PhD Annabelle Bédard

PhD Mikhail Sofiev

PhD Sylvie Arnavielhe

MD Josep M Antó

Judith Garcia-Aymerich

PhD Michel Thibaudon

Karl Christian Bergmann

MD Ruta Dubakiene

BSc Anna Bedbrook

MSc Gabrielle L Onorato

PhD I Annesi-Maesano

MD (11), D Laune PhD J L Pépin

MSc S Zeng

MD J Bousquet

PhD X Basagaña

Xavier Basagana

Interactions Between Air Pollution and Pollen Season for Rhinitis Using Mobile Technology: A MASK-POLLAR Study

come

Highlight box 1. What is already known about this topic?

The impact of air pollutants on the severity of allergic rhinitis symptoms, and its modification by pollen exposure is still a matter of debate. Real-life studies have never been conducted using an mHealth App.

2. What does this article add to our knowledge? An association between air pollutants and rhinitis symptoms is observed during the grass but not the birch pollen season, suggesting an interaction between air pollutants and grass pollen.

3. How does this study impact current management guidelines? This paper is of primary importance in the development of next-generation GRADE guidelines that will embed real-world-evidence, aerobiology and air pollution.

Key words

Allergic rhinitis, MASK, mobile health, ozone, POLLAR pollen, pollution, PM2.5

Introduction

Allergic rhinitis (AR) impacts quality of life and affects sleep, work productivity, school performance and daily activities [START_REF] Bousquet | Allergic Rhinitis and its Impact on Asthma (ARIA) 2008 update (in collaboration with the World Health Organization, GA(2)LEN and AllerGen)[END_REF]. Both pollen and outdoor air pollution exposure [START_REF] Jones | Ragweed pollen and allergic symptoms in children: Results from a three-year longitudinal study[END_REF][START_REF] Kim | Association between Pollen Risk Indexes, Air Pollutants, and Allergic Diseases in Korea[END_REF] are known to influence rhinitis symptoms. Several in vitro studies have suggested that traffic-related pollutants can disrupt pollen, a modification that causes more allergic symptoms [START_REF] Senechal | A Review of the Effects of Major Atmospheric Pollutants on Pollen Grains, Pollen Content, and Allergenicity[END_REF][START_REF] Schiavoni | The dangerous liaison between pollens and pollution in respiratory allergy[END_REF]. In particular, it has been suggested that particulate matter (PM) may interact with aeroallergens, promoting airway sensitization by modulating the allergenicity of airborne allergens [START_REF] Baldacci | Allergy and asthma: Effects of the exposure to particulate matter and biological allergens[END_REF], and that diesel exhaust carbon particles could interact with grass pollen allergens and trigger attacks of asthma [START_REF] Knox | Major grass pollen allergen Lol p 1 binds to diesel exhaust particles: implications for asthma and air pollution[END_REF]. Several epidemiological studies have investigated the additive effects of air pollution and pollen exposure on rhinitis symptoms, with conflicting findings [START_REF] Annesi-Maesano | Grass pollen counts, air pollution levels and allergic rhinitis severity[END_REF][START_REF] Villeneuve | Is outdoor air pollution associated with physician visits for allergic rhinitis among the elderly in Toronto, Canada?[END_REF] but to our knowledge, no study has investigated the effect of the interaction between air pollution and pollen exposure on rhinitis symptoms (i.e. how the relationship between air pollution and allergy symptoms is modified by pollen exposure or vice versa). A few studies have investigated the effect of this interaction on asthma symptoms. One study conducted in two Spanish citiesone with high levels of air pollution and one with low levels of pollution -reported strong positive associations between air pollutants (ozone, SO2 and PM10) levels and seasonal allergic asthma symptoms, only in the city with high levels of pollution [START_REF] Brito | Air pollution and seasonal asthma during the pollen season. A cohort study in Puertollano and Ciudad Real (Spain)[END_REF]. One study has investigated associations between grass and birch pollen concentrations with hospital admissions for asthma, reporting stronger associations in the case of high PM10 and ozone concentrations [START_REF] Guilbert | Relationships between aeroallergen levels and hospital admissions for asthma in the Brussels-Capital Region: A daily time series analysis[END_REF].

The two major pollutants reported in rhinitis are ozone and PM2.5.

Mobile technology may help to better understand the links between pollution and allergic diseases. MASK-air (Mobile Airways Sentinel NetworK) is an information and communication technology (ICT) system centred around the patient [START_REF] Bourret | MASK rhinitis, a single tool for integrated care pathways in allergic rhinitis[END_REF][START_REF] Bousquet | MACVIA-ARIA Sentinel NetworK for allergic rhinitis (MASK-rhinitis): the new generation guideline implementation[END_REF][START_REF] Bousquet | ARIA 2016: Care pathways implementing emerging technologies for predictive medicine in rhinitis and asthma across the life cycle[END_REF][START_REF] Bousquet | MASK 2017: ARIA digitally-enabled, integrated, person-centred care for rhinitis and asthma multimorbidity using real-world-evidence[END_REF][START_REF] Bedard | Mobile technology offers novel insights on control and treatment of allergic rhinitis. The MASK study[END_REF] and operational in 23 countries. It uses a treatment scroll list which includes all medications customized for each country as well as visual analogue scales (VASs) to assess global allergy, rhinitis, eye and asthma control. Over 29,000 users and 200,000 VAS days have been recorded. MASK can be used to investigate the relationship between outdoor air pollutants and rhinitis.

In the frame of the POLLAR (Impact of Air POLLution in Asthma and Rhinitis) project, the objective of our study was to investigate the interaction between air pollution and allergens and its impact on allergy symptoms.

Methods

Design of the study

POLLAR (Impact of Air POLLution in Asthma and Rhinitis) is a project of the European Institute of Innovation and Technology (EIT Health) that embeds environmental data into the data of MASK-air patients in order to deliver interconnected rhinitis management assistance during peak periods of air pollution and promote measures that help people deal with health and societal consequences of the condition [START_REF] Bousquet | POLLAR: Impact of air POLLution on Asthma and Rhinitis; a European Institute of Innovation and Technology Health (EIT Health) project[END_REF]. To achieve this, one of the objectives of POLLAR is to investigate the interaction between air pollution and pollen exposure and its impact on AR control. A series of analyses were carried out on the MASK-air data available for 2017 and 2018 in Northern and Central Europe. The aim was to assess the impact of the two major pollutants associated with rhinitis (ozone and PM2.5) on the severity of rhinitis during and outside the estimated pollen seasons. To do so, we linked the self-reported rhinitis symptoms obtained via an App with air pollution predictions obtained from the SILAM (System for integrated modelling of atmospheric composition (18)) database. We then conducted analyses separately by pollen seasons derived using Google Trends (GTs). We restricted the study to Northern and Central Europe (Austria, Belgium, Czech Republic, Denmark, Finland, France, Germany, Northern Italy, Lithuania, Netherlands, Poland, Sweden, Switzerland, United Kingdom) since these pollen seasons are better identified than those in Southern Europe. The two major pollen seasons included Betulaceae (i.e. birch) and grass (more details below).

Setting -study population and observations unit

All days recorded by the users of the MASK-air app between February 2017 and October 2018 were included with no exclusion criteria and according to methods previously described [START_REF] Bedard | Mobile technology offers novel insights on control and treatment of allergic rhinitis. The MASK study[END_REF][START_REF] Bousquet | Work productivity in rhinitis using cell phones: The MASK pilot study[END_REF][START_REF] Bousquet | Treatment of allergic rhinitis using mobile technology with real-world data: The MASK observational pilot study[END_REF]. The unit of observations used in the present study is person-days (i.e. each day of app use by any user). Some of the users used the app more than once a day to report symptoms. We carefully analyzed this issue and found that the highest recording should be used and this approach was used to eliminate duplicates/multiplicates [START_REF] Bedard | Mobile technology offers novel insights on control and treatment of allergic rhinitis. The MASK study[END_REF][START_REF] Bousquet | Work productivity in rhinitis using cell phones: The MASK pilot study[END_REF].

Days where the App was used in Northern and Central Europe were included in the analyses.

Areas were defined as those above the latitude of Lyon (Figure E1). Table E1 summarizes the number of days included in the study by country for each year.

Ethics

The Allergy Diary is CE1 registered. By using k-anonymity, the data were anonymized including the data related to geolocation [START_REF] Samreth | Geolocation with respect to personal privacy for the Allergy Diary app -a MASK study[END_REF]. An Independent Review Board approval was not required since the study is observational and users agree to having their data analysed (terms of use) [START_REF] Bedard | Mobile technology offers novel insights on control and treatment of allergic rhinitis. The MASK study[END_REF][START_REF] Bousquet | Work productivity in rhinitis using cell phones: The MASK pilot study[END_REF][START_REF] Bousquet | Treatment of allergic rhinitis using mobile technology with real-world data: The MASK observational pilot study[END_REF].

m-health data

Geolocalized users assess their daily symptom control in a participatory way using the touchscreen functionality on their smart phone to click on four consecutive VAS scores (i.e. global, nasal and ocular symptoms and asthma, quantifying the degree of symptoms from 0 to 100). Only global VAS for rhinitis was used in the study. Concomitantly to the report of VAS scores, users also input their daily medications using a scroll list which contains all countryspecific OTC and prescribed medications available for each country. The list has been populated using IMS data. Days reported by users were thus either considered as days with or without treatment. Information on gender, country of residence and year of birth was provided for all users, through the creation of their profile upon registration to MASK-air.

The present study is another MASK study. Some of the raw data used in other papers (up to December 2017) [START_REF] Bedard | Mobile technology offers novel insights on control and treatment of allergic rhinitis. The MASK study[END_REF][START_REF] Bousquet | Treatment of allergic rhinitis using mobile technology with real-world data: The MASK observational pilot study[END_REF] were used in this study, but the analyses differed.

Outcomes

We considered two outcomes to assess the interaction between pollen seasons and pollutants: "uncontrolled rhinitis" (as primary outcome) and VAS global (as secondary outcome).Uncontrolled rhinitis was defined, based on the previous data already available for MASK [START_REF] Bedard | Mobile technology offers novel insights on control and treatment of allergic rhinitis. The MASK study[END_REF][START_REF] Bousquet | Treatment of allergic rhinitis using mobile technology with real-world data: The MASK observational pilot study[END_REF] as days with "rhinitis high" according to the following criteria:

Exposure data

Health data from the mobile App was linked to environmental exposures (pollen and air pollution data) by date and geolocation (when access to this information was allowed by the user of the app).

Air pollution data

Pollutants (ozone and PM2.5 concentrations; spatial resolution: 12x12 km grid) for the whole of Europe were obtained from the SILAM (System for integrated modelling of atmospheric composition) database [START_REF] Kollanus | Mortality due to Vegetation Fire-Originated PM2.5 Exposure in Europe-Assessment for the Years 2005 and 2008[END_REF][START_REF] Sofiev | A dispersion system SILAM and its evaluation against ETEX data[END_REF][START_REF] Sofiev | Construction of the SILAM Eulerian atmospheric dispersion model based on the advection algorithm of Michael Galperin[END_REF] of the Finnish Meteorological institute (http://silam.fmi.fi).

SILAM is a global-to-meso-scale dispersion model that uses anthropogenic emission datasets, information on wild-land fires, meteorological data, as well as embedded emission computations for sea salt, pollen, wind-blown dust, and natural volatile organic compounds to produce the forecasts of several pollutants, including ozone and particulate matter with a diameter of less than 2.5 m (PM2.5). We used forecasts for the same day at noon at a 0.1 degree resolution.

