges uniformly to U (θ + h)B, when n tends to infinity, on the closed interval [-h, 0].

	More precisely, for any n ≥ 4, the following inequality
	holds
	sup
	θ∈[-h,0]

Table 1

 1 Sufficiency: maximal allowable delay h which satisfies[START_REF] Gomez | Necessary stability conditions for neutral-type systems with multiple commensurate delays[END_REF].

		n = 1 n = 2 n = 3	Expected
	Ex. 1 0.577 0.604 0.604	atan √ √ 3	3	0.604
	Ex. 2	-	1.600 1.603	1.603
	Table 2
	Necessity: estimated order N * given by (43).
		h = 0.1 h = 0.5 h = 1 h = 2
	Ex. 1	75	10 8	10 10		10 13
	Ex. 2	6	537	10 7		10 10

and n in N * , the following equation holds

(A.3) (iv) Bound: For any θ in [0, 1], the Legendre polynomials verify the following inequalities

(v) Differentiation: For any θ in [0, 1], the Legendre polynomials verify the following differentiation rule

(A.6) (vi) Bound on the derivative: For any θ in [0, 1], the Legendre polynomials verify the following inequality

(vii) Bonnet's recursion formula: For any θ in [0, 1] and k in N * , the Legendre polynomials satisfy

Proof : The properties of orthogonality (i), special values (ii)-(iii) and differentiation (v) are provided in [1, Formula 22.2.11, 22.4.5 and 22.7.15], respectively. For the boundedness property, especially (A.5), the proof can be found in [START_REF] Rainville | Special functions[END_REF]Theorem 61].

we get to

where ∆U (0) := lim →0 (U ( ) -U (-)). Then, an upper bound of the norm of U k 2 can be derived by the use of property (A.5) for θ ∈ [0, 1] and especially for θ = 1 2 , yielding

Thanks to Property 5 (ii), upper bound (E.1), inequality (2k-1) ≤ 2(k-2) and

= π, we have

where notation 3 := √ 2π(1 + ρπh)h |B| 2 is introduced. For the same reasons stated in the proof of Lemma 6 (i), for any integer N ≥ n and for all θ in [0, 1], the following upper bound is obtained

We conclude that the series Ũ2,n (θ) =

exists, converges to zero as n tends to infinity and that inequality (26) holds.