Temperatures

We used daily maximum temperature information from the dataset of NASA's Modern-Era Retrospective Analysis for Research and Applications (MERRA). MERRA is a NASA reanalysis dataset generated using version 5.2.0 of the Goddard Earth Observing System (GEOS-5). MERRA uses observations (such as weather station data and satellite irradiance) from multiple platforms to reanalyse these observations using numerical models based on physical rules of atmospheric motions and to generate meteorological records [START_REF] Rienecker | MERRA: NASA's Modern-Era Retrospective Analysis for Research and Applications[END_REF].

Urban vs rural area

Information on urban vs rural area by geolocation was obtained using the Global Human Settlement Layer data that classifies settlement typologies using data on population size, population and built-up area [START_REF] Florczyka | GHSL Data Package[END_REF]. Classes 30-23-22-21 were aggregated to form the urban domain, while classes 13-12-11-10 formed the rural domain.

Pollen season

The hypothesis of the study was that air pollution affects symptoms due to pollen exposure.

As explained above, we restricted the study to Northern and Central Europe since pollen seasons are better identified there compared to Southern Europe. The two major pollen seasons included Betulaceae (i.e. birch, alder, elm) and grass pollens. To identify the pollen seasons, we initially used the visual inspection of GTs separately for 2017 and 2018. This is supported by previous results

showing that GTs could be used in France to identify the Betulaceae (birch) and grass pollen seasons [START_REF] Bousquet | Google Trends and pollen concentrations in allergy and airway diseases in France[END_REF]. We used three terms that we had previously validated (allergy, disease; allergic rhinitis, subject; pollen, subject) [START_REF] Bousquet | Google Trends terms reporting rhinitis and related topics differ in European countries[END_REF]. We did not use asthma as a term since it is not very sensitive except during severe outbreaks [START_REF] Bousquet | Assessment of thunderstorm-induced asthma using Google Trends[END_REF].

Since there was an overlap between seasons in 2018, we used pollen countsonly available for a subset of the study population -to refine the GTs analyses. These were provided by M Thibaudon (RNSA) for France [START_REF] Bousquet | Differences in Reporting the Ragweed Pollen Season Using Google Trends across 15 Countries[END_REF], KC Bergmann for Germany [START_REF] Karatzas | Computational validation of the recently proposed pollen season definition criteria[END_REF] and R Dubakiene for Lithuania.

Statistical analysis

Generalized Estimating Equations models (accounting for repeated daily measures per user) with a binomial family and a logistic link were used to study the associations between pollutant concentrations and uncontrolled rhinitis (with controlled rhinitis as a reference). We used the interquartile range (IQR, i.e. 75 th percentile (3 rd quartile) minus 25 th percentile (1 st quartile)) as the unit of exposure (i.e. ozone and PM2.5) levels. The results of logistic regression models can therefore be interpreted as the odds ratio of uncontrolled rhinitis risk obtained when comparing those exposed to a pollutant at the third quartile (i.e. high exposure) to those exposed at the first quartile (i.e. low exposure). This unit is commonly used in epidemiological studies on air pollution [START_REF] Alvarez-Pedrerol | Impact of commuting exposure to traffic-related air pollution on cognitive development in children walking to school[END_REF].

Models were adjusted for daily treatment (i.e. day with/without treatment), gender, age (derived from the year of birth), and country of residence of the user. Analyses were stratified by pollen seasons and formal tests for interaction between air pollutants and pollen season were conducted by including the product term of pollution and pollen season in the same model. Separate analyses were conducted for 2017 and 2018 (no pooled or meta-analysis was performed given the differences in the assessment of pollen seasons between 2017 and 2018).

When evidence for an association was found between a pollutant and uncontrolled rhinitis, we further adjusted for temperature in a sensitivity analysis, to investigate potential confounding. The correlation between pollutant levels and temperature was calculated for each period/season, in order to investigate potential collinearity. In order to investigate potential confounding, we conducted twopollutant models (i.e. models that include ozone and PM2.5 simultaneously as exposures).

In order to investigate a potential lag-time in the effect of air pollution on allergic rhinitis control, especially in a cross-sectional study, we repeated the analyses of the associations between air pollution and uncontrolled rhinitis, using a 1-day lag (i.e. considering the air pollution levels of the user's location on the previous dayassuming the geolocation on the previous day was the same as the geolocation on the day the app was used). In order to investigate potential effect modification by urbanization, associations were stratified by rural and urban areas. As a sensitivity analysis, we repeated our main analyses by considering global VAS -which has been used in previous papers [START_REF] Bedard | Mobile technology offers novel insights on control and treatment of allergic rhinitis. The MASK study[END_REF][START_REF] Bousquet | Treatment of allergic rhinitis using mobile technology with real-world data: The MASK observational pilot study[END_REF] -as an outcome (instead of uncontrolled rhinitis), using mixed linear regression models (for continuous outcome). In order to investigate potential non-linear effects, we repeated our analyses of the associations between air pollutants and uncontrolled rhinitis, considering ozone and PM2.5 in quartiles, testing a linear trend.

Because pollen seasons were defined in 2018 only for a subset of the countries included in the 2017 analyses (Table E1), we checked as a sensitivity analysis whether associations observed in 2017 remain similar when we restrict our study sample to that subset.

Results

Demographic characteristics of the users and usage of the app

Both SILAM and GTs data were available for 36,440 days of VAS recorded by 3,323 Allergy Diary App users in 2017 and 2018 in Northern and Central European countries (Figure 1).

Approximately 5% of the users did not report their age or reported an age below 10. Users ranged in age from zero to 91 years (mean, SD: 34.7 ± 15.6 years). There were 54% of women and 46% of men. There were 16,797 (46%) days without treatment and 19,699 (54%) days with treatment. Uncontrolled rhinitis was observed in 20% of VAS days.

The number of days of VAS reported by user had a median [p25-p75] of 2 [START_REF] Bousquet | Allergic Rhinitis and its Impact on Asthma (ARIA) 2008 update (in collaboration with the World Health Organization, GA(2)LEN and AllerGen)[END_REF][START_REF] Jones | Ragweed pollen and allergic symptoms in children: Results from a three-year longitudinal study[END_REF][START_REF] Kim | Association between Pollen Risk Indexes, Air Pollutants, and Allergic Diseases in Korea[END_REF][START_REF] Senechal | A Review of the Effects of Major Atmospheric Pollutants on Pollen Grains, Pollen Content, and Allergenicity[END_REF][START_REF] Schiavoni | The dangerous liaison between pollens and pollution in respiratory allergy[END_REF][START_REF] Baldacci | Allergy and asthma: Effects of the exposure to particulate matter and biological allergens[END_REF][START_REF] Knox | Major grass pollen allergen Lol p 1 binds to diesel exhaust particles: implications for asthma and air pollution[END_REF] and varied by country of residence but not by gender or year of birth of the user. The daily global VAS score had a median [p25-p75] of 13 and varied by gender, country of residence, year of birth of the user as well as daily treatment (Table E2).

Estimation of the pollen season

For 2017, three patterns of pollen seasons were determined corresponding to the three areas depicted in Figure E1: 1) Continental and semi-continental climate (Germany, Austria, Belgium, Northern France, Netherlands, Poland, Switzerland, Czech Republic, Northern Italy), 2) Northern countries (Denmark, Finland, Lithuania, Sweden), and 3) the UK (Figure 2A). The Betulaeae and grass pollen seasons could not be easily distinguished and the analysis was completed with pollen counts. For most countries, two GT peaks were observed in 2017 (i.e. clear "birch" and "grass" peaks) with an in between pollen seasons and no pollen seasonsee examples in Figure 3A.

For 2018, five patterns of pollen seasons were determined: 1) Germany, France and Switzerland, 2) Belgium, Netherlands and Denmark, 3) the UK, 4) Austria and 5) Poland (Figure 2B). For most countries, three GT peaks were observed in 2018 (i.e. in addition to the "birch" and "grass" peaks, an intermediate peak was observed -see examples in Figure 3B). Since there was some uncertainty on the intermediate birch-grass pollen season, pollen counts from Germany, France and Lithuania were assessed and showed that grass pollens were retrieved during this intermediate season. Therefore, for all countries, the grass pollen and the intermediate seasons were studied jointly.

Associations between pollution and uncontrolled rhinitis

Strong significant associations were observed between ozone concentrations and uncontrolled rhinitis for both 2017 and 2018 overall (Table 1). These differences were not statistically significant on formal testing for interaction when associations were stratified by seasons. No association was observed during the birch season (odds ratios (ORs) per IQR increase in ozone levels: 1.10 (95% confidence interval: 0.95, 1.26) and 0.96 (0.87, 1.05), for 2017 and 2018 respectively) whereas positive associations were found during the grass pollen season (ORs: 1.25 (1.11, 1.41) and 1.14 (1.09, 1.21), for 2017 and 2018 respectively). Associations were stronger during the grass pollen season of 2017 than in 2018. In general, levels of ozone were highest during the grass pollen season and lowest outside the pollen season (Table 1).

No association was observed between PM2.5 and uncontrolled rhinitis for 2017 overall (Table 1).

When the association was stratified by seasons, no association was observed during the birch season whereas a strong positive association was found during the grass pollen season and a weak negative association was found during the no pollen season (ORs: 1.02 (0.96, 1.09), 1.21 (1.09, 1.34) and 0.96 (0.93, 0.99), respectively). Levels of PM2.5 were highest during the birch pollen season and lowest outside the pollen season (Table 1). A positive association was observed between PM2.5 and uncontrolled rhinitis for 2018 overall. When the association was stratified by seasons, similar positive associations were found during the birch and grass pollen seasons but no association was observed during the no pollen season (ORs: 1.11 (1.07, 1.15), 1.11 (1.04, 1.19), 1.00 (0.96, 1.04), respectively).

In order to investigate potential confounding by temperature in the associations found between air pollutants and uncontrolled rhinitis, we fitted additional analyses by considering both the pollutant level (ozone or PM2.5) and the maximal temperature as exposures in the regression models (Table 2).

We observed that, while strong positive associations were found quite consistently throughout the year between temperature and uncontrolled rhinitis, the positive associations found between pollutant levels and uncontrolled rhinitis weakened after accounting for temperature. The association between air pollutant levels (both ozone and PM2.5) and uncontrolled rhinitis during the grass pollen season remained statistically significant in 2017 but not in 2018 (Table 2). The associations found for 2018 between ozone and uncontrolled rhinitis during the "no pollen season", and between PM2.5 and uncontrolled rhinitis during the birch pollen season remained statistically significant. Strong correlations were found quite consistently throughout the year between pollutant levels and temperature. This may lead to unstable estimates when simultaneously included in the same model.

The results of the two-pollutant models (i.e. models that include ozone and PM2.5 simultaneously as exposures) were similar to the results of the single-pollutant models (Table 3).

When the main analyses were repeated by considering a one day-lag, the same tendency in the results was observed, with overall weaker associations (Table E3). When analyses were stratified by urban/rural areas, similar associations as the ones observed in the main analysis were found in urban areas (Table E4), while in rural areas, given the smaller sample size, most of the associations that were observed in the main analyses were no longer significant (Table E5).

When we repeated our analyses by considering global VAS as an outcome (instead of uncontrolled rhinitis), similar patterns of association were found (Table E6). When we repeated our analyses by considering ozone and PM2.5 in quartiles, the results were consistent with linear trends, with small departures that were not replicated in the different years (Table E7).

When we reran our 2017 analyses among the subset of countries for which pollen seasons were assessed in 2018, results remained similar (results not shown).

Discussion

The results of this study suggest a deleterious effect of air pollution (particularly ozone) on allergic rhinitis control, especially during the grass pollen season. Positive associations were found between air pollutants (ozone and PM2.5) and allergic rhinitis symptoms. Differences between pollen seasons were found, suggesting an interaction between air pollution and pollen exposure (i.e. that the deleterious effects of exposure to pollen are magnified by exposure to air pollutants). Results for 2017 and 2018 are consistent for ozone, suggesting that ozone increases rhinitis symptoms during the grass pollen season (and outside pollen seasons) but not during the birch pollen season. Results are less consistent for PM2.5: results for 2017 suggest that PM2.5 increases rhinitis symptoms during the grass pollen season but not during the birch pollen season, whereas results for 2018 suggest a similar deleterious effect of PM2.5 in both pollen seasons.

Strengths and weaknesses

The current study has many strengths including a large number of observations on daily symptoms and treatment use, with participants from multiple countries. Mobile technology is becoming an important tool for better understanding and managing AR and provides novel information that was not previously available.

Other strengths and limitations of MASK have already been reported [START_REF] Bedard | Mobile technology offers novel insights on control and treatment of allergic rhinitis. The MASK study[END_REF][START_REF] Bousquet | Treatment of allergic rhinitis using mobile technology with real-world data: The MASK observational pilot study[END_REF]. As for all studies using participatory data, potential biases include (i) the likelihood of sampling bias being present and (ii) the lack of generalizability of the study as most users have a moderate to severe disease. In particular, users of the MASK-air App might use it mostly when symptoms are present, thus producing missing data in days with mild or no symptoms. This bias is expected to attenuate (i.e. underestimate) the true associations [START_REF] Daniel | Using causal diagrams to guide analysis in missing data problems[END_REF].

As in previous studies using MASK-air data [START_REF] Bedard | Mobile technology offers novel insights on control and treatment of allergic rhinitis. The MASK study[END_REF][START_REF] Bousquet | Treatment of allergic rhinitis using mobile technology with real-world data: The MASK observational pilot study[END_REF], we were not able to conduct any longitudinal study as most users utilize the App intermittently, hence there is no clear pattern of treatment.

In the current study, we cannot ascertain that the users are allergic to a given allergen since this information is not available for all patients. The diagnosis of AR was not supported by a physician but was a response to the question: "Do you have allergic rhinitis? Yes/No". Some users with nonallergic rhinitis may therefore have responded "Yes" to the question but >95% of responders declared symptoms of AR by questionnaire. Precise patient characterization is impossible using an App, but every observational study using MASK has been able to identify days with poor control or criteria of severity [START_REF] Bedard | Mobile technology offers novel insights on control and treatment of allergic rhinitis. The MASK study[END_REF][START_REF] Bousquet | Treatment of allergic rhinitis using mobile technology with real-world data: The MASK observational pilot study[END_REF]. Potential unmeasured confounding arising from the limited amount of information available on users' profile (e.g. socioeconomic status) is another limitation of the present study and cannot be avoided due to privacy regulations.

We used MASK, a validated App for the study of allergic rhinitis [START_REF] Bedard | Mobile technology offers novel insights on control and treatment of allergic rhinitis. The MASK study[END_REF][START_REF] Bousquet | Treatment of allergic rhinitis using mobile technology with real-world data: The MASK observational pilot study[END_REF]. Thus, the data obtained from the patients are of high quality. We defined the severity of rhinitis according to MASK studies [START_REF] Bedard | Mobile technology offers novel insights on control and treatment of allergic rhinitis. The MASK study[END_REF][START_REF] Bousquet | Treatment of allergic rhinitis using mobile technology with real-world data: The MASK observational pilot study[END_REF] but other methods may have been used such as symptom-medication scores proposed for allergen immunotherapy [START_REF] Pfaar | Recommendations for the standardization of clinical outcomes used in allergen immunotherapy trials for allergic rhinoconjunctivitis: an EAACI Position Paper[END_REF]. Results remained similar when we considered VAS global -which has been used to assess AR severity in previous papers [START_REF] Bedard | Mobile technology offers novel insights on control and treatment of allergic rhinitis. The MASK study[END_REF][START_REF] Bousquet | Treatment of allergic rhinitis using mobile technology with real-world data: The MASK observational pilot study[END_REF] -as an outcome. Both VAS global and uncontrolled rhinitis have been found to be highly correlated with work productivity (assessed using a specific VAS score) in MASK-air App users (unpublished data), providing internal validity of our results.

We only checked the association with ozone and PM2.5 since these two pollutants appear to be the most relevant. Interestingly, results for both ozone and PM2.5 remained similar after mutual adjustment (i.e. when we conducted two-pollutant models). Other pollutants need to be tested in a further study. A multipollutant air quality index may be appropriate [START_REF] Olstrup | A Multi-Pollutant Air Quality Health Index (AQHI) Based on Short-Term Respiratory Effects in Stockholm, Sweden[END_REF].

The models used for assigning air pollution exposure had a spatial resolution of 12 by 12 km.

Although more spatially resolved models could have been used, our approach is reasonable given that we used a single geolocation point to assign exposure and subjects often move around wider areas around their residence or work address. Actually, without having data on time-activity patterns, using highly spatially-resolved models can introduce more error that using others with lower spatial resolution [START_REF] Sellier | Health effects of ambient air pollution: Do different methods for estimating exposure lead to different results?[END_REF]. Nevertheless, the nature of measurement error due to spatial averaging is of Berkson type [START_REF] Armstrong | Effect of measurement error on epidemiological studies of environmental and occupational exposures[END_REF]. This error is not expected to bias the exposure-health associations, although it makes it more difficult to detect associations [START_REF] Armstrong | Effect of measurement error on epidemiological studies of environmental and occupational exposures[END_REF].

Interpretation of the results

The results observed in this study show that ozone levels are associated with an increased AR severity during the grass pollen season. Positive associations were also found outside the pollen season. No association was found during the birch pollen season. Results are highly consistent across years and outcomes. Additional analyses were conducted without suggesting non-linear effects or lag effects (although the assumption that the geolocation on the previous day was the same as the geolocation on the day the app was used might not be true). Associations stratified by urban vs rural areas did not suggest any effect modification by urbanization (although rural areas represented less than 20% of the person days included in our study).

We observed stronger positive associations between air pollutantsparticularly ozoneand uncontrolled rhinitis in 2017 compared to 2018. A plausible explanation could be that pollen seasons were less clear in 2018 compared to 2017. This might have caused misclassification in the assessment of 2018 pollen seasons using GTs and thus could explain some of the inconsistencies found between 2017 and 2018, more particularly for the birch pollen season and the "no pollen season". Although in 2017 pollen seasons were assessed for a broader range of countries than in 2018, results for 2017 remained similar when we restricted our analyses to the subset of countries that were included in the 2018 analyses. This suggests that these inconsistencies are not caused by heterogeneity between geographical regions. Nevertheless, although GTs are not the most precise tool to assess pollen counts, they appear to be more related to symptoms during the pollen season than pollen counts [START_REF] Karatzas | New European Academy of Allergy and Clinical Immunology definition on pollen season mirrors symptom load for grass and birch pollen-induced allergic rhinitis[END_REF][START_REF] Karatzas | Google Trends reflect allergic rhinitis symptoms related to birch and grass pollen seasons[END_REF]. Further studies using pollen counts (that are not available for all our study population yet) are needed to complete our results.

These results confirm previous data suggesting that traffic-related pollutants can disrupt pollen, a modification that causes more allergic symptoms [START_REF] Senechal | A Review of the Effects of Major Atmospheric Pollutants on Pollen Grains, Pollen Content, and Allergenicity[END_REF][START_REF] Schiavoni | The dangerous liaison between pollens and pollution in respiratory allergy[END_REF]. However, this is the first study to provide daily assessment in a large number of patients using daily reports.

The respective role of temperature and ozone is still unclear. Temperature increases grass pollen counts [START_REF] Bruffaerts | Comparative long-term trend analysis of daily weather conditions with daily pollen concentrations in Brussels, Belgium[END_REF] and the effect of ozone may be related to increased temperature in one year. Although the associations we found weakened after accounting for maximal temperature, strong correlations were found between pollutant levels and temperature, quite consistently throughout the year. Given the high correlation, results from models including both terms may be unstable. Although we have found evidence suggesting a deleterious effect of air pollution (particularly ozone) on allergic rhinitis control, especially during the grass pollen season, we cannot rule out (at least partially) confounding by high temperature. Further studies are needed to disentangle the deleterious effects of air pollution and other meteorological factors (temperature, humidity, etc.) on allergic rhinitis control.

Conclusions

These results show the importance of air pollution and allergen concentrations, as well as their interaction, as predictors of intensity of rhinitis symptoms. They will form the base for the development of predictive models in the next stage of the POLLAR project. The results of MASKair have been used to propose a DG Santé awarded Good Practice for digitally-enabled, integrated, patient-centred care [START_REF] Bousquet | Guidance to 2018 good practice: ARIA digitally-enabled, integrated, person-centred care for rhinitis and asthma[END_REF] as we expect to embed aerobiology and air pollution data in care pathways for rhinitis and asthma. IQR: interquartile range, OR: odds ratio *per IQR increase in pollutants levels (ozone or PM2.5), adjusting for sex, age, country and treatment ** treating pollutants levels (ozone or PM2.5) as a continuous variable and the pollen season as a categorical variable ┼ In 2017, for most countries, two GT peaks were observed (i.e. clear "birch" and "grass" peaks) with an in between pollen seasons and no pollen season. In 2018, for most countries, three GT peaks were observed in 2018 (i.e. in addition to the "birch" and "grass" peaks, an intermediate peak was observed). Since there was some uncertainty on the intermediate birch-grass pollen season, pollen counts from Germany, France and Lithuania were assessed and showed that grass pollens were retrieved during this intermediate season. Therefore, for all countries, the grass pollen and the intermediate seasons were studied jointly.

1 .

 1 days with VAS global ≥ 50/100 or 2. days with VAS global ≥ 35 and use of INCS-containing medication or 3. days with VAS global ≥ 20 and use of at least 3 medications Days were defined as controlled rhinitis otherwise.

( 1 .

 1 04, 1.25), for 2017 and 2018 respectively) and no pollen season (ORs: 1.08 (1.01, 1.16) and 1.15

Figure 2A . 26 Figure 2B .

 2A262B Figure 2A. Assessment of pollen seasons by region/countries for 2017 using Google Trends

Figure 3A :Figure 3B :

 3A3B Figure 3A: Assessment of pollen seasons for 2017 using GTsexamples for France, Germany and the UK

Figure E1 .

 E1 Figure E1. Areas of the study

Table 2 .

 2 Associations between air pollutants (ozone and PM2.5) and uncontrolled rhinitis,

	Birch (n=4,371)	8.5 [5.1-14.0]	1.11 (1.07, 1.15)	<0.001
	Grass + intermediate seasons (n=4,960) ┼	7.5 [4.9-11.0]	1.11 (1.04, 1.19)	0.004	0.0001
	No pollen season (n=8,711)	6.3 [3.8-9.6]	1.00 (0.96, 1.04)	0.83

Table 3 .

 3 Associations between air pollutants (ozone and PM2.5) and uncontrolled rhinitis, considering ozone and PM2.5 simultaneously as exposures in regression models (i.e. two-Flow chart of the study population *including for days recorded in Italy and France only those above the latitude of Lyon

	pollutant models)
	Association between ozone, PM2.5 and uncontrolled rhinitis high
	Period/season	OR* (95%CI) for ozone P	OR** (95%CI) for PM2.5 P
	All 2017 (n=18,398)	1.17 (1.12, 1.23)	<0.001	1.00 (0.98, 1.02)	0.88
	Birch (n=2,133)	1.10 (0.95, 1.27)	0.19	1.03 (0.96, 1.10)	0.38
	Grass (n=4,397)	1.18 (1.03, 1.34)	0.02	1.14 (1.01, 1.28)	0.04
	Between-seasons (n=1,101)	1.06 (0.75, 1.48)	0.75	0.86 (0.65, 1.12)	0.26
	No pollen season (n=10,767)	1.07 (1.00, 1.15)	0.04	0.96 (0.93, 1.00)	0.03
	All 2018 (n=18,042)	1.13 (1.10, 1.17)	<0.001	1.08 (1.05, 1.10)	<0.001
	Birch (n=4,371)	0.93 (0.85, 1.03)	0.15	1.11 (1.07, 1.15)	<0.001
	Grass + intermediate seasons (n=4,960) 1.11 (1.01, 1.22)	0.03	1.09 (1.01, 1.17)	0.03
	No pollen season (n=8,711)	1.15 (1.10, 1.21)	<0.001	1.00 (0.96, 1.04)	0.87
	IQR: interquartile range, OR: odds ratio

*per IQR increase in ozone levels, adjusting for sex, age, country, treatment and PM2.5 levels **per IQR increase in PM2.5 levels, adjusting for sex, age, country, treatment and ozone levels Figure 1.

single VAS person days recorded in 2017 and 2018 51,094 single VAS person days recorded in Northern and Central Europe* in 2017 and 2018 with geolocation 36,440 single VAS person days (3,323 users) in Northern and Central Europe in 2017 and 2018 with available information on air pollution and pollen

	190,492 VAS recorded
	between February 2017
	and October 2018
	-4,287 duplicates or multiplicates (for days with > 1 VAS recorded)
	-66,076 days recorded without geolocation
	120,129 single VAS
	person days recorded in
	2017 and 2018 with
	geolocation -69,035 days recoded outside Northern or Central Europe
	-1,999 days without air pollution data (SILAM European database)
	-12,655 days without pollen data (Google Trends)
	18,398 days in 2017
	18,042 days in 2018

Table E1 .

 E1 Number of days included in the analyses by country and year 39 *NA: information not available (eg. located on borders etc.)

	Country		Number of days studied
		2017	2018	Total days	Total users
	Austria	847	1,435	2,319
	Belgium	270	311	581
	Czech Republic	136	0	136	11
	Denmark	279	170	449	51
	Finland	1,022	0	1,022
	France	684	2,635	3,319
	Germany	3,650	3,836	7,486
	Northern Italy	2,104	0	2,104
	Lithuania	4,004	0	4,004
	NA*	252	0	252	73
	Netherlands	851	3,998	4,849
	Poland	1,581	3,462	5,043
	Sweden	279	0	279	38
	Switzerland	598	1,687	2,285
	United Kingdom	1,804	508	2,312
	Total	18,398	18,042	36,440	3,323
	40
	41

Table E2 .

 E2 Usage of the app (number of days used and VAS global) by users' characteristics 43

		Number of days per user ± (n=3,323)
		N users (%)	Median [p25-p75]	P*
	Sex
	Male	1,514 (45.6)	2 [1-8]
	Female	1,806 (54.4)	2 [1-7]	0.32
	Country of residence
	Austria	174 (5.2)	1 [1-5]
	Belgium	74 (2.2)	1 [1-7]
	Czech Republic	7 (0.2)	14 [4-35]
	Denmark	53 (1.6)	3 [1-10]
	Finland	139 (4.2)	2 [1-4]
	France	354 (10.7)	2 [1-4]	0.0001
	Germany	428 (12.9)	3 [1-15]
	Northern Italy	129 (3.9)	3 [1-14]
	Lithuania	212 (6.4)	6.5 [2-25]
	Netherlands	535 (16.1)	2 [1-6]
	Poland	405 (12.2)	2 [1-8]
	Sweden	44 (1.3)	2 [1-7]
	Switzerland	487 (14.7)	1 [1-2]
	United Kingdom	230 (6.9)	3 [1-9]
	Year of birth
	≤1974	1,038 (31.3)	2 [1-9]	0.06
	1975-1988	1,120 (33.7)	2 [1-6.5]
	≥1989	1,162 (35.0)	2 [1-7]

VAS global (n=36,440) N person-days (%) median [p25-p75

			]	P*
	Sex
	Male	18,359 (50.4)	10 [0-28]
	Female	18,078 (49.6)	16 [3-44]	0.0001
	Country of residence
	Austria
	Belgium
	Czech Republic
	Denmark
	Finland
	France
	Germany
	Northern Italy
	Lithuania
	Netherlands
	Poland

grants and personal fees from Philips, grants from Fisher and Paykel, grants and personal fees from Agiradom, grants and personal fees from Astra Zeneca, grants from Mutualia, grants from Vitalaire, grants from Air Liquide Foundation, grants and personal fees from SEFAM, personal fees from JAZZ, personal fees from ITAMAR, outside the submitted work. D Laune has nothing to disclose. submitted work, and KYomed-Innov shares. Dr. Basagaña has nothing to disclose. Source of funding EIT Health (Grant ref: 18354 -Impact of POLLution on Asthma and Rhinitis), Fondation FMC-VIA-LR

Abbreviations


Silviu-Iulian Niculescu

Fazia Bedouhene

Timothée Schmoderer

Islam Boussaada
email: islam.boussaada@centralesupelec.fr

Insights on equidistributed real spectral values in second-order delay systems: perspectives in partial pole placement

Keywords: Time-delay differential equation, Stability and stabilization, Spectral method, Exponential decay, Partial pole placement, Coexisting-real-root-induced-dominancy tuning parameters

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

Introduction

The purpose of this paper is to study the asymptotic stability of the solutions of a dynamical system represented by a second-order linear differential equation of the form: y ′′ (t) + a 1 y ′ (t) + a 0 y(t) = u(t-τ 0 ), τ 0 ≥ 0, where τ 0 is a minimum delay due to physical constraints and u(t) is a forcing delayed proportionalderivative controller given by u(t) = -α 1 y ′ (t -τ 1 ) -α 0 y(t -τ 1 ), τ 1 ≥ 0.

The closed-loop system is represented by a second-order autonomous delay-differential equation (DDE) including five parameters: the parameters of the open-loop system a 1 , a 0 ∈ R, the controller gains α 1 , α 0 ∈ R and the sum of input and output channel delays τ = τ 0 + τ 1 ∈ R * + , see for instance [START_REF] Hale | Introduction to functional differential equations[END_REF]- [START_REF] Breda | Controlling delayed dynamics-advances in theory, methods and applications[END_REF] for further insights on time-delay systems and their qualitative properties.

If the interest in considering a delayed control law as above lies in its generality and simplicity (tuning only three parameters, including the delay) in real-time implementation [START_REF] Suh | Proportional minus delay controller[END_REF]- [START_REF] Boussaada | Inverted pendulum stabilization: Characterization of codimension-three triple zero bifurcation via multiple delayed proportional gains[END_REF], its main inconvenient concerns the fact that the closed-loop system becomes infinite-dimensional leading, in some cases, to unexpected dynamical behaviors with respect to changes in the system's parameters. Such a controller belongs to the class of low complexity controllers. For a more general discussion on such controllers as well as some of their applications, we refer to [START_REF] Seuret | Low-complexity controllers for timedelay systems[END_REF].

In this frame, it should be mentioned that the idea to use the delay as a controller parameter is not new and there exists several examples in the literature exhibiting its relevance. For instance, at the end of 1970, [START_REF] Suh | Proportional minus delay controller[END_REF] introduced the so-called "proportional-minus-delay" (PMD) controller and highlighted a few interesting properties. More precisely, similar to the PD controllers, it operates quick responses to input changes, but, surprisingly, it is less sensitive to high-frequency noise, and thus it can offer an alternative to the "classical" PD controllers by conserving its low-complexity character * . Such controllers have been implemented and have demonstrated their interest in various case studies [START_REF] Suh | Use of time-delay actions in the controller design[END_REF] including, among others, the well-known inverted pendulum [START_REF] Sieber | Extending the permissible control loop latency for the controlled inverted pendulum[END_REF], [START_REF] Boussaada | Inverted pendulum stabilization: Characterization of codimension-three triple zero bifurcation via multiple delayed proportional gains[END_REF] and, more recently, human balancing [START_REF] Balogh | Conditions for stabilizability of time-delay systems with real-rooted plant[END_REF]. Finally, as [START_REF] Michiels | Stability, control, and computation for time-delay systems: an eigenvalue-based approach[END_REF] points out in a different framework, the use of a delay in the controller led to a broader class of stabilizing second-order systems. In other words, there exists stabilizing delayed PD controller such that the delay-free PD controller does not stabilize the corresponding dynamical system. This idea represents additional motivation for our paper.

In this framework, the method that we propose consists in tuning the control parameters (the "gains" α 1 , α 0 and the "delay" τ ) in such a way that the solutions of the closed-loop system are asymptotically stable with a guaranteed exponential decay rate.

To achieve this general purpose, several methodologies exist with various objectives. On the one hand, a time-domain approach based on Lyapunov functions has proven its efficiency in estimating solutions decay; see e.g. [START_REF] Malisoff | Constructions of strict Lyapunov functions (Communications and Control Engineering Series[END_REF] and the references therein. On the other hand, in frequency-domain, recent studies [START_REF] Bedouhene | Real spectral values coexistence and their effect on the stability of time-delay systems: Vandermonde matrices and exponential decay[END_REF]- [START_REF] Mazanti | Multiplicity-induced-dominancy for delaydifferential equations of retarded type[END_REF] have illustrated that the so-called Multiplicity-Induced-Dominancy (MID) property can effectively be used to prescribe the decay rate of a DDE solution, [START_REF] Pinney | Ordinary difference-differential equations[END_REF]. In this approach, one assigns to the characteristic function of the system a single real root s 0 with maximal multiplicity. Then, one shows (under necessary and sufficient conditions) that any other root should have a real part less than s 0 , which proves the exponential stability for the solutions of the system. In other words, s 0 represents the spectral abscissa of the characteristic function.

The originality of the proposed approach is to assign a maximal number of simple real zeros of the characteristic function rather than a multiple zero. Then, we give necessary and sufficient conditions ensuring that the largest assigned root is negative and dominant, so that it is nothing but the corresponding spectral abscissa. We call this property: Coexisting-Real-Root-Induced-Dominancy (CRRID). The systematic analysis of the CRRID began in [START_REF] Bedouhene | Real spectral values coexistence and their effect on the stability of time-delay systems: Vandermonde matrices and exponential decay[END_REF], [START_REF] Amrane | On qualitative properties of low-degree quasipolynomials: Further remarks on the spectral abscissa and rightmostroots assignment[END_REF], where the asymptotic stability of the solutions of an n-order linear differential equation commanded by a delayed proportional-position-controller. Notice that the MID property mentioned above [START_REF] Boussaada | The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: When delay-systems characteristics meet the zeros of kummer functions[END_REF], [START_REF] Boussaada | Multiplicityinduced-dominancy in parametric second-order delay differential equations: Analysis and application in control design[END_REF] can be seen as the limit of the CRRID property when all the simple real roots tend to the same (real) value. Moreover, it is commonly accepted that multiple roots are sensitive with respect to small parameter changes; see for instance [START_REF] Michiels | An explicit formula for the splitting of multiple eigenvalues for nonlinear eigenvalue problems and connections with the linearization for the delay eigenvalue problem[END_REF] and references therein. Therefore, the CRRID property exhibits an interesting parametric robustness compared to the MID. As a matter of fact, the effectiveness of the CRRID property has already been emphasized through the problem of control of vibrations, see for instance [START_REF] Falcón-Prado | Active vibration control of axisymmetric membrane through partial pole placement[END_REF]. For other methods/approaches and ideas, we refer to [START_REF] Ramıérez | Design of proportional-integralretarded (PIR) controllers for second-order LTI systems[END_REF] and [START_REF] Da Silva | Controlling industrial deadtime systems: When to use a pid or an advanced controller[END_REF] (see also, [START_REF] O'dwyer | Handbook of PI and PID Controller Tuning Rules, 3rd[END_REF] for an overview on such topics).

The contribution of this work is twofold. First, we show that the CRRID property can successfully be applied to stabilize second-order linear systems represented by differential equations with a delayed proportional-minus-derivative controller. More precisely, we provide a necessary and sufficient condition that guarantees that the assigned real roots are negative and that the largest one is the dominant root of the characteristic function that is, the spectral abscissa of the dynamical system in closed-loop. Hence, we guarantee the exponential decay rate of the solutions of the system. To the best of the authors' knowledge, such a result represents a novelty in the literature. The proof makes use of the Stépán-Hassard formula [START_REF] Stépán | Retarded dynamical systems: stability and characteristic functions[END_REF], [START_REF] Hassard | Counting roots of the characteristic equation for linear delay-differential systems[END_REF] applied to an appropriate transformation of the characteristic function. Second, we apply our methodology to two case studies, showing that the proposed methodology can be applied to stabilize unstable (second-order) plants and to improve the solution decay rate. Furthermore, in the first case, the controller gains do not guarantee the closed-loop stability of the system in the delay-free case reinforcing thus the argument that the delay has a stabilizing effect (see, e.g., Remark 4.1). In the second case, the decay rate is guaranteed for "small" (controller) gains, and thus less sensitive to high-frequency noise (see, for instance, Remark 4.2).

The remaining of this paper is organized as follows: some preliminary results (functional Vandermonde-type matrices, Stépán-Hassard formula) are presented in Section 2. Next, the main results are derived and commented in Section 3. Two illustrative examples are proposed in Section 4 and some concluding remarks end the paper. The notations are standard.

Preliminaries

In this paper, we consider a dynamical system represented by a general second-order linear differential equation with a proportional-derivative-delay term of the form

y ′′ (t) + a 1 y ′ (t) + a 0 y(t) + α 1 y ′ (t -τ ) + α 0 y(t -τ ) = 0, (1)
under appropriate initial conditions belonging to the Banach space of continuous functions C([-τ, 0], R). In the Laplace domain, the corresponding characteristic function is a quasipolynomial defined by ∆ :

C × R + * → C: ∆(s, τ ) = s 2 + a 1 s + a 0 + e -τ s (α 1 s + α 0 ) , (2)
where (a 1 , a 0 , α 1 , α 0 ) ∈ R 4 , see for instance [START_REF] Michiels | Stability, control, and computation for time-delay systems: an eigenvalue-based approach[END_REF]. The degree of ∆, defined by the sum of the degrees of the polynomials plus the number of delays (one in our case), is four. The following result asserts that the maximal number of real roots counted with multiplicity is four, see [START_REF] Pólya | Problems and Theorems in Analysis: Series • Integral Calculus • Theory of Functions[END_REF].

Proposition 1 (Pólya-Szegö bound). Let ∆ be the quasipolynomial given by (2), and α, β ∈ R be such that α ≤ β. Denote by M α,β the number of roots of ∆ counted with multiplicities contained in the set {s ∈ C, α ≤ ℑ(s) ≤ β}. Then,

τ (β -α) 2π -4 ≤ M α,β ≤ τ (β -α) 2π + 4.
As a consequence of the previous theorem, setting α = β = 0, we conclude that the degree 4 of ∆ is a sharp bound for the number of real roots of the quasipolynomial ∆. Our analysis of the assignment of real roots to the quasipolynomial (2) will rely on the invertibility of a certain matrix, which because of its shape and properties, is called in the sequel functional Vandermonde-type matrix, similar to the ones studied in [START_REF] Boussaada | Characterizing the codimension of zero singularities for time-delay systems: A link with Vandermonde and Birkhoff incidence matrices[END_REF], [START_REF] Bedouhene | On pole placement and spectral abscissa characterization for time-delay systems[END_REF]; see Lemma 3 below. This result will be proved with the help of the multivariate functions G n : R n+1 × R + * → R defined recursively by G 0 (s k , τ ) = e -τ s k and

(s 1 -s n+1 )G n (s 1 , . . . , s n+1 , τ ) = G n-1 (s 2 , . . . , s n+1 , τ ) -G n-1 (s 1 , . . . , s n , τ ),
for n ≥ 1. The multivariate functions G n (•, τ ) are closely related to the functions F n (•, τ ) introduced and analyzed in [START_REF] Bedouhene | Real spectral values coexistence and their effect on the stability of time-delay systems: Vandermonde matrices and exponential decay[END_REF] and, thus, benefit from similar properties that we present without proof in Lemma 2 below.

Lemma 2 (Properties of the G n functions). For any real numbers s 1 , . . . , s n+1 and τ > 0, we have

1. G n (s 1 , . . . , s n+1 , τ ) ̸ = 0. 2. lim τ →0 1 (-τ ) n G n (s 1 , . . . , s n+1 , τ ) = 1 n! .
Lemma 3 (Invertibility of a structured functional Vandermonde-type matrix ). Let s = (s 1 , s 2 , s 3 , s 4 ), then the matrix

V (s, τ ) =     s 1 1 s 1 e -τ s 1 e -τ s 1 s 2 1 s 2 e -τ s 2 e -τ s 2 s 3 1 s 3 e -τ s 3 e -τ s 3 s 4 1 s 4 e -τ s 4 e -τ s 4     (3)
is invertible for any τ > 0 and any distinct real numbers s 1 , . . . , s 4 .

Proof. We give an explicit factorized form of the determinant υ(s, τ ) of V (s, τ ) and then we show that υ(s, τ ) ̸ = 0 for τ > 0 and s i ̸ = s j . Observe that for 1 ≤ k ≤ 4, we have e -τ s k = G 0 (s k , τ ) and thus s k e -τ s k = -G ′ 0 (s k , τ ), where the prime denotes the derivative with respect to τ . We start with the structured functional Vandermonde-type matrix V defined by (3) and we denote by L i , for 1 ≤ i ≤ 4, its i-th line. By replacing L i by L i -L i+1 , for 1 ≤ i < 4 and using the properties of the G n functions we get

υ = (s 1 -s 2 )(s 2 -s 3 )(s 3 -s 4 ) 1 0 G ′ 1 (s 1 , s 2 , τ ) G 1 (s 1 , s 2 , τ ) 1 0 G ′ 1 (s 2 , s 3 , τ ) G 1 (s 2 , s 3 , τ ) 1 0 G ′ 1 (s 3 , s 4 , τ ) G 1 (s 3 , s 4 , τ ) s 4 1 -G ′ 0 (s 4 , τ ) G 0 (s 4 , τ )
.

We replace L i by L i -L i+1 one more time, for 1 ≤ i ≤ 2, and factor out (s 1 -s 3 )(s 2 -s 4 ) to obtain

υ = (s 1 -s 2 )(s 2 -s 3 )(s 3 -s 4 )(s 1 -s 3 )(s 2 -s 4 ) 0 0 G ′ 2 (s 1 , s 2 , s 3 , τ ) G 2 (s 1 , s 2 , s 3 , τ ) 0 0 G ′ 2 (s 2 , s 3 , s 4 , τ ) G 2 (s 2 , s 3 , s 4 , τ ) 1 0 G ′ 1 (s 3 , s 4 , τ ) G 1 (s 3 , s 4 , τ ) s 4 1 -G ′ 0 (s 4 , τ ) G 0 (s 4 , τ )
.

To show that υ ̸ = 0, it suffices to prove that the latter matrix is invertible. Note that we can express the determinant

G ′ 2 (s 1 , s 2 , s 3 , τ ) G 2 (s 1 , s 2 , s 3 , τ ) G ′ 2 (s 2 , s 3 , s 4 , τ ) G 2 (s 2 , s 3 , s 4 , τ )
via the following quotient derivative

-G 2 (s 1 , s 2 , s 3 , τ ) 2 G 2 (s 2 , s 3 , s 4 , τ ) G 2 (s 1 , s 2 , s 3 , τ ) ′ .
The above expression holds for any τ > 0, thus its vanishing implies that there exists a real constant C (depending on s) such that

G 2 (s 2 , s 3 , s 4 , τ ) = CG 2 (s 1 , s 2 , s 3 , τ ).
By continuity, taking τ → 0 yields that C = 1, a contradiction, since

G 2 (s 2 , s 3 , s 4 , τ ) -G 2 (s 1 , s 2 , s 3 , τ ) = (s 4 -s 1 )G 3 (s, τ ) ̸ = 0.
Therefore, υ ̸ = 0 for any τ > 0 and any distinct numbers s k , k = 1, . . . , 4.

To prove the dominancy of the rightmost real spectral value, we will use the Stépán-Hassard formula [START_REF] Stépán | Retarded dynamical systems: stability and characteristic functions[END_REF], [START_REF] Hassard | Counting roots of the characteristic equation for linear delay-differential systems[END_REF] that we recall in the next theorem, in a particular context.

Theorem 4 (Stépán-Hassard formula). Let Q(x, q) = x n + n-1 k=0 b k x k + e -qx m k=0 β k
x k be a quasipolynomial function and 0 < ρ 1 ≤ . . . ≤ ρ r be the positive zeros (counted with multiplicities) of the real function R q (ω) = ℜ(i -n Q(iω, q)). For each 1 ≤ j ≤ r such that Q(iρ j , q) = 0, assume that the multiplicity of iρ j as a zero of Q is the same as the multiplicity of ρ j as a zero of R q (ω). Then the number of roots of the quasipolynomial Q(x, q) which lie in the half plane {ℜ(x) > 0}, counted with multiplicity, is given by the formula

Z = n -K 2 + 1 2 (-1) r sgn S (κ) (0) + r j=1 (-1) j-1 sgn S(ρ j ), ( 4
)
where K is the number of purely imaginary roots of ∆, counted by multiplicity, κ is the multiplicity of 0 as a root of Q, and S q (ω) = ℑ(i -n Q(iω, q)).

Main results

In this section, we analyze the properties of quasipolynomial functions saturating their number of real zeros, and we derive some results for the asymptotic stability of the solutions of the DDE (1). More precisely, first, in Proposition 5 we give necessary and sufficient conditions on the coefficients of the quasipolynomial (2) for the number of distinct real zeros to be maximal.

For the next results, we assume that the real roots are uniformly distributed. Second, in Proposition 7 we give a necessary and sufficient condition that guarantees the negativity of the real zeros of ∆. Third, in Theorem 8 we show that the coexistence of the maximum number of real spectral values implies the CRRID property for the quasipolynomial ∆, i.e. the rightmost real zero is the spectral abscissa. Finally, the combination of these results allows us to prove exponential stability with a guaranteed decay rate for the solutions of (1).

Assigning real roots of the characteristic function

In this subsection, we show that assigning the maximal number of distinct real roots to the quasipolynomial ∆ uniquely determines the coefficients a k and α k .

Proposition 5 (Coexisting real roots). For a given delay τ > 0, the quasipolynomial (2) admits 4 distinct real spectral values s 1 , . . . , s 4 if, and only if, the real coefficients a 1 , a 0 and α 1 , α 0 are respectively given by the following functions in τ and s = (s 1 , . . . , s 4 ):

a 1 (s, τ ) = -1 υ(s, τ ) det     s 2 1 1 s 1 e -τ s 1 e -τ s 1 s 2 2 1 s 2 e -τ s 2 e -τ s 2 s 2 3 1 s 3 e -τ s 3 e -τ s 3 s 2 4 1 s 4 e -τ s 4 e -τ s 4     , a 0 (s, τ ) = 1 υ(s, τ ) det     s 2 1 s 1 s 1 e -τ s 1 e -τ s 1 s 2 2 s 2 s 2 e -τ s 2 e -τ s 2 s 2 3 s 3 s 3 e -τ s 3 e -τ s 3 s 2 4 s 4 s 4 e -τ s 4 e -τ s 4     ,
and

α 1 (s, τ ) = -1 υ(s, τ ) det     s 2 1 s 1 1 e -τ s 1 s 2 2 s 2 1 e -τ s 2 s 2 3 s 3 1 e -τ s 3 s 2 4 s 4 1 e -τ s 4     , α 0 (s, τ ) = 1 υ(s, τ ) det     s 2 1 s 1 1 s 1 e -τ s 1 s 2 2 s 2 1 s 2 e -τ s 2 s 2 3 s 3 1 s 3 e -τ s 3 s 2 4 s 4 1 s 4 e -τ s 4     ,
where υ(s, τ ) is the determinant of the structured functional Vandermonde-type matrix V (s, τ ), introduced in Lemma 3.

Proof. Assume that (2) admits four distinct real roots s 1 , . . . , s 4 , which entails that the coefficients a 1 , a 0 and α 1 , α 0 satisfy the linear system

a 1 s k + a 0 + α 1 e -τ s k s k + α 0 e -τ s k = -s 2 k , for 1 ≤ k ≤ 4.
Thanks to the invertibility of matrix V (s, τ ), as ensured by Lemma 3, one deals with a Cramer system with respect to the coefficients a 1 , a 0 and α 1 , α 0 . So that, we easily compute these coefficients with the standard formulas.

The following corollary gives the closed-form expression of the coefficients a 1 , a 0 and α 1 , α 0 , in the case where the real zeros are equidisributed. Corollary 6. The quasipolynomial ∆ admits the equidistanced real roots s k = s 1 -(k -1)d, for d > 0 and 1 ≤ k ≤ 4, if and only if its coefficients are given by

a 0 (s 1 , d, τ ) = 6d 2 (e τ d -1) 2 + s 2 1 -ds 1 e τ d -5 e τ d -1 , (5)
a 1 (s 1 , d, τ ) = d e τ d -5 e τ d -1 -2s 1 , (6)
α 0 (s 1 , d, τ ) = -2de τ s 1 (e τ d -1) 2 3d -s 1 (1 -e -τ d ) , (7)
α 1 (s 1 , d, τ ) = -2de τ s 1 e -τ d e τ d -1 . ( 8
)
Note that the expressions of the coefficients a 1 , a 0 and α 1 , α 0 in Corollary 6 characterize the existence of four equidistant real roots s k = s 1 -(k -1)d, with d > 0 and k = 1, • • • , 4, for the quasipolynomial ∆. In practical applications, notice that a 1 and a 0 are fixed by the model. Hence, one has the solve (6) to determine σ := τ d. We inject the expression of σ into [START_REF] Sipahi | Stability and stabilization of systems with time delay: Limitations and opportunities[END_REF] and [START_REF] Diekmann | Delay equations. Functional, complex, and nonlinear analysis[END_REF] to express α 0 and α 1 as a function of s 1 , d, τ and a 1 , and in equation [START_REF] Els | Introduction to the theory and application of the theory of differential equations with deviating argument[END_REF] to determine the relation verified by d (as a polynomial of degree 2). The suitable solution of d is quickly determined via the positivity constraint of τ and d.

Remark.

Observe that the open-loop system P (s) = s 2 + a 1 s + a 0 needs not a priori to be stable when applying the retarded PD controller. Indeed, observe that a 1 (s 1 , d, τ ) is negative for τ < 1 d ln 5d-2s 1 d-2s 1 , which implies that P (s) has at least one root with a positive real part.

Assuring negativity of the real roots

The following proposition gives a necessary and sufficient condition for the coefficients of the quasipolynomial (2) to ensure that the rightmost real root is negative.

Proposition 7. For a fixed delay τ > 0, assume that the quasipolynomial ∆ admits four equidistanced real roots

s k = s 1 -(k -1)d, for d > 0 and 1 ≤ k ≤ 4.
Then, s 1 is negative if, and only if, there exits τ * > 0 such that :

a 1 (s 1 , d, τ * ) + s 1 -d = 0. ( 9
)
Furthermore, τ * necessarily satisfies:

τ * = 1 d ln 1 -4 d s 1 > 0.
Proof. By a direct calculation using the explicit form of a 1 given in Corollary 6, we obtain

a 1 (s 1 , d, τ ) + s 1 -d = -s 1 -4d 1 e τ d -1 . Then, a 1 (s 1 , d, τ * ) + s 1 -d = 0 if, and only if, τ * = 1 d ln 1 -4 d s 1 .
Finally, one immediately observes that τ * > 0 if, and only if, s 1 < 0.

Remark. Note that Proposition 7 is of importance in stabilization of unstable plants. Indeed,

• The existence of the mentioned τ * represents a guarantee for the negativity of the closedloop assigned real roots.

• Equation ( 9) is of importance for control design purposes. In particular, if the coefficient a 1 is fixed by the model and one wants to assign equidistanced real roots to stabilize the system, then the distance d > 0 needs to be chosen such that d> a 1 + 2s 1 .

The CRRID property

In this subsection, we prove the CRRID property for the quasipolynomial ∆. Namely, we have the following theorem.

Theorem 8. If the quasipolynomial ∆(•, τ ) admits the maximal number of four equidistanced real roots

s k = s 1 -(k -1)d, for 1 ≤ k ≤ 4, then ∀ s 0 ∈ C\{s 1 }, ∆(s 0 ) = 0 =⇒ ℜ(s 0 ) < s 1 .
Theorem 8 asserts that s 1 is a strictly dominant root of ∆, i.e. it is the spectral abscissa of the quasipolynomial. Moreover, by carefully studying the proof, we have the stronger result that the pair (s 1 , s 2 = s 1 -d) is a system of dominant roots for ∆, i.e.

∀ s 0 ∈ C\{s 1 , s 2 }, ∆(s 0 ) = 0 =⇒ ℜ(s 0 ) ≤ s 2 .
Our strategy to prove the dominancy of s 1 for ∆(•, τ ) relies on the application of the Stépán-Hassard formula [START_REF] Hassard | Counting roots of the characteristic equation for linear delay-differential systems[END_REF] on a transformed version of ∆. Let us introduce the following transformation of the complex plane:

ϕ : C -→ C z -→ dz + s 1 -d,
which translates the root s 1 -d to the origin and scales the distance between two consecutive real root to 1. We introduce the transformed quasipolynomial

Q(z, q) = 1 d 2 ∆ (ϕ(z), τ ) = z 2 + b 1 z + b 0 + e -qz (β 1 z + β 0 ) ,
where the new delay parameter is q = τ d. The following lemma assert that the normalized quasipolynomial Q(•, q) admits four distinct real roots, gives the explicit form of the coefficients b 1 , b 0 , β 1 , and β 0 , and asserts that the problem of proving that s 1 is dominant for ∆ is the same as proving that z = 1 is dominant for Q.

Lemma 9 (Properties of the normalized quasipolynomial). The normalized quasipolynomial Q(z, q) admits the real roots {1, 0, -1, -2}. The coefficients {b 1 , b 0 , β 1 , β 0 } of Q are given by b 0 (q) = 2 2e q + 1 (e q -1) 2 , b 1 (q) = -e q + 3 e q -1 ,

β 0 (q) = -2 2e q + 1 (e q -1) 2 , β 1 (q) = - 2 e q -1 .
The root s 1 of ∆ is dominant if, and only if,

z = 1 is a dominant root of Q(•, q).
Proof. The map ϕ(z) is an isomorphism of the complex plane, that maps {1, 0, -1, -2} to the real roots {s 1 , s 2 , s 3 , s 4 } of ∆ to. Thus, the coefficients {b 1 , b 0 , β 1 , β 0 } of Q satisfy the set of linear equations Q(z 0 ) = 0 for z 0 ∈ {1, 0, -1, -2}. Therefore, using Corollary 6 with s 1 = 1 and d = 1, we deduce that the coefficients {b 1 , b 0 , β 1 , β 0 } admit the closed forms given in the statement of the Lemma. Moreover, the linear transform ϕ is the composition of a translation and a positive scaling (in particular, it does not involve a rotation). Thus, any inequality between the real parts of the roots of ∆ is preserved. Hence, the dominancy of 1 for Q corresponds to the dominancy of s 1 for ∆.

To prove the dominancy of 1 for Q, we apply the Stépán-Hassard formula (see Theorem 4), which relies on the properties of the following parametrized real functions R q (ω) = ℜ (-Q(iω, q)) and S q (ω) = ℑ (-Q(iω, q)) , given by R q (ω) = ω 2 -b 0 (q) -cos(qω)β 0 (q) -sin(qω)β 1 (q)ω, (10) S q (ω) = sin(qω)β 0 (q) -cos(qω)ωβ 1 (q) -b 1 (q)ω.

In the remaining part of this section, we compute the different numbers involved in the Stépán-Hassard formula (4).

Number of imaginary roots of Q. Let ω ≥ 0 be such that Q(iω) = 0, then ω satisfies R q (ω) = 0 and S q (ω) = 0.

Using simple algebraic manipulations, we obtain that the expressions of cos(qω) and sin(qω) are given by cos

(qω) = - ω 2 (b 1 β 1 -β 0 ) + b 0 β 0 ω 2 β 2 1 + β 2 0 , sin(qω) = ω(ω 2 β 1 -β 0 β 1 + b 1 β 0 ) ω 2 β 2 1 + β 2 0 .
The relation cos(qω) 2 + sin(qω) 2 = 1, implies that ω is necessarily given by

ω 1 = 0 or ω 2 = ± 2b 0 -b 2 1 + β 2 1 = ±i.
Therefore, the only imaginary root of Q(•, q) is ω = 0, which, by definition of Q, is of multiplicity 1. Moreover, we have S ′ q (0) = 1 (e q -1) 2 e 2q -4(q -1)e q -2q -5 , which can be seen positive for any q > 0 with the following argument. Denote by f (q) the numerator of the right-hand-side of the above equality. Then, f ′′ (q) = 4e q (e q -(q + 1)) is positive for q > 0. Thus, f ′ (q) is increasing and, since f ′ (0) = 0, is positive. Finally, we deduce that f (q) is increasing and, since f (0) = 0, we conclude that f (q) is positive for q > 0. Hence, in the Stépán-Hassard formula we have K = 1, κ = 1, and sgn S (κ)

q (0) = 1.
Locating the real positive roots of R q . If ω > 0 is a root of (10), then (using the expressions of the coefficients given in Lemma 9) we get ω 2 + 2 2e q + 1 (e q -1) 2 (cos(qω) -1) + 2 e q -1 ω sin(qω) = 0.

Multiplying the last equation by q 2 , reduces the problem to the search of the positive zeros of the parameterized function F q : R + * → R defined by F q (ρ) = ρ 2 + A(q) (cos(ρ) -1) + B(q)ρ sin(ρ) = 0, where ρ = qω > 0, A(q) = 2q 2 2e q +1 (e q -1) 2 , and B(q) = 2q e q -1 . The following technical lemma gives the main properties of the functions A and B that we will use the next results.

Lemma 10 (Properties of the A and B functions). For all q > 0, we have B(q) < 2 and 6 > A(q) > 3B(q) > B(q) > 0.

The function ρq = B(q) + B(q) 2 + 8A(q) /2 is strictly decreasing and lim q→0 + ρq = 1+ √ 13.

Proof. Clearly we have A(q) > 0 and B(q) > 0. Moreover, B ′ (q) = -2 (e q -1) 2 (e q (q -1) + 1) < 0, thus B is decreasing and B(q) < lim q→0 + B(q) = 2. Second, we also obtain A ′ (q) < 0, which yields A(q) < lim q→0 + A(q) = 6. Third, we have A -3B = 2q (e q -1) 2 (2qe q -3e q + q + 3), the second factor is seen to be positive by differentiating it twice, implying that A -3B > 0. Consider the function ρq defined in the Lemma and whose expression is ρq = q e q -1 1 + √ 5 + 8e q . To show that ρq is decreasing, we show that it is the sum of two positive decreasing functions. Namely, we set ρq = ρ1 q + ρ2 q , with ρ1 q = q e q -1 and ρ2 q = q e q -1 √ 5 + 8e q . First, we obviously have that ρ1 q is decreasing. Second, we have d dq (ρ 2 q ) = -e q 4e q +9 (e q -1) 2 √ 8e q +5 < 0, hence ρ2 q is also decreasing. Thus, ρq < lim q→0 + ρq = 1 + √ 13

The results of Lemma 10 are illustrated by Figure 1.

Lemma 11. The parametrized transcendental function F q (ρ) admits at most one real positive zero ρ * q , and when it does, it is located within the open interval I = 0, 3π 2 .

2 6 1 + √ 13 q A(q) B(q) 3B(q) ρq Figure 1: Qualitative behavior of the functions A(q) (red) and B(q) (blue) involved in the definition of F q . We also illustrate the function ρq (dashed).

Proof. First, since A(q) > 0 and B(q) > 0, observe that for ρ > 0 we have

F q (ρ) ≥ ρ 2 -2A(q) -B(q)ρ.
The right hand side of the above inequality is a polynomial of degree 2 in the variable ρ, whose largest root is ρq defined in Lemma 10. Since lim q→0 ρq = 1 + √ 13 < 3π 2 , we deduce that, necessarily, all roots ρ * q (if any) of F q (ρ) must lie within I = 0, 3 2 π . Now, by studying the first derivatives of F q with respect to ρ we show that F q admits at most one positive root. The third derivative of F q with respect to ρ is given by: F (3) q (ρ) = -B(q)ρ cos(ρ) + (A(q) -3B(q)) sin(ρ).

Depending on whether the function C(q) = B(q)

A(q)-3B(q) is greater than 1, the equation tan(ρ) = C(t)ρ has either one or two solutions within the interval I. We separate the two cases.

If F

(3) q (ρ) has only one zero ρ 1 ∈ I. Using that F

(3) q (0) = 0 and

F (3) q 3π 2
= 3B(q)-A(q) < 0, we deduce that F ′′ q (ρ) is increasing on ]0, ρ 1 [ and decreasing on ρ 1 , 3π 2 . Observe that

F ′′ q 3π 2 = 2 + 3π 2 B(q) > 0 and denote f 2 (q) = F ′′ q (0) = 2 -A(q) + 2B(q).
(a) If f 2 (q) ≥ 0, then F ′′ q (ρ) is positive on I. Thus, F ′ q (ρ) is increasing and since F ′ q (0) = 0, we conclude that F ′ q is positive. Finally, F q (ρ) is increasing on I and, since F q (0) = 0, we conclude that F q does not admit a zero on I.

(b) If f 2 (q) < 0, then F ′′ q (ρ) admits a zero ρ 2 ∈]0, ρ 1 [. Thus, F ′ q (ρ) is decreasing on ]0, ρ 2 [ and increasing on ρ 2 , 3π 2 . Using that F ′ q (0) = 0 and F ′ q 3π 2 = A -B + 3π > 0, we conclude that F ′ q (ρ) admits a zero ρ 3 ∈ ρ 2 , 3π 2 .
Therefore, F q is decreasing on ]0, ρ 3 [ and increasing on ρ 3 , 3π 2 . Since F q (0) = 0 and F q 3π 2 = 9π 2 4 -3π 2 B(q) -A(q) > 0, we conclude that F q admits a unique zero ρ * ∈ ρ 3 , 3π 2 .

If F

(3) q (ρ) has two zeros ρ 1 , ρ 2 ∈ I. Then, F ′′ q (ρ) is decreasing on ]0, ρ 1 [, increasing on ]ρ 1 , ρ 2 [, and decreasing on ρ 2 , 3π 2 . In this case we necessarily have F ′′ q (0) < 0 and, using

F ′′ q 3π 2
> 0, we conclude that F ′′ q has a unique zero

ρ 3 ∈]ρ 1 , ρ 2 [. So F ′ q (ρ) is decreasing on ]0, ρ 3 [ and increasing on ]ρ 3 , 0[. Since F ′ q (0) = 0 and F ′ q 3π 2
> 0, the function F ′ q admits a unique zero ρ 4 ∈ ρ 3 , 3π 2 . Finally, F q is decreasing on ]0, ρ 4 [ and increasing on ρ 4 , 3π 2 . Since, F q (0) = 0 and F q 3π 2 > 0, we conclude that F q admits a unique zero ρ * ∈ ρ 4 , 3π 2 . To summarize the case distinction made above, we have proved that for any value of the parameter q, the function F q (ρ) admits at most a root ρ * > 0.

Therefore, for a given delay q * , there exists at most one ω * > 0, defined by ρ * = q * ω * , such that R q (ω * ) = 0. We now show that S q (ω * ) > 0. In the same fashion as for R q , we introduce the parametrized univariate function H q (ρ) = qS q ( ρ q ), which can be expressed as H q (ρ) = C(q)ρ + D(q)ρ cos(ρ) + E(q) sin(ρ), with C(q) = e q +3 e q -1 , D(q) = 2 e q -1 , and E(q) = -2q 2e q +1 (e q -1) 2 . The following lemma details the properties of the C, D, E functions that we will use in the next results.

Lemma 12 (Properties of the C, D, E functions). For all q > 0, we have C(q) > D(q) > 0 and 2D(q) + E(q) < 0.

Proof. It is immediate to obtain C(q) > 0 and D(q) > 0; moreover, C(q) -D(q) = e q +1 e q -1 > 0. Next, we have 2D(q) + E(q) = -2 (e q -1) 2 (2e q (q -1) + q + 2), which can be seen to be positive since the second factor is positive (proved using its second derivative).

The results of Lemma 12 are illustrated by Figure 2.

2 4 6 8 q C(q) D(q) -E(q)
2 Figure 2: Qualitative behavior the functions C(q) (red), D(q) (blue), and E(q) (black), involved in the definition of H q . Lemma 13. For any q > 0 and ρ ∈ 0, 3π 2 , we have H q > 0.

Proof. By a direct calculation, we obtain

H ′′ q (ρ) = -D(q)ρ cos(ρ) -(2D(q) + E(q)) sin(ρ).
The function J(q) = -D(q) 2D(q)+E(q) satisfies 0 < J(q) < 1, so we conclude that the equation tan(ρ) = J(q)ρ admits a unique solution in the interval I. Using H ′′ q (0) = 0 and H ′′ q 3π 2 = 2D(q) + E(q) < 0, it follows that H ′ q is increasing on ]0, ρ 1 [ and decreasing on ρ 1 , 3π 2 . Since, H ′ q (0) = C(q) + D(q) + E(q) > 0 and H ′ q 3π 2 = 3π 2 D(q) + C(q) > 0, we conclude that H ′ q is positive, and H q is increasing, since H q (0) = 0, it follows that H q (ρ) > 0.

Hence, in the Stépán-Hassard formula we either have r = 0 or r = 1 and sgn S q (ρ * ) = 1.

Applying the Stépán-Hassard formula. The previous preliminary results allow to compute the number of zeros of Q located in the right-half plane. Theorem 14. The root 1 is dominant for Q.

Proof. We apply the Stépán-Hassard formula (4) to obtain the number of zeros of Q in {ℜ(z) > 0}. If r = 0, then

Z 0 = 2 -1 2 + 1 2 (-1) 0 (1) = 1. If r = 1, then Z 1 = 2 -1 2 + 1 2 (-1) 1 (1) + (-1) 0 (1) = 1.
Thus, in both cases, the quasipolynomial Q admits only one root in the right-half plane. But, we already know that z = 1 is a root of Q. Therefore any other root z 0 ∈ C of Q satisfies ℜ(z 0 ) ≤ 0 and we conclude the dominancy of 1 for Q.

As a consequence of Theorem 14 and of Lemma 9, we conclude that s 1 is dominant for ∆(s, τ ) and that ∆ satisfies the CRRID property.

Asymptotic stability of the DDE

Collecting the results of the previous subsections give the following theorem.

Theorem 15. If the quasipolynomial ∆ given by (2) admits four equidistributed real zeros and ( 9) is satisfied, then the trivial solution of (1) is asymptotically stable with a decay rate equal to s 1 .

Illustrative examples

In this section, we illustrate the effectiveness of our approach by providing two examples. First, we show that the CRRID property can be applied via a delayed PD controller to successfully stabilize unstable plants. Second, we show that even the second-order open-loop plant is stable, one is able to improve its solution decay rate.

Stabilizing two unstable poles

To highlight the utility of using a retarded PD controller, we use the approach developed in the previous section to stabilize an unstable second-order plant free of delays. Consider the openloop system given by y ′′ (t) + a 1 y ′ (t) + a 0 y(t) = 0 and assume that a 1 < 0, which implies that its characteristic function has at least one unstable zero. Following the methodology introduced in the paper, we consider that we use a delayed PD controller u(t) = -α 1 y ′ (t -τ ) -α 0 y(t -τ ) and that the closed loop characteristic function

∆(s) = s 2 + a 1 s + a 0 + e -τ s (α 1 s + α 0 ) (11)
admits four equidistributed real roots s k = s 1 -(k -1)d, for 1 ≤ k ≤ 4 and d > 0. Therefore, ∆ satisfy the CRRID property, i.e. the largest real root s 1 is the dominant one, and thus the negativity of s 1 implies that the solutions of the closed-loop system are asymptotically stable.

The control parameters s 1 , d, and τ cannot be chosen arbitrarily. Indeed, a 1 must be given by Corollary 6 and its negativity implies that the control parameters have to be chosen such that it holds

s 1 > d 2 e τ d -5 e τ d -1 .
Hence, to guarantee that the spectral abscissa of the closed loop system s 1 is negative, we need to satisfy τ d < ln [START_REF] Els | Introduction to the theory and application of the theory of differential equations with deviating argument[END_REF]. The precise assignability region for s 1 as a function of τ > 0 and d > 0 is depicted in Figure 3. For a numerical illustration, we set a 1 = -2, a 0 = 51/8, s 1 = -1, and ln( 5)

-2 -1 1 τ d τ s 1
Negativity of s 1 and a 1 d = 1. Then, the formula for a 1 of Corollary 6 implies that necessarily τ = ln 9 5 giving the following values of the controller's gains α 0 = -3875/648, α 1 = -125/162. First, Figure 4 shows that the roots of the unstable open-loop system (given by 1 ± 1 2 i 43 2 ) are located in the right half of the complex plane, whereas the infinite number of roots of the closed-loop [START_REF] Sieber | Extending the permissible control loop latency for the controlled inverted pendulum[END_REF] system are located in the left half of the complex plane and are dominated by s 1 . Remark. In the numerical example above, it is easy to observe that the gain α 1 of the proposed controller satisfies the inequality |α 1 | < |a 1 |, and since a 1 < 0, the corresponding delay-free PD controller u(t) = -α 0 y(t) -α 1 y ′ (t) does not stabilize the corresponding system. In other words, the presence of the delay in the control law guarantees the stability of the closed-loop system reinforcing thus the stabilizing effect induced by the delay on the system's dynamics. For further discussions on such topics, we refer to [START_REF] Michiels | Stability, control, and computation for time-delay systems: an eigenvalue-based approach[END_REF], [START_REF] Sipahi | Stability and stabilization of systems with time delay: Limitations and opportunities[END_REF], and the references therein.

Improving decay rate

In this second example, we show that we can improve the decay rate of a stable second-order control system without instantaneous velocity and state observations. We consider the control problem

y ′′ (t) + 2ζωy ′ (t) + ω 2 y(t) = u(t), (12)
where ζ > 0 is the damping ratio and ω > 0 is the oscillator's natural frequency. In absence of a controller, i.e. u(t) = 0, the characteristic polynomial is ∆(s) = s 2 + 2ζωs + ω 2 , the roots of which are

s ± = ω -ζ ± i 1 -ζ 2 if 0 < ζ < 1, and s ± = ω -ζ ± ζ 2 -1 if ζ ≥ 1.
Hence, the spectral abscissa of ∆ is given by

γ 0 = -ζω if 0 < ζ < 1, ω -ζ + ζ 2 -1 if ζ ≥ 1.
In particular, we have γ 0 < 0 and the system is asymptotically stable. For many applications, we need a delay τ > 0 to access a good approximation of the velocity and state of the system. This suggests to use a controller u having a delayed proportional-minus-derivative structure, i.e.

u(t) = -α 1 y ′ (t -τ ) -α 0 y(t -τ ), τ > 0. ( 13
)
The closed-loop system defined by ( 12) and ( 13) has a characteristic function given by the four-degree quasipolynomial

∆(s) = s 2 + 2ζωs + ω 2 + e -τ s (α 1 s + α 0 ). (14)
To illustrate the results of Section 3.3, we consider the case where ∆ admits equidistributed real roots s k = s 1 -(k -1)d, with d > 0 and 1 ≤ k ≤ 4. One solves the system of transcendental equations ∆(s k ) = 0 for the control parameters (α 1 , α 0 , τ ) in terms of the system's physical parameters ζ and ω, as well as the assigned root s 1 and the distance between two consecutive roots d. More precisely, we obtain the following expressions:

τ = σ d , (15)
α 1 = - 1 2 (d -2s 1 -2ζω) e -τ (d-s 1 ) , α 0 = - 15 8 (d -2s 1 -2ζω) d - 2 3 s 1 - 2 5 ζω e -τ (d-s 1 ) , with σ = ln 5d -2s 1 -2ζω d -2s 1 -2ζω .
Moreover, we also get the following relation that links the distance d, the spectral value s 1 , and the parameters ζ and ω

1 2 d 2 -2(ωζ + s 1 )d + 2 3 (3ζ 2 -2)ω 2 + 4 3 (ζωs 1 ) + 1 3 (2s 2 1 ) = 0. (16)
Now, to achieve asymptotic stability, we need to choose the positive distance d and the negative root s 1 such that the positivity of the delay τ , given by ( 15), is guaranteed and such that the constraint ( 16) is satisfied. First, note that τ > 0 corresponds to choosing s 1 and d such that

5d -2s 1 -2ζω d -2s 1 -2ζω = 1 + 4d d -2s 1 -2ζω > 1,
which is equivalent to the condition

d -2s 1 -2ζω > 0. (17)
Second, solving [START_REF] Michiels | Stability, control, and computation for time-delay systems: an eigenvalue-based approach[END_REF] we infer that d is necessarily given by

d = 2s 1 + 2ζω + 2 3 6s 2 1 + 6ω 2 + 12ζωs 1 , which is real if, s 1 < ω -ζ -ζ 2 -1 or s 1 > ω -ζ + ζ 2 -1 , when ζ > 1 or s 1 is arbitrary when 0 < ζ ≤ 1. Furthermore, we have d > 0 if s 1 > ω -ζ - √ 2 1 -ζ 2 if 0 < ζ ≤ 1, s 1 > ω -ζ -ζ 2 -1 if ζ ≥ 1,
Summing up all the conditions to ensure the realness and the positivity of d, we obtain the assignment region for the spectral abscissa s 1 that is depicted in Figure 5. In particular, we observe that we can improve the negative spectral abscissa γ 0 only in the case ζ < 1.

1 - √ 2 ζ s 1 ω

Positive and real distance d

The open-loop spectral abscissa γ 0 It is interesting to compare our approach with the MID one. Adapting the results of [START_REF] Boussaada | Multiplicityinduced-dominancy in parametric second-order delay differential equations: Analysis and application in control design[END_REF], [START_REF] Mazanti | Multiplicity-induced-dominancy for delaydifferential equations of retarded type[END_REF] to our context, we see that the MID can be applied with a controller of the form (13) only when 0 < ζ < 1, in which case the assigned root of maximal multiplicity four is

s 0 = ω -ζ - √ 2 1 -ζ 2 .
This case illustrates the property that the MID is the limiting case of the CRRID when the distance d approaches 0. To apply the CRRID, we need to choose the spectral abscissa s 1 in such a way that the corresponding distance d is not zero. Hence, the spectral abscissa given by the CRRID is larger than the one given by the MID. However, the CRRID approach involves choosing the spectral abscissa in an open interval, whereas the MID approach assigns a unique value for the spectral abscissa given in terms of the corresponding delay, illustrating the robustness property of the CRRID. Moreover, the CRRID approach can be applied to systems with ζ > 1, even though the exponential decay rate will be worse than the one of the open-loop system. For a numerical illustration of our results (dominancy of s 1 and stability), we set ω = 1 and ζ = 1 5 . In this case, the assigned root s 1 must satisfy the bound s 1 > -1 5 -4 √ 3

5 to ensure the positivity of the distance d. Let us set s 1 = -1, which yields the following values of the other parameters

                   d = -
In Figure 6, we give the spectrum of the characteristic function [START_REF] Suh | Use of time-delay actions in the controller design[END_REF] with the above parameters and we illustrate the dominancy of the root s 1 with respect to all the other complex roots. 14) corresponding to the closed-loop system given by ( 12) and ( 13), with parameters given in [START_REF] Bedouhene | Real spectral values coexistence and their effect on the stability of time-delay systems: Vandermonde matrices and exponential decay[END_REF] Remark. Regarding the control problem (12)- [START_REF] Seuret | Low-complexity controllers for timedelay systems[END_REF], it is crucial to highlight that in the absence of the input delay, that is if τ = 0, then employing a standard filtered PD may lead to outcomes featuring dominant poles. Notably, one can strategically place the poles of the finite-dimensional closed-loop system anywhere in the complex plane by employing a "state-feedback plus a secondorder observer" type controller. This approach provides further flexibility when performing a finite-dimensional (delay-free) prescribed stabilization. However, the undoubted advantage of the approach proposed through this paper consists of the reduced complexity control that it suggests.

Remark. Practical control systems are often subject to operational constraints such as limited control capacity, i.e. limitations on input, state, and output variables. To emphasis the interest of using a delayed controller [START_REF] Seuret | Low-complexity controllers for timedelay systems[END_REF] in such scenarios we compare it to a classical PD controller v(t) = -β 1 y ′ (t) -β 0 y(t). Let us focus on the limitation on the controllers amplitude by defining the following constraint on the gains:

|β 1 | 2 + |β 0 | 2 < 1, (19)
replacing β i by α i in the case of the retarded PD controller u(t) given by [START_REF] Seuret | Low-complexity controllers for timedelay systems[END_REF]. Using a classical pole placement strategy, we assign the same spectral abscissa s 1 = -1 and we set the second root to s 2 = s 1 -d, which yields β 1 = 8 √ 15 15 ≈ 2.07 and β 0 = 8 √ 15

15 -8 5 ≈ 0.47. Hence, we observe that the gains of the PD controller v(t) do not satisfy the constraints [START_REF] Boussaada | The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: When delay-systems characteristics meet the zeros of kummer functions[END_REF]. On the other hand, the gains α 1 ≈ -0.14 and α 0 ≈ -0.54 of the delayed version u(t) satisfy such a constraint. Thus, similar to the discussion proposed in [33] † , we illustrate the positive effect of time-delay in constrained control systems. Furthermore, it should be emphasized that the decay rate is guaranteed for"small" (controller) gains, and thus less sensitive to high-frequency noise.

P3δ Software

Partial pole placement via delay action (P3δ) is an intuitive Python software application that facilitates the design of stabilizing feedback laws incorporating time-delays. The software makes use of quasipolynomials properties (in particular, the MID property as studied in [START_REF] Boussaada | The generic multiplicity-induced-dominancy property from retarded to neutral delay-differential equations: When delay-systems characteristics meet the zeros of kummer functions[END_REF], [START_REF] Mazanti | Multiplicity-induced-dominancy for delaydifferential equations of retarded type[END_REF]) to compute efficient control laws. Based on the results of this work, as well as reference [START_REF] Bedouhene | Real spectral values coexistence and their effect on the stability of time-delay systems: Vandermonde matrices and exponential decay[END_REF], the software has been enhanced to include the CRRID property for design control strategies. P3δ is available for free download at https://cutt.ly/p3delta, where installation instructions, video demonstrations, and the user manual can also be found. Interested readers may also contact directly the software authors.

Concluding remarks

This work focused on the stabilization of a generic second-order linear differential equation with a delayed PD controller via the assignment of the maximum number of real roots of the corresponding characteristic function. We provided the expression of the differential equation for which the corresponding quasipolynomial admits the maximal number of real roots (which is, in our case, equal to four). In the case where the real roots are equidistant, we determined a necessary and sufficient condition on those coefficients for the largest real root to be negative. And we showed that this root is always the spectral abscissa of the quasipolynomial. Therefore, we proved exponential stability of the second-order linear differential equation and thus, we provided a partial poles placement alternative to the MID property.

Figure 3 :

 3 Figure 3: Assignability region for the spectral abscissa of the closed loop system (11) as a function of the distance d and delay τ .

Figure 4 :

 4 Figure 4: Spectrum ditribution of (red) the closed-loop system (blue) the closed-loop system[START_REF] Sieber | Extending the permissible control loop latency for the controlled inverted pendulum[END_REF].

Figure 5 :

 5 Figure 5: Assignment region for s 1 ω to ensure the positivity (blue) and realness (red) of the distance d.

Figure 6 :

 6 Figure 6: Spectrum distribution of the characteristic function (14) corresponding to the closed-loop system given by (12) and (13), with parameters given in[START_REF] Bedouhene | Real spectral values coexistence and their effect on the stability of time-delay systems: Vandermonde matrices and exponential decay[END_REF]

† Partial results of this paper were previously presented in[START_REF] Schmoderer | On Quasipolynomials Real Roots Coexistence: Effect on Stability of Time-Delay Systems with Perspectives in Partial Pole Placement[END_REF], the current paper includes updated results with full proofs. The first author would like to thank Inria Saclay and IPSA Paris for their financial support. The authors are grateful to their collaborator Karim L. Trabelsi (IPSA Paris) for careful reading of the manuscript and for valuable comments.

† for controlling the integrator by using two-delay blocks

Acknowledgments.

The authors would like to thank the Associate Editor for handling the reviewing process and anonymous reviewers, whose suggestions and remarks have substantially improved the overall quality of the paper.